Analytical Ultracentrifugation VIII (Progress in Colloid and Polymer Science) (Progress in Colloid and Polymer Science)

Progress in Colloid and Polymer Science · Volume 131 · 2006

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IV

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ISSN 0340-255X ISBN-10 3-540-29615-8 ISBN-13 978-3-540-29615-7 DOI 10.1007/b96539 Springer Berlin, Heidelberg, New York

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Progr Colloid Polym Sci (2006) 131: V © Springer-Verlag 2006

PREFACE

The Symposia on Analytical Ultracentrifugation have had the tradition to take place biannually at universities or other institutions in Germany. This tradition was, for the first time, interrupted organizing the 14th International Symposium on Analytical Ultracentrifugation on March 1–2, 2005, at the École Polytechnique Fédérale de Lausanne in Switzerland under the auspice of the Faculté des Sciences de Base and the Institut des Sciences et Ingénierie Chimiques. On the other hand, this development characterizes the continuous extension of the symposium, which was initiated in 1978 as a national event in Germany, then increasingly opened for international participation, and now even placed in a neighboring country. The internationality is reflected by about 65 participants from leading groups in 10 countries, the majority of them coming traditionally from Germany, UK, and USA but also, for the first time, from China. Furthermore, collaborators of international companies such as Roche, Serono, BASF, and Bayer attended the symposium. The major goal of the symposium was to continue promoting the best use of analytical ultracentrifugation by establishing and renewing fruitful exchange of ideas and contacts between biologists, chemists, physicists, not yet experienced users and professional developers of the technique, young and senior scientists from academia and industry. A two days workshop with 35 participants preceded the symposium. 27 lectures were presented and discussed by globally leading scientists but also by junior staff, which is expected to ensure the future of analytical ultracentrifugation. Posters completed the presentations. The program of the symposium was organized in seven sessions revealing the wide potential of this fascinating and powerful technique such as • Instrumentation • Data analysis and modeling • Biological systems • Particles, colloids, synthetic macromolecules, interacting systems and included comparison with results obtained from other methods. 21 contributions were selected for publication in this special volume of Progress in Colloid and Polymer Science covering all topics of the symposium with the hope to provide a helpful collection of recent developments but also to show the still extendable potential of the technique. The EPFL, the Kontaktgruppe für Forschungsfragen (Novartis, Serono, Syngenta, Roche), the Swiss National Science Foundation, Beckman Coulter International, the Polymer Group of Switzerland, H.G. Müller-Bayer Industry Service, BASF AG, and the Canton de Vaud, Suisse, generously sponsored the 14th International Symposium on Analytical Ultracentrifugation. Thanks go to all the unnamed reviewers of the contributions, to Cristina Spillmann, Laurent Bourdillon, and Ronald Zbinden but, particularly, to Ingrid Margot for her assistance preparing the symposium and this volume. November 2005,

Christine Wandrey (EPFL, Switzerland) Helmut Cölfen (MPI-KGF, Germany)

Progr Colloid Polym Sci (2006) 131: VI–VII © Springer-Verlag 2006

CONTENTS

Instrumentation

Laue TM, Austin JB, Rau DA:

A Light Intensity Measurement System for the Analytical Ultracentrifuge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Bhattacharyya SK, Maciejewska P, Börger L, Stadler M, Gülsün AM, Cicek HB, Cölfen H:

Development of a Fast Fiber Based UV-Vis Multiwavelength Detector for an Ultracentrifuge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

Lavrenko VP, Lavrenko PN:

Automatic Analysis of Lebedev Interference Patterns . . . . . . . . . . . . . . . . . . . . .

23

Data Analysis and Modeling

Behlke J, Ristau O:

A New Possibility to Recognize the Concentration Dependence of Sedimentation Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

Genetic Algorithm Optimization for Obtaining Accurate Molecular Weight Distributions from Sedimentation Velocity Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

Zipper P, Durchschlag H, Krebs A:

Modeling of the Dodecameric Subunit of Lumbricus Hemoglobin. . . . . . . . . .

40

Durchschlag H, Zipper P, Krebs A:

Ab initio and Constrained Modeling of Phosphorylase . . . . . . . . . . . . . . . . . . . .

54

Brookes E, Demeler B:

Biological Systems

Salvay AG, Ebel C:

Analytical Ultracentrifuge for the Characterization of Detergent in Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

Chebotareva NA, Meremyanin AV, Makeeva VF, Kurganov BI:

Self-Association of Phosphorylase Kinase under Molecular Crowding Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

Mutowo P, Scott DJ:

Oligomerisation of TBP1 from Haloferax volcanii . . . . . . . . . . . . . . . . . . . . . . .

92

Clay O, Carels N, Douady CJ, Bernardi G:

Density Gradient Ultracentrifugation and Whole Genome Sequences: Fine-tuning the Correspondence . . . . . . . . . .

96

Stouffer AL, DeGrado WF, Lear JD:

Analytical Ultracentrifugation Studies of the Influenza M2 Homotetramerization Equilibrium in Detergent Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

107

Khan A, Hughes RK, Belfield EJ, Casey R, Rowe AJ, Harding SE:

Oligomerization of Hydroperoxide Lyase, a Novel P450 Enzyme in Plants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Particles, Colloids, Synthetic Macromolecules and Interacting Systems

Müller HG:

Determination of Particle Size Distributions of Swollen (Hydrated) Particles by Analytical Ultracentrifugation . . . . . . . . . . 120

VII

Cölfen H, Völkel A:

Application of the Density Variation Method on Calciumcarbonate Nanoparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

125

Cölfen H, Lucas G:

Particle Sedimentation in pH-Gradients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

Pavlov GM, Panarin EF, Korneeva EV, Gavrilova II, Tarasova NN:

Molecular Properties and Electrostatic Interactions of Linear Poly(allylamine hydrochloride) Chains . . . . . . . . . . . . . . . . . . . . . . . . . 133

Bourdillon L, Hunkeler D, Wandrey C:

The Analytical Ultracentrifuge for the Characterization of Polydisperse Polyelectrolytes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

Bourdillon L, Freitag R, Wandrey C:

Association and Temperature-induced Phase Transition Studied by Analytical Ultracentrifugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Lavrenko PN, Belyaeva EV, Volokhova DM, Okatova OV:

Aggregation of Dibutyl Phthalate Molecules in Decalin Solutions Evidenced by Hydrodynamic and Optical Measurements . . . . . . . . . . . . . . . . . . 157

Ras¸a M, Tziatzios C, Lohmeijer BGG:

Analytical Ultracentrifugation Studies on Terpyridine-end-functionalized Poly(ethylene oxide) and Polystyrene Systems Complexed via Ru(II) ions . . . . . . . . . . . . . . . . . . . . .

164

Author/Title Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

Keyword Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

Progr Colloid Polym Sci (2006) 131: 1–8 DOI 10.1007/2882_001 © Springer-Verlag Berlin Heidelberg 2006 Published online: 4 February 2006

INSTRUMENTATION

Thomas M. Laue Joseph B. Austin David A. Rau

A Light Intensity Measurement System for the Analytical Ultracentrifuge

Thomas M. Laue (u) · Joseph B. Austin · David A. Rau The Center to Advance Molecular Interaction Science, University of New Hampshire, Rudman Hall-379, Durham, New Hampshire 03824-3544, USA e-mail: [email protected]

Abstract Light intensity measurements are required by absorbance, fluorescence, turbidity and low-angle light scattering detectors in order to determine the concentration distributions encountered in analytical ultracentrifugation. By using four fast analog-to-digital converters operating in parallel, a data acquisition system has been developed for the analytical ultracentrifuge that can acquire light intensity readings from three detectors (e.g. photomultipliers, avalanche photodiodes, etc.) simultaneously. For each detector, up to forty thousand intensity readings are acquired during each rotor revolution, for up to ten revolutions. Software synchronizes data acquisition with the

Introduction Measurement of the radial concentration distribution in a sample as a function of time is central to analytical ultracentrifugation. Amongst the optical systems used to measure concentration distributions, absorbance, fluorescence and light scattering detectors require measurements of the light intensity either passing through or emanating from a sample. Data acquisition from the ultracentrifuge poses two particular problems. First, signal acquisition must be synchronized with the spinning rotor and, second, some means must be provided to isolate the signals from each sample. The standard Beckman Coulter XLI analytical ultracentrifuge solves the first problem by using a pair of small magnets on the rotor and a Hall effect sensor mounted on the chamber bottom to produce two closely

spinning rotor. The use of continuous light sources allows simultaneous acquisition of dozens of intensity readings from all of the samples. Data acquisition is fast, allowing rapid radial scanning of samples. This data acquisition scheme is used in the Aviv Biomedical AU-FDS fluorescence detection retrofit system for the Beckman XLI analytical ultracentrifuge. It also will be used for the updated XLI absorbance system, as well as the next generation of analytical ultracentrifuge. Keywords Absorbance detector · Analytical ultracentrifugation · Detector systems · Fluorescence detector · Instrumentation software

spaced pulses, one positive going and the other negative going (Fig. 1), with each turn of the rotor. A single TTL pulse is generated at the zero-crossing point between the positive and negative Hall effect pulses, and the leading edge of this TTL pulse is used to synchronize the XLI data acquisition systems. The second problem is solved by using a pulsed light source that is triggered when a sample is aligned with the detector. Timing of the trigger signal is accomplished in hardware using counters and a fast clock as described previously [1]. In this scheme, the rotor timing signal must be jitterfree to better than one part in 4000, or else data quality suffers from poor timing of the light pulses [2]. Propagation delays in the electronics that generate the rotor timing pulse, in the clocks that produce the strobe pulse, and in the light sources themselves, require that the synchronizing systems provide rotor speed-dependent timing

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Fig. 1 Rotor timing pulse. Data are shown as the A/D values (0–4095, corresponding to ±1.0 volts) as a function of the array index (solid line), for a span of ∼ 20◦ of rotation. The dotted line shows the AverageSignal (∼ 2047), with two steps above the AverageSignal. The first, wider step shows the value of the PulseDiscriminator (= AverageSignal + 200). For clarity, this step covers the range of data between the positive and negative peaks in the rotor timing signal. The second, higher step shows the value used for the PeakDiscriminator (= AverageSignal + 300). The spike in the data shows the position calculated for the midpoint between the two peaks. Data were simulated for a signal to noise ratio of 40 dB

adjustments [1, 2]. Additional complications arise from the frequency dependence of the signals and propagation delays, for which the synchronizing system must compensate [1]. Given this complexity, the hardware based strobe light system of the XLI is remarkably reliable. Despite its reliability, the current system limits the XLI absorbance data acquisition systems to one light pulse per rotor revolution. Furthermore, the maximum repetition rate for the XLI’s high intensity Xenon light source is 100 Hz, so that 10 ms must elapse between light bursts. At rotor speeds above 6000 rpm, the 10 ms hiatus limits the rate at which data may be acquired, and at 60 000 rpm the light is pulsed only once for every 10 rotor revolutions. While similar lamps are available with slightly higher repetition rates (300 Hz), it is not feasible to construct lamps of this intensity that will operate at suitably higher repetition rates (> 1000 Hz). Even if such lamps were available, data acquisition is slowed by the fact that only one sample is illuminated per light pulse. If the XLI’s absorbance system is being operated in the usual doublebeam mode, then a second light pulse is required to illuminate the reference channel. At 60 000 rpm, a minimum of 20 rotor revolutions is required to gather the data for one absorbance reading for each sample. If the user chooses to average multiple intensity readings in order to improve data precision, the time required per reading increases proportionally.

T.M. Laue et al.

As currently configured, the XLI performs a complete radial scan of one sample before resetting and starting to scan a second sample. At a radial spacing of 50 microns, and using one sample and one reference intensity per radial position, a radial scan of 1.3 cm length may be completed in 20–25 seconds, with the interval between scans being about 30 seconds when the time needed to reset between scans is included. For an experiment with seven samples, each sample will be scanned approximately every 3 1/2 to 4 minutes. This is a minimum time interval per scan. If the user chooses to acquire data at multiple wavelengths, or increases the number of intensity readings by either decreasing the radial spacing or increasing the number of intensity readings per radial position, the time interval between scans of a sample increases. The compromise between data quantity and data quality fundamentally limits the usefulness of the absorbance system for sedimentation velocity analysis. One means to speed up data acquisition is to use a continuous light source, and to separate the signals from the different samples by synchronizing the detector with the spinning rotor. In this sort of scheme, those portions of the detector signal corresponding to the moments when a particular sample is in the light beam are separated, e.g. by a de-multiplexing circuit. The analog absorbance system of the Beckman Model E used this scheme to acquire data from one sample at a time. More recently, the turbidity and schlieren systems have used an analogous scheme to acquire intensity data from all of the samples with each rotor revolution [3, 4]. The system presented here uses four parallel highspeed analog-to-digital (A/D) converters to acquire data simultaneously from four separate detectors. The signal from the XLI Hall effect detector is one of the signals, and light intensity data from the absorbance, fluorescence or turbidity optical systems are the other signals that may be digitized. The software both synchronizes data acquisition with the spinning rotor and isolates the signals from each sample. With this system, it is possible to average the data for all of the detectors over several rotor revolutions for each of the samples simultaneously. Furthermore, the software can be configured to acquire data from new rotor and cell designs easily.

Description Signals: There are four signal sources that require highspeed data acquisition: the rotor timing pulse and three light intensity detector voltages. A less expensive interface handles the lower speed analog and digital signals used to monitor device status (e.g. the presence of various optical systems, the fluorescence laser power and temperature, etc.) or control devices (e.g. fluorescence laser on/off, PMT voltages). In addition to the two interface cards, two RS-232 serial I/O ports are required. The first serial port

A Light Intensity Measurement System for the Analytical Ultracentrifuge

provides communication with the XLI and is used both to control its operation and to monitor its status. The second serial port controls the stepping motors used in the optical systems. Up to six stepping motors may be present: two for the fluorescence hardware (radial position and objective lens focus), two for the absorbance system (radial position and wavelength selector) and two for the refractive detectors (camera position and schlieren bar angle). Hardware and software are included to determine which optical systems are present so that the stepping motor functions are configured properly. The software controlling the instrument uses a low level hardware abstraction layer. That is, the software is written so that function calls from common programming languages (e.g. C++, Visual Basic) remain unchanged even if the underlying hardware changes. Hardware: Voltages from the four high speed sources are digitized using four 12-bit, 20 MHz A/D converters (PCI-DAS4020/12, Measurement Computing, Inc., Middleboro, MA, USA). This device also provides two 12-bit digital-to-analog (D/A) outputs and 24 digital inputoutput (DIO) lines. An internal first-in-first-out buffer memory accommodates computer latencies (e.g. hardware memory refresh, software interrupt handling) thus allowing sustained data acquisition at the full data rate. The ±1 volt input range was used for all four A/D channels. The digitized data streams are stored as interleaved values in the computer memory as: I1,1 , I2,1 , I3,1 , I4,1 , I1,2 , I2,2 . . . I1,n , I2,n , I3,n , I4,n , where the first index refers to which A/D produced the signal and the second index indicates which reading this is in the sequence. When fresh data are desired, the software triggers data acquisition from all four high-speed A/Ds. Once triggered, a sufficient number of data points (TotalCount) are gathered to cover several turns of the rotor, RevolutionsToAverage. The operating software adjusts the A/D clock frequency (ADClockRate) and acquisition period (TotalPeriod) so that each detector will acquire at least 10 000 values per rotor revolution (CountsPerRevolution). After each data acquisition period, fresh values ADClockRate, TotalPeriod and TotalCount are calculated so that rotor acceleration/deceleration may be accommodated. Following data acquisition, four separate data arrays are created from the interleaved data, one for each A/D channel. A separate device (PCI-DAS6025, Measurement Computing, Inc., Middleborough, MA, USA) is used as the low-speed interface and contains 16 channels of 12-bit A/D input (at 200 KHz), two 12-bit D/A outputs, 8 DIO lines and two 16-bit counters. All signals to and from the computer are connected to a custom-designed “system box” that contains the amplifiers, power supplies, digital signal buffers, a programmable clock and custom analog signal conditioning circuits to accommodate five optical systems (absorbance, fluorescence, turbidity, interference and schlieren). Signal conditioning includes programmable gain amplifiers that provide 1, 2, 4 or 8-fold gain for each of the light detec-

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tor signals, as well as a precision voltage source used to offset the signals so that zero light intensity corresponds to −1 volt and saturating light corresponds to +1 volt. All of the components in the signal pathways have a frequency response > 30 MHz. A single wiring harness connects the system box to a 100-pin connector on the XLI. A custom-designed “distribution box” within the XLI contains power supplies, the laser controller for the fluorescence optics, circuits to buffer the rotor timing pulse and vacuum signals, circuits to detect which optical systems are present, and connectors to direct the various signals to the different optical subsystems. Throughout, careful circuit layout is needed to minimize stray noise, crosstalk and ground loops. The rotor timing pulse: All data acquisition must be synchronized to an accurate rotor timing pulse. Previous data acquisition systems use the rotor timing pulse to trigger data acquisition hardware, with subsequent transfer of the data to the computer [1–3]. The system described here acquires data asynchronously, with post-acquisition processing used to extract the light intensity signals. One of the four high speed A/Ds monitors the rotor timing signal from the Hall effect sensor (Fig. 1). The signal is amplified to provide ±0.95 volts when a magnet passes over the sensor, and otherwise is nominally 0 volts. One magnet produces a positive going pulse and the other a negative going pulse, with each pulse occupying ∼ 18 degrees of the rotor revolution, and the two pulses being separated by ∼ 20 degrees. Because data acquisition is triggered asynchronously, rotor timing pulses may fall anywhere in the data acquisition time period. Furthermore, until ADClockRate, TotalPeriod and TotalCount are set correctly, an unknown number of rotor timing pulses may be present in the data. Thus, two situations arise: one where these parameters have not yet been determined and, more commonly, one where good estimates of these parameters exist. When the software is first started, the rotor speed is unknown (i.e. the XLI may or may not be operating when the software is started), and default values for ADClockRate, TotalPeriod, TotalCount and RevolutionsToAverage are used. The default values for these parameters are set to acquire 20 000 data points (e.g. TotalCount = 20 000) for a TotalPeriod covering seven rotor revolutions (RevolutionsToAverage +2) at 1000 rpm. These defaults allow the system to detect rotor spinning at low rpm (e.g. during initial rotor acceleration), while still gathering data at sufficient resolution to determine an approximate rotor speed should the rotor already be spinning at 60 000 rpm. Two other default values are set: the number of clock pulses per rotor revolution (CountsPerRevolution, as TotalCount divided by the number of revolutions) and the number of clock pulses (MagnetPulseWidth) corresponding to the width (approximately 18 degrees) of one of the two peaks from the Hall effect sensor.

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The software uses two values to discriminate a rotor timing pulse from background noise. The first (PulseDiscriminator) is the minimum signal amplitude required to consider the signal at any point (Signal[i]) in the data array to be part of a rotor timing pulse. The second (PeakDiscriminator) is the minimum amplitude of Signal[i] required to consider it the peak value. Comparisons are made as the absolute value of the difference between the Signal[i] and the average signal (AverageSignal, default = 2048, corresponding to 0 volts) calculated for a portion of the data that does not contain a rotor timing pulse (calculated as described below). The default values of PulseDiscriminator and PeakDiscriminator are 200 and 300, respectively. These values are stored in a database and may be changed if necessary. The rotor timing peak finding algorithm begins by determining whether or not there is rotor timing pulse at the start of the Signal array (i.e. data acquisition was triggered when a rotor timing pulse was occurring). Data are tested for Signal[i], for i = 0 to i = twice the MagnetPulseWidth to determine if the magnitude any data point exceeds the AverageSignal. If so, the starting index for peak finding (StartIndex) is set to one-half the CountsPerRevolution to avoid dealing with partial rotor timing pulses. Because StartIndex is guaranteed to be in a region of Signal[i] devoid of a rotor timing pulse, AverageSignal is calculated as the mean value of Signal[i] from i = StartIndex to i = StartIndex plus twice the MagnetPulseWidth. Beginning with StartIndex, Signal[i] is scanned to find the first data point having a magnitude exceeding PulseDiscriminator. At this point, StartIndex is set to i, and the portion of the Signal array from StartIndex to four times the MagnetPulseWidth (EndIndex) is searched to find the indices of the maximum and minimum values. To avoid false maxima or minima due to noise, data generated using the “reverse smoothing” algorithm of Roark [5] are searched. The values of Signal[i] corresponding to the indices for the maximum and minimum are tested to see that they exceed PeakDiscriminator, and the two indices are tested to make sure that they are not equal to either StartIndex or EndIndex (indicating that a partial rotor timing pulse is being tested). If the indices pass this test, then the index corresponding to the midpoint between them (calculated as the rounded integer of the minimum and maximum indices cast to be double precision numbers) is considered to be the position of a rotor timing pulse, and this index is added to an array (RotorPulseIndices). Once a pulse has been found, the StartIndex is set equal to the EndIndex and the peak-finding process is repeated so long as the EndIndex is less than TotalCount. Once the rotor timing pulses have been determined, the CountsPerRevolution is calculated as the rounded integer of the average difference between entries in the RotorPulseIndices array. From CountsPerRevolution, the rotor speed (as rpm) is determined as 60∗ CountsPerRevolution/ ADClockRate. The rotor speed calculated in this man-

T.M. Laue et al.

ner is accurate to better than ±5 rpm, with a precision of ±2 rpm. Synchronizing data acquisition to the spinning rotor: The magnets used to produce the rotor timing signal are incorporated in the “over-speed disk” that is glued to the base of each rotor. The over-speed disk is positioned so that the two magnets are approximately midway between the first and last rotor hole (e.g. between rotor holes 1 and 4 for a 4-hole rotor). The magnet position is not exact, and changes every time the over-speed disc is replaced. Consequently, the rotor timing pulse provides only an approximate position, and separate means must be provided to determine the exact relationship between the rotor timing pulse and a reference position on the rotor as it passes a detector (below). Once this angle, MagnetAngle, is established at a particular rotor speed, all other angles needed to acquire data from a particular channel are fixed. The fixed angles include: the offset angle, called the CellAngle, from the center of the first rotor hole to the center of each of the other rotor holes; the offset angle from the center of a rotor hole to the center of each channel in a centerpiece, called the ChannelAngle; and the angles separating the optical detectors, called the DetectorAngle. Values of the CellAngle are constant for a given rotor type, and are just 360∗CN/CT , where CN is the cell number (starting at zero) and CT is the total number of cell holes. ChannelAngle values are constant for any given type of cell, and the angular positions of the XLI optical tracks, measured clockwise from the fluorescence detector, determine the DetectorAngle values. A database is used to store the ChannelAngle values for each type of cell. Similarly, each type of detector has a DetectorAngle value stored in the database. It is possible to accommodate new cell designs or add new detectors without changing the software by defining and storing these fixed angles. Light intensity data typically are acquired over a small angle (e.g. 0.2 degrees, called the DataOffsetAngle) on either side of the channel center. The starting angle for data acquisition, StartingAngle, is simply the sum all of the fixed angles, plus the MagnetAngle less the DataOffsetAngle. Determining MagnetAngle: In order to produce highquality data, it is essential that the angle over which intensity data is gathered is held constant to within a few thousandths of a degree. Because of propagation delays in the electronics, signals arrive at the computer some time after they are produced. During the delay, the rotor continues to turn, so that the fixed geometric angles described above appear to vary with rotor speed [2]. The variation in angle is complex, particularly at rotor speeds <∼ 6000 rpm, due to the frequency dependence of the components in the various signal processing circuits. Accordingly, the MagnetAngle is determined empirically. Experience suggests that the MagnetAngle should be checked whenever the ro-

A Light Intensity Measurement System for the Analytical Ultracentrifuge

tor speed changes by 5000 rpm. The operating software monitors the rotor speed and, once the rotor speed is stable, determines the magnet angle for each optical system. For the fluorescence system [6, and in preparation], a special calibration cell has been developed (www.camis. unh.edu) that incorporates a two-degree long by 1 mm high by 1 mm deep groove. The groove is centered at a radial position of 5.85 cm with a ChannelOffset of zero degrees. This groove is filled with a dye so that a bright fluorescent signal is produced whenever the sample is in the excitation beam. The MagnetAngle is determined as the apparent angle needed to move the “image” of the groove so that it coincides exactly with the angle predicted from the fixed angles (above). A similar scheme is used for the absorbance system, except that the image used is that of the pair of intensity pulses that occur whenever the inner calibration holes (at 5.75 cm) of the counterweight pass over the absorbance detector. Intensity measurements: Intensity data are acquired for each channel starting from its StartingAngle and covering an angle equal to twice the DataOffsetAngle. To accomplish this, the StartingAngle and DataOffsetAngle must be converted to the corresponding number of A/D clock pulses (i.e. range of indices in the data arrays). The StartingAngle is converted to an index offset, Offset, which corresponds to the fraction of counts in one revolution: Offset = CountsPerRevolution∗ (StartingAngle/360). The starting indices for the data arrays are simply the value of Offset added to each of the values saved in the RotorPulseIndices array. Similarly, the range of indices, IndexRange, over which intensity data will be gathered is: IndexRange = CountsPerRevolution∗ (2∗DataOffsetAngle /360). Typically, the IndexRange is between 10 and 100, meaning that that many intensity readings will be obtained for each A/D channel for each rotor revolution. The software is configured to acquire data from three to ten (default five) rotor revolutions. Thus, each average intensity reading comprises 30 to 1000 individual intensity readings. The upper limit of averaging over ten rotor revolutions maintains relatively rapid radial scanning while providing good reduction of the stochastic noise. There is no reason, however, that the upper limit could not be set much higher (the software limit is 180 scans, and it could be set higher). Our experience is that only a small reduction in stochastic noise can be expected with increased averaging. However, future versions of the software will allow for more averaging. From the moment the A/Ds complete their data acquisition until the average intensities are stored for all of the samples requires < 20 milliseconds on a 1 GHz computer. The computations are done while the radial positions of the optical systems are changed, which requires 50 milliseconds or more, depending upon the distance moved. Thus, the calculations are not the rate limiting step in data acquisition.

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Intensity scans: For mechanical reasons, both the fluorescence and absorbance data acquisition systems scan from the bottom to the top of the cell. The radial increment for the scans is adjustable from 2–50 microns (default = 20 microns). After each step, the fast A/Ds are triggered and intensity data are acquired for the period determined by the RevolutionsToAverage. Using the scheme described above, raw intensity data are available for all of the detectors and all of the cells. However, not all of the data are useful, and some tests must be applied to determine which data to include in the output file for a sample. The operating software uses five layers of logic to determine whether to include the intensity data from an optical system in the output file for a particular sample. These logic layers may be posed as questions: 1) is the optical system in use, 2) are data being acquired for this sample using the particular optical system, 3) is the radial position in the correct range (e.g. for multi-channel equilibrium cells) for the sample, 4) is the wavelength setting one that is being used for the sample, and 5) are the gain settings for the detector the correct ones for this wavelength? If all of these conditions are true, then the average intensity, along with the standard deviation, is determined and saved in a temporary array. When the scan is completed, the data files are written to disc, and the data displayed in the chart for that sample. All disc and charting operations are done during the period required to reset the detector positions to their starting positions (∼ 5 seconds).

Methods Data simulation: The operating software for the data acquisition system includes a signal simulator which allows the gains, voltage offsets, amplifier slew rates and noise level to be set for each A/D channel. The rotor timing pulse signal is simulated as a single sine wave whose angular extent and amplitude may be set. If the detector is situated at a radial position that is between the top and bottom of a channel, intensity data are simulated initially as a square wave pulse centered at the ChannelAngle for that channel. The amplitude of the pulse is adjusted according to the optical properties (scattering factor, extinction coefficient or quantum yield) and concentration of the solution components in the channel, as well as the gain setting for the A/D channel. For the A/D channels, gain settings are ∗ simulated as: PGA · e((10 V/4095)−10) , where PGA is the gain setting of the programmable gain amplifier (1, 2, 4 or 8) for the A/D channel, V is the D/A setting (0–4095) corresponding to the PMT high-voltage setting. The rise and fall times of the square pulse are adjusted for the slew rate for the A/D channel, and white-noise with user-adjustable amplitude (in bits) is added to the signal. In simulation mode, then, the signal to noise ratio may be controlled by the user.

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Jitter determination: The effect of noise on synchronizing data acquisition was assessed using the root-meansquare (rms) variation in the difference between the indices determined for each rotor timing pulse (Fig. 1). Any variation in these positions from the correct value directly affects the portion of the signals used to acquire intensity data. Rotor timing pulses were simulated with a fixed noise level, and the average distance (in number of data points) calculated between rotor timing pulses for ten rotor pulses. The jitter was calculated as the variance (in number of data points) for a set of ten average distances, and is reported as parts per thousand of the ratio of the variance to the average distance.

Results and Discussion Synchronizing data acquisition: Figure 1 shows the signal for a single rotor timing pulse, along with the discriminator values and the array index determined to be the “rotor timing pulse.” The index for all of the intensity data during the rotor revolution will be an integer offset from this index. The number of indices, Ni, between this rotor timing pulse, Rtp[i], and the next pulse, Rtp[i+1] corresponds to one turn of the rotor (i.e. Ni = Rtp[i+1] − Rtp[i]). Hence, the angular position to the center of any sample (ChannelAngle) may be converted to an index, ChannelIndex, as: ChannelIndex = Rtp[i] + Ni∗ ChannelAngle/360 .

Fig. 2 Simulated raw intensity data acquired from ten revolutions of the rotor. Data were superimposed using the conversion from intensity array index to angle, as described in the text. For data shown here, the rotor timing signal had a signal to noise ratio of 40 dB, and the intensity data shown here have ±40 bits of noise. The dashed vertical line shows the angle calculated to be the center of the cell. The solid vertical line shows the angle calculated to be the center of the sample channel, and the two dotted vertical lines enclose the range of angles that will be used to calculate the average intensity

It was found that double precision math, followed by rounding to the nearest integer, was required to minimize jitter. Likewise, any index i between Rtp[i] and Rtp[i + 1], may be converted to an angle using: Angle = i − Rtp[i]/CountsPerRevolution · 360. Intensity data acquired over ten revolutions of the rotor, then superimposed by converting their indices to angles, are shown in Fig. 2. The variation in the position of the signal pulses is a consequence of the rotor timing signal having a signal to noise ratio of 40 dB. At 40 dB, the uncertainty in the rotor timing pulse position is 0.1 part per thousand (Fig. 3), corresponding to an angular uncertainty of ±37 millidegrees. Radial scans: Because data may be acquired for all samples simultaneously at each radial position, significantly more data may be acquired for each sample over the course of an experiment. The time required to acquire a complete scan of the samples depends on several factors: the number of revolutions over which data are to be averaged, the rotor speed, the time needed to move the optics to the next radial position, the number of radial data positions sampled, and the time needed to reset the radial scan mechanism to the starting position. At rotor speeds above 40 000 rpm, the system requires approximately 60 seconds to acquire data at 600 radial positions (e.g. at 20 micron intervals over a 1.2 cm radial distance),

Fig. 3 Jitter as function of the signal to noise ratio of the rotor timing pulse. The jitter is presented in parts per thousand, determined as the root mean square variation in the number of data points between rotor timing pulses (Fig. 1) divided by the average number of data points between rotor timing pulses. Previous work [1] determined that a jitter less than 0.25 parts per thousand provides sufficient precision to acquire data. For the present system, jitter is acceptable so long as the signal to noise ratio is greater than 20 dB

A Light Intensity Measurement System for the Analytical Ultracentrifuge

averaging the intensities from 5 rotor revolutions at each radial position (resulting in 50 or so intensity readings per sample), then resetting the optical position to start the next scan. Using the current XLA absorbance system, approximately 720 seconds is required to gather data from three samples at the same radial spacing and averaging 5 flashes of the lamp (i.e. resulting in 5 intensities per sample). Thus, the new system provides significant speed and signal processing advantages over the current XLA absorbance optics. The shorter scan times and the use of a continuous light source in the new data acquisition system results in significantly improved data acquisition from samples containing rapidly sedimenting material. If samples provide dramatically different signals (e.g. there is > 100-fold concentration difference in fluorescent dye), it will be necessary to use different gain settings in order to optimize the signal to noise. Due to the time needed for the photomultiplier tube to settle down after changing its gain, it is not practical to change the gain at each radial position. Consequently, a separate scan must be acquired for each gain setting needed. This means that the time interval between scans of any given sample will in-

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crease in direct proportion to the number of gain settings used in a protocol. Fluorescence intensity scans of green fluorescent protein are shown in Fig. 4a. The total signal from this sample is about 300 (on a scale from 0 to 4095) and the rms noise is about 5, yielding a signal to noise of 35 dB. For comparison, this is approximately the same signal to noise ratio that would be obtained for absorbance data with an optical density of 0.5 ± 0.01. The apparent molecular weight distribution for this sample is presented in Fig. 4b. The peak at a molecular weight of 29 570 (using an assumed partial specific volume of 0.73 ml/g) is in reasonable agreement with the protein’s sequence molecular weight of 30 800.

Conclusion The light intensity data acquisition system described here offers four advantages over current methods: 1) greater signal averaging, thus reducing stochastic noise; 2) intensity data may be acquired from all samples simultaneously,

Fig. 4 Fluorescence intensity data from a sedimentation velocity (50 000 rpm, 20.0 ◦ C) experiment. The sample is 40 nM green fluorescent protein (M = 30 800) in 100 mM KCl, 20 mM Tris, pH 7.5. a Scans were acquired at approximately 50 second intervals, with scans taken ∼ 30 minutes apart shown in this panel. For each scan, intensity data were acquired at 20 micron radial intervals over the range from 5.9–7.3 cm. At each radial position, data from 5 rotations were acquired, yielding 264 individual intensity readings for each sample. The average and standard deviation of these readings is saved in the output file and presented here. All data were acquired using 73% of the maximum photomultiplier tube voltage and an 8× gain setting for the programmable gain amplifier. The photomultiplier tube gain is highly nonlinear, so that a 90% voltage setting is about 2 orders of magnitude more sensitive than the 73% setting. Hence, these data were acquired at a moderate, not high, gain setting. b Molecular weight distribution calculated by Sedfit [7] for the data in panel A. The peak in the distribution falls within 4% of sequence molecular weight

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increasing the number of scans that may be acquired per experiment; 3) intensity data may be acquired from multiple detectors (e.g. fluorescence and absorbance) simultaneously, thus increasing the amount of information that may be obtained from a sample; and 4) where data are to be acquired (e.g. over which radii and at what angles) may be defined by the user, thus making it possible to use new centerpieces and rotors without changing the data acquisition software. This system is available commercially in the Aviv Biomedical AU-FDS (Lakeview, NJ) fluorescence detector retrofit for the XLI analytical ultracentrifuge. It will be used for the upcoming rapid-scan absorbance system

XLI retrofit, and it is anticipated that it will be used on the next generation of analytical ultracentrifuge. The full characterization of the AU-FDS optical system (e.g. sensitivity, radial resolution, scan times, etc.) will be presented in a separate publication. Acknowledgement The authors wish to thank the following individuals for their patient help and encouragement in making this system a reality: Rachel Kroe, Jack Correia, Edward Eisenstein, Jeff Hansen, Bo Demeler, Jack Aviv and Ash Tripathy. This research was supported in part by NIH grant R01GM6283601, NSF grant DBI-9876582, the Biomolecular Interactions Technology Center, the Center to Advance Molecular Interaction Technology and the Aviv Family Foundation.

References 1. Laue TM, Yphantis DA, Rhodes DG (1984) Anal Biochem 143:103–112 2. Laue TM, Domanik DM, Yphantis DA (1983) Anal Biochem 131:220–231

3. Scholtan W, Lange H (1972) Polymere 250:782–796 4. Mächtle W (1999) Biophys J 76:1080–1091 5. Roark DE (2004) Biophys Chem 2004 108:121–126

6. MacGregor IK, Anderson AL, Laue TM (2004) Biophys Chem 108:165–185 7. Dam J, Schuck P (2004) Methods Enzymol 384:185–212

Progr Colloid Polym Sci (2006) 131: 9–22 DOI 10.1007/2882_002 © Springer-Verlag Berlin Heidelberg 2006 Published online: 10 February 2006

Saroj Kumar Bhattacharyya Patrycja Maciejewska Lars Börger Manfred Stadler Akif Mehmet Gülsün Hasan Basri Cicek Helmut Cölfen

Saroj Kumar Bhattacharyya · Patrycja Maciejewska · Helmut Cölfen (u) Department of Colloid Chemistry, MPI Research Campus Golm, Max Planck Institute of Colloids and Interfaces, Am Mühlenberg 2, 14424 Potsdam, Germany e-mail: [email protected] Lars Börger · Manfred Stadler Polymer Research Laboratory, BASF AG, G201, 67056 Ludwigshafen, Germany Akif Mehmet Gülsün · Hasan Basri Cicek Department of Electrical and Electronics Engineering, Bilkent University, 06800 Bilkent Ankara, Turkey

INSTRUMENTATION

Development of a Fast Fiber Based UV-Vis Multiwavelength Detector for an Ultracentrifuge

Abstract The advantages of simultaneously detecting multiple wavelengths in ultracentrifugation experiments are obvious, especially for interacting systems. In addition, the detection of the wavelength dependence of turbidity opens up the possibility to obtain independent information on the particle size in addition to the usual sedimentation coefficient distribution for colloidal systems. We therefore made an effort to develop a fast UV/Vis detector, which is able to simultaneously detect the range from 200–800 nm. This is possible by the use of a modern CCD chip based generation of UV-Vis spectrometers, which translates the dispersed white light onto a CCD chip, where each pixel corresponds to a particular wavelength. In addition to the simultaneous detection of a large number of wavelengths in the range 200–800 nm, also with non integer values, these spectrometers are very fast. Current typical spectrum scan times with the necessary scan quality in the ultracentrifuge are in the range of 100 ms but this time can be significantly shortened down to 3 ms for higher light intensities and even down to 10 µs for a new generation of CCD chip based spectrometers. The introduction of a fiber based UV-Vis optics into a preparative XL-80K ultracentrifuge with the associated hardware developments

will be described as a first generation prototype. In this study, we use a wavelength dependent optical lens system instead of the necessary but more complex wavelength independent mirror optical system for a first check on possibilities and limitations of the optical system. First examples for biopolymers and latexes will be presented and compared to those obtained in the commercial XL-A ultracentrifuge. Already the fast detection enables completely new possibilities like the determination of a particle size distribution in a few minutes. Multiwavelength detection at constant position in dependence of time will be demonstrated, which is an important mode for the use of speed profiles for very polydisperse samples. Also, the use of radial multiwavelength scans will be demonstrated producing a three dimensional data space for monitoring the sedimentation via radial scans with multiwavelength detection. However, despite the advantages, the current problems with the detector will also be discussed including the main problem that much intensity is lost in the important UV range as a result of fiber coupling and bending. Keywords Analytical Ultracentrifugation · CCD based UV-Vis Spectrometer · Detector Development · Fiber Optics

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Introduction The development of optical detection systems for ultracentrifuges is as old as Analytical Ultracentrifugation itself and dates back to the development of Analytical Ultracentrifugation [1]. The classical detection systems are UVVis absorption optics, Rayleigh Interference optics and Schlieren optics as discussed in detail in the classical book of Lloyd [2]. Furthermore, fluorescence optics was developed for the platform of the Beckman Model E [3] and more recently for the platform of the Beckman XL-I ultracentrifuge [4, 5]. In addition, turbidity optics was developed for the characterization of particle size distributions of colloidal particles [6–9]. Despite some progress on the detector or illumination part of the optical systems concerning the transformation of photographic detection to CCD chip detection for Schlieren and Interference optics or the use of laser light sources [10–18] (for further references see also the references for multiplexer systems) and photomultiplier detection for the UV-Vis [19] and fluorescence optics [3, 4], the classical optical systems remained essentially unchanged for decades. Exceptions are the Flossdorf optics [20], which significantly increased the detected intensity for UV-Vis optical systems and the ultrasensitive Schlieren optics [21]. However, the demands of Analytical Ultracentrifugation have greatly improved with the advent of cheap and powerful computer resources. The most obvious change is the speed of data acquisition and the amount of experimental data becoming available from computer based detectors, which has found its commercial expression in the launch of the Beckman XL-I ultracentrifuge in the 90s of last century [19]. This has in turn catalyzed many methodic advances, which enable analyses with an accuracy not being even imaginable a decade ago [22–26]. However, it must clearly be stated that the development on the data evaluation side is by magnitudes more significant than the hardware development – especially that of optical detection systems for the Analytical Ultracentrifuge (AUC). This is understandable as hardware development is by factors slower than software development due to the delivery and workshop manufacturing waiting times. Nevertheless, there is an urgent need for improved multi detector development for the AUC [27], not only concerning detection speed but also detection capabilities. Examples for the benefit of multi detector use are manifold. They include determination of extinction coefficients [28], even in complicated mixtures [29] but also the possibility of global analysis by evaluation of multiple experiments [30–32] as well as increased experimental information from a single experiment with multiple species and different chromophores by multiwavelength analysis. The detection of multiple wavelengths from UV-Vis absorption optics proved especially useful for the analysis of biopolymers from sedimentation equilibrium [33] or sedimentation velocity [34], although in the

S.K. Bhattacharyya et al.

reported cases, the analysis was restricted to the three possible scanning wavelengths of the XL-I AUC under addition of information from the interference optical system. Thus, the potential benefit of multiwavelength analyses of hundreds of wavelengths is obvious not only for interaction analyses of complicated interacting polymer mixtures but also for colloidal systems as the wavelength dependence of the turbidity of colloidal samples contains information about their particle size according to the MIE scattering theory [35–38]. With the advent of global analysis approaches involving AUC data, multiwavelength analysis will become an especially important technique for all light absorbing samples of colloidal or polymeric nature. Parallel to the above developments in AUC methodology, there was also a significant development in the UVVis spectrometer technology. Modern UV-Vis spectrometers can use simple CCD line arrays for the simultaneous detection of a large number of wavelengths, which is essentially only limited by the CCD array pixel number as well as the quality of the applied diffraction grating for the dispersion of the incoming white light. These spectrometers do not only allow for the simultaneous detection of the entire UV-Vis wavelength range, but are furthermore very fast and cheap. Commonly available spectrometers like the USB2000 by Ocean Optics allow for scanning times of down to 3 ms for an entire UV-Vis spectrum [39]. More recent instruments like the Ocean Optics HR4000 [40] or the series of CCD and ICCD based spectrometers available from LOT-Oriel [41] are magnitudes faster and can scan UV-Vis spectra as fast as 10 µs. The cooled spectrometers give a quite broad linear dynamic range with the advantage of working with low dark current due to cooling [41]. However, although these fast and powerful UV-Vis detectors are available, they were not yet applied as detectors for AUC’s. One of the reasons is that these detectors are based on fiber optics and also operate with incoming white light. White light with wavelength dependent refractive indices makes the precise use of a refractive optics impossible – at least in the UV range, where the conventional achromat lenses cannot be applied or vice versa in the visible range if an expensive UV achromat lens system is applied. This is only one obstacle, with no parallel in previous AUC optics, which always operates with monochromatic light – or in case of the XL-A AUC with a torroidally curved diffraction grating producing monochromatic light. This means that in AUC history, the challenge of a multiwavelength optics has not been undertaken despite the potential benefits. This was the starting point for the cooperation between the Max-Planck Institute of Colloids and Interfaces and the polymer physics department of BASF R&D. This study presents the first results from this cooperation aiming to improve the AUC capabilities by the development of new detection systems. We have on purpose started with the use of conventional wavelength dependent refractive index based lens optics as:

Development of a Fast Fiber Based UV-Vis Multiwavelength Detector for an Ultracentrifuge

1. Ocean Optics as spectrometer supplier also offers optimized lens systems. They are available as plug & play components for a fiber based optical system so that potentially, a fiber optical plug and play system could be applied using the commercial spectrometer software as a first data acquisition software. With this system, first experiences can be made for the further detector development. 2. The mirror optics that use small focal length spherical and parabolic mirrors are not standard optical components, and are not yet available for this application. 3. Fiber optics are only very rarely applied in AUC detectors. Therefore, potentials and limitations of this optical transport medium for AUC detectors shall be explored. 4. The detector hardware needs to be developed. A lens based system is relatively compact but also the most distorting optical system for white light. Nevertheless, first developments can be made with this compact setup. The adaptation to the mirror based system can be done on the basis of these experiences if necessary. As a consequence, the present study can only be seen as the first step towards a fast fiber based UV-Vis multiwavelength detector. Here we will only use classical lens optics as a case study well knowing about the problem of chromatic abberation. It is clear that this design cannot be the final prototype but there is already much to be learned not only about the actually detected light intensity but potential problems that may come in way for using other components as fibers or fiber coupled spectrometers. We have therefore also included developments, which are independent of the fiber optical system or multiwavelength detector so that we report about a fiber based high power Xenon lamp as well as on an advanced multiplexer hard/software based on earlier hardware developments [42] currently available as a LABVIEW software. Although just being a snapshot of an ongoing development process, the present study will be able to show the possibilities but also the first limitations of a fast fiber based UV-Vis detector for the AUC.

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(b) Fibers: 600 µm UV/Vis fibers in four channel vacuum feed through, OZ optics, Canada for illumination optics. 600 µm UV/Vis patchcord from Oceanoptics, Duiven, The Netherlands. (c) Focuser: Receptacle style focuser from OZ optics, Canada with biconvex lens ( f = 15 mm) and SMA 905 connector (d) Linear actuator: Linear actuator model T-LA28-SV (shorter version) with necessary power supply from Zaber, Canada for precise movement of the detector arm. (e) Lenses and Prism: Biconvex lenses f = 20.6 mm for custom built collimation optics and f = 20 mm for collecting transmitted UV/Vis light and a right angled prism from Linos, Göttingen, Germany. (f) Slit: 25 µm slit from slit-lens assembly of commercial Beckman XL-I, Analytical Ultracentrifuge. (g) Light source: Xe-Flash lamp with mirror (model L-4633-01) with the relevant power supply from Hamamatsu Photonics GmbH, Herrsching, Germany. (h) Optical Bench: From Owis GmbH, Göttingen, Germany with possibility for x, y, z adjustment and angular adjustment for coupling the light from the flash lamp to fiber. (i) Optical stage: From OWIS GmbH, Staufen, Germany with possibility to fine-adjust in x and y direction for the precise adjustment of the detection optics part of the detector arm. (j) Electrical feedthrough: 18 pin electrical feedthrough for electrical connection of the linear actuator from Lemo Electronic GmbH, Munich, Germany. Samples for measurement: The proteins were obtained from Sigma-Aldrich, Germany, the latexes were prepared in house. For data generation with the designed detector the following samples were used: (1) BSA (∼ 1.0 g/L) in 0.1 MNaCl, pH = 7.5. (2) Cytochrome C (∼ 1.0 g/L) in 20 mM MOPS pH 7.2 containing 2 microM K3 Fe(CN)6 and 5 mg/mL dodecyl maltoside. (3) Polystyrene Latex with different particle sizes (150 nm, 190 nm, 240 nm, 430 nm, 450 nm & 660 nm) were mixed and prepared by dilution with water so that satisfactory intensity of the UV/Vis transmitted light can be detected. (4) For the experiment corresponding to Fig. 12, Polystyrene latex particles of sizes 190 nm, 305 nm and 605 nm were mixed in a ratio of 1 : 2 : 2 by wt. and diluted with water to suitable dilution for the AUC UV/Vis absorption optics.

Experimental For the detector development, we used the following commercial products: (a) Spectrometer: USB 2000 Fiber Optics Spectrometer (UV/Vis), Ocean Optics BV, Duiven, The Netherlands with SMA 905 fiber optics connection, 25 µm entrance slit, grating: 600 lines/mm, L2 detector collection lens with UV optimized coating and OFLV-350-1000 2-nd order filter installed and operable with the Ocean optics OOIBASE 32 software. For the UV optimized spectrometer, we have applied a 1200 line holographic grating in combination with a 50 µm slit as the entrance aperture (Fig. 5, component 3) and all other UV optimized compounds as described above.

Results and Discussion Hardware Developments Illumination System. The whole optical system was designed as a modular system, which permits the use of as many commercially available components as possible and simultaneously enables easy exchange of individual components. For illumination, a Hamamatsu L4633-01 flash lamp was used with a built in reflector which focuses the white light directly into an optical fiber. For further details, see [43]. This design enables a fourfold greater relative light intensity relative to the conventional Hamamatsu

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Fig. 1 Set-up of the illumination system for the fiber optics. (1) Lamp, (2) Holder with x, y, z positioning unit, (3) Fiber, (4) Power supply

L-2435 flash lamp, which is installed in the XL-I. However, the main intensity gain is obtained from the fact, that much of the light is coupled into an optical fiber, whereas in the design of the Beckman XL-A AUC, a pinhole in front of the flash lamp is applied as optical aperture, so that much of the light does not enter the optical detection system. A direct intensity comparison between the fiber coupled Xenon flash lamp and the XL-A design in the wavelength range 400–800 nm revealed a factor of 35–70 higher intensity at the end of the fiber set-up depending on the applied wavelength. However, the flash repetition speed of the flash lamp is relatively slow (only 100 Hz), which corresponds to a maximum flash repetition of 10 ms. Although this is the same speed as the lamp of the XL-A, it is not satisfactory in view of the fast spectrometer detection capabilities. If it is considered that a single flash is only about 4 µs long, but the flash repetition takes 10 ms, it is clear that the flash lamp is the time limiting component in the present set-up at rotor speed > 6000 rpm. The set-up of the illumination system for the fiber optics is simple. An optical bench (OWIS GmbH, Staufen, Germany) with an x, y, z positioning element is used to mount the lamp and the fiber as shown in Fig. 1. This set-up enables the optimum coupling of the light from the lamp into a fiber, which can be adjusted by observing the beam intensity exiting the fiber on a sheet of paper. Coupling is then further optimized using the intensity reading from the spectrometer. Vacuum Feedthroughs/Heatsink Modification. One of the most significant changes of the AUC hardware necessary for the fiber based UV-Vis optical system is the need to feed light and electricity into and out of the vacuum chamber. In the present set-up, this is solved by drilling 6 holes into the heatsink of the XL ultracentrifuge, which was done in the mechanical workshop of the BASF (see

S.K. Bhattacharyya et al.

Fig. 2 XL-heatsink with vacuum feedthroughs and mounted fibers. (1) Focuser (2) Vacuum feedthrough for electrical cables (3) Window for further optical system (4) vacuum feedthroughs for fibers (5) Vacuum feedthrough for electrical cables

Fig. 2). This has the significant disadvantage that the Beckman warranty is lost so that future designs will be based upon XL-I heatsinks. However, modified heatsinks of the shown type are operated already for several years in the AUC laboratory at BASF AG, Ludwigshafen, Germany without any problem so that they are considered by the authors to be safe. In addition, the present design in principle allows for the mounting of three detector arms of the type presented in this work so that up to three detectors can be run simultaneously. As vacuum feedthroughs, either a four fiber 600 µm vacuum feedthrough from Oz Optics, Ottawa, Canada was applied as shown in Fig. 2 or one fiber vacuum feedthroughs with SMA 905 connectors (Ocean optics B.V., Duiven, The Netherlands). Both feedthroughs have their specific advantages and disadvantages. Whereas the one fiber feedthrough with SMA 905 connector is a very easy and modular system, it looses light (∼ 5%) as it creates an additional two fiber couplings. The four fiber feedthrough does not have this disadvantage. It also uses only one feedthrough for four fibers and should be the system of choice. However, if a fiber breaks, the whole feedthrough has to be taken out and be sent for repair, which can be a significant disadvantage at least in the prototype design process, where many modifications of the optical system are carried out. The feedthroughs for electrical cables are less problematic. Here, we use LEMO Electronic, Munich, Germany electric feedthroughs for 18 cables, where one feedthrough is sufficient for the whole UV-Vis fiber optics. Optical Fibers. The choice of optical fibers is difficult and always a compromise. On one hand, they should enable the transmittance of as much light as possible, while on the other hand, the precision of the optical system is determined by the fiber diameter, as this is the smallest spot size, the light can be focused to. Another requirement is

Development of a Fast Fiber Based UV-Vis Multiwavelength Detector for an Ultracentrifuge

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that the fibers should have a high transmittance over the entire UV-Vis range (200–800 nm). We have decided to allow for the maximum light intensity, meaning that we use 600 µm or even 1000 µm fibers (Oz optics or Ocean optics). Another issue, which should not be neglected is the capability of the fibers to bend without too much light intensity loss, especially in the UV range, as the fibers have to be bent in the vacuum chamber in the present prototype. As connector system for the fibers, we have applied SMA 905 connectors, which appear to be the most universal for our purpose, but other connectors are also possible. The Detector Arm. The detector arm is the central element of the UV-Vis optical system, as it contains all essential optical components of the optical system itself. The detector arm is schematically shown in Fig. 3. The fiber coming from the lamp is coupled to a collimation system (2 in Fig. 3). This collimation system focuses the light onto the center of the cell. However, it has to be mentioned that the focusing is only achieved for the visible part of the white light and that unavoidable errors are made as long as lenses are used due to chromatic aberration. This can partially be compensated for by using achromatic lenses but they are only available for the visible or UV range respectively and not over the entire range between 200–800 nm, which is envisaged in our application. Therefore, we did not use achromats for this prototype. Instead, we have focused the light onto the center of the cell (spot size ca. 2 mm) for the reason that we were not able to make

Fig. 3 The detector arm. (1) 600 micron patch fiber UV/Vis (Ocean optics), (2) The collimating lens system (self built), f = 20.6 mm biconvex, (3) 90◦ Quartz prism, (4) Slit-lens assembly (from the XL-A) only one lens f = 20 mm biconvex, (5) Focuser, one biconvex lens (15 mm), (6) 600 µm patch fiber (Ocean optics). The light path is also shown schematically

Fig. 4 left. Photograph of the arm, (1) Detector arm, (2) Stepping motor (0.1 µm resolution at up to 4 mm/s), (3) Detection unit with x-y positioning screws. Right: Mounted arm in the rotor chamber

the light parallel, which is the desired situation and in addition needed as much light for detection as possible to investigate the principal capabilities of this first generation prototype. Earlier efforts were made for producing collimated light by using plano convex lenses in the collimation optics part. However, the desired collimated light could not be achieved by using commercially available lenses. The light, which has passed the centrifuge cell is then collected by a modified slit-lens assembly from a Beckman XL-A (4 in Fig. 3) with a 25 µm slit forming the aperture for the radial resolution. The camera lenses of the original system were replaced by a single biconvex lens (f = 20 mm). After the light has passed the slit, it is collected by a commercial focuser system (Ocean optics, with a f = 15 mm biconvex lens) and coupled back into a fiber to the spectrometer. The whole slit-lens assembly can be adjusted by an x-y table to collect the maximum light. The entire illumination and detection parts of the detector arm (Fig. 3) are connected so that they move as a unit along the ultracentrifuge cell. This is achieved by a Zaber (Zaber, Richmond, Canada) step motor (Model T-LA28-SV), which is very fast and precise (Fig. 4). It enables a scanning speed of up to 4 mm/s with an accuracy of 100 nm. In principle, the motor can scan a whole cell in only 3 s. The motor axis itself pushes a plate attached to a spring to ensure mechanical contact between the step motor and the moveable part of the detector arm. The whole detector arm is shown in Fig. 4 in the unmounted state and mounted in the ultracentrifuge. The UV-Vis Spectrometer. The most important part of the UV-Vis optics, besides the detector arm, is the detector itself. In the present study, we have applied a USB2000 spectrometer (Ocean Optics, Duiven, The Netherlands). There are many available spectrometer options (see http://

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oceanoptics.com/), the most important being the choice of the diffraction grating and the spectrometer aperture (see Fig. 5). We have applied the 1200 line holographic grating for the UV optimized spectrometer & 600 lines blazed grating at 400 nm for the UV/Vis in combination with a 25 µm slit (for UV/Vis) and a 50 µm (for UV) entrance slit as the entrance aperture (Fig. 5, component 3). The main feature of the USB2000 UV-Vis spectrometer is the use of a CCD line array detector (a 2048-element linear silicon CCD array), which allows for the simultaneous detection of the entire spectral range within an integration time as short as 3 ms. Practical integration times used in this study are in the 50–100 ms range so that the spectrometer is rate limiting rather than the lamp. The USBS2000 spectrometer has several vital components. They are shown in Fig. 5 and explained in the following. First, the fiber with the light coming from the sample cell is connected to the SMA905 connector (1) of the spectrometer. After passing a filter (2), the light passes the 25 or 50 µm slit (3) as the entrance aperture. It is then collimated by a collimating mirror (4), which reflects the light from the entrance aperture as a collimated beam towards the diffraction grating (5), which disperses the white light. The refracted light is subsequently reflected by a focusing mirror (6), which reflects and focuses the light onto the detector array. The light collection efficiency is further increased by a cylindrical lens in front of the CCD array (7). Before the light finally reaches the CCD detector (9), it passes through a longpass filter (8) for the elimination of second and third order effects. All spectrometer lenses were coated with a special coating for increased UV transmittance. The spectrometer data can then be directly read out by the commercial spectrometer software. However, this is only of very limited use for the application in an AUC. We therefore used the LABVIEW drivers supplied by Ocean Optics and programmed our own application software (see 2. Software developments).

Fig. 5 The USB2000 UV-Vis spectrometer. Figure courtesy of Ocean Optics and reproduced with kind permission from Ocean optics BV, Duiven, The Netherlands. For description of the components see text

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Multiplexer and Data Acquisition System. To control the firing of the lamp, a multiplexer system is needed. For this, a trigger signal per rotor revolution has to be generated. This is achieved in our case by polishing an inner sector shaped part of the rotor next to the overspeed disk. This polished part is detected by a reflection light gate similar to an earlier system developed for the model E AUC [42, 44] resulting in a TTL rectangular trigger pulse per rotor revolution. This TTL trigger signal then serves for the precise determination of the rotor speed and the exact time between two rotor revolutions. This time interval is divided into 1000 steps corresponding to an angular resolution of 0.36◦ on the spinning AUC rotor and the time delay of a cell passing the optical detection part to the trigger signal from the rotor has to be calculated by a so-called multiplexer. These units were described before as analog and partially already as digital devices to enable the use of multiplace rotors without wedge windows; [14–17, 45–55]. Multiplexer technology is also common standard in the XL-I AUC although for this centrifuge platform, it is difficult to get precise information on the position where to grab a TTL output signal from the multiplexer. As such signals have to be grabbed directly from the XL-I boards, it is recommended that the user supplies their own separate system for the generation of a TTL trigger pulse similar to the one used in this study. As we aim at a complete software control of the centrifuge system, we applied a National Instruments PCI6602 timer card, which can be programmed under the LABVIEW environment using the supplied virtual instrument drivers. The programming of the multiplexer is described under Software developments. Principal Operation Modes The new fiber based UV-Vis detection system was developed for two principal operation modes. One is the familiar radial mode, where the radial concentration variation in the sample cell is observed at various times, as already realized in all previous AUC’s (Radial mode). This mode, however is not suitable for very polydisperse systems, and only very recently, it became possible to combine the radial profiles obtained at various speeds for a sedimentation velocity experiment [56, 57]. For very polydisperse samples, the use of speed profiles is essential to resolve the complete sample over the entire colloidal range [7, 58]. This is achieved by observing the sample at a fixed position over time so that via use of a speed profile, all samples pass the detector at some stage and are detected (Time mode). This mode is not only interesting for colloids but also for polymers, for example for the detection of aggregates etc. In addition, it has to be stated that the time mode is a very universal mode, where several different samples can be combined in one run, simultaneously enhancing the effectivity of the AUC measurements.

Development of a Fast Fiber Based UV-Vis Multiwavelength Detector for an Ultracentrifuge

For each of the detection modes, a calibration scan has to be performed to find the correct sector position as well as for the calibration of the radius. The cell position calibration is done fully automatically, as is the radial calibration. An option is implemented to store the calibration file and use it if the same rotor and cells are used again, which is especially useful for fast sedimenting samples. Alternatively, an option can be selected to perform the radial calibration after the last scan of the experiment, which allows for instantaneous sample observation but with the disadvantage that the detector positions are not precisely known until the calibration step at the end of the experiment. Both modes are restricted in speed by the flash repetition speed of the flash lamp. Time Mode. In this mode, the detector is set fixed at a defined and known position in the cell. Usually, this is the center of the cell, but if a short experimental duration is desired, a position near the top of the cell can be selected, where care must be taken that the detection position is far enough from the meniscus to avoid optical and physical distortions of the sedimenting boundary. If the highest fractionation of the sample is desired, a position near the cell bottom can be used. As the detector is very fast, it is possible to use more than one detection positions, as the motor of the detection system can move 4 mm/s and the acquisition of a spectrum is in the ms range. With this, the movement of the sample can be followed between two or more precisely known positions, which greatly enhances the accuracy of measurements in the time mode, as already discussed for the dual beam mode [59]. Up to now, in the existing turbidity optics [8, 9], the sample is detected at a single position and the cells have to be completely filled in order to know the starting point of sedimentation, with the related inaccuracies in case of small remaining air bubbles etc. In the time mode, the scan time interval is set as well as the radial position for detection and the length of the entire experiment. This mode is not only useful for very polydisperse samples but also for very fast detection, as a sedimentation velocity experiment can be performed in just a few minutes by speeding up the centrifuge to 60 000 rpm with maximum acceleration provided that the samples sediment sufficiently fast. This suppresses diffusion to the maximum possible extent. Also, the time mode runs can be very useful screening experiments to find out proper centrifugation conditions for radial scans of an unknown sample. In addition it can be used as supplementary data to radial scan data in a global analysis approach [32], as the time mode can detect minor amounts of large aggregates, which maybe lost from detection during rotor acceleration. Radial Mode. The radial mode is familiar to every centrifuge user. The described multiwavelength detector essentially works as the XL-A AUC with the difference of

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a higher speed and a whole acquired UV-Vis spectrum for each radial point. However, if the user is just interested in 1–3 wavelengths for radial scans, for example to be compatible with existing XL-A evaluation programs, they can be selected in analogy to the XL-A with the difference, that for the CCD based detector, it makes no difference in detection speed how many wavelengths are detected. For all other applications, the wavelength range is selected in order to save only useful data and to reduce data storage space required. Radial scan intervals down to 100 nm can provide a factor 100 in the amount of radial points as compared to the XL-A, which could be a very useful feature for sedimentation equilibrium analysis. Likewise the fast scanning times allow many more radial scan to be acquired per experiment than the XL-A, even with the additional multiwavelength information. However, making the radial steps too small can over-sample the data because two neighbor radial points could be not independent from each other anymore depending on the radial optical resolution of the system. Software Development The software for centrifuge control, multiplexer and data acquisition was written using the LABVIEW 7.1 (National Instruments) programming environment. The software has the capability of ultracentrifuge and step motor control via a serial port, spectrometer and flash lamp triggering via the timer card, as well as data acquisition from the spectrometer. It is convenient to classify software options into three categories such as General Options, Cell Options and Calibration Settings. 1) General Options. Before starting an experiment, some important parameters of the experiment are adjusted. Rotor Adjustment: 4 hole or 8 hole rotor XL Settings Adjustment: Speed, temperature and time setting of ultracentrifuge Scan Mode Time Mode Scan Radial Mode Scan Scan Options Adjustment Acquire intensity data instead of absorbance: Default is absorbance data Filter scan intensities below a certain level: Only in radial mode scan Stop XL after last scan Make radial calibration after experiment Method Scan Options Adjustment Delayed Start: Time delay before the first scan Time between Scans: In Radial Mode only Number of Scans: In Radial Mode only Experiment Directory

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2) Cell Options. For both time mode and radial mode scan, after enabling desired cells, their experimental parameters should be adjusted, which differ in the two modes Common Options Wavelength Selection: Up to 4 wavelengths Saving Full Spectra Wavelength Range Time Mode Options Radius: Position in the cell Acquisition/Averaging time for one data point (number of lamp flashes) Radial Mode Options Rmin and Rmax: Minimum and maximum radial distance Radial Step: Radial step size between two detector readings Replicates: Number of data points collected at each wavelength Set All Settings Identical to Cell 1: Sets the parameters of cell 1 for each cell 3) Calibration Settings. Cell Calibration: used to set the angular cell positions with respect to the triggering pulse. Position: angle from the trigger pulse to the cell Length: size of the cell in angle units Shifting:provides to access two different positions in the cell e.g. reference and sample sector to allow for the correction of the sample for the reference solvent Angle Calibration: the positions of the cells can be determined with this property Radial Calibration: inner and outer radii for the cells are determined Step Motor Settings: Adjustment of Port, Minimum and Maximum positions of motor Calibrate Outer Position: Maximum outer position of the step motor determined with a sensor on arm Graph Settings: Determination of Scale Settings When the scan starts, the user can observe the experiment online for each cell. Some important information is also shown such as XL Status and Cell Information. It is also possible to change some experimental parameters during the scan such as XL speed, temperature and the intensity limit below which filtering is done. This is an important improvement with respect to the commercial XL-I, which only allows for parameter changes after stopping an ongoing scan. All the options above can be saved in different files for different categories so that the user only needs to adjust necessary parts for any other experiment. The user interface of the control system was adapted to the style of the existing XL-I centrifuge control user interface to maintain an easy operation by the majority of users,

S.K. Bhattacharyya et al.

already being familiar with the XL-I instrument control and data acquisition software. Basic System Performance The basic check if the optical system correctly images the ultracentrifuge cell and also provides sufficient radial resolution can be performed by scanning a slit. We used a 200 µm slit, which was placed as coating on a cell window (see Fig. 6 lower image). This window was used in an AUC cell with a 2 mm centerpiece with the slit facing the centerpiece and oriented perpendicular to the radial direction. Scanning this cell in the intensity mode yields a radial scan, which should show a 200 µm wide rectangular peak. Ideally, the transition from zero to maximum intensity should be very sharp. However, as the aperture slit for the optics is only 25 µm, a better optical resolution cannot be expected. In Fig. 6, the result of this experiment is shown for the XL-A (right) and our fiber detector (Fig. 6 left). For the experiment, we chose a radial step size of 3 µm, the step size for the XL-A was set to the finest possible value of 10 µm. It can be seen that the transition zone for 0–100% intensity is about 50 µm for the fiber optics detector. Due to the limited number of data points, the transition is more shallow for the XL-A optical system. This demonstrates that our basic optical system performance is at least as good as that of the XL-A, although we use a lens optics, which shows the chromatic aberration problem for white light.

Fig. 6 Resolution observed by placing a 200 micrometer slit at the centre of the cell. The figure in the top left shows observation for the fiber optics based detector. The figure at upper right shows the observation for the commercial XL-A. Bottom: A light microscopy image of the slit. Scale bar = 500 µm

Development of a Fast Fiber Based UV-Vis Multiwavelength Detector for an Ultracentrifuge

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Results and Discussion Time Mode. The first sample, which we used to test our optical system was bovine serum albumin (BSA) – a standard protein sample which shows an absorbance peak at 280 nm in the UV-Vis spectrum. We set the slit position to 6.1 cm close to the meniscus in order to enable a fast measurement in the time mode. Absorbance was determined by recording the intensity of the sample and applying the intensity recorded for the reference solvent in a separate calibration experiment. The experimental raw data are shown in Fig. 7. The tremendous amount of data becomes directly obvious. Also, it can be seen that the present lower wavelength limit of the fiber detector is 240 nm, which just shows the onset of the 230 nm BSA peak. This is a bad situation for investigations of proteins, where usually the 230 nm peak is the most intense one but tolerable for most other samples. If only one wavelength is picked out of the multiwavelength scans, the time dependent sedimentation can be visualized in a similar way to the traditional AUC scans and the data quality can be controlled (Fig. 8). The expected sigmoidal curve is shown and the data quality is comparable to that of the XL-A. Repetition of the experiment in time mode with the fiber optics and in the radial mode in the XL-A yielded comparable results in terms of the sedimentation coefficients for the identical sample run on both instruments as shown in Fig. 9. The noise level in both scans is about the same although for the fiber optics, there are many more data points, which is advantageous for the subsequent evaluation. Another obvious difference is the broadness of the experimental curve. The apparent sedimentation coefficient distribution, which is not corrected for diffusion broadening is considerably sharper from the fiber optics than from

Fig. 7 Sedimentation velocity experiment for BSA, temperature: 25 ◦ C, Rotor Speed: 50 000 rpm, Slit position: 6.1 cm, lowest detectable wavelength 240 nm. Spectra recording condition: 98 ms integration time and 10 averages

Fig. 8 Absorbance vs time plot for BSA 1 g/L at 280 nm. (Experimental conditions; Rotor speed: 5000 rpm, Slit position: 6.1 cm. Data recorded with 98 ms integration time and 10 averages). The data were not yet corrected for reference solvent

Fig. 9 Comparison of XL-A and Fiber optics. Slit position 6.1 cm, Meniscus 5.81 cm, Fiber optics: 1 s time interval and time mode scan, XL-A: Continuous radial scan 0.05 cm, 2averages. For fiber optics: 14 100 points and for XL-A: 250 points. The data were recorded at wavelength 280 nm. Equation used to convert ‘r’ values to ‘s’: s = (1/ω2 t) ln(r/rm ); variables have their usual meaning

the XL-A. This effect is attributed to the fact that the detector was placed very close to the meniscus so that diffusion broadening of the boundary could not yet set in. Whereas the sedimentation coefficient distributions get sharper with time in the radial mode, as diffusion goes with the square root of time but sedimentation is directly proportional; this is not observed in the time mode as long as the detector is placed at a position, where diffusion broadening could not yet set in to a significant extent. On the other hand, the fractionation capability of the AUC is not fully exploited with that setting. A slightly tilted lower plateau is observed for the data from the fiber optics. The reason for this is not yet clear, but it has to be stressed that we do not yet use a reference solvent correction nor a correction for variations in the flash lamp intensity. In addition, this tilting of the lower plateau is not regularly observed. Another comparison between the time mode of the fiber optics detector and the radial mode in the XL-A was made for a mixture of three latex standards (150 nm, 190 nm & 240 nm). The results are shown in Fig. 10.

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Fig. 10 Sedimentation velocity of a latex mixture (Polystyrene latex mixture with particle sizes 150 nm, 190 nm and 240 nm). Left: Fiber optics, time mode scan (slit position 6.1 cm, Recording conditions, wavelength: 450 nm, Integration time: 100 ms, 10 averages). Right: XL-A, radial mode scan, continuous (0.005 cm and 2 averages)

It can be seen that the 3 components are resolved in both ultracentrifuges. Also, the detected concentration of the individual species is comparable for both ultracentrifuges. The data quality of the fiber optics detector is, however, much higher than that of the XL-A. The reason maybe the fact that the flash lamp intensity is lower in the visible range, which degrades the data quality. For the fiber optics, the UV-intensity is significantly decreased due to fiber bending (see discussion below) so that the data quality is better in the visible range for the present prototype. However, for the fiber optics, a significant tilting of the upper plateau is observed at the early times. Such tilting is always observed in case of turbid samples and could have to do with the significantly enhanced stray light for these samples. In the slit-lens assembly of the XL-A, an aperture is installed, which limits stray light from the sample to +/− 4◦ . This aperture is not yet realized for the present fiber optics prototype. Indicative for stray light is also the fact that just the upper plateau is tilted but not the lower plateau, where all turbid latices have already sedimented. The turbidity of colloidal dispersions is often a problem for their analysis by AUC. This is caused by the fact that the scattering of particles is size dependent according to the MIE theory where a small amount of large particles scatters more light than a large amount of small particles. This in turn affects the apparent light absorption and turbidity and makes the absorption optics sensitive to size dependence analysis. Therefore, a so called coupling technique was developed for the determination of particle size distributions with a turbidity detector, where the same sample is investigated at a high concentration to detect the small particles and at a low concentration for the detection of the larger particles [60]. This obstacle is omitted with the new multiwavelength detector, as the short wavelengths are especially sensitive, so that small particles could be detected at short wavelengths and the larger ones at the more insensitive large wavelengths in

a single experiment. This effect is demonstrated in Fig. 11 for a binary latex mixture, which is displayed for three different wavelengths. The drastic effect of the wavelength dependence of the detected signals is clearly visible. When for example the wavelengths 430 & 660 nm are combined, the two different latexes are detected with a comparable sensitivity. Another feature of the new detector is its speed. It is no problem to perform a whole sedimentation velocity experiment during the acceleration of the AUC to 60 000 rpm with maximum speed provided that the samples sediment fast enough. This is shown in Fig. 12 where the experimental raw data and the differential distribution being proportional to the particle size and sedimentation coefficient distribution are displayed. In only 3 minutes, the whole sedimentation coefficient distribution is determined. The huge advantage of such experiments beside the speed is the fact that diffusion broadening is suppressed to the maximum possible extent so

Fig. 11 Latex mixture (150 & 240 nm) observed at three different wavelengths (430, 450 & 660 nm). Experimental conditions: Rotor speed: 10 000 rpm, Temperature: 25 ◦ C, Slit position: 6.1 cm. Data collection conditions: 100 ms averaging time, 10 averages

Development of a Fast Fiber Based UV-Vis Multiwavelength Detector for an Ultracentrifuge

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Fig. 12 Fast sedimentation velocity Experiment with a Polystyrene Latex Mixture of sizes 190 nm, 305 nm and 605 nm in 1 : 2 : 2 mixing ratio by wt.; data collected at 350 nm; Left: Experimental raw data. Right: Derivative curve was generated performing differentiation after 5 point FFT smoothing. Slit position: 6.1 cm

Fig. 13 Upper left: Raw data for a total radial Scan (Sample: 175 nm Polystyrene Latex, 10 000 rpm, 25 ◦ C). Upper right: radial scan at a fixed wavelength. Lower left: Radial scan at a fixed wavelength (Zoomed view), lower right: Radial scan after cutting off noisy regions. Note that the radius is expressed in radius points corresponding to the steps of the detector movement and no real radial calibration was performed yet

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that a diffusion correction can become obsolete for many samples. However, the evaluation of such experiments has to be improved. With this, we can adapt the experiences with the turbidity detector where similar problems were faced. Radial Mode. The data amount of the radial mode is much higher than that of the time mode, as a whole spectrum can be saved for every radial point. A 1 cm solution column can in principle be scanned with 100 nm resolution, which corresponds to 105 data points per scan. For radial scans with the detector prototype, we first used the spectrometer software supplied with the spectrometer and visualized the data with a data viewer written by B. Demeler, UTHCSA, San Antonio. In the following we demonstrate how the useable data in a single scan can be extracted for the example of a 175 nm latex sample (Fig. 13). First of all, all experimental data are read in for a particular scan. If the whole cell is scanned as in this example, the first view along the cell from top to bottom just shows noise from the cell part before the sector starts (Fig. 13 upper left). Therefore, the plot has to be transformed from the wavelength display to a radial display at a selected wavelength (Fig. 13 upper right), which is a familiar data display for every AUC user. In the zoomed view (Fig. 13 lower left) the meniscus can be seen. Also top and bottom of the cell are clearly visible and define the data regions, which have to be cut off. This step is eliminated in the new data acquisition software, where the radius is calibrated before the experiment, so that only the useful data range is scanned. Also, noisy wavelength regions can be clipped until the final scan is obtained as shown in Fig. 13 lower right. If this scan is turned appropriately, the three dimensional nature of a scan becomes obvious (Fig. 14). Not only the sigmoidal radial sedimentation velocity profile is shown, but also the wavelength dependence of

Fig. 14 Radial Scan for a 175 nm polystyrene latex sample (10 000 rpm, 25 ◦ C)

Fig. 15 Radial scan for a 1.0 g/L solution of Cytochrome C (Speed 50 000 rpm)

turbidity, which can even be used to evaluate the colloid particle size independently. It is clear that such scan contains much more information than the conventional two dimensional radial scan. Especially fitting algorithms should greatly profit from the additional wavelength dimension. Another example for a single radial scan is presented in Fig. 15 for Cytochrome C. Even although the typical 230 & 280 protein absorption peaks could not be detected without significant noise in this example, Fig. 15 already shows the information increase of a single multiwavelength scan as compared to a radial scan at a single wavelength as performed in the commercial XL-I. The use of the introduced multiwavelength detection can easily be imagined for interacting systems with different chromophores. Problems. One of the big current limitations of the detector is the low light intensity in the UV region, which is the most important part of the spectrum – at least for

Fig. 16 UV-Vis spectrum detected without fiber bending on an optical bench and with fiber bending in the AUC

Development of a Fast Fiber Based UV-Vis Multiwavelength Detector for an Ultracentrifuge

many biopolymers. This is mainly caused by the necessity to bend the fibers upon installation of the detector arm in the ultracentrifuge (see also Fig. 4). Fiber bending reduces the light especially in the UV region < 250 nm. When the detector arm is mounted on an optical bench without fiber bending, there is still an intensity of more than 200 counts registered down to 225 nm, which would enable the investigation of proteins. Fiber bending, however, causes intensity loss of a factor of three around 250 nm and about a factor of two for the rest of the spectral range (Fig. 15). Therefore, the construction of the arm has to be modified to avoid fiber bending.

Conclusion and Outlook In this work, we could show a first prototype of a fiber based multiwavelength detector for the XL preparative ultracentrifuge. First measurements demonstrate that the detector is very fast with present typical scan times of only 100 ms/spectrum and delivers a data quality comparable to that of the XL-A. Due to the speed of the detector, two experimental modes are possible – the well known radial scanning mode as well as the time mode, where the detector is set at a fixed position and the sedimentation is observed with time. These modes extend the possibility of AUC with absorption optics detection to very polydisperse samples. Sedimentation velocity experiments as fast as two minutes become possible now, where just the centrifuge is accelerated to 60 000 rpm and immediately slowed down again. The detector design was made in a modular way so that all components can be easily exchanged. Therefore, advantage of new developments for example in lamp or spectrometer performance can be fully taken. For example, Ocean optics has very recently released the HR4000 spectrometer, which can scan as fast as 10 µs. This already allows the use of a continuous lamp

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with a pulsed spectrometer so that a very high light intensity could be realized in future designs on that basis. As commercially available compounds are applied, the price for the detector is also moderate. One drawback of the present detector is the high light intensity loss upon fiber bending and the use of lens optics with the associated chromatic aberration problem. Although the latter appeared to be of minor importance, we will develop a mirror optical system for this detector in the future, which enables defined and equal light beams at all wavelengths. In addition, mirrors can help to eliminate the necessity for fiber bending. Nevertheless, the present detector prototype is a cheap and modular add on, which transforms a preparative ultracentrifuge into an analytical machine. The now available multiwavelength data add a further dimension to ultracentrifuge analyses allowing for future use of global multiwavelength fits, evaluation of the particle size via the wavelength dependence of turbidity, a turbidity detection with multiple sensitivity, rapid sedimentation velocity experiments and many more. Nevertheless, the software basis for these applications still needs to be developed. Acknowledgement We thank the BASF AG, Ludwigshafen and the Max-Planck-Society for financial support of this work. We also thank Dr. Walter Mächtle, BASF, for useful discussions and Dr. Borries Demeler, UTHCSA, San Antonio for writing a data visualization program to display the multiwavelength data. The mechanical workshop of the BASF is acknowledged for the modification of the XL heatsink and for building the basic detector arm. Andreas Kretzschmar, mechanical workshop MPI-KGF, is acknowledged for multiple modifications of the detector arm. Henryk Pitas, electrical workshop MPI-KGF, is thanked for help with electrical problems. Dr. K. Tauer (MPI-KGF) is acknowledged for the latex samples and Dr. Neil Robinson (UTHCSA) for the cytochrome samples. Finally, we thank Antje Völkel for helpful assistance and Prof. Dr. Dr. h.c. Markus Antonietti for the overall support of this project.

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Ultracentrifugation in Biochemistry and Polymer Science. Royal Society of Chemistry, Cambridge, p 63 Börger L, Lechner MD, Stadler M (2004) Progr Colloid Polym Sci 127:19 Mächtle W (1999) Progr Colloid Polym Sci 113:1 Flossdorf J, Schillig H, Schindler KP (1978) Makromol Chem 179:1617 Giebeler R (1992) The Optima XL-A: A New Analytical Ultracentrifuge with a Novel Precision Absorption Optical System. In: Harding SE et al. (ed) Analytical Ultracentrifugation in Biochemistry and Polymer Science. Royal Society of Chemistry, Cambridge, p 16 Flossdorf J (1980) Makromol Chem 181:715 Cölfen H, Borchard W (1994) Ultrasensitive Schlieren Optical System. In: Bonner RF et al. (ed) Progress in Biomedical Optics, Vol 2136. SPIE Bellingham, Washington, p 307 Schuck P (2000) Biophys J 78:1606 Stafford WF (1992) Anal Biochem 203:295 Johnson ML, Correia JJ, Yphantis DA, Halvorson (1981) Biophys J 36:575 McRorie DK, Voelker PJ (1993) Self-Associating Systems in the Analytical Ultracentrifuge. Beckman Instruments California (http://www.beckman.com/Literature/ BioResearch/362784.pdf) Demeler B, van Holde KE (2004) Anal Biochem 335:279 and Ultrascan Website: http://www.ultrascan.uthscsa.edu/ Mächtle W (1991) Progr Colloid Polym Sci 86:111

28. Voelker P (1995) Progr Colloid Polym Sci 99:162 29. Cölfen H, Pauck T, Antonietti M (1997) Progr Colloid Polym Sci 107:136 30. Schuck P (2003) Anal Biochem 320:104 31. Cölfen H, Völkel A (2003) Eur Biophys J 32:432 32. Global analysis is also possible with the SEDPHAT software by Peter Schuck. See the following Website for further details: http://www.analyticalultracentrifugation.com/sedphat/sedphat.htm 33. Vistica J, Dam J, Balbo A, Yikilmaz E, Mariuzza RA, Rouault TA, Schuck P (2004) Anal Biochem 326:234 34. Balbo A, Minor KH, Velikovsky CA, Mariuzza RA, Peterson CB, Schuck P (2005) Proc Natl Acad Sci USA 102 1:81 35. Gledhill RJ (1962) J Phys Chem 66:458 36. Bateman JB, Weneck EJ, Eshler DC (1959) J Colloid Sci 14:308 37. Heller W, Bhatnagar HL, Nakagaki M (1962) J Chem Phys 36:1163 38. Hosono M, Sugii S, Kusudo O, Tsuji W (1973) Bull Inst Chem Res, Kyoto Univ 51:104 39. http://oceanoptics.com/technical/ engineering/USB2000%20OEM% 20Data%20Sheet.pdf 40. http://oceanoptics.com/technical/ engineering/OEM%20Data% 20Sheet%20–%20HR4000.pdf 41. www.lot-oriel.com/ccd 42. Cölfen H (1994) Bestimmung thermodynamischer und elastischer Eigenschaften von Gelen mit Hilfe von Sedimentations-gleichgewichten

43.

44. 45.

46. 47. 48. 49. 50. 51. 52. 53. 54.

55. 56. 57. 58. 59. 60.

in einer Analytischen Ultrazentrifuge am Beispiel des Systems Gelatine/Wasser. Verlag Köster, Berlin Visit the following website: http://usa.hamamatsu.com/en/ products/electron-tube-division/lightsources/xenon-flash-lamps.php Cölfen H, Borchard W (1994) Anal Biochem 219:321 Laue TM (1981) PhD Dissertation, Univ Of Connecticut, Storrs, CT, USA Rockholt DL, Royce CR, Richards (1976) Biophys Chem 5:55 Laue TM, Domanic RA, Yphantis DA (1983) Anal Biochem 131:220 Yphantis DA, Laue TM, Anderson IA (1983) Anal Biochem 143:95 Mächtle W, Klodwig U (1979) Makromol Chem 180:2507 Mächtle W, Klodwig U (1976) Makromol Chem 177:1607 Sedlack U, Lechner MD (1995) Progr Colloid Polym Sci 99:136 Ortlepp B, Panke D (1991) Progr Colloid Polym Sci 86:57 Schindler KP (1980) Avail NTIS. Report (Order No. PB81-167140):157 Flossdorf J, Schillig H, Schindler KP (1980) J Phys E: Scientific Instruments 13:647 Kuhnert R, Boedel E, Stegemann H, Wastl G (1973) CZ-Chem Tech 2:441 Müller HG (2004) Progr Colloid Polym Sci 127:9 Stafford WE, Braswell EH (2004) Biophys Chem 108:273 Müller HG (1989) Colloid Polym Sci 267:1113 Mächtle W (1999) Biophys J 76:1080 Mächtle W (1988) Angew Makromol Chem 162:35

Progr Colloid Polym Sci (2006) 131: 23–28 DOI 10.1007/2882_016 © Springer-Verlag Berlin Heidelberg 2006 Published online: 22 February 2006

INSTRUMENTATION

Victor P. Lavrenko Peter N. Lavrenko

Automatic Analysis of Lebedev Interference Patterns

Victor P. Lavrenko (u) · Peter N. Lavrenko Institute of Macromolecular Compounds, Russian Academy of Sciences, 199004 St. Petersburg, Russia e-mail: [email protected]

Abstract We developed a system for acquisition and analysis of interference patterns from an analytical ultracentrifuge or a polarising diffusiometer. The system is the first of its kind to be based on highly sensitive Lebedev optical system. We also present a novel algorithm for automatic extraction of interference curves from photographs. The algorithm is fast, robust in the face of

Victor P. Lavrenko (u) Present address: Department of Computer Science, University of Massachusetts, Amherst, MA 01003, USA

Introduction Optical interferometers play an important part in analysis of ultracentrifugation and diffusion processes. In recent years there has been increased interest in developing methods for automated acquisition of data from these interferometers, as well as in algorithms for their analysis by a digital computer. Computerized processing is valuable because it enables rapid scrutinizing of very large amounts of experimental results – something that is not possible with manual investigation. Several promising approaches [1–5] have been proposed and developed by Laue, Stafford, Mächtle, Cölfen, Harding, and others for the widely used Schlieren and Rayleigh optical systems. For example, the authors use a modified Walsh analysis [6] of light intensities across the interference envelope to obtain fringe phases and the magnitudes of associated transformations. However, previously proposed approaches are closely tied to the particular interferometer for which they were developed, and cannot be readily used with the Lebedev interferometer. In this paper we describe a system for automatic acquisition and analysis of data from the Lebedev polarising interferometer [7]. The Lebedev interferometer represents one of the most sensitive optical systems. It is sim-

optical noise and may have potential applications in other domains.

Keywords Digital recording · Optical noise · Polarising interference patterns

ple, convenient to use and enjoys a near-optimal balance of resolving power to noise. The Lebedev interferometer differs from the Rayleigh system and from similar approaches [9, 10] in that the interfering beams of light are not isolated from each other. Instead, both systems of rays pass simultaneously through the whole field, allowing the use of very small separations between the rays. The principle is schematically represented below for one arbitrary beam of polarised light.

Scheme 1

V.P. Lavrenko · P.N. Lavrenko

24

The interferometer is made up of two similarly oriented plane-parallel plates of Iceland spar of equal thickness. The plates are cut at a 45 degree angle to the proper crystal optical axis to provide the spar twinning, a. A half-wave plate λ/2 is placed between them to rotate the light polarisation direction for 90 degree. The ordinate value of the interference fringe, y, is proportional to the difference in the refractive indices in a cell, y ∼ (n 1 − n 2 ). A reader interested in a more detailed description of the system, or in comparing its properties with other interferometers will find a plethora of relevant material in [11]. For years the Lebedev optical scheme has been successfully applied to the analysis of sedimentation and diffusion of numerous polymers [8]. However, all these studies involved manual data acquisition and analysis of optical images. Manual analysis of interference patterns is an extremely tedious process. A photograph of an interference pattern in Fig. 1 serves to illustrate this.

based on the Lebedev polarising interferometer [8, 11]. The interferometer operates with the spar twinning of 0.10 cm and the Babbine compensator fringe interval, b, of 0.167 cm. Interference photographs were captured using a digital camera. We obtained satisfactory results using a very simple cameras, such as the Logitech QuickCam 2 web-cam and the Olympus C-310 digital camera. We used an additional short-focus lens to project the interference pattern from the compensator through the optics of the digital camera and onto its sensor. The camera sensor was aligned in such a way that rows of pixels were parallel to the radial direction of the ultracentrifugation. The camera was controlled by a personal computer through a USB cable, and could be programmed to capture interference images at set time intervals. Each image was then passed to the analytical algorithm, which automatically digitizes the interference curves and computes a set of standard summary statistics. In the following section we will provide a detailed description of that algorithm. Our algorithm may appear somewhat unusual to a reader familiar with pattern analysis, therefore, we will start discussing the challenges that motivated our choice and precluded simpler solutions.

Motivation

Fig. 1 The fragment of a striped pattern of Lebedev interference optics as recorded for a concentration boundary formed between decalin (left) and decalin solution of dioctyl phthalate at solute concentration c = 0.99 wt%. The recording time is 15 min

The photograph shows a set of five bell-shaped curves. Analysis of such a pattern involves the following three steps: 1. selecting the curve which exhibits the lowest level of noise, 2. digitising the curve, i.e. converting it to a set of {x, y} values, 3. computing a number of standard statistics for the curve, such as the location and height of the peak, the centre of mass, variance around the centre, etc. The second step of this process is particularly tedious when performed by hand. Therefore, this paper will focus in this step. We will first describe the experimental setup, which has been constructed for the automatic acquisition of interference photographs. Then we present a novel algorithm for digitizing the photographs obtained thereof.

Experimental Setup We used the MOM 3180 analytical ultracentrifuge manufactured by the Hungarian Optical Works [12]. The centrifuge was equipped with the Tsvetkov optical device

Figure 1 shows a pattern of the Lebedev interference optics, reflecting a concentration boundary between two liquids. Human eyes looking at the image will easily recognize a series of bell-shaped curves stretching from left to right. The curves are of nearly identical shape and can be thought of as the same curve repeated several times at different vertical positions. Our goal is, through several steps illustrated in Figs. 2–5, to construct a numerical representation of the curve – a set of points {x, y} similar to the one presented in Fig. 6. We must note that the exact scale of {x, y} values is somewhat arbitrary, our primary interest concerns the general shape of the curve and how it evolves over time. Extracting numerical values {x, y} from a photograph of the curve is a simple task for a human, but represents a formidable challenge for a digital computer as was already mentioned by Laue and others [2–4]. The apparent simplicity stems from the fact that humans are endowed with an incredibly powerful visual cortex, which quickly extracts salient patterns, filters out noise and fills in the missing portions of an image – all performed in a blink of an eye without a human realizing it. By comparison, the computer’s capabilities are extremely primitive. To illustrate shortly this claim, we would like to point out that any computer working with the photograph in Fig. 1 must explicitly deal with the following difficulties that humans will hardly notice: 1. There is no sharp boundary to distinguish the curve and the background – the computer will see shades of grey

Automatic Analysis of Lebedev Interference Patterns

2. 3. 4. 5.

where a human makes a judgement call as to what is black and what is white. The curves themselves are thick, and it is not trivial to identify their centre. Parts are missing for some of the curves, for example, the topmost curve is missing its base line, whereas the bottom curve is missing its peak. Optical noise produces small dark and light spots in random locations. Lighting intensity varies significantly over the photograph, for example the “white” background between the curves in the upper-right portion of the image is actually darker than the “black” curves in the lower-left portion.

Overall, a digital computer is presented with an input that is quite different from the human perception of the photograph. A computer sees a surface formed by light intensities, similar to the one shown in Fig. 2.

Fig. 2 The surface of the lighting intensity for the fragment presented in Fig. 1 as obtained with using the vidicon tube (for visual convenience, the image was arbitrarily turned left for an optimal angle). The x-scale ranges from 0 to 120 pixel columns, and y(×0.1)-scale ranges from 0 to 80 pixels

Darker portions of an image (e.g. curves) correspond to troughs, whereas lighter portions (space between curves) are represented by elevated regions. Recognizing a curve in this environment is quite challenging, particularly in the right portion of the figure where the distinction between curves and the background (troughs and peaks) is almost completely washed out. Below we will propose an automatic procedure that attempts to deal with the difficulties outlined above and ultimately converts a photograph (Fig. 1) into a corresponding numerical representation (Fig. 6).

25

Algorithm Intuitively, if we wanted to extract a numerical representation of a photographed curve we would need to: (1) find any initial point {x 0 , y0} that lies on the curve, and (2) “trace” the curve by finding adjacent points {x i , yi } until we hit the image boundary. Unfortunately, this simple procedure will run into substantial difficulties because of the noise inherent in the photographs. It is our experience that the procedure frequently gets stuck if there is a light spot on a curve, and will sometimes “jump” to a neighbouring curve if there is a dark region between the two curves. Simple smoothing and filtering techniques did not alleviate the problem. Therefore, we focused on a completely different solution. Instead of directly extracting the points {x i , yi } of the target curve, we will seek the points { dx i , dyi } that roughly correspond to partial derivatives at the ith point along the target curve. Note that dx i will be constant if the points are spaced out at equal intervals. Once we have obtained the derivatives, we can easily reconstruct the target curve by performing numerical integration over { dx i , dyi }. Now we turn our attention to extracting the partial derivatives { dx i , dyi } from the original photograph. Suppose that {x, y} is a point that lies on one of the curves in Fig. 1. A key observation behind our approach is that the partial derivative { dx, dy} at the point {x, y} defines a vector tangential to the target curve, and therefore tangential to the perceived boundary separating the dark regions of the photograph (the curves) from the lighter areas (the background). In other words, in order to find the partial derivatives { dx i , dyi } of the target curve, we need to find vectors that are tangential to the illumination boundaries in the photograph. The photograph is represented as an m × n illumination matrix V where Vj,i represents light intensity at the ith pixel of the jth line in the digitized photograph. Intensities are given in arbitrary units. As aforementioned, this matrix can be represented as an intensity surface (Fig. 2), where elevation is proportional to light intensity at that point. Accordingly, boundaries between light and dark regions will correspond to steep slopes in the intensity surface, and vectors {dx,dy} are precisely the vectors that are tangential to these slopes. This observation leads to a procedure for obtaining the tangent vectors: 1. For each coordinate ( j, i) in the digitized photograph we compute the numerical gradient: G xj,i = Vj,i+1 − Vj,i y

G j,i = Vj+1,i − Vj,i . Together, the matrices G x and G y define a vector field. A fragment thereof is shown in Fig. 3. Each gradient vector {G x ,G y } j,i is rooted at the point ( j, i) and points in the direction of the steepest rise in the light intensity surface, i.e. perpendicular to the boundaries between light and dark.

V.P. Lavrenko · P.N. Lavrenko

26

3. In the next step, we average the tangent vectors by column: dx i = dyi =

Fig. 3 Fragment of the vector field of the light intensity gradient (with horizontal x-axis in column units and vertical y-axis in pixel units, ≈ 20 µm/column and ≈ 9 µm/pixel, respectively) as obtained with using the program applied to the surface shown in Fig. 2. Vectors are directed to higher illumination, and their random disordering is resulted from the optical noises

m 1
x Tj,i m

1 m

j=1 m


y

Tj,i .

j=1

The averaging is motivated by the fact that the photograph contains several identical curves shifted vertically. Since the curves are identical, all non-zero partial derivatives {dx,dy} in the same column should ideally be the same. Averaging serves as a noisereduction step: whatever irregularities are present in any individual curve are unlikely to repeat within the same column and will hopefully wash out. The resulting set of partial derivatives {dx,dy} is shown in Fig. 5.

Fig. 5 Fragment of the vector field with the envelope ∂ f/∂x(x) function obtained by summation of the tangential vectors given in Fig. 4. The axes definitions are the same as in Fig. 3 Fig. 4 Fragment of the field of the vectors tangential to those given in Fig. 3. The axe definitions are the same as in Fig. 3

4. Finally, we recover the target curve from the partial derivatives by performing discrete integration: 2. Since we are interested in the vectors tangential to the boundaries, we rotate each gradient vector {G x , G y } j,i by 90 degrees. The result is a tangent vector field {T x , T y }, displayed in Fig. 4, where each vector points along the boundary between light and dark. Note that we can rotate each vector either clockwise or counterclockwise: in either case we get a vector tangential to illumination slope. We choose the direction of rotation so as to ensure that the resulting vector has a positive x. x-component Tj,i

xi =

i


dx j

j=1

yi =

i


dy j .

j=1

The result is shown in Fig. 6. Computing the summary statistics from the target curve is a trivial task.

Automatic Analysis of Lebedev Interference Patterns

Fig. 6 The desired f(x) distribution function with the required parameters as calculated by integration of the function given in Fig. 5. The axe definitions are the same as in Fig. 3

Discussion The algorithm proposed in the previous section is suited to extract a digitized curve from an interference photograph such as shown in Fig. 1. The algorithm is believed to be advantageous, specifically due to the following features: 1. The algorithm is extremely fast. It can process thousands of interference photographs per hour on a Pentium-4 personal computer. The speed results from the fact that all intermediate steps can be expressed as simple matrix operations. 2. The algorithm requires no interaction with the user and can be incorporated into a completely autonomous system for on-line acquisition and analysis of centrifugation and diffusion experiments. 3. The algorithm is robust to optical noise: we have demonstrated its ability to extract a curve from a photograph with non-uniform lighting, optical blemishes, washed-out boundaries and missing curve portions. 4. The algorithm takes advantage of all curves in the image. A typical curve extraction algorithm would rely on a single dark line in Fig. 1. Our algorithm leverages redundancy between curves; it effectively collates portions of broken curves (top-most and bottom-most curves in Fig. 1). The resolving power of an optical system can be evaluated by the ratio d/b, where d is an optical width of the interference fringe as the distance between the equal-density lines with an intensity equal to half of the maximum darkness of the fringe [11]. We have evaluated the average d/b value of 0.35 for the curve shown in Fig. 1 and a precision of ∆ y ≈ 0.2 fringes for the fringe displacement. The precision value for the final curve in Fig. 6 is approxi-

27

mately five times higher. Hence, the procedure developed in this paper leads to increased accuracy when localizing the fringe position. This effect is similar to the one attained for Rayleigh optics by Stafford [4] who has reached an accuracy of about 0.01 fringe in the blank corrected measurements. In addition to increased accuracy, the specific summation procedure proposed in our algorithm has the effect of smoothing out the fringe contour. The problem addressed in this paper bears a lot of similarity to the problem solved by Yphantis et al. [3]. However, the algorithm we propose is fundamentally different from the Fourier approach used by Yphantis et al. The authors in [3] perform a single-frequency Discrete Fourier Transform (DFT) on each column of the target interference photograph to determine the “phase” of the underlying intensity profile. Fringe displacement is computed as a cumulative sum of phase differences between adjacent columns. The algorithm proposed in this paper exhibits two important differences from the Fourier method [3]: (1) Single-frequency DFT requires the operator to select the target frequency. Yphantis et. al. [3] estimate this frequency as the number of pixels separating two fringes in an image. This step has to be done manually and the authors recommend that optimal target frequency be re-estimated for each new experiment. Contrarily, our algorithm makes no assumptions about the harmonic nature of the pixel column. Not relying on a harmonic representation gives our algorithm two important advantages. First, the algorithm requires no manual intervention or re-calibration from run to run. Second, the algorithm can be applied to images that cannot be processed with DFT, e.g. images with nonuniform spacing between the curves, or optically-distorted images where the middle of the image is magnified and contains fewer curves than the edges (lens effect). (2) The DFT method is limited to tracking curves with gradient no greater than 0.5 fringe per column of pixels. The limitation stems from the fact that Fourier analysis is performed on each column of pixels independently of its neighbours. In contrast to that, our algorithm uses both adjacent rows and adjacent columns to compute the tangent vector at each point in the image. This tangent vector can have any orientation, so technically there is no limit on the gradient of the curve to be tracked.

Conclusions We have presented a novel idea for digitising noisy interference images to extract intrinsic sedimentation and diffusion data from Lebedev interference optical systems by using gradient averages of intensity recording. The novel method of gradient vector summation was applied to eliminate some optical defects of the interference patterns and to provide a smoothing analysis. We are presently investigating applications of the algorithm to band images recorded with other interference schemes.

V.P. Lavrenko · P.N. Lavrenko

28

References 1. Rowe AJ, Wynne Jones S, Thomas DG, Harding SE (1992) In: Harding SE, Rowe AJ, Horton JC (eds) Analytical ultracentrifugation in biochemistry and polymer science. The Royal Society of Chemistry, Cambridge, p 49 2. Laue TM (1992) In: Harding SE, Rowe AJ, Horton JC (eds) Analytical ultracentrifugation in biochemistry and polymer science. The Royal Society of Chemistry, Cambridge, p 63 3. Yphantis DA, Lary JW, Stafford WF, Liu S, Olsen PH, Hayes DB, Moody TP, Ridgeway TM, Lyons DA, Laue TM (1994) On line Data Acquisition for the Rayleigh Interference Optical System of the Analytical Ultracentrifuge. In: Modern Analytical Ultracentrifugation: Acquisition and

4.

5.

6.

7.

Interpretation of Data for Biological Synthetic Polymer System. Birkhauser-Boston, Boston, p 209 Stafford WF (1994) Boundary Analysis in Sedimentation Velocity Experiments. Methods in Enzymology. In: Johnson ML, Brand L (eds) Numerical Computer Methods, 240, Part B. Academic Press, Orlando, p 478–501 Börger L, Lechner MD, Stadler M (2004) Prog Colloid Polym Sci 127:19–25 Beauchamp KG (1975) Walsh functions and their applications. Academic Press, Inc. London Lebedev AA (1930) Polarization interferometer and its applications. Rev d’ Opt 9:385–417

8. Tsvetkov VN (1989) Rigid-Chain Polymers. Plenum, New York 9. Brillouin M (1903) Compt Rend 137:786 10. Frenkel SY, Lavrenko PN (1983) Measurement of refractive index gradients. In: Ioffe BV (ed) Refractometric Methods in Chemistry. Khimiya, Leningrad, p 276–305 (in Russian) 11. Lavrenko P, Lavrenko V, Tsvetkov V (1999) Prog Colloid Polym Sci 113:14–22 12. Görnitz E, Linow K-J (1992) In: Harding SE, Rowe AJ, Horton JC (eds) Analytical ultracentrifugation in biochemistry and polymer science. The Royal Society of Chemistry, Cambridge, p 26

Progr Colloid Polym Sci (2006) 131: 29–32 DOI 10.1007/2882_003 © Springer-Verlag Berlin Heidelberg 2006 Published online: 4 February 2006

Joachim Behlke Otto Ristau

Joachim Behlke (u) · Otto Ristau Max Dellbrück Center for Molecular Medicine, Robert-Rössle-Str. 10, 13125 Berlin, Germany e-mail: [email protected]

D A T A A N A L Y SI S A N D MO D E L I N G

A New Possibility to Recognize the Concentration Dependence of Sedimentation Coefficients

Abstract The binding constants of self-associating proteins can be determined from sedimentation velocity runs using the numerical solution of the Lamm equation. This procedure requires good starting values of sedimentation and diffusion coefficients or their possible concentration dependencies. We found, based on fits of synthetic data files, a connection between the concentration dependence of sedimentation coefficients (ks ) and the estimated

Introduction Analytical ultracentrifugation is a powerful tool to study self-association of proteins. This technique is especially important to study enzymes that regulate their biological function though changes of their quaternary structure e.g. in [1, 2]. The classical approaches to study selfassociation events are using the sedimentation equilibrium or the concentration dependent analysis of weight average sedimentation coefficients, the so-called isotherm method [3]. An additional method to obtain association constants is based on the shape analysis of sedimentation velocity profiles [4–7], which can be performed rapidly from a single sedimentation velocity run. A disadvantage of this technique compared to a conventional sedimentation equilibrium approach is the fit of multiple parameters (sedimentation and diffusion coefficients) while suitable start values should be available. Moreover, one has to consider that the latter parameters can depend on the concentration. The concentration dependence of sedimentation coefficients which is described in the equation sc = s0 /(1 + ks c) has been shown by different laboratories [8–11]. The influence of not correctly consid-

radius position at the cell base (rb ). Deviation of the estimated rb value from the expected one is indicating the occurrence of ks that can be estimated by the program LammNum as a prerequisite for the calculation of reliable binding constants. Keywords Sedimentation velocity · Lamm equation · Binding constants · Concentration dependence of sedimentation coefficients

ered ks values to other parameters e.g. the estimated apparent position of bottom radius (rb ) has not been investigated yet. In this short communication we describe the influence of ks on rb and the possibility to predict a concentration dependence of s from the estimated rb values.

Methodical Approach When analyzing sedimentation velocity profiles of single species with or without association it is important to examine the curves whether a distinct concentration dependence of s has to be considered. In order to find a reliable criterion between ks and other hydrodynamic parameters we have simulated concentration gradients with different ks values using the finite element method based on the approach developed by Schuck [5]. Taking the law of mass action into account cw T

= cM +

i=n
i=2

ciM K iw

(1)

30

J. Behlke · O. Ristau

with cw T the total weight concentration, cM the monomer concentration or K iw the weight related equilibium constant and using the relations for spherical shaped complexes. si = i 2/3 sM Di = i

−1/3

(2) DM

for the constituent sedimentation or diffusion coefficients follows i=n  2/3 w i−1 1+ i K i cM i=2 (3) s(cw ) = sM i=n  w i−1 1+ K i cM

Results The fit procedure of synthetic data files for a monomerdimer equilibrium with a monomer molecular mass of 32.1 kDa and a ks value of 0.01 L/g yields a small discrepancy with respect to the dimer association constant when the concentration dependence of s is ignored (Fig. 1). This becomes more apparent for substances with lower or higher affinity to form self-associates. However, when the ks value is elevated up to 0.025 L/g and is not considered in the fit procedure the obtained difference from the ex-

i=2

1+ D(cw ) = DM

i=n 

i−1 i 2/3 K iw cM

i=2 i=n 

1+

i=2

(4) i−1 iK iw cM

The concentration dependence of sedimentation coefficient s(cw ) (Eq. 3) is approximately described by the dependence of the monomer sedimentation coefficient on the total weight concentration according to (Eq. 5). 1+ s(cw ) = sM /(1 + kS cw T)

i=n 

i−1 i 2/3 K iw cM

i=2 i=n 

1+

i=2

.

(5)

i−1 K iw cM

Data Acquisition

Fig. 1 Estimated dimer association constants for an assumed protein (M = 32.1 kDa, loading cw = 1 g/L). The fit procedure of synthetic data files (sedimentation velocity profiles generated with ks = 0.01 L/g) with the program LammNum (7) but without consideration of ks . The molecular weight was kept constant during the fit. The dotted line represents identity of observed with expected binding constants

We have generated a series of data files with the parameters s = 3 S, D = 8.44 F, ( f/ f 0 = 1.2) and dimer association constants of 100, 13.5, 4.5, 1.5, 0.5, 0.15 (L/g) and zero (no association). The ks values (Eq. 5) vary from 0.01, 0.025, 0.05, 0.075 until 0.1 L/g. The loading concentration is cw = 1 g/L and the molecular mass M = 32.1 kDa. Curve Fitting Synthetic data files were fitted using the program LammNum [7] that also allows estimating the radius position at meniscus and cell bottom. Each data set was fitted from the meniscus up to the crossing point of the concentration profiles. This radius position was used as starting bottom radius for fit. The steep bottom region, which is very likely described inaccurate by Eq. 5, was ignored. In all fits the molecular mass of 32.1 kDa was kept constant.

Fig. 2 Estimated dimer association constants for the same protein as in Fig. 1 and with the same fit conditions, but the data files were generated with ks = 0.025 L/g, which again was ignored in the fit procedure. As in Fig. 1 the dotted line shows agreement of observed and expected binding constants

A New Possibility to Recognize the Concentration Dependence of Sedimentation Coefficients

31

Table 1 Estimated hydrodynamic parameters for a self-associating protein (monomer-dimer) with a molecular mass of 32.1 kDa (kept constant during the fit). The dependence of sedimentation constant on ks (Eq. 5) was not considered during the fit. The loading concentration was cw = 1 g/L

sideration of ks . This is also true for non-interacting single species.

ks (L/g)

Our results are showing a relationship between ks the parameter which describes the strength of concentration dependence of sedimentation coefficients and the estimated apparent bottom radius position rb for sedimentation velocity experiments. To identify a distinct ks value is an important and necessary step in the fit procedure when estimating binding constants from sedimentation velocity runs. Neglecting of ks can lead to considerable divergence in the estimated parameters, e.g. binding constants etc. Our program LammNum allows the estimation of rb and therefore has the possibility to indicate a distinct concentration dependence of s. This can be used to obtain more reliable binding constants. When in praxis the estimated apparent bottom radius is distinct lower than the radius found in the 3000 rpm run the simultaneous estimation of ks may be necessary. In our analysis we have considered a broad range of ks data between 0.01 and 0.1 L/g and we have to discuss which of the values are realistic. As known from the literature one can expect nearly 0.01 L/g [10, 11] for globular proteins. More extended molecules possesses a clearly higher value [8] due to the increased surface. This seems to be valid also for globular proteins with a tail of unordered structure in the N- or C-terminal sequence of polypeptides, which usually can not be recognized in the X-ray crystal structure [12]. Furthermore, one has to expect a ks > 0.01 L/g for contributions of net charge when the chosen solvent conditions are far from the isoelectric point of a protein and not sufficient neutral salts are present. For all these cases the above mentioned procedure is helpful to estimate reliable parameters from sedimentation velocity runs using the program LammNum [7].

0.01

0.025

0.05

0.075 0.1

Bottom 7.20 radius (cm)

7.1967

7.193

7.188

7.183 7.182

Meniscus 6.2 radius (cm)

6.2

6.2

6.2

6.2

Sediment const. (S)

3.0

2.956

2.89

2.795

2.719 2.655

Diffusion const. (F)∗

8.44

8.317

8.14

7.864

7.650 7.470

Binding constant (L/g)

100

147

301

2673

4620

Frictional ratio

1.2

1.218

1.244

1.288

1.324 1.356

∗

0.00

6.2

1.1 × 109

F = 10−7 cm2 s−1

pected dimer association constant can vary up to multiple orders of a magnitude (Fig. 2). As we could demonstrate furthermore not only is the association constant different from the actual ones but also other hydrodynamic parameters vary when the real ks value in (Eq. 5) is ignored, (see Table 1). It is also striking that a substance unspecific parameter, e.g. the position of bottom radius is estimated incorrectly. This observation led to the conclusion that the estimated bottom radius could be an indicator for a distinct concentration dependence of sedimentation coefficients. As shown in Table 1 the estimated rb value is found to be to low when analyzing a self-associating protein without con-

Discussion

References 1. Behlke J, Heidrich K, Naumann M, Müller E-C, Otto A, Reuter R, Kriegel T (1998) Biochemistry 37:11989 2. Golbik R, Naumann M, Otto A, Müller E-C, Behlke J, Reuter RR, Hübner G, Kriegel TM (2001) Biochemistry 40:1083 3. Frigon RP, Timasheff SN (1975) Biochemistry 14:4559

4. Claverie J-M, Dreux H, Cohen R (1975) Biopolymers 14:1685 5. Schuck P (1998) Biophys J 75:1503 6. Schuck P (2003) Analytical Biochem 320:104 7. Behlke J, Ristau O (2005) LAMMNUM: A Program to Study Self-Associating Macromulecules in Sedimentation Velocity Experiments.

In: Scott DJ, Harding SE (eds) Modern Analytical Ultracentrifugation: Techniques and Methods. Royal Society of Chemistry UK p 122–132 8. Rowe AJ (1992) The Concentration Dependence of Sedimentation. In: Harding SE, Rowe AJ, Horton JC (eds) Analytical Ultracentrifugation in Biochemistry and Polymer Science p 394–406

32

9. Adams ET Jr (1992) Sedimentation Coefficients of Self-Associating Species. In: Harding SE, Rowe AJ, Horton JC (eds) Analytical Ultracentrifugation in Biochemistry and Polymer Science p 407–427 10. Teller DC (1973) Characterization of Proteins by Sedimentation Equilibrium

J. Behlke · O. Ristau

in the Analytical Ultracentrifuge. In: Hirs CHW, Timasheff SN (eds) Methods in Enzymology, Vol XXVII. Academic Press, New York, p 346 11. Kumosinski TF, Pessen H (1985) Structural Interpretation of Hydrodynamic Measurements of Proteins in Solution through

Correlations with X-Ray Data. In: Hirs CHW, Timasheff SN (eds) Methods in Enzymology, Vol 117. Academic Press, New York, p 154 12. Pikuleva IA, Tesh K, Waterman MR, Kim Y (2000) Arch Biochem Biophys 373:44

Progr Colloid Polym Sci (2006) 131: 33–40 DOI 10.1007/2882_004 © Springer-Verlag Berlin Heidelberg 2006 Published online: 15 February 2006

Emre Brookes Borries Demeler

Emre Brookes (u) The University of Texas at San Antonio, Dept. of Computer Science, San Antonio, USA e-mail: [email protected] Borries Demeler The University of Texas Health Science Center at San Antonio, Dept. of Biochemistry, San Antonio, USA

D A T A A N A L Y SI S A N D MO D E L I N G

Genetic Algorithm Optimization for Obtaining Accurate Molecular Weight Distributions from Sedimentation Velocity Experiments

Abstract Sedimentation experiments can provide a large amount of information about the composition of a sample, and the properties of each component contained in the sample. To extract the details of the composition and the component properties, experimental data can be described by a mathematical model, which can then be fitted to the data. If the model is nonlinear in the parameters, the parameter adjustments are typically performed by a nonlinear least squares optimization algorithm. For models with many parameters, the error surface of this optimization often becomes very complex, the parameter solution tends to become

Introduction A commonly used approach for determining observable parameters from experimental sedimentation data involves building a mathematical model describing the transport of macromolecular species in the ultracentrifugation cell. In such a model, every feature of the system under study which results in an observable signal such as a change in absorbance is described by an adjustable parameter in the model. The parameters are adjusted by a fitting method that seeks to minimize the difference between the model and the experimental data in a least squares sense: Min

n


[yi − Yi (P)]2

(1)

i

Here, yi represents each experimental data point i, and Yi (P) represents the corresponding data point simulated

trapped in a local minimum and the method may fail to converge. We introduce here a stochastic optimization approach for sedimentation velocity experiments utilizing genetic algorithms which is immune to such convergence traps and allows high-resolution fitting of nonlinear multi-component sedimentation models to yield distributions for sedimentation and diffusion coefficients, molecular weights, and partial concentrations. Keywords Analytical ultracentrifugation · Genetic algorithms · Sedimentation velocity analysis

by the model as described by parameter vector P. Models that are linear with respect to their parameters can be fitted by a generalized linear least squares approach in a single iteration, while models that are nonlinear with respect to their parameters have to be fitted in an approach that iteratively adjusts all parameters to an optimal value. The optimal parameter combination Pˆ represents those parameters that minimize Eq. 1. For highly nonlinear problems, models with many parameters, and parameters that exhibit a high degree of correlation, the error surface described by Eq. 1 can be very complex and contain many local minima which tend to trap the optimization process. For such cases, conventional gradient descent methods such as Gauss-Newton, Levenberg-Marquardt, and quasi-Newton commonly will fail to converge to the global minimum, and will provide less than optimal solutions for the fit. Other problems associated with direct fitting of experimental data relate to the selection of the correct model

34

function. Here, user input is required which introduces an unavoidable bias into the approach. To circumvent such caveats, other methods not relying on nonlinear least squares fitting have been explored. Not unexpectedly, each method exhibits both advantages and shortcomings. We will briefly review three popular methods for the analysis of sedimentation velocity experiments. Graphical transformations of sedimentation velocity data were introduced by van Holde and Weischet [1] and later refined [2]. This approach yields model-independent, diffusion-corrected sedimentation coefficient distributions. However, diffusion coefficients and molecular weights, or frictional parameters can not be reliably obtained by this method. In another model-independent approach, Stafford [3] introduced time-differencing of sedimentation velocity profiles to transform sedimentation velocity data to g(s) profiles. The main advantage of this approach is its capability to eliminate time-invariant noise contributions. Here too, molecular weights and diffusion coefficients are unavailable and sedimentation distributions are not deconvoluted from diffusion without further nonlinear least squares fitting. The C(s) method [4] avoids a multi-dimensional nonlinear least squares fit by linearizing the problem over the sedimentation coefficient domain using a constant frictional ratio f/ f 0 to provide corresponding diffusion coefficients. In the latter approach, a linear combination of finite element solutions to the Lamm equation [5] are fitted to the experimental data. The NNLS algorithm [6] is used to fit the linear coefficients of each term. The method finds all non-negative contributions, the remaining terms are set to zero. In an iterative approach, the frictional coefficient (which makes this problem nonlinear) can be fitted as well, reducing the problem to a wellconditioned one-dimensional search. Because a frictional coefficient is available, diffusion coefficients can also be obtained from the fit and can be used to determine molecular weights. This approach works well for those systems where all components have very nearly the same frictional properties, and can be equally well described by a single frictional ratio. Such cases may be present when all components in the system are globular proteins. However, when components of appreciably different frictional ratios are present, such as mixtures of globular proteins with different length aggregates, fibrils, extended DNA molecules, random coil conformations, or mixtures of any of the above, and complexes with intermediate frictional properties, a single frictional ratio will yield suboptimal results and is insufficient to describe the entire system accurately. When parameters from such a system are determined with the C(s) method, the determined molecular weights will be unreliable if only a single frictional coefficient is used to represent all components. Therefore, for such systems it is important to allow the frictional ratio or the diffusion coefficients to be adjusted independently for all components. Table 1 summarizes the results obtained from the analysis of a non-interacting mixture (Fig. 1) of a linear

E. Brookes · B. Demeler

Table 1 Comparison between sedimentation velocity methods. Data analysis parameters from the finite element and C(s) fitting method, and the van Holde–Weischet method. Molecular weights (Mw ) in parentheses indicate theoretical molecular weight Model

s20,W D20,W f/ f0 Mw (kD) Species (×10−13 ) (×10−7 )

Finite Element Fit, RMSD: 4.60 × 10−3

5.43 1.71

C(s) Fit, RMSD: 5.19 6.26 × 10−3 2.45 van Holde– Weischet

5.35 1.87

2.28 10.2

3.1 1.29

3.67

2.29∗

5.45

1.75∗

N/A N/A

N/A N/A

128.8 (130.7) 14.6 (14.4) 76.65 (130.7) 39.67 (14.4) N/A N/A

DNA Lysozyme DNA Lysozyme DNA Lysozyme

∗

Frictional ratios are corrected for the partial specific volume of DNA and lysozyme, respectively

208 basepair DNA molecule and a small globular protein (lysozyme), with three different approaches: 1. a nonlinear least squares fitting approach of finite element solutions to the Lamm equations (overlays are shown in Fig. 1), 2. the C(s) method, and 3. by the van Holde–Weischet method (results from all three methods are combined in Fig. 2). Results from the nonlinear least squares fitting with finite element solutions of the Lamm equation allow parameters from both components in the system to be adjusted independently, and are in excellent agreement with the known molecular weights of the two components. As expected, two very different frictional ratios are observed. While the van Holde–Weischet method results in sedimentation coefficients closely matching the nonlinear least squares fitting

Fig. 1 Sedimentation velocity data of a 208 bp linear DNA fragment and lysozyme fitted to a finite element solution of a twocomponent non-interacting model. Such a mixture is representative of a system that exhibits two very different frictional ratios in a single experiment. Experimental datapoints are represented by open circles, the finite element solution is shown as continuous lines. Parameters for this fit are shown in Table 1

Genetic Algorithm Optimization

Fig. 2 Sedimentation velocity analysis of the two-component system shown in Fig. 1 by the C(s) method (regularization with F-ratio of 95%, solid line, no regularization: line with upside-down triangles, RMSD = 0.0063), the van Holde–Weischet analysis (line with filled circles) and the direct boundary fit with the finite element method (stars, RMSD = 0.0046). Due to the difference in frictional ratio, the C(s) method fails to provide reliable sedimentation coefficient distributions, while the van Holde–Weischet method approximates more closely the sedimentation coefficients observed in the finite element direct boundary fitting method

results, molecular weight information or shape information is not available. When the C(s) method is used in conjunction with an adjustable frictional ratios, the ratio obtained can by design only represent the best average between the two species, and is therefore too high for lysozyme, and too low for DNA. When analyzed with no regularization, the C(s) method will introduce artifactual peaks around the lysozyme peak, and when regularization with an F-ratio of 0.95 is used, the peak is strongly broadened and the peak appears off-center. Furthermore, the C(s) analysis results in an RMSD 36% higher than the RMSD from the finite element fit. When the frictional ratio is combined with the sedimentation coefficient distribution, this value results in incorrect molecular weights for both species. Even after correction with the appropriate partial specific volumes the molecular weight of lysozyme is predicted too high, while the molecular weight of the DNA molecule is predicted too low. Therefore, we conclude that for systems involving components with different frictional ratios the C(s) method is not the appropriate approach. In addition, using a constant frictional term for the description of all species in the cell further complicates the interpretation of the results when molecules with different densities are examined (for example, mixtures of nucleic acids and nucleic acid binding proteins). From the results shown in Table 1 it may appear that the nonlinear least squares fitting approach with finite element solutions of the Lamm equation will provide the best information possible. In the example shown here, a bimodal

35

fit clearly results in a satisfactory solution, but this result will not be observed in the general case. The same approach will almost always fail for cases where more than 2 or 3 components are fitted and all relevant parameters are adjusted simultaneously (data not shown). The reason for this failure can be traced to the nonlinear least squares fitting approaches which are very sensitive to initial parameter guesses and the presence of local minima in the error surface of the fitting function. The problem becomes especially apparent when many nonlinear parameters are present, or the parameters are highly correlated. Such circumstances result in very complex error surfaces and often prevent convergence at the global minimum. Our goal is therefore to provide a method that combines a modelindependent approach with the generality of directly and independently fitting each parameter’s sedimentation and diffusion coefficient with an optimization method robust enough to reliably identify global minima in the error surface described by a multi-dimensional parameter space.

Proposed Method Here we propose an alternative approach to the global parameter optimization by nonlinear least squares fitting by introducing stochastic methods for parameter estimation. Stochastic methods do not rely on gradient descent in the error surface to find the best-fit solution, but instead introduce a random parameterization of the search space which allows the methods to probe for solutions by randomly placing parameter values within a constrained domain. Although these methods are computationally more expensive, they allow the solution to escape or ignore local minima and provide a higher likelihood of finding the global minimum. Several approaches employing stochastic parameter estimation have been explored, among them are genetic algorithms (GA) [7–9], Monte Carlo methods and simulated annealing methods [10, 11]. For our study, we chose to explore Monte Carlo methods and GAs. A Monte Carlo approach is based on randomly selecting parameters from the search space, simulating a model function based on these parameters, and evaluating the fitness function. However, there is no particular bias towards any solution, so if the search space is very large, the computational effort can be prohibitive because the entire space needs to be evaluated. When GAs are employed, a bias is introduced to the generic Monte Carlo approach by implementing a selection process analogous to natural selection in evolution. An initial random population of individuals is simulated, and each individual’s fitness is evaluated and the population is allowed to evolve. Parameter vectors are treated as genes which can exchange or modify parameters (bases) by crossover with other parameter vectors or by mutation, insertion or deletion operators. Multiple populations (demes) can evolve independently, or experience a controlled migration rate, which allows for exchange of

36

E. Brookes · B. Demeler

parameter information among multiple demes. Evolution of the best fit parameter combination within a population is controlled by a multi-generational selection process, which favors survival of individuals with a better fit. The survival pressure, migration rate, crossover frequency, mutation, insertion and deletion probability can be independently controlled by random number operators, and each probability rate needs to be optimized for best efficiency. The fitness function is given by the l 2 -norm of Eq. 1:   n
1  (2) [yi − Yi (P)]2 . n i

Description of the Algorithm The algorithm proceeds as follows: To limit the search to a reasonable domain, the search space is initialized with a model-independent approach. We found that the van Holde–Weischet analysis [1, 2] best serves this requirement by providing a lower and upper limit of the sedimentation space, and by providing an approximate number of species in the system. Using the number of estimated components and the sedimentation coefficient range, the search space is partitioned accordingly, and a set of random individuals are initialized with sedimentation coefficients randomly selected between the limits projected by the van Holde–Weischet method. In order to form a complete parameter set for a given species j, a diffusion coefficient is needed as well. We chose to initialize diffusion coefficients based on a reasonable range of frictional coefficient ratios κ j as a parameter to the sedimentation coefficient s for each species in the system. Equation 3 shows the formula used to calculate the diffusion coefficient for species j:   9s j κ j ν¯ η −1/2 −1 D j (s, κ j ) = RT(Nκ j 6πη) (3) 2(1 − ν¯ ρ) and 1.0 ≤ κ j ≤ κmax , where R is the gas constant, T the temperature in Kelvin, N is Avogadro’s number, η is the viscosity of the solvent, ρ is the density of solvent, and rˆ is the partial specific volume of the molecule. Here, κ j is assigned a random value in a reasonable range for the system under study. For globular proteins, we suggest values between 1.0 and 3.0, for linear DNA fragments κmax values as high as 15 are appropriate. The size of this initial set can vary between 50–500 individuals in each deme, and several demes can evolve simultaneously, depending on computational resources. Each pair of sedimentation and diffusion coefficients is used to calculate a finite element solution for the experimental conditions [12] with a unity concentration factor. The NNLS algorithm [6] is used to solve Eq. 4, where C is the vector of coefficients c j for each species j, and L is the matrix of Lamm equations used to model

each individual component. The values of c j correspond to the relative concentration of each species in the linear combination that forms an individual in the deme. CL(s j , κ j ) ≈ B .

(4)

A fitness value is then computed for each individual in the population using the l 2 -norm shown in Eq. 2. At this point, the first generation is completed and the GA is used to calculate the next generation. For each individual, progeny is generated by applying the GA operators on the parameter sets defining the individuals. Selection takes place by preferentially discarding individuals which display a poor fit, and maintaining the best individuals in the population. It is important to avoid dominance of one particular individual by always maintaining a range of fitness in the population, which will assure diversity in the parameter pool. The next generation is created by applying the GA operators to the selected individuals. Each operator is controlled by a probabilistic rate constant. Here, the deletion operator may delete a species from an individual, or a new species is added to an individual. Individuals containing fewer species are assigned a higher selection rate than individuals with more species and the same fitness value in order to reflect the lower computational cost of reproduction. Mutation operators may change the value of any one species’ sedimentation or diffusion coefficient, and the crossover operator allows two parental individuals to randomly exchange information encoding one or more species. The fitness of the individuals in the new generation is again evaluated, and the process is repeated until convergence is approached. Once the solution reaches an optimum, a nonlinear least squares optimization routine can be applied to quickly find the estimated global minimum in the vicinity of the GA solution. Once the solution is in the vicinity of the global minimum, any gradient descent will perform much more quickly in locating the global minimum, since the error surface is generally well conditioned in this much reduced parameter space. To alleviate the computational demand for solving this stochastic problem, we have implemented the GA on a parallel architecture using SUSE Linux 64 on a 44-node opteron cluster at the Department of Biochemistry at the University of Texas Health Science Center at San Antonio [13].

Results and Discussion A typical problem that may be faced in a laboratory is the analysis of a DNA binding protein complex, with some free DNA and free protein present in the mixture. If the association kinetics are sufficiently slow, such a system can be simulated with a noninteracting model and will present a situation where all components will have different frictional ratios: A large frictional ratio for the free DNA, a globular frictional ratio for the free protein, and an intermediate frictional ratio for the complex. To ver-

Genetic Algorithm Optimization

37

ify the capability of the proposed method, we simulated such a system and added noise of comparable quality to that observed in the XL-A analytical ultracentrifuge to the solution. The parameters for this 3-species system are listed in Table 2. Data analysis results from the van Holde–Weischet method, the C(s) method, and the GA optimization are presented in Fig. 3. As can be seen from this Figure, the results from the GA are nearly identical with the parameters used in the original simulation. We further compared the results from different speed simulations (20, 40 and 60 krpm, data not shown) and from globally fitting all three speeds simultaneously. Table 3 lists the results from these experiments and indicates the errors between the simulated parameters and the parameters determined with the GA optimization. The results suggest that the highest speed provides the most reliable information for the partial concentration and sedimentation parameters, as long as enough signal for the fastest component can be collected. Adding slower speeds to a global multi-speed fit further improved the optimized paTable 2 Parameters used for the simulation of the 3- and 8-species velocity experiments. For the 8-species system, parameters were chosen to determine if components with the same molecular weight, but different shapes (species 6 and 8) and species with closely spaced sedimentation and diffusion coefficients (species 3 and 4) can be resolved. Also, the components were broadly spaced in s and molecular weight, to determine the limits of the method in its ability to resolve multiple species, even if they are closely spaced Model

Partial s concentration

3 species, 1 0.3 3 species, 2 0.2 3 species, 3 0.8 8 species, 1 0.1 8 species, 2 0.2 8 species, 3 0.3 8 species, 4 0.4 8 species, 5 0.3 8 species, 6 0.2 8 species, 7 0.1 8 species, 8 0.15

4.238 ×10−13 5.943 ×10−13 9.023 ×10−13 2.481 ×10−13 3.707 ×10−13 4.587 ×10−13 5.063 ×10−13 1.053 ×10−12 7.827 ×10−13 1.294 ×10−12 1.760 ×10−12

D

7.343 ×10−7 1.622 ×10−7 1.872 ×10−7 1.080 ×10−6 6.423 ×10−7 4.416 ×10−7 3.147 ×10−7 9.651 ×10−8 1.514 ×10−7 1.502 ×10−7 3.405 ×10−7

Molecular f/ f 0 Weight (kD) 50

1.203

198

3.765

298

2.754

20

1.112

50

1.375

90

1.644

140

1.994

945

3.435

450

2.809

750

2.389

450

1.249

Fig. 3 Sedimentation coefficient distributions reported by various data analysis methods when applied to the simulated 3-species system shown in Table 2 (60 krpm). Shown are the target values from the simulation (stars), the van Holde–Weischet analysis (dotted line), the C(s) analysis without regularization applied (solid line) and the results from the genetic algorithm (vertical bars). Residual bitmaps for the C(s) fit and the genetic algorithm fit are shown in insert on top left. The characteristic diagonal indicating systematic deviations that can be seen in the C(s) analysis bitmap is absent in the genetic algorithm fit, indicating that a more appropriate fit is obtained with the genetic algorithm that allows for a variation in the frictional ratio

rameters. To further test the capabilities of this method, we attempted to identify the components of an 8-species system, where the components cover a large range of molecular weights. Simultaneously, we wanted to challenge the resolving power of this method by simulating a system where two components may share the same molecular weight, but have markedly different frictional ratios. Such a condition may be presented by a system undergoing a conformational isomerization, an example may be a folded protein with a fraction of the protein in an unfolded or improperly folded state. A system displaying both heterogeneity in sedimentation and in frictional properties may be presented by aggregating proteins forming fibrils and other extended structures. The simulation parameters for the 8-component system are listed in Table 2, the results for the fit and errors are shown in Table 4, and a comparison of the partial concentrations and calculated molecular weights with those from the target distribution is shown in Fig. 4. Here, the results suggest that the method can resolve well 8 components with disparate frictional ratios extending over a large sedimentation coefficient range. While sedimentation coefficients and partial concentrations are reproduced most reliably, the resolution of diffusion coefficients can be impaired when species with similar sedimentation coefficients but diverging diffusion coefficients are present. In all cases we compared evolutions spanning 100 generations and 500 individuals

38

E. Brookes · B. Demeler

Table 3 Parameters obtained from the average of 25 runs of 100 generations of the GA optimization for the simulated 3-species system shown in Table 2. The same system was simulated with a speed of 20 krpm (GA fit RMSD: 5.74 × 10−3 ), 40 krpm (GA fit RMSD: 5.79 × 10−3 ), and 60 krpm (GA fit RMSD: 5.84 × 10−3 ). We show here the results from the 60 krpm experiment and the results from the global fit of multiple speeds (20, 40 and 60 krpm, GA fit RMSD: 4.05 × 10−3 ). As can be seen from these results, not only is the overall percent error for parameter determinations smaller for a global multi-speed analysis, but the RMSD of the fit is smaller as well, indicating a better fit. The highest increase in accuracy results from the improvements in the estimation of the diffusion coefficients. We speculate that the longer run times in the low speed runs improve the signal for the diffusion and hence provide additional information to the global fit allowing a higher accuracy for diffusion determinations. Each run took approximately 7 minutes using 1 CPU Parameter

60 krpm

% Error

20/40/ 60 krpm

% Error

Concentration 1: Concentration 2: Concentration 3: Sed. Coeff. 1:

0.2946 0.2047 0.8009 4.228 ×10−13 5.915 ×10−13 9.021 ×10−13 7.239 ×10−7 1.899 ×10−7 1.890 ×10−7

− 1.80% + 2.35% − 0.11% − 0.24% ×10−13 − 0.47% ×10−13 − 0.02% ×10−13 − 1.42% ×10−7 + 17.01% ×10−7 + 0.96% ×10−7

0.2989 0.1998 0.8010 4.236

− 0.37% − 0.10% + 0.13% − 0.05%

5.934

− 0.15%

9.016

− 0.08%

7.358

+ 0.20%

1.6250

+ 0.19%

1.875

0.16%

Sed. Coeff. 2: Sed. Coeff. 3: Diff. Coeff. 1: Diff. Coeff. 2: Diff. Coeff. 3:

and 410 demes. At this point a well isolated solution had emerged in all cases, and further iterations did not improve the fit. It should be noted that the RMSD values from all individuals in each iteration provide a statistical sampling of the parameter space which can be used in place of Monte Carlo analysis results. An example of such a parameter distribution is shown in Fig. 5 for the sedimentation coefficient distributions of the 8-component system. We further observed a trend in the speed of convergence. For the case of the 3 component system, we noticed that convergence was most rapidly obtained in the 20 krpm and the global 20/40/60 krpm fits, followed by the 40 krpm fit, and then trailed by the 60 krpm fit. While all fits resulted in a convergence with essentially the same RMSD, the relative performance at different speeds suggests that different experimental speeds result in varying signal strengths that can be distinguished by the GA. The performance of a GA is measured by the rate of convergence to a target solution. Tuning the parameters controlling the GA has a major impact on performance. Let N be the popu-

Table 4 Parameters obtained from the GA optimization for the 8-species system shown in Table 2 simulated with 60 krpm. At 60 krpm the errors for the sedimentation coefficient are much smaller than for the diffusion coefficient, indicating a stronger signal from sedimentation than diffusion. The resolution of the method is limited for diffusion coefficients when the species sediment very close together, even when the diffusion coefficients are far apart (species 8, see discussion). This multi-deme run took approximately 10 hours using 44 CPUs Species

Concentration

s

D

1

0.100379 (− 0.38%) 0.198768 (- 0.62%) 0.306232 (+ 2.08%) 0.394280 (− 1.43%) 0.198518 ( − 0.74%) 0.301060 (+ 0.35%) 0.099526 (− 0.47%) 0.151501 (+ 1.00%)

2.49 × 10−13 (+ 0.36%) 3.69 × 10−13 (− 0.46%) 4.61 × 10−13 (+ 0.50%) 5.06 × 10−13 (− 0.06%) 7.80 × 10−13 (− 0.34%) 1.05 × 10−13 (+/− 0.0%) 1.29 × 10−13 (− 0.46%) 1.75 × 10−13 (− 0.62%)

1.062 × 10−6 (− 1.67%) 6.190 × 10−7 (− 3.63%) 4.190 × 10−7 (− 5.12%) 3.290 × 10−7 (+ 4.54%) 1.370 × 10−7 (− 9.51%) 1.000 × 10−7 (+ 3.62%) 1.600 × 10−7 (+ 6.52%) 5.340 × 10−7 (+ 56.83%)

2 3 4 5 6 7 8

lation size times the number of generations. N is the total number of individuals tested throughout the run and is approximately proportional to the total running time. We evaluated the effect of several factors impacting the performance of the GA. First, we compared the performance of a small population and a large number of generations to the performance of a large population with a small number of generations. At the extremes, a population size of one with N generations would be useless, and a population size of N with one generation would be equivalent to random guessing. Neither case is optimal as can be seen from the results presented in Table 5. Next, we considered crossover and mutation rates together. In crossover, parts from two good individuals are taken and combined to create a new individual for the next generation. Mutation is applied by taking one individual and adding randomness to one or more parameters in this individual to create a new individual for the next generation. Naively, one may want high rates for both crossover and mutation operators, but our total population size is restricted, so high rates of mutation would hide any crossover benefit. Preliminary results indicate that crossover likely provides a benefit in finding solutions to sedimentation velocity experiments. From this data it is clear that the GA method affords remarkable resolution in partial concentration, sedimentation and diffusion coefficient determination for sedimentation velocity experiments. Therefore, the method

Genetic Algorithm Optimization

39

Fig. 4 Molecular Weight distribution obtained from the genetic algorithm when fitting the simulated 8-component system listed in Table 2. The simulated target values are represented by stars, the values derived from the genetic algorithm optimization are shown as vertical lines. The vertical position of the stars and the height of the lines correspond to the partial concentration of each species. As can be seen from this graph, the genetic algorithm faithfully reproduces the number of components and the partial concentration of each component in the system, and in all but one cases closely matches the molecular weight of the target. The two targets at 450 000 dalton represent two species with the same molecular weight, but different frictional ratios and sedimentation coefficients. In this case, only one of the species could be resolved correctly Table 5 Performance observed for a fixed N (N = population size × generations). Targeted is the 3-species velocity experiment. For each population size and generations pair, 25 separate GA runs were performed with different random seeds. The lowest RMSD individual is chosen from each run giving 25 individuals from which the average and best RMSD are reported. The first case of a single generation can be considered a pure Monte Carlo method since no GA operators are applied. An intermediate population and generation size results in the most consistent performance improvement Population size Generations

Average RMSD Best RMSD

1000 100 50 20 10 5 2

0.01173 0.00946 0.00932 0.00919 0.00919 0.00947 0.04487

1 10 20 50 100 200 500

0.00965 0.00902 0.00906 0.00903 0.00901 0.00901 0.01776

excells at resolving molecular weights, even for systems with a relatively large number of components. In addition, GAs gain an increase in accuracy by globally fitting experiments conducted at multiple speeds. Insertion and

Fig. 5 Monte Carlo parameter distributions derived from the 8-component genetic algorithm fit for the sedimentation coefficient distribution. Here the sedimentation coefficient distribution from the best-fit 5000 individuals are shown. The best definition is for well separated components in the center of the distribution which have the largest amplitude and the narrowest parameter spread. The best fit parameter combination is represented by the tips of each peak, and corresponds to an RMSD of 5.84 × 10−3 . The parameters at the bottom of the distribution are derived from those individuals with an RMSD similar to the RMSD of the worst individual in the top 5000 individuals. The area under each peak is exactly 5000 individuals, since all top 5000 individuals showed 8 species

deletion operators effectively and automatically control the selection of the appropriate model, removing the user bias associated with the selection of a fitting model, providing a model-independent and general approach for fitting sedimentation velocity experiments. Overall, the reproduced accuracy and resolution is unmatched by any other method available to us.

Future Outlook We are currently investigating if GAs can also be applied to determine self-association properties and equilibrium constants, and if GAs can automatically determine the appropriate model for an interacting system. Future work in this area will focus on further reducing the search space and optimizing the GA itself. While the potential of this method for resolving individual species in a sedimentation velocity experiment are obvious from the presented data, GA calculations are computationally intensive and are best performed through parallel computation. Optimization of the method itself remains a requirement before this analysis method can be adopted on a routine basis. Although a poorly optimized algorithm will eventually arrive at the global optimum, convergence rates may vary drastically. Considering the computational expense of Lamm equation evaluations, a well-tuned GA is an important requirement. Our preliminary findings indicate that factors such as population size, the number of demes and generations, the

40

E. Brookes · B. Demeler

rates for mutation, crossover, deletion, insertion, migration, and the method chosen for initialization all effect the convergence rate. Thus, we can consider these factors as parameters in a second optimization problem. Since the GA is a stochastic process, each parameter vector must be tested on multiple target systems. It is quite possible that performance tuning will suggest different search conditions for different systems, and that these parameters are dependent on the number of species, s and D distributions, and partial concentrations. Each parameter needs to be evaluated with many trials using different random seeds. We are continuing to explore this parameter space. We further propose that this method could easily be extended to also include related experiments in a global analysis such as a combined analysis of sedimentation velocity and

equilibrium experiments, and dynamic light scattering experiments. Here, the experiment can be described with a nonlinear least squares fitting model containing the same parameters used in the sedimentation velocity experiment (D and the ratio of s/D, which is proportional to molecular weight). In summary, we conclude that the GA approach excells in avoiding local minima convergence traps. From our analysis we conclude that the only limitation of the method is the signal contained in the data. As soon as changes in concentration needed for the determination of a parameter are masked by experimental noise, the limit of resolution has been reached. Highly correlated parameters such as multiple diffusion coefficients for molecules sedimenting with similar sedimentation coefficients have also proven to be difficult to resolve.

References 1. van Holde KE, Weischet WO (1978) Biopolymers 17:1387–1403 2. Demeler B, van Holde KE (2004) Anal Biochem 335(2):279–288 3. Stafford WF (2000) Methods Enzymol 323:302–25 4. Schuck P (2000) Biophys J 78(3):1606–19 5. Lamm O (1929) Ark Mat Astron Fys 21B:1–4 6. Lawson CL, Hanson RJ (1974) Solving Least Squares Problems. Prentice-Hall, Inc., Englewood Cliffs, New Jersey

7. Holland JH (1975) Adaption in Natural and Artificial Systems. University of Michigan Press 8. Goldberg DE (1989) Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley 9. Koza JR (1992) Genetic Programming: On the Programming of Computers by Means of Natural Selection. MIT Press, Cambridge, MA 10. Kirkpatrick S, Gelatt Jr CD, Vecchi MP (1983) Science 220:671–680

11. Cerny V (1985) J Opt Theory Appl 45/1:41–51 12. Cao W, Demeler B (2005) Modelling analytical ultracentrifugation experiments with an adaptive space-time finite element solution of the Lamm equation. Biophys J 89(3):1589–1602 13. Demeler B (2005) UltraScan version 7.1 – A Software Package for Analytical Ultracentrifugation Experiments. The University of Texas Health Science Center at San Antonio, Department of Biochemistry, San Antonio TX, 78229 USA http://www.ultrascan.uthscsa.edu

Progr Colloid Polym Sci (2006) 131: 41–54 DOI 10.1007/2882_005 © Springer-Verlag Berlin Heidelberg 2006 Published online: 10 February 2006

Peter Zipper Helmut Durchschlag Angelika Krebs

Peter Zipper Physical Chemistry, Institute of Chemistry, University of Graz, Heinrichstrasse 28, 8010 Graz, Austria Helmut Durchschlag (u) Institute of Biophysics and Physical Biochemistry, University of Regensburg, Universitätsstrasse 31, 93040 Regensburg, Germany e-mail: [email protected] Angelika Krebs Structural and Computational Biology Programme, European Molecular Biology Laboratory, Meyerhofstrasse 1, 69117 Heidelberg, Germany

D A T A A N A L Y SI S A N D MO D E L I N G

Modeling of the Dodecameric Subunit of Lumbricus Hemoglobin

Abstract For modeling the lowresolution shape of the dodecameric subunit of Lumbricus terrestris hemoglobin, experimental smallangle X-ray scattering (SAXS) data and ab initio modeling approaches using a genetic algorithm or simulated annealing have been applied. In addition to the use of strict ab initio approaches, procedures which additionally include available structural information concerning symmetry and shape in the form of constraints or templates have been employed to improve the results. Templates for the subunit were preferably derived from SAXS-based models for the native hexagonal bilayer (HBL) complex that were biased by electron microscopic reconstructions. The obtained subunit models were carefully examined by variation of different selection and averaging methods and other checks such as surface renderings of the models. The findings were quantified by prediction of scattering profiles, I(h) and p(r), and structural

Introduction The hexagonal bilayer (HBL) appearance of the annelid hemoglobins represents an extraordinary type of quaternary structure observed among hemoglobins [1, 2]. Their extracellular occurrence is predicated on the large molecular size that prevents loss through membranes. The most extensively studied HBL hemoglobin is that of the common earthworm Lumbricus terrestris (L.t. Hb). This

and hydrodynamic parameters (V , RG , dmax , s, D). The best matching models for the subunit were also scrutinized by comparing them to a model derived from currently available crystallographic data. The following results could be obtained: (i) The obtained parameter predictions for the dodecameric subunit are satisfactory, if compared to the SAXS data (consensus model, profiles and molecular parameters) or the results from hydrodynamic studies. (ii) The comparison between solution and crystal data of the dodecameric subunit, however, unequivocally proves a different behavior of the subunit in solution and the crystalline state. Keywords Advanced modeling techniques · Dodecameric subunit · Extracellular hemoglobin · Hydrodynamics · Small-angle X-ray scattering

giant, multisubunit molecule has a sedimentation coefficient of about 60 S and its molecular mass is likely to be 3.5 MDa. Although, in the past, a broad range of masses has been reported for L.t. Hb, recent considerations [3] provide evidence for the validity of the given mass for the HBL assembly consisting of 144 globins and 36 nonglobin linker chains. The assembled chains form a molecule with D6 symmetry [4, 5], and the validity of the previous “bracelet” model [6, 7] has been

42

proven meanwhile. 12 subassemblies of globin chains are arranged in two hexagonal layers, and, as a consequence of this assemblage, a central cavity is formed in the particle center. The linker subunits are required for HBL formation [8, 9]. Relevant information concerning the three-dimensional (3D) structure of L.t. Hb and other annelid hemoglobins mainly emerged from electron microscopy (EM) including 3D reconstructions [1, 10–15], small-angle X-ray scattering (SAXS) [16–20], and X-ray crystallography [21]. The dissociation behavior of L.t. Hb was investigated at extremes of pH and in the presence of several dissociating agents (e.g., [6, 7, 22–24]). A variety of dissociation and re-association experiments resulted in the observation of a 9–10 S species which was considered to represent one-twelfth protomer, an approx. 200 kDa linker-free subassembly [25, 26]. Again, EM [15, 27], SAXS [28] and crystallographic studies [29, 30] concentrated on the 3D structure of the dodecameric subunit, in particular to identify it as the putative major dissociation product. Because of its special architecture, L.t. Hb has already been subject of several modeling attempts. Previous SAXS studies aimed at the elucidation of beaded consensus models for the native HBL L.t. Hb [19] and its dodecameric subunit [28], using conventional (trial-anderror) modeling approaches. More recent studies tried to apply a variety of different modeling approaches, in order to combine and compare the results from quite different solution and nonsolution techniques. Such approaches are indispensable links between different methods, eventually enabling the critical comparison and the transfer of molecule parameters from one method to another. As a first approximation, wholebody approaches have been applied to the HBL complex of L.t. Hb [31–33] and the dodecameric subunit [32, 34, 35] as well. These simple approaches already proved the compatibility and reliability of the basic parameters applied (e.g., radius of gyration, volume, axial ratio, sedimentation coefficient, in addition to external parameters such as molar mass and partial specific volume), however, revealed slight overestimations of predicted s values [31, 32, 34]. The HBL complex has also been subject of detailed multibody approaches [20, 31, 32, 36–39], employing primarily the information from solution techniques (SAXS, hydrodynamics) and EM. In particular, several SAXS-based ab initio methods were applied, to test their capability to predict the low-resolution particle shape and structural and hydrodynamic parameters of this complex protein, utilizing either strict ab initio conditions or the advantage of additionally adopting realistic constraints or templates. Amongst the advanced modeling programs available, approaches using a genetic algorithm (GA; program DALAI_GA [40, 41]) or simulated annealing (SA; program DAMMIN [42, 43]) turned out to be especially helpful for the reconstruction of 3D solution structures [32, 38]. A crucial problem encountered with all approaches which model protein shapes in solution is the

P. Zipper et al.

problem of particle uniqueness [32, 38, 44]. For the dodecameric subunit of L.t. Hb only one concise modeling attempt has been performed that compared the effectiveness of a few DAMMIN results [32]. No detailed modeling attempt of the subunit or inclusion of crystallographic data has been tried up to now. The present study tries to model the dodecameric subunit of L.t. Hb on the basis of experimental SAXS data for the subunit [28] in a detailed way, considering numerous aspects that are of importance for ab initio modeling studies. First, the most relevant approaches (GA, SA) were applied, varying input parameters, possible constraints and EM-biased templates as well as averaging procedures over a wide range. The obtained models were controlled by surface renderings and a quantitative comparison of predicted and observed molecular parameters. These parameters comprise aspects usually considered in SAXS and analytical ultracentrifugation (AUC) as well. Second, since just now the crystallographic data for the subunit [30] became available, the comparison of models derived from the 0.26 nm resolution crystal structure and those from solution scattering turned out to be a further challenging task, to validate or invalidate the models under consideration. In this context, again, various input variables and models can be tested and the influence of hydration contributions can be taken into account. The significance of the project is focused on a variety of useful modeling aspects rather than on biological data. Low-resolution shapes and molecule parameters are predicted by ab initio modeling approaches and the results obtained under various input conditions will be tested and compared to other techniques including crystallography. It is clear a priori that the anticipation of data by ab initio modeling can only provide rough shape estimates, unless accessory information is included. In particular, the results will show that the combined use of solution scattering and hydrodynamic data, in connection with 3D crystallographic information, allows various interesting aspects to be addressed.

Methods All modeling approaches described and used in the following are based on scattering curves derived from SAXS experiments or calculated from crystallographic data. The models obtained by GA or SA were used to anticipate structural and hydrodynamic parameters, after a variety of preceding steps such as selection of appropriate models, superimposing of models, and averaging procedures for the generation of proper consensus models. In some SA calculations, appropriate templates, ultimately based on 3D EM reconstructions of the native HBL complex, were used. Finally, the results obtained from the obtained consensus models were compared to available reference data.

Modeling of the Dodecameric Subunit of Lumbricus Hemoglobin

Sources of Data Scattering profiles and structural and hydrodynamic parameters of L.t. Hb and its dodecameric subunit were taken from previous papers [19, 20, 28]: SAXS data originate from measurements performed at protein concentrations, c, ranging from 5 to 53 mg/ml; the experimental scattering curves had been extrapolated to c = 0 and evaluated by the program ITP [45]; sedimentation coefficients descend from runs performed at varying c (0.04–12 mg/ml). EM-biased templates were derived from [39]. The atomic coordinates of the dodecamer were obtained from the Protein Data Bank (PDB) [46]. For calculations, primarily the following data were used: atomic coordinates, scattering intensity, I(h) [where h = (4π/λ) sin θ, θ = half the scattering angle, and λ the wavelength of the incident X-rays], pair-distance distribution function (PDDF), p(r) [where r = distance between pairs of scatterers and p(r) ≡ 0 for r values greater than the maximum particle diameter], molar mass, M, radius of gyration, RG , hydrated volume, V , maximum particle diameter, dmax , translational diffusion coefficient, D, and sedimentation coefficient, s. Modeling by Ab Initio Approaches Particularly substantial low-resolution models can be obtained by a GA or SA. (i) The program DALAI_GA [40, 41] uses an iterative fitting of scattering curves by a GA. The GA gradually explores a discrete search space and evolves convergent bead models fitting the target experimental profile upon Debye calculation of the beaded models. Modeling refinement is achieved by reducing the bead size together with the search space in successive cycles. (ii) The program DAMMIN [42, 43] is based on SA. It starts with a search space filled with densely packed dummy atoms and assigns the atoms either to the protein or the solvent. By SA an appropriate particle configuration is finally obtained that matches the experimental SAXS curve satisfactorily. For the calculation of scattering intensities of the models, the DAMMIN approach uses spherical harmonics. As primary input for the experimental data, a GNOM file [47] is required. The program may be used in the strict ab initio sense or allowing the input of additional structural information (e.g., use of symmetry constraints, expected anisometry, or a template to replace the default spherical search space). For further details the reader is referred to previous papers [20, 32, 36, 38, 39]. Selection and Averaging of Models For the alignment and superposition of best-matching 3D models as well as the subsequent averaging process, special auxiliary procedures are required.

43

In context with the application of ab initio approaches for the shape determination based on solution scattering data, usually the program package DAMAVER [44] is applied. The package comprises several subprograms and is based on the precedent program SUPCOMB [48]. In an automated analysis, initially all models are pairwise aligned and a respective dissimilarity measure is acquired for each pair. Finally all obtained model pairs are compared: a reference model is selected owing to the smallest average dissimilarity and outliers are discarded. All models remaining in the analysis are then superimposed onto the reference model and the resultant assembly is mapped onto a densely packed grid of equally sized spheres but varying occupancy factors. These factors are then exploited for the ensuing filtering of the averaged model, by neglecting grid points characterized by low occupancy. By contrast, our program DAMMIX [32] creates averaged models where either different densities of equally sized spheres or different sphere volumes (radii) are used as a probability measure; RG and in the second case also V of the models are identical to the mean values of the models used for averaging. Sorting out spheres of low probability can serve as appropriate filtering rule, concomitantly standardizing the sum of sphere volumes to the value of V , if required. While in the DAMAVER procedure the individual models become re-oriented (after shifting and rotation procedures) until an optimal overlapping of the models is achieved, the DAMMIX procedure only uses a rotation around the z axis. This implies that the DAMAVER approach is ideal for averaging pure ab initio models without any additional information, while DAMMIX is rather qualified for the generation of averaged template-biased models. Modeling of Hydration Since both SAXS curves and hydrodynamic data inherently contain already hydration contributions, no further precautions have to be adopted for applying SAXS-based modeling approaches. In principle, the program GASBOR [49, 50], a SA program similar to DAMMIN, can model water molecules and, thus, simulate hydration. Since, however, only dummy waters are used, no biophysically relevant model of a hydrated protein is obtained by this approach (cf. separate paper in this issue). A more realistic allocation of bound water is achieved, if atomic coordinates in conjunction with accurate surface calculation programs such as SIMS [51] and special hydration algorithms are applied. Our hydration programs HYDMODEL or HYDCRYST [52–56] assign specific hydration numbers, for example those obtained by Kuntz [57], to definite amino acid residues on the protein surface, so that hydrated models of biochemical relevance are achieved.

44

P. Zipper et al.

Parameter Predictions

Computational Requirements

For predicting scattering and hydrodynamic parameters of the models, in general the bead radii were upscaled by appropriate factors to account for the packing density of the beads. For averaged models, additionally, a rescaling of the coordinates of the beads turned out to be useful. Calculation of I(h) and p(r) profiles and molecular parameters from the coordinates of spheres were performed by programs based on algorithms described elsewhere [45, 58]. The I(h) curves of averaged models were calculated by means of Debye’s formula, applying the upscaled bead radii and rescaled coordinates; for calculating the I(h) curves of individual DAMMIN models, however, the size of the bead radii had to be reduced by the same empirical factor as in previous studies [39] to optimize the fit to the experimental I(h) and to the model scattering curves as provided directly by DAMMIN. The necessity for this different treatment is a consequence of the fact that the DAMMIN program uses spherical harmonics instead of Debye’s formula for the internal calculation of scattering curves; the resulting slight discrepancies in the tail ends of the I(h) functions calculated by the different algorithms can usually be reduced by downscaling the bead radii when applying Debye’s formula. The p(r) functions of all models were calculated as outlined by Glatter [58] from the coordinates and the upscaled radii of the beads, if applicable taking into account additionally the rescaling of bead coordinates. Hydrodynamic parameters (s, D) were obtained by applying version 13 of the program HYDRO [59, 60]. However, the dimensions of several arrays had to be increased and the input routine of the program was adapted, to allow handling of models consisting of a multitude of spheres and to ease the input steps. Our expression for the interaction tensor of overlapping spheres of unequal size [61–63] is already implemented in the used version of the program [64].

For our model calculations, personal computers with processors Intel Pentium IV (3.2 GHz and 2 GB RAM) or AMD 3000+ (2.2 GHz and 1 GB RAM) were used; for surface renderings a SGI workstation was applied. The CPU time required for the ab initio calculations depended on the program and input variables used: typical CPU times for DALAI_GA and DAMMIN varied between 2 h and 1 d. HYDRO runs with about 6800 beads took about 2 1/2 h; the calculation times for models with smaller sphere numbers were essentially lower.

Results and Discussion SAXS Experiments: Information in Reciprocal and Real Space Figure 1a illustrates the experimental scattering intensities I(h) of the native HBL complex of L.t. Hb and of its dodecameric subunit [17, 19, 28]. These scattering curves, which have been normalized to a value of I(0) = 1, represent the starting points for all subsequent creations of

Visualization of Models All obtained 3D models can be visualized by the program RASMOL [65] or any other 3D molecular graphics program. Further structural features of the models obtained can be obtained by surface representations of the models, preferably applying programs from the CCP4 package [66]. For this purpose, electron density maps were generated from the sphere coordinates and threshold levels were chosen as to furnish all models with the same volume [20]; the electron densities obtained were displayed by the program VOLVIS [67].

Fig. 1 Experimental SAXS profiles of L.t. Hb: scattering intensity, I(h), and distance distribution function, p(r), of the native HBL complex (dashed lines) and its dodecameric subunit (solid lines). For convenience, the scattering intensities are normalized to I(0) = 1, whereas the areas under the p(r) functions are proportional to the molecular mass of the particles under consideration

Modeling of the Dodecameric Subunit of Lumbricus Hemoglobin

SAXS models for the subunit and the HBL complex. Obviously, the decay of I(h) at small angles is much more pronounced for the HBL complex than for the subunit. Considering the inverse relation between particle size (real space) and the decay of scattering intensity (reciprocal space) [45], the two curves clearly reflect the different size of the two objects of interest. Moreover, the distinct submaxima in the I(h) curve of the complex suggest that this particle is rather globular and symmetric, whereas the dodecameric subunit appears to be much less isometric than the whole complex. The information in the reciprocal space, the scattering intensity I(h), can be converted to the real space by applying a Fourier transformation [45]. This procedure yields a pair-distance distribution function, p(r), which allows a much more realistic (physical) visualization of particle dimensions in terms of a distance distribution profile and a particle diameter. Figure 1b clearly reveals that all distances found for the subunit of L.t. Hb are located at small r values while those for the complex exhibit a broad distribution and, consequently, dmax of the entire HBL complex (about 30 nm) exceeds that of the subunit (about 10 nm) by far. Owing to the proper scaling of the curves, the enormous difference between the masses of the complex and the subunit is convincingly illustrated.

45

SAXS-based Modeling of the Dodecameric Subunit

Fig. 2 Various SAXS models of the dodecameric subunit of L.t. Hb presented in different views. A The trial-and-error consensus model consists of beads of equal density but different volumes (bead radii rb ≤ 0.661 nm) reflecting different probabilities of bead positions. The different probabilities are additionally expressed by different shades of gray (low probability: light gray; high probability: dark gray). The image on the right-hand side of the first row depicts a central slab of the side view. B and C The selected DALAI_GA models are the result of two different runs and correspond to bead radii of 0.95, 0.75, and 0.55 nm

The first SAXS models for the dodecameric subunit of L.t. Hb had been generated by trial-and-error procedures, starting from 1/12 of a previously established trial-anderror SAXS model for the entire HBL complex [17, 28]: a total of about 300 different bead models with bead radius rb = 0.661 nm were generated; eventually, 22 of them were selected for further analysis, because of their good matching the scattering behavior of the dodecameric subunit. The selected models were superimposed and averaged to yield a consensus model for the subunit. While in the original consensus model [28] the different probabilities of positions of the beads were taken into account by different weightings (i.e. different densities) of the equalsized beads, in the consensus model shown in Fig. 2A the different probabilities are represented by different volumes of beads of equal density. This way of expressing the probabilities was chosen with regard to the subsequent application of the program HYDRO [59, 60], used for the prediction of hydrodynamic parameters. HYDRO cannot make use of different densities of beads, whereas it can handle beads of different size. For better visualization, in the graphic presentation of the consensus model (Fig. 2A) the different probabilities of positions are additionally reflected by different shades of gray: the lighter the color of a bead, the lower is its probability. Mean values for the calculated structural parameters RG , V , dmax , and the hydrodynamic parameters D and s of the selected trialand-error models are presented in Table 1, together with

the values of the corresponding parameters as obtained from the averaged structure represented by the consensus model in Fig. 2A. A comparison of I(h) and p(r) profiles of the consensus model with the experimentally obtained functions is shown in Fig. 3. The good agreement between experimental and model functions (in the case of I(h) only for h ≤ 2 nm−1 ) reveals the consensus model as well as the underlying trial-and-error models to be nearly equivalent in scattering with the native particles. An interesting feature of these models is the kinked appearance that can be observed particularly in side views and corresponding central slabs (Fig. 2A). This is clearly related to the domed overall shape of the models. It should be noted that models lacking such a domed shape gave much poorer fits to the experimental scattering functions [28]. To study the efficiency of advanced modeling approaches [32, 38], we first applied the GA as implemented in the program DALAI_GA [40, 41]. In this context several runs were performed. Here we present the results obtained from two runs applying a background-corrected experimental I(h) curve as input, together with an oblate ellipsoid as initial search space and rb = 1.15 nm as initial bead radius. In each of these runs, the bead size was stepwise reduced down to rb = 0.55 nm; the upper limit of the analyzed h range of the experimental scattering curve, h max , was changed simultaneously according to the relation h max ≤ π/(2rb ). The models corresponding

46

Fig. 3 Comparison of the experimental scattering profiles, I(h) and p(r), of the dodecameric subunit of L.t. Hb with the scattering profiles calculated from various SAXS models. The black lines signify the experimental profiles and the filled black circles the profiles of the consensus model. The colored open circles mark the averaged functions of DAMMIN models derived from modified experimental scattering curves, and the colored lines represent the scattering curves calculated for averaged DAMMIN models: blue: P1U; green: P3U:O; dark green: P3U:P; red: P3O; brown: P1T; the averaging process was performed by DAMAVER (solid lines) or by DAMMIX (dashed lines). The codes P1U, P3U:O, P3U:P, and P3U correspond to the nomenclature as used in the DAMMIN approach [42] and symbolize the absence/presence of different symmetry constraints and shape bias, while the code P1T points to a template-biased model (see text)

to rb varying between 0.95 and 0.55 nm are presented in the lower part of Fig. 2B and C. These models also exhibit a kinked shape, quite similar to the trial-and-error models. In this context it has to be mentioned that models generated by applying the DALAI_GA approach can be considered to be strictly ab initio, because, apart from the guess of the initial search space, no other a priori information is used by the program. The structural and hydrodynamic parameters of the selected DALAI_GA models are listed in Table 1. The I(h) curves calculated from these models were found to be virtually identical with the experimental curves, in the range below the respective h max . A similar, however somewhat less perfect coincidence was also proven by a comparison of calculated and experimental p(r) functions. For these reasons, the I(h) and p(r) functions of the DALAI_GA models are not shown in Fig. 3.

P. Zipper et al.

A plethora of further ab initio models for the dodecameric subunit were created by applying the SA approach implemented in the program DAMMIN [42, 43]. Owing to the extended input facilities of this program, models were generated without any symmetry constraint and shape bias (coded as P1U), with C3 symmetry as constraint but without any a priori shape bias (coded as P3U:O and P3U:P, respectively, depending on the oblate or prolate shape of the resulting model), and with C3 symmetry as constraint and preference of oblate shape as bias (coded as P3O). The majority of the ab initio DAMMIN models were obtained making use of the option for the automatic subtraction of a constant background from the experimental scattering curve. In Table 1, these models are tabulated separately from models that were derived without background subtraction. In addition to the mean values and standard deviations of the structural and hydrodynamic parameters of the various sets of ab initio models, Table 1 also presents the values that were obtained from the corresponding averaged structures generated by applying the program package DAMAVER [44] which performs a superposition and filtered averaging of a selection of models. To superimpose the models, each model, except the one which is chosen as reference model by the program, is rotated appropriately to achieve an optimal overlap. While a rescaling of the nominal bead radii rb by a factor of 1.105 (to account for the packing density of the beads) was applied with all ab initio DAMMIN models, an additional rescaling of the bead coordinates was performed in the case of the averaged models, to match the mean RG of the set of models used for averaging. A close inspection of the tabulated values for RG , V , dmax , D, and s reveals a high extent of agreement between the values obtained for different types of models (P1U, P3U:O, P3U:P, P3O), irrespective of whether individual/averaged models are considered or the data are based on the original/modified experimental scattering curve. The largest deviations are encountered between the volumes, V , of the models derived from the original experimental scattering curve and those from the modified curve. Here the application of the background correction of the scattering curve obviously favors V values that are more similar to the experimentally observed V . Smaller deviations can also be observed between the mean values for D and s of the individual P3U:P models and the values for the corresponding averaged structures. Surface representations of the averaged ab initio DAMMIN models, based on the modified experimental scattering curve, are displayed in Fig. 4. All models are shown in four views corresponding to successive 90◦ rotations around the vertical axis. The averaged I(h) and p(r) curves of the ab initio DAMMIN models derived from the modified experimental scattering curve and the I(h) and p(r) profiles of the corresponding averaged structures are presented in Fig. 3. The constant background that had been automat-

Ab initio DAMMIN models derived from the original experimental scattering curve g,f Selected models without symmetry constraint and shape bias (code P1U) Filtered average (obtained by DAMAVER) of the selected P1U models h Selected prolate models with symmetry C3 and no a priori shape bias (code P3U:P) Filtered average (obtained by DAMAVER) of the selected P3U:P models h Selected models with symmetry C3 and shape bias oblate (code P3O) Filtered average (obtained by DAMAVER) of the selected P3O models h Ab initio DAMMIN models derived from the modified experimental scattering curve e,f Selected models without symmetry constraint and shape bias (code P1U) Filtered average (obtained by DAMAVER) of the selected P1U models h Selected oblate models with symmetry C3 and no a priori shape bias (code P3U:O) Filtered average (obtained by DAMAVER) of the selected P3U:O models h Selected prolate models with symmetry C3 and no a priori shape bias (code P3U:P) Filtered average (obtained by DAMAVER) of the selected P3U:P models h 23

13

12

19

16

6

1,548±171 0.3–0.325 1,369 0.325 1,351±245 0.30–0.35 1,325 0.325 1,326±249 0.30–0.35 1,299 0.325

1,177±109 0.325 1,177 0.325 1,075±202 0.30–0.35 1,083 0.325 1,111±203 0.30–0.35 1,102 0.325

53–275 0.95–0.55

6

Consensus model: unfiltered average (obtained by DAMMIX) of the selected models Ab initio DALAI_GA models Selected models e,f d

206±2 0.661 265 0.661

33

Nb ,rb (nm)

Experimental Trial-and-error models Selected models

Nm

252±5 252

3.766

257

3.766 3.766±0.002

257±4

266

3.765 3.766±0.002

265±4

214

3.764

3.765±0.001

214±4

210

3.763 3.764±0.002

210±4

229

3.763 3.763±0.003

228±2

257±4

249

249±2

255±10

V (nm3 )

3.763±0.001

3.79±0.01

3.739

3.740±0.005

3.74±0.01

RG (nm)

11.8

11.5±0.1

11.8

11.6±0.1

11.5

11.6±0.1

11.9

11.5±0.2

11.4

11.5±0.2

11.9

11.7±0.1

10.5±0.3 12.1±0.2

10.2±0.2 11.4±0.2 10.7 c 11.1

10.6±0.0

c

c

dmax b (nm)

Table 1 Comparison of experimental and model parameters of the dodecameric subunit from Lumbricus terrestris hemoglobina

4.84

4.75±0.05

4.77

4.76±0.02

4.82

4.80±0.03

4.82

4.78±0.04

4.97

4.77±0.07

4.85

4.84±0.04

10.09

9.90±0.10

9.93

9.92±0.05

10.05

10.00±0.06

10.03

9.96±0.09

10.34

9.93±0.15

10.10

10.09±0.09

9.82±0.13

10.04

4.82

4.71±0.06

10.09±0.02

4.84±0.01

9.18–9.44

D (10−7 cm2 /s) s (10−13 s)

Modeling of the Dodecameric Subunit of Lumbricus Hemoglobin 47

5,125 0.18–0.57

1,755 0.24–0.57

179±1 0.60 483 0.60 168–309 0.60 900 0.35

48

11

1,357±243 0.30–0.35 1,066 0.35

Nb ,rb (nm)

36

Nm

3.740

3.740 4.152

255 m 333 255 m

m

m

250

267

3.798

4.074

267

267

3.798 3.798

267±2

259

3.766

3.798±0.008

258±5

V (nm3 )

3.766±0.002

RG (nm)

11.5 m

11.6 m 12.8

12.5

12.1

11.8±0.2

12.8

12.4±0.3

11.8

11.6±0.1

dmax b (nm)

4.75 m

4.79 m 4.28

4.42

4.77

4.52±0.03

4.09

4.77±0.02

4.78

4.77±0.03

9.89

m

9.98 m 8.91

9.22

9.93

9.43±0.06

8.52

9.94±0.05

9.95

9.94±0.06

D (10−7 cm2 /s) s (10−13 s)

Experimental SAXS and hydrodynamic data for the oxygenated dodecamer are taken from [28]. The SAXS data (radius of gyration, RG , hydrated volume, V , and maximum diameter, dmax ) refer to the dodecamer (obtained from urea dissociation) at infinite dilution (c = 0) and represent mean values and standard deviations of the results derived from the I(h) and p(r) functions obtained from two independent measurements yielding a molar mass of M = 190 ± 19 kg/mol. The hydrodynamic data refer to dodecamers obtained from urea and SiW dissociation, respectively, and represent sw,20 values derived from sedimentation velocity experiments at c = 0.04, 4.0, and 12.4 mg/ml. The corresponding model parameters were calculated for bead models consisting of Nb beads of radius rb . The data represent the mean values and standard deviations of Nm selected models or refer to averaged structures obtained by superimposing Nm models b If not stated otherwise, for models the value corresponding to the maximum vertex-to-vertex distance of the beads is given c Value corresponding to the maximum center-to-center distance of the beads d For the calculation of parameters, the probabilities of bead positions, resulting from superimposing the selected models, were taken into account by different bead volumes; the nominal bead radius rb given in column 3 only applies to positions of probability = 1 e For generating these models, the experimental I(h) function was modified by subtraction of a constant background f The nominal bead radii rb given in column 3 were upscaled by a factor of 1.105 to account for the packing density of the beads g For generating these models, the experimental I(h) function was not modified by subtraction of a constant background h For the calculation of parameters, the coordinates of the beads were rescaled to match the mean RG of the models used for averaging i The templates were derived from 8 EM-biased DAMMIN models of the HBL complex of L.t. Hb [39]. Each template contains 1/12 of the HBL model, starting at angular position 22.5 or 30◦ j The nominal bead radius rb given in column 3 was upscaled by a factor of 1.183 to account for the packing density of the beads in these models k These averaged structures are composed of equal-sized beads; only beads occurring in at least f min models were selected. The mean values and standard deviations of parameters were obtained by varying f min from 12 to 24, thereby applying separate optimized scaling factors for the coordinates and radii of the beads to match the mean RG and V of the 48 models used for averaging l The crystal data for the dodecamer were obtained from the Protein Data Bank (file 1X9F.PDB). For surface calculations by SIMS [51], the following input parameters were used: dot density ddot = 100 nm−2 , rprobe = 0.14 nm, water volume Vw = 0.0245 nm3 , a smoothing probe sphere, rsm = 0.04 nm, and a centerto-center distance criterion for the spacing between water and AA spheres. Hydration calculations by HYDMODEL [52–54] used the hydration numbers given by Kuntz [57] for preferentially bound water molecules, in combination with a hydration factor of f K = 1 m Parameter calculated after rescaling the coordinates and radii of the beads by two slightly different factors to match the experimental values for RG and V

a

Reduced crystal structure, hydrated (AA residues + heme groups + 3,370 H2 O obtained by SIMS and HYDMODEL)

Models derived from the crystal structure l Reduced crystal structure, anhydrous (AA residues + heme groups)

Selected models with symmetry C3 and shape bias oblate (code P3O) Filtered average (obtained by DAMAVER) of the selected P3O models h DAMMIN models biased by templates g Selected models without symmetry constraint (code P1T) i,j Unfiltered average (obtained by DAMMIX) of the selected P1T models d,j,h Filtered averages (obtained by DAMMIX) of the selected P1T models k Filtered average (obtained by DAMAVER) of the selected P1T models j,h

Table 1 (continued)

48 P. Zipper et al.

Modeling of the Dodecameric Subunit of Lumbricus Hemoglobin

Fig. 4 Surface representations of averaged DAMMIN models of the dodecameric subunit of L.t. Hb. The models are shown in four different views corresponding to successive 90◦ rotations around the vertical axis. The dark-gray images comprise ab initio models (P1U, P3U:O, P3U:P, P3O) and represent filtered averages produced by DAMAVER. The light-gray images illustrate models biased by templates; they correspond to filtered averages created by DAMAVER or to unfiltered averages produced by DAMMIX. The encoding items for the models are the same as given in the legend to Fig. 3. The DAMAVER package determined an optimum rotation for each of the P1T models by comparison with the model that was adopted as reference, disregarding any information about the angular position of the template applied, and without any restrictions about the rotation axes. In the approach based on DAMMIX, the rotation of each model was strictly depending on the angular position (22.5 or 30◦ ) of the underlying template and amounted to ±3.75◦ around the z direction

ically subtracted from the experimental scattering curve by DAMMIN was, of course, re-added to the scattering curves of the ab initio DAMMIN models presented in Fig. 3. A second line of DAMMIN models for the dodecameric subunit was created by using more detailed a priori shape information, imported in the form of templates ultimately based on 3D EM reconstructions of the native HBL complex of L.t. Hb [19]. Figure 5 outlines the various steps of this approach. The first two steps, the derivation of templates for the entire HBL complex from 3D reconstructions obtained from cryo-EM and the use

49

Fig. 5 Creation of models for the HBL complex of L.t. Hb models based on templates. A Template for the native HBL complex as derived from 3D EM reconstructions. B and C Sectors containing 1/12 of the EM-biased DAMMIN model of the complex can be used as templates for the DAMMIN analysis of the scattering curve of the dodecameric subunit. The patterns drawn in red and blue represent two alternative sectors. The angular positions of the two sectors are 22.5◦ (red) or 30◦ (blue), respectively. D and E Selected DAMMIN models of the dodecameric subunit and the underlying templates. The superpositions of the templates (light colors) with the resulting DAMMIN models (dark colors) of the dodecameric subunit are shown in two different views and as central slabs

of these templates for generating EM-biased DAMMIN models of the complex have already been described in detail elsewhere [39]. Eight selected DAMMIN models from the previous study, consisting of beads with rb = 0.6 nm on hexagonal lattice points, were now used to prepare appropriate templates for the dodecameric subunit. For this purpose, from each of these previously established models two slightly different sectors were taken and used as templates, each of them representing exactly 1/12 of the original DAMMIN model. One sector (drawn in Fig. 5 in red) was taken starting at angular position 22.5◦ , whereas the start position of the other sector (drawn in blue) was at 30◦ . In this way a total of 16 different templates were established, and each template was applied in at least three separate runs of the DAMMIN program. Finally, 48 different template-biased DAMMIN models for the dodecameric subunit resulted from applying this approach to the original experimental scattering curve. These models (coded as P1T) were generated without any symmetry constraint1 . Each of the P1T models represents a subset of the beads contained in the underlying template, 1

The simultaneous use of a template and a special symmetry constraint requires that the template fulfils the selected symmetry conditions strictly [39]. If, on the other hand, a symmetrical template but no symmetry constraint is applied, the resulting model will certainly not reflect the symmetry of the template perfectly.

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without any additional beads not already occurring in the template.2 Surface representations of averaged structures derived from the P1T models are shown in the lower part of Fig. 4. The first row of these images corresponds to a filtered average as obtained by application of the DAMAVER package, the second row to an unfiltered average computed by our program DAMMIX. The appearance of the structures obtained by the two averaging approaches is obviously diverging. This is mainly caused by the differences in the procedures used for superimposing the individual models. In the resulting unfiltered averages the different probabilities of the beads were again expressed by different bead volumes, to enable the calculation of structural and hydrodynamic parameters. Alternatively, also special filtered averages were generated by applying the DAMMIX approach. For these averages, only beads occurring in at least f min P1T models were selected; the different probabilities of these selected beads were neglected and not taken into account in the calculation of parameters. To study the influence of filtering more thoroughly, the value of f min was varied systematically between 12 and 24. Figure 6 presents visualizations of the three limiting DAMMIX averages corresponding to f min = 1, 12, and 24, respectively. The mean values of the structural and hydrodynamic parameters of the individual P1T models and the parameters calculated from the various averaged structures based on these models are again compiled in Table 1. An inspec2

In principle templates for the dodecameric subunit can also be derived directly from EM-based templates for the whole HBL complex; this alternative approach was only used in a few pilot tests performed, however, with HBL templates consisting of beads with rb = 0.8 nm, which disables a direct comparison with the results presented in this study.

Fig. 6 DAMMIX averages of P1T models for the dodecameric subunit of L.t. Hb, studied at varying filtering conditions. A unfiltered average, f min = 1 (different probabilities are expressed by different shades of gray and by different bead radii); B filtered average, fmin = 12; C filtered average, fmin = 24

P. Zipper et al.

tion of the data discloses a good agreement of the mean values for D and s of these models with the results obtained for the various ab initio DAMMIN models based on the modified experimental scattering curve; the mean value for V only agrees with the corresponding data for the P1U models, whereas the mean values for RG and dmax are distinctly higher than the results obtained for the ab initio models. Among the various averaged structures derived from the P1T models, only the filtered average obtained by application of the DAMAVER package yields the closest agreement with the mean values for D and s of the underlying models, whereas the unfiltered average generated by DAMMIX exhibits the largest deviations from the mean values; smaller deviations are observed, however, in the case of the set of filtered DAMMIX averages. It should be mentioned, that preliminary tests with a few template-biased models derived from the modified experimental scattering curve yielded results (data not shown) similar to those reported in Table 1, however, the volumes of these models were found to be systematically higher by a few percent. Scattering curves I(h) and p(r) functions of the P1T models are included in Fig. 3. For convenience, as I(h) values of the individual models, the data provided by DAMMIN were adopted in this case. The averaged I(h) and p(r) functions excellently agree with the experimental functions. On the other hand, the I(h) and p(r) functions calculated from the various averaged structures deviate considerably; the most pronounced deviations from the experimental functions are observed with the unfiltered averaged structure generated by DAMMIX. Models of the Dodecameric Subunit Based on the Crystal Structure The recently published crystal structure of the dodecameric subunit of L.t. Hb [30] and the atomic coordinates deposited in the Protein Data Bank (PDB ID: 1X9F) provided the basis for a critical check of the physical relevance of the various models derived from SAXS. For this purpose, several models of the anhydrous and the hydrated protein were constructed from the original crystal structure. These models are visualized in Fig. 7 in three orthogonal views, to allow a better recognition of the shape and structural details such as the location of heme groups and waters. The CPK model of the anhydrous protein (A) was obtained by omission of the 264 water molecules observed in the crystal structure. Since the number of atoms exceeded the limit imposed by the program HYDRO [59, 60], the CPK model of the anhydrous protein was converted to a model (B) in which each amino acid (AA) residue is represented by a single bead placed at the center of gravity of the residue; the size of each bead corresponds to the molecular volume of the respective residue. In a similar way also each heme group is represented by a single bead; for convenience these beads

Modeling of the Dodecameric Subunit of Lumbricus Hemoglobin

Fig. 7 Models of the crystal structure of the dodecameric subunit of L.t. Hb, shown in top, side, and bottom views. A Original crystal structure (atoms shown in CPK colors), anhydrous model. B Reduced crystal structure (AA residues in gray, heme groups in black), anhydrous model. C Central slabs of the anhydrous reduced crystal structure shown in B. D Reduced crystal structure, hydrated model (obtained by means of the programs SIMS and HYDMODEL; water molecules are displayed in green)

are drawn in black color. Central slabs of the reduced model of the anhydrous docecameric subunit (C) unveil further details of its structure (e.g. the different appearances in the top and bottom views). The model for the hydrated protein (D) was obtained in a multi-step procedure: first the coordinates of about 100,000 potential water positions were obtained by applying the program SIMS [51] to the anhydrous crystal structure; a subsequent analysis of these potential positions by means of our program HYDMODEL [52–54], applied to the reduced anhydrous model (B) and using Kuntz hydration numbers [57], eventually resulted in the selection of 3,370 positions as coordinates of hydration waters (green beads). In Fig. 8 the I(h) and p(r) curves calculated for the anhydrous and hydrated models based on the crystal structure of the dodecameric subunit are compared to the experimental curves and the functions obtained for the SAXS consensus model of the subunit. This comparison evidently reveals a distinct discrepancy in the particle dimensions, suggesting the crystal structure to correspond to larger dmax and RG values than obtained from SAXS. The parameters tabulated in Table 1 reflect this discrepancy in a quantitative manner. In an attempt to improve the fit to the experimental I(h) and p(r) functions, the reduced hydrated model was downscaled appropriately to match the experimental values for RG and V . As follows from Fig. 8, in this way the divergent behavior of the I(h) and

51

Fig. 8 Comparison of the experimental scattering profiles, I(h) and p(r), of the dodecameric subunit of L.t. Hb (open black circles) and the SAXS consensus model (solid black circles) with the scattering profiles of models derived from crystallographic data of the subunit (lines). The dot-dashed lines mirror an anhydrous model describing the crystal structure built from AA residues and the solid lines render the corresponding hydrated model; the dashed lines indicate a rescaled hydrated model

p(r) functions could be reduced considerably, however, significant deviations from the experimental functions still persist. Figure 9 presents superpositions of the reduced anhydrous model (grey beads) derived from the crystal structure with a gamut of averaged SAXS models (red beads) based on trial and error or the DAMMIN approach. For generating the superpositions, the SAXS models were aligned appropriately by means of the programs DAMSEL and DAMSUP from the DAMAVER suite. The first impression gained from the figure is that the shape of nearly all SAXS models considered is in fair accord with the crystal structure of the dodecameric subunit. However, a closer inspection, especially of the central slabs related to the top and side views of the particles, unveils characteristic differences. On the whole, the SAXS models exhibit a less domed shape than the model based on the crystal structure, mainly owing to a less pronounced cavity on the bottom side of the models. This cavity is least expressed in the models P3U:O and P3O and in the trialand-error model, and appears to be most pronounced in the models P1U and P1T. The model P3U:P represents an exception per se because of its divergent structure, which is, however, quite obscured by the superposition

52

P. Zipper et al.

Conclusions

Fig. 9 Comparison of the reduced crystal structure of the dodecameric subunit of L.t. Hb with various models derived from SAXS. All images represent superpositions of the reduced anhydrous model (gray beads) excerpted from the crystallographic information with averaged SAXS models (red beads) based on trialand-error or DAMMIN approaches. The averaged trial-and-error models in the first row are compared to the averaged DAMMIN models beneath which have been encoded as listed in the legend to Fig. 3

with the crystal structure model. That none of the SAXS models matches the crystal structure more perfectly is not unexpected in view of the mentioned discrepancies observed with the I(h) and p(r) functions and the particle dimensions. Superpositions of the downscaled reduced anhydrous model with the SAXS models (not shown here) differ from the superpositions shown in Fig. 9 to some extent because of differences in the alignment and orientation of the SAXS models; the general appearance of these superpositions is, however, quite similar to that observed in Fig. 9.

Exploiting the information of experimental SAXS curves, ab initio modeling programs allow the low-resolution shape of proteins to be predicted with a high degree of probability. The present study shows that this is also valid for the dodecameric subunit of L.t. Hb, a subassembly of a complex of high structural hierarchy (HBL complex). Both types of approaches applied, GA (program DALAI_GA) and SA (program DAMMIN), turned out to be effective in estimating the 3D solution shape of the subunit and predicting structural and hydrodynamic molecular parameters. In agreement with the results of previous modeling trials on several proteins [32, 38], the use of strict ab initio conditions is clearly inferior to approaches which implement additional structural information. In particular, the performance of DAMMIN calculations with symmetry constraints and shape bias as well as computations using EM-biased templates yielded less ambiguous models. However, appropriate selection criteria for the models chosen and averaging procedures have to be applied, to avoid less relevant models. To judge the reliability of the obtained models, these were compared to the previously established SAXS trial-and-error model [17, 28] and to the 3D subunit structure derived from crystallographic data [30]. In the latter case, however, appropriate hydration contributions have to be taken into account, to make the molecular parameters comparable to the data observed in solution scattering or hydrodynamic studies. As an essential outcome, the comparison between solution and crystal data reveals a different behavior of the dodecameric subunit in solution and the crystalline state, respectively. This may be taken unequivocally from the differences in the models presented in Fig. 9, irrespective of the constraints for the DAMMIN modeling procedure. In particular, the cavity in the particle center found by crystallography could not be simulated sufficiently by any of the SAXS-based DAMMIN models. Evidently, the crystallographic models are more domed than the SAXS-based models. It is therefore not surprising that the rather speculative attempts to reduce the differences in the p(r) functions of the models by a homogeneous downscaling of the models derived from the crystal structure, applying the same scaling factor to the x, y and z coordinates, were unable to shift the maximum of the p(r) function sufficiently to lower r values, i.e. to the position found by SAXS experiments and also observed with the SAXS-based models. Alternative attempts with inhomogeneous downscaling, applying in pilot tests a lower scaling factor to the z coordinates as compared to the factor acting on the x and y coordinates, improved the results only slightly. From the obvious failure of mere downscaling approaches in reducing the observed discrepancies one must conclude that additional modifications of the crystallographic models will be required to succeed. At the moment we can only speculate about the nature of these neces-

Modeling of the Dodecameric Subunit of Lumbricus Hemoglobin

sary modifications (presumably rotation and realignment of parts of the crystallographic models) which must decrease the size and the cavity of the models, thereby shifting the maximum of the p(r) function from about 5.7 to 4.6 nm (Fig. 8). To achieve this goal, further systematic investigations, outside the scope of this paper, have to be performed. In this context, however, also questions concerning the ultimate reliability of experimental data and calculation modes should be addressed. Of course, the investigation of biological material is always, unavoidably, afflicted with the problem of absolute sample purity and stability. In the case of the L.t. Hb subunit, no disturbances have been reported [28]. Both SAXS and hydrodynamic data have been obtained after extrapolating the data to c = 0 and using high-tech equipment and modern evaluation procedures. Similarly, the applied modeling approaches used the highest standard of available computer programs. Therefore any explanations of the differences in the p(r) functions by the assumption of errors in the processing of experimental and/or model data can be ruled out definitely. In the past, numerous attempts have been made to compare the results from solution techniques and crystallography, although this intention must a priori remain tentatively, because of the investigation in different aggregation states (solution or crystal). In the course of such investigations, essentially agreement between the structure in solution and the crystal has been established [68–70]. A close inspection of the results in molecular terms, however, revealed minor, sometimes only marginal differences, e.g. in the values for RG , obtained either experimentally by SAXS or calculated from the atomic coordinates, for volume and surface area, or in the tail-end regions of SAXS

53

curves. Remaining discrepancies were ascribed to different flexibility of certain protein parts (e.g. termini and surface loops), different particle anisometry, different extent of preferential hydration and salt binding, in addition to varying protein-protein interactions and unavoidable aggregation/irradiation effects (influenceable, however, by variation of c). In addition, the results, may be influenced by a different extent of surface roughness (rugosity) of the models, a problem which usually crops up if SAXS and diffraction data are to be compared quantitatively [33, 70]. Because of the complex effects, a clear-cut answer concerning the differences between solution and crystal data for the subunit of L.t. Hb cannot be given at present. Since reasonable SAXS-based models for the whole HBL complex have already been established [19], modeling an entire HBL complex from the dodecameric subunit model, would also be a further challenging modeling task. Putting side by side the resulting models and the models obtained for the native HBL complex revealed a basic agreement in the overall structure [32], the quantitative comparison of data, however, disclosed a major discrepancy, owing to the fact that HBL complexes built alone from models of the dodecameric subunit lack the scaffold of linker chains which form the central part of the native HBL complex. Acknowledgement The authors are much obliged to several scientists and institutions for use of their computer programs: to D.I. Svergun for DAMMIN, the DAMAVER suite, GNOM and SUPCOMB, to P. Chacón for DALAI_GA, to Y.N. Vorobjev for SIMS, to J. García de la Torre for HYDRO, to R.A Sayle for RASMOL, to the SERC Daresbury Laboratory for the CCP4 suite, and to the Research Foundation of the State University of New York for VOLVIS, respectively. A.K. thanks the Austrian Academy of Sciences for support (APART fellowship).

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Progr Colloid Polym Sci (2006) 131: 55–73 DOI 10.1007/2882_017 © Springer-Verlag Berlin Heidelberg 2006 Published online: 17 February 2006

Helmut Durchschlag Peter Zipper Angelika Krebs

Helmut Durchschlag (u) Institute of Biophysics and Physical Biochemistry, University of Regensburg, Universitätsstrasse 31, 93040 Regensburg, Germany e-mail: [email protected] Peter Zipper Physical Chemistry, Institute of Chemistry, University of Graz, Heinrichstrasse 28, 8010 Graz, Austria Angelika Krebs Structural and Computational Biology Programme, European Molecular Biology Laboratory, Meyerhofstrasse 1, 69117 Heidelberg, Germany

D A T A A N A L Y SI S A N D MO D E L I N G

Ab initio and Constrained Modeling of Phosphorylase

Abstract The joint use of smallangle X-ray scattering (SAXS) and hydrodynamic data permits biologically useful reconstructions of protein structures to be determined. Lowresolution shapes of proteins can be obtained by SAXS-based modeling approaches, among them the ab initio approaches being the most recent and challenging ones. The programs DAMMIN and GASBOR have been applied to C. callunae starch phosphorylase in a case study, to test in a systematic manner the principles governing the evaluation strategies of the approaches applied. Therefore, emphasis was laid on the elaboration of modeling aspects rather than on biological details. Optimum results concerning the predictions of particle shapes and molecule properties have been obtained by utilizing tight constraints for modeling, such as symmetry and anisometry information. The use of pure ab initio conditions yields rather moderate shape and parameter predictions. Application of erroneous constraints generally leads to unrealistic particle shapes, although the parameter predictions may be satisfactory. The usage of the program DAMMIN turned out to be superior to application of the program GASBOR, whether the

latter approach was used in the reciprocal- or real-space version. For hydrodynamic modeling, a modified version of the program HYDRO was adopted. By recourse to known crystallographic 3D structures for phosphorylases from other sources, SAXS profiles of anhydrous proteins can be modeled. Procedures for the addition of individual water molecules to anhydrous protein envelopes based on the atomic coordinates yield biologically relevant models for hydrated phosphorylases. This requires the usage of advanced surface calculation programs such as SIMS and of appropriate hydration algorithms such as those implemented in our programs HYDCRYST and HYDMODEL. The resulting SAXS profiles and structural and hydrodynamic parameters of the hydrated proteins can be compared with the data obtained by solution scattering.

Keywords Advanced modeling techniques · Hydration · Smallangle X-ray scattering · X-ray crystallography · Hydrodynamics

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Introduction Glycogen phosphorylase (GP) catalyzes the first step in the intracellular degradation of glycogen, to give the product glucose-1-phosphate. The activity of the enzyme is regulated by the interconversion between two alternative structural states (T and R, designating tense and relaxed structures of the enzyme), an event that is modulated by allosteric interactions and reversible phosphorylation. The nonphosphorylated form (phosphorylase b) is activated by adenosine monophosphate (AMP), whereas the phosphorylated form (phosphorylase a) is independent of AMP. The properties of GP and the structural mechanism of its control have been reviewed previously [1–6]. In the absence of effectors, GP is a dimer composed of approx. 90–100 kDa subunits. Comprehensive results are available for various data of the enzyme from various sources, including solution scattering and hydrodynamic data. In particular, the enzyme from rabbit muscle attracted interest over many years. Among the techniques applied to this enzyme, analytical ultracentrifugation (AUC) (e.g. [7–17]) and small-angle X-ray scattering (SAXS) [14, 18] played a major role. More recent investigations concentrated on bacterial phosphorylases from various sources (reviewed in [6]), including the enzyme from Escherichia coli and Corynebacterium callunae. The E. coli enzyme (MalP) shows a clear preference for maltodextrins, whereas the enzyme from C. callunae (StP) seems to prefer starch as a substrate. MalP as well as StP have been subjected to SAXS investigations, however, only the latter one resulted in an unambiguous model for the solution structure and yielded well-defined results for the SAXS parameters [19]. Overall, the shape model for StP differs slightly from that of rabbit GP, both in size and axial dimensions [19]. Hydrodynamic data for StP are not available to date, contrary to the GPs from many other sources. The crystal structure of several GPs has been determined to a high resolution. Crystallographic investigations are primarily concerned with the rabbit enzyme [2, 3, 20], MalP from E. coli [21–23], and the enzyme from yeast [24, 25]. Crystallization attempts to solve the 3D structure of StP have not yet been successful. The X-ray diffraction analyses of T to R transitions of GPs have provided explanations for the molecular mechanism of regulation, attributing the allosteric mechanism to an intimate connection between tertiary and quaternary structure [3, 20]. The 3D models obtained by crystallography resemble the consensus model for StP as obtained by SAXS that established a slightly elongated, flat ellipsoid with 2-fold symmetry in solution [19]. Quite recently, StP was used for probing the efficacy of different ab initio modeling procedures based on SAXS data [26]. A comparison of approaches based on simulated annealing (SA) [27], a genetic algorithm [28, 29] or Monte-Carlo simulation [30] proved the extraordinary

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practicability of the available programs applying SA. In particular, the possibility to use either strict ab initio conditions or adopt constraints or templates turned out to be a major advantage of the available SA-based programs. The present paper uses StP for detailed modeling attempts, comprising either simple ab initio conditions or the usage of realistic and unrealistic assumptions concerning the constraints. This emerged as an especially challenging task, since frequently the information concerning size, shape and particle symmetry is unknown or even may be wrong. Insofar it was of interest whether the modeling calculations can derive acceptable results concerning particle shape and parameter predictions also under complicated premises. This intention was tested in a systematic manner, applying different available SA approaches, namely the programs DAMMIN [27] and GASBOR [31, 32], under varying input and calculation options and applying different procedures for averaging the obtained model structures. Use of surface representation techniques can be used for additionally controlling the obtained particle shapes. However, the ultimate decision concerning the usability of applied programs and constraints is the comparison of predicted and observed structural and hydrodynamic properties that usually are obtained by SAXS and AUC studies. This also implies a critical examination of experimental and calculated scattering profiles, both in reciprocal and real space. The critical comparison of the SAXS models obtained for the solution shape (StP), on the one hand, and those derived from crystal structures (rabbit phosphorylase and MalP), on the other, arouses additional interest. In this context, however, we should remember the existing slight differences in the molecular features of the enzymes from different sources. The influence of hydration contributions, i.e. of water molecules bound preferentially to the protein surface, has to be considered as well.

Methods X-ray data can be used advantageously as the general basis for efficient modeling approaches to describe the structure of biopolymers in solution or the solid, preferably crystalline state, respectively. This requires knowledge of either low-resolution SAXS data (one-dimensional scattering curves) to obtain eventually the size and shape of molecules in solution, or information from high-resolution diffraction analyses (atomic coordinates) to derive precise 3D images of crystal structures. Usually, in SAXS investigations, models consisting of a certain number of spheres (“bead models”) are constructed, to illustrate the spatial distribution of the masses of the biopolymers under consideration. The fine structure of molecules can be modeled by integrating a great number of beads in the modeling calculations, for example by characterizing atoms or amino acid (AA) residues of proteins by many beads of constant or variable size.

Ab initio and Constrained Modeling of Phosphorylase

In the context of scattering and hydrodynamic modeling, a variety of multibody approaches and applications cross our mind, among them the ab initio approaches [27– 32] being the most challenging attempts. The SA approaches DAMMIN [27] and GASBOR [31, 32], which will be used in the following, permit the calculation of such bead models which, in turn, after various intermediate steps, allow the prediction of structural and hydrodynamic parameters of hydrated molecules that can be compared to existing experimental data.

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diameter, dmax , translational diffusion coefficient, D, and sedimentation coefficient, s. In connection with scattering and hydrodynamic modeling approaches, two essential points should be mentioned explicitly. (i) Both SAXS and hydrodynamic techniques usually yield data which already inherently contain hydration contributions [35]. (ii) The interpretation of SAXS data and their evaluation in terms of molecular parameters can be performed either in the reciprocal or the real space, applying either I(h) or p(r); the two functions are connected via Fourier transformation ([36, 37]; cf. also separate paper in this issue).

Sources of Data SAXS data (scattering profiles and parameters as well as the characteristics of the best-fit trial-and-error model) of C. callunae StP were taken from a previous paper [19]. For the establishment of hydrodynamic reference data, consensus values were derived from a critical comparison of s and D values found in the literature for phosphorylases from various sources [7–17]. Needless to say, the enzymes from different sources possess slightly different properties, with respect to number and composition of AA residues as well as mass and shape of the particles. For example, the enzymes from rabbit, E. coli and C. callunae possess 842, 796 and 796 AA residues per subunit, respectively, as follows from AA sequence data stored in the SWISS-PROT data bank [33] (entries P00489 for rabbit GP, P00490 for E. coli MalP, Q8KQ56 for C. callunae StP). The atomic coordinates and molar masses of rabbit and E. coli phosphorylase were obtained from the Protein Data Bank (PDB) [34]: accession codes 3GPB for rabbit GP and 1AHP for E. coli MalP. The molar masses derived from the PDB files, however, are slightly smaller than those retrieved from the AA sequence data stored in the SWISSPROT data bank [33], since a few AA residues are not resolved in the crystal structure analysis (3GPB: only 833 of a total of 842 AA residues per subunit have been detected) or are slightly different from the composition given in the SWISS-PROT data bank. For the calculations of parameters of models, hetero atoms and also the crystallographically found water molecules were neglected. The latter aspect is obviously justified, since previous considerations [35] revealed that the number and position of these waters generally do not match the situation valid for solution studies. For the calculations with StP we used the mass as derived from SAXS [19]. For the calculations to be performed, essentially the following information was applied: atomic coordinates, scattering intensity, I(h) (where h = (4π/λ) sin θ, θ = half the scattering angle, and λ the wavelength of the incident X-rays), pair-distance distribution function (PDDF), p(r) (where r = distance between pairs of scatterers and p(r) ≡ 0 for r values greater than dmax ), molar mass, M, partial specific volume, v, radius of gyration, RG , hydrated volume, V , amount of hydration, δ1 , maximum particle

Ab initio and Constrained Modeling Approaches As outlined previously [26, 38], the SA approaches in the form of the programs DAMMIN [27] and GASBOR [31, 32] are particularly useful for the ab initio modeling of particle shapes because of their suitability to use the programs in the strict ab initio sense or by inclusion of various constraints concerning symmetry and shape specifications. The program DAMMIN [27, 39] starts with a predefined search space (usually a sphere of a diameter slightly exceeding dmax ) filled with densely packed dummy atoms and allocates the atoms either to the protein or the solvent. SA is used as a global minimization algorithm that finally achieves a particle configuration matching the experimental SAXS data (scattering profiles and molecule parameters) as far as possible. The program uses the entire scattering curve for fitting, and spheres of equal size for modeling. Compactness and connectivity restrictions are imposed on the search, thus minimizing the number of ambiguous models. To calculate scattering curves of the models, the DAMMIN approach uses spherical harmonics instead of Debye’s formula which is conventionally adopted in bead modeling studies. SAXS experiments usually abstain from the usage of contrast variation techniques. Then the DAMMIN program can act as a strict ab initio method, unless particle symmetry (point group) or shape (unknown/oblate/prolate) or any kind of template (e.g., obtained from electron microscopic or atomic data) is implemented as additional structural information (expert mode). In this case, the program can advantageously be used as a constrained modeling approach. As primary input for the experimental data, a GNOM file [40] is required. Usually, a constant is subtracted from the experimental scattering intensities to consider the influence of the internal particle structure [37]. The DAMMIN program, however, allows consideration of such a background correction (BC) or suppression of any correction (NBC). The program GASBOR [31, 32] represents another SA approach, similar to DAMMIN. However, contrary to DAMMIN, the GASBOR procedure uses a chain-like

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ensemble of dummy residues to simulate the protein backbone, and dummy water molecules to consider hydration associated with the protein. Scattering intensities of the obtained sphere models are calculated by means of Debye’s formula. Currently, two versions of GASBOR are available, using either the reciprocal-space [31] or the realspace [32] information. For both GASBOR versions the input of experimental data is possible by a GNOM file, and the accounting for a constant background of experimental scattering curves (BC) is enabled, at least in the reciprocalspace version. The real-space version that fits p(r) instead of I(h) allows a substantially faster generation of models than the reciprocal-space version. The real-space version, therefore, could be a valuable tool for screening a large number of models, provided that the accuracy achieved is the same as that obtained in the reciprocal-space program. All modeling approaches, including the SA procedures DAMMIN and GASBOR, call for the performance of multiple runs for each condition chosen. Comparing the resulting models additionally requires visualizing, aligning, superimposing, averaging and filtering the models obtained. In defiance of all possible precautions, however, all shape reconstructions lack an ultimate perfection; all models obtained share the problem of deficient uniqueness [26, 38, 41]. Subsequent Processing of Models For the processing of the models provided by DAMMIN [27] and GASBOR [31, 32], the program package DAMAVER [41] is highly qualified. The program set consists of the programs DAMSEL, DAMSUP, DAMAVER and DAMFILT, and uses SUPCOMB [42]. The shared access to this program suite allows aligning and comparing all the models under construction, averaging the aligned models, filtering the averaged models, and, finally, building the most probable model. The whole process is automated. In the first step, the selected models are pairwise aligned and a measure of dissimilarity is obtained for each pair of models. When all pairs of independent models have been tested in this way, the dissimilarities of all models are compared, outliers are discarded, and from the remaining models the one with the lowest average dissimilarity is chosen as reference model. The other ultimately selected models are then superimposed onto the reference model. The assembly of aligned models is averaged by mapping their dummy atoms onto a densely packed grid of beads. The occupancy factors of the obtained grid points can be used for filtering the averaged model at a given cut-off volume by neglecting points of too low occupancy. Averaging procedures have also been described for other types of ab initio approaches [29, 30, 43]. In context with the generation of averaged models from the SA procedures under consideration, averaging was also

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executed by in-house, purpose-written software. In the present study, the program DAMHEXMIX [26] was used which maps the models to be averaged onto a grid of hexagonal cells; the edge length of the cells is an adjustable parameter. DAMHEXMIX is similar to the program DAMMIX [26] which has been designed for direct averaging of models. With both programs, the different probabilities of the beads in the averaged models can be expressed by different bead volumes (radii) or by different densities of equally sized beads. In both cases, the values for RG of the averaged models correspond to the mean value of the models used for averaging, whereas a similar statement concerning the V values is restricted to averaged models of the first kind (models with variable bead radii). Preferably with models of the second kind (models with variable bead densities), if desired, a filtering process and a standardization of the overall volume can be launched by sorting out spheres of low likelihood. Since DAMHEXMIX, similarly to DAMMIX but unlike the DAMAVER suite, does not provide tools for performing an appropriate alignment of the models prior to averaging (the only operation implemented in DAMHEXMIX is a rotation around the z axis), the models were properly aligned by means of DAMSEL and DAMSUP from the DAMAVER package before DAMHEXMIX was applied. Previous trials have shown that averaging is a helpful and legitimate tool for accentuating the peculiarities of structural models [26, 44]. Similarly, the combined use of different averaging procedures, for example with respect to filtering aspects, may be applied for discriminating between different ab initio and constrained models. Modeling of Hydration In principle, all ab initio approaches that are based on SAXS data consider the contributions of hydration, since the target experimental profiles already involve both the protein and the water molecules bound to it. While the SA program DAMMIN [27] mimics the shape of the hydrated proteins by a number of dummy spheres of appropriate size which already include the contributions of hydration, the program GASBOR [31, 32] simulates their solution structure by anhydrous protein dummies to which water dummies are added, thereby simulating some kind of “water shell”. As has been shown previously [35], and will be proven in the following, the resultant models are unacceptable with respect to their biological relevance: the water molecules are placed on the protein surface neither in the form of a shell nor at biologically realistic surface sites. Because of the obvious deficiencies in biological relevance of water molecules in the ab initio models, we kept an eye out for more efficient hydration algorithms that are able to allocate definite amounts of water to definite AA residues of the protein surface [45]. This re-

Ab initio and Constrained Modeling of Phosphorylase

quires knowledge of the exact protein surface topography, on the one hand, and of appropriate amounts of water bound to specific sites on the protein surface, on the other. To comply with these requirements, the atomic coordinates, the AA sequence and specific hydration numbers for the AA residues must be known, in addition to the availability of advanced surface calculation and hydration programs. The accessible surface area of proteins can be calculated analytically by modern surface calculation programs such as MSROLL [46] or SIMS [47], which use one or two probe spheres, respectively, for analyzing the surface area by means of a rolling-ball mechanism. Essentially, both programs create an accurate, smooth molecular dot surface. Because of the high homogeneity of the dot distributions obtained, usage of the SIMS program is a good choice for the following hydration calculations. The inhouse programs HYDCRYST (for models based on atomic coordinates) or HYDMODEL (for reduced models using AA coordinates; in these models each AA residue is expressed by a single sphere of appropriate volume) employ special hydration algorithms for assigning specific hydration numbers to definite amino acid residues on the protein surface [35, 48, 49]. For these purposes, the following general procedure is applied. From the atomic coordinates of the crystal structure, only the atoms representing the protein moiety are selected. Application of any of the surface calculation programs to this anhydrous (“dry”) protein model results in a great number of surface points, Ndot , and respective normal vectors. From this collection of points and vectors, a pool of potential positions of water molecules bound to the protein surface is derived. By means of special hydration algorithms, implemented in HYDCRYST or HYDMODEL, preferential sites for bound water molecules can be selected. The number of water molecules, Nw , assigned to each AA residue can be adopted from the values given by Kuntz [50], but have to be triggered slightly to achieve realistic levels of hydration. The hydration algorithms have been checked for various molecules and conditions [35, 45, 48, 49, 51– 54]. Such investigations also revealed the elucidation of several fine tuning parameters required for optimum results, among them the choice of the water volume, Vw , and the scaling factor acting on the Kuntz hydration numbers, f K , being the most relevant ones. The results concerning the hydration obtained by application of HYDCRYST and HYDMODEL may differ slightly owing to the different representation of the AA residues, either by a number of small beads or by single larger spheres. To avoid such discrepancies, in cases where the number of protein atoms and assigned water molecules exceeds the maximum number of beads that are allowed in subsequent calculations, the water molecules assigned by HYDCRYST may be combined also with the reduced anhydrous protein model (in AA coordinates); this alternative approach was used, for instance, in the prediction of hydrodynamic parameters

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of rabbit GP and MalP by means of the program HYDRO (see below). Visualization of Models The obtained models were visualized by the program RASMOL [55]. Surface representations of the models utilized programs from the CCP4 package [56]; electron density maps were created and threshold levels were chosen as described [44]. The electron densities obtained were displayed by the program VOLVIS [57]. Prediction of Molecule Parameters The calculation of SAXS profiles, I(h) and p(r), from the coordinates of the spheres and the computation of structural parameters were performed by means of Debye’s formula and the algorithms described previously [37, 58]. Hydrodynamic parameters (s, D) were obtained by applying version 13 of García de la Torre’s program HYDRO [59, 60]; our ad hoc expression for the interaction tensor of overlapping spheres of unequal size [61] is implemented in this version of the program [62]. The version used by us was additionally adapted to handle models of up to about 10 000 beads. Further details concerning the prediction of structural and hydrodynamic parameters can be found in previous papers (e.g. [26, 44]). Computational Requirements For the model calculations, personal computers with processors Intel Pentium IV (3.2 GHz and 2 GB RAM) or AMD 3000+ (2.2 GHz and 1 GB RAM) were used; for surface renderings a SGI workstation was applied. The CPU times required depended substantially on the programs and input variables used: typical CPU times for DAMMIN runs varied between 10 and 14 h; GASBOR runs in the reciprocal-space version amounted up to 2 d, while runs in the real-space version were finished after approx. 1/10 of the time needed for the reciprocal-space version; HYDRO runs with about 8100 beads required about 4 h.

Results and Discussion The experimental SAXS curves of C. callunae StP [19] were used as data basis for the solution modeling approaches on phosphorylases. The previous SAXS investigations have established a simple model for the solution structure of the protein dimer and provided several molecular properties such as M, RG , V and dmax . The trial-and-error model obtained as well as the models derived from the crystallographic analyses of rabbit GP and E. coli MalP can be used as reference structures for a comparison with the ab initio reconstructions to be performed.

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SAXS-based Modeling of C. callunae StP by DAMMIN The DAMMIN approach [27] was used in a systematic manner, varying the input variables over a wide range. In particular, handling of scattering background (BC or NBC procedures), symmetry constraints (point groups P1– P6, corresponding to no symmetry or space groups C2 to C6 , respectively), shape biases (unknown/oblate/prolate and subgroups unknown/along/across) and averaging procedures (DAMAVER, DAMHEXMIX) have been tested in their capability to predict reasonable particle models and/or structural and hydrodynamic parameters. The models obtained were processed (aligned, compared, superimposed, averaged and filtered) as outlined in the Methods section, to generate ultimately various models for hydrated StP. Figure 1 comprises a variety of realistic models that were obtained by applying the program DAMMIN as a pure ab initio approach (code P1U) or after imposing plausible constraints (code P2, in consideration of the dimeric structure of the particle). Both the ab initio and the constrained modeling procedures yield particle shapes in accord with the previously mentioned reference structures. Among the P2 models, assumption of oblate or unknown particle shapes is superior to the choice of a prolate bias. Irrespective of the known information on symmetry and particle shape, we additionally scrutinized the possibility to use erroneous symmetry and shape constraints, at least in a few examples. Figure 2 summarizes a collection of several unrealistic models obtained after using codes P3–P6. Most of these structures exhibit a rather curious

Fig. 1 Top and side views of acceptable reconstructions of the StP structure, obtained by DAMMIN modeling without any constraint (P1U) or with C2 symmetry as constraint and plausible assumptions of additional shape biases [P2: unknown (U), oblate (O) or prolate (P) shape; in the two latter cases the direction of the C2 axis could be defined as unknown (U), along (L) or across (C)], adapted from [26]. All models were derived from the background-corrected (BC) experimental scattering curve

Fig. 2 Top and side views of unacceptable reconstructions of the StP structure, obtained by DAMMIN modeling with unreasonable symmetry constraints (C3 , C4 , C5 or C6 ) and assumptions of additional shape biases (P3–P6: unknown, oblate or prolate shape). All models were derived from the background-corrected (BC) experimental scattering curve

Fig. 3 Surface representations of averaged DAMMIN models of StP in two orthogonal views. The selected models were derived from experimental SAXS curves by applying or neglecting a background correction (BC, NBC) and were constructed on the basis of C2 symmetry and different shape biases; averaging (usually over six individual models) was performed by DAMHEXMIX or the DAMAVER suite

Ab initio and Constrained Modeling of Phosphorylase

appearance; only a few of them resemble the correct structure, oblate biases hit the proper shape most nearly. For the performance of further calculation steps, averaged DAMMIN models were created by different averaging approaches (DAMHEXMIX and DAMAVER). The resultant averaged models (cf. [26]) provide the basis for the ensuing surface renderings and hydrodynamic predictions by HYDRO. The surface rendering procedure applied to the most realistic models (P2U and P2O) reveals a crude overall shape (Fig. 3), strictly speaking, a smoothed oblate spheroid, essentially devoid of any structural peculiarities. Notable differences between the models derived from BC- or NBCtreated scattering curves cannot be observed. The critical comparison of experimental I(h) and p(r) SAXS profiles with those calculated for various realistic DAMMIN models (P2U and P2O) shows excellent agreement between experimental and calculated

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profiles up to high angles (Fig. 4). This obviously shows the highly professional efficiency of the DAMMIN approach for fitting target SAXS profiles (BC: panels a and b). Discrepancies in the functions, however, can be observed when the experimental curves are compared with the functions of averaged models (dashed and dotdashed lines). This is in accord with previous considerations [26, 41, 44, 63] that averaging procedures may enhance the most persistent features of the bead models and shape reconstructions. However, the scattering profiles of the averaged models do not necessarily result in good fits to the experimental SAXS profiles. At first glance, neglect of a background subtraction (NBC: panels c and d) does not seem to affect the quality of the accordance of observed and predicted curves. The comparison of the experimental I(h) curve with the curves for the averaged models, however, discloses an enhanced incongruity, obviously caused by disregarding the con-

Fig. 4 Comparison of experimental and model SAXS profiles of StP: a,c scattering intensities I(h); b,d distance-distribution functions p(r). The selected models were derived from experimental SAXS curves by using a preceding background-correction step (BC models; panels a and b) or disregarding such a correction (NBC models; panels c and d). The experimental functions (solid black lines) comply widely with the functions calculated for various BC and NBC models. The filled black circles denote the trial-and-error model obtained earlier [19] and the open circles the averaged functions for various DAMMIN models based on C2 symmetry and absence/presence of shape constraints (P2U: blue circles; P2OL: green circles; P2OC: red circles). Usually the functions of six individual DAMMIN models of each type were averaged; the resulting standard deviations are very small. The colored lines represent the SAXS profiles of the averaged structures of these models, obtained by DAMHEXMIX (averaged models of unequal beads: dashed lines) or the DAMAVER suite (averaged models of equal-sized beads: dot-dashed lines). For the lines the same colors are used as applied to the circles; the green lines in b and d are nearly identical to the red lines

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Table 1 Comparison of experimental and model parameters of StP from C. callunae. Models were obtained by the program DAMMIN, applying ab initio (P1U) or constrained (P2–P6) modeling conditions; models were derived from the experimental scattering curves by using a background correction (BC) or disregarding such a correction (NBC). Averaging of models was performed by employing the DAMAVER suite or our program DAMHEXMIX. For comparison, the model data of the previous trial-and-error model [19] have been added Nm

Experimental c Trial-and-error model DAMMIN models 3 (BC, unconstrained): P1U DAMMIN models (BC, symmetry constraint C2 ) e No shape bias: P2U 6 No shape bias, averaged (DAMAVER) No shape bias, 4 averaged (DAMHEXMIX) Shape bias oblate 1 (unknown): P2OU Shape bias oblate 6 (along): P2OL Shape bias oblate (along), averaged (DAMAVER) Shape bias oblate 4 (along), averaged (DAMHEXMIX) Shape bias oblate 6 (across): P2OC Shape bias oblate (across), averaged (DAMAVER) Shape bias oblate 4 (across), averaged (DAMHEXMIX) Shape bias prolate 1 (unknown): P2PU Shape bias prolate 1 (along): P2PL Shape bias prolate 1 (across): P2PC

Nb

rb a (nm)

RG (nm)

V (nm3 )

dmax b (nm)

4612

0.248

3.89 ± 0.02 3.87

290 ± 14 293

11.8 ± 0.0 4.3 ± 0.2 11.9 d ,12.4 4.67

8.7 ± 0.3 8.83

D (10−7 cm2 /s)

s (10−13 s)

1860 ± 20 0.3

3.897 ± 0.001 284 ± 3

12.5 ± 0.1 4.66 ± 0.05

8.81 ± 0.09

1823 ± 6 1149

3.897 ± 0.001 278 ± 1 278 3.897 f

12.7 ± 0.1 4.61 ± 0.03 12.8 f 4.68 f

8.73 ± 0.05 8.85 f

2655–8081 0.6–0.3 g

3.890 ± 0.002 279 ± 0

1860

0.3

3.898

13.0 ± 0.0 4.43 ± 0.12 4.58 h 13.0 4.59

8.37 ± 0.22 8.66 h 8.68

1838 ± 9

0.3

3.897 ± 0.001 281 ± 1

12.8 ± 0.1 4.59 ± 0.01

8.69 ± 0.01

1840

0.3

3.897 f

12.8 f

8.81 f

0.3 0.35

284

281

4.66 f

2342–6461 0.6–0.3 g

3.892 ± 0.002 281 ± 0

12.5 ± 0.0 4.52 ± 0.07 4.60 h

8.56 ± 0.13 8.70 h

1845 ± 11 0.3

3.897 ± 0.001 282 ± 2

12.6 ± 0.1 4.61 ± 0.01

8.73 ± 0.05

1846

3.897 f

12.6 f

8.84 f

0.3

282

4.67 f

2360–8068 0.6–0.3 g

3.892 ± 0.002 282 ± 0

13.0 ± 0.0 4.52 ± 0.08 4.62 h

8.56 ± 0.14 8.74 h

1842

0.3

3.895

281

12.7

4.63

8.75

1820

0.3

3.898

278

12.8

4.60

8.71

1835

0.3

3.897

280

12.8

4.63

8.77

tribution of the scattering background in the high-angle regime. Apart from the visual inspection of models and the comparison of SAXS profiles, the assessment of structural and hydrodynamic parameters of all the models is the crucial point. Table 1 demonstrates that the calculated parameters (RG , V , dmax , D, s) of all models, especially those of type P1 and P2 (Fig. 1), are in reasonable agreement with the experimental data set (indeed, the values for dmax and D slightly overestimate the experimental values), irrespective of the mode of approach (ab initio or based

on constraints). The conformity of the values also holds for different averaging procedures and, except the values for V , for different handlings of the background correction. Amazingly, the models of biochemically unrealistic appearance (codes P3–P6; Fig. 2), which are based on intentionally erroneous constraints, yield a fair agreement as well; in the case of P4–P6 models, the discrepancies in some parameters (especially in V ), however, are more pronounced, presumably caused by differences in the packing density of the spheres in models of different symmetry.

Ab initio and Constrained Modeling of Phosphorylase

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Table 1 (continued) Nm

Nb

rb a (nm)

DAMMIN models (BC, symmetry constraint C3 ) e No shape bias: P3U 1 1780 0.3 Shape bias oblate: P3O 1 1794 0.3 Shape bias prolate: P3P 1 1775 0.3 DAMMIN models (BC, symmetry constraint C4 ) e No shape bias: P4U 1 1732 0.3 Shape bias oblate: P4O 1 1612 0.3 Shape bias prolate: P4P 1 1704 0.3 DAMMIN models (BC, symmetry constraint C5 ) e No shape bias: P5U 1 1432 0.3 Shape bias oblate: P5O 1 1442 0.3 Shape bias prolate: P5P 1 1453 0.3 e DAMMIN models (BC, symmetry constraint C6 ) No shape bias: P6U 1 1572 0.3 Shape bias oblate: P6O 1 1516 0.3 Shape bias prolate: P6P 1 1581 0.3 DAMMIN models (NBC, symmetry constraint C2 ) e No shape bias: 6 1579 ± 6 0.3 P2U No shape bias, 1243 0.325 averaged (DAMAVER) No shape bias, 4 2479–7590 0.6–0.3 g averaged (DAMHEXMIX) Shape bias oblate 7 1570 ± 13 0.3 (along): P2OL Shape bias oblate 1571 0.3 (along), averaged (DAMAVER) Shape bias oblate 4 2278–7749 0.6–0.3 g (along), averaged (DAMHEXMIX) Shape bias oblate 6 1571 ± 13 0.3 (across): P2OC Shape bias oblate 1237 0.325 (across), averaged (DAMAVER) Shape bias oblate 4 2332–7073 0.6–0.3 g (across), averaged (DAMHEXMIX) a

RG (nm)

V (nm3 )

dmax b (nm)

D (10−7 cm2 /s)

s (10−13 s)

3.898 3.897 3.896

272 274 271

12.5 12.7 12.5

4.50 4.55 4.56

8.51 8.61 8.62

3.899 3.900 3.897

265 246 260

12.4 13.0 12.4

4.62 4.17 4.62

8.74 7.89 8.74

3.890 3.888 3.897

219 220 222

12.6 12.6 12.4

4.24 4.24 4.26

8.01 8.03 8.06

3.896 3.891 3.898

240 232 241

12.3 12.7 12.5

4.54 4.22 4.41

8.59 7.99 8.34

3.897 ± 0.002

241 ± 1

12.7 ± 0.2 4.67 ± 0.05

241

13.0

3.894 ± 0.002

241 ± 0

3.896 ± 0.001

240 ± 2

13.0 ± 0.0 4.44 ± 0.08 4.55 h 12.8 ± 0.1 4.65 ± 0.01

240

12.9

3.889 ± 0.002

240 ± 0

3.896 ± 0.002

240 ± 2

12.6 ± 0.0 4.57 ± 0.08 4.67 h 12.8 ± 0.2 4.65 ± 0.02

240

12.9

240 ± 0

13.0 ± 0.1 4.52 ± 0.07 4.62 h

3.897

3.896

3.896

f

f

f

3.891 ± 0.002

f

f

f

4.74

4.71

4.67

f

f

f

8.84 ± 0.09 8.97

f

8.40 ± 0.15 8.61 h 8.80 ± 0.02 8.91

f

8.65 ± 0.15 8.84 h 8.79 ± 0.04 8.84

f

8.55 ± 0.14 8.73 h

For the calculation of parameters, in general the nominal bead radii rb were upscaled by a factor of 1.105 to account for the packing density of the beads b If not stated otherwise, for bead models the value corresponding to the maximum vertex-to-vertex distance of the beads is given c Experimental values were adopted from the literature: SAXS data were taken from [19]; the molar mass of StP as estimated from SAXS is M = 170 kg/mol, based on ν = 0.729 cm3 /g [14]; for hydrodynamic reference data, consensus values derived from phosphorylases of different sources were used d Value corresponding to the maximum center-to-center distance of the beads e BC signifies that a background subtraction was performed and NBC denotes no background subtraction f Parameter calculated after rescaling the coordinates of the beads to match the mean R of the models used for averaging G g The edge length of the hexagonal lattice cells (voxel size) is given instead of r b h Value referring to the model with voxel size 0.3 nm

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SAXS-based Modeling of C. callunae StP by GASBOR Because of its conceptual formulation, viz. separate modeling steps for protein and water moieties, the GASBOR approach [31, 32] seems to be more target-oriented for the modeling of hydrated protein structures. Therefore, we performed a detailed study for testing this approach in its reciprocal-space and real-space versions, in order to derive the optimum input and evaluation conditions. Accordingly, all P1–P2 models were computed by means of the reciprocal-space version (BC) and compared systematically. Again, symmetry constraints, shape biases and averaging procedures were varied. Furthermore, to obtain rapid estimates of models, also the real-space version was tried out. Figure 5 presents a gallery of selected models, illustrating the results for various model types of the reciprocaland real-space GASBOR runs. The overall shapes are similar to those obtained by the DAMMIN approach (Fig. 1). In the first instance, oblate spheroids are obtained. Except for a few examples (e.g. model P2PL, not shown), the shapes of the less time-consuming real-space runs are similar to the reciprocal-space runs. The most striking result, however, is the highly unrealistic distribution of the water dummies on the protein surface. As may be learnt from the accumulations of red water beads in Fig. 5, the dummies are placed quite unreasonably on the protein envelope, frequently accumulated in crevices or around poles of the particle. From the biological point of view and physical considerations as well, such an appearance is unacceptable. A homogeneous distribution of water molecules would rather realize the perception of a water shell, although this assumption is still missing the true placement of preferentially bound waters [35]. In the GASBOR program, the small water molecules are rather used as filling material for modulating the models than for elaborating biologically realistic hydrated models. To confine the multitude of models obtained, grand averages of reciprocal-space and real-space models of all constrained combinations used were created, applying the DAMAVER suite to already DAMAVERaged GASBOR models (Fig. 6). As a result, the overall impression of an oblate spheroid still remains. Comparing the experimental SAXS profiles, I(h) and p(r), with the functions derived from the various reciprocal-space GASBOR models (Fig. 7) reveals agreement of I(h) up to h about 1 nm−1 . In the h range between 1 and 4 nm−1 the agreement is half-decent only for the models representing the averages of the scattering curves of the ab initio models, calculated by GASBOR (black diamonds) or by means of Debye’s formula (blue diamonds). The scattering curves corresponding to averaged structures (red and green diamonds; colored lines) deviate considerably. The most obvious discrepancy in the p(r) profiles is observed for functions derived from the grand averages of

Fig. 5 Selected models of StP as obtained by applying the reciprocal-space or real-space GASBOR approaches to the experimental I(h) and p(r) functions, respectively. The symmetry constraints and shape biases used in the GASBOR analyses are given below the images. Beads representing the protein are displayed in gray, and beads symbolizing dummy water atoms in red. Each model is shown in three different views corresponding to 90◦ rotations around the horizontal or vertical axis. The models presented in the figure were selected by DAMSEL as reference models for the purpose of averaging over usually five models of the same type by means of the DAMAVER suite

the ab initio models (black and blue solid lines). In principle, a similar situation holds for the real-space GASBOR models (Fig. 8), the mentioned discrepancies between experimental curves and the profiles of the models, however, are much more expressed. Again, the key point for examining the validity of model calculations is the quantitative comparison of experimental and calculated molecule parameters. Table 2

Fig. 6 Grand averages of GASBOR models of StP, obtained by applying the DAMAVER suite once more to the already DAMAVERaged GASBOR models corresponding to the various combinations of constraints and biases. A: grand average of all reciprocal-space GASBOR models; B: grand average of the real-space GASBOR models except the models coded P2PL. The averaged structures consist only of protein beads; the beads representing dummy water atoms in the original GASBOR models were discarded in the DAMAVER process

Ab initio and Constrained Modeling of Phosphorylase

65

Fig. 7 Comparison of I(h) (a) and p(r) (b) curves derived from reciprocal-space GASBOR models of StP with the experimentally obtained scattering functions (open black circles). The blue, red and green diamonds with error bars represent averages of Debye scattering curves (a) or p(r) functions (b) calculated from the protein coordinates of 50 ab initio models, after rescaling the bead radii to match the experimental V (blue diamonds), or of 10 averaged models (generated by means of the DAMAVER suite), after rescaling the bead radii and coordinates to match the experimental V and RG values (red diamonds) or the experimental V and the mean RG of the 50 ab initio models (green diamonds). The black diamonds (a) represent the average of the scattering curves calculated by GASBOR for the 50 ab initio models. Scattering functions calculated from the coordinates of the grand average of the 50 ab initio models (obtained by DAMAVERaging over 10 averaged models) are shown as solid lines. The black lines, representing the functions calculated from the original grand average, are nearly completely overlaid by the functions calculated after rescaling the bead radii to match the experimental V (blue lines). For the red lines, bead radii and coordinates were rescaled to match the experimental V and RG values; rescaling bead radii and coordinates to match the experimental V and the mean RG of the 50 ab initio models gave the green lines

Fig. 8 Comparison of I(h) (a) and p(r) (b) curves derived from real-space GASBOR models of StP with the experimentally obtained scattering functions (open black circles). The blue, red and green diamonds with error bars represent averages of Debye scattering curves (a) or p(r) functions (b) calculated from the protein coordinates of 45 ab initio models, after rescaling the bead radii to match the experimental V (blue diamonds), or of 9 averaged models (generated by means of the DAMAVER suite), after rescaling the bead radii and coordinates to match the experimental V and RG values (red diamonds) or the experimental V and the mean RG of the 45 ab initio models (green diamonds). The black diamonds represent the averages of the I(h) and p(r) functions provided by GASBOR for the 45 ab initio models. Scattering functions calculated from the coordinates of the grand average of the 45 ab initio models (obtained by DAMAVERaging over 9 averaged models) are shown as solid lines. The black lines, representing the functions calculated from the original grand average, are nearly completely overlaid by the functions calculated after rescaling the bead radii to match the experimental V (blue lines). For the red lines, bead radii and coordinates were rescaled to match the experimental V and RG values; rescaling bead radii and coordinates to match the experimental V and the mean RG of the 45 ab initio models gave the green lines

reveals the problems encountered with the GASBOR approach convincingly. The anticipation of most of the parameters under analysis (dmax , D, s) gave overestimates: the predicted values hit the experimental reference values to a lesser extent than the DAMMIN approach, regardless of the model constraints applied. The problem observed with the GASBOR data is presumably caused by the concept of a beaded chain and artificial water dummies for

consideration of protein hydration. The chain structure of the molecule and the neglect of the water dummies in the subsequent calculations require rigorous scaling procedures which also might contribute to enhanced discrepancies. The usage of both water and protein dummies in the case of the GASBOR approach is clearly opposed to the DAMMIN practice, where only one type of dummies is used to simulate the hydrated protein.

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Table 2 Comparison of experimental and model parameters of StP from C. callunae. Models were obtained by the program GASBOR, applying ab initio (P1U) or constrained (P1O, P1P, P2) modelling conditions; models were derived from reciprocal-space or real-space versions of the program; for the reciprocal-space approach a background correction (BC) of the experimental scattering curves was used. Averaging of models was performed by employing the DAMAVER suite Nm

Nb

rb a (nm)

Experimental c Reciprocal-space GASBOR models, no symmetry constraint No shape bias: P1U 5 1605 ± 0 0.19 d No shape bias, 1406 0.325 f averaged Shape bias oblate: P1O 5 1605 ± 0 0.19 d Shape bias oblate, 1405 0.325 f averaged Shape bias prolate: P1P 5 1605 ± 0 0.19 d Shape bias prolate, 1406 0.325 f averaged Reciprocal-space GASBOR models, symmetry constraint C2 No shape bias: P2U 5 1604 ± 0 0.19 d No shape bias, 1406 0.325 f averaged Shape bias oblate 5 1604 ± 0 0.19 d (unknown): P2OU Shape bias oblate 1404 0.325 f (unknown): averaged Shape bias oblate 5 1604 ± 0 0.19d (along): P2OL Shape bias oblate 1785 0.3 f (along), averaged Shape bias oblate 5 1604 ± 0 0.19 d (across): P2OC Shape bias oblate 1785 0.3 f (across), averaged Shape bias prolate 5 1604 0.19 d (unknown): P2PU Shape bias prolate 1786 0.3 f (unknown), averaged Shape bias prolate 5 1604 ± 0 0.19 d (along): P2PL Shape bias prolate 1785 0.3 f (along), averaged Shape bias prolate 5 1604 ± 0 0.19 d (across): P2PC Shape bias prolate 1785 0.3 f (across), averaged Reciprocal-space GASBOR models, grand averages 0.19 d Averages of the mean values 10 g 1604 ± 0 for all types of models Grand average 10 g 2957 0.25 h of all averaged models

RG (nm)

dmax b (nm)

D (10−7 cm2 /s)

s (10−13 s)

3.89 ± 0.02

11.8 ± 0.0

4.3 ± 0.2

8.7 ± 0.3

3.836 ± 0.008 3.836 e

12.8 ± 0.2 12.8 e

4.86 ± 0.02 4.74 e

9.19 ± 0.03 8.96 e

3.829 ± 0.005 3.829 e

12.7 ± 0.1 12.9 e

4.80 ± 0.03 4.68 e

9.08 ± 0.05 8.86 e

3.833 ± 0.008 3.833 e

12.7 ± 0.2 12.8 e

4.86 ± 0.02 4.74 e

9.19 ± 0.04 8.96 e

3.834 ± 0.005 3.834 e

12.6 ± 0.3 13.0 e

4.85 ± 0.03 4.70 e

9.17 ± 0.05 8.90 e

3.828 ± 0.005

12.6 ± 0.1

4.82 ± 0.03

9.12 ± 0.05

3.828 e

13.0 e

4.68 e

8.86 e

3.835 ± 0.006

12.9 ± 0.3

4.83 ± 0.02

9.14 ± 0.04

3.835 e

13.0 e

4.71 e

8.91 e

3.827 ± 0.009

12.6 ± 0.2

4.82 ± 0.02

9.12 ± 0.05

4.69 e

8.87 e

4.87 ± 0.03

9.21 ± 0.06

4.74 e

8.96 e

4.88 ± 0.02

9.23 ± 0.04

4.75 e

8.99 e

4.86 ± 0.03

9.20 ± 0.06

4.73 e

8.96 e

4.85 ± 0.03

9.17 ± 0.05

4.77 e

9.03 e

3.827

e

3.834 ± 0.007 3.834

e

3.828 ± 0.005 3.828

e

3.834 ± 0.004 3.834

e

3.832 ± 0.003 3.832

e

12.7

e

12.7 ± 0.4 12.9

e

12.4 ± 0.1 12.9

e

12.8 ± 0.2 12.9

e

12.7 ± 0.1 12.6

e

Ab initio and Constrained Modeling of Phosphorylase

67

Table 2 (continued) Nm

Nb

Real-space GASBOR models, no symmetry constraint No shape bias: P1U 5 1605 ± 0 No shape bias, averaged 1406 Shape bias oblate: P1O 5 1605 ± 0 Shape bias oblate, averaged 1405 Shape bias prolate: P1P 5 1605 ± 0 Shape bias prolate, averaged 1406 Real-space GASBOR models, symmetry constraint C2 No shape bias: P2U 5 1604 ± 0 No shape bias, averaged 1786 Shape bias oblate 5 1604 ± 0 (unknown): P2OU Shape bias oblate 1405 (unknown): averaged Shape bias oblate 5 1604 ± 0 (along): P2OL Shape bias oblate 2319 (along), averaged Shape bias oblate 5 1604 ± 0 (across): P2OC Shape bias oblate 1787 (across), averaged Shape bias prolate 5 1604 ± 0 (unknown): P2PU Shape bias prolate 1786 (unknown), averaged Shape bias prolate 5 1604 (along): P2PL Shape bias prolate 754 (along), averaged Shape bias prolate 5 1604 ± 0 (across): P2PC Shape bias prolate 1786 (across), averaged Real-space GASBOR models, grand averages 1604 ± 1 Averages of the mean values 9g for all types of models except type P2PL Grand average of all averaged 9 g 2850 models except type P2PL

rb a (nm)

RG (nm)

dmax b (nm)

D (10−7 cm2 /s)

s (10−13 s)

0.19 d 0.325 f 0.19 d 0.325 f 0.19 d 0.325 f

3.821 ± 0.007 3.821 e 3.825 ± 0.004 3.825 e 3.829 ± 0.007 3.829 e

12.8 ± 0.2 12.8 e 12.7 ± 0.3 12.9 e 12.9 ± 0.2 13.1 e

4.87 ± 0.01 4.76 e 4.83 ± 0.01 4.73 e 4.85 ± 0.02 4.71 e

9.21 ± 0.02 9.00 e 9.14 ± 0.03 8.95 e 9.18 ± 0.04 8.91 e

0.19 d 0.3 f 0.19 d

3.830 ± 0.008 3.830 e 3.820 ± 0.012

12.9 ± 0.2 13.2 e 12.5 ± 0.2

4.87 ± 0.03 4.72 e 4.83 ± 0.02

9.20 ± 0.05 8.94 e 9.14 ± 0.04

0.325 f

3.820

0.19 d

3.809 ± 0.012

0.275 f

3.809

0.19 d

3.817 ± 0.008

0.3 f

3.817

0.19 d

3.822 ± 0.006

0.3 f

3.822

0.19 d

3.830 ± 0.013

0.4 f

3.830

0.19 d

3.828 ± 0.012

0.3 f

3.828

0.19 d

3.822 ± 0.007

0.25 i

3.822

e

e

e

e

e

e

e

12.8

e

12.4 ± 0.1 12.6

e

12.8 ± 0.3 13.2

e

12.5 ± 0.3 12.8

e

12.8 ± 0.2 12.0

e

12.9 ± 0.2 13.2

e

12.7 ± 0.2

13.0

e

4.71

e

4.87 ± 0.03 4.78

e

4.80 ± 0.04 4.72

e

4.86 ± 0.01 4.72

e

4.58 ± 0.03 4.52

e

4.88 ± 0.02 4.76

e

4.85 ± 0.03

4.78

e

8.91

e

9.22 ± 0.06 9.05

e

9.08 ± 0.07 8.92

e

9.20 ± 0.02 8.94

e

8.67 ± 0.06 8.55

e

9.22 ± 0.05 9.00

e

9.18 ± 0.05

9.05

e

With all models, the bead radii rb were upscaled by appropriate factors to match the experimental volume of V = 290 nm3 For bead models the value corresponding to the maximum vertex-to-vertex distance of the beads is given c Experimental values were adopted from the literature: SAXS data were taken from [19]; the molar mass of StP as estimated from SAXS is M = 170 kg/mol, based on ν = 0.729 cm3 /g [14]; for hydrodynamic reference data, consensus values derived from phosphorylases of different sources were used d For the calculation of parameters the nominal bead radii r were upscaled by a factor of about 1.846 to match the experimental V b e Parameter calculated after rescaling the coordinates of the beads to match the mean R of the models used for averaging G f For the calculation of parameters the nominal bead radii r were upscaled by a factor of about 1.128 to match the experimental V b g Number of different types of models included h For the calculation of parameters the nominal bead radii r were upscaled by a factor of about 1.144 to match the experimental V b i For the calculation of parameters the nominal bead radii r were upscaled by a factor of about 1.158 to match the experimental V b a

b

68

Models of Rabbit GP and E. coli MalP Based on the Crystal Structures The availability of the high-resolution crystal structures of rabbit GP [2, 3, 20] and E. coli MalP [21–23] and of the respective atomic coordinates offers the possibility to use this type of information for designing anhydrous 3D protein structures, and, together with surface calcu-

Fig. 9 Space-filling models for anhydrous GP from rabbit muscle (A) and for anhydrous (B,C) and hydrated (D,E) MalP from E. coli in three different views corresponding to successive 90◦ rotations around two perpendicular axes. In A and B, the basic protein atoms derived from the crystal structures (Protein Data Bank files: 3GPB.pdb1 (A) and 1AHP.pdb1 (B)), however, without ligands and crystallographically found waters, are shown in the usual CPK colors (C in light gray, O in red, N in light blue, S in yellow); in C each AA residue is represented by a single bead placed at the mass center of gravity of the AA residue and having a volume identical to the sum of volumes of the constituent atoms. The hydrated models were obtained by usage of the surface calculation program SIMS (rprobe = 0.145 nm, ddot = 300 nm−2 ) followed by application of the hydration algorithm implemented in our programs HYDCRYST (D: Nw = 3556, calculated using Vw = 0.0244 nm3 , Kuntz hydration values, fK = 2.5) and HYDMODEL (E: Nw = 3521, calculated using Vw = 0.0244 nm3 , Kuntz hydration values, f K = 2.0). The hydration waters are displayed in green

H. Durchschlag et al.

lation and advanced hydration programs, for constructing hydrated protein models as well (Fig. 9). The anhydrous models of the enzymes from both sources (panels A and B) reveal oblate particles with twofold symmetry and are, in general terms, of similar appearance. Panels B and C show space-filling models of MalP, with beads representing either the atoms or the AA residues. Application of the surface calculation program SIMS [47] together with the hydration algorithm HYDCRYST [35, 49] unravels the finding of a water sheath (panel D), with many water molecules covering the protein envelope. Owing to some hydrophobic patches on the protein surface, however, the protein is not completely engulfed by the water molecules. A similar coverage of the protein surface by water molecules is achieved when AA-residue-based models are used instead of models derived from the atomic coordinates (panel E). Modeling water molecules to AAresidues-based protein models requires use of the hydration program HYDMODEL [48, 49]. Attaching water molecules to the anhydrous 3D structures of rabbit GP and E. coli MalP by means of our special hydration approaches yields similarly hydrated models. It is quite obvious that their appearance rather reflects biochemical relevance than the previously mentioned GASBOR models.

Fig. 10 Comparison of I(h) (a) and p(r) (b) curves derived from hydrated models of rabbit muscle GP (solid lines) and E. coli MalP (dashed lines) with the experimentally obtained functions for StP from C. callunae (open circles). The calculated SAXS profiles are based on the hydrated models (as shown in panels D and E of Fig. 9, for instance, for MalP). The electron densities of the beads were taken into account appropriately

Ab initio and Constrained Modeling of Phosphorylase

69

In Fig. 10 the I(h) and p(r) curves deduced from the hydrated models of rabbit GP and E. coli MalP based on their crystal structures are compared with the corresponding experimental profiles of C. callunae StP. Despite some slight disagreements between observed and predicted profiles, the overall impression is that of reasonable conformity. In Tables 3 and 4 the calculated structural and hydrodynamic parameters of rabbit GP and E. coli MalP are listed in dependence on the most important input variables, rw , Vw and f K . In spite of a substantial influence of f K on the number of bound water molecules (Nw varying between approx. 2500 and 4500), the predicted values for RG , D and s differ only insignificantly. The anticipated values for s yield slight overestimates if compared to the consensus values of phosphorylases from various sources, in particular when the crystal data for rabbit GP have been applied for such a comparison. The discrepancies observed in scattering and hydrodynamic data for the three enzymes under consideration have to be considered in the light of the a priori given differ-

ences. There are subtle distinctions in the enzyme sources (C. callunae StP, rabbit GP, E. coli MalP) and in the molecular size and shape of the particles as well as in the mode of their investigation (solution scattering of C. callunae StP, crystallographic studies of rabbit GP and E. coli MalP). Consequently, the two bacterial enzymes of similar size (C. callunae StP, E. coli MalP) exhibit a more pronounced agreement in the SAXS profiles and molecule parameters. Superpositions of anhydrous models of MalP from E. coli, deduced from the crystallographic data, with the most important models of C. callunae StP, figured out on the basis of SAXS in conjunction with modeling studies, again cast a positive light on the relevance of the obtained solution models (Fig. 11). The trial-and-error model (panel A), the DAMMIN models (panels B and C) and the GASBOR models (panels D and E) exhibit a rough overall conformity of all the models. A critical peer assessment, however, discloses slight differences with respect to some mass clusters at certain protein positions, both on the surface and the molecule interior. This may be recognized from the fig-

Table 3 Comparison of calculated parameters for hydrated rabbit GP models as obtained by the hydration program HYDCRYST on the basis of the crystal structure (3GPB), together with several input parameters (rw , Vw , f K ). Potential positions of hydration waters were derived by the preceding surface calculation program SIMS applied to the crystal structure of the anhydrous apo-enzyme, using ddot = 300 nm−2 and rprobe = 0.14, 0.145, and 0.15 nm, respectively; the number of potential water positions varied between 208 000 and 219 000 fK

Crystal structure, anhydrous b 0 0 Reduced anhydrous model b Hydrated models (HYDCRYST) e rw = 0.14 nm, 1.0 Vw = 0.0220 nm3 1.5–4.0 rw = 0.145 nm, Vw = 0.0244 nm3

rw = 0.15 nm, Vw = 0.0270 nm3

a

Nw

0 0 2939 3972 ± 301

1.0

2792

1.5–4.0

3710 ± 258

1.0

2609

1.5–4.0

3440 ± 245

Nb a

RG (nm)

13 558 1666

3.859 3.835

16 497 4605 f 17 530 ± 301 5638 ± 301 f 16 350 4458 f 17 268 ± 258 5376 ± 258 f 16 167 4275 f 16 998 ± 245 5106 ± 245 f

3.950 c 3.922 d 4.000 ± 0.014 c 3.969 ± 0.014 d 3.930 c 3.934 d 3.967 ± 0.010 c 3.980 ± 0.013 d 3.900 c 3.941 d 3.922 ± 0.007 c 3.988 ± 0.014 d

c d

V (nm3 )

D (10−7 cm2 /s)

s (10−13 s)

234.1 234.0

4.67

9.78

4.53

9.69

4.51 ± 0.00

9.66 ± 0.01

4.51

9.65

4.49 ± 0.00

9.61 ± 0.01

4.50

9.62

4.47 ± 0.01

9.56 ± 0.01

298.6 298.6 321 ± 7 321 ± 7 302.2 302.2 325 ± 6 325 ± 6 304.5 304.5 327 ± 7 327 ± 7

Total number of beads (protein + water) The maximum diameter, dmax , of the anhydrous model (vertex-to-vertex distance) amounts to 12.96 or 13.07 nm, as revealed from the atomic or AA coordinates, respectively c Value calculated by HYDCRYST, using coordinates, radii and electron densities of the beads d Value calculated by HYDRO, using coordinates and radii of the beads e For the calculations, M = 192.234 kg/mol (obtained from the PDB file) and ν = 0.729 cm3 /g [14] were used. For comparison, the values for maximum hydration based on the values by Kuntz [50], and for the hydration of accessible AA residues obtained from HYDCRYST, are given: Nw = 4258 (δ1 = 0.399 g g−1 ) and, depending on rw , Nw,acc = 3951–4045 (δ1 = 0.370–0.379 g g−1 ) f The N water beads located by HYDCRYST are attached to the reduced anhydrous model w b

70

H. Durchschlag et al.

Table 4 Comparison of calculated parameters for hydrated E. coli MalP models as obtained by the hydration programs HYDCRYST and HYDMODEL on the basis of the crystal structure (1AHP), together with several input parameters (rw , Vw , f K ). Potential positions of hydration waters were derived by the preceding surface calculation program SIMS applied to the crystal structure of the anhydrous apoenzyme, using ddot = 300 nm−2 and rprobe = 0.14, 0.145, and 0.15 nm, respectively; the number of potential water positions varied between 184 000 and 191 000 fK

Crystal structure, anhydrous b 0 Reduced anhydrous model b 0 Hydrated models (HYDCRYST) e rw = 0.14 nm, 1.0 Vw = 0.0220 nm3 1.5–4.0 rw = 0.145 nm, Vw = 0.0244 nm3

rw = 0.15 nm, Vw = 0.0270 nm3

Nw

0 0 2670 3724 ± 319

1.0

2569

1.5–4.0

3481 ± 261

1.0

2434

1.5–4.0

3241 ± 228

Hydrated models (HYDMODEL) e rw = 0.145 nm, 1.0 Vw = 0.0244 nm3 1.5–4.0

Nb a

RG (nm)

12 732 1592

3.830 3.843

15 402 4262 f 16 456 ± 319 5316 ± 319f 15 301 4161 f 16 213 ± 261 5073 ± 261f 15 166 4026 f 15 973 ± 228 4833 ± 228f

3.929 c 3.927 d 3.989 ± 0.018c 3.982 ± 0.016d 3.906 c 3.937 d 3.951 ± 0.013c 3.991 ± 0.016d 3.875 c 3.946 d 3.901 ± 0.008c 4.001 ± 0.015d

2611

4203

3669 ± 309

5261 ± 309

c d

V (nm3 )

D (10−7 cm2 /s)

s (10−13 s)

219.9 219.9

4.73

9.52

4.54

9.13

4.51 ± 0.00

9.07 ± 0.01

4.52

9.10

4.49 ± 0.01

9.03 ± 0.01

4.50

9.06

4.47 ± 0.01

8.99 ± 0.01

4.57

9.19

4.51 ± 0.02

9.07 ± 0.05

278.5 278.5 302 ± 7 302 ± 7 282.6 282.6 305 ± 6 305 ± 6 285.6 285.6 307 ± 6 307 ± 6

3.906 g 283.6 3.943 d 3.956 ± 0.015g 309 ± 8 4.003 ± 0.018d

a

Total number of beads (protein + water) The maximum diameter, dmax , of the anhydrous model (vertex-to-vertex distance) amounts to 13.37 nm, as revealed from the atomic or AA coordinates, respectively c Value calculated by HYDCRYST, using coordinates, radii and electron densities of the beads d Value calculated by HYDRO, using coordinates and radii of the beads e For the calculations, M = 180.782 kg/mol (obtained from the PDB file) and ν = 0.729 cm3 /g [14] were used. For comparison, the values for maximum hydration based on the values by Kuntz [50], and for the hydration of accessible AA residues obtained from HYDCRYST, are given: Nw = 3916 (δ1 = 0.390 g g1 ) and, depending on rw , Nw,acc = 3540–3628 (δ1 = 0.353–0.362 g g1 ) f The N water beads located by HYDCRYST are attached to the reduced anhydrous model w g Value calculated by HYDMODEL, using coordinates, radii and electron densities of the beads b

ure by comparing the model derived from crystallography (gray beads) with the SAXS-based models (red beads). The occurrence of more distant bead spacings in the models derived from crystal data is in agreement with the finding of elevated dmax values (Table 4), in comparison with the maximum distances of the solution models found by DAMMIN or GASBOR (Tables 1 and 2). More experiments, outside the scope of this study, would be required to elucidate the causes of the remaining discrepancies.

Conclusions The application of advanced modeling techniques presents the rational basis for relating the results obtained from so-

lution scattering and hydrodynamic techniques, on the one hand, and from solution techniques and crystallography, on the other. This holds both for the comparison of model structures and molecular parameters. Of course, the combined use of different techniques of structural analysis is much more efficient than the application of a single method. SAXS-based ab initio modeling of protein structures is the latest, but a highly promising development in this field [39, 64–67]; this especially holds when it is coupled to hydrodynamic analyses [26, 38]. For obvious reasons, the expedient application of these modern techniques requires a systematic investigation of the scientific settings and fine adjustments of the input variables under a variety of conditions, together with probing different procedures for averaging the obtained models. Moreover, biomolecules

Ab initio and Constrained Modeling of Phosphorylase

Fig. 11 Comparison of the crystal structure of MalP from E. coli with various models of C. callunae StP derived from SAXS. All images are shown in two different views corresponding to a 90◦ rotation around the horizontal axis and as central slabs. They represent superpositions of the anhydrous model (gray beads), excerpted from the crystallographic information, with SAXS models (red beads) based on trial-and-error [19] (A) or SA approaches (B: average of DAMMIN models P2U (BC); C: average of DAMMIN models P2OL (BC); D: average of reciprocal-space GASBOR models P2PU (BC); E: average of real-space GASBOR models P2U; all averages were obtained by the DAMAVER suite)

of quite different size, shape and substructure have to be analyzed critically. In the present study, this was performed for the enzyme phosphorylase and the most prominent programs in this area, DAMMIN [27] and GASBOR [31, 32] in the SAXS field, and HYDRO [59, 60] in hydrodynamics. The results of the solution data may also be compared to those derived from crystallography, if appropriate contributions of hydration are taken into account [35, 48, 49]. The following conclusions can be drawn: (i) Biologically reasonable low-resolution shapes of proteins can be obtained by the SAXS-based ab initio approaches DAMMIN and GASBOR. With both programs quite correct low-resolution shapes of hydrated particles are obtained. Since both SA programs exploit dummy beads, the localization of the individual dummies (either protein or water beads), however, are of no biological significance.

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(ii) The most efficient shape predictions can be obtained when constrained modeling is applied, provided that the applied constraints are reasonable. This modus operandi is clearly superior to the performance of a strictly ab initio approach. Use of erroneous constraints may lead to unrealistic shape predictions. (iii) Application of DAMMIN and GASBOR in combination with HYDRO, together with the known potential of SAXS and AUC knowledge, allows various molecule properties to be anticipated correctly. Most astonishingly, the prediction of structural and hydrodynamic parameters may also be quite correct, if wrong modeling constraints are applied. Of course, the correct choice of conditions is superior to the usage of ill-conditioned constraints. (iv) The agreement of experimental data and calculated parameters is no ultimate proof of the correctness of a model under consideration. This unequivocally follows from the possible coincidence of data in the case of wrong models, which however fit the target SAXS profiles very well. This problem is also well known in SAXS analyses: usually only models which are equivalent in scattering are obtained [37]; in context with the application of ab initio modeling, this phenomenon is called the problem of uniqueness [41]. (v) Both for the shape reconstructions and the subsequent parameter predictions, the program DAMMIN is superior to the program GASBOR. Application of the reciprocal-space version of GASBOR exceeds the real-space version with respect to the accuracy of the data obtained. (vi) The comparison of the low-resolution SAXS-based shapes and parameters of the proteins with highresolution 3D structures and parameters derived therefrom also corroborates the practicability of the applied scattering and hydrodynamic modeling approaches. Hydrated models of biological significance, however, can only be obtained by employing advanced surface calculation programs (such as SIMS [47]) in connection with appropriate hydration algorithms (such as HYDCRYST or HYDMODEL [35, 48, 49]). Acknowledgement The authors are much obliged to several scientists and institutions for use of their computer programs: to D.I. Svergun for DAMMIN, GASBOR, the DAMAVER suite, GNOM and SUPCOMB, to Y.N. Vorobjev for SIMS, to J. García de la Torre for HYDRO, to R.A Sayle for RASMOL, to the SERC Daresbury Laboratory for the CCP4 suite, and to the Research Foundation of the State University of New York for VOLVIS, respectively. A.K. thanks the Austrian Academy of Sciences for support (APART fellowship).

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Progr Colloid Polym Sci (2006) 131: 74–82 DOI 10.1007/2882_006 © Springer-Verlag Berlin Heidelberg 2006 Published online: 22 February 2006

Andrés G. Salvay Christine Ebel

Andrés G. Salvay Instituto de Física de Líquidos y Sistemas Biológicos, Universidad Nacional de La Plata, c.c. 565, B1900BTE La Plata, Argentina Andrés G. Salvay · Christine Ebel (u) Laboratoire de Biophysique Moléculaire, Institut de Biologie Structurale, UMR 5075 CEA-CNRS-UJF, 41 rue Jules Horowitz, F-38027 Grenoble Cedex 01, France e-mail: [email protected]

BIOLOGICAL SYSTEMS

Analytical Ultracentrifuge for the Characterization of Detergent in Solution

Abstract We have characterized the hydrodynamic behaviour in H2 O and D2 O of two detergents, dodecyl-β-Dmaltoside (DDM) and octaethylene glycol monododecyl ether (C12 E8 ), by analysing sedimentation velocity profiles obtained with interference optics in terms of continuous particle distribution and non-interacting species. The analysis in H2 0 provides values for the sedimentation and diffusion coefficients that give aggregation numbers of 130 and 115 for DDM and C12 E8 , respectively, close to that given in the literature. The analysis of the number of interference fringes as a function of detergent concentration gave values

Introduction The characterization of the association states of membrane proteins in solution is rather difficult and often the subject of controversy. For example, the native state of rhodopsin [1–4] and of ethidium multidrug resistance protein (EmrE) [5–9] were recently the object of debates; proteins have associated with them detergent and lipids in unknown but sometimes large amounts: see for reviews: [10, 11]. As recent examples, about 4 gram per gram in the case of EmrE [6, 9]; 0.5 gram per gram in the case of renal Na+ -, K+ -APTase [12]. Also, protein-detergent complexes coexist with detergent micelles of similar sizes. The effect on protein self-association of changing the solvent composition is poorly understood [13–18], and these studies participate to the more general purpose of understanding membrane protein folding and oligomerization [19–21].

for the critical micelle concentration and refractive index increments that are also in good agreement with published values. The values of the partial specific volumes of DDM and C12 E8 , obtained by sedimentation in H2 O and D2 O – where C12 E8 floats – are identical to those obtained from density measurements. These results indicate that sedimentation velocity has good potential for proper characterisation of detergent-solubilized proteins. Keywords Analytical ultracentrifugation · Detergents · Membrane · Proteins

Analytical ultracentrifugation (AUC), including sedimentation velocity (SV) is an essential technique for studying the association and interactions of macromolecular assemblies, because it is firmly based on rigorous hydrodynamic and thermodynamic theory and benefits from recent progress in data analysis [22–24]. In the last years, SV has become a reliable technique for determination of mass and hydrodynamic radius for solutions containing one or two types of non-interacting macromolecules [25– 28]. For interacting systems, recent data treatments allows SV to provide reliable values for the association constants and kinetics of association/dissociation [28–31], and, in the case of weak interactions, second virial coefficientsrelated to weak inter-particle potentials and characterizing potentially good crystallization conditions [32]. Analysis of SV experiments in terms of size distribution is extremely powerful for evaluation of sample polydispersity and concentration dependency phenom-

Analytical Ultracentrifuge for the Characterization of Detergent in Solution

ena [33–35]. Using solvents of different densities gives the density properties of the particle (related to composition) [36, 37]; the potential of SV was recently described in detail in H2 O/D2 O solvents: for a heterogeneous polyelectrolyte polymer, the effective charge was determined [38]. Using different optics such as absorbance and interference in SV allows in theory the characterization of multi-component systems if the partners have different spectral properties, which was recently investigated [39]. For the study of membrane proteins, these new approaches are of major interest. Some years ago, different detergents, which either absorb UV light or were monitored using a trace of fluorescent dye, were studied mainly by equilibrium sedimentation [40]. But the analysis was not appropriate for further studies of the detergent associated with membrane proteins. The analysis of SV in terms of size distribution, using both absorbance and interference optics, has the potential to evaluate the amount of amphiphile bound to the proteins. This is because the absorbance of detergent and proteins is different: for example, most detergents do not absorb light at 280 nm. On the other hand, the interference optics depends on the concentration of all partners of the complex. Also, the concentration of “free” detergent can be estimated by the amount of detergent micelles: in a number of experimental studies, this parameter – while essential for thermodynamic analysis – in general is far from being known, because of the protocols used for protein purification and the difficulty of proper dialysis with detergents. SV, through measurements in solvents of different densities, by using H2 O and D2 O, can be used to evaluate the density of the particles which can give insight to their compositions. The preliminary step is to check if the detergent itself can be properly described. Difficulties can arise from the equilibrium between monomer and micelle, and from their low buoyancy. Also, a slightly different behaviour in D2 O – a different aggregation number, for example – can-

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not be a priori excluded. Finally, most of the detergents do not absorb UV light and have to be detected using interference optics, which provides a signal superimposed with systematic noise. We will demonstrate here that the concentration in the ultracentrifuge and the partial specific volume of two detergents can be determined with accuracy by SV with interference optics. The detergents investigated here – dodecyl-β-D-maltoside (DDM) and octaethylene glycol monododecyl ether (C12 E8 ) – have been chosen because they are commonly used for the solubilization and stabilization of membrane proteins and assemblies. Their chemical structures are given in Scheme 1. Also, their buoyant properties differ: DDM will sediment in dilute aqueous solutions, while C12 E8 has a density close to water, and is therefore almost invisible to the centrifugal field.

Theoretical Background For each homogeneous ideal solute the sedimentation, i.e. the evolution of the weight concentration, c, with time, t, and radial position, r, is given by the Lamm equation: (∂c/∂t) = −1/r∂/∂r[r(csω2r − D∂c/∂r)] ,

(1)

ω being the angular velocity of the rotor. The sedimentation coefficient, s, and the diffusion coefficient, D, are related to other properties of the macromolecule: buoyant molar mass, Mb , and hydrodynamic radius, RH , and to the solvent viscosity, η◦ : D = kT/6πη◦ RH , s = Mb /NA 6πη◦ RH ,

(2) (3)

k being Bolzman’s constant, T the temperature, NA Avogradro’s number. Mb is related to the molar mass, M, and partial specific volume, v¯ , of the particle and to the solvent density, ρ ◦ : Mb = M(1 − ρ ◦ v¯ ) .

(4)

In the case of a detergent micelle, M is related to the aggregation number Nagg through the molar mass of the monomer Mmonomer : Nagg = M/Mmonomer .

(5)

Experiments performed in D2 O solvents can provide v¯ determination, if the macromolecule assembly (Nagg in the case of detergent) and shape (RH ) is unaffected by isotopic substitution. The Svedberg relation in the D2 O solvent, with the corresponding index D, is: sD = MbD /NA 6πη◦D RH ,

(6)

If RH does not change,RMb , the ratio of the buoyancy molar masses in D2 O and H2 O solvents, reduces to: Scheme 1 Chemical Structure of DDM and C12 E8

RMb = MbD /MbH = s D η◦D /sη◦ .

(7)

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Because of the substitution of labile H for D, the mass of the particle increases to MD in the D2 O solvent, while isotopic substitution does not change its volume. As a consequence, the partial specific volume in D2 O can be expressed as a function of v¯ : v¯ D = v¯ M/MD , then, from ◦ v) = M((M /M)− Eq. 4: MbD = MD (1 − (M/MD )ρD ¯ D ◦ ◦ ρD v¯ ), and RMb = ((MD /M) − ρD v¯ )/(1 − ρ ◦v¯ ), provides a relation allowing to extraction of v¯ from RMb : ◦ v¯ = (MD /M − RMb )/(ρD − RMb ρ ◦ ) .

(8)

Thus, RMb decreases slowly from 1 (¯v close to 0) to zero ◦ ) and more and more rapidly as v approaches (¯v = 1/ρD ¯ ◦ < v < 1/ρ ◦ , R ◦ 1/ρ (RMb infinitely negative). For 1/ρD ¯ Mb is negative and the macromolecule floats in D2 O and sediments in H2 O. The closer to 1 is the value of v¯ , the more accurate its estimate from RMb . In addition, uncertainties on MD /M entail small errors on v¯ for v¯ close to 1. We have shown previously that, when RMb is close to 1 (low values of v¯ ), the values of v¯ derived from it strongly depend on the MD /M ratio. Conversely, uncertainties on MD /M entail smaller errors on v¯ when RMb values move farther away from 1. For example, the values of v¯ calculated from RMb values of 0.9, 0.75 and 0.3 with MD /M values of 1.000 or 1.016 – corresponding to 0 and 100% exchange of the exchangeable H for bovine serum albumin and 0 and 84% exchange of the exchangeable H for white egg lysozyme – differ by 0.07, 0.04 and 0.02 ml g−1 , respectively. It is therefore most important to use good estimates of MD /M when measuring low partial specific volumes [38].

Materials and Methods All experiments were performed at 20 ◦ C. Preparation of the Solutions DDM (Anatrace) was dried as a power over P2 O5 during three days. Solutions were prepared by its weighted dissolution in H2 O and D2 O. C12 E8 (Anatrace), obtained as a solution at 100 mg per ml of H2 O, was diluted in H2 O and mixtures of H2 O and D2 O. The weights of detergent and water were measured with a Mettler AE240 balance with a precision of 10−5 g. Density Measurements Solution and solvent densities, ρ and ρ ◦ (g ml−1 ), were measured with a DMA5000 density meter (Anton Paar, Graz, Austria) with a precision of 2 × 10−6 g ml−1 . The partial specific volume v¯ (ml g−1 ) was obtained from the dependency of ρ–ρ ◦ with the detergent concentration c (g ml−1 ), calculated from the weights of detergent and water in the solution and ρ: ∂ρ/∂c = (1 − v¯ ρ ◦ ) .

(9)

Analytical Ultracentrifugation Sedimentation velocity experiments were performed on a Beckman Coulter XL-I analytical ultracentrifuge at a rotor speed of 42 000 rpm (the maximum rotor speed recommended for the standard central pieces that we used). Typically the sample and solvent (400 µl each) were loaded into a double sector cell of optical path 1.2 cm equipped with sapphire windows. The experiments were performed over more than 16 hours and sedimentation profiles of the detergents solutions acquired using interference optics every 2 minutes. SV data were analysed with the c(s) analysis of the program Sedfit (version 8.9) available at www.analyticalultra centrifugation.com [35]. The c(s) method deconvolutes the effects of diffusion broadening, which results in highresolution sedimentation coefficient distributions. This is done by assuming a relationship between the sedimentation and diffusion coefficients, through reasonable values of v¯ , of the frictional ratio f/ f ◦ , of ρ ◦ and of the solvent viscosity η◦ . f/ f ◦ links the value of RH to the minimum theoretical value, R◦H , corresponding to a non-hydrated ◦ . A value of 1.25 corresponds to sphere: RH = f/ f ◦ RH a globular usually hydrated particle. The analysis is based on finite element solutions of the Lamm equation (Eq. 1) and algebraically accounts for the systematic noise of the experimental data. Further details are given in the web site. Typically, we considered 60 experimental profiles corresponding to a total of 16 hours of sedimentation. During analysis, the Lamm equation was simulated for 200 particles in the range [0.01S, 4S] for DDM and C12 E8 , and [−1S, −0.01S] for C12 E8 in D2 O/H2 O mixtures. We supplied, for the detergents, the v¯ values determined by us via densimetry. We also supplied f/ f ◦ = 1.25 and the experimental values of ρ ◦ and η◦ , the latter being measured using a AMVn viscosity meter (Anton Paar, Graz, Austria): ρ ◦ = 0.998282 g ml−1 and η◦ = 1.00 cP for H2 O; ρ ◦ = 1.105166 g ml−1 and η◦ = 1.239 cP for D2 O. For mixtures of 98, 75, 50, and 0% D2 O, the densities are 1.101301, 1.076002, 1.051299, and 0.998282 g ml−1 and the viscosities are 1.232, 1.178, 1.120, and 1.00 cP, respectively. All the c(s) distributions were calculated with a regularization procedure (confidence level of 0.7). The non-interacting species model analysis of Sedfit was also used, which evaluates, for each species, independent values of the sedimentation and diffusion coefficients, without assumptions about the shape of the molecules or assemblies. For the sedimentation of DDM solutions, six samples at different concentrations were globally analysed with a two non-interacting species model (monomer and micelles), using the program Sedphat (http://www.analytical ultracentrifugation.com/sedphat/sedphat.htm). This analysis evaluates globally for each species the values of the sedimentation and diffusion coefficients, while providing

Analytical Ultracentrifuge for the Characterization of Detergent in Solution

for each sample the absorbance or fringe shift characterizing the species.

The refractive index increments ∂n/∂c (ml g−1 ) were calculated from the number of fringes J of interference of the sedimenting species measured in the analytical ultracentrifuge at different detergent concentrations c (g ml−1 ), according to ∂n/∂c = (∂J/∂c)(λ/Kl) ,

Table 1 Characterization of DDM and C12 E8 by sedimentation velocity Parameter

Determination of the Refractive Index Increments from Analytical Ultracentrifugation

(10)

with λ the wavelength of the laser (675 × 10−7 cm), K = 1 the magnifying coefficient, and l the optical path of the cell (in cm).

Results and Discussion Characterization of DDM and C12 E8 in H2 O Partial Specific Volumes from Density Measurements. Figure 1 shows the results of the density measurements. The solution density increases linearly with the detergent concentration. The v¯ values are determined from the slopes of the plot. Knowledge of the absolute concentration of the detergent is thus required. For DDM, it was estimated from the weight of the dried detergent powder. For C12 E8 , we used the nominal value given for the commercial solution. Values for v¯ of 0.82 ± 0.01 and 0.97 ± 0.02 ml g−1 were obtained for DDM and C12 E8 respectively, considering 5% incertitude on the concentrations. The values of v¯ are close to those reported for these detergents in the literature [11] (see Table 1).

Fig. 1 Determination of v¯ . The density increment ρ–ρ◦ , with ρ and ρ◦ the solution and solvent densities, is plotted as a function of the detergent concentration. The lines connecting the experimental points are the linear regressions of the data

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Mmonomer (Da) v¯ (ml g−1 ) v¯ (ml g−1 ) v¯ (ml g−1 ) s0 (S) ks (ml g−1 ) D0 (F) M (kDa) Nagg Nagg ˚ RH (A) CMC (mg ml−1 ) CMC (mg ml−1 ) ∂n/∂c (ml g−1 ) ∂n/∂c (ml g−1 )

Method (our work) or reference

density SV in H2 0 and D2 0 [11] SV, c(s) SV, c(s) SV, IS SV, from s0 and D0 SV, from M [11] SV, from D0 SV, c(s) [11] SV, c(s) [41]

DDM

C12 E8

511 538 0.82 ± 0.01 0.95 ± 0.02 0.815 ± 0.015 0.95 ± 0.01 0.81–0.837 3.12 ± 0.02 0.0 ± 0.1 6.34 ± 0.01 67 ± 2

0.973 0.73 ± 0.01 11.6 ± 1.0 5.61 ± 0.05 62 ± 4

130 ± 4 110–140 34 ± 1 0.11 ± 0.02 0.092 0.143 ± 0.001 –

115 ± 8 90–120 38 ± 2 0.12 ± 0.04 0.048 0.121 ± 0.02 0.134

Index 0 is given for values extrapolated to infinite dilution. SV: sedimentation velocity. c(s): size distribution analysis. IS: independent species analysis.

Sedimentation Velocity of DDM and C12 E8 in H2 O: General Behavior. The sedimentation of DDM and C12 E8 in H2 O for 8 hours at 42 000 rpm can be compared in Fig. 2, panels 1A and 2A. However, for most of the sample and particularly for C12 E8 , the analysis was made considering a total of 16 hours typically. DDM sediments as a nice boundary while C12 E8 sediments very slowly and without depletion at the meniscus position. This is expected, since the density of C12 E8 is close to that of the solvent. Size Distribution Analysis of DDM and C12 E8 in H2 O. The c(s) analysis was used first to describe the detergent solution in term of particle distributions. For DDM at 1.8 mg ml−1 , one major peak is obtained at 3.1S, which may reasonably be attributed to the detergent micelle (Fig. 2, Panels 1A-1C). The barely distinguishable signal below 0.1S in Fig. 2, Panel 1C may be monomer DDM. For DDM at 0.3, 0.6, 0.8, 1.8, 3.6, and 4.7 mg ml−1 , the same behavior is found, with one major peak corresponding to with the well-defined micelle (Fig. 2, Panel 1D). For C12 E8 at 1.8 mg ml−1 only one contribution at 0.7S is observed and is attributed to the detergent micelle (Fig. 2, Panels 2A–2C). The sedimentation of the monomer must be so slow that not distinguished from a slight baseline drift (considered by the program as time independent noise). For C12 E8 at 0.2, 1.8, 4.8, 9.5, and 14.4 mg ml−1 , the same behavior with one peak corresponding to the micelle is found (Fig. 2, Panel 2D).

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Fig. 3 Concentration dependence of the sedimentation of DDM and C12 E8 . The inverse of the sedimentation coefficient s is plotted as a function of the detergent concentration. This dependence is analyzed by the equation s−1 = s0−1 + (ks /s0 )c, where s0 is the sedimentation coefficient extrapolated to zero concentration, c is the concentration (g ml−1 ), and ks is the concentration dependence coefficient (ml g−1 ). The lines connecting the experimental points are the linear regressions of the data

the concentrations in the range of investigation (up to 4.7 mg ml−1 ). All peaks in the c(s) distributions (Fig. 2, Panel 1D) are around 3.1S and the linear fit of s −1 (c) = s0−1 + ks s0−1 c (Fig. 3) gave s0 = 3.12 ± 0.02S and ks = 0.0 ± 0.1 ml g−1 . The explanation for the difference in the behavior of the two detergents is not clear.

Fig. 2 Sedimentation velocity in H2 O of 1: DDM, and 2: C12 E8 . Panels: A: selection of experimental data (dots) and fit to the data (continuous line) corrected for all systematic noises, for the detergents at 1.8 mg/ml. The last profiles correspond to 8 hours of sedimentation at 42 000 rpm; B superposition of the difference between the experimental and fitted curves; C corresponding c(s) distributions; D superimposition of the c(s) distributions for concentrations in the range 0.2–5 mg ml−1 for DDM and 0.2–15 mg ml−1 for C12 E8 . The sedimentation coefficient of C12 E8 decreases with increasing concentration (this effect is not observed for DDM)

Concentration Dependence of s. For C12 E8 the value of the sedimentation coefficient decreases from 0.72 S at the lower concentration of 0.2 mg ml−1 to 0.62S at the larger one of 14.4 mg ml−1 (Fig. 2, Panel 1D). This can be understood considering weak repulsive interparticle interactions related to excluded volume effects. Fitting s −1 (c) = s0−1 + ks s0−1 c (shown in Fig. 3) provided a limiting value of s at infinite dilution, s0 , of 0.73 ± 0.01S and a ks of 11.6 ± 1.0 ml g−1 . For DDM the value of the sedimentation coefficient is essentially not altered by

Analysis in Terms of Independent Species. Because the sample is homogeneous, the sedimentation velocity profiles were analyzed in terms of one (for C12 E8 ) – micelles – or two (for DDM) – monomers and micelles – non-interacting particles, providing independent estimations, for the micelle, of the sedimentation and apparent diffusion coefficients s and Dapp for the micelle. Due to the large effects of non-ideality mentioned above, through concentration dependencies of s (and in a minor way of D), on the shape boundary, Dapp is unrelated to the value of D at the considered concentration [32]. The extrapolation to infinite dilution of Dapp – which for C12 E8 was found to decrease when c increases, as shown in Fig. 4 – provides for the micelle values for the diffusion coefficient at infinite dilution (D0 ) reported on Table 1. The Molecular Characteristics of DDM and C12 E8 Micelles. The hydrodynamic radius RH , the molecular weight M and related aggregation number of the micelle Nagg , were derived from s0 and D0 (Eqs. 2 and 3) and are given in Table 1. The values for Nagg are close to those reported in the literature [11].

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for ∂n/∂c of 0.143 ml g−1 for DDM and 0.121 ml g−1 for C12 E8 are close to the value of 0.134 ml g−1 reported for C12 E8 and generally used for DDM [41]. This indicates that the estimation of the fringe number – and thus concentration – is precise and accurate, even at low detergent concentration. Global Analysis of DDM at Various Concentrations. Because of the nice superimposition of the c(s) distribution for DDM in the range 0.2–5 mg ml−1 , we used the data from these six sets of SV and analyzed them globally with a model for two non-interacting species, using the program Sedphat. The analysis – Fig. 6 and Table 2 – provides for the detergent monomer a sedimentation coefficient of 0.112S (the molar mass of the monomer was fixed) and, for the micelle, s = 3.15S, M = 67 kDa (Nagg = 130 ˚ These values are virmolecules of DDM) and RH = 34 A. Fig. 4 Concentration dependence of the apparent diffusion coeffitually identical to those shown in Table 1, which are decient Dapp of DDM and C12 E8 termined from s0 and D0 . The analysis provides for each SV experiment, the numbers of interference fringes corMicelle Concentration. The number of interference fringes responding to the monomer and the micelle species. If the obtained from the area under the peaks of the c(s) distribution is related to the micelle concentration. It increases linearly with the detergent concentration (Fig. 5). The slope of the linear regression gives the value of the refractive index increment ∂n/∂c (Eq. 10) and the critical micelle concentration (CMC) can be estimated from the intercept with the concentration axis. The values, reported in Table 1, compare well with those from the literature. Values for CMC of 0.11 mg ml−1 for DDM and of 0.12 mg ml−1 for C12 E8 are obtained, compared with values of 0.09 and 0.05 mg ml−1 , respectively, reported in [11]. Our estimates

Fig. 5 Concentration of the micelles in the ultracentrifuge. The concentrations are measured in terms of a number of fringes of interference. The lines connecting the experimental points are the linear regressions of the data. The insert shows the intersection of the lines with the concentration axis that determine the CMC of detergents

Fig. 6 Sedimentation of 1 0.3, 2 0.6, 3 0.8, 4 1.8, 5 3.6, and 6 4.7 mg DDM per ml of H2 O. The panels show: A the experimental interference data (dots) and fits to the data (continuous line) corrected for all systematic noises. The last profile corresponds to 8 hours of sedimentation at 42 000 rpm; B the difference between the experimental and fitted curves. The global analysis was undertaken with the program Sedphat using a two non-interacting species model (monomer and micelles). This analysis evaluates globally for each species the values of the sedimentation and diffusion coefficients, while providing for each sample the fringe shift J characterizing the species

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A.G. Salvay · C. Ebel

Table 2 Global analysis of DDM in H2 O s1 (S)

M1 (kDa) Nagg

˚ s2 (S) RH1 (A)

˚ M2∗ (Da) RH2 (A)

3.15

67

34

511

130

0.112

7

Six sets of sedimentation profiles of DDM between 0.3 and 4.7 mg ml−1 in H2 0 were analyzed globally using the program Sedphat (available at analyticalultracentrifugation.com) in terms of two independent species. Species 1 represents the detergent micelle and species 2 the monomer. The molar mass of the monomer was fixed (underlined value)

former was imprecise, between 0 and 0.4 fringes, the latter gave precise values that were, again, virtually the same as those determined by the c(s) distribution for the micelle. This global analysis in term of non-interacting species is not appropriate for C12 E8 , for which non-ideal behaviour is observed. Sedimentation Velocity in H2 O and D2 O Partial Specific Volume of DDM from SV in D2 O and H2 O. For DDM at 1.6 mg ml−1 in D2 O, one peak is obtained in the c(s) analysis around 1.5S, which is attributed to the detergent micelle (Fig. 7, Panels 1A–1C). For DDM at 0.3, 0.8, 1.6, and 5.0 mg ml−1 , the same behaviour is found (Fig. 7, Panel 1D). The global analysis using Sedphat with a single non-interacting species model was also performed for this series of DDM samples. The analysis – not shown – provided for the micelles a value of sD = 1.55 S. From the ratio RMb of 0.61 calculated with Eq. 7, i.e. the ratio of the buoyant mass in D2 O and H2 O assuming the structure remains unchanged in the two solvents, and assuming 7H exchanged with D for each DDM in D2 O, we obtain (Eq. 8) v¯ = 0.815 ± 0.015 ml g−1 , i.e. virtually the same value as that obtained by density. Partial Specific Volume of C12 E8 from SV in H2 O/ D2 O Mixtures. Figure 7, Panels 2A–2C, show the flotation of 2 mg ml−1 of C12 E8 in 98% D2 O. One peak is obtained in the c(s) analysis around −0.5S, which as attributed to the flotation of the detergent micelle. Because of the small value of the sedimentation coefficient, the preciseness of the measurement is questionable. For this reason, the sedimentation of C12 E8 at 2 mg ml−1 was studied in 98, 75, 50, and 0% D2 O (Fig. 7, Panel 2D): the c(s) analysis shows one unique peak for each solvent condition, at −0.5, −0.22, 0.02, and 0.72S correspondingly. The plot of the product of the sedimentation coefficient with the solvent viscosity of C12 E8 in the different solvent mixtures is plotted as function of the solvent density ρ (Fig. 8). The linear relationship demonstrates the high precision of the estimation of the sedimentation coefficients, despite their

Fig. 7 Sedimentation velocity in D2 O of 1: DDM, and 2: C12 E8 . Panels A: selection of experimental data (dots) and fits to the data (continuous line) corrected for all systematic noises, for the detergents at 1.6 mg ml−1 (DDM) and 2 mg ml−1 (C12 E8 ). The last profiles correspond to 8 hours of sedimentation at 42 000 rpm; B superposition of the difference between the experimental and fitted curves; C corresponding c(s) distribution; D superimposition of the c(s) distributions for concentrations in the range 0.2–5 mg ml−1 (DDM), and for concentration 2 mg ml−1 (C12 E8 in 98, 75, 50, and 0% D2 O – left to right respectively in the Panel 2D-)

rather low values. The density of the detergent ρdet can be estimated as the density of the solvent −1.05651 g ml−1 – corresponding to a nul contrast: the density of the particle is the same of the solvent and is invisible to the centrifugal field (the complications related to H/D exchange are reasonably neglected because this density is close to 1, see the theoretical background). Assuming ρdet = 1/¯v, we obtain v¯ = 0.95 ml g−1 , a value identical to the value determined by density measurements and reported in Table 1. The same value is obtained from the RMb ratio of −0.88 (Eq. 7), considering 1 H/D exchange in D2 O.

Analytical Ultracentrifuge for the Characterization of Detergent in Solution

Fig. 8 Sedimentation of C12 E8 in mixtures of D2 O and H2 O. The product of the solvent viscosity η with the sedimentation coefficient s for 2 mg ml−1 of C12 E8 in different solvent mixtures is plotted as function of the solvent density ρ. The line connecting the experimental points is the linear regression of the data. The intersection of the line with the abscissa determines the reciprocal of v¯ for C12 E8

Conclusion Analytical ultracentrifugation was used to investigate the hydrodynamic behavior of two detergents in solution: DDM and C12 E8 . The analysis of the SV data in terms of continuous distribution allows the determination of the value of the sedimentation coefficients in a rather precise way, even at low detergent concentrations, close to the CMC: 0.3 mg ml−1 DDM; 0.2 mg ml−1 C12 E8 (see Figs. 2 and 3). The analysis in term of 1 or 2 non-interacting species gave values for the aggregation number and dimension of the micelle. These values are compatible with those given in the literature. This emphasizes the new poten-

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tial of AUC related to the recent development of new data treatments that take into account sedimentation and diffusion in the transport process and evaluate the systematic noise. This is of particular importance for sedimentation profiles obtained using interference optics as is needed in the case of detergents, for which absorbance optics cannot be used: detergents often do not absorb UV light, as is the case with DDM or C12 E8 , or absorb it too much as is the case with detergent with aromatic rings (such as “Triton”). From SV analyses, the number of fringes corresponding to the micelles is also determined with precision. This is demonstrated by the nice linear relationship between the concentration and fringes in Fig. 5, and by the fact that the CMC extrapolated from SV, 0.11 mg ml−1 for DDM and 0.12 mg ml−1 for C12 E8 are comparable with values reported in the literature. Even at low detergent concentration close to the CMC, the concentrations of micelles in solution can be determined with accuracy. The results of SV analysis performed in H2 0 and D2 O solvents give values for the partial specific volumes of the detergent that are those that we determine experimentally by density measurements, and close to the value given in the literature, despite the small and sometime negative values of the sedimentation coefficients. Again, this indicates a rather robust analysis at least in terms of sedimentation coefficients. Future work in this area will concern the study of other uncharacterized detergents, and of course of membrane proteins in solution. From the results of the present work, sedimentation velocity should be appropriate to quantify the amounts of bound and free detergent, allowing the determination of the association states of the proteins within the complex, which is often a difficult task. Acknowledgement This work was supported by the CNRS (postdoctoral fellowship to Andrès Salvay), CEA and Université Joseph Fourrier, Grenoble, France.

References 1. Chabre M, Cone R, Saibil H (2003) Nature 426:30 2. Fotiadis D, Liang Y, Filipek S, Saperstein DA, Engel A, Palczewski K (2003) Nature 421:127 3. Jastrzebska B, Maeda T, Zhu L, Fotiadis D, Filipek S, Engel A, Stenkamp RE, Palczewski K (2004) J Biol Chem 279:54663 4. Medina R, Perdomo D, Bubis J (2004) J Biol Chem 279:39565 5. Ubarretxena-Belandia I, Baldwin JM, Schuldiner S, Tate CG (2003) Embo J 22:6175 6. Butler PJ, Ubarretxena-Belandia I, Warne T, Tate CG (2004) J Mol Biol 340:797

7. Gottschalk KE, Soskine M, Schuldiner S, Kessler H (2004) Biophys J 86:3335 8. Ma C, Chang G (2004) Proc Natl Acad Sci USA 101:2852 9. Winstone TL, Jidenko M, le Maire M, Ebel C, Duncalf KA, Turner RJ (2005) Biochem Biophys Res Commun 327:437 10. Moller JV, le Maire M (1993) J Biol Chem 268:18659 11. le Maire M, Champeil P, Moller JV (2000) Biochim Biophys Acta 1508:86 12. Cohen E, Goldshleger R, Shainskaya A, Tal DM, Ebel C, le Maire M, Karlish SJ (2005) J Biol Chem 280:16610

13. Stanley AM, Fleming KG (2005) J Mol Biol 347:759 14. Fleming KG, Ackerman AL, Engelman DM (1997) J Mol Biol 272:266 15. Fleming KG, Ren CC, Doura AK, Eisley ME, Kobus FJ, Stanley AM (2004) Biophys Chem 108:43 16. Sturgis JN, Robert B (1994) J Mol Biol 238:445 17. Fisher LE, Engelman DM, Sturgis JN (2003) Biophys J 85:3097 18. Josse D, Ebel C, Stroebel D, Fontaine A, Borges F, Echalier A, Baud D, Renault F, Le Maire M, Chabrieres E, Masson P (2002) J Biol Chem 277:33386

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19. DeGrado WF, Gratkowski H, Lear JD (2003) Protein Sci 12:647 20. Popot JL, Engelman DM (1990) Biochemistry 29:4031 21. Engelman DM, Chen Y, Chin CN, Curran AR, Dixon AM, Dupuy AD, Lee AS, Lehnert U, Matthews EE, Reshetnyak YK, Senes A, Popot J-L (2003) FEBS Lett 555:122 22. Ebel C (2004) Progr Colloid Polym Sci 127:73 23. Lebowitz J, Lewis MS, Schuck P (2002) Protein Sci 11:2067 24. Rivas G, Stafford W, Minton AP (1999) Methods 19:194 25. Schuck P (1998) Biophys J 75:1503 26. Demeler B, Behlke J, Ristau O (2000) Methods Enzymol 321:38

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27. Philo JS (2000) Anal Biochem 279:151 28. Stafford WF (2000) Methods Enzymol 323:302 29. Stafford WF, Sherwood PJ (2004) Biophys Chem 108:231 30. Dam J, Velikovsky CA, Mariuzza RA, Urbanke C, Schuck P (2005) Biophys J 89:619 31. Dam J, Schuck P (2005) Biophys J 89:651 32. Solovyova A, Schuck P, Costenaro L, Ebel C (2001) Biophys J 81:1868 33. Schuck P (2000) Biophys J 78:1606 34. Schuck P, Perugini MA, Gonzales NR, Howlett GJ, Schubert D (2002) Biophys J 82:1096

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Progr Colloid Polym Sci (2006) 131: 83–92 DOI 10.1007/2882_007 © Springer-Verlag Berlin Heidelberg 2006 Published online: 10 February 2006

Natalia A. Chebotareva Alexey V. Meremyanin Valentina F. Makeeva Boris I. Kurganov

Natalia A. Chebotareva (u) · Alexey V. Meremyanin · Valentina F. Makeeva · Boris I. Kurganov A.N. Bach Institute of Biochemistry, Russian Academy of Sciences, Leninskii pr. 33, 119071 Moscow, Russia e-mail: [email protected]

BIOLOGICAL SYSTEMS

Self-Association of Phosphorylase Kinase under Molecular Crowding Conditions

Abstract Self-association of phosphorylase kinase (PhK) has been studied using analytical ultracentrifugation and dynamic light scattering under the conditions of molecular crowding arising from the presence of high concentrations of osmolyte. Sedimentation velocity analysis shows that in accordance with the predictions of molecular crowding theory, trimethylamine N-oxide (TMAO) greatly favours self-association of PhK induced by Mg2+ and Ca2+ . On the contrary, proline suppresses this process, probably, due to its specific interaction with PhK. We have also established that α-crystallin, a protein possessing chaperone-like activity, counteracts the self-association of PhK under mo-

Introduction Phosphorylase kinase (PhK; EC 2.7.1.38) catalyzing phosphorylation and activation of glycogen phosphorylase b (Phb) plays a key role in the cascade system of regulation of glycogen metabolism [1–3]. The PhK molecule is a hexadecamer with subunit composition (αβγδ)4 and molecular mass of 1320 kDa [4]. The oligomeric state of the native enzyme is dependent on the protein, Ca2+ , and Mg2+ concentrations. Ca2+ and Mg2+ ions stimulate PhK activity by inducing tertiary and subsequent quaternary structural changes in the enzyme molecule [5, 6]. In the absence of Ca2+ and Mg2+ , the enzyme exists in the monomeric and dimeric forms with s20,w = 23 S (that corresponding to a molecular mass of 1320 kDa) and s20,w = 36.5 S, respectively [4, 7]. However, the addition

lecular crowding conditions. Using dynamic light scattering we have shown that the increase in the light scattering intensity accompanying self-association of PhK is due to the formation of particles having hydrodynamic radius of hundreds of nanometers. The hydrodynamic radius of the start associates (seeds of association) was found to be approximately 80 nm. TMAO facilitates the formation of the associates of larger size whereas proline and α-crystallin suppress self-association of PhK. Keywords Analytical ultracentrifugation · Phosphorylase kinase · Self-association · Crowding · Dynamic light scattering

of 0.1 mM Ca2+ and 10 mM Mg2+ results in the appearance of higher order associates [7, 8]. Molecular crowding has a pronounced effect on macromolecular interactions, the rate, and the reaction equilibrium of the biochemical processes in living systems [11– 16]. Crowding influences the conformation and degree of association of macromolecules. In response to stress (osmotic, chemical, or thermal) many living organisms accumulate the high concentrations of osmolytes. Osmolytes are the small organic molecules, such as polyols, some amino acids, and methylamines. They protect proteins under stress conditions by stabilization of the protein structure [17–21]. For instance, the tissues of sharks and rays contain urea in high concentration. It is known that urea causes denaturation of the proteins, but this effect could be compensated through supporting of the high

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concentration of the counteracting osmolytes, especially trimethylamine N-oxide (TMAO), betaine, and sarcosine [17–21]. Among organisms, concentrating proline, are those that have adapted to live in high salt environments, and they use the osmolyte to offset the high external osmotic pressure [19]. Using high concentrations of osmolytes in vitro permits the simulation of molecular crowding conditions [7, 22–24]. Proline as a crowding-agent is worthy of special attention, because some authors suggest that proline is able to interact with non-native forms of proteins [25]. Previously we showed that self-association of PhK was greatly stimulated by high concentrations of the natural osmolyte trimethylamine N-oxide (TMAO) [7, 23]. Sedimentation velocity studies of self-association of PhK in 40 mM Hepes, pH 6.8, containing 0.1 mM Ca2+ and 10 mM Mg2+ at 20 ◦ C showed that the formation of very large associates took place under molecular crowding conditions. Species with sedimentation coefficients of 23 700 S were registered at a rotor speed of 2000 rpm for PhK in the presence of 1 M TMAO. The self-association of PhK is a reversible process. Addition of 1 mM EGTA causes a substantial disintegration of PhK associates formed in the presence of TMAO [7]. The major protein of the mammalian lens α-crystallin exhibits chaperone-like properties, including the ability to prevent the precipitation of denatured proteins and to increase cellular tolerance to stress [26–31]. α-crystallin is an aggregate assembled from two polypeptides (A and B), each with a molecular weight around 20 kDa [29]. αB-crystallin is also found in skeletal muscle [32–34]. The three-dimensional structure of α-crystallin remains unknown. Determination of the true molecular weight of α-crystallin has not yet been achieved not only because of its polydispersity but also because of the large variations in the size of the protein from different preparations. Depending on the isolation procedure, average molecular weights of α-crystallin range from 280 kDa [35] to in excess of 10 MDa [36]. The reasons for the variability are not yet clear; also isolation conditions can have an effect [29]. For example, isolation of α-crystallin at 5 ◦ C yields macromolecules with an average molecular weight of about 800 kDa, whereas isolation of α-crystallin at 37 ◦ C yields of an average molecular weight 320 kDa [37]. The main goal of this paper was to examine the effect of molecular crowding arising from high concentrations of osmolytes (TMAO and proline) on the self-association of PhK and the ability of α-crystallin to counteract the enhancement of self-association of PhK under the action of the crowding agent.

Materials and Methods The self-association of PhK under molecular crowding conditions arising from the presence of high concen-

N.A. Chebotareva et al.

trations of osmolytes (TMAO and proline) and its interaction with α-crystallin has been studied in 40 mM Hepes, pH 6.8, containing 0.1 mM Ca2+ , 10 mM Mg2+ , and 10 mM NaCl, using analytical ultracentrifugation. Sedimentation velocity experiments were carried out in a Model E analytical ultracentrifuge (Beckman), equipped with a photoelectric scanner, a monochromator, and computer on line; 12 mm double sector cells were used. Sedimentation profiles were recorded by measuring the absorbance of the enzyme at 280 nm. The time interval between scans was 4 min. The sedimentation coefficients were estimated from a differential sedimentation coefficient distribution (c(s) versus s), which was determined using the SEDFIT program [38, 39]. The c(s) analysis was performed with regularization at a confidence level of 0.68 and a floating frictional ratio. The sedimentation coefficients were corrected for solvent density and viscosity (20 ◦ C, water) in the standard way. DLS measurements were performed using a photon correlation spectrometer (PhotoCor Instruments Inc., USA). A Coherent (USA) 31-2082 He-Ne-Laser (632.8 nm, 10 mW) served as the light source. Signals were recorded with a photon counting system PhotoCor-PC 3 and further processed by a digital flexible correlator PhotoCor-FC. Autocorrelation functions were automatically recorded on 288 channels, logarithmically spaced in time, and analyzed using DynaLS software (Alango, Israel). The temperature was controlled by PhotoCor-TC to within ±0.1 ◦ C. Kinetics of the association of PhK were studied in 40 mM HEPES, pH 6.8, containing 10 mM NaCl at 25 ◦ C by dynamic light scattering (DLS). The buffer was placed in a cylindrical cell and preincubated for 3 min. To study the association process, aliquots of TMAO, proline, α-crystallin, and PhK were added into the cell. The process was initiated by the addition of an aliquot of Ca2+ and Mg2+ to a final volume of 0.5 ml. When studying the kinetics of association of PhK, the scattered light was collected at a scattering angle of 90◦ . For spherical particles the diffusion coefficient (D0 ) and hydrodynamic radius (Rh ) are connected by the Stokes-Einstein relationship: D0 = kB T/6πηRh , where kB is Boltzmann’s constant, T is the absolute temperature, and h is the solvent viscosity. Assuming a globular form for PhK association products the following relation between the sedimentation coefficients of monomer (s1 ) and n-mer (sn ; n = 2, 3, 4. . .) was used: sn = s1 n 2/3 , where 2/3 is the Mark-HouwinkKuhn-Sakurada coefficient for a sphere. PhK was isolated from rabbit skeletal muscle according to Cohen [4] using ion-exchange chromatography on DEAE-Toyopearl as the final step of purification [40]. Purity of enzyme was confirmed by SDS-PAGE electrophoresis [41]. Preparations of PhK in 25 mM Naglycerol β-phosphate buffer, pH 7.05, containing 1 mM EDTA, 0.5 mM 2-mercaptoetanol, and 50% glycerol were

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stored for 3 weeks at −20 ◦ C. Before the experiments PhK was dialyzed against 40 mM Hepes buffer, pH 6.8. The protein concentration was determined spectrophotometrically at 280 nm using an extinction coefficient 1.24 cm2 mg−1 for PhK [4] and extinction coefficient 0.85 cm2 mg−1 for α-crystallin [42]. TMAO, proline, α-crystallin, and Hepes were purchased from Sigma Chemical Co. (USA). The reagents were used without additional purification. When preparing the solutions containing osmolyte, the initial pH value of osmolyte solution in 40 mM Hepes buffer was adjusted so that the final pH value was 6.8. The relative viscosities of the solutions containing osmolytes (TMAO and proline) were measured using Anton Paar viscosimeter at 25 ◦ C. Refractive indices of proline solutions were measured using a refractometer (IRF-22, Russia).

Results Self-association of PhK Studied by Analytical Ultracentrifugation The differential sedimentation coefficient distributions c(s) for a self-associating system of PhK (in 40 mM Hepes, pH 6.8, containing 0.1 mM Ca2+ , 10 mM Mg2+ , and 10 mM NaCl at 15 ◦ C) in the absence (A) and in the presence (B) of 0.5 M TMAO are shown in Fig. 1. In the absence of TMAO the enzyme (0.3 g/l) exists mainly in the monomeric, dimeric, trimeric, and tetrameric forms with s20,w = 23.0 ± 1.5, 37 ± 2, 49.0 ± 2.5 and 64.0 ± 2.2 S (Fig. 1A). In the presence of 0.5 M TMAO one can see the change in the ratio of peak areas and the shift of peak positions to higher s with average s20,w = 38.0 ± 2.2, 61.0 ± 1.9, 106.0 ± 1.8, 131.0 ± 1.5, 148 ± 1, 218 ± 1 and 327 ± 1 S (Fig. 1B). It is worth noting that according to molecular crowding theory compact states of macromolecules are favoured over asymmetric ones [43]. This allowed us to use the relationship sn = s1 n 2/3 to characterize the dimension of large oligomeric forms. The peak with s20,w = 327 S corresponds to a 53-mer of PhK. It is worth noting that there are no data about influence of α-crystallin on association of native proteins. It was of special interest to study the effect of α-crystallin on PhK association. Figure 2 shows the interaction of PhK (0.3 g/l) and α-crystallin (0.3 g/l). Interactions can be detected through the change in the ratios of the peak areas and shifts in the peak positions to lesser s values. Analysis of the sedimentation behaviour of the PhK and α-crystallin mixture was performed under different conditions (different concentrations of α-crystallin, rotor speed, and temperatures), but the system is too complex to calculate the stoichiometry of the complex formed. We can only suggest that PhK binds with α-crystallin. In order to check whether α-crystallin is able to counteract the enhancement of self-association of PhK in the

Fig. 1 Self-association of PhK (0.3 g/l) under molecular crowding conditions arising from the presence of 0.5 M trimethylamine N-oxide (TMAO). Differential sedimentation coefficient distributions c(s) were obtained: A in the absence of TMAO, B in the presence of 0.5 M TMAO, and C in the presence of 0.5 M TMAO and α-crystallin (0.3 g/l) (40 mM Hepes, pH 6.8; 0.1 mM Ca2+ , 10 mM Mg2+ ; and 10 mM NaCl; 15 ◦ C). Rotor speed was 20 000 rpm

presence of the crowding-agent, we studied the effect of αcrystallin on the PhK association in the presence of TMAO (Fig. 1C). Figure 1C shows the c(s) distribution of PhK (0.3 g/l) in the presence of 0.5 M TMAO and α-crystallin (0.3 g/l). As can be seen, from the comparison of the data

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Fig. 2 The c(s) distributions for a self-associating system of PhK (0.3 g/l; curve 1), α-crystallin (0.3 g/l; curve 2) and their mixture at the same concentrations (curve 3)

presented in Figs. 1B and C under molecular crowding conditions α-crystallin shifts the equilibrium towards the smaller oligomeric forms with the sedimentation coefficients of 17.0 ± 1.4, 35.0 ± 1.9, and 54.0 ± 2.3 S, which may be transformed to standard conditions: s20,w = 23.0 ± 1.9, 48.0 ± 2.6, and 72 ± 3 S. Figure 3 shows the effect of proline on the self-association of PhK (0.3 g/l) (40 mM Hepes, pH 6.8; 0.1 mM Ca2+ , 10 mM Mg2+ , and 10 mM NaCl; 20 ◦ C). The c(s) distributions obtained in the absence of proline (A) and in the presence of 0.15 M proline (B), 0.3 M proline (C), and 0.6 M proline (D) show that the addition of proline to the PhK solution does not result in the appearance of new large associates. On the contrary, proline suppresses selfassociation of PhK, probably due to its specific interaction with the monomeric form of PhK. With increasing concentration of proline, the equilibrium state shifts toward a monomeric state: s20,w = 23.0 ± 1.9 S (A), 20.0 ± 0.6 S (B), 19.0 ± 0.6 S (C and D). It was interesting to study the effect of proline on PhK self-association under molecular crowding conditions. Figure 4B shows the effect of 0.6 M proline on PhK self-association in the presence of 0.5 M TMAO. Comparison of c(s) distributions obtained without proline (Fig. 4A) and in the presence of 0.6 M proline (Fig. 4B) demonstrates that the equilibrium state shifts toward smaller oligomeric forms. Thus, the effect of proline counteracts the effect of molecular crowding. Figs. 4B and 4C demonstrate the counteracting effects of proline and αcrystallin under the same conditions. There are only two main peaks with s20,w = 15.4 ± 0.3 and 23.3 ± 1.5 S in the presence of α-crystallin. Thus, α-crystallin, a protein possessing the chaperone-like activity, and proline suppress the PhK self-association under molecular crowding conditions.

Fig. 3 The effect of proline on the self-association of PhK (0.3 g/l). The c(s) distributions were obtained: A in the absence of proline, B in the presence of 0.15 M proline, C in the presence of 0.3 M proline, and D in the presence of 0.6 M proline (40 mM Hepes, pH 6.8; 0.1 mM Ca2+ , 10 mM Mg2+ ; and 10 mM NaCl; 20 ◦ C). Rotor speed was 30 000 rpm

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of the protein particles whose formation results in the enhancement of turbidity of the enzyme solution. Curve 1 in Fig. 5A shows the dependence of the light scattering intensity of the PhK solution (0.3 g/l) after the addition of Ca2+ and Mg2+ . The more marked increase in the light scattering intensity in the presence of 0.5 M TMAO (curve 2) means that TMAO favours self-association of PhK. The estimation of the values of the hydrodynamic radius of the particles formed in the process of PhK association suggests that associates of larger size are formed in the presence of this osmolyte (Fig. 5B). In Fig. 5B only the radii of the PhK associates are shown. The radius of the PhK molecule is 13 ± 1 nm. The distinctive feature of self-association of PhK is the fact that the size of primary associates (in the following, we shall denote them as “start associates”) exceeds substantially the dimension of the PhK molecule. The hydrodynamic radii

Fig. 4 The effect of proline and α-crystallin on the self-association of PhK (0.3 g/l) under molecular crowding conditions arising from the presence of 0.5 M TMAO. The c(s) distributions for the PhK self-association were obtained: A in the absence of proline, B in the presence of 0.6 M proline, C in the presence of α-crystallin (0.6 g/l) (40 mM Hepes, pH 6.8; 0.1 mM Ca2+ , 10 mM Mg2+ ; and 10 mM NaCl; 20 ◦ C). Rotor speed was 30 000 rpm

Self-association of PhK Studied by Dynamic Light Scattering As was shown by us previously [7], TMAO-stimulated self-association of PhK in the presence of Ca2+ and Mg2+ is accompanied by an increase in apparent absorbance in the visible region of the spectrum. The study of selfassociation of PhK by DLS gives information on the size

Fig. 5 The effect of TMAO on the self-association of PhK (0.3 g/l). The dependences of the light scattering intensity (I) on time (A) and the dependences of the hydrodynamic radius (Rh ) on time (B) were obtained in the absence (1; points are the experimental data and the solid curve is drawn in accordance with Eq. 1) and in the presence of 0.5 M TMAO (2)

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of the start associates determined by extrapolation of the Rh values to zero time in the absence and in the presence of TMAO are equal to 82 ± 3 and 117 ± 4 nm, respectively. It should be noted that DLS does not allow particles with intermediate size, i.e., with size between the values corresponding to the PhK molecule and the start associates, to be registered because of low concentrations of the intermediate forms. The increase in the light scattering intensity in the course of the self-association of PhK is probably connected with the sticking together of start associates. It has been assumed [44, 45] that the self-association of PhK results in the decrease of the specific activity of the enzyme. Therefore it is of special interest to find cellular factors capable of preventing association of PhK. According to recent data obtained by B.I. Kurganov and

coworkers (personal communication) heat-induced protein aggregation is considered to be a process involving the sticking together of start aggregates having a hydrodynamic radius of several tens of nanometers. Since there is an analogy between the self-association of PhK and the aggregation of unfolded states of the proteins, we decided to test the effect of α-crystallin, a protein with a well-studied chaperone-like activity, on the selfassociation of PhK in the region where large associates of PhK are formed (Rh > 82 nm). As can be seen from Fig. 6, sufficiently high concentrations of α-crystallin completely suppress self-association of PhK induced by Ca2+ and Mg2+ . The addition of α-crystallin results in the reduction of the increment in the light scattering intensity (Fig. 6A) and the formation of the particles of lesser size (Fig. 6B). When the concentration of α-crystallin is

Fig. 6 The effect of α-crystallin on the self-association of PhK (0.3 g/l). The dependences of the light scattering intensity increment (I − I0 ) on time (A) and the hydrodynamic radius (Rh ) on time (B) were obtained at various concentrations of α-crystallin: 0 (1), 0.3 g/l (2), 0.6 g/l (3); solid curve is drawn in accordance with Eq. 7 and 0.9 g/l (4). I0 is the value of the light scattering intensity at t = 0

Fig. 7 The effect of α-crystallin on the kinetics of PhK selfassociation under molecular crowding conditions arising from the presence of 0.5 M TMAO. The dependences of the light scattering intensity increment (I − I0 ) on time (A) and the dependences of the hydrodynamic radius (Rh ) on time (B) were obtained at various concentrations of α-crystallin: 0 (1), 0.3 g/l (2), and 0.9 g/l (3)

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Discussion The results obtained by sedimentation velocity analysis show that TMAO greatly favours the self-association of PhK. This is consistent with the general notion about molecular crowding as a factor favouring protein oligomerisation and the formation of supramolecular structures [15, 46–49]. The suppression of these processes by proline is unexpected and is probably due to the specific interaction of proline with PhK. The specific interaction of proline with PhK is supported by the fact that proline inhibits oligomerisation even at relatively low concentrations. The results of the study of self-association of PhK by DLS show that the increment in the light scattering intensity accompanying self-association of PhK is connected with the formation of particles having hydrodynamic radius of hundreds of nanometers. The dependence of the hydrodynamic radius on time obtained for Ca2+ and Mg2+ -induced self-association of PhK (0.3 g/l; curve 1 in Fig. 5B) obeys the power law, which was established for diffusion-limited colloidal aggregation [50]: Rh = Rh,0 (1 + Kt)1/df

Fig. 8 The effect of proline on the self-association of PhK (0.3 g/l). The dependences of the light scattering intensity (I) on time (A) and the dependences of the hydrodynamic radius (Rh ) on time (B) were obtained at various concentrations of proline: 0 (1); 0.08 M (2), 0.15 M (3)

rather high (0.9 g/l), only start associates are registered (Rh = 100 ± 4 nm). It should be noted that the hydrodynamic radius of α-crystallin was found to be 10.5 ± 0.2 nm. Since TMAO facilitates self-association of PhK, it is of interest to elucidate whether α-crystallin is able to counteract such an action of this osmolyte. As can be seen from Fig. 7, α-crystallin retains its ability to suppress selfassociation of PhK in the presence of TMAO. We studied also the influence of proline on the size of the particles formed in the course of self-association of PhK. In agreement with the results of sedimentation analysis, proline suppresses self-association of PhK as detected by the reduction of the increment in the light scattering intensity (Fig. 8A) and by diminishing the size of the particles formed in the course of PhK association (Fig. 8B). It is worth noting that the size of the start aggregates remains the same in the presence of proline.

(1)

where Rh,0 is the hydrodynamic radius of a seed particle, df is the fractal dimension, and K is a constant. The fractal dimension is a characteristic of aggregates which are formed as a result of unordered interactions (random aggregation). The mass of an aggregate formed in such a way (M) is connected with its effective radius (R) by the following relationship: M ∼ Rdf . The following values of parameters were obtained for the dependence of Rh on t under discussion: Rh,0 = 82 ± 3 nm and df = 1.78 ± 0.04. To make an estimate of the number of the PhK molecules forming start associates, we can use an empirical relationship between the molecular mass (Mr , kDa) and the hydrodynamic radius (Rh , nm) of the protein [51]: Mr = (1.68 Rh )2.3398

(2)

If we take into account the molecular mass of the PhK molecule (1320 kDa), the start aggregate with Rh,0 = 82 ± 3 nm should contain 76 ± 7 PhK molecules. For aggregation proceeding in a diffusion-limited regime a universal fractal dimension of 1.8 is observed [52–54]. Thus, one can assume that the selfassociation of PhK registered by DLS is a process involving the interaction of start associates and that this process is diffusion-limited. This means that each collision of the particles results in their sticking. In other words, the sticking probability is equal to unity. Such a mechanism of association results in the formation of associates of limitless size (to be more specific, large-sized associates prone to precipitate). It should be noted that if the growth of the associate proceeds by the attachment of an enzyme molecule to the initial nucleus the hydrodynamic radius would approach a limiting value at rather high values of

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time as free PhK is depleted. It is evident that this mechanism is not realised in the case of PhK association. Enhancement of self-association of PhK in the presence of TMAO may be explained in terms of the thermodynamic nonideality produced by the osmolyte. From the theory of absolute reaction rate, the second order rate constant (k) characterising the interaction of two particles (Pi + Pj → Pi+ j ) is expressed as follows [55]: k=

kB T  = c = K ; K = = h ci c j

(3)

where kB is Boltzmann’s constant, T is the absolute temperature, K = is the constant of equilibrium between the initial components (Pi and Pj , concentrations ci and c j respectively) and the activated complex (P = , concentration c= ). In the presence of the crowding-agent the equilibrium constant K = is considered as an apparent equilibrium constant:  = γi γ j (4) K = = K 0 = , γ =

where K 0 is the true equilibrium constant, γi and γ j are the thermodynamic activity coefficients for the particles Pi and Pj , and γ = is the thermodynamic activity coefficient for the activated complex. Thermodynamic activities (a) and concentrations (c) are connected by the following relationships: ai = γi ci , a j = γ j c j , and a= = γ = c= . The reaction rate constant k measured in the presence of the crowding-agent is described by the expression: γi γ j (5) k = k0 Γ = k0  = γ where k0 is the rate constant in the absence of the crowding-agent and Γ is the nonideality factor (Γ = γi γ j /γ = ). The thermodynamic activity coefficient for the solute of m-th kind is determined by the fraction of the total volume available to this solute [55]: νtotal , (6) γm = νm where νtotal is the total volume and νm is the volume available to the solute m. It is evident that γm exceeds unity inasmuch as νm < νtotal . Because the size of the activated complex is higher than that for the particle Pi or Pj , the value of ν = (the volume available to the activated complex) is less than the values of νi or ν j (the volumes available to the particles Pi and Pj , respectively). Therefore the thermodynamic activity coefficient of the activated complex (γ = ) is greater than the thermodynamic activity coefficients of the particles Pi and Pj (γi and γ j ). Since the values of the thermodynamic activity coefficients exceeds unity, the product γi γ j appears to be greater than γ = and consequently k > k0 . This means that thermodynamic nonideality produced by osmolyte should result in the acceleration of association of the protein particles.

Such considerations explain the enhancement of the rate of self-association of PhK in presence of TMAO. The action of proline on the rate of self-association of PhK is opposite to that revealed by TMAO. In contrast to TMAO, proline hampers association of PhK molecules. Samuel et al. [25] have shown that proline prevents aggregation during lysozyme refolding. Other crowing-agents such as ethylene glycol, glycine, glycerol, and sucrose were ineffective in preventing protein aggregation. The authors suggest that proline inhibits protein aggregation by binding to folding intermediates and trapping the folding intermediates into “aggregation-insensitive” states. It was demonstrated [25] that proline forms supramolecular assemblies in the solution. The results of the study of interaction of supramolecular assemblies with fluorescence probe (1-anilino-8-naphthalene sulfonic acid) are indicative of hydrophobic character of such assemblies. Owing to their hydrophobic properties, supramolecular assemblies of proline interact with the nonpolar portions of the unfolded states of the proteins and consequently prevent the aggregation process. The ability of proline to reduce the rate of selfassociation of PhK may be explained by direct incorporation of proline into the start associates. The start associates modified in this way would have a lower sticking probability than the initial start associates. Such a change in the reactivity of PhK associates implies the transition from a regime of diffusion-limited association to a regime of reaction-limited association [56]. The ability of α-crystallin to suppress the self-association of PhK is surprising, if one takes into account that, according to generally accepted opinion, α-crystallin binds to the unfolded states of proteins. One can propose that Ca2+ and Mg2+ , the cofactors which are necessary for the enzymatic activity of PhK, induce a partially unfolded state of the PhK molecule acquiring an ability to bind α-crystallin. As in the case of proline, α-crystallin is probably incorporated into the start associates and reduces their propensity for sticking together. Deceleration of selfassociation of PhK is connected with the transition of the association process from the regime of diffusion-limited association into the regime of reaction-limited association (the sticking probability becomes less than unity). If the reaction-limited regime is operative, the time-course of the hydrodynamic radius obeys the following relationship [56]: Rh = Rh,0 exp(Kt) ,

(7)

where K is a constant. As can be seen from Fig. 6B, at sufficiently high concentrations of α-crystallin (namely 0.3 g/l) the experimental dependence of Rh on t is satisfactorily described by Eq. 7. The results of the study of self-association of PhK by two methods (sedimentation velocity and dynamic light scattering) indicates that there are two stages of protein as-

Self-Association of Phosphorylase Kinase under Molecular Crowding Conditions

sociation. The first stage corresponds to the formation of associates with lesser size than that of the start associates. The second stage includes the formation of superassociates from the start associates. The initial association of PhK is a step-wise process as evidenced by the data presented in Fig. 1B. A set of oligomeric forms with s20,w = 38, 61, 106, 131, 148, 218 and 327 S was obtained. The results of the sedimentation velocity analyses of PhK self-association carried out by us [7] at lower Mg2+ concentration (2 mM) also indicate that initial association of PhK is characterized by the formation of discrete oligomeric forms with s20,w = 189 and 385 S and lesser values. It should be noted that the empirical relationship (2) between the molecular mass of the protein associate and its hydrodynamic radius is valid only for initial associates of PhK including the start associates. Although the initial associates of PhK really exist, the formation of the start associates proceeds substantially as

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a process of cooperative association and the concentrations of the intermediate forms are too low to be registered by DLS. In contrast to analytical ultracentrifugation, dynamic light scattering is a very suitable method for the detection of the superassociates which are formed by the sticking together of start associates. Judging from the size of the start associate, it contains about 80 PhK molecules. It is significant that the crowding-agent (TMAO) exerts a similar effect on both stages of PhK self-association. TMAO favours the formation of prestart associates as well as the formation of superassociates. Proline and α-crystallin exerts a similar counteracting effect on all the stages of PhK association. The results obtained imply that proline and αcrystallin bind directly to the PhK molecule. Acknowledgement This study was funded by the Russian Foundation for Basic Research (grant 05-04-48691), Programs “Molecular and Cell Biology” of the Presidium of the Russian Academy of Sciences, the Program for the Support of the Leading Scientific Schools in Russia (grant 813.2003.4), and by INTAS (grant 03-51-4813).

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Progr Colloid Polym Sci (2006) 131: 93–96 DOI 10.1007/2882_008 © Springer-Verlag Berlin Heidelberg 2006 Published online: 15 February 2006

BIOLOGICAL SYSTEMS

Prudence Mutowo David J. Scott

Oligomerisation of TBP1 from Haloferax volcanii

Prudence Mutowo · David J. Scott (u) National Centre for Macromolecular Hydrodynamics, School of Biosciences, University of Nottingham, Sutton Bonington, Leicestershire LE12 5RD, UK e-mail: [email protected]

Abstract This study considers the oligomeric of TBP1 from the extreme halophilic archaeaon Haloferax volcanii. This protein is in the first step of the cascade of binding events leading to transcription initiation. Sedimentation velocity shows that at

3M KCl, the protein appears to be in several oligomeric forms. Keywords Halophilic proteins · Protein-DNA interactions · Haloferax volcanii · archaea

Introduction

Materials and Methods

Fundamental to all life on earth are the interactions of proteins with DNA. These occur in sequence and nonsequence specific manner, and although diverse, general themes for these interactions have occurred. In particular these interactions are salt sensitive: increases in salt concentration destabilises the DNA/protein complex [11]. This is due to the fact that upon complex formation ions are released from the biomolecules. Hence addition of salt to the buffer will cause the equilibrium to be shifted towards the complex separating into its component parts. Halophilic organisms are able to thrive in salt concentrations that range from 2–5 M [17]. In the extreme, halophiles are able to live in environments that have 250–300 gl − 1 NaCl. In fact such is their dependence of these high levels of salt that at levels lower than these concentrations, these organisms die. We consider the oligomeric state using analytical ultracentrifugation of the TATA-binding protein (TBP1) from the extreme halophile Haloferax volcanii. This organism is an obligate halophiles that has an internal salt concentration of between 1.4–4.0 M KCl. This presents a serious challenge to the standard model of protein/DNA ion dependency. As a first step to solving this problem we present sedimentation velocity results on the oligomerisation of TBP1.

Protein Cloning Polymerase Chain Reaction: Conventional PCR was carried out in a PCR mix containing 20 nmol Nde1 forward and BamH1 reverse primers in a mix with 10 µl buffer, 5 µl enhancer, 2 µl MgSO4 , 3 µl dNTPs, 0.5 µl chrosomosomal DNA, 1 µl Pfx and was made up to 100 µl with deionised water. The PCR amplicons were analysed by electrophoresis of 10 µl of reaction mix in 1% agarose gels at 100 volts for one hour. Fragment selection was carried out and fragments of the desired size were cleaned up using the wizard PCR clean up kit (Promega, UK). Restriction digest: a 30 µl restriction digest containing 3 µl 10 X buffer, 10 µl of the PCR fragment, 0.3 µl BSA, 10 µl deionizer water and 1 µl each of the forward and reverse primers was set up and left to digest for 2 hours. The product was cleaned up with the Wizard cleanup kit. The vector was phosphatased by adding 10 µl 10x buffer, 2 µl shrimp alkaline phosphatase and 43 µl of vector DNA. The mixture was incubated at 37 ◦ C for half an hour and then cleaned up with the wizard clean up kit. Vector-insert ligation: 4 µl vector, 1 µl 10x buffer, 1 µl T4 ligase and 4 µl deionised water were set us as the main mixture for three different reactions containing 0 µl insert (as control), 4 µl and 0.4 µl insert respectively. The mixture was left on ice water overnight. 3 µl of DNA were

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added into competent cells and placed on ice for 30 minutes. The mixture was heat shocked at 42 ◦ C for 45 seconds and placed immediately on ice. SOC media, 250 µl was added to the tubes and incubated at 37 ◦ C for 60 minutes. The mixture was plated out on lactose broth plates containing 50 µg/ml kanamycin and placed at 37 ◦ C. Agarose gel preparation and running: agarose gel prepared and ran as per Protocol 1 Sambrook and Russell Vol 1 Section 5.4–5.10). Transformation of competent cells: plasmid DNA 1 µl was added to 20 µl of competent Escherichia Coli cells and the mixture was incubated on ice for 30 minutes. The mixture was then transferred to a 42 ◦ C water bath for 1 minute and then straight onto ice for 2 minutes. SOC media (250 µl) was added to the mixture and incubated at 37 ◦ C for an hour. The media was plated onto lactose broth plates containing the correct antibiotics and incubated overnight at 37 ◦ C. For expression of protein in the E. coli BL-star strain (Invitrogen, UK) carrying the pRARE2 plasmid (Novagen, UK), a similar procedure was carried out and the transformants plated out on LB plates containing chloramphenicol and kanamycin. Protein Expression Expression of protein in Star Rosetta strain: lactose broth media (500 ml) containing kanamycin and chloramphenicol was inoculated with a colony of the transformed Star Rosetta strains. The media was grown to an optical density of 0.6 at a wavelength of 600 nm the mixture was induced with 1 mM IPTG and grown for three hours. At time points 0, 1, 2 and 3 hours respectively, aliquots of media were taken, spun down in a microfuge and resuspended in 1X TAE buffer. The aliquots were denatured at 95 ◦ C and run on a gel. After three hours, the main culture was spun down at 3000 rpm for half an hour, the supernatant was discarded and the pellet resuspended in 50 ml Phosphate buffered saline (PBS) and stored at −20 ◦ C. Protein purification by His tag column: frozen protein sample was thawed and sonicated at 10 micron amplitude using two thirty second blasts. The mixture was then spun at 20 000 rpm for forty five minutes. The supernatant was passed over a prepared His Tag column which is then washed out with 100 mM imidazole and 2 M imidazole. Aliquots from each step were run on a 12% acrylamide gel. Concentration of protein sample: protein sample was concentrated down by placing sample in 50 ml centricon tube with an 18 kDa molecular weight filter. The sample was spun at 3000 rpm for 45 minutes and protein retained in the filter was collected. Protein purification via gel filtration column: concentrated protein in buffer (50 mm Tris HCl pH 8.0, 2 M KCl), was passed through a gel filtration column equilibrated with buffer. Dialysis of protein sample: protein sample (2 ml) was placed in a dialysis tubing (molecular weight cut off

P. Mutowo · D.J. Scott

10–12 kDA, pore size 6.3 mm diameter) that had been previously soaked in distilled water. The tube was then placed in a beaker containing dialysis buffer. The reaction was allowed to proceed overnight at 4 ◦ C. Protein Purification Concentrated protein in buffer (50 mm Tris HCl pH 8.0, 2 M KCl), was passed through a gel filtration column equilibrated with buffer. Analysis of Protein Using AUC Analytical ultracentrifugation experiments were carried out on a Beckman XL-I analytical ultracentrifuge. Sedimentation velocity determinations were carried out using a Beckman Optima XL-A analytical ultracentrifuge on an AnTi 60 rotor set to run at 20 ◦ C. Data was obtained at 40 000 rpm using Epon two channel centrepieces, with loading concentrations of 2.0, 1.0, and 0.5 mg ml−1 , absorbance was recorded at 280 nm. Data was analysed using the program SEDFIT, and apparent sedimentation coefficient distributions were estimated by the Maximum Entropy method of Schuck [1] (2000). Sample buffer throughout was 50 mm Tris HCl pH 8.0, 2 M KCl.

Results and Discussion Figure 1 shows the c(s) results for three loading concentrations for TBP1. As can clearly be seen, the sedimentation coefficient distributions do not resemble a single

Fig. 1 Sedimentation coefficient distribution derived using the c(s) method in SEDFIT. As can clearly be seen, TBP1 is not a single species. The expected sedimentation coefficient for the monomer is 1.4 s

Oligomerisation of TBP1 from Haloferax volcanii

species. The two major peaks at around 2.5 S and 5 S. Calculations of the sedimentation coefficient using HYDROPRO [2] and the homologous X-ray crystal structure from Pyroccocus woesei (PDB code: 1AIS) showed that the monomer had a sedimentation coefficient of around 1.4 S. This therefore indicates that the 2.5 S peak is at least a dimer. The 5 S peak being twice the sedimentation coefficient of the slower species does not readily correspond to a dimer. The study over several protein concentrations showed minimal shifts in peak sedimentation value, but some shift in concentration, indicating some reversible multimerisation is taking place. The multimerisation of TBPs is something of a controversy. Fried and co-workers found that TBP from Sacchromyces cerevisiae was present in solution in a variety of multimeric forms including tetramers and octamers [3, 4]. Pugh and co-workers hypothesised that TBP dimerisation was part of an auto-inhibition mechanism [5–7] whereby dimerisation across the DNA binding interface prevented promoter binding. Additionally, they also showed slow off-rate kinetics [5] which they proposed to be part of a regulatory feedback mechanism. However, it should be pointed out that the former studies were carried out by analytical ultracentrifugation, and the latter by tritium exchange at several orders of magnitude lower concentrations. This gives rise to the possibility that the disparity between the studies is due to the concentration of TBP used and that the monomeric state is only populated at low (nanomolar) concentrations. It should also be noted that the crystal structures of apo-TBPs are dimeric, but in the heterocomplex with DNA they are monomeric [8]. Given that TBP may be present in quite low concentrations in the cell [8, 9], it may well be monomeric; this, however, does not take account in anyway to the contribution to dimerisation from crowding: something that will be a significant effect in molar salt concentrations [10]. Ladbury and co-workers found that the TBP from the archaeon Pyroccocus woesei was monomeric at the concentrations used in both their analytical ultracentrifugation and isothermal titration calorimeter studies [11, 12]. The results presented here can be interpreted in several ways. One is that Haloferax TBP1 is present in a variety of oligomeric species naturally. Another is that as Haloferax has four TBPs in its genome (T.E. Batstone and D.J. Scott, unpublished results), then this TBP is simply not active at salinities of 3 M KCl, and one of the others is. The third unknown quantity is that the protein is expressed with an N-terminal hexahistidine tag for ease of purification. It is unknown as yet the contribution that this makes to the oligomerisation. Finally, we have only measured TBP1 at one salt type, hence it is unknown as to what the contribution of changing salt type is. In the cell, the predominant salt is KCl, however other salts are present at lower (mM) concentrations. Mono and divalent salts are required in all organisms because they act as co-catalysts in enzymatic reactions.

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The most commonly found ions are of Mg2+ , Na+ , Cl− , Zn2+ , etc. The participation of these ions in DNA protein interactions are vital for the propagation of life especially in transcription initiation where a DNA/protein complex needs to be formed prior to the recruitment of the RNA polymerase enzyme. In general, DNA protein interactions occur at conditions of a salt concentration of less than 0.5% (http://esapub.esrin.esa.it/sp/sp1231/chap1.3.pdf). At salt levels higher than these, the following problems are encountered: The presence of a high concentration of solute ions is generally devastating to proteins and other macromolecules: (i) it causes aggregation or structural collapse of proteins because of enhancement of hydrophobic interactions; (ii) it interferes with essential electrostatic interaction within or between molecules because of charge shielding; and (iii) it reduces the availability of free water below that required to sustain essential biological processes because of salt ion hydration. There exists a network of relationships among DNA, protein, water and associated ions involved in the formation of DNA/protein complexes. DNA is on average more highly hydrated than protein because it has a phosphate backbone [13]. The minor groove of DNA does not have a hydroxyl group and thus is barely hydrated. Local hydration of DNA occurs when the minor groove of DNA traps water molecules bound to adenine, thus effecting local hydration. This means that the hydration properties of DNA are affected by the shell of counter ions and divalent cations around it [14]. One strategy proposed to explain how halophiles cope with high salt concentration is that of compatible solutes. These are small organic compounds that are accumulated in the cell interior by some halophiles. These compounds are accumulated in the cytoplasm thus protecting the intracellular environment from the extreme exterior. A typical case is that seen in halobacteria that accumulate glutamate in their cell interior in order to maintain an osmotic balance with the cell exterior. This strategy works very well up to a concentration of osmolyte of 2 M, thereafter in becomes less effective. In general this limits these particular life forms to environments where the salinity is less than around 2 M [15]. Given that the halophilic archaea have been known to survive extreme salt conditions by either balancing the internal and external salt concentrations, and also by evolutionary modification of cell macromolecules to function in extreme salt conditions. Possible evolutionary adaptations that may have occurred in proteins include the increase of certain residues on the protein surface in order to counteract the effect of high salt concentrations. Glutamate and aspartate have been proposed to be some of these residues that are evolutionary preferred by halophiles over the more hydrophobic amino acids [16].

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The results presented here show that TBP1 is soluble, but in a variety of oligomeric states. Further work will decide whether these oligomers are artifactual or a real indication of an underlying mechanism of transcription regulation.

Acknowledgement The authors would like to thank Ken Davis for invaluable assistance in running the analytical ultracentrifuge, and Tom Batstone for in-depth discussion of the transcription mechanisms in the archaea. This work was supported by The University of Nottingham, The Nuffield Foundation and the Royal Society (UK).

References 1. Schuck P (2000) Size-distribution analysis of macromolecules by sedimentation velocity ultracentrifugation and lamm equation modeling. Biophys J 78(3):1606–1619 2. Garcia de la Torre J, Huertas ML, Carrasco B (2000) Calculation of Hydrodynamic Properties of Globular Proteins from Their Atomic-Level Structure. Biophys J 78(2):719–730 3. Daugherty MA, Brenowitz M, Fried MG (2000) Participation of the amino-terminal domain in the self-association of the full-length yeast TATA binding protein. Biochemistry 39(16):4869– 4880 4. Khrapunov S, Pastor N, Brenowitz M (2002) Solution structural studies of the Saccharomyces cerevisiae TATA binding protein (TBP). Biochemistry 41(30):9559–9571 5. Coleman RA, Pugh BF (1997) Slow dimer dissociation of the TATA binding protein dictates the kinetics of DNA binding. Proc Natl Acad Sci USA 94(14):7221–7226

6. Jackson-Fisher AJ et al. (1999) A role for TBP dimerization in preventing unregulated gene expression. Mol Cell 3(6):717–727 7. Weideman CA et al. (1997) Dynamic interplay of TFIIA, TBP and TATA DNA. J Mol Biol 271(1):61–75 8. Kosa PF et al. (1997) The 2.1-A crystal structure of an archaeal preinitiation complex: TATA-box-binding protein/transcription factor (II)B core/TATA-box. Proc Natl Acad Sci USA 94(12):6042–6047 9. Littlefield O, Korkhin Y, Sigler PB (1999) The structural basis for the oriented assembly of a TBP/TFB/promoter complex. Proc Natl Acad Sci USA 96(24):13668–13673 10. Zimmerman SB, Minton AP (1993) Macromolecular crowding: biochemical, biophysical, and physiological consequences. Annu Rev Biophys Biomol Struct 22:27–65 11. Bergqvist S et al. (2002) Reversal of halophilicity in a protein-DNA

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interaction by limited mutation strategy. Structure (Camb) 10(5):629–637 Bergqvist S et al. (2004) Heat capacity effects of water molecules and ions at a protein-DNA interface. J Mol Biol 336(4):829–842 Csermely P (2001) Water and cellular folding processes. Cell Mol Biol (Noisy-le-grand) 47(5):791–800 Becker M et al. (1999) The double helix is dehydrated: evidence from the hydrolysis of acridinium ester-labeled probes. Biochemistry 38(17):5603–5611 Oren A (1999) Bioenergetic aspects of halophilism. Microbiol Mol Biol Rev 63(2):334–348 Britton KL et al. (1998) Insights into the molecular basis of salt tolerance from the study of glutamate dehydrogenase from Halobacterium salinarum. J Biol Chem 273(15):9023–9030 Dennis P et al. (1997) Evolutionary divergence and salinity-mediated selection in halophilic archaea. Microbiol Mol Biol Rev 61(1):90–104

Progr Colloid Polym Sci (2006) 131: 97–107 DOI 10.1007/2882_009 © Springer-Verlag Berlin Heidelberg 2006 Published online: 15 February 2006

Oliver Clay Nicolas Carels Christophe J. Douady Giorgio Bernardi

Oliver Clay · Nicolas Carels · Giorgio Bernardi (u) Laboratory of Molecular Evolution, Stazione Zoologica Anton Dohrn, Villa Comunale, 80121 Napoli, Italy e-mail: [email protected] Christophe J. Douady Équipe d’Hydrobiologie et Ecologie Souterraines & Plateforme d’Ecologie Moléculaire, Laboratoire d’Ecologie des Hydrosystèmes Fluviaux, Université Claude Bernard Lyon 1, UMR CNRS 5023, 69622 Villeurbanne Cedex, France

BIOLOGICAL SYSTEMS

Density Gradient Ultracentrifugation and Whole Genome Sequences: Fine-tuning the Correspondence

Abstract Since its introduction in the 1950’s, analytical ultracentrifugation (AUC) of DNA in CsCl and other salt density gradients at sedimentation equilibrium has remained an elegant way to gain insight into the variation of base composition (GC, guanine + cytosine %) among and within animal and plant chromosomes, and into functional correlates of GC. Absorbance profiles of routine preparations of DNA in CsCl are essentially GC histograms of fixed-length sequence fragments (≈ 15–100 kb). This correspondence has been amply illustrated by genome sequences obtained over the past 5 years. Both AUC and sequencing have now generated large amounts of data that can be jointly mined. The dialogue between these two approaches should render tractable some tenacious problems

Introduction The base composition of a DNA molecule is primarily its GC level, the molar ratio of GC (guanine-cytosine) base pairs in the DNA. If one considers just one strand, GC is the percentage of nucleotides that are G or C, and not A or T. This is a most fundamental property of DNA. Numerous functional and evolutionary correlates of its variation along chromosomes, in taxa ranging from bacteria to human, are now known [1]. CsCl gradient density ultracentrifugation of DNA was introduced in 1957. Its principle is well summarized in the original paper [2]: “A solution of a low-molecular

of CsCl profile analysis, such as the correct treatment of concentration dependence for heterogeneous DNA. We focus on how absorbance profiles of a species’ DNA vary as one changes the scale of one’s observation (molecular weight), and dissect this scale-dependence into the contributions from its two main sources (diffusion, sequence effects). Our understanding of heterogeneous DNA in CsCl gradients can profit from the comparison of results from AUC and whole-genome sequencing, and the insights gained should prompt more strategic AUC analyses of DNA. Keywords Analytical ultracentrifugation · Sedimentation equilibrium · Base composition · Evolution · Long-range correlations

weight solute [e.g., CsCl] is centrifuged until equilibrium is closely approached, [resulting] in a continuously increasing density along the direction of centrifugal force. Consider the distribution of a small amount of a single macromolecular species [e.g., DNA] in this density gradient. The initial concentration of the low-molecular-weight solute, the centrifugal field strength, and the length of the liquid column may be chosen so that the range of density at equilibrium encompasses the [buoyant] density of the macromolecular material. The centrifugal field tends to drive the macromolecules into the region where the sum of the forces acting on a given molecule is zero. (The [buoyant] density of the macromolecular material is here

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defined as the density of the solution in this region.) This concentrating tendency is opposed by Brownian motion, with the result that at equilibrium the macromolecules are distributed with respect to concentration in a band of width inversely related to their molecular weight.” Soon after the introduction of this technique, which was conceived with labelling in mind (see [3] for a historical account), it was discovered that there is an important, simple and accurate empirical link between analytical ultracentrifugation (AUC) and GC [4–6]. This link has been used routinely since then, yet it still awaits a full quantitative, physicochemical explanation. In a CsCl density gradient, at sedimentation equilibrium, the GC of a macromolecule of DNA is linearly related to its (time-averaged) buoyant density. The buoyant density of that DNA is, in turn, practically a linear function of its radial position: the density gradient is effectively linear over the distances from the ultracentrifuge axis that are of interest. The positional distribution of the DNA macromolecules in a gradient is scanned by an analytical ultracentrifuge and reported as an absorbance profile. A DNA sample consisting of molecules or fragments that have different base compositions (GC) will have, at sedimentation equilibrium, a profile of finite width. To a good approximation, valid for most species’ DNA and especially if the molecular weight is high, this equilibrium profile is just the GC distribution of the molecules. Corrections that improve the accuracy of the match are based on tractable, well-known effects experienced by macromolecules in solution. Indeed, there are basically three components: DNA, salt, and water, although DNA macromolecules can have different sequences that make them behave in solution, or interact, in different ways by adopting anomalous configurations or aggregating. The problem is not so much in understanding the individual effects involved, but in assessing their relative importance and cross-influencing, which encumbers a modular treatment. Subtle but persistent impediments to perfect matching may involve DNA concentration effects and/or aggregation, anomalous sedimentation of repetitive DNA, and methylation. Molecular weight polydispersity, where present, can add further complexity, although in many situations it is unproblematic: current DNA extraction protocols typically produce narrow molecular weight distributions, so in calculations one can simply use the mean. GC and its contrasts are of interest because they correspond to functionally and evolutionarily telling genome properties. For example, in mammals and birds the GCrichest regions of a genome have the highest gene densities and expression levels, the most interior locations in the nucleus at interphase and the earliest replication in S-phase, preferentially open chromatin and short-intron genes, and more frequent and longer CpG islands (reviewed in [1]). The analytical ultracentrifuge is likely to soon become more refined, precise and versatile (see, e.g., [7]), and

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whole-genome sequencing should become cheaper and faster (see [8–10]). Such technical advances promise to allow more accurate and varied comparisons between GC distributions of genomic DNA sequences and their CsCl absorbance profiles. It now seems, therefore, the right time to address remaining open problems in our understanding of how heterogeneous DNA behaves while it is being ultracentrifuged in CsCl gradients. In this paper, we use results from density gradient AUC and entirely sequenced genomes to specify how sequences and absorbance profiles should dovetail. Proper dovetailing permits consistency checks that can tell us if our ideas about profile formation are correct.

Experimental At sedimentation equilibrium in a CsCl density gradient, the GC level of a molecule or fragment of double-stranded DNA is related, with a few exceptions that are listed below, to its (time-averaged) buoyant density ρ in the gradient [4– 6]. The linear equation that relates these two quantities is [11] ρ − 1.660 g cm−3 GC = × 100% . (1) 0.098 Because there are exceptions to this rule, buoyant density (also called “density” in early publications), rather than GC, is often chosen as the quantity of interest in AUC studies. GC is, however, the ultimate object of our investigations. Buoyant density in CsCl is, in turn, a simple function of the distance r of the molecule from the axis of an analytical ultracentrifuge [12], 2 ). ρ = ρm + κω2 (r 2 − rm

(2)

Here, ρm and rm are, respectively, the buoyant density in CsCl and radial position of a suitable marker, such as bacteriophage 2C (which has a very high ρm , 1.742 g/cm3 because of its modified bases). ω is the angular speed and κ is a constant that depends on the details of the ultracentrifuge cell and the rotor (κ ≈ 4.2 × 10−10 for Beckman models E and XL-A). Since the differences in radial position are very small compared to the distances from the 2 ≈ 2r (r − r ), i.e., the nonlinaxis, we can write r 2 − rm m m earity in Eq. 2 is negligible, and the gradient is essentially linear. For a generic, high molecular weight DNA sample (e.g., 50–100 kilobases or kb) in which we can neglect diffusion of the molecules or fragments, their radial distribution in the CsCl gradient is, after a linear calibration, just their GC distribution. At lower molecular weights, diffusion visibly broadens the radial distribution. Indeed, small DNA molecules or fragments will continue to diffuse appreciably around their expected positions in the gradient at sedimentation equilibrium, although the average number of molecules or fragments at any given position will not

Density Gradient Ultracentrifugation and Whole Genome Sequences: Fine-tuning the Correspondence

change, i.e., at a macroscopic scale there will be no more net motion. The absorbance profile that reports the radial distribution thus corresponds, via Eqs. 1 and 2, to the GC distribution of the DNA molecules, if their molecular weight is high and they do not contain satellite or modified DNA. For this correspondence to hold, however, certain standard experimental conditions must be met (see, e.g., [13] and [14] for partial lists). Among the conditions, we mention that the average concentration should be neither too low, because the absorbance of DNA is then easily confounded with a possibly irregular “baseline” that may include minor contaminants, nor too high, because the response is no longer linear when DNA crowds excessively at the band center. If CsCl solution is excluded from the region and/or light is blocked, this can locally deform the otherwise linear gradient and/or flatten the profile. Absorbances are measured at 260 nm and the standard speed is 44 000 rpm.

The Shaping of the DNA Absorbance Profile: Physicochemical Contributions We begin with a historical absorbance profile: the first view of a mammalian genome’s GC distribution ever published. Figure 1 shows the positional distribution of fragments of DNA from a calf’s thymus DNA in a CsCl gradient, at sedimentation equilibrium, reproduced from the original paper that introduced density gradient ultracentrifugation [2]. The speed of the rotor (44 700 rpm) was similar to those used in routine CsCl analyses today; only the molecular weight (presumably around 5 kb) was not as high. Such equilibrium profiles can be obtained after about 24 hours. An introduction to the method’s physical foundations can be found in the book by van Holde et al. [15].

Fig. 1 Equilibrium profile of the cow genome in CsCl, obtained in 1957 when CsCl gradient AUC was introduced. Molecular weight is presumably less than or around 5 kb, other conditions are standard. Reproduced with permission from Meselson et al. [2]

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The caption accompanying the 1957 plot concluded with the sentence: “The skewness in the resultant band indicates heterogeneity in effective density”. Such marked heterogeneity of the effective density, or buoyant density, is generally pronounced in mammals and birds, and results largely from GC heterogeneity among the chromosomal fragments represented in the sample. In the special case of cow, the profile heterogeneity is further exaggerated by the very high percentage (≈ 25%) of highly repetitive satellite DNA in the bovine genome, as was discovered later [16]. In October 2004, the first draft of the cow genome sequence was placed in the public domain (http://genome. gov/12512874). The full assembly of the large scaffolds and the publication describing the sequence have not yet appeared. If one allows for some inaccuracies due to gaps, the GC levels of the sequenced Hereford cow’s 5 kb segments (or, rather, of the presently available scaffolds’ segments) can be fetched from an annotation database that now exists for this genome and plotted as a histogram, as is shown in Fig. 2. In this way we can again see the cow genome’s GC distribution, 47 years and some $53 million after the scan of Fig. 1. We now go into technical detail. When one overlays (after converting to GC units) an experimental curve such as the one in Fig. 1, which represents collections of similarly sized fragments of a genome, by its sequence-derived counterpart, such as the histogram in Fig. 2, one finds that the absorbance profile is wider than the histogram. The two main reasons for this difference in cow are well known: highly repetitive DNA [16], and diffusion. Highly repetitive DNA is present only in modest amounts in sequence scaffolds: heterochromatic regions such as centromeres are not (and will not soon be) tar-

Fig. 2 GC histogram of 5 kb segments of the recently sequenced cow genome. The draft sequence’s GC levels were obtained from the gc5base.txt table of the UCSC annotation database for cow, http://genome.ucsc.edu (ratio sumData/validCount). Bins of 1% GC were used

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geted for sequencing, since they rarely contain genes and are more difficult to sequence. Where scaffolds do contain such DNA, the sequences report the correct GC. In an ultracentrifuged sample, on the other hand, a fair number of the molecules will consist of (largely unsequenced) highly repetitive DNA, which can extend over long tracts of unstable length, expanding or contracting even from one generation to the next as a result of mechanisms such as slippage and out-of-register homologous recombination. Such satellite DNA will often (but not always) appear as visible peaks or bumps within or next to the main-band profile: since the many molecules of a particular satellite are identical or almost identical, their GC level will be overrepresented in the absorbance profile. The amount of a particular satellite in a genome, i.e., the height of its peak or bump in the profile, will often change quickly over evolutionary timescales, sometimes differing visibly within genera (as in kangaroo rats [17]). Some of the repetitive satellite DNA may also band anomalously, i.e., be found at another position in the gradient than expected on the basis of its GC (as in guinea pig [18]). Such abnormal banding occurs presumably because the repetitiveness (and/or methylation) of the satellite fragments alters their buoyant density. Diffusion is an effect that is irrelevant to GC distributions obtained from sequences, whereas in a CsCl gradient the DNA molecules undergo random Brownian motion around their expected equilibrium positions, and thus broaden the absorbance profile. When molecules or fragments are identical, diffusion broadens the profile in inverse square proportion to the fragments’ lengths. To approach an understanding of how diffusion acts on DNA in real situations, we begin by considering the natural DNA that gives the simplest CsCl profile: a preparation of intact, identical copies of a bacteriophage such as phage lambda. Such a sample of lambda DNA will have no intermolecular GC heterogeneity (the molecules will typically all have the same sequence, gb:lamcg), and no polydispersity in molecular weight (M = 48.5 kb). Because of the simplicity of the scenario they permit, phages have been often used as models for studying macromolecules’ behavior in density gradients. The absorbance profile of such a homogeneous, monodisperse DNA sample will report a Gaussian distribution of molecules’ positions. The width (standard deviation) of that Gaussian profile will depend on diffusion, but also to some extent on the amount of DNA that was loaded into the ultracentrifuge cell: virial effects will cause an additional broadening of the profile when the overall DNA concentration is high. The virial effect is absent, by definition, at infinite dilution, and in phages it has been observed to increase exponentially with increasing concentration. We can therefore summarize the situation for homogeneous DNA samples by saying that their absorbance profiles are well described by √ a Gaussian distribution with standard deviation σ = a/M e Bc , where M is the molecular weight, c is a measure of over-

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all concentration or of the amount of DNA loaded, such as the maximum absorbance (optical density), a is a factor that takes solvation into account, and B can depend on the species analyzed [19, 20]. Vertebrate genomes or chromosomes are much larger than those of intact phages, and are therefore represented only by their fragments in DNA samples: during the experimental preparation of DNA, a very large macromolecule such as a cow chromosome is inevitably sheared into fragments. Where a genome is heterogeneous in GC, there will then be intermolecular GC heterogeneity in the sample, and thus broad profiles, which will be further broadened by diffusion and/or satellites. Local narrowing can also occur: if some but not all DNA aggregates during the approach to equilibrium, the aggregating DNA will attain a higher molecular weight than the rest of the DNA, so an effective polydispersity can develop, with the aggregating DNA forming a more highly peaked “subprofile” (see [21] for an example in mouse). Several other factors can also play a role in shaping or shifting profiles, such as DNA methylation, electric effects, pressure effects, local deformations of the CsCl gradient where DNA is crowded, or light bending by the gradient. Most of these factors hardly distort the relatively broad profiles of vertebrates. At infinite dilution, there is no concentration dependence, and the total profile variance of a heterogeneous sample is then the GC distribution’s variance plus the dif2 fusion variance σdiffusion = a/M [20]. In other words, we have a convolution of an (often non-Gaussian) distribution representing the GC heterogeneity and a Gaussian distribution of unit area (also called a filter, kernel, or point spread function) representing the diffusion broadening. Once the constant a is known, the standard deviation of the GC distribution can be calculated from the absorbance profile and an estimate of the sample’s molecular weight M. With some inevitable numerical inaccuracies, we can then even extract the full GC distribution by deconvolving, i.e., “peeling off” the Gaussian of unit area that represents the diffusion. In real situations, dilutions are finite, and when we wish to take concentration effects into account we can no longer carry over the solution from the homogeneous case: virial effects and possible aggregation can act simultaneously but in opposite directions (broadening or narrowing the profile) and in different parts of the profile (tails or center). The remarks we have made so far pertain to ways in which DNA macromolecules’ behavior in a density gradient shape their absorbance profile. The shaping entails a molecular weight dependence that is especially strong for small fragments, but becomes weak or negligible for long fragments. It has been understood in its rudiments for over three decades. This rudimentary understanding often permits accurate extraction of the underlying GC distribution. In most cases where the entire genome sequence is known, one finds that by neglecting concentration de-

Density Gradient Ultracentrifugation and Whole Genome Sequences: Fine-tuning the Correspondence

pendence, and by assuming a Gaussian point spread function whose width is inversely related to the square root of the fragments’ mean length, one obtains estimates of the underlying GC distribution that differ only very slightly from the sequence-derived GC distribution (cf. [14]). In some species or conditions, however, the profile and/or its concentration dependence depart visibly from expectations. Another contribution to the profile’s final shape is of practical as well as molecular-biological and evolutionary interest: the GC distribution, which entails a molecular weight dependence that is intrinsic to the genome under study.

The Shaping of the DNA Absorbance Profile: GC Contributions At the sequence level, molecular weight corresponds to fragment length or, if one views the fragment as part of a chromosome sequence, to segment or window length. Thus a GC distribution that has been deduced from an absorbance profile will correspond to a sequence-derived GC histogram. If the sample consists of heterogeneous but monodisperse fragments of chromosomal DNA, the histogram will be a histogram of fixed-length segments or windows along the chromosome(s). Sequence heterogeneity can be investigated experimentally by CsCl gradient ultracentrifugation over two orders of magnitude (≈ 3–300 kb), using a simple principle: intramolecular heterogeneity becomes intermolecular heterogeneity when one regards smaller molecules or fragments. Thus, much of the intragenomic GC heterogeneity in mammals was well understood already in the 1970’s, even though no chromosomal regions (and only a handful of genes) had known sequences at that time. The investigations using AUC led to the following picture. Sequences of mammalian DNA (GC and AT) do not resemble runs of independent coin tosses (heads and tails), but have a remarkable organization. Within relatively short (kb) regions, GC levels are already long-range correlated. At larger scales (> 100 kb), GC levels are organized in a mosaic as one travels along a chromosome, in which GC-rich segments alternate with GC-poorer segments. The pieces of the mosaic, which are much more homogeneous in GC than the whole genome and persist in this relative homogeneity over long distances, from ≈ 300 kb to several megabases, are called isochores ([21–23]; their discovery, functional correlates and evolution are reviewed in [1]). Intragenomic heterogeneity at a given scale, such as 10 or 70 kb, can be quantified by the width of the CsCl absorbance profile or, more precisely, by the standard deviation of its underlying GC distribution. Thus, plotting the width (standard deviation) of a genome’s GC distributions versus the relevant segment size (molecular weight)

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gives an immediate visual impression of that genome’s GC heterogeneity at all scales. Plots of this type are shown in Fig. 3, on double-logarithmic scales. The slope of the loglog plot for human is around −0.15 (already much less steep than for an uncorrelated sequence) at low molecular weights < 10 kb, and then gradually flattens out, finally reaching a horizontal plateau that remains constant for molecular weights greater than about 70–100 kb. In other words, the narrowing of a mammalian CsCl profile, as molecular weights are increased, slows down and reaches a constant width from about 70–100 kb onwards. In fact, at 100 kb a mammalian profile (and not just its standard deviation) is practically indistinguishable from one at 300 kb. This observation was first made using AUC [21] and was recently confirmed with genome sequences [24]. The original observation was a key element in deducing the existence of isochores [21], since it is exactly what one would expect if chromosomes are organized into long regions  300 kb within which heterogeneity is remarkably low compared to the genome-wide heterogeneity, and it is very far from what one would expect if no similar large-scale structure were present.

Fig. 3 GC standard deviations of entirely sequenced animals at different molecular weights. Vertebrates are shown by solid lines, from top to bottom: dog, human, chicken (square markers), Tetraodon pufferfish, zebrafish (circular markers). Protostomes (dashed lines and circular markers, from top to bottom): Anopheles gambiae mosquito, Drosophila yakuba fly, and Caenorhabditis elegans nematode worm. Sequences were retrieved from the UCSC genome browser (http://genome.ucsc.edu, GoldFasta files). Logarithmic scales are used for both axes, since both random DNA (dotted line at bottom left, log interval slope of −1/2) and powerlaw correlated DNA (slope less steep than −1/2) give straight lines on such plots. “Complete” vertebrate sequences still contain gaps, including ribosomal and heterochromatic DNA, so the plotted values are approximate. The grey region indicates the interval (15–100 kb) in which most AUC data points will be found. For clarity only the human plot is extended past this interval

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Implications of Long-range Correlations in Isochores The detection and characterization of long-range correlations or “long memory” in sequences or time series is an active field of research that has interested not just biologists, hydrologists, physicists, mathematicians and statisticians, but also econometrists and financial modellers, and several methods exist for estimating such correlations’ characteristic parameters (see, e.g., [25] or Chapter 8 of [26]). We here mention only the detection and estimation method that is natural in the context of CsCl gradient AUC. Long-range positive, serial autocorrelations in GC that are present in DNA can be detected by a linear decrease in the log-log plot of inter-segment standard deviation versus segment length, with a slope that is distinctly less steep than −1/2 over one or two orders of magnitude (see, e.g., [25, 27–29]). A slope of −1/2 is what one would expect for a random sequence of uncorrelated or independent nucleotides; a less steep slope but again following a straight line is what one would expect for a serial autocorrelation that decreases as a power of the intervening distance d (proportional to d −α ). The closer the slope is to 0, the higher is the long-memory parameter, i.e., the more serious is the departure from a statistical scenario of independent and identically distributed (i.i.d.), or random nucleotides. In particular, no short-range (Markovian) dependence can be assumed in such cases: the dependence is irreparably longrange. The fact that one cannot invoke familiar textbook scenarios has far-reaching implications. For one thing, the independence assumption is a pillar on which most of traditional statistics is based. When this assumption falls, many standard tests for large-scale DNA properties or contrasts become invalid, and the modified tests that are applicable (see Section 8.6 of [25] for examples) have lower apparent statistical power, at a given length scale. Similarly, traditional models used to reconstruct phylogenies from sequences typically assume independence among nucleotides. Using only analytical ultracentrifugation in CsCl, one can already use standard deviations of a species’ profiles, obtained at different molecular weights, to obtain log-log plots such as those shown in Fig. 3, and then use the slopes to estimate bulk long-range memory parameters (e.g., an estimate of the Hurst exponent H is the slope plus 1). One can, furthermore, experimentally obtain compositional fractions or Gaussian components [23]. By plotting their standard deviations individually against molecular weight, one observes that the standard deviation and long-memory parameters (non-independence) of mammalian DNA are systematically higher for GCrich isochores than for GC-poor isochores, at molecular weights up to around 70 kb. All of this can be done in principle, and largely also in practice, without knowing any

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sequences ([22, 23]; see [27] for a discussion of the concordance with sequence results). One consequence of long-range dependence that is directly visible during routine sequence analyses is statistical self-similarity, which holds in the “linear” range of the log-log plot, i.e., at scales up to about 70–100 kb. Indeed, moving-window GC scans of chromosomal regions obtained using one window size look, statistically, very similar to those obtained using a much smaller or much larger window size, if one correctly resizes the vertical axis (which one can do by consulting the standard deviation versus window plot). Thus, it is often impossible to deduce anything about the scale (window size/molecular weight) by just looking at the scans.

Comparing Species when the Molecular Weights are Different The general features described above are common to eutherian mammals, but the quantitative details can be quite different among individual species. If we had a collection of DNA samples from different mammals, all samples having the same molecular weight, we would have main band profiles with different modes, means, standard deviations (as shown in Fig. 3) and/or asymmetries. In other words, different mammals have different GC distributions. These differences can be phylogenetically informative, and/or tell us about base compositional shifts that occurred during mammalian evolution, such as the narrowing of the profile in a rodent lineage that led to mouse and rat. We would often like to compare different species’ genomes by comparing their GC distributions. We may have one DNA sample for each of the species, but then find that those samples’ molecular weights are different. This situation is not uncommon, since samples collected in the wild, preserved under different conditions by different investigators, and injected into the ultracentrifuge cell can end up with average molecular weights varying from about 10 kb (or less) to 100 kb. If we can accurately measure the average molecular weight of each sample, for example by pulsed-field gel electrophoresis, we can then estimate the GC distributions and/or their standard deviations. As mentioned above, a GC distribution of fixed-length fragments or segments of mammalian DNA (< 100 kb) will be narrower if the fragments are long than if they are short. Because of this molecular weight dependence, it can be difficult to compare profiles from two different species if one species is represented by, say, a 70 kb sample and the other species is represented only by a 10 kb sample. Ideally, one would like to predict what a profile would look like if the same species’ sample had instead some other, standard molecular weight. The plots of Fig. 3, obtained from recently sequenced genomes, show that this is in general a difficult task. Indeed, for mammals, other vertebrates, or even protostomes such as insects and ne-

Density Gradient Ultracentrifugation and Whole Genome Sequences: Fine-tuning the Correspondence

matodes there is no general rule that would allow us to reliably “convert” a species’ profile obtained at 10 kb to the same species’ profile at 70 kb. Conversely we might be tempted to shear the 70 kb down to about 10 kb and then centrifuge the smaller fragments, but Fig. 3 shows that interspecific comparisons at 10 kb would no longer allow much resolution: valuable large-scale information that can differ markedly among genomes is destroyed by such shearing. The best we can offer, to shed some light on the possible molecular weight dependencies of different taxa, is a “calibration” of the plot of standard deviation versus molecular weight: one traces the molecular weight dependencies for species represented by several samples of different molecular weights [21, 23] or by a whole-genome sequence. Both the utility and the limitation of such a calibration are seen in Fig. 3. In the molecular weight range that is of most interest for CsCl work (≈ 15–100 kb), it is a promising sign that the lines traced by different vertebrate species do not cross, but fan out. In other words, if we have one fish genome’s GC distribution for only 10 kb fragments and another fish genome’s GC distribution for only 70 kb, both obtained via ultracentrifugation, then we can guess (but only guess, until we have more sequence- or AUCderived traces of fishes as calibrating guidelines) the angle at which the traces of Fig. 3 would pass through each of those two data points. The fact that zebrafish has narrower profiles than pufferfish, at a given molecular weight > 1 kb, is of interest since the genomes of these two bony fishes are comparable, and indeed the difference between their heterogeneities has long been known from AUC (see [30] and references therein). Similarly, the observation that the warm-blooded vertebrates have distinctly higher GC heterogeneities than the cold-blooded vertebrates exemplifies a well-documented difference of functional and evolutionary significance, and several lines of reasoning point to an explanation in terms of thermal stability ([31]; reviewed in [1]). The difference between the two dipteran insects, mosquito and fly, is also interesting, since they have diverged considerably in their genomes’ compositional properties [32], during the 250 million years or so since their lineages separated [33]. Figure 3 also shows that there is no hierarchy separating all vertebrates from all insects or nematodes: some large-scale genomic GC differences obviously evolved independently in fishes and insects. This example illustrates the danger of excessively widening the taxonomic range of one’s comparison: a homoplasy, i.e., a convergent or apparently convergent evolution of a trait (in this case GC heterogeneity) will mislead phylogenetic reconstructions if they are based on that trait. Zebrafish is obviously not more related to a worm than to pufferfish: the evolutionary distances over which one is trying to directly compare GC data are too wide. To reduce the chances of being misled, one can include the profile mean or

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mode as a second trait (giving a 2D plot, if molecular weights are similar; see [34] for an example), restrict one’s phylogenetic range, and sample taxa densely within that range.

Concentration Dependence and the Approach to Equilibrium The good agreement, after applying established corrections, between CsCl absorbance profiles and genome sequence-based histograms also indicates where theoretical improvements would be welcome. One remaining unsolved problem is how heterogeneous equilibrium profiles change as different amounts, i.e., concentrations of DNA are loaded. Another interesting problem would be to describe how heterogeneous pre-equilibrium profiles change as they approach equilibrium. A solution to the first problem, if indeed a generally applicable solution exists, would be of much help for extracting accurate GC distributions from AUC profiles. A solution to the second problem might be useful in providing a rough but “realtime” estimate of a sample’s molecular weight, without needing to resort to separate pulsed-field gels or sedimentation velocity runs, or for double-checking molecular weight estimates obtained via these other routes (see Appendix). It would be particularly useful for detecting and monitoring, in real time, any unexpected aggregation of DNA into high-molecular weight clusters, for example as a satellite band or crowded main band center is being formed. Not all calculations for homogeneous samples can be easily transferred to heterogeneous samples, and indeed the exponential concentration dependence of homogeneous DNA cannot be simply ported. Aggregation of DNA is often likely to enter the picture: it narrows (instead of broadening) a profile when DNA concentration is raised. In fact, if only the virial effect, observed for phages, were active, one might be tempted to try a folding (generalized convolution) operation, in which again a Gaussian spread function, this time with a width that increases exponentially with (local) concentration, spreads a heterogeneous genome’s GC distribution. The most flattened region of the profile would then be the modal region near the band center. This is not observed, and instead the modal region is often narrower than one would expect from the corresponding genome sequence. It is not yet clear if such observations are generally best explained by aggregation of crowded DNA, or if they are almost as often caused by other effects, or by unsequenced satellite DNA that may be present in the profile but not in the sequence-derived GC distribution. For a complete treatment one must probably return to first principles. Concerning the second problem, the approach to equilibrium of heterogeneous DNA, one might begin by considering a simple “toy” genome model consisting of two

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well-spaced components or peaks, such as those of phages lambda and 2C. Once the CsCl gradient is well established, the two components’ approaches to equilibrium will each be exponential (see Appendix). Their trapezoidlike pre-equilibrium profiles will appear, first superimposed, and then gradually narrow into the two equilibrium peaks. It is however unlikely that this simple problem of two well-separated, narrow bands can be satisfactorily generalized to the continuous wide band or GC distribution of a mammal. Rather than tinkering an approximate solution from off-the-shelf pieces, it would again seem indicated to begin afresh from first principles, where one has a firm grip on the assumptions one is making at every step of a derivation. The calculations of [35] treat the approach to equilibrium in the special case where dynamic changes of the DNA/CsCl solvation are negligible, and one possible path might begin from there.

Conclusion and Perspectives A raw AUC profile of DNA in a CsCl density gradient, and its underlying GC distribution, change when a DNA sample is substituted by a sample from the same species but having a different molecular weight. The way in which the profile changes can report functionally relevant, statistical properties of the genome and its genes, a fact that renders CsCl profiles especially useful when a genome has not been sequenced. Base compositional information can also help in reconstructing or confirming phylogenetic relationships (see [34] for a GC-based study of rodents). AUC-derived GC distributions can be phylogenetically informative, although satellite DNA contributions may need to be discounted: such highly repetitive DNA can band anomalously and, even when it does not, its rapid changes usually amount to noise except at the population or species levels. Many genomes exhibiting substantial GC heterogeneity at the 50–100 kb level have recently been sequenced. Such sequences have amply confirmed earlier rigorous deductions from AUC [14, 24, 27, 36–38] and now point the way to refined post-processing of CsCl absorbance profiles. Where there are subtle but unexpected differences between AUC profiles and the GC histograms from scans of whole-genome sequences, they can indicate gaps in the sequence or gaps in our understanding of macromolecules’ collective behavior in density gradients. CsCl profiles can be simulated or produced in silico from a genome sequence, incorporating facts and hypotheses to account for different factors that affect experimental profiles. Sequence-AUC comparisons can then be designed to fine-tune the hypotheses. Acknowledgement sions.

We thank Gabriel Macaya for helpful discus-

Appendix Historical and Theoretical Details Compositionally Homogeneous DNA. Almost all of the important quantitative physico-chemical and biophysical work on salt density gradient problems was done between 1957 and the mid-1970’s, and progress in deriving appropriate formulae came to an almost complete halt when molecular cloning began. Most early applications of CsCl density AUC involved DNA samples that were homogeneous in buoyant density and monodisperse in molecular weight, as is the case for preparations of intact complete phage DNAs (or nearly homogeneous and monodisperse, as for some bacterial DNAs), so that most of the early theoretical treatments also focussed on such DNA. The studies estimated, for example, molecular weight from band widths, i.e., from profile standard deviations, and calculated how various factors relevant to a routine AUC run would influence the position, shape and width of a band at sedimentation equilibrium. The factors included electric fields (DNA as a polyelectrolyte) [39], nonideality/solvation [40], virial effects/concentration dependence [19], pressure/compressibility [41, 42], methylation [43], other modifications of DNA [11], the presence of highly repetitive DNA [18, 44], and light bending [45]. Absorbance profile shapes depend primarily on the molecular weight, on the GC heterogeneity present, and on any anomalous banding behavior that may affect some but not the rest of the DNA. In the case of a homogeneous sample the molecular weight M is the most obvious factor. The expected profile is then roughly Gaussian, and the variance has the form 2 σtotal = a/M ,

(3)

where a is a proportionality constant (see [40] for details). Profiles’ shapes also depend on the average concentration of DNA, i.e., on the amount of DNA loaded, all other things being equal. This concentration dependence was quantitatively analyzed by Schmid and Hearst [19, 46], who found that the Gaussian profiles of all phages tested had widths that increased exponentially with increasing concentration c, i.e. they had the form
(4) σtotal = a/M e Bc . This behavior is what one would expect from an unusually strong virial effect. The widening of the profile is highly reproducible when the same phage is analyzed (Gabriel Macaya, personal communication describing unpublished data). On the other hand, some phages’ profiles widen more rapidly with increasing concentration than others, i.e., B is species-specific. How a homogeneous sample of macromolecules approaches its equilibrium distribution in density gradients

Density Gradient Ultracentrifugation and Whole Genome Sequences: Fine-tuning the Correspondence

has been the topic of several articles. One detailed modelling effort [47] assumed some fixed functional forms a priori in order to extend a diffusion-free solution that had, in essence, been obtained earlier from the Lamm equation in another context via the method of characteristics [48]. The articles also included an elegant study for DNA in the special case of a pre-formed gradient [49]. The results derived in that study allow one to estimate a homogeneous sample’s molecular weight by observing the banding of the DNA during the last hours of its approach to equilibrium: once the CsCl gradient has been established, the width and mean of a band’s profile approach their equilibrium values exponentially [50]. A log-linear plot of the remaining difference versus time then yields a straight line, whether one is observing the standard deviation or the mean. In the former case, the molecular weight can be calculated from the straight line’s slope. The calculation takes solvation (non-ideality) into account [50], and the results for phages agree favorably with molecular weights calculated via sedimentation velocity and/or now by whole-phage sequencing. The approach to equilibrium in density gradients is, interestingly, the topic also of two much more recent studies [35, 51], and a solution, for the limiting case of no interaction between DNA/CsCl and water, is now included in the program SEDFIT (http://www.analyticalultracentrifugation.com). Compositionally Heterogeneous DNA. The conceptual simplicity of homogeneous samples, exemplified by the short genomes of bacteriophages, prompted some excellent theoretical work. Such short genomes are, however, rare among many of the species of current interest, and the larger chromosomes of prokaryotes and especially of eukaryotes are invariably broken into fragments during routine DNA extraction. In addition, genomes of eukaryotes such as deuterostome or protostome animals or angiosperm plants are characterized by marked intragenomic contrasts in GC. As a result, samples of total nuclear DNA from those genomes exhibit substantial intermolecular (inter-fragment) heterogeneity, so they are represented by wider main bands in CsCl at equilibrium. Indeed, the original 1957 paper on CsCl gradient AUC [2] already showed absorbance profiles of a phage, and of calf, mentioning that in calf “the skewness in the resultant band indicates heterogeneity in [mean buoyant] density”, and that such “density heterogeneity may be compositional or structural in origin”. A detailed discussion of heterogeneous DNA was given by Sueoka two years later [52]. We assume here for simplicity that we are analyzing a sample in which none of the DNA bands anomalously, i.e., the time-averaged position of each of the jittering DNA molecules is where Eqs. 1 and 2 predict. If some DNA does band anomalously, its GC must be replaced by its effective GC in the following, i.e., by the GC to which its buoyant density would correspond; the effective GC can even be negative in the case of long

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poly-A repeats [44, 53]. We also assume here that we are near the theoretical limit of infinite dilution, so that there are no virial or other concentration-dependent effects. The absorbance profile of the DNA is then a convolution of the true GC distribution, after converting to appropriate units, and a Gaussian point spread function (i.e., a kernel, or filter) that broadens the GC distribution via diffusion. The 2 of the band is therefore just the sum total variance σtotal 2 due to true GC heterogeneity, plus of the variance σGC 2 the variance σdiffusion caused by the Brownian motion of molecules. When molecular weight is constant (monodisperse sample, fixed-length fragments or molecules), the latter variance is inversely proportional to the molecules’ common length or molecular weight M: 2 2 σtotal = σGC + a/M .

(5)

When molecular weight is not constant (polydisperse sample, variable-length fragments or molecules), the point spread function is in general not Gaussian. Indeed, our point spread function corresponds simply to the absorbance profile of a sample that is homogeneous in GC. If that sample is homogeneous but polydisperse, its absorbance profile is the weighted superposition of the profiles that would be obtained for each of the different molecular weights present. Even if our sample is heterogeneous and polydisperse, but exhibits no correlation between molecular weight and GC, a convolution can still be assumed. The resultant point spread function is then a weighted sum (or integral) of Gaussians, and a suitably modified version of Eq. 5 holds. The molecular weight M of a DNA sample can be determined via an independent method such as sedimentation velocity or (more recently) pulsed-field gel electrophoresis. If one knows the proportionality factor a in Eq. 5, one can then quickly calculate the GC heterogeneity. The best estimate of this factor is that given by Schmid and Hearst [20], who included solvation (non-ideality) in their treatment, checked their formula using several phages of known lengths, and described its use for heterogeneous DNA. Since a is only weakly dependent on GC, since the GC levels of heterogeneous genomes’ fragments are usually between 30% and 70%, and since other variables such as temperature remain standard for AUC runs, we hardly sacrifice any accuracy by treating a as a constant. For standard deviations in units of GC% we obtain for a, at 25◦ and other standard conditions, the value (44.5 ± 0.5) kb [14], which is at least as accurate as typical estimates of molecular weights from pulsed-field gels. The above estimates and observations can be theoretically justified only at infinite dilution. This simplifying assumption is, however, unrealistic in practical AUC situations. AUC runs are not done at infinite dilution of the DNA, and not even at high dilutions: maximal absorbances (optical densities) should not be less than about 0.25. The

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estimate of a discussed above is therefore a rough approximation, which we use only because at present no more accurate alternative is available (see below, and main text). The roughness of the approximation can depend on the genome of interest and/or on the part of the profile considered. As Sueoka’s early calculations [52] showed, the CsCl profile of calf thymus, i.e., of the bovine genome, was wider than those of certain bacteria and phages because of true heterogeneity among the (time-averaged) buoyant densities of the calf’s DNA fragments, and not trivially because of any molecular weight differences among the samples being compared. It was not until 14 years later, however, that the CsCl profile of the cow genome could be quantitatively explained in terms of its genomic GC. Indeed, its heterogeneity comes in part from the GC heterogeneity of single-copy DNA, but also in part from the massive amounts (25%) of highly repetitive satellite DNA that are present in the cow genome [16]. Such satellite DNA can be cryptic, and can easily broaden a profile, especially when it bands anomalously. Other examples in point are the guinea pig profile [18] and the mouse profile that is still included in standard biology textbooks to illustrate the concept of satellites. In summary, at infinite dilution a solution for a homogeneous sample can be transferred to the case of a heterogeneous sample: the solution of the homogeneous problem becomes the point spread function for the heterogeneous problem. The convolution principle that allows this guarantees that the variances of the spreading and of the GC distribution will add up to give (in units of equivalent GC%) the total profile variance. Deconvolving a CsCl profile might seem the most direct way to obtain or extract a GC distribution from an absorbance profile. In view of the current resolution limits of commercially available analytical ultracentrifuges, however, it is both numerically easier and more instructive

to work in the other direction. For example, our experience has shown that a truncated exponential distribution convolved with a Gaussian is typically a very good fitting function for GC distributions or (with a wider Gaussian, to accommodate diffusion) raw CsCl profiles of a mammal’s total DNA. Satellites are then often visible as bumps that locally deviate from the best fit [14, 34]. Similarly, when a genome sequence is available in its entirety (not yet common, even when the sequence is declared “finished”), its GC distribution can be convolved with model spread functions and the resulting distribution can then be compared with experimentally obtained CsCl profiles. Partial sequences allow rough comparisions. In a convolution/deconvolution problem, the spreading or broadening must be the same at all parts of the GC distribution or profile. A natural generalization of such a problem is a corresponding folding/unfolding problem [54], where this requirement is relaxed. In a folding model, the spread function and its width (extent of broadening) could be allowed to depend directly on the GC (in our case), and therefore indirectly on any other variables that are unambiguously specified by the GC. Folding models might therefore be useful in some situations where convolution models are too restrictive, although they can unfortunately not be applied to solve two problems for heterogeneous DNA discussed in this paper, concentration dependence and the approach to equilibrium. In profiles of heterogeneous DNA, aggregation and possibly other effects can counteract the flattening effect of concentration that is observed for species with homogeneous DNA such as bacteriophages. Thus, the homogeneous case cannot be generalized via folding to yield the concentration dependence for a heterogeneous species such as human. Similarly, no stage in the approach to equilibrium admits a folding model: GCrich and GC-poor DNA molecules are not much quicker or slower in separating from each other than in approaching equilibrium.

References 1. Bernardi G (2004) Structural and evolutionary genomics: Natural selection in genome evolution. Elsevier, Amsterdam etc. 2. Meselson M, Stahl FW, Vinograd J (1957) Proc Natl Acad Sci USA 43:581 3. Holmes FL (2001) Meselson, Stahl, and the replication of DNA. Yale University Press, New Haven London 4. Sueoka N, Marmur J, Doty P (1959) Nature 183:1429 5. Rolfe R, Meselson M (1959) Proc Natl Acad Sci USA 45:1039 6. Marmur J, Doty P (1959) Nature 183:1427

7. MacGregor IK, Anderson AL, Laue TM (2004) Biophys Chem 108:165 8. Collins FS, Green ED, Guttmacher AE, Guyer MS (2003) Nature 422:835 9. Powledge TM (2004) Genome Biology (Research News) 18 Nov 04 10. Smith C (2005) Nature 435:991 11. Schildkraut CL, Marmur J, Doty P (1962) J Mol Biol 4:430 12. Ifft JB, Voet DM, Vinograd J (1961) J Phys Chem 65:1138 13. Thiery JP, Macaya G, Bernardi G (1976) J Mol Biol 108:219 14. Clay O, Douady CJ, Carels N, Hughes S, Bucciarelli G, Bernardi G (2003) Eur Biophys J 32:418

15. van Holde KE, Johnson WC, Ho PS (1998) Principles of physical biochemistry. Prentice-Hall, Upper Saddle River NJ 16. Filipski J, Thiery JP, Bernardi G (1973) J Mol Biol 80:177 17. Mazrimas JA, Hatch FT (1972) Nature New Biol 240:102 18. Corneo G, Ginelli E, Soave C, Bernardi G (1968) Biochemistry 7:4373 19. Schmid CW, Hearst JE (1969) J Mol Biol 44:143 20. Schmid CW, Hearst JE (1972) Biopolymers 11:1913 21. Macaya G, Thiery JP, Bernardi G (1976) J Mol Biol 108:237

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22. Hudson AP, Cuny G, Cortadas J, Haschemeyer AE, Bernardi G (1980) Eur J Biochem 112:203 23. Cuny G, Soriano P, Macaya G, Bernardi G (1981) Eur J Biochem 115:227 24. Pavlíˇcek A, Paˇces J, Clay O, Bernardi G (2002) FEBS Lett 511:165 25. Beran J (1994) Statistics for long-memory processes. Chapman & Hall/CRC, Boca Raton etc. 26. Zivot E, Wang J (2003) Modeling financial time series with S-PLUS. Springer-Verlag, New York NY 27. Clay O, Carels N, Douady C, Macaya G, Bernardi G (2001) Gene 276:15 28. Clay O (2001) Gene 276:33 29. Li W, Holste D (2005) Phys Rev E 71:041910 30. Bucciarelli G, Bernardi G, Bernardi G (2002) Gene 295:153 31. Bernardi G, Bernardi G (1986) J Mol Evol 24:1

32. Jabbari K, Bernardi G (2004) Gene 333:183 33. Kulathinal RJ, Bettencourt BR, Hartl DL (2004) Science 306:1553 34. Douady C, Carels N, Clay O, Catzeflis F, Bernardi G (2000) Mol Phylogenet Evol 17:219 35. Schuck P (2004) Biophys Chem 108:187 36. Lander ES, Linton LM, Birren B et al. (2001) Nature 409:860 37. Bernardi G (2001) Gene 276:3 38. Bernaola-Galván P, Oliver JL, Carpena P, Clay O, Bernardi G (2004) Gene 333:121 39. Yeandle S (1959) Proc Natl Acad Sci USA 45:184 40. Schmid CW, Hearst JE (1971) Biopolymers 10:1901 41. Vinograd J, Hearst JE (1962) Fortschritte der Chemie organischer Naturstoffe 20:372 42. Szybalski W (1968) Methods Enzymol 12:330

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43. Kirk JT (1967) J Mol Biol 28:171 44. Wells RD, Blair JE (1967) J Mol Biol 27:273 45. Hearst JE, Vinograd J (1961) J Phys Chem 65:1069 46. Hearst JE, Schmid CW (1973) Methods Enzymol 27:111 47. Dishon M, Weiss GH, Yphantis DA (1971) Biopolymers 10:2095 48. Fujita H (1956) J Am Chem Soc 78:3598 49. Hearst JE (1965) Biopolymers 3:1 50. Schmid CW, Hearst JE (1972) Biopolymers 11:1765 51. Minton AP (1992) Biophys Chem 42:13 52. Sueoka N (1959) Proc Natl Acad Sci USA 45:1480 53. Sober HA (ed) (1968) Handbook of biochemistry: Selected data for molecular biology. CRC, Cleveland Ohio 54. Roe BP (1992) Probability and statistics in experimental physics. Springer-Verlag, New York etc.

Progr Colloid Polym Sci (2006) 131: 108–115 DOI 10.1007/2882_010 © Springer-Verlag Berlin Heidelberg 2006 Published online: 15 February 2006

Amanda L. Stouffer William F. DeGrado James D. Lear

Amanda L. Stouffer · William F. DeGrado · James D. Lear (u) Department of Biochemistry & Biophysics, School of Medicine, University of Pennsylvania, Philadelphia, PA 19104-6059, USA e-mail: [email protected] Amanda L. Stouffer · William F. DeGrado Department of Chemistry, University of Pennsylvania, Philadelphia, PA 19103, USA

BIOLOGICAL SYSTEMS

Analytical Ultracentrifugation Studies of the Influenza M2 Homotetramerization Equilibrium in Detergent Solutions

Abstract Though much progress has been made in the field of membrane protein folding, there is still much to learn about the association of transmembrane helices. Equilibrium analytical ultracentrifugation (EAUC) has been an important method of determining free energies of association for these types of systems. The M2 protein from the Influenza A virus, its transmembrane region and variants of that region represent over half of the thermodynamic data currently available for membrane proteins. Here, we consider the technical details of the EAUC methods used to study the stability of M2 in detergent solutions. Density-matching of detergent-buffer solutions yields precise values of the monomer-tetramer dissociation constant, and expressing these values in

Introduction Equilibrium analytical ultracentrifugation sedimentation experiments have for some time now been used to determine molecular weights of membrane proteins solubilized in detergent micelles (e.g. [1, 2]). To account for the contribution of the detergent to the buoyant molecular weight, it is usually assumed that the partial specific volume of the protein-detergent complex (ν c ) can be approximated as the weighted average of partial specific volumes of the detergent (ν d ) with weight fraction δd and the protein (ν p ):
 ν c = ν p + δd ν d (1 + δd )

(1)

terms of peptide/detergent mole fraction units gives constant values over a modest range of experimentally relevant detergent concentrations. Furthermore, we have extended the EAUC method to calculate the number of detergent molecules bound to both monomer and tetramer species by employing a range of 2 H2 O buffer compositions. Determination of peptide-bound detergent not only reinforces the importance of density matching, but it opens the door to future analytical ultracentrifugation experiments involving membrane proteins in alternative detergents and lipid bicelles. Keywords Analytical ultracentrifugation · Micelle number · Membrane protein · Proton channel · Influenza

With this assumption, the equation relating the protein molecular weight to the observed buoyant molecular weight (Mb) is:   (2) Mw = Mb 1 − ν p ρ + δd (1 − ν d ρ) By matching the solvent density to that of the detergent, the contribution from the detergent vanishes (see Fig. 1) and the molecular weight of the protein can be determined by fitting of equilibrium concentration gradients to the usual equation:  2    ω

2 2 Abs = A exp 1 − ν p ρ Mw r − r o + E (3) 2RT

AUC Studies of M2

Fig. 1 2 H2 O is used to adjust the density of a 50 mM Tris-HCl pH 7, 0.1 M NaCl buffer to equal that of a DPC (here 15 mM) detergent solution. Interference scans show that density matched solutions are sensitive to slight changes in 2 H2 O composition. A positive interference slope (a) indicates too little 2 H2 O, and the micelles sediment (bottom left). When micelles are evenly distributed throughout the centrifuge compartment (bottom center), the interference slope is equal to zero (b) and the detergent buffer solution is density matched. A negative interference slope (c) indicates too much 2 H2 O, and the micelles float (bottom right). We have determined that, for 15 mM DPC, the density match is accurate to ±0.00025 gms/cc, introducing a negligible level of uncertainty in determining molecular weights

in which Abs is the absorbance at radius position r, A
is the absorbance at ro , ω = angular velocity (radians/sec), R is the gas constant (8.3144 × 107 dyne-cm K−1 ), T the temperature (K) and E is the baseline absorbance corresponding to zero concentration. In a natural extension of this idea [3], it has been demonstrated that EAUC data in detergents can also be fit to a monomer-nmer equilibrium. Under density-matched conditions, as with the single species case, one need not take explicit account of the number of detergent molecules bound to the monomer and nmer species. The contributions of the monomer and oligomer are related in an equation similar to Eq. 3 using the equilibrium dissociation constants appropriate to each monomer-nmer equilibrium considered:  2   ω 2 2 Mw r − r o Abs = εcol exp 2RT   

cno ω2 2 2 n Mw r − r o + E εn l exp + (4) K dn 2RT n where co is the concentration of monomer at ro , ε is the extinction coefficient, l is the optical path length, and K dn is the monomer-nmer dissociation constant. The choice of units used to define the extinction coefficient and equilibrium constant is important [4, 5]. The centrifugation equilibrium state is established by a balance of gravitational

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and chemical potentials throughout the centrifuge cell. So, the units chosen to represent the concentration gradient of the sedimenting species must also reflect the chemical potential gradient. For peptides wholly confined to the aqueous phase, aqueous volume concentration units are most appropriate because, particularly at low concentrations, the chemical potential is dominated by entropy and varies linearly with log c. For peptides wholly confined to the micellar phase the chemical potential can similarly be expected to vary with log X where X is the mole ratio of peptide in the micellar (constant chemical potential) detergent phase. These are two extremes of more general analyses [5–7] which introduce non-linear variation of protein-detergent complex chemical potentials with log detergent concentration to account for data over wider ranges of detergent concentrations. Because the M2 protein, a proton selective channel, is the target of the anti-Influenza A drug amantadine [8], its structure (e.g. [9, 10]), dynamics (e.g. [11, 12]), and function (e.g. [13, 14]) have been extensively researched. M2 is a 97 amino acid protein with a single transmembrane (TM) helix that homo-tetramerizes. Both the full-length and the TM helix alone have been shown to exist in a reversible and cooperative monomer-tetramer equilibrium in dodecyl phosphocholine (DPC) detergent micelles, providing a useful model system for thermodynamic studies of membrane protein folding. In this paper, we focus on our measurements of monomer-tetramer equilibrium constants for the Udorn strain of Influenza A, M2TM peptides in DPC micelles [15–18]. In particular, we consider the precision of the measurements of the wild-type peptide, the appropriateness of peptide/detergent mole ratio concentration units, and explore the use of non-density matched detergent conditions to infer differences in the number of detergent molecules bound to the monomer and the tetramer. Multiple measurements in carefully densitymatched, 15 mM DPC using a peptide partial specific volume (0.789) calculated from the weight averaged amino acid residue values gave a pK d4 in mole ratio units of 6.25 ± 0.2 for the M2TM peptide. Provided that a monomer-tetramer-octamer equilibrium is included in the analysis of data covering peptide/detergent ratios above 0.01, this value shows no systematic variation when the detergent nominal concentrations are varied between 5 and 30 mM. An increase of pK d4 near to the 1.5 mM DPC critical micelle concentration (CMC) can be attributed to an increasing level of higher order aggregation at higher peptide/detergent ratios. From non-density matched experiments, we find that the M2 monomer and tetramer bind different numbers of detergent molecules and do not simply dissolve in average size DPC micelles. Moreover, because the buoyant molecular weight varies with solvent density, we can infer from global curve-fitting a peptide partial specific volume about 5% less than the value previously calculated from the peptide sequence. Using this

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new value in curve-fitting resulted in an approximately 5% systematic reduction in fitted pK d4 ’s.

Methods Peptide Synthesis and Sample Preparation M2TM (residues 22-46,C-terminally amidated) was synthesized on an Applied Biosystems 433A peptide synthesizer (Perkin-Elmer). Solid phase peptide synthesis was carried out using Fmoc chemistry, a 25% DMSO/75% N-methyl pyrolidinone solvent, and free amino acids (activated in situ by HBTU/HOBt/DIEA) on a PAL-PEG resin with a substitution level of 0.65 m mol e/g. Cleavage from resin and deprotection of amino acid protecting groups were carried out in a mixture of Trifluoroacetic acid/Triisopropyl silane/H2 O (90 : 5 : 5 v/v/v) at room temperature under nitrogen for 4 h. The resin was filtered off and the peptide was precipitated with cold hexane/ether (1 : 1 v/v). Peptides were purified by RP-HPLC on a semi-preparative C4 column (Vydac) using a linear gradient of buffer B (6 : 3 : 1 2propanol/acetonitrile/H2O) and buffer A (0.1% aqueous TFA). Elution occurred at ∼ 83% buffer B. Identity of peptides was confirmed by MALDI-TOF (Perspective Biosystems) mass spectrometry. Purity was then assessed by analytical HPLC using a linear A/B gradient. Analytical Ultracentrifugation Sedimentation equilibrium experiments were performed using a Beckman XL-I analytical ultracentrifuge at 25 ◦ C on M2TM wild type peptide solubilized in dodecylphosphocholine (DPC, Avanti Polar Lipids) or n-octyl glucoside (n-OG, Sigma), micelles at concentrations above their respective CMC of 1.5 mM or 22 mM. Density-matched experiments were carried out at a buffer density adjusted to equal that of the DPC as determined by interference at equilibrium for solutions containing various percentages of deuterium oxide (2 H2 O). For all samples, buffer densities were measured in a Paar densitometer. Samples were prepared by dissolving the desired amount of peptide in a 1 : 1 isopropanol/H2O (ε@280 nm = 5853 M−1 cm−1 ) solution and removing the organic solvent by high vacuum centrifugation. The resulting peptide films were kept under high vacuum overnight. A solution of detergent and 2 H2 O in a 50 mM Tris-HCl (pH = 7.5) and 0.1 M NaCl buffer was added to each vial and samples were vortexed until they became clear. A typical data set for density matched and non-density matched buffers (constant detergent concentration) include the peptide/detergent ratios 1 : 150 and 1 : 250. For density matched samples of various detergent concentrations peptide/DPC ratios ranged from 1 : 50 to 1 : 300. This range was required to maintain a peptide absorbance (at 280 nm) within the measurable limits of the centrifuge.

A.L. Stouffer et al.

The extinction coefficient of M2TM in the experimental buffer (6122 M−1 cm−1 ) was determined by absorption measurements of samples prepared to known concentration. The peptide monomer molecular mass and partial specific volume were calculated using the program SEDENTRP [19] modified to incorporate the most recently published values of amino acid residue specific volumes [20]. Computed values were then corrected for partial deuteration of the amide nitrogens, amines, alcohols, and guanidinium groups for the buffer 2 H2 O composition. Data was collected for each sample being brought to equilibrium at three different rotor speeds (40 000, 45 000, and 48 000 rpm). Data Analysis For the density-matched experiments, data were globally fit to a monomer-tetramer (-octamer) equilibrium model (Eq. 4) encoded using Igor Pro (Wavemetrics). Baselines were constrained to be equal for individual samples spun at different speeds. For samples spun using different density solvents, the equilibrium model was modified to allow for differing buoyant molecular weights of monomer and tetramer with the difference being related to the difference in number of bound detergent molecules. Because the detergent chemical potential is independent of its concentration above the CMC1 , this difference (as well as the individual binding stochiometries) was assumed to be independent of radial distance in the cell. The specific equation employed was a simple variant of Eq. 5:  2   ω 2 2 Mb1 r − ro Abs =εco l exp 2RT    ω2 cno 2 2 Mb2 r − ro + E l exp + εn (5) K dn 2RT where the buoyant molecular weights of the monomer (Mb1 ) and nmer (Mb2 ) are related to the component quantities by: (6) Mb1 = m 1 Mwd + Mw p − (m 1 υd Mwd + υ p Mw p )ρ Mb2 = m 2 Mwd + n Mw p − (m 2 υd Mwd + nυ p Mw p )ρ (7) with m 1 and m 2 equal to the number of detergents bound to the monomer and tetramer, the p and d subscripts referring to the peptide and detergent molecular weights (Mw) and partial specific volumes (υ). n is the peptide aggregation number (4 in our case) and ρ is the solvent density corresponding to the 2 H2 O concentration. Because the solvent density variation can be used to define the peptide υ p [21, 22], this variable was allowed to vary globally in fitting along with the monomer-tetramer pK d4 , and the number of molecules bound to each species. The detergent 1

This has been experimentally verified for sodium decanoate (E. Vikingstad (1979) Journal of Colloid and Interface Science 72:68–74).

AUC Studies of M2

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υ was held constant at the reciprocal of the experimentally determined matching solvent density. To simplify the curve-fitting computations, peptide Mw and υ values were held at the values corrected to the density-match 2 H2 O concentration because parameters from fits using values calculated for 20% and 80% 2 H2 O differed insignificantly. Since M2TM is insoluble in the aqueous buffer, the peptide is assumed to be partitioned into the detergent phase. Therefore, and as discussed in more detail below, all tetramerization constants, except where otherwise indicated, were expressed in terms of peptide to detergent mole ratio units.

Results Density-matched Experiments; Measurement Precision Our recently published work [18] involved determining the effect of amino acid sequence changes on the free energy of association of the A/Udorn M2TM peptide in dodecyl phosphocholine micelles. Somewhat surprisingly, we found such small effects that the precision of our methods became of the utmost concern. Consequently, we first compiled all of the data we had available on the A/Udorn transmembrane sequence residues (22–46). The first published value [16] was determined in 15 mM DPC at pH 7.5 in 50 mM Tris-HCl, 0.1 M NaCl buffer density matched with 2 H2 O and reported in units of peptide solution molarity which we have converted to mole ratio units (and corrected to a slightly larger, more accurate extinction coefficient). In later work [17], it was determined that the tetramerization equilibrium was sensitive to pH only be-

low ∼ 7 so all subsequent experiments were also done at pH 7.5. Table 1 summarizes values of pK d4 determined in separate experiments, but analyzed with the same fixed Mw and υ p . The average of all these values is 6.25 ± 0.12, well within the typical fit-estimated uncertainty and supports the calculation of free energy differences accurate to ±0.3 to 0.4 kcal/mole, about 1/4 to 1/5 times the range of ∆∆G’s reported in our work. Density Variation Experiments; Determination of Bound Detergent A simplified view of the solubilization of membrane proteins is that the protein dissolves in a detergent micelle. Applying this view to analytical ultracentrifugation analysis, Tanford and Reynolds [1] suggested that there must be a sufficient number of micelles present to avoid proteinprotein interactions induced by “artificial crowding”. In fact, there is no reason to think that the protein binds the same number of detergent molecules that are in a detergent micelle. Binding of the protein can be expected to either add or displace detergents from the micelle [5]. With this more rigorous view, one becomes curious as to the difference in the number of detergents bound to the monomer and tetramer in the M2 equilibrium. Consequently, following earlier work with Apo-A1, a single species protein [2], we undertook a series of experiments in nondensity matched buffers to explore this issue. Our original

Table 1 Comparison of pK d4 (negative log of the dissociation constant for a monomer/tetramer equilibrium) values determined in our laboratory since 1999 for M2TM using different preparations of peptide and detergent and fresh density matched buffers. Error as reported from the global fitting program is ±0.2 to 0.3 MF3 Date of Experiment

pK d4

Aug 99 a Aug 02 Mar 03 b Sep 03 Jul 04 Aug 04 Apr 05 May 05

6.45 6.17 6.19 6.22 6.15 6.22 6.42 6.15

a

The previously reported value [16] was corrected for an extinction coefficient of 6121.8 and put in MF3 units. Inclusion of a monomeroctamer equilibrium in the original report had no significant effect on the value. b Reported in [18]

Fig. 2 Radial absorbance profiles (under equilibrium conditions at 40 K, 45 K, and 48 K rpm) were collected for samples with M2TM to DPC ratios of 1/150 and 1/250 containing 20, 30, 50, 70, and 80% 2 H2 O buffers with 15 mM DPC. Data (points) were globally fit (lines) to a monomer-tetramer-equilibrium: A while fixing the peptide partial specific volume to 0.789 and the pK d4 to the average of that measured in density-match experiments and B while floating both pK d4 and the peptide partial specific volume. C Same as B except baselines were unconstrained. In all cases, the number of detergent molecules bound to the monomer and tetramer were allowed to vary globally

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Table 2 Summary of parameters used in curve-fits of Fig. 2 Parameter

Figure 2A fit

Figure 2B fit

Figure 2C fit

Peptide ν m1 m2 pK d4 = − log(K d4 (MF3 ))

0.7899 (fixed) 2±4 98 ± 7 6.2 (fixed)

0.752 ± 0.007 40 ± 10 99 ± 12 6.7 ± .2

0.748 ± .007 53 ± 9 113 ± 13 6.7 ± .2

intent was to hold the peptide parameters, including the pK d4 , constant in order to better define the detergent binding parameters (m 1 and m 2 in Eq. 5). Figure 2A shows the results covering densities ranging well above (1.0884) and below (1.0249) the matching value (1.0580). As is evident from the highly periodic residuals autocorrelation function, the fit was too poor to draw any firm conclusions (although for completeness, we report the parameter values in Table 2). However, by allowing the peptide partial specific volume as well as the pK d4 to vary globally in fitting, a significantly improved, though still problematic, fit (Fig. 2B) resulted. By relaxing the common baseline constraint, we obtained a better fit to the data with slightly different parameters (Fig. 2C). The parameter values for all fits are shown in Table 2. Density-matched Experiments in Different Detergent Concentrations All of the above-described experiments were done in 15 mM DPC and the K d4 results reported in peptide/deter-

Fig. 4 Radial absorbance profiles and curve fits for M2TM in 25 mM n-octyl glucoside (n-OG). A M2TM to n-OG ration of 1/250 was used in buffers containing 20, 30, 40, 60, 70, and 80% 2 H O. In fitting, the peptide partial specific volume was held fixed 2 at the value determined in DPC, but that for n-OG was allowed to vary. The fitted value of 0.871 ± 0.006 is within experimental error of the published value [29] of 0.867

gent mole ratio units. This choice was based on a simple model, which assumes the peptide chemical potential, at DPC concentrations well above the CMC, varies linearly with the peptide/detergent mole ratio. To determine if these units are appropriate to this system, measureTable 3 Comparison of pK d4 and pK d8 values calculated from curve fitting using the noted peptide partial specific volumes and concentration units. pK d8 values for DPC concentrations higher than 10 mM are not reported because they were indeterminant in curve fitting DPC concentration (mM)

Fig. 3 Data and fits of M2 tetramerization equilibrium in 5 mM (A) and 25 mM (B) DPC density-matched buffers. C pK d4 values computed with peptide partial specific volume of 0.748 and with units of peptide/detergent mole ratio (open circles), and peptide molar concentration units (solid circles). Points are plotted against the logarithm of the micellar detergent concentration (detergent concentration – CMC in mM units). Error bars are from centrifuge data curve fits. Lines are linear fits with slopes of −2.77 ± 0.16 (right axis) and −0.21 ± 0.19 (left axis)

5 10 15 20 25 30

pK d4 from pep/det mole ratio for 0.7899

pK d4 from pep/det mole ratio for 0.748

pK d4 from pep moles/liter for 0.748

5.9 ± .7/14 ± 1 (pK d8 = 14.0 ± 1) 6.6 ± 0.3 (pK d8 = 14.8 ± .5) 6.4 ± 0.3 6.2 ± 0.2 6.6 ± 0.2 6.4 ± 3

6.1 ± 0.4 (pK d8 = 13.6 ± 0.4) 6.0 ± 0.1 (pK d8 = 13.0 ± .2) 5.7 ± 0.1 5.8 ± 0.1 5.9 ± 0.2 6.05 ± 0.2

13.45 12.04 11.14 10.86 10.85 10.62

AUC Studies of M2

ments were made at 5, 10, 15, 20, 25, and 30 mM DPC in order to compare the consistency of K d4 expressed in either aqueous peptide concentration or mole ratio units. Figures 4A and 4B show data and fits at 5 and 25 mM DPC concentrations and Fig. 3C shows the dependence of the differently defined pK d4 s on the micellar detergent concentration. Table 3 summarizes the results of all experiments. Data at 5 and 10 mM DPC were fit to a monomertetramer-octamer equilibrium model for reasons discussed below. Non-density Matched Experiment in n-Octyl Glucoside Measurement of M2 tetramerization in n-OG micelles was undertaken to investigate an unpublished report [23] based on thiol-disulphide exchange measurements [24] that this particular detergent promoted much stronger tetramerization than DPC. Accordingly, we employed the same series of different 2 H2 O composition buffers in an experiment where n-OG (25 mM) was substituted for DPC. Results are shown in Fig. 4. Confirming the earlier-mentioned report, stronger tetramerization (pK d4 = 10.4 ± 0.7) was observed and the bound detergent numbers (m 1 = 46 ± 30, m 2 = 75 ± 8) were reasonable.

Discussion Measurements of a monomer-nmer equilibrium for small, hydrophobic peptides in detergent solutions present a challenge to equilibrium analytical ultracentrifugation experiments. Our results show that by using highly purified peptides with careful density matching of a chemically homogenous detergent, reasonably reproducible equilibrium constants can be obtained. Using global data fitting protocols for radial absorbances collected for two peptide concentrations at multiple speeds while constraining baselines to be identical for individual samples, we find a precision adequate to discriminate free energy changes of ±0.4 kcal/mole. Thus our previously published data [18, 25], exhibiting small (0 to 3.1 kcal/mol) changes in stability of single site mutants of M2TM are justified. The accuracy of the pK d4 values, however, may be less certain because of the difference in the peptide partial specific volume found here from the calculated value. Our experiments in non-density-matched solutions proved refractory to fitting using the value of the peptide partial specific volume (0.7899) calculated from the Kharazov values [20] of amino acid residue partial specific volumes. Using amino acid residue specific volumes from either the Cohn and Edsall [26] values included with Sedentrp or the consensus values published by Perkins [27] gave calculated (2 H2 O-corrected) υ p ’s of 0.786 and 0.779 respectively, still too large to significantly improve data fitting. Our measured density values were

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eliminated as a possible source of error because they were in nearly perfect agreement with those calculated for our buffers. However, by allowing the peptide υ p to vary freely in fitting we obtained a significantly improved fit with a peptide υ p of 0.748 ± 0.007. The ∼ −5% difference from the sequence-calculated values is beyond the expected accuracy of the calculation [20] and remains unexplained, a source of uncertainty in the absolute pK d4 values. The fitted values for the number of detergent molecules bound to the monomer and tetramer indicate that the monomer displaces about 10 molecules of detergent from the typical DPC micelle number of 65 (experimentally determined in our system [17]) and that coalescence of four monomeric M2-detergent complexes into a tetramer releases about 100 detergent molecules (25 per monomer). Following the analysis of Josse et al. [7], the dependence of the molar concentration unit equilibrium dissociation constant K on detergent micelle concentration c is given by Eq. 8: ∆N ∂ ln K = ∂ ln c n

(8)

where ∆N is the difference in the number of monomeric detergent molecules bound to the two species involved in the reaction and n is the micelle number. This equation was derived on the assumption that the chemical potential of the monomeric detergent is related to the chemical potential of the detergent micelle via the micellization equilibrium and that this in turn varies linearly with the bulk concentration of detergent micelles. For a dimerization, if the number of detergents solvating a dimer is exactly half the number solvating two monomers, ∆N/n would be −1 and the equilibrium constant would be equivalent to one occurring in a phase of constant chemical potential. That is, it would be constant if expressed in peptide/micelle activity (∼ concentration) ratio units. This is, in fact, what is reported for dimerizations of micelle-solubilized human serum paraoxonase (−1.1) [7] and glycophorin A (−1.0) [28]. Extending this analysis to tetramerization, it is evident that ∆N/n should be −3 for the same ideal case. Our observed variation of molar concentration unit pK d4 ’s with the log of the micelle concentration (Fig. 3) gave ∆N/n = −2.8 ± 0.2. This is significantly higher than the range of values (−0.8 to −2.3) the equation of Josse et al. [7] would predict from our experimentally estimated solvation numbers, but is consistent with the assumption of constant detergent monomer chemical potential used in the analysis of the centrifugation data. Put another way, at detergent concentrations well above the CMC, release of more or less detergent upon M2TM monomer association has little or no influence on the tetramerization equilibrium. Why this should be is beyond the scope of this paper, but it is certainly an interesting subject to pursue in the future.

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Because the peptide-bound micelle numbers are qualitatively as large as those of the pure detergent, and because the peptide Mw is small, the detergent will contribute significantly to the peptide’s molecular weight under nondensity-matched conditions. This lends support to the importance of careful density matching to obtain reproducible pK d4 ’s in these systems. With the new υ p for M2TM, fitting of density-matched data obtained in experiments at different DPC concentrations indicated a pK d4 for 15 mM DPC of 5.9 ± 0.2, slightly but significantly lower than our previously reported values. We also noted that the mole ratio unit pK d4 returned by monomer-tetramer fits of data at 5 mM DPC concentration (not shown) was much higher (7.2) than that expected from the average of values obtained from similar fits at higher DPC concentrations. We attributed this to the high peptide to detergent ratios that must be used to detect radial absorption at such low detergent concentrations (1 : 50 and 1 : 100 for 5 mM DPC). At peptide to detergent ratios higher than 1/150, it was previously observed [17, 18] that M2TM begins to form higher order aggregates. Our measured micelle numbers in fact predict that at peptide/detergent ratios above .01, there would be insufficient detergent to maintain a constant degree of tetramer solvation. One possible consequence of this would be higher order peptide aggregation. We therefore re-fit the 5 and 10 mM data to a monomer-tetrameroctamer equilibrium (as the simplest higher order aggregation model). Good fits (e.g., Fig. 3A) were obtained with monomer-octamer pK d8 ’s (MF7 units) of 13.3 ± 0.3, corresponding to the presence of about 10% octamer at the highest peptide/detergent mole ratio (0.0018) achieved in those two experiments. Since lower mole ratio levels were attained in the higher DPC concentration experiments, inclusion of the octamer had no effect on the fitting results so no pK d8 ’s are reported for those experiments. The relative constancy of mole ratio concentration unit pK d4 ’s with DPC concentration indicates that these units are appropriate to this system. For our purpose of comparing free energies of dissociation among peptide variants, this work shows that measurements over a modest range of detergent concentrations well above the CMC can safely be compared using mole ratio units. However, a much wider range would be needed for the type of analysis necessary to explore the detailed thermodynamics of peptide-detergent interactions [6, 28]. The detergent n-OG has a reported partial specific volume of 0.867 [29] which is much lower than the detergents used in our previous centrifugation experiments. The

high density of the glucoside head group prevents density matching of glucoside detergents with 2 H2 O. Although density-matching can be achieved using dense solvent additives other than 2 H2 O [30], such additives can significantly affect any equilibria being studied [31]. Therefore, we chose to use the non-density matched experiment [2] to measure the pK d4 of M2TM in n-OG. For this detergent, the monomer and tetramer bound micelle numbers are both less than the reported n-OG micelle number of ∼ 90. Although the monomer micelle number is highly uncertain, the tetramer micelle number is still less than 4× that of the monomer, again indicating that tetramerization is accompanied by the release of some detergent. Membrane proteins make up about 30% of all genome reading frames and are very frequently the targets for drugs and toxins. Therefore, it is of great importance that we continue to develop new methods to study their functions and structures. Due to the added complication of solubilizing hydrophobic proteins along with the difficulty of measuring their biophysical properties in their natural environment, this is not a simple task. Here, we have shown that EAUC is a reliable and reproducible method for determining the free energy of formation of oligomeric transmembrane helices in detergent micelles at densitymatched conditions. Additionally, a series of non-density matched buffers was used to estimate the number of detergent molecules bound to each species of protein in a multi state equilibrium. The M2 monomer-tetramer equilibrium appears to be dominated by transmembrane interactions because dissociation constants expressed in mole ratio units are reasonably constant over about a 20-fold range of detergent concentrations. However, at peptide concentrations exceeding the detergent’s capacity to fully solvate all of the species, higher order aggregation can be expected to complicate the interpretation of experiments so such conditions should be avoided. Though this has not been an aim of our work, it is becoming increasingly evident (e.g., the n-OG result shown here and the work of Fisher et al. [6]) that the detergent plays an important role in modulation of the peptidepeptide interactions. The methods we have discussed here appear to be adaptable to experiments in many other detergent and bilayer environments. We look forward to extending our studies of M2 into this important area. Acknowledgement ALS thanks the National Organizing Committee and the International Scientific Committee of the 14th International Symposium on Analytical Ultracentrifugation for travel funding. This work was supported by NIH Grant GM-54623.

References 1. Tanford C, Reynolds JA (1976) Biochim Biophys Acta 457:133–170 2. Noy D, Calhoun JR, Lear JD (2003) Anal Biochem 320:185–92

3. Fleming KG (2000) Methods Enzymol 323:63–77 4. Lear JD, Gratkowski H, Degrado WF (2001) Biochem Soc Trans 29:559–64

5. Fleming KG (2002) J Mol Biol 323:563–571 6. Fisher LE, Engelman DM, Sturgis JN (2003) Biophys J 85:3097–31105

AUC Studies of M2

7. Josse D, Ebel C, Stroebel D, Fontaine A, Borges F, Echalier A, Baud D, Renault F, Le Maire M, Chabrieres E, Masson P (2002) J Biol Chem 277:33386–33397 8. Hay AJ, Wolstenholme AJ, Skehel JJ, Smith MH (1985) Embo J 4:3021–3024 9. Pinto LH, Dieckmann GR, Gandhi CS, Papworth CG, Braman J, Shaughnessy MA, Lear JD, Lamb RA, DeGrado WF (1997) Proc Natl Acad Sci USA 94:11301–11336 10. Nishimura K, Kim S, Zhang L, Cross TA (2002) Biochemistry 41:13170–13177 11. Smondyrev AM, Voth GA (2002) Biophys J 83:1987–1996 12. Zhong Q, Newns DM, Pattnaik P, Lear JD, Klein ML (2000) FEBS Lett 473:195–198 13. Mould JA, Li HC, Dudlak CS, Lear JD, Pekosz A, Lamb RA, Pinto LH (2000) J Biol Chem 275:8592–8599

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14. Lear JD (2003) FEBS Lett 552:17–22 15. DeGrado WF, Gratkowski H, Lear JD (2003) Protein Sci 12:647–665 16. Kochendoerfer GG, Salom D, Lear JD, Wilk-Orescan R, Kent SB, DeGrado WF (1999) Biochemistry 38:11905–11913 17. Salom D, Hill BR, Lear JD, DeGrado WF (2000) Biochemistry 39:14160–14170 18. Stouffer AL, Nanda V, Lear JD, DeGrado WF (2005) J Mol Biol 347:169–179 19. Laue TM, Stafford WF 3rd (1999) Annu Rev Biophys Biomol Struct 28:75–100 20. Kharakoz DP (1997) Biochemistry 36:10276–10285 21. Gohon Y, Pavlov G, Timmins P, Tribet C, Popot JL, Ebel C (2004) Anal Biochem 334:318–334 22. Edelstein SJ, Schachman HK (1967) J Biol Chem 242:306–311

23. Cristian L (2005) (personal communication) 24. Cristian L, Lear JD, DeGrado WF (2003) Protein Sci 12:1732–1740 25. Howard KP, Lear JD, DeGrado WF (2002) Proc Natl Acad Sci USA 99:8568–8572 26. Cohn E, Edsall J (1943) pp 155–176, 370–381. Reinhold, New York 27. Perkins SJ (1986) Eur J Biochem 157:169–180 28. Fleming KG, Ren CC, Doura AK, Eisley ME, Kobus FJ, Stanley AM (2004) Biophys Chem 108:43–49 29. Shire SJ (1994) In: Schuster TM, Laue TM (eds) Modern Analytical Ultracentrifugation. Birkhauser, Boston, MA, p 261–297 30. Lustig A, Engel A, Tsiotis G, Landau EM, Baschong W (2000) Biochim Biophys Acta 1464:199–206 31. Eisenberg H (2000) Biophysical Chemistry 88:1–9

Progr Colloid Polym Sci (2006) 131: 116–120 DOI 10.1007/2882_021 © Springer-Verlag Berlin Heidelberg 2006 Published online: 10 February 2006

Anuja Khan Richard K. Hughes Eric J. Belfield Rod Casey Arthur J. Rowe Stephen E. Harding

Anuja Khan (u) · Arthur J. Rowe · Stephen E. Harding National Centre for Macromolecular Hydrodynamics, University of Nottingham, School of Biosciences, Sutton Bonington LE12 5RD, UK e-mail: [email protected] Richard K. Hughes · Eric J. Belfield · Rod Casey John Innes Centre, Norwich Research Park, Norwich NR4 7UH, UK

BIOLOGICAL SYSTEMS

Oligomerization of Hydroperoxide Lyase, a Novel P450 Enzyme in Plants

Abstract The oligomeric state of fatty acid hydroperoxide lyase (HPL), of molar mass ∼ 55 kDa is uncertain and it has been reported as a trimer or tetramer in vivo. The enzyme has been found to be bi-functional and is active even in the absence of detergent. The association with detergent is known to stabilise the binding of the enzyme to its substrate and the enzyme is more active. No high resolution structure of any plant P450 is available so far because of difficulty in crystallising the protein. We employ analytical ultracentrifugation to characterise the oligomeric state of an E. coli-expressed recombinant HPL from Medicago truncatula (HPL-F) under different solution conditions. Sedimentation velocity analyses show that HPL-F (under detergent-free conditions) is largely a monomer with a sedimentation coefficient s20,w of ∼ 4.1 S (a value expected from the molar mass of the

Introduction Fatty acid hydroperoxide lyase (HPL) is a membrane associated cytochrome P450 enzyme found in plants. It is a member of the P450 subfamily CYP74B [1]. HPL is associated with the generation of an array of oxylipins and aldehydes that play an important role in oxylipin metabolism, plant defence and is associated with developmental pathways in plants [2–4]. Some of the by-products of this pathway act as antimicrobial toxins and defend the plant against pathogen attack [5]. The C12 oxo-acid

monomer). The effects of protein concentration, and detergent micelles on the oligomeric state of detergentfree HPL-F are reported for the first time. With increase in protein concentration only traces of dimers can be detected. However, HPLF in association with detergent is a mixture of oligomers, which are not in reversible equilibrium with each other. These studies have important implications as they show that the oligomeric state of HPL-F changes with micellar association, both of which are related to the activity of the protein. They also show the virtue of combining sedimentation velocity with sedimentation equilibrium in the ultracentrifuge for the study of enzyme-detergent systems. Keywords Analytical ultracentrifugation · Oligomerization · Oxylipin metabolism

product of HPL is the precursor of the wound signal “traumatin” associated with wound healing in plants. The short chain aldehydes and their reduced derivative alcohols (byproducts of the oxylipin pathway) are important volatile constituents responsible for the characteristic odour of fruits, vegetables, green leaves and are of immense biotechnological importance [6–8]. The enzyme cleaves the C − C bond in the hydroperoxides (HPOs), which are generated by the oxygenation of polyunsaturated fatty acids. The final by-product of the oxylipin pathway depends on where the C − C bond is cleaved in the HPO.

Oligomerization of Hydroperoxide Lyase, a Novel P450 Enzyme in Plants

The study of membrane proteins is often very difficult as the protein fails to retain its “native structure” when isolated from associated membrane and this results in loss of biological activity. However the presence of detergent in solution is known to stabilise the protein structure and hence membrane proteins are usually studied in detergent which mimics the membrane environment in vivo. Because of the problems associated with isolation of such proteins in suitable form for crystallisation studies, high resolution structures are not available. The oligomeric state of HPL is uncertain and it has been reported as a trimer or tetramer in vivo in higher plants [5, 9–11]. The structure of any plant P450 cytochrome is not currently available and there is a clear need for a full biochemical and structural analysis. The HPL under study is a recombinant enzyme from Medicago truncatula (HPL-F) expressed in E. coli at the John Innes Centre, Norwich. Studies have shown that HPL-F is a bi-functional enzyme and retains its biological activity both in the presence and absence of detergent [12]. The detergent is known to activate the protein but the mechanism by which this is done is unknown. Recent biochemical studies show that under detergent-free conditions, HPL-F has very different binding kinetics to the enzyme in buffer containing detergent and salt [12]. The difference in activity arises from the difference in conformational (oligomeric) state of HPL-F in the presence and absence of detergent. Recent studies have shown that in the absence of detergent the activity is retained but is lower than when present in association with detergent. In this study we employ both sedimentation velocity and sedimentation equilibrium in the analytical ultracentrifuge to probe the state of oligomerisation in detergentfree and detergent containing solution conditions, thus facilitating an insight into the complex detergent-enzyme interactions in relation to functionality.

Materials and Methods Materials Hydroperoxide lyase (HPL-F) from Medicago truncatula was expressed in E. coli [12]. The protein was prepared in two batches. Batch one HPL-F was in 0.1 M sodium phosphate, pH 6.5, batch two was HPL-F in 50 mM potassium phosphate buffer containing 0.9 M sodium chloride, 50 mM glycine, 5% (w/v) glycerol and 0.1% (v/v) Emulphogene (polyoxyethylene 10 tridecyl ether). All chemicals for buffer preparation were supplied by Sigma Aldrich (Dorset, UK). Sedimentation Velocity Sedimentation analysis was employed to characterise the oligomeric state of HPL-F in detergent-free solution

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under the effect of increasing protein concentration, ionic strength and pH. Sedimentation velocity studies in the Beckman XL-A/I (Palo Alto, USA) were carried out on various HPL-F concentrations (0.6–15.0 mg/ml) at a rotor speed of 45–50 000 rpm, at 20 ◦ C. The study was performed in two sets, one investigating the low concentration range (0.6–2.6 mg/ml) and the other on the high concentration range (5–15 mg/ml). Scans were recorded by interference and absorbance optics every 3.5 min. The interference scans were necessary for the high protein concentration experiments, because with absorbance optics the concentration use was limited to the Lambert-Beer range. The oligomeric state of HPL-F at concentration 0.5 and 1 mg/ml in association with detergent (emulphogene 1.6 mM) was studied using a Beckman XL-A at 45 000 rpm at 20 ◦ C. Absorbance scans were recorded at 232 nm (to keep absorbance within Lambert-Beer range) every 3.5 min. The use of interference optics was avoided as the detergent present in association with the protein may contribute towards the signal and introduce uncertainty in the sedimentation analyses. In the present work, the concentration of emulphogene used (1.6 mM) was well above the critical micelle concentration of the detergent (0.125 mM) [13]. The raw sedimentation velocity data were analysed using the program SEDFIT [16]. The least squares g(s) model within SEDFIT were employed for the analysis of the sedimentation data and for the determination of the apparent (i.e. not corrected for the effects of diffusion) sedimentation coefficient. Although the sedimentation analysis of HPL-F in detergent-free solution was relatively straightforward the analyses of the detergent-protein complexes was rather difficult due to structural heterogeneity of the protein in presence of detergent. The following procedure was therefore adopted: the c(s) model was employed with the regularization [16] switched off so that the resultant “spikes” when overlayed on top of the least squares g(s) distribution of the same data, greatly facilitated the identification of the peaks in the least squares g(s) plot. The peaks in the g(s) distribution which coincided with the c(s) distribution represented the approximate sedimentation coefficient of the different detergent solubilised oligomers in solution. The proportion of each of these oligomeric forms of HPL-F was estimated by fitting multiple Gaussians via the ROBUST fitting algorithm within the software package pro-Fit (Quantum Soft, Switzerland) to the least squares g(s) profiles. Sedimentation Equilibrium The weight average molecular weight (molar mass) Mw , of HPL-F under detergent-free conditions was also investigated by sedimentation equilibrium in the analytical ultracentrifuge. HPL-F samples (0.5, 1.0 mg/ml) in detergent-free buffer were run in a Beckman Optima XLA at 20 000 rpm at 20.0 ◦ C. Scans were taken at 280 nm

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every 2 h until equilibrium was reached (after 24 h). The raw data was analysed using the MSTARA algorithm in MSTAR [14]. This procedure permits the evaluation of the apparent weight average molecular weight Mw,app over the whole distribution of macromolecular solute in an ultracentrifuge cell (from cell meniscus to cell base), using the M ∗ function of Creeth and Harding [17]. Because of the low HPL-F concentrations used, it was assumed that nonideality effects were negligible and Mw,app ∼ Mw . Other Parametric Calculations The partial specific volume, v of HPL-F (20 ◦ C) was determined from its amino acid sequence via SEDNTERP [15]. The v of HPL-F in absence of detergent in 0.1 M sodium phosphate buffer, pH 6.5 was 0.744 ml/g. The relative viscosity and density of the detergent-buffer was measured using a viscometer and an Anton-Paar density meter respectively in order to determine the detergent contribution towards the partial specific volume of the detergent-protein complex. The measured partial specific volume for the detergent was 0.945 ml/g. The association of the HPL-F with the detergent micelle in solution complicates the hydrodynamic interpretation of the sedimentation process and therefore it is crucial to understand how much detergent was bound per gram of the monomer protein. However for this, the partial specific volume of the protein-micelle complex has to be known. As no direct experiment was performed, and to avoid uncertainties associate with density matching (especially partial deuteration) using D2 O, a theoretical approach was employed. The method adopted for evaluation of the bound detergent is as follows: If “x” grams of detergent is bound per gram of monomer of protein of molecular weight Mp , then the molecular weight of the monomer-micelle complex Mc is Mp (1 + x), The partial specific volume of complex, vc , can then be expressed as

A. Khan et al.

protein monomer were entered into a spreadsheet calculation and the calculated sedimentation coefficient, sc (Eq. 3) for the complex was matched with that the experimental s-value for the monomer-micelle complex.

Results and Discussion Oligomeric State of Detergent-Free HPL-F The apparent distribution of the sedimentation coefficient (as obtained from SEDFIT analyses) for both low and high HPL-F concentration range show that there is only one major species in solution with weight average sedimentation coefficient of ∼ 3.6 S (s20,w = 4.1 S). An example is shown in Fig. 1, with of trace amounts of low molecular weight contaminant. The weight average sedimentation coefficient was seen to remain roughly constant over the entire concentration range studied. At high protein concentration (∼ 5 mg/ml) trace amounts of a larger species of s20,w ∼ 6.3 S were detected, with a weight fraction of no more than ∼ 8.5%. The value of 6.3 S is consistent with a dimeric species, assuming monomer and dimer as both globular. The proportion of this species did not go up with increase in concentration indicating that there is no associative reaction within this system. A corresponding weight average molecular weight, Mw , of (55 ± 2) kDa was determined from sedimentation equilibrium studies (MSTARA analysis) on HPL-F at 1 mg/ml (Fig. 2), in excellent agreement with the sequence molecular weight for the monomer (∼ 56.8 kDa). From these observations we could conclude that the HPL-F preparation without detergent was largely mono-

vc = {vp (1/(1 + x)) + vd (x/1 + x)} (1) (vp + xvd ) , (2) vc = 1+x where vd is the partial specific volume of detergent, 0.945 ml/g. If the sedimentation coefficient of complex is sc and sedimentation coefficient of protein is sp , then a relation can be obtained linking the respective vs and sedimentation coefficient. sc Mc (1 − vc ρ) rp = , (3) sp Mp (1 − vp ρ) rc where rp and rc are the radii of the monomer of protein and monomer-detergent complex. The s20,w for the HPL-F monomer in association with micelle was experimentally determined in sedimentation velocity studies. Appropriate values for detergent bound to

Fig. 1 Apparent distribution of the sedimentation coefficient (expressed in terms of the so-called least squares g(s) distribution of Schuck [16]) for HPL-F at a rotor speed = 40 000 rpm, protein concentration = 1 mg/ml. The plusses represent the experimental data and the dashed line a Gaussian fit to the main peak

Oligomerization of Hydroperoxide Lyase, a Novel P450 Enzyme in Plants

Fig. 2 Plot of M ∗ (r) versus ξ(r), where ξ(r) is the normalized squared radial position. ξ(r) = (r 2 − a2 )/(b2 − a2 ) where r is the radial position in an equilibrium solute distribution and a, b the corresponding radial positions at the cell meniscus and base respectively. M ∗ (ξ → 1) = Mw,app , the apparent weight average molecular weight over the whole distribution in the ultracentrifuge cell (from meniscus to cell base) [17]. Extrapolated Mw,app = (55 ± 2) kDa. Data obtained in the Beckman-Optima XL-A at 20 000 rpm, at a scanning wavelength of 280 nm, rotor temperature = 20.0 ◦ C. Loading concentration ∼ 1 mg/ml

meric. Kinetic data has further supported the view that the monomer was the active species in solution and was responsible for the low enzyme activity observed in the presence of the substrate [12]. Oligomeric State of HPL-F in Detergent-Buffer Usually the sedimentation analyses of detergent-bound protein complexes are very complex and interpretation of such data is difficult. However HPL-F in association with detergent was a heterogeneous mixture of different oligomers, and the association with the micelles allowed a very distinct separation of the different oligomeric forms. The partial specific volume of the monomer-micelle complex vc of 0.854 ml/g was determined by applying Eqs. 1 and 2. An approximate calculation of the detergent bound (in grams) per gram of protein monomer was 1.25 (∼ error ± 20%). Analyses of sedimentation velocity data by SEDFIT have shown that detergent bound oligomers sediment at ∼ 3.2, 5.3, 7.0, 9.4, 12.1, 14.8, 16.0 S (uncorrected to standard conditions) as shown in Fig. 3 for a total loading concentration of 1 mg/ml. The detergent bound to each of the oligomers will affect their buoyant density and we may have a resulting different partial specific volume for each of them: however, in our calculations we use only the average partial specific value for the monomer-micelle complex. The c(s) peak at ∼ 1 S corresponded to the sedimentation coefficient of the detergent micelle. It was

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Fig. 3 Overlay of the c(s) and g(s) distribution data for sedimentation velocity analysis of HPL-F at a loading concentration of 1 mg/ml. The spikes represent c(s) peaks used to identify the s values of each species present

found that decreasing the HPL-F concentration 1 mg/ml to 0.5 mg/ml revealed no net decrease in the proportion of the higher sedimentation coefficient material relative to the lower sedimentation coefficient material, an observation that was strongly indicative of a non-reversibly associative system. We estimated the proportion of each detergent associated-oligomer in solution by fitting multiple Gaussian to the least squares g(s) distribution via the ROBUST fitting procedure in the spreadsheet package pro-FIT (see above) and it appears that the largest proportion of detergent associated-oligomer present in solution is a trimer. However the sedimentation coefficient of the trimer and the tetramer are very close and the association with detergent complicates the identification of the exact oligomeric form in such a heterogeneous mixture. Our finding can be related to the observation that HPL-F purified from a number of higher plants [5, 9–11] has been reported to be either trimeric or tetrameric in vivo. Even though the trimers are the most abundant species in solution whether or not they can be related to the high enzyme activity of HPL-F is yet to be ascertained.

Discussion Sedimentation velocity and equilibrium analyses of the HPL-F in the presence and absence of detergent has allowed us to probe its oligomeric state in solution. Our findings indicate that under detergent-free conditions HPL-F was almost exclusively a monomer (s20,w = 4.1 S) and, by contrast, a heterogenous mixture of oligomers in association with detergent. This behaviour may mimic the membrane association of the protein in vivo. Our studies appear to have shown that the monomer of HPL-F under detergent-free conditions does not form high oligomers at high protein concentration, except for trace dimers which are not in reversible equilibrium with the

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monomer. Similarly in the presence of detergent, though a number of distinct oligomers of the protein are formed they are not in reversible equilibrium with each other. The detergent micelle encapsulation of the n-mer (oligomer) seems to separate the oligomers quite distinctly during the

sedimentation process. Our observations explain that the micelle association is favourable to the functioning of the protein as it maximises the chances of oligomer formation which can be directly related to the increased catalytic activity [12].

References 1. Grechkin AN, Mukhtarova LS, Hamberg M (2003) FEBS Lett 549:31–34 2. Staswick P (1999) Plant Physiol 121:312 3. Weiler EW, Albrecht T, Groth B, Xia ZQ, Luxem M, Liss H, Andert L, Spengler P (1993) Phytochem 32:591–600 4. McConn M, Browse J (1996) The Plant Cell 8(3):403–416 5. Matsui K, Miyahara C, Wilkinson J, Hiatt B, Knauf V, Kajiwara T (2000) Biosci Biotechnol Biochem 64:1189–1196 6. Hatanaka A (1993) Phytochemistry 34:1201–1218

7. Hatanaka T, Ono S, Hotta H, Satoh F, Gonzalez FJ, Tsutsui M (1996) Xenobiotica 26:681–694 8. Gatfield D, Walker K, Ketchum REB, Hezari M, Goleniowski M, Barthol A, Croteau (1999) R Arch Biochem Biophys 364:273–279 9. Shibata Y, Matsui K, Kajiwara T, Hatanaka A (1995) Plant Cell Physiol 36:147–156 10. Husson F, Belin JM (2002) J Agr Food Chem 50:1991–1995 11. Itoh A, Vick BA (1999) Biochim Biophys Acta 1436:531–540 12. Hughes RK, Belfield EJ, Muthusamay M, Khan A, Rowe AJ, Harding SE, Fairhurst SA,

13. 14. 15.

16. 17.

Bornemann S, Thorneley RNF, Casey R (2006) Biochem J (in press) Helenius A, Simons K (1975) Biochim Biophys Acta 415:29–79 Cölfen H, Harding SE (1997) Eur Biophys J 25:333–346 Laue TM, Shah BD, Ridgeway TM, Pelletier SL (1992) In: Harding SE, Rowe AJ, Horton JC (eds) Analytical Ultracentrifugation in Biochemisry and Polymer Science. Royal Society of Chemistry, Cambridge, p 90–125 Schuck P (2000) Biophys J 78:1606–1619 Creeth JM, Harding SE (1982) J Biochem Biophys Meth 17:25–34

Progr Colloid Polym Sci (2006) 131: 121–125 DOI 10.1007/2882_011 © Springer-Verlag Berlin Heidelberg 2006 Published online: 4 February 2006

Hans Georg Müller

Hans Georg Müller (u) Bayer Industry Services, Da 5, 41538 Dormagen, Germany e-mail: [email protected]

PARTICLES, POLYMERS AND INTERACTING SYSTEMS

Determination of Particle Size Distributions of Swollen (Hydrated) Particles by Analytical Ultracentrifugation

Abstract The classic idea of a particle is that of a hard particle for example a hard sphere. Deviations from this idea may refer to differences in shape. It may have the shape of an ellipsoid or a cylinder. These cases have in common, that the density of the particles is still well defined: It is the density of the solid material these particles consist of. In industrial practice however a great number of dispersions consists of particles, which have absorbed solvent, due to the fact, that they contain a few percent of soluble polymers. In the case of an aqueous dispersion then they are called hydrophilic. In these cases the density of these particles is unknown, it is between the density of the dry particle and that of the solvating medium. Applying the well-known ultracentrifugation method for determining particle size distributions from sedimentation velocity using the dry density results then in

Introduction The ultracentrifugation method developed by Scholtan and Lange [1] and since widely adopted [2–4] for the determination of particle size distributions is one of the best methods available for use in the submicron particle size range. In a round robin test [5] of available methods it was found to give the best results. Recently it has been further improved for the determination of

apparent diameters, which are smaller than the real diameters. To overcome this problem the degree of hydration has to be taken into account. So it is possible to calculate from the apparent diameter the diameter of the unswollen, compact particles as well as the diameter of the hydrated, swollen particles. The degree of hydration (or swelling) can be determined by preparative ultracentrifugation by pelleting the material and determining the weight of the wet material and after drying of the dry substance. To achieve equilibrium hydration pelleting is carried out at low concentrations and the particles are allowed to swell back at rest in the serum of the dispersion for several hours. Several examples are given. Keywords Particle size distribution · Hydrated particles · Degree of swelling · Analytical ultracentrifugation · Ultracentrifugation

very broad particle size distributions via interference optics [6]. Samples containing nine individual components were correctly analysed (Fig. 1). The high performance of this method is underlined by the fact, that it has not been necessary to couple measurements of different cells [4, 7, 8] and different concentrations for this result, but this analysis has been conducted in one cell in one run.

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Fig. 1 Particle size distribution of a mixture of nine calibration latices (used for electron microscopy) in differential and integral distribution as a test for the performance of the method to determine particle size distributions by analytical ultracentrifugation

The baseline separation of two components, differing in diameter by only 10%, is further evidence of the high resolving power of the method (Fig. 2). It should be mentioned here, that the physical basis of this method is Stoke’s law and the application of the light scattering theory of Mie, which is valid for absorbing and non absorbing particles, as has been described in [1]. Experiments are carried out at low concentrations as a rule lower than 0.5% so lack of extrapolation of s – values to zero concentration does not lead to remarkable errors in particle diameter.

Fig. 2 Resolving power of the method: Result of a mixture differing in 10% of diameter in differential and integral distribution

For the given examples it has been taken for granted that the particles investigated are hard spheres. Deviations from this may refer to differences in shape, they may have the shape of an ellipsoid or a cylinder as has been discussed in [9] or they may be flexible chains [10]. These cases have in common, that the density of the particles often is still well defined: It is the density of the solid material these particles consist of. In industrial practice however a great number of dispersions consists of particles, which are spheres as a rule but have absorbed solvent, due to the fact, that they contain a few percent of soluble polymers. In the case of an aqueous dispersion this may be polyacrylic acid for example in this case particle become hydrophilic. In these cases the density of these particles is unknown, it is between the density of the dry particle and that of the solvating medium. Unlike the situation at proteins [11–13] it is not possible to calculate the degree of hydration. This means that the density difference between particle and dispersing medium is reduced, resulting in a reduced sedimentation velocity. So calculating the particle diameter of such particles using the dry density instead of the unknown density of the swollen particles results in apparent diameters, which are smaller than the real diameters. This is caused by the fact that the force of friction and the force of buoyancy are affected by the hydration in different ways. This apparent diameter is a pure number and has no physical significance.

Determination of Particle Size Distributions of Swollen (Hydrated) Particles by Analytical Ultracentrifugation

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The force of friction K f on the other hand increases when particles swell.

for example at 1% and the particles are allowed to swell back at rest in the serum of the dispersion for several hours to achieve equilibrium swelling. The weight of the compact particles is achieved after drying of the swollen particles at vacuum. This simple method implies a principal error because Q m has the same value as Q v only if the density of the particles and of the solvent (most often water) is the same. In most cases the density of the investigated particles is not far from 1 due to the organic nature of the polymeric particles. If this is not the case, the degree of volume swelling can be computed from the mentioned quantities, the (dry) density of the polymer ρp and the density of the solvent ρs from Eqs. 10 or 11.   ρp m k mq − mk + (10) Qv = m k ρp ρs

K f = 3πdq ηu

or

Theory The following discussion is made on the basis of particle diameters instead of sedimentation coefficients, because this is our primary interest. The force of sedimentation K s in a field with the centrifugal acceleration ω2 r depends on the density difference between particle and dispersing medium ∆ρ and on the particle volume (π/6)dk3. Except for the case of preferential interaction this force is independent on a possible swelling of the particles. Therefore for the sake of simplicity the particles are considered as dry, compact particles with the corresponding dry density and diameter dk , k is used as abbreviation for compact. K s = ω2 r∆ρ(π/6)dk3

(1)

(2)

where dq is the diameter of the swollen particle, η the viscosity of the surrounding medium and u the sedimentation velocity. Both forces are in an equilibrium, so Ks = Kf

(3)

ω2 r∆ρ(1/18)dk3 = dq ηu

(4)

If the difference between swollen and compact diameters is neglected, an apparent diameter da results:
da = (18uη)/(ω2r∆ρ) (5) The compact and swollen diameter are related with each other by the degree of volume swelling Q v : Q v = (dq /dk )3

(6)

Then follows from Eq. 4: 
dk = 6 Q v (18uη)/(ω2r∆ρ) 
dq = Q v (18uη)/(ω2r∆ρ)

(7) (8)

From Eqs. 5, 7 and 8 can mathematically be concluded that it is da < dk < dq , the lack of physical significance of da has already been discussed.

Qv = 1 +

ρp ρs



mq − mk mk

 (11)

As can be seen from Eq. 10 if the densities of the polymer and solvent are the same, the degree of volume swelling and mass swelling become equal. Another approximation is due to the fact that the interstitial volume is neglected, which in the case of spheres in the hexagonal package of highest density is 1.33. In the case of swollen particles it is an open question whether this interstitial volume should be taken into account or not, because this volume will be occupied by swollen material instead of solvent. The dry density of the particles may be taken from literature or it may be determined by extrapolation of the measured density of the dispersion and the density of the serum to a concentration of 100% as described in [1]. Another way, which is applied for copolymers, is the calculation of particle density and refractive index by assuming volume additivity.

Examples

The first example in Fig. 3 is the particle size distribution of a black nanodispersion in water consisting of Practical Aspects polyethylendioxythiophen (PEDT). The median d50 apFor reasons of simplification the degree of volume swelling parent diameter of the mass distribution (calculated withQ v is substituted by the degree of mass swelling Q m de- out taking swelling into account) was determined with fined by 9 nanometers. cp. the lower part of Fig. 3. The degree of mass swelling has been determined with 7.7. Taking this Q m = m q /m k (9) into account results in compact diameters in the middle m q is the weight of the swollen particle and m k is the of Fig. 3 with a median diameter of 13 nm and a swollen weight of the dry compact particle. diameter in the upper part of Fig. 3 of 26 nm. To determine the weight of the swollen particles The second example in Fig. 4 is the distribution of preparative pelleting is carried out at low concentrations grafted latex dispersed in propylencarbonate as solvent.

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Fig. 3 Differential particle size distributions of a polyethylenedioxythiophen dispersion in water: Apparent (lower picture), compact and swollen diameters

H.G. Müller

Fig. 5 Differential particle size distributions of a polyol dispersed in water: Apparent (lower picture), compact and swollen diameters

The median d50 apparent diameter (calculated without taking swelling into account) was determined to be 0.271 µm cp. the lower part of Fig. 4. The degree of mass swelling has been determined with 3.3. Taking this into account results in compact diameters in the middle of Fig. 4 with a median diameter of 0.331 µm and swollen diameters in the upper part of Fig. 4 with a median value of 0.492 µm. The third example in Fig. 5 is the distribution of a polyol dispersion in water. The median d50 apparent diameter (calculated without taking swelling into account) was determined to be 0.100 µm cp. the lower part of Fig. 5. The degree of mass swelling has been determined with 2.7. Taking this into account results in a compact median diameter in the middle of Fig. 5 of 0.118 µm and a swollen diameter in the upper part of Fig. 5 of 0.165 µm.

Results and Discussion

Fig. 4 Differential particle size distributions of a grafted latex, dispersed in propylenecarbonate: Apparent (lower picture), compact and swollen diameters

In the case of swollen particles the ultracentrifugal method results in meaningless apparent diameters if the degree of swelling is not taken into account cp. Eq. 5. If this is done, compact diameters and swollen diameters can be calculated, the precision of determining Q w in the described way is about ±5%. The compact diameters calculated following Eq. 7 are in accordance with diameters determined by electron microscopy.

Determination of Particle Size Distributions of Swollen (Hydrated) Particles by Analytical Ultracentrifugation

Of greater importance however are the swollen diameters determined by Eq. 8 because they represent the particles, which are actually present in the dispersion. The degree of swelling besides the mass concentration is an important factor for the viscosity of a concentrated dispersion. The product of solid content of the sample c
(given in per cent) and Q w is a measure for the volume in the dispersion, which is occupied from particles. If this value exceeds 100%, which is quite often the case, particle interact, which makes viscosity increase sharply. The third example reveals however a problem for this method. For this bimodal distribution it is unlikely that the small particle fraction has the same degree of swelling as the bigger one. The problem can be solved by fractionating the sample corresponding to particle size by preparative ultracentrifugation. In this way the fraction of smaller particles is easily obtained, with some more work after redispersing the pellet also the fraction of the bigger particles. Then the degree of swelling can be determined for both fractions and taken into account separately. The special case of particles, which are partly soluble has been discussed in [14].

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Another way to determine Q v has been described in [15] based merely on analytical ultracentrifugation, a very fine and direct method which requires a high capacity of analytical ultracentrifuges however and a complete removal of the swollen gel from an analytical cell. Hydration of particles can also be measured via solvent spin relaxation by NMR [16]. Another way for the determination of the degree of swelling could be the comparison between the average diameter determined by photon correlation spectroscopy (PCS) and the apparent diameter by analytical ultracentrifugation. Photon correlation spectroscopy should measure the swollen (hydrodynamic) particle diameter and so following Eq. 8 it should be possible to determine the degree of swelling by Eq. 12. 2 Q v = dPCS /da2

(12)

If possible of both methods the same average value of the diameter e.g. the arithmetic mean of the mass distribution should be inserted into Eq. 12. Preliminary results show that values obtained by Eq. 12 differ in numerical values from that by ultracentrifugation but they show the same trend.

References 1. Scholtan W, Lange H (1972) Kolloid-Z u Z Polymere 250:782 2. Mächtle W (1984) Makromol Chem 185:1025 3. Müller HG (1989) Colloid Polym Sci 267:1113–1116 4. Müller HG (1997) Progr Colloid Polym Sci 107:180–188 5. Lange H (1995) Part Part Syst Charact 12:148–157 6. Müller HG (2004) Progr Colloid Polym Sci 127:9–13 7. Mächtle W (1992) Makromol Chem Makromol Symp 61:131–142 8. Mächtle W (1999) Biophys J 76:1080–1091 9. de la Torre JG (1992) Sedimentation coefficients of complex biological particles. In: Harding SE, Rowe AJ,

Horton JC (eds) Analytical Ultracentrifugation in Biochemistry. Royal Society of Chemistry, Cambridge, p 333–345 10. Freire JJ, de la Torre JG (1992) Sedimentation coefficients of flexible chain polymers. In: Harding SE, Rowe AJ, Horton JC (eds) Analytical Ultracentrifugation in Biochemistry. Royal Society of Chemistry, Cambridge, p 346–358 11. Laue TM, Shah BD, Ridgeway TM, Pelletier (1992) Computer aided interpretation of analytical sedimentation data for proteins. In: Harding SE, Rowe AJ, Horton JC (eds) Analytical Ultracentrifugation in Biochemistry. Royal Society of Chemistry, Cambridge, p 90–125

12. Durchschlag H (1986) Specific volumes of biological macromolecules and some other moleculs of biological interest. In: Hinz H-J (ed) Thermodynamic Data for Biochemistry and Biotechnology. Springer, Berlin, p 45–128 13. Durchschlag H (2003) In: Hinz H-J (ed) Landolt-Börnstein new series, biophysics – proteins, vol VII/2A. Springer, Berlin, Heidelberg, New York, p 4/1–157 14. Müller HG, Schmidt A, Kranz D (1991) Progr Colloid Polymer Sci 86:70–75 15. Lange H (1986) Colloid Polymer Sci 264:488–493 16. Adalsteinsson T, Dong W-F, Schönhoff M (2004) J Phys Chem B 180(52):20056

Progr Colloid Polym Sci (2006) 131: 126–128 DOI 10.1007/2882_012 © Springer-Verlag Berlin Heidelberg 2006 Published online: 10 February 2006

Helmut Cölfen Antje Völkel

Helmut Cölfen (u) · Antje Völkel Department of Colloid Chemistry, Max Planck Institute of Colloids and Interfaces, MPI Research Campus Golm, 14424 Potsdam, Germany e-mail: [email protected]

PARTICLES, POLYMERS AND INTERACTING SYSTEMS

Application of the Density Variation Method on Calciumcarbonate Nanoparticles

Abstract The simultaneous determination of particle size and density distributions by the so called density variation method via measurement of the same sample in H2 O and D2 O proved to be of great value for latex systems. However, many colloids of practical interest are inorganic or inorganic-organic hybrid colloids. Their density is usually much higher than that of the solvents so that the density variation method appears of limited applicability. In addition, these systems are usually charged – a complication, which was so far not yet considered in the theory for the density variation method. In this work, we apply this method to the determination of the particle size and

Introduction The precise determination of nanoparticle size distributions is an issue for Analytical Ultracentrifugation (AUC) since the invention of the technique in the 20’s of the last century [1–4]. However, according to the well known Svedberg equation, the determined particle size is strongly dependent on the particle density. This problem is also an issue for macromolecules so that Schachman et. al. introduced sedimentation equilibrium measurements of the same sample in similar solvents of varying density (e.g. H2 O & D2 O), to obtain not only the molar mass of the sample but also its density [5] – a big advantage if only very small sample amounts are available. This approach was extended to sedimentation velocity experiments with latices by Mächtle [6] and Müller [7] and resulted in two main equations for the determination of the particle dens-

density of CaCO3 precursor particles, which form superstructures in order to elucidate their crystal modification. Interestingly, the particle densities can be determined rather reasonably, whereas the particle size is much more influenced by the nonideality.

Keywords Amorphous calcium carbonate · Density variation method · Particle size distribution · Particle density distribution · Sedimentation velocity

ity and size: s1 ηs1 ρs2 − s2 ηs2 ρs1 ρp = s η −s η
1 s1 2 s2 18 (s2 ηs2 − s1 ηs1 ) dp = ρs1 − ρs2

(1) (2)

With s = sedimentation coefficient, ρ = density and η = viscosity with the indices 1 resp. s1 for solvent 1 and 2 resp. s2 for solvent 2. Polymer latices are an ideal system for this technique [8] as their densities are rather close to those of the dispersion medium. For such samples, also density gradients proved of great value, which can access densities up to 2 g/ml at the maximum, but potentially suffer from the problem of preferential solvation in one of the gradient forming solvents [9]. For inorganic or inorganic-

Application of the Density Variation Method on Calciumcarbonate Nanoparticles

organic hybrid colloid samples density gradients are likely to fail due to their high density. Nevertheless these samples are often composed of several species with different densities so that the density measurement in a density oscillation tube would just yield the average density, which is of no significance in this case. Therefore, the determination of density distributions is of special importance for these systems as it is the only possibility to learn about different dispersed species in these mixtures. However, the obstacle of the inorganic or organic-inorganic hybrid systems is their high density so that separate density measurements by density gradients are often not possible. We therefore explored the capability of the density variation method for an organic-inorganic colloid with charges. The investigated nanoparticles are insoluble hybrid colloids of CaCO3 and poly(styrene sulfonate) (PSS, 70 000 g/mol), which are amorphous precursor particles undergoing a complex self assembly process [10]. This process yields so-called mesocrystals, which are colloidal crystals selfassembled from non-spherical building units [11]. Electron microscopy indicated that the whole assembly process is started from amorphous precursor particles so that the task of this study is to verify amorphous precursor particles directly in solution. Amorphous CaCO3 (ACC) is difficult to analyze and detect as it can readily crystallize. Common detection techniques are X-ray or electron diffraction as well as FTIR spectroscopy. All these techniques require a drying step for sample preparation where the sample can crystallize. Therefore, a solution technique is needed to address this problem. Here we explore in how far density information from AUC can be used to detect ACC.

Experimental The mineralisation was performed by a slow CO2 gas diffusion technique described by Addadi et. al. [12] and the detailed experimental procedure was described elsewhere [13]. In general, all crystallizations of CaCl2 in the presence of different concentrations of poly(styrenesulfonate) were carried out in glass bottles, which were put into the same closed desiccator at room temperature. At first, a stock solution of CaCl2 (10 mmol/L) was freshly prepared in boiled double distilled water resp. heavy water and bubbled with N2 overnight. 5 mL of the H2 O resp. D2 O solutions at different concentration of poly(styrenesulfonate) (1, 0.1 g/L) and CaCl2 (5, 1.25 mmol/L) were prepared from the above CaCl2 solution under vigorous stirring. Then, the bottles were covered with Parafilm and punched with three needle holes and placed in a larger desiccator with two small bottles (10 mL), with crushed ammonium carbonate, covered with Parafilm with three needle holes. After crystallisation for about 2 hours, the bottles were taken out from the desiccator and the solutions were prepared for AUC measurements.

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The samples were measured with an interference optics with H2 O resp. D2 O as reference solvent in a Beckman Coulter Optima XLI Analytical Ultracentrifuge (Palo Alto, CA) at 60 000 rpm and 25 ◦ C. The sedimentation coefficient distributions were evaluated by Sedfit 8.9 by Peter Schuck [14] and then transferred to the algorithm for the determination of particle size and density distributions (Eqs. 1 and 2).

Results and Discussion The apparent particle size and density distributions resulting from the evaluation of the sedimentation coefficient distributions obtained in H2 O and D2 O are shown in Fig. 1. The determined apparent particle sizes are very small, between 2–7 nm. Wide angle X-ray diffraction (WAXS) yielded a particle size of ca. 20–30 nm, atomic force microscopy (AFM) ca. 50 nm in the finally assembled mesocrystals [10]. Dynamic light scattering (DLS) yielded two modes corresponding to particle sizes of 13 and 180 nm [10] showing a typical behaviour found for polyelectrolytes [15]. This already hints at serious problems in the determination of particle sizes for the charged nanoparticles via transport properties in solution and is also well known in AUC of polyelectrolytes, which can yield significantly too small apparent sedimentation coefficients for charged particles due to charge and non-ideality [16]. However, as the particles are investigated under the same conditions in chemically almost identical solvents, one might expect that charge effects on the sedimentation coefficients cancel out. This is indeed essentially true for Eq. 1, which is used for the determination of the particle density as the sedimentation coefficients are found

Fig. 1 Apparent particle size and density distribution of ACC precursor particles from the density variation method for samples with different CaCO3 /PSS ratios

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H. Cölfen · A. Völkel

in nominator and denominator. In Eq. 2, however, which is used for the determination of the particle size, this is not fulfilled. Here, the sedimentation coefficients are only found in the nominator. If the sedimentation coefficients are found too low because of charge effects, their difference in the two solvents will also be lower and consequently the resulting particle size. From this consideration, the density variation method should only yield correct densities but incorrect particle sizes in case of charged or highly non-ideal samples, if these effects cannot be corrected by the usual extrapolation to infinite dilution, which is not possible in our case as the reactant concentration also influences the particle synthesis. In Fig. 1, the particle densities at the peak maxima of the particle size distribution are: A) 1.47 g/ml, B) 1.44 g/ml, C) 1.46 g/ml D) 1.58 g/ml, being lower than the poly(styrene sulfonate) density of 1.61 g/ml. In addition, the density of some larger particles can get as low as 1.2 g/ml, indicating a large amount of bound water. The density variation over 0.4 g/ml indicates a compositional variation of the amorphous CaCO3 , where the smaller particles contain less water and/or a higher relative amount of PSS. As the density variation method is expected to yield the density of the dry, compact particles, this finding indicates that this view has to be extended to physically or chemically bound components to yield the average density as the obtained particle density is certainly lower than that of the individual components under exclusion of bound solvent. Ballauff et al. reported a density of 1.49 g/ml for amorphous CaCO3 on basis of Small Angle X-Ray Scattering (SAXS) [17]. This value is remarkably close to the density values for the maxima in the differential particle size distributions, which reflect the majority of the particles. Also, the density is lower than that of any crystalline CaCO3 polymorph. The respective CaCO3 polymorph densities are: Calcite (2.71 g/ml), Aragonite (2.93 g/ml), Vaterite (2.54 g/ml) and Ikaite

(CaCO3 ∗ 6H2 O; 1.77–1.8 g/ml) reflecting a density decrease for increasing instability of the polymorphs. Therefore, it can be concluded that the species, which are detected in Fig. 1 are amorphous CaCO3 . It is interesting to reflect on possible error sources of these measurements. According to Eq. 1, the only factor, which can amplify a possible error in the sedimentation coefficients due to non-ideality is the solvent density. If we consider an experimental temperature of 25 ◦ C, the density of H2 O is 0.99704 g/ml ≈ 1 g/ml, that of D2 O is 1.1047 g/ml ≈ 1.1 g/ml, which is a difference of only 10%. The solvent viscosities do not count as they are treated equally in nominator and denominator. Therefore an error in the sedimentation coefficient is not amplified to a significant extent under these conditions, which makes the particle density determination via the solvent density variation rather reliable.

Conclusion In conclusion, the density variation method can be advantageously applied even to charged and highly non-ideal colloids. Whereas the particle size information is only apparent and can be largely erroneous, the particle density distribution can be determined with reasonable precision and a maximum error of 10%. This accuracy is usually more than sufficient to judge the nature of the colloid. The suggested technique can prove of great value for the characterization of amorphous precursor particles in crystallization reactions but also has potential to determine the organic content in organic-inorganic hybrid particles etc. For reacting systems like the one investigated in this study, it has to be assured that the time at which the samples are prepared and investigated is exactly identical, which is no problem if multi-hole rotors are used. Acknowledgement We thank the Max-Planck-Society for financial support of this work and Dr. Tong Xin Wang for the preparation of the CaCO3 /PSS hybrid colloids.

References 1. Svedberg T, Rinde H (1924) J Am Chem Soc 46:2677 2. Rinde H (1928) PhD thesis, University of Upsala, Sweden 3. Nichols JB (1931) Physics 1:254 4. Nichols JB, Kramer EO, Bailey ED (1932) J Phys Chem 36:326 5. Edelstein SJ, Schachman HK (1967) J Biol Chem 242(2)306:11 6. Mächtle W (1984) Makromol Chem 185:1025–1039 7. Müller HG, Herrmann F (1995) Progr Colloid Polym Sci 99:114–119 8. Mächtle W (1992) In: Harding SE, Rowe AJ, Horton JC (eds) Analytical

ultracentrifugation in biochemistry and polymer science. The royal society of chemistry, Cambridge, England, p 147 9. Mächtle W, Börger L (2005) Analytical Ultracentrifugation of Polymers and Nanoparticles Springer Laboratory Series, Springer Verlag, in press 10. Wang TX, Cölfen H, Antonietti M (2005) J Am Chem Soc 127(10):3246–3247 11. Cölfen H, Antonietti M (2005) Angew Chem Int Ed 44:5576– 5591

12. Albeck S, Weiner S, Addadi L (1996) Chem Europ J 2:278–284 13. Wang TX, Cölfen H, Antonietti M (2005) Chem Eur J submitted 14. Schuck P (2000) Biophys J 78:1606 15. Förster S, Schmidt M, Antonietti M (1990) Polymer 31(5):781–792 16. Budd P (1992) In: Harding SE, Rowe AJ, Horton JC (eds) Analytical ultracentrifugation in biochemistry and polymer science. The royal society of chemistry, Cambridge, England, p 593 17. Bolze J, Peng B, Dingenouts N, Panine P, Narayanan T, Ballauff M (2002) Langmuir 18:8364–8369

Progr Colloid Polym Sci (2006) 131: 129–133 DOI 10.1007/2882_013 © Springer-Verlag Berlin Heidelberg 2006 Published online: 10 February 2006

Helmut Cölfen Gordon Lucas

Helmut Cölfen (u) · Gordon Lucas Department of Colloid Chemistry, Max Planck Institute of Colloids and Interfaces Research Campus Golm, MPI Research Campus Golm, 14424 Potsdam, Germany e-mail: [email protected]

PARTICLES, POLYMERS AND INTERACTING SYSTEMS

Particle Sedimentation in pH-Gradients

Abstract A new method is explored, where charged colloidal particles are sedimented in a pH-gradient, and an application in following the sedimentation velocity behavior of colloidal gold particles with cationic resp. anionic charge in a pH-gradient is described. A procedure to analyze this type of experiment is developed, where the effects of the changing pH and other parameters resulting from a pH-gradient onto sedimentation are considered. This qualitative

Introduction The production of stable colloidal dispersions is a large and important area of colloid chemistry, as already reflected in the classical textbooks of colloid science [1]. Colloidal particles can be stabilized against coagulation in a number of ways, including electrostatically, where the charge on the surface of the particle is used to repel against coagulation. The amount of charge on the surface of colloidal particles is dependent on pH, and therefore the stability against coagulation of electrostatically stabilized colloidal dispersions is pH dependent. A method to investigate the stability of electrostatically stabilized colloidal particles with pH is therefore of high interest for colloid chemistry. Up to now, the determination of the ζ-potential is the standard method to access the particle charge and thus its stability but it is a one point measurement, which also only yields average values if the particles exhibit a charge distribution. In addition, the ζ-potential is defined as the potential at the shear plane, which also does not correctly display the particle surface charge. Although ζ-potentials are widely determined, this quantity is based on one of the three electrokinetic

analysis allows an understanding of charge effects on the sedimentation behavior of colloids in a preformed pH-gradient and suggests ways for future quantitative evaluation of such data towards the determination of particle charge distributions. Keywords Sedimentation equilibrium · Density gradient ultracentrifugation · Sedimentation velocity · Stability of colloids · Particle charge

effects: electrophoresis, electroosmosis and the streaming potential [2], where the common determination of the ζ-potential via the determination of electrophoretic particle mobilities can be complicated by electroosmotic flow in a capillary [3]. Recent sedimentation equilibrium studies on charged colloids revealed their average particle surface charge [4]. In addition, recent sedimentation velocity studies on model TiO2 particles with varying surface charge indicate the potential of AUC to even determine particle surface charge distributions if a global analysis approach can be applied [5]. However, such global analysis relies on the independent determination of a particle size distribution free from charge effects, which cannot be achieved by simple salt addition as commonly performed in the case of polyelectrolytes, because this endangers particle destabilization and aggregation. Therefore, a method where a range of pHs can be investigated in one experiment is highly desirable as the continuous pH variation not only principally allows for the determination of surface charge distributions but also will indicate the stability limits for electrostatically stabilized colloids by pH-induced aggregation. A new method based on pH-gradient ultracentrifugation will be described here, whereby the stability

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Fig. 1 The concept of sedimenting colloidal particles in a pHgradient. Upon particle sedimentation, their surface charge decreases by pH change and the particles start to aggregate. It is also possible that the particles charge up upon the pH change related to sedimentation and get stabilized (not shown)

of colloidal dispersions at a range of pH’s can be investigated with one experiment. The colloidal particles are sedimented in a sedimentation velocity experiment through a pH-gradient, and the sedimentation behavior of the particles at various posititions in the pH-gradient is measured, and this information used to infer information about the stability of the colloidal dispersion at different pH values (Fig. 1). The effects of radial increasing solution density and viscosity of a pH-gradient are taken into account by calculating the apparent hydrodynamic diameter of the colloids from the obtained sedimentation coefficient. We are therefore able to at least qualitatively address particle stability issues with the presented method, although we only work with average sedimentation coefficients in the present study for the sake of simplicity and to better work out the underlying effects.

Experimental A Beckman XL-I analytical ultracentrifuge (Beckman Coulter, Palo Alto, CA) was used simultaneously applying Rayleigh interference and UV-Vis absorption optics. dn/ dc was measured with an NFT Scanref (Göttingen) at λ = 632.8 nm. Density measurements were performed with the Anton Paar DMA 5000 density meter. All chemicals were purchased from Aldrich unless otherwise stated: Poly(sodium 4-styrene sulfonate) (PSS) standard MW ca. 344 000 g/mol, Mw /Mn = 1.05 (in house) and 5 nm gold particles (ρ = 18.88 g cm−3 produced in house) were used.

H. Cölfen · G. Lucas

run was then stopped, and the tempered colloidal gold particles to be analyzed were inserted carefully into the top of the cell, taking care not to disturb the pre-formed gradient and ensuring that the density of the overlaid dispersion (0.9973 g cm−3 for the gold dispersion) was lower than the solution density at the top of the pH gradient to avoid convection problems, where the polymer solution density at the detected concentration could be calculated via a calibration. A normal sedimentation velocity experiment in the band centrifugation mode was then performed at 15 000 RPM and 25 ◦ C on these colloidal particles as they sediment in the pH gradient. In order to show that the pH-gradient survives this interruption of stopping the machine to insert the particles to be analyzed, the interference optics was used to detect the gradient before and after this stop/start procedure. This treatment is justified when the concentration of the gold particles is very low (µM concentrations) so that they do not contribute to the refractive index signals in a notable manner. Hence, the detected fringe shift almost exclusively corresponds to the PSS gradient. The interference scans for the changes in a pH gradient upon stopping and restarting the ultracentrifuge are shown in Fig. 2. The interference optics in the analytical ultracentrifuge can be used to follow the concentration profile of the pH gradient material using ∆n = Jλ/a with n = refractive index, J = fringe shift, λ = light wavelength, and a = optical pathlength. In this case, the polyacid poly(sodium 4-styrene sulfonate) (PSS) was used as the pH-gradient forming material. The PSS concentration can be determined using the measured refractive index increment dn/ dc of 0.18 ml/g (PSS 344 000 g/mol), where the effect of counterion condensation on dn/ dc has been neglected. This is justified because both the PSS and the gold colloids were not used as their salts and pH gradients generally have to be performed at low salt concentrations [6]. From

Results and Discussion 0.1 Sedimentation Velocity of Colloids in a pH Gradient The creation and detection of a pH gradient using analytical ultracentrifugation has been described in a previous paper [6]. To perform a sedimentation velocity experiment in a pH gradient, two steps were performed. Firstly, a pH gradient was produced in a sedimentation equilibrium experiment with 5 g/l PSS at 50 000 RPM and 25 ◦ C. The

Fig. 2 Proof that the pH gradient survives the stop/start procedure required for the technique of sedimenting particles in a pH-gradient. Before restarting the centrifuge, the gold particles were added

Particle Sedimentation in pH-Gradients

Fig. 2 it is clear, that although the pH-gradient is less steep after an interruption, it does indeed survive, and can be used to study the sedimentation behavior of particles in a pH-gradient. The pH-gradient does however change with time, as usually a lower speed is used for the subsequent sedimentation velocity run than for the formation of the pH gradient, and this change is followed with the interference optics in the analytical ultracentrifuge and taken into account for the calculation of the apparent particle diameters with respect to local solution densities and viscosities. Nevertheless, a very fast sedimentation velocity experiment would be desirable, where the changes in the pH gradient are negligible with respect to particle sedimentation. The new generation of fast multiwavelength UV/Vis detectors would be ideally suited for this type of experiment, as they allow for sedimentation velocity experiments as fast as 2 min during maximum acceleration to 60 000 RPM [7], but they are currently not yet combined with the Rayleigh interference optics being necessary for the here described experiments. Two different types of gold particles were sedimented in a pH-gradient made from 5 g/l 344 000 g/mol PSS. The gold particles were stabilized with the stabilizers 4-carboxythiophenyl and the pKa -value of the carboxy group was calculated to be 4.2 and 4-dimethylaminopyridin (DMAP, pK a = 9.2 [8]). The anionic stabilizer 4-Carboxythiophenyl is a good stabilizer at higher pH values, whereas DMAP stabilizes the gold particles at lower pH values due to quaternization of the aminonitrogen and the resulting charge introduction. The sedimentation of the gold particles was followed using the UV-Vis Absorption optics in the analytical ultracentrifuge (λ = 520 nm), whereas the pH gradient was followed with the interference optics. Typical sedimentation velocity profiles during such an experiment are shown in Fig. 3.

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Figure 3b shows that the changes in the pH gradient are only minor after about 20 min. The sedimentation velocity profiles were analysed using the following procedure. Firstly, a weight average sedimentation coefficient was calculated using the formula: 1 d ln rbnd , ω2 dt where s is the sedimentation coefficient of the sedimenting particles, ω is the angular velocity, rbnd is the distance from the axis of rotation of the middle of the sedimenting band normalized for the meniscus position, and t is the time. An apparent particle diameter (dapp ) was then calculated from this sedimentation coefficient, in order to take into account the increasing density and viscosity of the solution towards the bottom of the cell with increasing PSS concentration. It was assumed that there was no electrostatic interaction between the gold particles and PSS for this analysis, which should be met for the anionic gold particles but is unlikely for cationic particles. As a consequence, the particle diameters have only apparent character.
18ηs dapp = , ρp − ρs s=

where η is the viscosity of the solution, ρp is the density of the sedimenting particles, and ρs is the density of the solution. The density and viscosity of the PSS solutions was measured for a concentration series (at 25 ◦ C) and found to obey the following empirical regression functions: η = −0.0781c2PSS + 0.8373cPSS + 0.9791 (0 g/L < cPSS < 5 g/L) ρs = 0.0004cPSS + 0.9978 (1 g/L < cPSS < 5 g/L) ,

Fig. 3 Typical sedimentation velocity profiles of 4-carboxythiophenyl stabilized gold colloids sedimented in a 5 g/l PSS pH-gradient. a UV/Vis absorption traces at 520 nm selectively detecting the gold colloids (c = 1.3 × 10−5 wt − %) and b selective detection of PSS and conversion of its concentration gradient into the pH gradient at 25 ◦ C and 15 000 RPM after the gradient was formed at 50 000 RPM

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where cPSS is the concentration of PSS (344 000 g/mol) in g/L. The results of this analysis for the sedimentation of the two types of gold dispersions are presented in Fig. 4. The different effects on the apparent particle size calculated by this method are as follows: 1. Primary Charge effect [9]: When charged bodies are sedimented in the ultracentrifuge, the drag caused by the sedimentation of the associated counterions needed for electrical neutrality results in the calculation of a smaller apparent particle size. As the charge on the particle is dependent on pH, dapp will be affected at different points in a pH-gradient in different ways. 2. Aggregation: When the particle charge is decreased, the stabilization gets worse and the apparent particle size increases due to aggregation. This is especially evident, if the apparent particle size is found bigger than that of the primary nanoparticles (Fig. 4b). 3. PSS volume exclusion. In the pH gradient, high PSS concentrations are generated. Thus nonideality arises due to volume exclusion [10]. Therefore, the sedimentation of the particles is slowed down and a lower apparent particle size is calculated. The gold particles that were stabilized by 4-carboxythiophenyl (Fig. 4a) show a minimum apparent particle size at pH 5.5, whereas the gold particles stabilized by 4-dimethylamino-pyradin (DMAP) show a maximum particle size at pH 6.4. The results can be explained in terms of the effect of the stabilizers on particle aggregation at different pH values, from the primary charge effect and volume exclusion effects from the PSS gradient material. Gold particles stabilized by 4-carboxythiophenyl are expected to be more protected against coagulation at higher pHs, when the carboxy group is not protonated and thus negatively charged. This trend is seen between pH 5.1 and 5.5, where as the pH increases, a lower value for dapp is measured, indicating an increasing particle surface charge and thus particle repulsion as well as an increasing pri-

H. Cölfen · G. Lucas

mary charge effect resulting in a lower apparent diameter with increasing pH. Interestingly, already at pH 5.1, the apparent diameter is about 20% lower than the real particle diameter due to the negative particle surface charge. Between pH values 5.6 and 6.3 however, the value for dapp increases with increasing pH against the expectation from the pure charge consideration. The pH increase is correlated to a decrease in PSS concentration and therefore decreasing excluded volume effects. However, at pH 6.3 the apparent particle size is found to be larger than the 5 nm primary particle size. This effect can be explained in terms of particle aggregation, which is the only phenomenum, which could result in a higher value for dapp than that of the initial particles. However, why the gold particles aggregate with increasing particle charge is not clear yet and against the expectations, so that also an artifact must be taken into account for this particular data point. As pH gradients work at low ionic strengths, counterion condensation is unlikely to be responsible for this effect. The gold particles stabilized by DMAP, on the other hand, seemingly show increased stability against coagulation with decreasing pH, as the amino group is protonated. This is seen between pH values 5.7 to 6.2 in Fig. 4b, where dapp decreases with decreasing pH. Primary charge effect and volume exclusion also decrease the apparent particle size with decreasing pH and coupled increased PSS concentration. Nevertheless, the particle size is always higher than that of the primary gold nanoparticles (5 nm) hinting at aggregate formation and/or interaction with the PSS at all pH values despite charge repulsion, volume exclusion and primary charge effect, which decrease the apparent particle size. Above pH 6.3, dapp decreases with increasing pH value, a trend which cannot be explained by stability against coagulation, the primary charge effect or volume exclusion. The increase of the DMAP particle size with decreasing pH > 6.3 strongly hints at an interaction of the DMAP (positive) with PSS (negative) and thus particle ag-

Fig. 4 The results of the analysis for the sedimentation of the two types of gold dispersions sedimented in a PSS pH-gradient. a Gold stabilized by 4-Carboxythiophenyl b Gold stabilized by DMAP. The dashed horizontal line indicates the size of the primary 5 nm particles

Particle Sedimentation in pH-Gradients

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gregation because the PSS concentration increases with decreasing pH. Therefore, positively charged particles are not suited for a sedimentation study in a pH gradient with a negatively charged polymeric gradient forming material.

Conclusion A new ultracentrifugation method based on particle sedimentation in a pH gradient was explored. These sedimentation velocity experiments in a pH gradient, offer the potential ability to investigate the stability of electrostatically stabilised colloids with pH, where a range in pH values can be investigated in one experiment. It has been shown here, that such an experiment can be carried out in two stages – firstly, a sedimentation equilibrium run is performed to build up the pH-gradient, and then a sedimentation velocity run of colloidal particles is carried out in this pH gradient. Although the pH-gradient is disturbed by the transition between the sedimentation equilibrium run and sedimentation velocity run, it survives enough to investigate the stability of colloids in a certain pH range. However, the experiments in the XL-I AUC are not fast enough to maintain a sedimentation velocity experiment without notable changes in the pH gradient. This situation can be improved in the future with fast UV/Vis detectors [6]. The sedimentation velocity behavior of the colloid is influenced from the primary charge effect and from volume exclusion caused by the pH-gradient material, as well as the colloids stability against coagulation at different pHs. The potentially possible calculation of sedimentation coefficient distributions resp. apparent particle size

distributions should allow for a detailed insight into the role of the above effects. However, a drawback is that the polymeric gradient forming material can also interact with the charged particles leading to aggregation (PSS interaction with DMAP). This undesired effect can only be suppressed, if low molar mass gradient materials are used, which is possible [6], although the adjustment of the pH range is a problem. But even then, an interaction with the gradient forming counterion can occur resulting in a particle density change. In how far such density change is relevant for high density particles like gold remains to be explored but for the safe application of pH gradients, only particles with a common charge to the gradient forming material should be investigated, where the repulsion forces open the way for particle surface charge determination. Volume exclusion effects need to be quantified in order that pH-gradient ultracentrifugation offers a potential future way for the determination of particle charge distributions. Therefore, the present study can only be a first explorative report on the possibilities of pH gradients. Their potential is high as not only charge dependent effects can be studied for particles, but also, pH dependent transformations like those of macromolecules or microgels as well as macromolecular interactions can be investigated. Nevertheless, the challenge remains to establish pH gradient forming materials over a wide pH range, which are no polymers with the described aggregation problems. Acknowledgement We acknowledge financial support from the Max-Planck Society and the DFG (SFB 448). Dr. Subramanya Mayya is gratefully acknowledged for the colloidal gold dispersions.

References 1. Everett DH (1988) Basic Principles of colloid science. Royal Society of Chemistry, Cambridge 2. Evans DF, Wennerström H (1999) The Colloidal Domain: Where Physics, Chemistry, Biology and Technology meet. Wiley, New York

3. Hunter RJ (1981) Zeta Potential in Colloid Science. Academic Press, New York 4. Rasa M, Philipse AP (2004) Nature 429:857 5. Cölfen H, Tirosh S, Zaban A (2003) Langmuir 19:10654 6. Lucas G, Börger L, Cölfen H (2002) Prog Colloid Polym Sci 119:11

7. Bhattacharyya SK, Macziejewska P, Börger L, Gülsün AM, Cicek HB, Cölfen H (2005) Progr Colloid Polym Sci: this issue 8. http://daecr1.harvard.edu/pKa/ N_Heterocycles.GIF 9. Pedersen KO (1958) J Phys Chem 62:1282 10. Rivas G, Fernandez JA, Minton AP (1999) Biochemistry 38:9379

Progr Colloid Polym Sci (2006) 131: 134–140 DOI 10.1007/2882_014 © Springer-Verlag Berlin Heidelberg 2006 Published online: 10 February 2006

G.M. Pavlov E.F. Panarin E.V. Korneeva I.I. Gavrilova N.N. Tarasova

G.M. Pavlov (u) Foke Institute of Physics, St. Petersburg University, Ul’yanovskaya str. 1, 198504 St. Petersburg, Russia e-mail: [email protected] E.F. Panarin · E.V. Korneeva · I.I. Gavrilova · N.N. Tarasova Institute of Macromolecular Compounds, Russian Academy of Sciences, Bol’shoi pr. 31, 199004 St. Petersburg, Russia

PARTICLES, POLYMERS AND INTERACTING SYSTEMS

Molecular Properties and Electrostatic Interactions of Linear Poly(allylamine hydrochloride) Chains

Abstract The efficiency of two azoinitiators in the polymerization of allylamine salts in water and organic solvents was studied comparatively. Hydrodynamic and molecular properties of poly(allylamine hydrochloride) in 0.1 M NaCl were investigated in the molecular mass range 18 < MsD × 10−3 (g/mol) < 65. Molecular mass relationships were obtained for intrinsic viscosity ([η]), translation diffusion coefficient (D0 ) and velocity sedimentation coefficient (s0 ): [η] = 7.65 × 10−3 M0.8±0.1 , D0 = 2.41 × 10−4 M−(0.59±0.05) ,

Introduction During the last decades, synthetic polycations attracted increasingly attention of researchers dealing with the development of gene vectors capable of transporting genetic materials into target cells [1, 2]. In this case soluble interpolyelectrolyte complexes were formed between nucleic acids and linear polycations. The complexes do not contain any supplementary biospecific component and are not recognized by the immune system. Moreover, polycations are used in various technologies as components of films, for eliminating electrostatic charges, and in the layer-by-layer technique [3, 4]. These applications justify the interest of researchers in the preparation and study of polycations, in particular poly(allylamine hydrochloride) (PAAH). Poly(allylamine hydrochloride) is a cationic polyelectrolyte containing a primary amino-group. The presence of

s0 = 2.77 × 10−15 M0.41±0.05 . Hydrodynamic data were interpreted by using the concept of electrostatic short- and long-range interactions. The equilibrium rigidity of poly(allylamine hydrochloride) chains in 0.1 M NaCl and structural and electrostatic contributions to it were quantitatively evaluated. It was shown that in pure water the conformation of poly(allylamine hydrochloride) chains is close to rod-like. Keywords Poly(allylamine hydrochloride) · Hydrodynamic properties · Electrostatic interactions

this group makes it possible to synthesize many new polymers for use in biomedicine and ecology (new sorbents for binding metal ions, polymer carriers of biologically active compounds, etc.) [5–8]. For example, PAAH complexes well with DNA and is an agent for its transport into the cell [9, 10]. In this connection, the method of PAAH synthesis by direct homopolymerization of allylammonium salts is of interest due to the possibility to obtain the polymer from a one-step process. Although PAAH arouses much interest, information about molecular and equilibrium characteristics of this polymer obtained under various conditions is rather scanty in the literature [11]. Results of previous systematic work on the synthesis and study of water-soluble polymers of pharmacological and medical importance were continuously reported by the authors [12–15]. In the frame of this work the PAAH molecules were studied by methods of macromolecular hydrodynamics.

Molecular Properties and Electrostatic Interactions of Linear Poly(allylamine hydrochloride) Chains

Materials The possibility of obtaining PAAH by polymerization in water in the presence of a water-soluble azoinitiator 2,2-azo-bis-amidinepropane dichloride (Scheme 1) was demonstrated in [6]

135

monium salts can not only be synthesized with a watersoluble azoinitiator but also by using Scheme 2 in butanol. Hydrodynamic investigations were carried out on the samples obtained with Scheme 1. It should be noted that poly(allylammonium phosphate) does not dissolve in water. Poly(allylammonium phosphate) was dissolved with excess of hydrochloric acid with subsequent isolation of hydrochloride by precipitation with acetone:

Scheme 1

In this case the polymer yield attained 80–85%. The efficiency of two azoinitiators: Scheme 1 and 2,2
-azobisisobutyronitrile (AIBN) (Scheme 2) was compared

The procedure of dissolving was repeated several times. The purity of the samples was tested by GPC-analysis, low molecular weight components were not observed.

Methods Diffusion in 0.1 M NaCl Scheme 2

in the polymerization of monoallylammonium salts (phosphates, chlorides, and sulphates). Performing polymerizations in water with Scheme 1 yielded up to 60–80% conversion but was depending on the nature of the counterion. Contrarily, using Scheme 2 yields up to 45% were achieved only in organic solvents (Table 1). Highest yields were obtained polymerizing chlorides and phosphates in water with a guanidine azoinitiator (Scheme 1). When Scheme 2 was used, the most suitable solvent was butanol. Comparably very low yields were obtained under these conditions for chloride and sulphate of monoallylammonium. Generally, it is demonstrated that polyallylam-

Isothermal translational diffusion was studied at 26 ◦ C by the classical method of forming a boundary between the solution and the solvent [16, 17]. The diffusion boundary was created in a glass cell of length h = 30 mm along the beam path at an average solution concentration c of 3 × 10−4 g/cm3 (Fig. 1). The average time for boundary

Table 1 Polymerization of monoallylammonium salts with azoinitiators Counterion

Solvent

Initiator T ◦ C Time Yield Type Weight (h) weight % %

PO4 3− PO4 3− Cl− PO4 3− PO4 3− PO4 3− PO4 3− Cl− SO4 2− SO4 2−

H2 O H2 O H2 O H2 O/methanol butanol butanol tert-butanol butanol methanol methanol

I I I I II II II II II II

0.6 2.9 3.5 2.7 9.8 8.0 9.8 5.0 3.0 20.0

55 50 50 60 60 60 60 60 60 60

15 41 42 90 72 82 90 90 40 24

26.3 57.3 80.0 7.6 33.0 47.0 5.4 <1 <1 <2

Fig. 1 Interference images of the diffusion boundary formed between the sample 7 solution (c = 0.057 × 10−2 g/cm3 ) and solvent (0.1 M NaCl) at time ti after the boundary formation: a t1 = 0.5 h, b t2 = 4 h, c t3 = 12 h. Spar twinning of polarizing interferometer is 10 mm

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formation was 20–30 min. Diffusion measurements were carried out in solutions at the greatest dilution when the Debye factor (c[η]) characterizing the solution dilution was c[η] < 0.02. The values of the diffusion coefficient D obtained at these concentrations were assumed to be the values extrapolated to zero concentration. The optical system used for recording the solution-solvent boundary in diffusion analyses was a Lebedev’s polarizing interferometer [16, 17]. Translational diffusion coefficients were calculated from the equation σ 2 = σ02 + 2Dt

(1)

where σ 2 is the dispersion of the diffusion boundary calculated from the maximum ordinate and from the area under the diffusion curve, σ02 is the zero dispersion characterizing the quality of boundary formation, and t is the diffusion time. Sedimentation in 0.1 M NaCl Velocity sedimentation was investigated running the Analytical Ultracentrifuge MOM 3170 (Hungary) at a rotor speed of 40 000 rpm at 25 ◦ C. A double-sector cell (h = 12 mm) was used for artificial boundary experiments. The optical system used for recording the sedimentation boundary analyses was the Lebedev’s polarizing interferometer [16, 17]. The apparent sedimentation coefficients sapp were calculated from the radial displacement r of the maximum of the sedimentation curve over time t. The concentration dependence of sapp was studied for four samples (2, 8, 10, and 12 in Table 2), in the concentration range (0.05–0.40)×10−2 g/cm3 . The reciprocal apparent sedimentation coefficients linearly depends on the concentration in the limit of zero concentration as −1 = s0−1 (1 + ks c + . . .) sapp

(2)

−1 for PAAH in 0.1 M NaCl. Fig. 2 Concentration dependences of sapp Curve number correspond to sample numbers in Table 2

with ks being the Gralen coefficient and s0 defining the limiting sedimentation coefficient (Fig. 2). At sapp < 2 S the concentration dependence is virtually not observed at c < 0.2 × 10−2 g/cm3 . Intrinsic Viscosity in 0.1 M NaCl and in Pure Water Viscosity measurements were carried out using an Ostwald viscometer. The respective flow times t0 and t were measured at 25 ◦ C for solvent and polymer solutions with relative viscosities ηr = t/t0 in the range 1.2–2.5. All flow times were long enough to make kinetic energy corrections unnecessary. The intrinsic viscosity, [η], was extrapolated according to the Huggins equation: ηsp /c ≡ (ηr − 1)/c = [η] + kH [η]2 c + . . . ,

(3)

where ηsp is specific viscosity and kH is the Huggins parameter. The buoyancy factor (1 − υbar ρ0 ) was measured picnometrically in pure water. The partial specific volume was

Table 2 Hydrodynamic and molecular characteristics of poly(allylamine hydrochloride) in 0.1 M NaCl at 25 ◦ C N

D0 107 cm2 /s

s0 1013 s

MsD 10−3 g/mol

[η] cm3 /g

A0 1010 g cm2 s−2 K−1 mol−1/3

DP∗

(L/κ −1 )

1 2 3 4 5 6 7 8 9 10 11 12

7.5 6.6 6.8 6.5 6.3 5.0 4.25 4.3 4.3 3.9 3.95 3.2

1.6 1.4 1.7 1.6 1.9 1.7 1.8 1.85 2.1 2.3 2.8 2.4

18.5 18.5 21.5 21.5 26.0 29.5 37.0 37.5 42.5 51.0 61.5 65.0

25 – 20 25 30 36 44 39 54 59 52 57

3.76 – 3.34 3.42 3.77 3.31 3.24 3.16 3.68 3.66 3.77 3.21

198 198 230 230 278 316 396 401 455 546 658 696

52 52 60 60 73 83 104 105 119 143 173 183

∗

degree of polymerization

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137

calculated as υbar = 0.715 cm3 /g. The influence of 0.1 M NaCl on the υbar value was neglected [18, 19]. All hydrodynamic values were measured at 25 ◦ C or reduced to this temperature. At 25 ◦ C the solvent (0.1 M NaCl) had the following characteristics: density ρ0 = 0.999 g/cm3 and viscosity η0 = 0.8974 cP.

Results and Discussion Fig. 3 Scaling dependencies (type of Kuhn-Mark-HouwinkSakurada plots) a 1 – D0 , 2 - s0 in 0.1 M NaCl, b 1 – [η] in 0.1 M NaCl, 2 – [η] in pure water

Molecular Mass and Relationships of Kuhn-Mark-Houwink-Sakurada Type To obtain molecular characteristics of PAAH, the hydrodynamic investigations were carried out in 0.1 M NaCl. Under such conditions primary polyelectrolyte effects are suppressed and transport properties are primarily determined by macromolecular characteristics. The values of [η], D0 , and s0 obtained in 0.1 M NaCl, the molecular mass (MM), and the Mandelkern-Flory-Tsvetkov-Klenin hydrodynamic invariant A0 [20, 21] are listed in Table 2. The molecular mass MsD was calculated by Svedberg’s equation. MsD = (RT/(1 − υbar ρ0 ))(s0/D0 ) = R[s]/[D]

(4)

The hydrodynamic invariant was calculated following the relationships A0 = R[s][η]1/3 M −2/3 = (R[D]2 [s][η]/100)1/3

(5)

and the sedimentation parameter [22] was estimated using the relationship βs = NA [s]ks M −2/3 = NA (R−2 [D]2 [s]ks )1/3 1/3

(6)

where R is the universal gas constant, NA is the Avogadro number, T is the Kelvin temperature, the intrinsic diffusion coefficient [D] ≡ D0 η0 /T , and the intrinsic sedimentation coefficient [s] ≡ s0 η0 /(1 − υbar ρ0 ). The following average values of parameters that do not depend on MM of PAAH were obtained: Huggins parameter kH = 0.4 ± 0.2; and refractive index increment ∆n/∆c = (0.16 ± 0.01) cm3 /g. The average value of the hydrodynamic invariant is A0 = (3.5 ± 0.2) × 10−10 g cm2 s−2 K−1 mol−1/3 , and the average value of the sedimentation parameter is βs = (1.1 ± 0.2)107 mol−1/3 (2, 8, 10, and 12 samples in Table 2). Correlating the hydrodynamic characteristics with the MM of the sample series resulted in the following relationships (Fig. 3) for linear PAAH molecules in 0.1 M NaCl at 25 ◦ C: [η] = 7.65 × 10−3 M0.8±0.1 D0 = 2.41 × 10

−4

−(0.59±0.05)

M

s0 = 2.77 × 10−15 M0.41±0.05

(7a) (7b) (7c)

The exponents bη (Eq. 7a), bD (Eq. 7b), and bs (Eq. 7c) are in good correlation with each other according to |bD | = (bη + 1)/3 bs = (2 − bη )/3

(8a) (8b)

The Eq. 7a relating [η] to MM is in good agreement with an analogous equation obtained previously [11] for the molecular mass range 27 < Mw 10−3 < 180 in 0.2 M NaCl: [η] = 7.18 × 10−3 M0.815 , where MM were determined by the light scattering method. Equilibrium Rigidity and its Structural and Electrostatic Components Qualitative estimation of PAAH coils in solutions may be carried out using the concept of normalized scaling relations [23, 24] in the system of coordinates [η]ML vs. M/ML shown in Fig. 4, where ML is the mass of unit length. From the position of the [η]ML values in Fig. 4 it may be concluded that PAAH chains occupy more volume in solution than the chains of neutral flexiblechain poly(vinylformamide) [15] but less volume than the chains of neutral rigid-chain cellulose nitrate [25–27].

Fig. 4 Double logarithmic plot of [η]ML vs. M/ML for: 1 – PAAH in 0.2 M NaCl [11], 2 – PAAH in 0.1 M NaCl data from Table 2, 3 – poly(vinylformamide) in 0.2 M NaCl [15], 4 – cellulose nitrate in ethyl acetate [25–27]

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The greater volume occupied by the macromolecule with the same contour length can be due to higher equilibrium chain rigidity and/or to excluded volume interactions which for uncharged chains are related to the thermodynamic quality of the solvent. Furthermore, the obtained value of bη is large as compared to similar values observed for linear uncharged aliphatic polymers in the range of degrees of polymerization considered here. This can be the reflection of both electrostatic short- and long-range interaction in PAAH chains. Electrostatic short-range interactions are responsible for the value of statistic Kuhn segment length (or persistent length); the electrostatic long-range interaction may be correlated to the thermodynamic quality of solvents for neutral chains. Polyelectrolyte theory [28, 29] uses the screening radius of Debye-Hueckel κ −1 as the measure of length. It is defined as the distance by which the action of an electric field of a separate charge placed in the medium containing other charges is spread: κ −1 = (8πλB µ)−1/2

(9)

where λB = ( e2 /4π ∈0 ∈ kT) is the Bjerrum length characterizing the screening action of the solvent, e is the elementary charge, ∈0 is the electrical constant, ∈ is the dielectric permittivity of the medium, k is the Boltzmann constant, µ = (1/2)Σn i Z i2 is the ionic strength of solution, n i is the number of i-th ions per unit volume, and Z i is the charge of the i-th ion in e units. For solutions of salts consisting of monovalent ions µ is C × 10−3 × NA cm−3 where C is the molar salt concentration. For aqueous 0.1 M NaCl solution at 25 ◦ C it is λB = 0.71 nm and κ −1 = 0.96 nm. The charges located along the chain at a distance l > κ −1 induce electrostatic long-range interactions. For the contour lengths of PAAH molecules studied here it is L  κ −1 (Table 2) and, hence, the high value of bη may be explained by electrostatic long-range interactions in PAAH chains, i.e. by their coil expansion. The parameter ε characterizing the coil expansion of chains was calculated from equations [30, 31] ε = (2bη − 1)/3 ε = 2bD − 1 .

(10a) (10b)

For the system studied here the ε-values are in the range of 0.18 < ε < 0.20. Electrostatic volume effects were taken into account on the basis of the Gray-Bloomfield-Hearst theory [32] for translational friction of wormlike chains [s]P0 NA = (3/(1 − ε)(3 − ε)) (1+ε)/2

× (ML

/ A(1−ε)/2 ) × M (1−ε)/2

+ (ML P0 /3π) × [ln A/d − (3 A/d)

(11) −1

− ϕ(ε)]

where P0 = 5.11 is the Flory hydrodynamic parameter, ML is the mass of unit length of the polymer chain, A is the

length of the statistical segment, and d is hydrodynamic diameter of the chain, ϕ(ε) = 1.431 + 2.64ε + 4.71ε2 [33]. Viscometric data were also analyzed on the basis of Eq. 11 replacing the variables [12, 34] [s]NA P0 = (M 2 Φ0 /[η])1/3

(12)

where Φ0 = 2.87 × 1023 mol−1 is the Flory viscosity parameter [17, 35]. The dependencies of s0 and (M 2 /[η])1/3 on M (1−ε)/2 are shown in Fig. 5. For ML = 3.7 × 109 g/cm the following numerical values are obtained for the length of the statistical segment A and the hydrodynamic chain diameter d: from translational friction data A f = (3.3 ± 1) nm, d f = (0.8 ± 0.5) nm; from viscometric data Aη = (2.6 ± 0.6) nm, dη = (0.4 ± 0.2) nm. A = 3 nm and d = 0.6 nm have been considered as average values. The data obtained in [11] for solvents of different ionic strengths was processed in the same way and the results are summarized in Table 3. In the case of polyelectrolytes, the concept of chain rigidity additivity is considered [28, 29]. The equilibrium

Fig. 5 Plots of s0 and (M 2 /[η])1/3 vs. M (1−ε)/2 used for evaluating the equilibrium rigidity of the chain A and the hydrodynamic diameter d from data on sedimentation (1) and intrinsic viscosity (2) Table 3 Total (Atotal ), electrostatic (Ael ), and structural (Astr ) Kuhn segment lengths, hydrodynamic diameter obtained for the poly(allylamine hydrochloride) chains in aqueous solution of different ionic strengths from intrinsic viscosity data (Eqs. 11 and 12) cNaCl M

Atotal , nm d, nm

Ael , nm

Astr , nm

Reference

0.05 0.1 0.2 0.5 1.0

3.3 2.6 2.5 2.1 2.2

1.31 0.65 0.33 0.13 0.07

2.0 2.0 2.2 2.0 2.1

[11] this work [11] [11] [11]

0.25 0.40 0.27 0.21 0.08

Molecular Properties and Electrostatic Interactions of Linear Poly(allylamine hydrochloride) Chains

139

rigidity of the chain is the sum of structural and electrostatic components: A = Astr + Ael

(13)

The charges located along the chain at distance l < κ −1 cause additional electrostatic short-range interaction leading to increased chain rigidity by the value Ael . This value depends on linear charge density and the ionic strength of the solution. A linear polyelectrolyte is characterized by a dimensionless parameter Q el which is the product of the Bjerrum length and the linear charge density (Z/L): Q el = (λB Z)/L. The general expression for Ael obtained by Odijk [36] contains the correction factor ( f 2 ) taking into account the condensation of counterions on the polyelectrolyte chain ( f 2 < 1) and a certain function of the macromolecule contour length normalized to the DebyeHueckel length F(L/κ −1 ). If L/κ −1  1, the function F(L/κ −1 ) becomes equal to unity. Hence, Ael = (Q 2el /2λB κ 2 ) f 2 F(L/κ −1 )

(14)

and in the case of Q el > 1 this expression simplifies as: Ael = κ −2 /2λB

(15)

In our case it is Q el ≈ 2.8 and the evaluation of the electrostatic component of the statistical segment length leads to the value of Ael = 0.65 nm. Therefore, the structural component Astr becomes ≈ 2.3 nm. This value is characteristic for uncharged aliphatic macromolecules without voluminous substituents in side chains [12, 15, 16, 33]. As a first approximation, poly(vinylformamide) chains can be considered as a linear uncharged analog of PAAH:

This water-soluble linear polymer has been studied previously [15] and the length of the statistical segment A has been obtained A = 2 nm. This value is in good agreement with the value of the structural component of PAAH chains equilibrium rigidity in 0.1 M NaCl. Intrinsic Viscosity in Water In pure water, PAAH molecules in spite of their low degree of polymerization exhibit strong polyelectrolyte effects. This is illustrated by the dependence of ηsp /c on c, shown in Fig. 6a. The data obtained in water were processed according to Fuoss’ empirical relationship [37, 38]: c/ηsp = 1/[η] + Bc0.5

(16)

(Fig. 6b), and the corresponding values of intrinsic viscosity were established. Here the slop B has the dimension

Fig. 6 a Plot of ηsp /c on c for PAAH samples in pure water. b Fuoss-plot c/ηsp on c1/2 corresponding to Fig. 6a. Numbers of curves in Figs. 6a and 6b correspond to numbers in Table 2

(g/cm3 )1/2 . The values of [η] obtained by Fuoss extrapolation exceed several times the values obtained in 0.1 M NaCl (Table 2). This indicates that the volume occupied by macromolecules in pure water considerably increases. This increase due to chain uncoiling is caused by high linear charge density of PAAH chains but may also reflect the influence of the electrostatic atmosphere around the molecules, which is determined by the Debye length, on the viscosity data. The correlation of [η] with MM (Fig. 4b) gives the following scaling ratio for PAAH in pure water: [η] = 14.2 × 10−3 M1.4±0.4

(17)

The value of the exponent bη indicates that PAAH chains become much more rigid in pure water. Assuming that for PAAH chains in pure water the effects of both electrostatic short- and long-range interactions are present, the statistic segment length is calculated as A ≈ 40 nm. This value exceeded by more than one order of magnitude the value obtained in 0.1 M NaCl. In this case the number of statistical segments in PAAH chains varies in the range of 1 ≤ (L/ A)H2 O ≤ 4. This means that the conformation of PAAH chains in water approaches rodlike conformation.

Conclusions Carefully selected and optimized synthesis conditions (initiator, solvent, and counterion) led to acceptable yields of PAAH salts. Hydrodynamic characteristics of PAAH in 0.1 M NaCl were studied, MM were determined and ratios relating hydrodynamic characteristics to the MM in the range 18 < MsD × 10−3 (g/mol) < 65 were established. The experimental results obtained in the study of hydrodynamic characteristic of PAAH at different ionic strengths and in pure water can be interpreted quantitatively separating the contributions of electrostatic shortand long-range interactions and relating them the to chains size. The analysis of hydrodynamic data in the framework of the Gray-Bloomfield-Hearst theory allowed estimating

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the Kuhn statistical segment length and the hydrodynamic diameter of PAAH chains. It was concluded that the longrange effect on coil extension prevails over short-range effects. The Kuhn segment length presents a value separated from the electrostatic component. Including literature data it was demonstrated that the structural component of equilibrium rigidity does not depend on the ionic strength of the solution. For all ionic strengths Astr ≈ 2 nm was calculated.

Acknowledgement This work is carried out with partial financial support of the Ministry of industry, science, and technologies of Russian Federation, grant N1823.2003.3. The authors are sincerely grateful to C. Wandrey for drawing our attention to an excellent experimental work on the study of dilute solution properties of poly(allylammonium chloride) in aqueous sodium chloride solutions [11], which, unfortunately, was not known to the authors. The authors also thank Gorbunova O. P., and Arpidov Yu. A. for participating in the experimental part of this work.

References 1. Wu G, Wu C (1985) J Biolog Chem 262:4429 2. Kabanov A, Kabanov V (1995) Bioconjugate Chem 6:7 3. Decher G, Schlenoff J (eds) (2003) Multilayer thin films. Wiley, New York 4. Kim B-S, Lebedeva O, Kim D, Caminade A-M, Majoral J-P, Knoll W, Vinogradova O (2005) Europ Polym Congress Abstracts:194, Moscow 5. Zubov V, Kumar V, Masterova M, Kabanov V (1979) J Macromol Sci Chem 13:111 6. Harada S, Hasegawa S (1984) Makromol Chem Rapid Comm 15:27 7. Masterova M, Andreeva L, Zubov V, Polak L, Kabanov V (1976) Vysokomolekul soedin 18A:1957 8. Kreindel M, Andreeva L, Kaplan A, Golubev V, Masterova M, Zubov V, Polak L, Kabanov V (1976) Vysokomolekul soedin 18A:2233 9. Wolfert M, Dash P, Nazarova O, Oupicky D, Seymour L, Smart S, Strohalm J, Ulbrich K (1999) Bioconjugate Chem 10:993 10. Kasiyanenko N, Kopyshev A, Obuchova O, Nazarova O, Panarin E (2002) Zh Phys Chem 76:2021 11. Ochiai H, Handa M, Matsumoto H, Moriga T, Murakami I (1985) Makromolek Chemie 186:2547 12. Pavlov G, Panarin E, Korneeva E, Kurochkin E, Baikov V, Uschakova V (1990) Makromolek Chemie 191:2889

13. Pavlov G, Ivanova N, Korneeva E, Michailova N, Panarin E (1996) J Carbohydrate Chem 15:417 14. Solovskij M, Panarin E, Gorbunova O, Korneeva E, Petuhkova N, Michailova N, Pavlov G (2000) Eur Polym J 36:1127 15. Pavlov G, Korneeva E, Ebel C, Gavrilova I, Nesterova N, Panarin E (2004) Vysokomolekul Soedin 46A:1732 16. Tsvetkov V, Eskin V, Frenkel S (1970) Structure of Macromolecules in Solutions. Butterworths, London 17. Tsvetkov V (1989) Rigid-chain Polymers. Hydrodinamic and optical Properties in Solution. Consultants Bureau, New York, London 18. Pavlov G, Korneeva E, Harding S, Vichoreva G (1998) Polymer 39:6951 19. Pavlov G, Finet S, Tatarenko K, Korneeva E, Ebel C (2003) Europ Biophysics J 32:437 20. Mandelkern L, Flory P (1952) J Chem Phys 20:212 21. Tsvetkov V, Klenin S (1953) Dokl Akad Nauk SSSR 88:49 22. Pavlov G (1997) Europ Biophysics J 25:385 23. Pavlov G, Harding S, Rowe A (1999) Progr Colloid Polym Sci 113:76 24. Pavlov G (2005) Vysokomolekul soedin 47:1872 25. Newman S, Loeb L, Conrad C (1953) J Polym Sci 10:463

26. Hunt M, Newman S, Scheraga H, Flory P (1956) J Phys Chem 60:1278 27. Pavlov G, Kozlov A, Martchenko G, Tsvetkov V (1982) Vysokomolekul soedin 24B:284 28. Mandel M (1993) Some properties of polyelectrolyte solutions and the scaling approach. In: Hara M (ed) Polyelectrolytes: science and technology. Dekker, New York, p 1–75 29. Barrat J-L, Joanny J-F (1996) Adv Chem Phys Ed by Prigogine I, Rice S 94:1 30. Ptitsyn O, Eizner Yu (1958) Zh Phys Chem 32:2464 31. Volkenstein M (1959) Configurational statistics of polymeric chains. Ed Academy Sci, Moscow, Leningrad 32. Gray G, Bloomfield V, Hearst J (1967) J Chem Phys 46:1493 33. Pavlov G (2002) Progr Colloid Polym Sci 119:149 34. Pavlov G, Selunin S, Shildiaeva N, Yakopson S, Efros L (1985) Vysokomolekul soedin 27A:1627 35. Yamakawa H (1971) Modern theory of polymer solutions. Harper and Row, New York 36. Odijk T (1977) J Polym Sci Polym Phys 15:477 37. Fuoss R, Strauss U (1948) J Polym Sci 3:602 38. Tanford C (1963) Physical chemistry of macromolecules. Wiley, New York

Progr Colloid Polym Sci (2006) 131: 141–149 DOI 10.1007/2882_018 © Springer-Verlag Berlin Heidelberg 2006 Published online: 15 February 2006

Laurent Bourdillon David Hunkeler Christine Wandrey

Laurent Bourdillon · Christine Wandrey (u) Institut des Sciences et Ingénierie Chimiques, Laboratoire de Médecine Régénérative et de Pharmacobiologie, École Polytechnique Fédérale de Lausanne, Station 15, CH-1015 Lausanne, Switzerland e-mail: [email protected] David Hunkeler AQUA+TECH Specialties S.A., Chemin du Chalet-du-Bac 4 CP 28, CH-1283 La Plaine, Switzerland

PARTICLES, POLYMERS AND INTERACTING SYSTEMS

The Analytical Ultracentrifuge for the Characterization of Polydisperse Polyelectrolytes

Abstract High molar mass polyelectrolytes, which are polydisperse concerning the molar mass, the charge density, and the chain architecture were characterized by hydrodynamic methods, in particular analytical ultracentrifugation. The samples of interest were a series of copolymers differing in the degree of branching at constant chemical composition/charge density and highly branched polyelectrolytes of various charge density/chemical composition. Combining synthetic boundary and sedimentation velocity experiments, and running at various speeds, allowed to identify the degree of homogeneity of the various distributions of the samples. It was

Introduction High molar mass polyelectrolytes are essential in the maintenance of water as a resource. They are also widely applied for industrial solid/liquid separation processes such as flocculation, flotation, dewatering, coagulation, or retention. The improvement of the polyelectrolyte performance in these application fields has significance for global sustainable development as it is not only the water quality, per se, which is influenced, but also the downstream burdens linked with the water life cycle [1–6]. The overall environmental impact can be reduced significantly if the flocculants result in higher sludge dry material levels requiring less transport capacity. Moreover, the increase of the efficiency will certainly lead to decreased process costs. The prerequisite of entire process optimization, however, is detailed knowledge of the interrelations between

illustrated that homogeneous batches of highly branched high molar mass polyelectrolytes can be synthesized with a negligible fraction of crosslinked molecules. Surprisingly, the sedimentation coefficient distribution of the main fraction was relatively narrow indicating a tight branching distribution. Overall, AUC delivers a variety of detail information, which cannot be obtained by other methods either due to lack of sensitivity or non-resolution of polydispersity. Keywords Analytical ultracentrifugation · Cationic polyelectrolytes · Polydispersity · Heterogeneous mixtures · Sedimentation velocity

polyelectrolyte constitution, polyelectrolyte solution behavior, and application performance based on comprehensive characterization. As a working hypothesis it has been proposed that the charge density of the polymer, linked to the degree of branching and its distribution, is a key driver in obtaining higher sludge dry material levels, and a more sustainable management of the water resources. Due to the wide field of applications, today a polyelectrolyte producer/supplier is obliged to offer a series of products designed for specific separation processes and technologies. For example, wastewater from a multitude of sources strongly varying in the composition requires appropriate selection of polymers in order to be efficient and economical. Typically, acrylamide copolymers have molar masses in the order of 106 –107 g/mol. It was found, and confirmed, that not only the molar mass, the type of charge and the charge density affect the

142

polyelectrolyte performance but also the chain architecture. For example, nonlinear flocculants are reported to have higher dewatering rate and dry material in the treated sludge for the same polymer dose or comparable dewatering rate and dry material for lower polymer dose [7, 8]. Therefore, product optimization needs to be carried out primarily towards two parameters the charge density and degree of branching but maintaining a high molar mass. Recent results of modeling and simulations considering the polyelectrolyte characteristics as well as the medium quality by including particle charge, particle concentration, particle size, pH of the medium, and ionic strength support this strategy [9]. Coarse-grain molecular models indicate that the floc density, and hence dry material, is, indeed, optimized at intermediate-to-high branching levels such as 2–4 long chain branches per molecule. The characterization of technical products of very high molar mass, varied composition of neutral and charged monomer units, branched copolymer chains presents a particular challenge. If produced, as it is typically the case, by radical polymerization, such technical products are expected to be polydisperse concerning the molar mass, the chemical composition, and the chain architecture. The successful characterization can contribute to both the optimization of the synthesis technology and the specific application process if the sample characteristics obtained are correlated with synthesis conditions and application performance. In this paper attention is paid to acrylamide-based copolymers of the general chemical structure presented in Fig. 1. Technological optimization aims at synthesizing, reproducibly, well defined branched copolymers, with a narrow branching distribution, though it also has to avoid cross-linking yielding less soluble and/or insoluble fractions, which do not or only inefficiently contribute to the intended separation processes. AUC was used to identify in particular aggregated, associated, and completely soluble portions in addition to measuring the sedimentation coefficients and sedimentation coefficient distribution. The

L. Bourdillon et al.

paper does not intend to present a complete characterization of all synthesized samples. Rather it will focus on data that cannot be obtained by any other method or that support the interpretation and understanding of results from other methods.

Experimental Polyelectrolyte Synthesis Two series of cationic copolymers were synthesized from the neutral monomer acrylamide (AM) and the cationic monomer acryloyloxyethyltrimethylammonium chloride (AEAC) in addition to copolymers for general viscosity studies. Series 1 was polymerized from the same chemical batch composition though with three different amounts of the cross-linker methylene-bis-acrylamide (MBA). For series 2 the recipe to obtain branching was maintained constant while the monomer batch composition was varied. All copolymers were synthesized by inverse-emulsion polymerization. Details of the synthesis have been described in previous communications [10–12]. Due to the similar monomer reactivity ratios of the two monomers the constancy of the average copolymer composition did not pose difficulties. It should be noted that the terminology in the subsequent discussion reflects, qualitatively, the branching level. A “+” suffix to a polymer name corresponds with one long-chain branch per molecule, “++” to two long-chain branches, etc. Sample Preparation For all measurements the polyelectrolyte samples isolated from the emulsion by repeated precipitation and redissolution in acetone and water, respectively, were dissolved in water, then mixed with 0.1 M NaCl solution (volume ratio 1 : 1), and dialyzed against 0.05 M NaCl solution. The dialysate was used for dilution during viscosity and density measurements and served as solvent in the double sector cells in the analytical ultracentrifuge. For the final concentration the sample humidity was considered and corrected. Methods The intrinsic viscosity, [η], of the isolated and redissolved copolymers was determined at 20 ± 0.1 ◦ C using a Viscologic TI1 capillary viscometer, capillary 0.58 mm (Sematech, Nice France). It was obtained according to Huggins, Eq. 1 [13] and Schulz-Blaschke, Eq. 2 [14]. η ηsp η −1 = 0 = [η] + kH [η]2 c c c ηred = [η] + kSB [η]2 ηsp

ηred =

Fig. 1 General chemical structure of the acrylamide/acryloyloxyethyltrimethylammonium chloride (AM/AEAC) copolymers

(1) (2)

The Analytical Ultracentrifuge for the Characterization of Polydisperse Polyelectrolytes

143

The partial specific volume, ν, of the copolymers in aqueous salt solution was calculated from density measurements performed with a High-Precision Digital Density Measuring System DMA60/DMA602 at 20 ± 0.02 ◦ C (Anton Paar K.G. Graz, Austria) according to Eq. 3 [15]. ρ = ρ0 + (1 − νρ0 ) c.

(3)

The copolymer composition was analyzed by argentometric titration of aqueous polymer solutions of known solid content using a 736 GP Titrino (Metrohm, Switzerland). The homogeneity, aggregation tendency and sedimentation behavior were evaluated by sedimentation velocity and synthetic boundary experiments. All concentration profiles were detected using an OPTIMA XL-I analytical ultracentrifuge (Beckman Coulter, Palo Alto, USA). Specifically, at 20 ◦ C synthetic boundary experiments were performed between 3000 and 10 000 rpm whereas for the velocity experiments the velocity was varied between 3000 and 35 000 rpm. An 8 hole titanium rotor was used equipped with two synthetic boundary cells and five double sector cells with solutions of different concentrations in the concentration range of 5 × 10−4 to 2 × 10−4 g/ml. For the synthetic boundary experiment always the highest and lowest concentration were investigated.

Results and Discussion General Problems of Polyelectrolyte Characterization Prior to the discussion of analytical ultracentrifugation results, data from the literature, as well as those obtained from complementary techniques, will be presented in order to demonstrate the complexity of the characterization topic. Then results from preceding viscosity studies will briefly be summarized, which serve to justify the experimental conditions selected for the studies here. The problematic of the characterization becomes obvious from the literature data plotted in Fig. 2a. For linear copolymers of the same chemical structure as studied here, AM/AEAC copolymers, Mark-Houwink-Kuhn-Sakurada (MHKS) plots obtained from viscometry and light scattering measurements are shown [16, 17]. The differences of the two studies, which were both performed in 1 M NaCl at 25 ◦ C are clearly visible. Assuming that the solution behavior is not so different for 25 or 30 mol% cationic monomer content in the copolymer the appropriate relationships have been used to calculate the molar mass for one viscosity value. For example, for an intrinsic viscosity value [η] = 500 ml/g average molar mass values of Mw = 2.56 × 106 g/mol [16] and Mw = 5.07 × 106 g/mol [17] are obtained. This presents a considerable difference for the calculation of the molar mass from viscosity experiments. Moreover, despite all measurements have been performed in NaCl of the

Fig. 2 a Mark-Houwink-Kuhn-Sakurada (MHKS) relationships of AM/AEAC copolymers of different molar composition. Viscosity and light scattering measurements were performed in 1 M NaCl at 25 ◦ C — [16], - - - [17]. The numbers indicate mol% of the cationic component. b Exponent of the MHKS relationship as a function of the acrylamide copolymer composition. Acryloyloxyethyltrimethylammonium chloride — — [16], —
— [17]; methacryloyloxyethyltrimethylammonium chloride - - -
- - - [18]; acrylaminopropyltrimethylammonium chloride - - -- - - [17]

•

same ionic strength, and the fact that the samples covered a similar molar mass range determined by light scattering, the exponents of the relationships as a function of the copolymer composition also differ considerably. Figure 2b demonstrates the composition dependence of the exponent for the copolymer type studied here and two other acrylamide copolymers of very similar chemical structure, with all of them being linear [16–19]. Differences over a wide range are clearly visible, in particular comparing the two curves of the AM/AEAC copolymers. The results are difficult to explain. Polydispersity effects might be one reason the experimental conditions during the synthesis and characterization another one.

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ported [10]. However, this may change for different copolymer compositions. The interpretation of [η] is supported by plotting of the Huggins and Schulz-Blaschke constants, kH and kSB , respectively, as a function of the ionic strength in Fig. 3b. At low ionic strengths, 5 × 10−3 to 5 × 10−2 mol/l, the viscosity constants only slightly change though rapidly increase if the ionic strength exceeds about 0.05 M NaCl. It is noteworthy, that the effect is much stronger for branched molecules, though may be different if the composition is varied. Not only [η] but also the partial specific volume changes if the ionic strength due to addition of low molar mass salt varies. As an example, for the linear copolymer containing 40 mol% cationic monomer units plotted in Fig. 3 the following values have been obtained (molar NaCl concentration/partial specific volume in ml/g): 0.005/0.717; 0.05/0.745; 0.1/0.804; 0.5/0.854. Influence of Chain Branching

Fig. 3 Influence of the ionic strength on a the intrinsic viscosity of linear and branched AM/AEAC copolymers containing 40 mol% cationic monomer. — — linear, - - -×- - - branched (+), - - - + - - branched (+ + ++), T = 20 ◦ C. b on the Huggins and SchulzBlaschke constants of linear and branched AM/AEAC copolymers containing 40 mol% cationic monomer. — — kH linear, —
— kSB linear, - - -×- - - kH branched (+), - - - + - - - kH branched (+ + ++), T = 20 ◦ C

•

•

The aforementioned discussion points to the next problem, which is the ionic strength employed for the characterization. As it is visible from Fig. 3a, the intrinsic viscosity decreases as the ionic strength increases. More importantly, the ionic strength dependency was found to be partially different for linear and branched copolymers. Whereas slightly branched and homogeneous polyelectrolytes are more extended than linear molecules at low ionic strength, probably due to stronger end group effects in the branched molecules, the behavior changes at higher ionic strength where the branched molecules were found to become more salt sensitive. Higher counterion activity in pure water was proved for branched polyelectrolytes compared to linear of the same chemical structure [20]. The branching effects seem to dominate the end group effects at higher ionic strength. In addition sample heterogeneity effects have been re-

As a conclusion from the intrinsic viscosity experiments the characterization of the branched samples of the two series of interest was performed in 0.05 M NaCl solution. Table 1 presents a summary of characteristics of all samples investigated more in detail. The first series served to study the influence of cross-linker dosage amount on the product quality and application performance. The “+” symbol refers to the cross-linker equivalent [10]. Constant average chemical composition was proved by argentometric titration as it is indicated in the second column of Table 1. The intrinsic viscosity extrapolated from Huggins and Schulz-Blaschke plots first increases with branching up to two branches per molecule (++), though is at a minimum for the thrice-branched copolymer (+ + +). Contrarily, for this sample kH and kSB have the highest values of this series indicating reduced solubility. All measurements were performed with solution concentrations < 5 × 10−4 g/ml, i.e. below the overlap concentration of the samples. The instantaneous cross-linker concentration present in the polymerizing batch primarily determines the chain architecture. An appropriate dosage technology, i.e. continuous, is required to produce a homogeneous batch. Dependent on the instantaneous availability of the crosslinker the architecture in the batch may range from linear completely soluble chains to cross-linked less or nonsoluble chains or even compact or gel networks. It is expected that also the chemical composition, i.e. the content of charged monomer units, affects the solubility of the branched molecules. It is well known, and was practiced, that synthetic boundary experiments at low speed may be used to determine particle/compact/impurity portions in nonabsorbing polymer samples by the interference optical system of the AUC [21]. Applying this technique here did

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Table 1 Molecular characteristics and solution parameters of linear and branched acrylamide/acryloyloxyethyltrimethylammonium chloride copolymers (AM/AEAC) obtained from potentiometric titration, dilution viscometry, densitometry, and analytical ultracentrifugation in 0.05 M NaCl at 20 ◦ C Sample

AEAC mol%

[η]H 10−2 ml/g

kH

[η]SB 10−2 ml/g

kSB

ν ml/g

s0 S

ks 10−2 ml/g

ks /[η]SB

B B(+) B(++) B(+++) E(++++) G(++++)

52.6 53.0 51.0 53.9 27 8

19.8 19.6 22.8 17.7 24.6 11.3

0.25 0.26 0.39 0.64 0.33 0.65

20.2 19.9 23.4 18.8 25.2 11.5

0.23 0.20 0.23 0.28 0.22 0.47

0.759 0.775 0.761 0.780 0.760 0.774

7.4 7.3 8.7 7.1 10.3 11.4

27.1 28.5 37.0 17.8 41.7 21.0

1.34 1.44 1.58 0.95 1.66 1.82

not detect any diminution of the fringe plateau after solvent layering at 3000 and 5000 rpm for the linear and lightly-branched samples B and B(+), whereas in particular for the thrice-branched B(+ + +) a fraction started to sediment in the double sector cells already at 3000 rpm, and was observed after layering at 5000 rpm in the synthetic boundary cells. Figure 4 shows corresponding interference scans for synthetic boundary cells. Sample B(+) did not reveal any diminution of the concentration plateau at 5000 rpm and underwent normal sedimentation at 35 000 rpm (Fig. 4a). Sample B(+ + +) behaved differently as it is visible from Fig. 4b, which shows a fast sedimenting fraction at 5000 rpm. After about 10 min the fringe plateau decreased by a bout 0.3 fringes but no depletion at the meniscus occurred. Further, and differently to Fig. 4a, the fast sedimenting fraction, which can be estimated as about 30% of the sample, occupies a considerable volume at the bottom of the solution sector up to about 7.1 cm of the radial position. As it is visible in Fig. 4c, the bottom fraction moves only very slightly at 10 000 rpm and still no meniscus depletion is monitored. In Fig. 5, which shows only this bottom fraction at various velocities in a double sector cell containing a higher amount of polymer solution than the synthetic boundary cell, this fraction even ranges up to about 7.0 cm of the radial position. Comparing all cells the bottom layer thickness corresponded to the initial amount of copolymer in the solution sector of the cells. Further increase of the velocity seems to compress this fraction to 7.1 cm. From such behavior gel-like consistency, rather than a compact cross-linked structure or compact single molecules, may be concluded. At 35 000 rpm the main fraction started to sediment and form a zero plateau at the meniscus position. For the moderately branched B(++) this fraction (data not shown here) was much less pronounced than for the more branched B(+ + +). Employing Sedfit [22] the sedimentation coefficient distributions of both fractions, the fast sedimenting and the main fraction, were calculated. The sedimentation coefficient at zero concentration, s0 , for the main fraction was extrapolated from the apparent sedimentation coefficients

Fig. 4 Interference scans comparing B(+) and B(+ + +). a B(+), c = 4.98 × 10−4 g/ml in 0.05 M NaCl, 35 000 rpm, scan delay 3 min, T = 20 ◦ C. b B(+ + +), c = 4.99 × 10−4 g/ml in 0.05 M NaCl, 5000 rpm, scan delay 1 min, c B(+ + +), c = 4.99 × 10−4 g/ml in 0.05 M NaCl, 10 000 rpm, scan delay 1 min. T = 20 ◦ C

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Fig. 5 Bottom fraction of B(+ + +) monitored at different velocities, c = 4.99 × 10−4 g/ml in 0.05 M NaCl, a 5000 rpm, b 10 000 rpm, c 25 000 rpm, d 35 000 rpm. T = 20 ◦ C. The different fringe values of the plateau are the consequence of no meniscus depletion for velocities below 35 000 rpm

at the maxima of the distribution curves employing Eq. 4. 1 1 1 = + ks c sc s0 s0

(4)

The s0 value of the fast fraction presents only an estimate due to the uncertainty from the broad distribution. All values are listed on Table 1. With values of about 2000 S the fast sedimentation exceeds the main fraction by about factor 200. To monitor the early sedimentation of the heavy fraction required running a very low velocity, preferably at no more than 3000 rpm. Due to insufficient passing of the solvent through the capillary of the synthetic boundary cell at such velocity the fast fraction was better observed in the normal double sector cell. Figure 6 shows the original sedimentation profiles and the corresponding Sedfit evaluation. s0 of the main fraction had the highest value for B(++) and lowest for B(+ + +) and thus corresponds to [η]. The higher kH and kSB of the total for the thrice-branched B(+ + +) may result from the contribution of the crosslinked fraction of about 30%. Influence of the Charge Density in Highly Branched Samples The goal of the syntheses was to produce high/optimum degree of branching but to avoid cross-linking and less soluble fractions, which might not be active in separation processes and other technical applications. With an optimized technology, particularly cross-linker dosage, the highly branched (+ + ++) samples of different composition were produced and characterized. The characterization results of two selected samples are also listed on Table 1. For these samples no fast sedimenting fraction

Fig. 6 Initial sedimentation at 3000 and 5000 rpm monitored in the double sector cells and corresponding Sedfit evaluation, c = 3.97 × 10−4 g/ml in 0.05 M NaCl, scan delay 1 min, T = 20 ◦ C. a 3000 rpm, b 5000 rpm

The Analytical Ultracentrifuge for the Characterization of Polydisperse Polyelectrolytes

Fig. 7 Synthetic boundary experiment to determine the homogeneity of sample G(+ + ++). c = 1.71 × 10−4 g/ml, 5000 rpm, scan delay 5 min, T = 20 ◦ C

147

The ratios are listed in Table 1. For random coils and compact spheres a value of about 1.6 is expected whereas for strongly asymmetric/rod-like molecules values < 1 have been reported [24]. With the exception of B(+ + +) all ratios allow the conclusion of symmetric molecules. Due to polydispersity, heterogeneity, and the specific chain architecture of the samples this information has to be considered as an estimate. Moreover, the ratio was theoretically derived for linear polymers. To which extent it is influenced by other chain architecture and charges was not quantified. Finally, Fig. 9 presents sedimentation coefficient distributions for comparable finite concentrations, which reveal, somewhat surprisingly, relatively narrow distributions of the main fractions in B(++), B(+ + +), and E(+ + ++) though a broader one in G(+ + ++). This can be explained by a high level of AM monomer units (over 90 mol%) in the G-series. For direct comparison the different partial specific volume and ks value of the samples have to be considered. On the basis of the hydrodynamic parameters summarized in Table 1, and employing Eqs. 5–7, the molar mass of all samples was estimated. 
[η] 1/2 3/2 3/2 Msη = (R/ A0 ) [s] (5) 100 1/2

Fig. 8 Plot of the reciprocal of the sedimentation coefficient sc vs. concentration for highly branched samples — — E(+ + ++) and —— G(+ + ++)

•

as for the B series was observed. The sample loss during the synthetic boundary experiment was < 10% and of different quality than for B, not gel-like. The interference patterns of the loss are shown in Fig. 7. Extrapolation to t0 yielded the initial plateau [21]. No substance became visible at the bottom at this time. Compared to all B higher [η] and s0 were obtained for the lower charged E(+ + ++) sample. For the very reduced cationic G(+ + ++) sample despite the lower [η] a higher s0 was extrapolated. Figure 8 shows the sc extrapolations for E(+ + ++) and G(+ + ++), which yield differences of the slope and consequently different sedimentation concentration dependence regression parameters, ks , given in Table 1. G(+ + ++) containing only a low molar fraction of charged monomer units sediments differently than higher charged molecules.

Mks = (N A /βs )3/2 [s]3/2 ks √ Msη = 9πN A 2 [s]3/2 (kSB [η])1/2

(6) (7)

All three equations contain [s] ≡ s0 η0 /(1 − νρ0 ) with s0 the sedimentation coefficient, η0 the viscosity of the solvent, and (1 − νρ0 ) the buoyancy term. It is combined with A0 = 3.4 × 10−10 (g cm2/K · s2 mol1/3 ) the hydrodynamic invariant of Mandelkern-Flory-TsvetkovKlenin, and the intrinsic viscosity in Eq. 5 [25–27], with

Chain Conformation and Molar Mass A useful indication of the possible conformation of the polyelectrolytes in 0.05 M NaCl solution can be estimated from the Wales-van Holde parameter, ks /[η] [23].

Fig. 9 Sedimentation coefficient distributions. From top to the bottom: B(++) c = 4.1 × 10−4 g/ml, B(+ + +) c = 3.79 × 10−4 g/ml, E(+ + ++) c = 4.4 × 10−4 g/ml, G(+ + ++) c = 4.3 × 10−4 g/ml; 35 000 rpm, T = 20 ◦ C

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Conclusions

thesized without significant portions of cross-linked/less soluble fractions. The combination of synthetic boundary and sedimentation velocity experiments performed at discrete velocities enabled determining the homogeneity of the samples, which in some cases consisted of two fractions. Surprisingly, the two fractions identified had extremely different sedimentation coefficients. No continuous broad distribution was monitored, as it is usual for many technical polymers. In all cases the main fraction was relatively narrow distributed. This confirms that despite polymerization up to complete monomer conversion the s0 distribution could be produced narrow as a consequence of a specific dosage regime during the synthesis. The utility of AUC is best demonstrated in the characterization of lower charged, highly branched macromolecules. In such cases, for 27 and 8 mol% polycations, the molar mass increases, which can be anticipated due to higher acrylamide contents, is offset by reduced solubility. Therefore, the AUC points out a tradeoff and practical limit in the degree of branching of cationic polyelectrolytes, which corresponds well to the optima observed in practice. Direct comparison of the samples is limited due to the influence of the heterogeneity in the about 50 mol% cationic B-series, which contributes to [η] and ν but not to s0 of the main fraction. In addition, different chemical composition/charge density is expected to influence the general solution behavior and the type of, for example, the KHKS relationship. It has further to be investigated to which extent the charge electrostatic influence and chain branching superimpose. Therefore, general predictions are still difficult and are limited to the range of the samples studied here. Nevertheless, good correlation between molecular characteristics and application performance concluded thereof was confirmed in application experiments.

Overall, analytical ultracentrifugation was found to provide a variety of details characteristic of technical polydisperse polyelectrolytes, which allow a better correlation with application performance. It was confirmed that highly branched polyelectrolytes of high molar mass can be syn-

Acknowledgements The Swiss National Science Foundation (FNS) and the Innovation Promotion Agency (CTI) is gratefully acknowledged for the financial support (grants, 21-64996.01, 200020103615/1, and 6824.1 IWS-IW9). Special thank is due to Vesela Malinova, Hugues Seitert, and Ricardo Losada for their experimental support, and to Georges Pavlov for his helpful discussion.

Table 2 Molar masses estimated on the base of sedimentation velocity and intrinsic viscosity results Sample

Msη 10−6 [25] g/mol

Mks 10−6 [26] g/mol

Msη 10−6 [27] g/mol

B B(+) B(++) B(+++) E(++++) G(++++)

2.9 3.1 4.0 3.0 5.3 4.6

2.9 3.3 4.5 2.6 6.0 5.4

3.2 3.2 4.5 3.7 5.9 7.4

βs = 1.25 × 107 (mol−1/3 ) the sedimentation parameter and ks in Eq. 6 [28, 29], and, finally, with kSB the SchulzBlaschke constant and the intrinsic viscosity extrapolated from Eq. 2 in Eq. 7 [30]. R is the gas constant and N A is Avogadro’s number. A0 and βs are common values for flexible macromolecules [25–29]. Table 2 presents the results of the calculations. The values are in accordance with the qualitative predictions from the copolymer synthesis. The molar masses of all three calculations reveal good agreement for the more homogeneous samples B, B(+), B(++), and E(+ + +). Contrarily, for the samples, which have been identified to contain a crosslinked fraction, B(+ + +), or to behave hydrodynamically different, G(+ + ++), the deviations of the values are more pronounced. From the calculations may be also concluded, due to the similar values obtained, that all three equations are likewise suited to estimate the molar mass of cationic copolymers investigated here on the basis of sedimentation velocity experiments and viscosity measurements.

References 1. Nalco Water Handbook (1988) Kemmer FN (ed) 2nd Ed, McGraw-Hill 2. Chen CY (1993) Water Sci Technol 28:1–7 3. Hocking MB, Klimchuk KA, Lowen S (1999) J Macromol Sci-Rev Macromol Chem Phys C39:177–203

4. Mortimer DA (1991) Polym Internat 25:29–41 5. Ayol A, Dentel SK, Filibeli A (2005) J Environ Eng-ASCE 131:1132–1138 6. Rebitzer G (2005) Doctoral Thesis EPFL 7. Farinato R, Hawkins P (1988) US Pat 5,807,489, Sept 15

8. Farinato R (1999) US Pat 5,879,564 March 9 9. Ulrich S, Laguecir A, Stoll S (2004) J Nanoparticle Res 6:595–603 10. Hernandez Barajas J, Hunkeler D, Wandrey C (2004) Polym News 29:239–246

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11. Hernandez Barajas J, Wandrey C, Hunkeler D (2001) US Pat 6,294,622 B1, Sept 25, (2003) US Pat 6,617,402 B2, Sept 9, (2003) US Par 6.667,374 B1, Dec 23 12. Hernandez Barajas J, Hunkeler D (1997) Polymer 38:437 13. Huggins ML (1942) J Am Chem Soc 64:2716 14. Schulz GV, Blaschke F (1941) J Prakt Chem 158:130, ibid (1941) 159:146 15. Wandrey C, Bartkowiak A, Hunkeler D (1999) Langmuir 15:4062–4068 16. Mabire F, Audebert R, Quivoron C (1984) Polymer 25:1317

17. Griebel T, Kulicke WM (1992) Makromol Chem 193:811 18. Griebel T, Kulicke WM, Hashemzadeh (1991) Colloid Polym Sci 269:113 19. Wandrey C, Görnitz E (1995) Polym News 20:377–384 20. Wandrey C (1997) Polyelektrolyte-Makromolekulare Parameter und Elektrolytverhalten. Cuvillier, Göttingen 21. Wandrey C, Görnitz E (1992) Acta Polym 43:320–326 22. Schuck P (2000) Biophys J 78:1606 23. Wales M, Van Holde KE (1954) J Polym Sci 14:81–86

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24. Harding SE, Berth G, Ball A, Michell JR, Garcia de la Torre J (1991) 16:1–15 25. Mandelkern L, Flory P (1952) J Chem Phays 20:212 26. Tsvetkov V, Klenin S (1953) Dokl Akad Nauk SSSR 88:49 27. Tsvetkov VN (1989) Rigid-chain polymers. Consultants Bureau, New York 28. Pavlov GM, Frenkel SYa (1995) Progr Colloid Polym Sci 99:101 29. Pavlov GM (1997) Europ Biophys J 25:385 30. Linow KJ, Philipp B (1968) Faserforsch Textiltech 19:509

Progr Colloid Polym Sci (2006) 131: 150–157 DOI 10.1007/2882_019 © Springer-Verlag Berlin Heidelberg 2006 Published online: 15 February 2006

Laurent Bourdillon Ruth Freitag Christine Wandrey

Laurent Bourdillon · Ruth Freitag · Christine Wandrey (u) Institut des Sciences et Ingénierie Chimiques, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland e-mail: [email protected] Ruth Freitag University of Bayreuth, Chair for Process Biotechnology, 95440 Bayreuth, Germany

PARTICLES, POLYMERS AND INTERACTING SYSTEMS

Association and Temperature-induced Phase Transition Studied by Analytical Ultracentrifugation

Abstract Sedimentation velocity studies revealed concentration and temperature dependent aggregation of stimuli-responsive pre-associated polymers. The sedimentation coefficient monotonically increases with concentration and temperature while confirming pre-association prior to the phase separation. The aggregation process was proved as completely reversible by monitoring reproducible concentration signals before aggregation and after re-dissolution. Moreover, absorbance concentration profiles identified a temperature dependent portion of polymer not

Introduction Stimulus-responsive materials, in particular such based on synthetic polymers, become increasingly interesting for a variety of applications including drug delivery, tissue engineering, and surface design [1–3]. Stimulus-responsive polymers are able to change their properties, in particular the molecular conformation, in response to small changes in external parameters, for example, the temperature, the pH, the ionic strength, or solvent quality. Thereby, the property change can be abrupt or gradually. Also response to radiation and magnetic fields is known. Especially interesting are materials with completely reversible property changes. The most investigated and reported polymer is poly (N-isopropylacryl-amide) (PNIPAM) [4], a temperatureresponsive polymer, for which a lower critical solution temperature (LCST) of about 31 ◦ C in water was reported with a critical solution temperature remaining below 35 ◦ C over a wide range of PNIPAM concentrations. The phase

participating in the aggregation process. Sedimentation equilibrium experiments allowed estimating the concentration influence on the molar mass. The study demonstrates for thermo-responsive polymers the suitability of analytical ultracentrifugation to analyze details of the association/aggregation process of stimuli-responsive polymers. Keywords Aggregation · Analytical ultracentrifugation · Molecule association · Phase separation · Stimuli-responsive polymers

separation was concluded to take place by association of the polymer molecules into larger aggregates formed by intermolecular hydrogen bonding and nonpolar bonds. Alternatively, this phenomenon was also ascribed to the fact that the polymer is more ordered in dilute solution than in the concentrated phase and that this ordering is due to the relative strong hydrogen bonds formed between water and the polymer. As the temperature is raised, these hydrogen bonds become weaker and the solution becomes unstable. Also simultaneous action of both mechanisms was not excluded to have an influence on the observation of an LCST for PNIPAM [5]. To shift the critical solution temperature, on the one hand, the medium conditions can be modified, on the other hand, modification of the polymer backbone by chemical reaction or copolymerization may serve the same purpose. A particular situation exists when into the modified polymer double functionality was introduced such as in the case of photo-responsive N-isopropylacrylamide cotelomers, which contain after copolymerization a certain

Association and Temperature-induced Phase Transition Studied by Analytical Ultracentrifugation

molar percentage of pendent light-responsive azobenzene groups [6]. The azo group in such copolymers can exist in the trans and cis configuration. The trans configuration is thermodynamically stable. Illumination at 305 nm reversibly switches the molecule into the cis. The back isomerization occurs spontaneously albeit slowly (t1/2 about 13 h) in the dark or within minutes after illumination with visible light. The trans configuration is more hydrophobic than the cis one and, therefore, the temperature-sensitivity of the corresponding PNIPAM copolymers can be influenced by illumination at the different wavelength. Differences in the critical temperature of up to 2 ◦ C have been observed between copolymers with azo groups predominantly in the trans or the cis configuration. Moreover, such azo groups have previously been shown to give rise to molecular interaction (“stacking”) [7]. Such an effect has previously been postulated to explain peculiarities in the solubility and phase separation of light-responsive cotelomers. An appropriate mechanism was proposed though not yet confirmed [6]. Various analytical techniques have been used to study and identify the phase transition of stimulus-responsive polymers including turbidity measurements, differential scanning calorimetry (DSC), light scattering (LS), viscometry, various spectroscopic methods (fluorescence, NMR, IR), electron microscopy (EM), surface tension measurements, and analytical ultracentrifugation (AUC). A long list of references could be added but will be abandoned here. For such a list it is referred to [8]. Surprisingly, the latter, AUC, was rarely used despite offering a spectrum of possibilities to study molecule and particle size and their distributions over a wide range of experimental conditions. In the past, AUC served to determine for PNIPAM sedimentation coefficients as a function of the temperature [5], to study the properties of novel ionic NIPAM copolymers [9], and, only recently, to monitor complexation of PNIPAM with poly(methacrylic acid) [10]. Therefore, it is the intention of this paper to discover the possibilities of analytical ultracentrifugation for analyzing many details of the association/aggregation/precipitation process of complex stimuli-responsive polymer systems rather than to investigate just the phase transition of a specific stimuli-responsive polymer. The study addresses the concentration dependence of the temperature-induced aggregation process, the mass of the associates/aggregates, the reversibility of the separation process, and the fraction of molecules, which remains in solution under various conditions.

Experimental Material Cotelomers of N-isopropylacrylamide (NIPAM) and 4-(phenylazo)phenylacrylamide (PAPAM), telo(NIPAMco-PAPAM), were synthesized by free radical chain trans-

151

Fig. 1 Chemical structure of the cotelomer of N-isopropylacrylamide (NIPAM) and 4-(phenylazo)phenylacrylamide (PAPAM), telo(NIPAM-co-PAPAM)

fer copolymerization as has been described recently [6, 8]. The general chemical structure is shown in Fig. 1. Aqueous copolymer solutions were prepared in the concentration range of 0.16 wt % to 0.25 wt % at temperatures below the critical solution temperature and adjusted to pH 4. Methods The association and aggregation behavior of the cotelomer as a function of the temperature were investigated by sedimentation velocity experiments. All concentration profiles were detected using both optical systems (UV/Vis optics and Rayleigh interference optics) of an OPTIMA XLI analytical ultracentrifuge (Beckman, Palo Alto, USA). Specifically, in the temperature range of 25 to 33 ◦ C the velocity was varied between 2200 and 10 000 rpm while detecting, simultaneously, interference fringes and absorbance at 340 and 600 nm. SEDFIT [11] and SEDANAL [12] were used to evaluate the concentration profiles of interference and absorbance scans at 600 nm. A minimum of 20 scans was monitored with a delay between 1 and 10 min. The polymer solutions were placed in double sector cells and stored at 4 ◦ C. Before starting the AUC run, the appropriate temperature was adjusted in the AUC. The same cell filling was used for experiments at different temperatures after re-dissolving the polymer at low temperature. Sedimentation equilibrium was studied at 27 ◦ C. A sixchannel centerpiece with an optical path length of 12 mm was filled with 100 µl of the same solutions as used for the

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Table 1 Summary of experiments Experiment

1 2 3 4 5 6 7 8 9 10 11 12

◦C

Temperature

Run velocity rpm

Polymer/conc. Nr./wt%

Scan delay (comment) min

30.8 27 29 31 33 33 25 27 29 31 33 27

3000 3000/5000 3000 3000 3000 2500 10 000 5000 5000 3500 2200 1000

1/0.25 1/0.25 1/0.25 1/0.25 1/0.25 1/0.25 2/0.16, 2/0.16, 2/0.16, 2/0.16, 2/0.16, 2/0.16,

3 10/3 6 4 2 (run too fast) 1 (I only) 5 (I and 600 nm only) 4 4/7 2/5 (I and 600 nm only) 3 (I only) equilibrium, 46 h

0.2, 0.25 0.2, 0.25 0.2, 0.25 0.2, 0.25 0.2, 0.25 0.2, 0.25

sedimentation velocity experiments. The rotor speed was 1000 rpm for 46 h. The equilibrium concentration profiles were monitored by both optical systems of the AUC. SEDFIT [11] and a procedure described in [12] were employed for data evaluation. Table 1 lists the conditions for all experiments performed.

imum of the sedimentation coefficient distribution curves calculated using SEDFIT, plots 1–5 in Figs. 3a and b, and plots 1–4 in Fig. 3c. For c = 0.25% at T = 33 ◦ C no sedimentation coefficient was extracted due to a very broad and heterogeneous distribution curve ranging up to

Results and Discussion Influence of Temperature and Concentration Figure 3 shows the temperature dependent sedimentation coefficient distributions for three concentrations of telo(NIPAM-co-PAPAM), for which the critical solution temperature was reported in the range of 29 to 32 ◦ C [8]. It has to be noted, that different experimental methods delivered differing phase transition temperatures. In the case of the polymer type studied here differences up to 5 ◦ C were reported comparing results from turbidity measurements and microcalorimetry [6]. The influence of both experimental parameters is clearly visible in Fig. 3. A considerably stronger temperature effect is observed at higher polymer concentration (Fig. 3c). Due to the differences of the sedimentation coefficients the experiments were carried out varying the run velocity between 2200 rpm, for 33 ◦ C, and 10 000 rpm for 25 ◦ C. Figure 2 demonstrates the reliability of the data evaluation for the highly polydisperse system by evaluating the same data set with SEDFIT and SEDANAL as well. Very similar distribution curves were obtained. The concentration and temperature dependencies are explicitly presented in Fig. 4. The apparent sedimentation coefficients plotted there were taken from the max-

Fig. 2 Evaluation of sedimentation velocity data by SEDFIT [11], ls − g∗ (s), and SEDANAL [12], g(s∗ ) (Experiment 3)

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153

Fig. 4 Influence of temperature and concentration on the apparent sedimentation coefficient sapp . a sapp = f(T) for three different concentrations • 0.16 wt %,
0.2 wt %,  0.25 wt %; b sapp = f(c) ◦ 25 ◦ C,  27 ◦ C, ♦ 29 ◦ C,  31 ◦ C,  33 ◦ C

Fig. 3 Apparent sedimentation coefficient distributions ls − g∗ (s) for different polymer concentrations and temperatures. Concentrations: a c = 0.16 wt %, b c = 0.20 wt %, c c = 0.25 wt %. Temperature curves: 1 25 ◦ C, 2 27 ◦ C, 3 29 ◦ C, 4 31 ◦ C, 5 33 ◦ C. Run velocity: 10 000 rpm for 1 in a–c; 5000 rpm for 2 and 3 in a–c; 3500 rpm for 4 in a–c; 2200 rpm for 5 in a and b

about 14 000 S. The applicability of the evaluation is confirmed by acceptable fitting (rms < 0.05) also visualized in Fig. 2 [11]. Monotonic dependency is obvious for all conditions. Whereas at 25 ◦ C only slightly higher apparent sedimentation coefficient are observed with increasing concentration, the differences become much more pronounced if the temperature increases. The increase of the apparent

sedimentation coefficient with the polymer concentration is contradictory to the behavior of single molecules for which, in general, the apparent sedimentation coefficients increase with dilution. Moreover, the values of the apparent sedimentation coefficient at 25 ◦ C, below the critical solution temperature are much too high for single cotelomer molecules, for which molar masses below 5000 g/mol have been reported [8]. From such high apparent sedimentation coefficients large molecule associates/clusters have to be concluded to be present in the clear solution below the critical solution temperature. The formation of molecule associates due to interpolymeric interaction of the hydrophobic azobenzene groups was suggested and presented as mechanistic possibility in the precipitation of cotelomers recently [6], and may serve as an explanation. Though the values of the apparent sedimentation coefficients confirm large clusters rather than dimers or assemblies of only a few molecules. Further-

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more, higher molecule density in the solutions seems to be favorable for clustering.

Reversibility and Reproducibility of the Association/Aggregation Monitoring concentration signals before aggregation and after re-dissolution proved the reproducibility of the association and precipitation process. In the case of precipitation the polymer precipitate was visible on the bottom of the solution sector in the double sector cell after removing the cell from the rotor. Shaking the cell first yielded turbidity but after cooling a clear solution was recovered for which a homogeneous fringe signal high over the radius was obtained at the beginning of the next velocity run indicating a homogeneous system. Most importantly, entire reproducibility was proved repeating one run at 31 ◦ C after having performed two runs at 27 ◦ C and 29 ◦ C with the same sample in between. In this case good reproducibility was proved for the interference scan, the absorbance scans at 340 nm and 600 nm as well as the calculated apparent sedimentation coefficient (see data in Fig. 6). The scans at 340 nm exceeded two absorbance units and have, therefore, not directly been considered for the reproducibility proof. Instead, they served to identify the residual portion of molecules, which remained in solution and did not participate in the precipitation under the experimental conditions studied here. Nevertheless, for all concentrations, the signal heights monotonically increased with the temperature for the interference and absorbance scans at 600 nm.

Polymer Fraction Participating in the Aggregation/Precipitation The telo(NIPAM-co-PAPAM) has its maximum absorbance at about 340 nm [6, 8]. While the interference scan and the absorbance at 600 nm detect complete sedimentation in Fig. 5 the signal at 340 nm does not approach zero at the meniscus position during the sedimentation process. This absorbance signal is assumed to identifying molecules, which did not participate in the sedimentation. Contrarily to the intensity of the interference and the absorbance at 600 nm the absorbance intensity at 340 nm decreased with increasing temperature. It correlates with the concentration while being higher for higher concentrations. The assumption of a temperature dependent fraction of cotelomer remaining in solution is supported by the observation that the intensity of the associating/precipitating fraction increases whereas the intensity of the signal related to the molecules remaining in solution decreases. Figure 6 presents a summary of the signal height changes as well as the temperature dependent residual fraction remaining in solution indicated by the residual

Fig. 5 Sedimentation velocity experiment at 29 ◦ C, 3000 rpm, c = 0.25 wt %. a Interference scans, b absorbance scans at 600 nm, c absorbance scans at 340 nm. Scan delay 6 min (Experiment 3)

absorbance at 340 nm. The latter is restricted to one concentration. To avoid too long scan delays, which would result when three cells at two wavelengths have to be scanned over the entire radial distance, the experiments 1–6 were restricted to one concentration. No additional information was expected varying the concen-

Association and Temperature-induced Phase Transition Studied by Analytical Ultracentrifugation

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itative trends are obvious in Fig. 6 without quantitative information.

Estimation of the Mass of Associates/Aggregates The mass of the associates/aggregates was estimated below the phase transition temperature from sedimentation equilibrium experiments at 27 ◦ C evaluating equilibrium distributions obtained after 46 h. Due to the polydispersity of the system the “low-speed” or “non-meniscus depletion” equilibrium technique was employed, which was reported to be more sensitive than the meniscus depletion method in the cases of highly polydisperse or associating systems [13, 14]. Figure 7 presents molar mass distributions for the same three concentrations studied in velocity experiments. The concentration influence is clearly visible indicating a considerable shift of the curve maxima to higher molar masses with the concentration while also the polydispersity increases. In addition to the data evaluation by SEDFIT, a procedure described in [15, 16] was used to calculate the apparent weight- and z-average of the molar masses, Mw, app and Mz,app . For this purpose the distribution curves were normalized over the radius with the meniscus (x = 0) and bottom (x = 1) [17, 18]. The resulting c(x)-profiles were subsequently fitted by a sum of positive exponential functions with n = 3 [16] c(x) =

n


ki exp(ai x)

with ai > 0 .

(1)

i=1

The molar masses were the calculated as  1 cb − cm 1 ki (exp ai − 1) = Mw,app = λ c0 λ c0

(2)

Fig. 6 Temperature-dependent signal changes • 0.16 wt %,
0.2 wt %,  0.25 wt %, ♦ 0.25 wt %; (full symbols Experiments 7– 11, empty symbols Experiments 1–6): a increase of the interference signal; b increase of the absorbance signal at 600 nm; c decrease of the absorbance signal at 340 nm indicating the molecular fraction

tration. For the highest concentration c = 0.25 wt % the absorbance at 600 nm exceeded two absorbance units at 31 ◦ C. The quantitative evaluation of the solution fractions requires a concentration calibration of the absorbance signal at 340 nm. Nevertheless, valuable qual-

Fig. 7 Molar mass distribution obtained from sedimentation equilibrium using SEDFIT [11]. Concentrations (wt%): broken line 0.16, dotted line 0.2, and solid line 0.25

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and

Table 2 Molar masses calculated from equilibrium concentration profiles with ν = 0.7 ml/g

1 cb ( d(ln c)/ dx)b − cm ( d(ln c)/ dx)m λ cb − cm 1 ki ai (exp ai − 1)  = λ ki (exp ai − 1)

Mz,app =

(3)

with

  2 /RT λ = (1 − νρ0 ) ω2 rb2 − rm

(4)

(ρ0 : density of the solvent, ω: rotor speed in rad/s, R: gas constant, T : Kelvin temperature). Figure 8 shows the very good fit results with fits exactly matching the experimental curves. The quality of the fits is confirmed by the very low χ values. The molar masses calculated for the three concentrations are summarized in Table 2.

Concentration wt%

Mw,app (×10−6 )

Mz,app (×10−6 )

χ2

0.16 0.20 0.25

15.1 10.8 5.2

54.4 64.2 106.5

1.69 × 10−6 2.08 × 10−6 7.66 × 10−7

For Mw,app and Mz,app an increase is expected with the concentration considering the results from velocity experiments. However, this is confirmed only for Mz,app , for which the concentration dependent values are very close to the appropriate curve maxima in Figure 7. Contrary, Mw,app decreases with increasing concentration. Similar findings were reported for constant concentration but increasing the temperature when other copolymers of NIPAM were investigated by sedimentation equilibrium [9]. The progressive formation of aggregates at higher temperature was provided as explanation there, and may also be the reason here. The calculation of Mz,app , which depends on the concentration only in terms of the differences between cell bottom and meniscus concentration, cb and cm , and on the slope of the radial concentration distribution (Eq. 3), is less sensitive to errors in the cell loading concentration. For other NIPAM copolymers with a phase transition temperature of 32 ◦ C weight average molar masses obtained from light scattering have been reported, which covered a range of four orders of magnitude, 106 to 1010 , when increasing the temperature from 27 to 33 ◦ C. Though already at 28 ◦ C Mw increased to 108 [9]. Surprisingly, the radial concentration curves overlap at the cell bottom position and terminate at approximately the same concentration. It can be speculated that at a sufficiently high concentration aggregation occurs. An indication for this may be concluded from the equilibrium absorbance scans at 600 and 340 nm (not shown here). Both detect a thin bottom fraction, which is difficult to see from the interference optics. This implies the coexistence of three fractions, soluble molecules homogeneously distributed over the radius, associated/clustered molecules detected by equilibrium distribution, and aggregates collected at the bottom. Nevertheless, from all the presented experimental findings it may be concluded that all three are in a dynamic equilibrium.

Conclusions Fig. 8 Normalized equilibrium concentration profiles from a 6channel experiment matched by fit curves (Experiment 12). Top: calculation of Mw,app , bottom: calculation of Mz,app

Analytical ultracentrifugation was proven as a valuable technique to study trends and details of the temperatureinduced association/aggregation/ precipitation of temperature-responsive NIPAM-copolymers possessing addi-

Association and Temperature-induced Phase Transition Studied by Analytical Ultracentrifugation

tional photo-responsive functionality by UV/Vis absorbing groups in the comonomer unit. Despite not being suited to determine precisely the critical solution temperature or the critical solution concentration, molecular dimensions and their changes upon variation of external parameters are accessible. Advantages are predominantly expected for more complicated molecules such as those used here to demonstrate principal applications. Information from AUC experiments is not restricted to temperature response. Likewise preparations at different pH or in solvent mixtures can be investigated. Even photoinduction appears to be possible directly in the AUC applying the appropriate wavelength. The separation of molecules from the solution during the sedimentation experiment might be evaluated as disadvantageous compared

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to light scattering, which also provides molecular dimensions. However, higher sensitivity and multiple use of the same cell filling compensate this putative drawback. As a final conclusion, a comprehensive analysis of phase transition processes of polymers will always require the combination of various sophisticated techniques. Analytical ultracentrifugation should be more considered as one of these. Acknowledgement The Swiss National Science Foundation is gratefully acknowledged for the financial support (Grants 2164996.01 and 200020-103615/1). Special thank is due to Arnaud Desponds for preparing the polymer samples and technical assistance, and Eckhard Görnitz for providing the program to fit the equilibrium distribution curves. Helmut Cölfen and Peter Schuck is acknowledged for helpful discussion.

References 1. Hoffman AS (1997) In: Park K (ed) Controlled drug delivery. ACS, Washington DC, p 485 2. Galaev IY, Mattiasson B (1999) Trends Biotechnol 17:335 3. Chen GH, Hoffman AS (1995) Nature 373:49 4. Schild H (1992) Progr Polym Sci 7:163 5. Heskins M, Guillet JE (1968) J Macromol Sci-Chem A2:1441 6. Desponds A, Freitag R (2003) Langmuir 19:6261 7. Hilbrig F, Freitag R (2003) J Chromatogr B 790:79

8. Desponds A (2003) Doctoral Thesis EPFL: 2737 9. Hahn M, Görnitz E, Dautzenberg H (1998) Macromolecules 31:5616 10. Burova TV, Grinberg NV, Grinberg VY, Lalinina EV, Lozinsky VI, Aseyev VO, Holappa S, Tenhu H, Khokhlov AR (2005) Macromolecules 38:1292 11. Schuck P (2000) Biophys J 78:1606 12. Stafford W (1992) Anal Biochem 203:295; http://www.rasmb.bbri.org/ rasmb/ms_dos/sedanal-stafford/ 13. Creeth JM, Harding SE (1982) J Biochem Biophys Methods 7:25

14. Harding SE, Vårum KM, Stokke BT, Smidsrød O (1991) Adv Carbohydr Analysis 1:3 15. Wandey C, Görnitz E (1992) Acta Polymer 43:320 16. Görnitz E, Hahn M, Jaeger W, Dautzenberg H (1997) Progr Colloid Polym Sci 107:127 17. Fujita H (1975) Foundations of Ultracentrifugal Analysis. Wiley, New York 18. Lechner MD, Mächtle W (1992) Macromol Chem Macromol Symp 123/124:85

Progr Colloid Polym Sci (2005) 000: 158–164 DOI 10.1007/2882_020 © Springer-Verlag Berlin Heidelberg 2006 Published online: 17 February 2006

Peter N. Lavrenko Elena V. Belyaeva Darya M. Volokhova Olga V. Okatova

Peter N. Lavrenko (u) · Elena V. Belyaeva · Darya M. Volokhova · Olga V. Okatova Institute of Macromolecular Compounds, Russian Academy of Sciences, 199004 St. Petersburg, Russia e-mail: [email protected]

PARTICLES, POLYMERS AND INTERACTING SYSTEMS

Aggregation of Dibutyl Phthalate Molecules in Decalin Solutions Evidenced by Hydrodynamic and Optical Measurements

Abstract Hydrodynamic and optical investigations were performed on the binary dibutyl phthalate (DBP)-decalin liquid mixtures with the DBP content varying from 0 to 100%. This system may be suitable for modelling of the lowmolecular-weight polymer behaviour in solution. Macro-characteristics of the system, such as medium viscosity and diffusivity, were evaluated in addition to the molar refraction and the flow birefringence of the system. A non-monotonous function of the

Introduction Investigations of the polymer molecule properties by the classical methods of molecular hydrodynamic and optics, as carried out in dilute solutions, assume usually an additivity of the appropriate effects of the solvent and the high molecular weight compound [1]. The solvent is considered as a continuous homogeneous dispersion media. However, this is only a rough approximation in the investigation of oligomers or low molecular weight polymers which, being liquid, can be studied over a wide concentration range [2]. Here, a binary liquid mixture of the organic substances can be used as a reasonable model system. This is particularly true for di-n-butyl phthalate (DBP), C6 H4 (COOC4 H9 )2 . The presence of the two butyl arms in the DBP molecule permits us to consider this compound as an intermediate between the solid non-deformable particles of the individual compounds and the polymer chain molecules. The second component, decalin, C10 H18 , was here chosen because of different but high enough own viscosity and optical anisotropy of the molecules.

molecular diffusivity of the mixture components was revealed. The lowspeed ultracentrifugation data yield the dimensions of the mobile units varying with the DBP molar fraction in the mixture. The characteristics of the extreme composition are evaluated and compared with those known for other systems. Keywords Additivity · Dibutyl phthalate-decalin mixture · Molecular diffusivity · Optical anisotropy · Physical properties · Refraction

There are series of works on fluidity and molecular mobility in the mixtures of various liquids based on data on mobility of the isotopic labels and on spin echo effects in the magnetic field with an impulse gradient [3–5]. However, in the latter case, the results can depend on the duration of the diffusion recording time [6]. Therefore, in the present work, additivity of the physical properties of the binary mixture of DBP and decalin has been investigated by the methods of molecular physics. Results of the low-speed ultracentrifugation, free diffusion, viscometry, flow birefringence, and refractometry methods are compared with the available literature data. The question discussed in this work can also arise in investigations of a polymer dissolved in the mixed binary solvents.

Experimental Materials and Solutions. Decalin of chemical pure grade (with cis-trans-form ratio of 0.7 : 0.3 as obtained from

Aggregation of Dibutyl Phthalate Molecules

APOLDA, Germany) and di-n-butyl phthalate (DBP) of technical grade were used as received. Density, viscosity, and refractive index of decalin at 24 ◦ C were −2 g/cm s, and n = ρ024 = 0.883 g/cm3 , η24 D 0 = 2.386 × 10 1.4768, respectively. For neat DBP the values were ρ024 = −2 g/cm s, and n = 1.040 g/cm3 , η24 D 0 = 16.398 × 10 1.4903. Mixtures were prepared at ambient temperature by stirring for a few hours. Methods. The hydrodynamic properties of a DBP-decalin mixture were studied at 24 ◦ C with varying the DBP weight fraction from 0 to 1. Refractive index was measured using IRF 454B refractometer (Russia). A liquid density was evaluated with a pycnometer. Low-speed analytical ultracentrifugation experiments were performed with an 3180 analytical ultracentrifuge from the Hungarian Optical Works MOM equipped with earlier described interference optics [7]. The rotor rotation frequency was 10 × 103 rpm with an accuracy of ±20 rpm. The temperature of the rotor (24 ◦ C) was maintained with an absolute accuracy of ±0.1 ◦ C and with a constancy of regulation of ±0.05 ◦ C. A synthetic boundary cell with a light path h (1.2 cm) and quartz windows were used. The double-sector polyamide centerpieces and gaskets have a good chemical compatibility with both of the liquids under investigation. A few long-time experiments were also performed also using the standard convection diffusion device [1]. Both the methods were equivalent in the results. The pictures were recorded with the Tsvetkov interference detector system [7] operating at wavelength λ = 546.1 nm with spar twinning a = 0.0209 cm and compensator fringes’ interval b = 0.0966 cm. A concentration boundary was formed between the two mixtures with different DBP concentrations, c1 and c2 , keeping the same difference, ∆c ≡ c1 − c2 = 5 wt %, for mixtures of various DBP content. The results were related to the average DBP concentration, c¯ ≡ (c1 + c2 )/2. The refractive increment, dn/ dc, was estimated from the interference curve area, Q, by dn/ dc = (λ/abh)Q/∆c. Viscosity measurements were performed in a dilute regime (c < 1/[η], where [η] is the intrinsic viscosity) using an Ostwald viscometer with an average velocity gradient of 14 s−1 (DBP) and 82 s−1 (decalin) in a water bath at 24.0 ± 0.01 ◦ C. The kinetic energy correction was negligible. The flow birefringence experiments were performed at 21 ◦ C in a co-axial cylindrical titanium dynamo-optimeter with an inner rotor radius, r, of 1.5 cm, length, l, of 4.0 cm in the light beam direction, and with a gap, ∆r, of 0.035 cm. One device was used for investigations of all the mixtures and the individual compounds. The measurements were carried out with varying the flow rate gradient in the gap, g, (g = 2πνr/∆r), through the variation of the rotor revolution frequency, ν(s −1 ), within the laminar regime (ν < νlim ). Here νlim is the upper gradient limit for

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a laminar flow in a liquid with viscosity η and density ρ defined by νlim = 41.3(r/∆r 5 )1/2 × (η/ρ). Birefringence ∆n induced in the flow shear field was measured by the visual compensation optical system [1] with the penumbral plate. A thin mica plate with an optical path difference, ∆λ/λ, of 0.038 (λ is the light wave length) was used as a Brice-type compensator. The ∆n was calculated using ∆n = (∆λ/l) sin 2θ, where θ is the azimuth angle of the compensator as fixed at given g. The light source was a Hg-lamp (λ = 546.1 nm). The ∆n value thus obtained was related to the effective shear stress, ∆τ(∆τ = gη).

Results and Discussion Experimental Data Fairly large and positive (in sign) birefringence ∆n was observed in the DBP-decalin mixture subjected to the flow field. Independently of the DBP fraction, ∆n was found to be a linear function of the shear rate g (Fig. 1) that is a typical result for a real solution. This permitted us to characterize the effect with the relation ∆n/g. For the individual liquids, decalin and DBP, as illustrated by points 6 and 1 in Fig. 1, we obtained ∆n/g = (0.6 ± 0.1) × 10−12 and (22.5 ± 0.5) × 10−12 s, respectively, which are in a good accordance with the literature data [1]. Figure 2 presents the physical properties of the DBPdecalin system of various compositions as a function of the DBP weight fraction, w, in terms of the specific values of density (1), refractive index (2), refractive increment (5), Maxwell constant (3), viscosity (4), and the mutual diffusion coefficient (6). So, specific density of the mixture was calculated by (ρ/ρw = 0 ) − 1, where ρ and ρw = 0 is a density of the mixture and neat decalin, respectively. Similar expressions were used for the other parameters represented

Fig. 1 Flow birefringence of liquid mixture DBP-decalin, ∆n, as a function of flow rate gradient, g. The DBP weight fraction w = (1) 1, (2) 0.75, (3) 0.50, (4) 0.25, (5) 0.10, and (6) 0. The lines represent a best linear fit

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Fig. 2 The specific values of density ρ (curve 1, ×250), refractive index n (curve 2, ×4000), Maxwell constant ∆n/g (curve 3), viscosity η (curve 4, ×5), refractive index increment dn/ dc581 (points 5, ×5), and the mutual diffusion coefficient D (smoothly connected points 6, ×10) in DBP-decalin system at 24 ◦ C as a function of the DBP weight fraction

in Fig. 2 in individual scale (the magnification coefficient being given in the figure legend). Figure 3 illustrates typical interference curves as obtained during two diffusion processes with the minimum and maximum values of the DBP content in the DBPdecalin mixture. An average, the DBP weight fraction in the two touched liquids is here w = 0.025 (a) and 0.975 (b). Easy to see a faster spreading of the a curves with time in comparison with the b curves. This reflects different diffusion mobility (diffusivity) of the kinetic units in these mixtures. Additionally, the same concentration jump over the boundary range (∆w = 0.05) was especially provided in both experiments. However, one can see the lower area of the a curves in comparison to the b curves. This is explained by non-additivity of refractive indexes of the mixture components in contrast to the molar refraction of a mixture, which is the additive value (as discussed below). The interference curves in Fig. 3 are sufficiently symmetric. They were, therefore, treated in a Gaussian approximation with the “high-area” method [1, 7]. The dispersion, σ 2 , of the ∂c/∂r(r) distribution was evaluated by σ 2 = (a2 /8)[argerf(aH/Q)]−2, where H is the maximum ordinate of the curve, r the radial distance, and argerf means an argument of the probability integral. Results are presented in Fig. 4, where σ 2 is plotted against reduced time, t/η (η being an average viscosity of the two touched liquids). The slope of the function depends obviously on the DBP average concentration. It is interesting to notice the non-monotonous behavior here: points 3 and 4 fall lower than points 5 meaning lower diffusivity in the mixtures at intermediate the DBP content. The time-dependence of σ 2 was well approximated by linear function that is typical for an ensemble of homogeneous particles responsible for the mass transport in a diffusion experiment. The mutual diffusion coefficient, D,

Fig. 3 Profile of the boundary between solution (left) and solvent (right) and its change in time (from left to right) during the mutual free diffusion of the DBP and decalin molecules. Here δ is the displacement of interference fringe, b the fringe interval, and r the radial distance. The average DBP concentration in the mixture c = 2.50 (curves a) and 97.5 wt % (b). In both the experiments, the concentration difference ∆c = 5 wt %. The overall scan time is (a) 24 and (b) 60 min. Every second curve is shifted to the right in proportion to the recording time

Fig. 4 Dispersion of concentration boundary σ 2 vs reduced time, t/η (η being viscosity of the liquid) for DBP in decalin solution at 24 ◦ C. The average concentration of DBP in the mixture is (1) 2.50, (2) 25, (3) 50, (4) 75, and 97.5 wt % (5). The lines represent a best linear fit

Aggregation of Dibutyl Phthalate Molecules

was then calculated by D = (1/2)σ/σt. Lamm correction for the field effect was negligible as estimated earlier [2]. Thus, for DBP in decalin solution (w → 0) we have obtained D = (30 ± 1) × 10−7 cm2 /s, which is comparable with the literature data. In particular, for α-bromonaphthaline, bromo-benzene, and carbon tetrachloride in decalin solution, at the same temperature, the D × 107 value of 44, 61, and 74 cm2 /s are available [8, 9]. In turn, for decalin in DBP solution (w → 1) we have D = (10.0 ± 0.5) × 10−7 cm2 /s. Theoretically, the mutual diffusion coefficients, D, in very dilute solutions coincide with the self-diffusion coefficients, D∗ , of the individual components in the DBPdecalin mixture, ∗ DDBP (wDBP → 0) = lim D

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Fig. 5 Concentration dependence of the reduced viscosity ηsp /c for (1) DBP in decalin solution and for (2) decalin in DBP solution. The lines represent a best linear fit

w→0

∗ Ddec (wdec → 0) = lim D , w→1

where wi is the fraction of the i-component in the mixture. However, within the extrapolation range, the D∗ (w) dependence is usually very sharp [10], and we can not determine the self-diffusion coefficients of neat DBP and decalin from our experimental data. The literature data tabulated in [11] lead to D∗ = 4.5 × 10−7 cm2 /s for neat DBP. Hence, in the DBP-decalin mixture, the self-diffusion coefficient of DBP molecules is varying from 4.5 × 10−7 to 30 × 10−7 cm2 /s with the decalin fraction increasing from 0 to 1. As was mentioned above, an area of the a curves in Fig. 3 is noticeably lower that of the b curves in spite of the fact that both the curve collections were obtained at equal ∆w. Hence, the refractive index increment for the mixture is a function of the DBP fraction, w, which is illustrated by points 5 in Fig. 2. The w-dependence of the mixture dynamic viscosity is presented here by curve 4. Similarly, the mixture kinematic viscosity is a monotonous function of the DBP fraction changing from 2.702 × 10−2 cm2 /s to 15.77 × 10−2 cm2 /s. Over the ranges of infinite dilution of one of the components (w → 0 and w → 1) the viscosity data were formaly, in the usual way, used for determination of the intrinsic viscosity, [η], from the concentration dependence of the inherent viscosity, ηsp /c, presented in Fig. 5. It was approximated by a linear function in accordance with the Huggins equation ηsp /c = [η] + k [η]2 c where k is the Huggins coefficient. [η] = 0.40 ± 0.01 (k = 13) and [η] = −3.50 ± 0.03 cm3 /g (k = 0.7) values were obtained for DBP in decalin solution and decalin in DBP solution, respectively. They were used to calculate the hydrodynamic parameter A0 defined by A0 = (Dη/T)(M[η]/100)1/3 , where T is the Kelvin temperature and η the medium viscosity. Note the formal manner of the calculation because

an invariance of the D(M[η])1/3 product was established for macromolecules and was explained latter theoretically under assumption that size of the solute particles exceeds many times that of the solvent molecules [1, 12]. For DBP in decalin solution (w → 0) and for decalin in DBP solution (w → 1) we obtained ADBP = (2.3 ± 0.2) × 0 −10g cm2 /K s2 mol1/3, = (−0.9 ± 0.3) × 10 10−10 and Adec 0 respectively. They are lower than the values typical for large particles (macromolecules). On the other hand, similar results were found earlier for oligomers [13, 14]. Polarisability and Anisotropy of Polarisability Figure 2 shows the refractive index of the DBP-decalin mixture, n, as a monotonous increasing function of w (curve 2) but it is not linear. Let us analyse the molar refraction, R, which is related to the refractive index and is a measure of the average electronic polarisability of the molecules, α, being almost independent of temperature, pressure, and aggregate state of the substance. For the DBP-decalin system we used the LorenzLorentz expression E  = E + (4π/3)P , developed for a real electric field, E  , in an isotropic liquid with own polarizability P which is subjected to an external field E. Molar refraction of the liquid with refractive index n is then given by R = (n 2 − 1)/(n 2 + 2)(M/ρ) = (4/3)πNA α

(1)

where M is molecular weight, ρ the liquid density, and NA the Adogadro’ number. For a binary system, we have to replace M and ρ in Eq. 1 by the average values of M and ρ defined by M = MDBP x + Mdec (1 − x) ρ = ρDBP x + ρdec (1 − x) , where x is the molar DBP fraction.

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Fig. 6 (1) Molar refraction, R, and (2) Maxwell constant, ∆n/gηn, of the DBP-decalin system as a function of the DBP molar fraction

We used the experimental n(w) dependence to calculate the R values. They are plotted versus x in Fig. 6 (curve 1). A linear function of R on x provides an evidence of additivity of the DBP and decalin molar refractions, that is usual when a dissolution process is not accompanied by changes in polarizability of the components. In turn, the refraction additivity provides an explanation of non-linear functions n(w) and dn/ dc(w) presented by curve 2 and points 5 in Fig. 2. Hence, the mixture components (DBP and decalin) keep their polarizability during the mixing procedure, and polarizability of the mixture is a sum of those of the components. Figure 2 shows also synchronicity in the ∆n/g(w) and η(w) functions (curves 3 and 4, respectively). In result, the ∆n/gη ratio (the shear optical coefficient) is a value increasing almost linearly with the DBP weight fraction. In turn, Maxwell constant, ∆n/gηn, plotted versus x in Fig. 6 (points 2) shows slight deviation up from the straight line connecting start and end points. In other words, for the system under investigation, the molecular ∆n/gη parameters of the components are not strongly additive, and the average anisotropy of the mixture polarizability is a value increasing non-monotonically with increasing x. When the average polarizability of the molecules remains unchanged, this result may reflect the non-monotonic change in the geometric form asymmetry of the particles which are orienting in the flow field. Translational Mobility Physical properties of ideal solution (which is formed from the component without significant thermal effects or changes in volume) depends, as known, on the molar fraction of a component only. In particular, viscosity of a mixture, η, is given by [15] ln(η/η2 ) = −( f 1 v2 / f 22 )c1 , where η2 , v2 , and f 2 is viscosity, specific volume, and relative free volume of the component 2, and f 1 and c1 are

P.N. Lavrenko et al.

relative free volume and concentration of the component 1, respectively. A non-linear function of ln η on w, found for the experimental data presented above, is an evidence of nonideality of the system under investigation [16], for which 2 a f dec / f DBP ratio is a decreasing function of cdec being not invariant to the components ratio in the mixture. Mobility of a particle in a liquid, which is inversely proportional to the own frictional coefficient (that is proportional to the D value) is first of all determined by the viscosity of the medium η, whereas the product D × η is usually insensitive to this medium parameter. However, for the DBP-decalin system we see an extremum in the Dη(w) dependence. The reduced diffusion coefficient, Dη, reaches a minimum value at w = 0.3 to 0.4 (about x = 0.2). We turn now to the analysis of the effect. For hard particles (molecules), the known diffusion mechanism is close to the mechanism of the sphere translation in a viscous liquid (Stokes). We used the SutherlandEinstein equation D = (kT/6πηRh )(1 + 3π/βRh )/(1 + 2π/βRh ) , where kT is thermal energy, η viscosity of the surrounding medium, Rh radius of a particle, β the coefficient of slipping of the particle surface in relation to the surrounding medium. When slipping is absent then β = 0 and Rh = kT/6πηD

(2)

In presence of slipping (and/or in the case of draining particles) β > 0, and the probable value of Rh is defined by [17] Rh = kT/4πηD

(3)

Table 1 Hydrodynamic and optical properties of binary DBPdecalin system at 24 ◦ C (xDBP is the DBP molar fraction, D the mutual diffusion coefficient, dn/ dc the refractive increment, and Rh the hydrodynamic radius) xDBP

D× 107 (cm2 /s)

dn/ dca581 (cm3 /g)

Rhb ˚ (A)

0 0.0126 0.1420 0.3318 0.5983 0.9509 1

(30x→0 ) 28 ± 1 13.2 ± 0.2 10.7 ± 0.2 11.8 ± 0.2 9.9 ± 0.1 (10x→1 )

– 0.010 0.017 0.022 0.018 0.023 –

(3.1–4.6)x→0 3.2–4.8 5.5–8.2 4.7–7.0 2.5–3.7 1.5–2.2 (1.3–2.0)x→1

a b

Experimental error ±0.001. The values calculated using Eqs. 2 and 3.

Aggregation of Dibutyl Phthalate Molecules

163

The Rh values, calculated under these assumptions, are given in Table 1. For DBP in decalin solution (w → 0) and for decalin in DBP solution (w → 1), Eq. 3 leads to RhDBP = (4.6 ± 0.2) × 10−8 cm and Rhdec = (2.0 ± 0.2) × 10−8 cm, respectively, which are well correlated with the geometric dimensions of the DBP and decalin molecules. Comparability of these values imposes a constraint on the consideration of the surrounding medium as the continuous one for any of the moving components. An average linear dimension of the kinetic unit, responsible for the mass transport in phenomenon of the mutual diffusion within the intermediate w range, is dependent on the system composition (curve 1 in Fig. 7). Its value has been found to be maximum at x = 0.2. Origin of such unusual (extreme) Rh (w) dependence lies obviously in different geometric and dipole structure of the DBP and decalin molecules because, as known, a geometric form and the orientational relations are of high importance in mobility of the liquids with the complicated and non-spherical molecules [16]. The similar size distribution of the mobile units was mentioned earlier in alcohol-containing systems and was referred to the association phenomena risen by the dipoledipole interactions. For example, the properties of the binary ethanol-carbon tetrachloride mixture [18] are pre-

Fig. 7 The average hydrodynamic radius of the mobile unit, Rh , as a function of the solute molar fraction. Curve (1) is obtained for the DBP-decalin system. Curves (2) and (3) represent the literature data on the (2) ethanol-carbon tetrachloride and (3) cyclohexane-carbon tetrachloride mixtures found in [18]

sented in Fig. 7 by curve 2. Note that in absence of the alcohol component, for instance, in the cyclohexane-carbon tetrachloride system, molecular dimensions are the additive values (curve 3). Similarly, in the dioxane-methanol system, mass distribution of the associates is characterized by the extremum (n ≈ 8) at the molar alcohol fraction of 0.5, whereas these distributions look like monotonic functions at lower and higher content of alcohol [19]. In accordance with the literature 13 C NMR data, the DBP molecules are rotating isotropically in a liquid phase [20], and the effective form of a molecule in neat DBP is close to spherical. However, some of the thermodynamic parameters show an extreme distribution even for an ensemble of the spherical particles of different sizes [21–23]. Therefore, it is prematurely to distinguish the mechanism responsible for molecular mobility observed in the DBP-decalin system (association, solvation, and role of the butyl arms).

Conclusions We may conclude that the hydrodynamic and optical investigations have indicated an additivity of the molar refraction and a slight deviation (from additivity) of Maxwell constant of the binary DBP-decalin system with varying composition. At the same time, molecular diffusivity in the medium was found to be a non-monotonic function of the mixture composition. Namely, in accordance with the diffusion-viscosity data, average dimensions of the kinetic unit in the system under investigation reach a maximum at the DBP molar fraction of 0.2 (in average, one DBP molecule per 4 decalin molecules). This aggregation effect is explained by the specific interactions between the DBP and decalin molecules at the given composition that lead to forming the bulky kinetic units of the 3 to 5 molecules of the system components. The life time of these units exceeds evidently the time of the particle thermal jump from one equilibrium position to another in the continuous liquid medium. Acknowledgement The authors express their gratitude to I. Kolomiets and L. Andreeva for fruitful discussion.

References 1. Tsvetkov VN, Eskin VE, Frenkel SY (1971) Structure of macromolecules in solution. Nat Lending Library, Boston (England) 2. Lavrenko P, Cornell A, Lavrenko V, Pogodina N (2004) Prog Colloid Polym Sci 127:67 3. Carman PC, Stein LH (1956) Trans Faraday Soc 52:619

4. Farrar TC, Becker ED (1971) Pulse and Fourier Transform NMR. Acad Press, New York 5. Skirda VD, Volkov VI (1999) Russ J Phys Chem 73:362 6. Skirda VD, Sevryugin VA, Maklakov AI (1983) Dokl Acad Sci USSR 269:638

7. Lavrenko P, Lavrenko V, Tsvetkov V (1999) Prog Colloid Polym Sci 113:14 8. Handbook of Chemist (1964) Khimiya, Moscow 3:926 (in Russian) 9. Reed R, Prausnitz D, Sherwood T (1982) Properties of Gases and Liquids. Khimia, Leningrad (in Russian)

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10. Maklakov AI, Skirda VD, Fatkullin NF (1987) Self-diffusion in polymer solutions and melts. Kazan Univ, Kazan (in Russian) 11. Pimenov GG, Zgadzai OE (1985) Vysokomolek Soedin Ser B 27:899 12. Tsvetkov VN, Lavrenko PN, Bushin SV (1984) J Polym Sci Polym Chem Ed 22:3447 13. Yamada T, Koyama H, Yoshizaki T (1993) Macromolecules 26:2566

P.N. Lavrenko et al.

14. Lavrenko P, Yevlampieva N, Dardel B (2004) Prog Colloid Polym Sci 127:61 15. Ferry JD, Stratton R (1960) Koll-Z 171:107 16. Frenkel YI (1945) Kinetical theory of liquids. Acad Sci USSR, Moscow (in Russian) 17. Dullien FAL (1963) Trans Faraday Soc 59:856 18. Hammond BR, Stokes RH (1956) Trans Faraday Soc 52:781

19. Durov VA, Tereshin OG (2003) Russ J Phys Chem 77:1210 20. Azancheev NM, Maklakov AI, Zikova VV (1981) Russ J Struct Chem 22:50 21. Lisnyanskii LI, Vuks MF (1964) Russ J Phys Chem 38:645 22. Tsvetkov VG, Rodnikova MI, Kaumova DB, Kuznetsova IA (2004) Russ J Phys Chem 78:1342 23. Rusanov AI (2004) Dokl Chem 397:640

Progr Colloid Polym Sci (2006) 131: 165–171 DOI 10.1007/2882_015 © Springer-Verlag Berlin Heidelberg 2006 Published online: 15 February 2006

Mircea Ras¸a Christos Tziatzios Bas G. G. Lohmeijer Dieter Schubert Ulrich S. Schubert

Mircea Ras¸a · Bas G. G. Lohmeijer · Ulrich S. Schubert (u) Laboratory of Macromolecular Chemistry and Nanoscience, Eindhoven University of Technology & Dutch Polymer Institute, Den Dolech 2, 5600 MB Eindhoven, The Netherlands e-mail: [email protected] Christos Tziatzios · Dieter Schubert Institute for Biophysics, Johann Wolfgang Goethe-University Frankfurt, Max von Laue-Str. 1, 60438 Frankfurt am Main, Germany

PARTICLES, POLYMERS AND INTERACTING SYSTEMS

Analytical Ultracentrifugation Studies on Terpyridine-end-functionalized Poly(ethylene oxide) and Polystyrene Systems Complexed via Ru(II) ions

Abstract In this paper we study the solution properties of several metallo-supramolecular assemblies containing polystyrene (PS) and poly(ethylene oxide) (PEO) blocks linked via Ru2+ ions: PS20 − [Ru2+ ] − PEO70 , PS20 − [Ru2+ ] − PS20 , and PEO225 − [Ru2+ ] − PEO225 . Sedimentation equilibrium measurements were performed to determine the molar mass distribution and state of association, and the sedimentation velocity measurements were used to

Introduction Analytical ultracentrifugation is a well known technique mostly used for characterization of proteins and protein complexes, but also of synthetic macromolecules [1–4]. In spite of its capabilities, AUC is still relatively little used for characterization of systems like synthetic metal-containing polymers or inorganic colloids. AUC has already been proposed for investigating the solution properties of metallosupramolecular architectures [5], followed by several papers [6–11] in which both experimental and theoretical problems, specific for these systems, were discussed. The importance of AUC studies on supramolecular assemblies, both for characterization and for applications (e.g. in the field of nanotechnology [12]), certainly merit further investigations. The goal of this paper is to characterize metallosupramolecular block copolymers [13–15] containing terpyridine-end-functionalized polystyrene (PS) and poly (ethylene oxide) (PEO) blocks linked via Ru2+ ions. We are mainly interested in determining the molar mass and state of association in solution. Molar mass determinations

obtain the molar mass distribution, sedimentation coefficient distribution, their average values, and average diffusion coefficient. The results obtained from the two types of experiments are compared and discussed. Keywords Association · Analytical ultracentrifugation · Assemblies · Metallo-supramolecular · Molar mass · Sedimentation

using more common techniques (like gel permeation chromatography or nuclear magnetic resonance) have some disadvantages or limitations for such systems [16]. Concerning the methods used for studying the association in solution, they are in most cases inferior to the use of AUC with respect to both precision and reliability [5]. The molar mass distribution was determined from the AUC data and further used to calculate average values. The results are discussed. Alternatively, the sedimentation coefficient spectrum was analyzed. The average diffusion coefficient was also obtained from data analysis, as an alternative to the more frequently used dynamic light scattering method [17] which was difficult to apply to our systems because of absorption and weak scattering intensities at low polymer concentration. Analytical ultracentrifugation offers several approaches for studying the sedimenting species. Two of them were used in this paper, the so-called sedimentation equilibrium and sedimentation velocity measurements. In the first case, the sedimentation-diffusion equilibrium profile is analyzed. In the second case, the entire process of sedimentation is studied by acquiring and analyzing a set

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of intermediate profiles. The fundamental equation which describes both types of profiles (transient and equilibrium ones) is the Lamm equation [4, 18]. Recent computer programs (see http://www.bbri.org/RASMB/rasmb.html) ease the interpretations of the data by calculating numerical solutions of this equation. These solutions can be integrated over the distribution of molar mass or sedimentation coefficient to yield the final function which describes the experimental profiles. The determination of the spectrum of either the sedimentation coefficient or molar mass can be done using the linear least square method. We used the Sedfit program (see [18] and also http://www.analyticalultracentrifugation.com) for data analysis. In addition, we used the AUC program Discreeq (by Peter Schuck) as well as the commercially available fitting program Table Curve 2D (Systat Software). More details concerning the advantages and disadvantages of AUC in comparison with other techniques, past results obtained on synthetic polymers and polymer complexes, and methods for analyzing the data are presented in [16].

Experimental Details

M. Ras¸a et al.

two complexes were prepared in the presence of ammonium hexafluorophosphate only. Information about the complexes and the synthesis is given elsewhere [13, 19] and their schematic representation is shown in Fig. 1. In view of the possible association in solution, the above mentioned complexes (molecularly dissolved) will be referred to as the monomers. The stability of the solutions was checked before ultracentrifugation by measuring the absorption spectrum in time, with a Lambda 45 spectrometer (Perkin-Elmer Instruments). No systematic changes in the spectra were observed within five days in the case of PS20 − [Ru2+ ] − PEO70 and PS20 − [Ru2+ ] − PS20 . After five days, a decrease of 7–10% was observed at 488 nm, but no further changes could be noticed on a longer time scale. The AUC measurements seem not to be affected by that. On the contrary, changes in the absorption spectrum of PEO225 − [Ru2+ ]− PEO225 were observed within 2 days, and later on the color of the solution even changed completely. The complex could be kept stable during sedimentation (max. 36 h) by adding 1% vol. mercaptoethanol. In all cases, the absorption peak corresponding to the Ru bis-complex occurs at 488 nm at which the AUC measurements were performed (except at higher compound concentrations). In this way only the complex was selected for analysis.

Samples Tetrahydrofuran (THF) solutions of PS20 − [Ru2+ ]− PEO70 , PS20 − [Ru2+ ] − PS20 , and PEO225 − [Ru2+ ]− PEO225 were prepared at room temperature. The first complex was synthesized in the presence of two different types of salt: ammonium hexafluorophosphate (NH4 PF6 ) and ammonium tetraphenyl borate (NH4 B(C6 H5 )4 ). The other

Measurements The sedimentation experiments were performed with an Optima XL-A analytical ultracentrifuge (BeckmanCoulter). The sedimentation cells contain double sector titanium centerpieces as well as gaskets which have a good chemical compatibility with organic solvents like THF.

Fig. 1 Schematic representation of a the PS20 − [Ru2+ ] − PEO70 diblock copolymer and b the PEO225 − [Ru2+ ] − PEO225 homo complex

Analytical Ultracentrifugation Studies on Terpyridine-end-functionalized Poly(ethylene Oxide)

Sample preparation also avoided materials which can be dissolved by THF or could potentially provide metal ions to the solution. Most of experiments were performed under ideal conditions, i.e. at high salt and low polymer concentrations, so that the electrostatic interactions between sedimenting species as well as the macroscopic internal field resulted from the spatial separation of free ions and polymers are negligible. Apart from this, we also performed sedimentation equilibrium measurements at higher polymer concentrations. Sedimentation velocity experiments were performed at 42 000 and 60 000 rpm. The salt concentration was 20 mM and the polymer concentration 25 µM. The same type of salt as used for preparation was added to the solutions, i.e., tetrabutyl ammonium hexafluorophosphate and tetrabutyl ammonium tetraphenyl borate, respectively (see Experimental Details). The PS homo complex was also measured at 100 mM salt, but the results were unchanged. All measurements were performed at 20 ◦ C and the sample volume was 300 µL. Before starting the measurements, we waited one hour after the desired temperature was attained, in order to assure a sufficient temperature stabilization of the samples. 80–100 intermediate profiles were acquired starting with the early sedimentation profiles and continuing until the upper plateau disappeared. No preliminary runs were performed prior to these measurements. Sedimentation equilibrium measurements were performed at speeds between 30 000 and 45 000 rpm, also at 20 ◦ C. The salt concentration was 20 mM (for the salt type, see Experimental Details and the paragraph above) and the polymer concentration was 10 µM. The sample volume was 200 µL. The window gaskets were cut to avoid light transmission artifacts close to the bottom of the cell. The approximate bottom of the cell was estimated from intensity scans (performed at 650 nm). Data interpretation requires accurate values for the partial specific volume, which was calculated according to the equation: v=

Ma v a + Mb v b , Ma + Mb

(1)

167

where indices a and b correspond to the two terpyridineend-functionalized polymer blocks PEO70 − [Ru and PS20 −[Ru, and Ma and Mb are their calculated molar masses. Their respective partial specific volumes are known from previous measurements [C. Tziatzios (2004) unpublished results] and were determined using the so-called “buoyant density method” [6]. The same method was used to determine the partial specific volume for the PEO225 − [Ru2+ ] −PEO225 complex. The results are presented in Table 1. Strictly speaking, these values are valid for THF solutions at high salt concentrations, as in our experiments. The calculated molar masses of the complexes, including two salt ions, are also given in Table 1.

Results and Discussions Equilibrium Measurements In the absence of interactions and association between complexes and in the absence of internal fields, the concentration profile at equilibrium is given by the well known exponential describing the ideal sedimentation behavior:
2  ω (1 − v¯ ρ)M 2 2 (r − r0 ) + b , (2) c(r) = c(r0 ) exp 2RT where M is the molar mass of the monodisperse species, ρ is the solvent density, ω the angular velocity of the rotor, T the absolute temperature, R the gas constant, r the radial coordinate, r0 a reference coordinate, and b the free baseline parameter. Equation 2 yields an average molar mass Mav if it is used to fit the profiles of polydisperse species. This average value differs from the well defined number average molar mass Mn and weight average molar mass Mw . Frequently, reversible association or aggregation of the complexes occurs in solution, leading to the formation of dimers, trimers, etc. A sum of exponentials of the type of Eq. 2, one for each species, can be used in that case to describe the experimental profile (see also [6]). The behavior of the sample synthesized in the presence of tetraphenyl borate salt differed from that of the others and will be described later. The equilibrium profiles

Table 1 Calculated molar mass Mc and partial specific volume v of the samples subjected to AUC measurements. The average molar mass Mav , number average molar mass Mn , and a rough estimation of weight average molar mass Mw , were obtained from the sedimentation equilibrium measurements Sample

Mc (g/mol)

v (mL/g)

Mav (g/mol)

PS20 −[Ru2+ ]−PEO70 (NH4 PF6 salt) PS20 −[Ru2+ ]−PEO70 (NH4 B(C6 H5 )4 salt) PS20 − [Ru2+ ] − PS20 PEO225 − [Ru2+ ] − PEO225

6050 6396 5020 20 700

0.868 0.868 0.923 0.840

7600 15 400 6750 22 000

Mn (g/mol)

Mw (g/mol)

7770

7980

–

–

6880 22 800

7080 23 500

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of the other three samples were first analyzed by fitting the data according to Eq. 2. Both the determination coefficient of the fit r 2 and the distribution of residuals indicate good fits. An example for the PS20 − [Ru2+ ] − PS20 complex, measured at 30 000 rpm, is given in Fig. 2. The fit was performed with TableCurve 2D by using the inverse variance as weight for each data point. The fit results are given in Table 1. The average molar masses determined from profiles at speeds between 30 000 and 45 000 rpm were in agreement with each other. We also used the Table Curve 2D and Discreeq programs to fit the data with a sum of exponentials, corresponding to different combinations of monomer, dimer, and trimer but the determination coefficient was not better than that of a single exponential fit. In this case, the discrepancy between the calculated and experimentally determined masses can be explained by the insufficient accuracy of the values used for the partial specific volume (mainly for PS containing complexes, where the discrepancy was significant). The profiles measured at 40 000 rpm were used to obtain the molar mass distribution c(M) from the Sedfit program (Fig. 3). According to the Sedfit help, the area under the c(M) curve gives the loading concentration, which, in our measurements is in the form of absorbance. Absorbance can be written as A ∝ nlσ , where n is the number density of identical species, l is the optical path, and σ the absorption cross section. The absorption in our case is due to the complexed ligand, but the polydispersity is due to the polymer blocks which do not absorb at 488 nm. Consequently, the absorption cross section is constant for such

Fig. 2 Fit to the equilibrium profile of the PS20 − [Ru2+ ] − PS20 complex with Eq. 2. Data was obtained at 30 000 rpm, and the average molar mass was Mav = 6300 ± 50 g/mol. The fit determination coefficient was r 2 = 0.9994

M. Ras¸a et al.

Fig. 3 Molar mass distribution obtained from equilibrium profiles using the Sedfit program

polydisperse species at low concentrations, and the only variable in the expression of the absorbance is the number density. In this particular situation, mass  the average molar  obtained by integration M = Mc(M) dM/ c(M) dM represents the number average molar mass Mn (shown in Table 1). The c(M) distribution determined from sedimentation equilibrium has the advantage of not being influenced by the friction coefficient of the species, as in the case of sedimentation velocity analysis [18]. The results of Fig. 3 also show no detectable association of the complexes (no separate peaks corresponding to monomer, dimer, etc.). It is straightforward to calculate Mw (Table 1) and PDI (of the order of 1.03), which would indicate a rather low polydispersity. However, the width of the distributions is significantly affected by the so-called regularization procedure used in the determination of c(M), as described in [18]. Thus, the Mw and PDI values are not reliable. The regularization is controlled by a parameter which may vary in the Sedfit program between 0.5 and 1. The results presented in Fig. 3 were obtained at a regularization parameter of 0.99. The effect of salt concentration was studied in [16]. In the case of the PS20 − [Ru2+ ] − PEO70 complex, we performed simultaneous sedimentation equilibrium measurements at six different values of polymer concentration, between 12.4 µM and 124 µM, at 20 mM salt (Table 2). The average molar mass was determined in each case using Eq. 2. First, the molar mass increased with the compound concentration, a fact which may be assigned to slight association. At 49.6 µM Sedfit indicates a small second peak in the determined molar mass distribution c(M), visible at lower values of regularization parameter (0.6–0.75). The subsequent decrease may indicate non-ideal sedimentation behavior. The PS20 − [Ru2+ ] − PEO70 complex synthesized in the presence of tetraphenyl borate obviously shows association: the average molar mass is much larger than the

Analytical Ultracentrifugation Studies on Terpyridine-end-functionalized Poly(ethylene Oxide)

Table 2 Polymer concentration cp , wavelength λ and determined average molar mass of the PS20 − [Ru2+ ] − PEO70 complex cp (µM)

λ (nm)

Mav (g/mol)

12.4 24.8 49.6 74.4 99.2 124.0

488 490 520 540 560 570

7400 7800 8600 8300 8200 7700

169

ments in separate experiments. The molar mass distribution was determined in this case also using Sedfit and compared with that obtained from equilibrium sedimentation. Sedfit uses an average friction factor f relative to the friction factor f 0 of a sphere having the same volume as the sedimenting species. The relative friction factor f/ f 0 was optimized in a nonlinear fit procedure and the results are given in Table 3. The distributions are presented in Fig. 5 and the average mass in Table 3. A comparison between the molar mass distributions obtained from sedimentation equilibrium and sedimentation velocity measurements is made here because, as already mentioned, there are important differences between the two determinations: in the first case the determination is independent on the friction factor, while in the latter the friction factor is required and polydispersity or error in the average friction factor may strongly influence the determination (see [18] and the Sedfit help at http://www.analyticalultracentrifugation.com). In addition, only one profile and the analytical solution of the Lamm equation is used in the first case while a large number of profiles but a numerical solution in the second case. In the case of PS20 − [Ru2+ ] − PEO70 and PS20 − [Ru2+ ] − PS20 , the distributions are comparable

Fig. 4 Contribution of monomers, dimers, and tetramers to the equilibrium profile of the PS20 − [Ru2+ ] − PEO70 complex synthesized in the presence of ammonium tetraphenyl borate. The fit to the data was done with a sum of exponentials given by Eq. 2, corresponding to the three types of aggregates

expected mass. The Discreeq program best fitted the data for a monomer (39%) – dimer (23%) – tetramer (38%) combination (Fig. 4). The fit with this combination led to a sum of the squared residuals which amounted to only 15% of that using Eq. 2. Sedimentation Velocity Measurements The samples prepared with ammonium hexafluorophosphate were subjected to sedimentation velocity measure-

Fig. 5 Molar mass distribution obtained from sedimentation velocity profiles using the Sedfit program

Table 3 Number average molar mass, average friction ratio f/ f 0 , number average sedimentation coefficient Sn , and average diffusion coefficient D obtained from the sedimentation velocity measurements. Sample

Mn (g/mol)

f/ f 0

Sn (S)

D (×10−10 m2 s−1 )

PS20 − [Ru2+ ] − PEO70 (NH4 PF6 salt) PS20 − [Ru2+ ] − PS20 PEO225 − [Ru2+ ] − PEO225

7000 6710 21 800

1.53 1.28 2.41

1.42 1.28 2.17

2.16 2.58 0.89

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case of sedimentation velocity measurements, only its width being influenced by the friction factor (thus its average is a reliable value). The last step was to determine the average translational diffusion coefficient (Table 3) from kT D= f . (3) f0 f0 Alternatively it could be obtained from the Svedberg equation [4], in which we used the average values Sn and Mn determined from the corresponding distributions: RT Sn D= (4) Mn (1 − v¯ ρ) A very good agreement between the values obtained with Eqs. 3 and 4 was observed. The coefficient f 0 in Eq. 3 was calculated using:   3Mn v¯ 1/3 , (5) f 0 = 6πη 4πNA where η is the solvent viscosity and NA is Avogadro’s number. The Mn values from Table 1 were used. Instead, the diffusion coefficient can be expressed in terms of average sedimentation coefficient (see Ref. [20]). We have also performed DLS measurements on PS20 − [Ru2+ ] − PEO70 and PS20 − [Ru2+ ] − PS20 solutions in order to determine D. However, reliable data could not be obtained so far because of weak scattering intensities at low concentrations and significant absorption effects even at 650 nm. Fig. 6 Normalized c(M) distributions obtained from equilibrium and velocity sedimentation, respectively, for a PS20 − [Ru2+ ] − PS20 and b PEO225 − [Ru2+ ] − PEO225

Conclusions

(an example is shown in Fig. 6a), while in the case of PEO225 − [Ru2+ ] − PEO225 (Fig. 6b) only the number average values are close to each other (the same value of 0.99 for the regularization parameter was used in all cases). Even though the widths are not accurate, the regularization parameter was the same in all cases and the result of Fig. 6b may confirm that the distribution obtained from sedimentation velocity measurements is more susceptible to errors. It is possible in this case that the determination using an average friction coefficient is not accurate. The sedimentation coefficient spectra c(S) were also determined using Sedfit. It is shown in [20] that the average value over the distribution S = Sc(S) dS/ c(S) dS is, in general, the weight-average sedimentation coefficient. However, following the discussion of the previous section for the particular situation of the samples investigated in this paper, this average seems to be the number-average sedimentation coefficient Sn . The determined average values over c(S) are given in Table 3. c(S) determination is actually the robust determination in the

We performed AUC characterization of several metallosupramolecular polymers in solution, using both sedimentation equilibrium and velocity measurements. The molar mass distribution was first determined from equilibrium measurements for PS20 − [Ru2+ ] − PEO70 , PS20 − [Ru2+ ] − PS20 and PEO225 − [Ru2+ ] − PEO225 prepared with NH4 PF6 salts. It was shown that the integration of the distribution yields the number average molar mass Mn . The number average molar mass values obtained from sedimentation equilibrium and velocity measurements agree with each other. The fits to the equilibrium profiles of these samples, assuming ideal sedimentation behavior, gave average molar masses in agreement with Mn . No evidence for association in these systems came out from our analysis. The average friction factors and diffusion coefficients were also determined. The PS20 − [Ru2+ ] − PEO70 complex synthesized in the presence of tetraphenyl borate shows significant association, best described in terms of monomer, dimer, and tetramer assemblies. In the case of samples prepared with NH4 PF6 salts, the results obtained at increasing polymer concentration suggest slight association and, in addition, non-ideal behavior.

Analytical Ultracentrifugation Studies on Terpyridine-end-functionalized Poly(ethylene Oxide)

Acknowledgement The discussions we had within the Sedfit user’s group (see http://bilbo.bio.purdue.edu/mailman/listinfo/sedfit) were veryhelpful andwe are grateful for the suggestions received. This work

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was financially supported by the Nederlandse Organisatie for Wetenschappelijk Onderzoek (NWO – through the VICI award), the Dutch Polymer Institute (DPI), and the Fonds der Chemischen Industrie.

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12. Wouters D, Schubert US (2004) Angew Chem Int Ed 43:2480 13. Lohmeijer BGG, Schubert US (2002) Angew Chem Int Ed 41:3825 14. Gohy J-F, Lohmeijer BGG, Schubert US (2002) Macromol Rapid Commun 35:555 15. Gohy J-F, Lohmeijer BGG, Varshney SK, Décamps B, Leroy E, Boileau S, Schubert US (2002) Macromolecules 235:7427 16. Rasa M, Lohmeijer BGG, Schubert US (2005, in preparation) 17. Dhont JKG (1996) An Introduction to Dynamics of Colloids. Elsevier, Amsterdam 18. Schuck P (2000) Biophysical J 78:1606 19. Lohmeijer BGG (2004) PhD Thesis. Eindhoven University of Technology Press, Eindhoven 20. Schuck P (2003) Anal Biochem 320:104

Progr Colloid Polym Sci (2006) 131: 172 © Springer-Verlag 2006

Austin JB → Laue TM Behlke J, Ristau O: A New Possibility to Recognize the Concentration Dependence of Sedimentation Coefficients 29 Belfield EJ → Khan A Belyaeva EV → Lavrenko PN Bernardi G → Clay O Bhattacharyya SK, Maciejewska P, Börger L, Stadler M, Gülsün AM, Cicek HB, Cölfen H: Development of a Fast Fiber Based UV-Vis Multiwavelength Detector for an Ultracentrifuge 9 Bourdillon L, Freitag R, Wandrey C: Association and Temperature-induced Phase Transition Studied by Analytical Ultracentrifugation 150 Bourdillon L, Hunkeler D, Wandrey C: The analytical ultracentrifuge for the characterization of polydisperse polyelectrolytes 141 Brookes E, Demeler B: Genetic Algorithm Optimization for Obtaining Accurate Molecular Weight Distributions from Sedimentation Velocity Experiments 33 Börger L → Bhattacharyya SK Carels N → Clay O Casey R → Khan A Chebotareva NA, Meremyanin AV, Makeeva VF, Kurganov BI: Self-Association of Phosphorylase Kinase under Molecular Crowding Conditions 83 Cicek HB → Bhattacharyya SK Clay O, Carels N, Douady CJ, Bernardi G: Density Gradient Ultracentrifugation and Whole Genome Sequences: Fine-tuning the Correspondence 97 Cölfen H, Lucas G: Particle Sedimentation in pH-Gradients 129 Cölfen H, Völkel A: Application of the Density Variation Method on Calciumcarbonate Nanoparticles 126 Cölfen H → Bhattacharyya SK

AUTHOR/TITLE INDEX DeGrado WF → Stouffer AL Demeler B → Brookes E Douady CJ → Clay O Durchschlag H, Zipper P, Krebs A: Ab initio and Constrained Modeling of Phosphorylase 55 Durchschlag H → Zipper P Ebel C → Salvay AG Freitag R → Bourdillon L Gavrilova II → Pavlov GM Gülsün AM → Bhattacharyya SK Harding SE → Khan A Hughes RK → Khan A Hunkeler D → Bourdillon L Khan A, Hughes RK, Belfield EJ, Casey R, Rowe AJ, Harding SE: Oligomerization of Hydroperoxide Lyase, a Novel P450 Enzyme in Plants 116 Korneeva EV → Pavlov GM Krebs A → Durchschlag H Krebs A → Zipper P Kurganov BI → Chebotareva NA Laue TM, Austin JB, Rau DA: A Light Intensity Measurement System for the Analytical Ultracentrifuge 1 Lavrenko PN, Belyaeva EV, Volokhova DM, Okatova OV: Aggregation of Dibutyl Phthalate Molecules in Decalin Solutions Evidenced by Hydrodynamic and Optical Measurements 158 Lavrenko PN → Lavrenko VP Lavrenko VP, Lavrenko PN: Automatic Analysis of Lebedev Interference Patterns 23 Lear JD → Stouffer AL Lohmeijer BGG → Ras¸a M Lucas G → Cölfen H Maciejewska P → Bhattacharyya SK Makeeva VF → Chebotareva NA Meremyanin AV → Chebotareva NA Mutowo P, Scott DJ: Oligomerisation of TBP1 from Haloferax volcanii 93

Müller HG: Determination of Particle Size Distributions of Swollen (Hydrated) Particles by Analytical Ultracentrifugation 121 Okatova OV → Lavrenko PN Panarin EF → Pavlov GM Pavlov GM, Panarin EF, Korneeva EV, Gavrilova II, Tarasova NN: Molecular Properties and Electrostatic Interactions of Linear Poly(allylamine hydrochloride) Chains 134 Ras¸a M, Tziatzios C, Lohmeijer BGG: Analytical Ultracentrifugation Studies on Terpyridine-end-functionalized Poly(ethylene oxide) and Polystyrene Systems Complexed via Ru(II) ions 165 Rau DA → Laue TM Ristau O → Behlke J Rowe AJ → Khan A Salvay AG, Ebel C: Analytical ultracentrifuge for the Characterization of Detergent in Solution 74 Scott DJ → Mutowo P Stadler M → Bhattacharyya SK Stouffer AL, DeGrado WF, Lear JD: Analytical Ultracentrifugation Studies of the Influenza M2 Homotetramerization Equilibrium in Detergent Solutions 108 Tarasova NN → Pavlov GM Tziatzios C → Ras¸a M Volokhova DM → Lavrenko PN Völkel A → Cölfen H Wandrey C → Bourdillon L Wandrey C → Bourdillon L Zipper P, Durchschlag H, Krebs A: Modeling of the Dodecameric Subunit of Lumbricus Hemoglobin 41 Zipper P → Durchschlag H

Progr Colloid Polym Sci (2006) 131: 173–174 © Springer-Verlag 2006

KEY WORD INDEX

Absorbance detector 1 Acrylamide copolymers 141 Acryloyloxyethyltrimethylammonium chloride 142 Additivity 158 Advanced modeling techniques 41, 55 Aggregation 150 Allylamine salts, polymerization 134 Annelid hemoglobins, hexagonal bilayer 41 Archaea 93 Association/aggregation, reversibility/reproducibility 154

Gold particles, 4-carboxythiophenyl stabilized 131 Haloferax volcanii 93

Base composition 97 Binding constants 29

Influenza 108 Instrumentation software 1 Interference patterns 23 N -Isopropylacrylamide (NIPAM) 151

CaCO3 precursor particles 126 CCD based UV-Vis spectrometer 9 Colloids, stability 129 Corynebacterium callunae 56 Crowding 83 Cytochrome C 20 Cytochrome P450 enzyme 116 DAMMIN 55 Degree of swelling 121 Density gradient ultracentrifugation 129 Density variation method 126 Detector systems/development 1, 9 Detergents 74 Dibutyl phthalate-decaline mixture 158 Diffusiometry 23 Digital recording 23 Dodecameric subunit 41 Dodecyl-β-D-maltoside (DDM) 75 Dynamic light scattering 83 Electrostatic interactions 134 Equilibrium analytical ultracentrifugation (EAUC) 108 Escherichia coli 56 Evolution 35, 97

Halophilic proteins 93 Heatsink modification 12 Hemoglobin, extracellular 41 Hepes 83 Heterogeneous mixtures 141 Hexagonal bilayer 41 Hydrated particles 121 Hydration 55 Hydrodynamics 41, 55, 134 Hydroperoxide lyase (HPL) 116

Lamm equation 29, 34, 76 Lebedev optical system 23 Long-range correlations 97, 102 Lumbricus terrestris, hemoglobin 41 Lysozyme 34 M2 protein 108 Maltodextrins 56 Mark-Houwink-Kuhn-Sakurada 137, 143 Medicago truncatula, hydroperoxide lyase (HPL-F) 116 Membrane 74 Membrane protein 108 Metallo-supramolecular assemblies, polystyrene/poly(ethylene oxide) 165 Methylene-bis-acrylamide (MBA) 142 Micelle number 108 Molecular diffusivity 158 Molecule association 150 Monoallylammonium salts, azoinitiators 135 Nanoparticle size distributions 126

Fatty acid hydroperoxide lyase (HPL) 116 Fatty acids, polyunsaturated 116 Fiber optics 9 Fluorescence detector 1 GASBOR 55 Genetic algorithms 33, 35 Glucose-1-phosphate 56 Glycogen phosphorylase 56, 83

Octaethylene glycol monododecyl ether 75 Optical anisotropy 158 Optical fibers 12 Optical noise 23 Oxylipin metabolism 116 P450 116 Particle charge 129

Particle density distribution 126 Particle size distribution 121, 126 PEO 165 pH-gradients, charged colloidal particles 129 Phase separation 150 4-(Phenylazo)phenylacrylamide (PAPAM) 151 Phosphorylase 55 Phosphorylase kinase 83 Polarising interference patterns 23 Poly(allylamine hydrochloride) 134 Poly(ethylene oxide), terpyridine-end-functionalized 165 Poly(N-isopropylacryl-amide) (PNIPAM) 150 Poly(sodium 4-styrene sulfonate) 130 Poly(styrene sulfonate) 127 Poly(vinylformamide) 139 Polydispersity 141 Polyelectrolytes, cationic 134, 141 – polydisperse 141 Polyethylene-dioxythiophen 124 Polyol dispersion 124 Polypeptides 84 Polystyrene latex mixture 18 Polystyrene systems, terpyridine-end-functionalized 165 Proline 84 Propylenecarbonate 124 Protein folding 108 Protein structure, reconstructions 55 Protein-DNA interactions 93 Proteins 74 Proton channel 108 Pyroccocus woesei 95 Refraction 158 Rotor timing pulse 3 Ru(II) 165 SAXS 41, 55 Sedimentation coefficients, concentration dependence 29 Sedimentation equilibrium 97, 129 Sedimentation velocity 29, 33, 126, 129, 141 Self-association 83 Small angle X-ray scattering 41, 55 Star Rosetta strains 94 Starch phosphorylase 55 Stimuli-responsive polymers 150 Stochastic methods 35 Swelling, degree 121 Swollen particles, degree of hydration 121

174

TATA-binding protein 93 TBP1 93 Telo(NIPAMco-PAPAM) 151 Temperature-induced phase transition 150 Transmembrane helices 108 Traumatin 116

Trimethylamine N-oxide (TMAO) 84 Urea 83 UV-Vis multiwavelength detector 9 UV-Vis spectrometer, CCD based 9 Vacuum feedthroughs 12 Vertebrate genomes 100

Volume swelling 123 Wound healing, plants 116 X-ray crystallography 55 Zebrafish 103