Multiscale Approaches to Protein Modeling

Multiscale Approaches to Protein Modeling

Andrzej Kolinski Editor

Multiscale Approaches to Protein Modeling

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Editor Andrzej Kolinski Department of Chemistry University of Warsaw ul. Pasteura 1 02-093 Warszawa Poland [email protected]

ISBN 978-1-4419-6888-3 e-ISBN 978-1-4419-6889-0 DOI 10.1007/978-1-4419-6889-0 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2010934732 © Springer Science+Business Media, LLC 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

Thanks to enormous progress in sequencing of genomic data, presently we know millions of protein sequences. At the same time the number of experimentally solved protein structures is much smaller, ca. 60,000. This is because of large cost of structure determination. Thus, the theoretical in silico prediction of protein structures and dynamics is essential for understanding the molecular basis of drug action, metabolic and signaling pathways in living cells, and designing new technologies in the life science and material sciences. Unfortunately, a “brute force” approach remains impractical. Folding of a typical protein (in vivo or in vitro) takes milliseconds to minutes, while the state-of-the-art all-atom molecular mechanics simulations of protein systems can cover only a time period of nanoseconds to microseconds. This is the reason for the enormous progress in the development of various multiscale modeling techniques applied to protein structure prediction, modeling of protein dynamics and folding pathways, in silico protein engineering, model-aided interpretation of experimental data, modeling of macromolecular assemblies, and theoretical studies of protein thermodynamics. Coarse-graining of the proteins’ conformational space is a common feature of all these approaches, although the details and the underlying physical models span a very broad spectrum. This book contains comprehensive reviews of the most advanced multiscale modeling methods in protein structure prediction, computational studies of protein dynamics, folding mechanisms, and macromolecular interactions. The presented approaches span a wide range of the levels of coarse-grained representations, various sampling techniques, and a variety of applications to biomedical and biophysical problems. It was our intention to provide a collection of comprehensive reviews that could be used as a reference book for those who just are beginning their adventure with biomacromolecular modeling but also as a valuable source of more detailed information for those who are already experts in the field of biomacromolecular modeling and in related areas of computational biology or biophysics. Proteins are linear copolymers composed of amino acids. Important ideas of polymer physics inspired the field of protein modeling. Chapter 1 explains some basic concepts of polymer conformational statistics and dynamics of chain molecules in context of simple lattice models. This chapter demonstrates how

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these ideas could be employed in protein modeling. Chapter 2 describes application of a lattice-based protein model to the very challenging problem of protein docking. Chapter 3 provides a comprehensive overview of various coarse-grained protein-like and protein models. This chapter describes (among other approaches) probably the most rigorous system of physics-based reduced modeling of proteins. Coarse-grained, multiscale, protein modeling requires specific designs of interaction schemes. Chapters 4–6 provide in-depth overviews of various level force-fields for the reduced representations of protein conformational space, including knowledgebased statistical potentials. Chapters 7 and 8 (but also, in part, Chapters 3–5 and 12) describe a variety of applications of reduced models in the study of protein dynamics, folding pathways, molecular mechanisms of mechanical unfolding, and protein interactions. Chapter 9 gives an overview of the most effective sampling strategies in a reduced, although unrestricted conformational space. Chapters 10 and 11 present a very efficient philosophy of a conformational search, where the target structures are assembled from fragments excised from already known protein structures. These strategies proven to be very effective in the large-scale, automated in silico structure prediction. Chapter 12 describes a multiscale method, based on a high-resolution lattice model, for modeling protein folding pathways. Chapters 13 and 14 discuss the most important ideas and techniques of comparative modeling – the most effective and the most popular method for theoretical prediction of protein structures. These chapters provide also reviews of the model-quality assessment methods. The contributing authors are world-wide recognized experts. Some of them (Bujnicki and Zhang) are leaders in the field of protein structure prediction, as assessed by the recent (CASP6–CASP8) community-wide experiments in a blind structure prediction. Others also developed very successful methods for the protein structure prediction (Scheraga, Liwo, Feig, and Kihara). Several of the authors of this book developed very efficient coarse-grained interaction schemes for protein models based on either an evolutionary knowledge approach (Jernigan and Scheraga have built theoretical foundations of this class of approaches, but others also contributed significantly: Feig and Micheletti) or a physics-based approach (Scheraga, Liwo, Feig, and Irback). Among the authors are also the world top leaders of comparative modeling (Bujnicki, Zhang, Tramontano, and Kihara) and automated structure prediction (Zhang and Bujnicki) – the structure prediction server created by Zhang is the best till date. The book presents also the state-of-the-art methods of evaluation of quality of the theoretical protein models (Tramontano and Kihara). Recently, a significant progress has been achieved in multiscale modeling of protein dynamics and folding mechanisms. The authors of the chapters dealing with this class of problems are also world-class leaders (Scheraga, Liwo, Irback, Feig, Cieplak, Jernigan, and Micheletti). The conformational search strategies are crucial in protein modeling. Developers of the most efficient computational techniques and strategies are also among the authors (Hansmann, Scheraga, and others). Warsaw, Poland

Andrzej Kolinski

Contents

1 Lattice Polymers and Protein Models . . . . . . . . . . . . . . . . Andrzej Kolinski

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2 Multiscale Protein and Peptide Docking . . . . . . . . . . . . . . . Mateusz Kurcinski, Michał Jamroz, and Andrzej Kolinski

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3 Coarse-Grained Models of Proteins: Theory and Applications . . . . . . . . . . . . . . . . . . . . . . . Cezary Czaplewski, Adam Liwo, Mariusz Makowski, Stanisław Ołdziej, and Harold A. Scheraga 4 Conformational Sampling in Structure Prediction and Refinement with Atomistic and Coarse-Grained Models . . . . . . Michael Feig, Srinivasa M. Gopal, Kanagasabai Vadivel, and Andrew Stumpff-Kane 5 Effective All-Atom Potentials for Proteins . . . . . . . . . . . . . . Anders Irbäck and Sandipan Mohanty 6 Statistical Contact Potentials in Protein Coarse-Grained Modeling: From Pair to Multi-body Potentials . . . . . . . . . . . Sumudu P. Leelananda, Yaping Feng, Pawel Gniewek, Andrzej Kloczkowski, and Robert L. Jernigan 7 Bridging the Atomic and Coarse-Grained Descriptions of Collective Motions in Proteins . . . . . . . . . . . . . . . . . . . . Vincenzo Carnevale, Cristian Micheletti, Francesco Pontiggia, and Raffaello Potestio 8 Structure-Based Models of Biomolecules: Stretching of Proteins, Dynamics of Knots, Hydrodynamic Effects, and Indentation of Virus Capsids . . . . . . . . . . . . . . . . . . . . . Marek Cieplak and Joanna I. Sułkowska 9 Sampling Protein Energy Landscapes – The Quest for Efficient Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . Ulrich H. E. Hansmann

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Contents

10 Protein Structure Prediction: From Recognition of Matches with Known Structures to Recombination of Fragments . . . . . . . . . . . . . . . . . . . Michal J. Gajda, Marcin Pawlowski, and Janusz M. Bujnicki

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11 Genome-Wide Protein Structure Prediction . . . . . . . . . . . . Srayanta Mukherjee, Andras Szilagyi, Ambrish Roy, and Yang Zhang

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12 Multiscale Approach to Protein Folding Dynamics . . . . . . . . . Sebastian Kmiecik, Michał Jamroz, and Andrzej Kolinski

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13 Error Estimation of Template-Based Protein Structure Models . . Daisuke Kihara, Yifeng David Yang, and Hao Chen

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14 Evaluation of Protein Structure Prediction Methods: Issues and Strategies . . . . . . . . . . . . . . . . . . . . . . . . . Anna Tramontano and Domenico Cozzetto

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Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contributors

Janusz M. Bujnicki Laboratory of Bioinformatics and Protein Engineering, International Institute of Molecular and Cell Biology, Warsaw, Poland; Laboratory of Bioinformatics, Institute of Molecular Biology and Biotechnology, Faculty of Biology, Adam Mickiewicz University, Poznan, Poland, [email protected] Vincenzo Carnevale Institute for Computational Molecular Science, Temple University, Philadelphia, PA, USA, [email protected] Hao Chen Department of Biological Sciences, College of Science, Purdue University, West Lafayette, IN, USA, [email protected] Marek Cieplak Institute of Physics, Polish Academy of Sciences, Warsaw, Poland, [email protected] Domenico Cozzetto Department of Biochemical Sciences, “Sapienza” University of Rome, Rome, Italy, [email protected] Cezary Czaplewski Faculty of Chemistry, University of Gda´nsk, Gda´nsk, Poland; Baker Laboratory of Chemistry and Chemical Biology, Cornell University, Ithaca, NY, USA, [email protected] Michael Feig Department of Biochemistry and Molecular Biology, Michigan State University, East Lansing, MI, USA; Department of Chemistry, Michigan State University, East Lansing, MI, USA, [email protected] Yaping Feng Department of Biochemistry, Biophysics, and Molecular Biology, Iowa State University, Ames, IA, USA; L.H.Baker Center for Bioinformatics and Biological Statistics, Iowa State University, Ames, IA, USA, [email protected] Michal J. Gajda European Molecular Biology Laboratories, Hamburg Outstation, Hamburg, Germany, [email protected] Pawel Gniewek L.H.Baker Center for Bioinformatics and Biological Statistics, Iowa State University, Ames, IA, USA; Laboratory of Theory of Biopolymers, Faculty of Chemistry, University of Warsaw, Warsaw, Poland, [email protected]

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Srinivasa M. Gopal Department of Biochemistry and Molecular Biology, Michigan State University, East Lansing, MI, USA, [email protected] Ulrich H. E. Hansmann Department of Physics, Michigan Technological University, Houghton, MI, USA, [email protected] Anders Irbäck Computational Biology & Biological Physics, Department of Theoretical Physics, Lund University, Lund, Sweden, [email protected] Michał Jamroz Faculty of Chemistry, University of Warsaw, Warsaw, Poland, [email protected] Robert L. Jernigan Department of Biochemistry, Biophysics, and Molecular Biology, Iowa State University, Ames, IA, USA; L.H.Baker Center for Bioinformatics and Biological Statistics, Iowa State University, Ames, IA, USA, [email protected] Daisuke Kihara Department of Biological Sciences, College of Science; Department of Computer Science, College of Science; Markey Center for Structural Biology, Purdue University, West Lafayette, IN, USA, [email protected] Andrzej Kloczkowski Department of Biochemistry, Biophysics, and Molecular Biology, Iowa State University, Ames, IA, USA; L.H.Baker Center for Bioinformatics and Biological Statistics, Iowa State University, Ames, IA, USA, [email protected] Sebastian Kmiecik Faculty of Chemistry, University of Warsaw, Warsaw, Poland, [email protected] Andrzej Kolinski Faculty of Chemistry, University of Warsaw, Warsaw, Poland, [email protected] Mateusz Kurcinski Faculty of Chemistry, University of Warsaw, Warsaw, Poland, [email protected] Sumudu P. Leelananda Department of Biochemistry, Biophysics, and Molecular Biology, Iowa State University, Ames, IA, USA; L.H.Baker Center for Bioinformatics and Biological Statistics, Iowa State University, Ames, IA, USA, [email protected] Adam Liwo Faculty of Chemistry, University of Gda´nsk, Gda´nsk, Poland; Baker Laboratory of Chemistry and Chemical Biology, Cornell University, Ithaca, NY, USA, [email protected] Mariusz Makowski Faculty of Chemistry, University of Gda´nsk, Gda´nsk, Poland; Baker Laboratory of Chemistry and Chemical Biology, Cornell University, Ithaca, NY, USA, [email protected]

Contributors

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Cristian Micheletti Scuola Internazionale Superiore di Studi Avanzati, Trieste, Italy; Democritos CNR-IOM and Italian Institute of Technology (SISSA Unit), Trieste, Italy, [email protected] Sandipan Mohanty Jülich Supercomputing Centre, Forschungszentrum Jülich GmbH, Jülich, Germany, [email protected] Srayanta Mukherjee Center for Computational Medicine and Bioinformatics, University of Michigan, Ann Arbor, MI, USA; Center for Bioinformatics, University of Kansas, Lawrence, KS, USA, [email protected] Stanisław Ołdziej Baker Laboratory of Chemistry and Chemical Biology, Cornell University, Ithaca, NY, USA; Laboratory of Biopolymer Structure, Intercollegiate Faculty of Biotechnology, University of Gda´nsk and Medical University of Gda´nsk, Gda´nsk, Poland, [email protected] Marcin Pawlowski Laboratory of Bioinformatics and Protein Engineering, International Institute of Molecular and Cell Biology, Warsaw, Poland, [email protected] Francesco Pontiggia Department of Biochemistry, Brandeis University, Waltham, MA, USA, [email protected] Raffaello Potestio Scuola Internazionale Superiore di Studi Avanzati, Trieste, Italy, [email protected] Ambrish Roy Center for Computational Medicine and Bioinformatics, University of Michigan, Ann Arbor, MI, USA; Center for Bioinformatics, University of Kansas, Lawrence, KS, USA, [email protected] Harold A. Scheraga Baker Laboratory of Chemistry and Chemical Biology, Cornell University, Ithaca, NY, USA, [email protected] Andrew Stumpff-Kane Department of Biochemistry and Molecular Biology, Michigan State University, Michigan, USA, [email protected] Joanna I. Sułkowska Institute of Physics, Polish Academy of Sciences, Warsaw, Poland; CTBP, University of California, Gilman Drive 9500, La Jolla, San Diego, CA, USA, [email protected] Andras Szilagyi Center for Bioinformatics, University of Kansas, Lawrence, KS, USA; Institute of Enzymology, BRC, Hungarian Academy of Sciences, Budapest, Hungary, [email protected] Anna Tramontano Department of Biochemical Sciences, “Sapienza” University of Rome, Rome, Italy; Istituto Pasteur – Fondazione Cenci Bolognetti, “Sapienza” University of Rome, Rome, Italy, [email protected] Kanagasabai Vadivel Department of Biochemistry and Molecular Biology, Michigan State University, East Lansing, MI, USA, [email protected]

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Yifeng David Yang Department of Biological Sciences, College of Science, Purdue University, West Lafayette, IN, USA, [email protected] Yang Zhang Center for Computational Medicine and Bioinformatics, University of Michigan, Ann Arbor, MI, USA; Center for Bioinformatics, University of Kansas, Lawrence, KS, USA, [email protected]

Chapter 1

Lattice Polymers and Protein Models Andrzej Kolinski

Abstract The size of conformational space of chain polymers is enormous. Much has been learned about polymer structure, thermodynamics, and dynamics by theoretical considerations and numerical study of simple lattice models. Self-avoiding random walks on a lattice provide a good approximation for the excluded volume effect and nature of the coil–globule transition. Semiflexible polymers on a lattice exhibit two-state collapse transition that captures some essential features of the allor-none folding transition of small globular proteins. More complex, decorated with some structural details, lattice polymers provide a very powerful means for study of protein dynamics and thermodynamics and protein structure prediction.

1.1 Reduced Models of Chain Molecules The torsional rotations, only around the main-chain backbone bonds, make the conformational space of chain molecules enormous in size (Flory 1969). For a chain containing N single bonds, the number of conformations is in the range of qN , where q is approximately equal to the number of distinct low-energy regions of the rotational potential. For a polyethylene chain, q would be 3. Obviously, when N is hundreds or many thousands, a detailed conformational analysis becomes impractical. Impractical are also detailed all-atom computer simulations, unless only very local conformational changes require examination. Thus, in order to make the problem tractable, simplified models have often been designed and studied (Milik et al. 1990; Kolinski and Skolnick 1996), either from statistical analyses or/and by computer simulations. As it will become apparent later, the statistical analysis itself is of rather limited utility and in typical cases requires quite drastic simplifications. Usually, it is difficult to estimate a priori the effect of such simplifications on the final results.

A. Kolinski (B) Faculty of Chemistry, University of Warsaw, Warsaw, Poland e-mail: [email protected]

A. Kolinski (ed.), Multiscale Approaches to Protein Modeling, C Springer Science+Business Media, LLC 2011 DOI 10.1007/978-1-4419-6889-0_1, 

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Let us consider two extremely simple models of polymers, one for idealized conformational statistics and the second for the first level of approximation for chain dynamics. These models can be solved rigorously by simple analytical considerations (Flory 1969). The first is the freely jointed chain (sometimes it is also called “the random flight model”). The freely jointed chain (see Fig. 1.1) consists of n segments of equal length l. Mutual orientations of the segments are completely uncorrelated. It is well known that for a sufficiently large number of segments, the mean-square end-to-end distance of such a chain scales with the number of segments as = l2 n. This result closely resembles the central formula obtained for a Brownian particle theory, where the mean-square displacement is proportional to time. It is also easy to show that the mean-square radius of gyration (a quantity that is easier to measure experimentally than the ) is related to as <S2 > = /6. The distribution of the end-to-end distance and distribution of the segment density is Gaussian. Such an ideal polymer random coil is frequently called the Gaussian chain, although the freely jointed chain is not uniquely Gaussian since other types of chains can also follow Gaussian statistics. Fig. 1.1 An example of the freely jointed chain

The simplifications of the physical properties of real polymers assumed in the freely jointed chain model are essentially of two types. First, the correlations between the chain segments, especially between those that are close to one another along the chain contour, are an important property of polymers and strongly depend on their chemical structure. As long as these correlations extend only to a distance small in comparison with the chain length, it is relatively straightforward to generalize the model by introducing various approximation of the local chain stiffness related to sometimes complex profiles of the rotational potential energy. All the short-range (short distance along the chain contour) correlations do not change the general picture. For all such ideal models = Cl2 n, and the value of the prefactor C depends on the shape of the rotational potential and the temperature. Approximations of the second type are much more significant and much more difficult to deal with. Namely, all ideal chains neglect the effective interactions between the chain segments that are far away from one another along the chain

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but close to each other in space. On the most trivial level, the fact that two segments cannot occupy the same element of space must be taken into account. A rigorous analytical treatment of such “real” chains is not possible, although approximate theories exist (de Gennes 1979). Probably, the most famous is Flory’s mean-field theory (Flory 1953). The theory assumes that a balance between intramolecular interactions and those with solvent defines the average coil size. A quasi-chemical approximation is employed and an average Gaussian density of segments is assumed. The resulting formula describes the chain dimension as a function of temperature: α 5−α3 = const. (1−/T) n1/2

(1.1)

where α is the so-called expansion factor and is defined as α 2 = /

(1.2)

with denoting the ideal chain dimensions. Note that for T =  the chain dimensions become identical with the dimensions of the ideal chain. Thus, idea behind Flory’s “theta” () temperature closely resembles the Boyle temperature for real gases. At temperatures below , the chain undergoes a transition to a dense globular state, and this transition is somewhat similar to the gas–liquid transition of small molecule systems. However, the transition for flexible polymers is continuous and has most of the features of a secondorder phase transition (Kolinski et al. 1987b). At high temperatures (see Eq. (1.1)) ∼ n6/5 , and the average chain dimensions are much larger than for an equivalent ideal chain. Interestingly, despite a rather poor estimation of chain entropy and internal energy, Flory’s theory gives quite an accurate estimation of the free energy and conformational properties of chain molecules. Such a cancellation of errors is quite typical of mean-field-type theories. Ideal chain statistics provides a zero-order picture of the protein denatured state, while Flory’s theory is a zero-order approximation for the folding (or collapse) transition. The approximation is quite crude for several reasons. First, protein chains are relatively stiff polymers and the limit of infinitely long chains is hardly satisfied even for large proteins (Creighton 1993). Second, proteins are heteropolymers with highly specific patterns of intramolecular interactions (Branden and Tooze 1991). Even in the random coil state, there is a significant extent of residual structure. Thus, the mean-field theory is hardly applicable. We will address these issues later in more detail. Somewhat analogous to ideal chain statistics, models for ideal chain dynamics were designed. Probably the best known of these is the Rouse model (Rouse 1953), shown in a schematic fashion in Fig. 1.2. It assumes that a flexible polymer chain can be represented as a chain of points joined by harmonic springs of equal strength. This model is analytically solvable. The results are quite interesting. For short times, when the average displacements of chain segments must be small in comparison with the coil size a single segment moves according to

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Fig. 1.2 Schematic drawing of beads-and-springs Rouse chain

(r)2 ∼ t1/2

for

l2 < (r)2 <<S2 >

(1.3)

while at longer times the “regular” diffusion is recovered and the mean-square displacement of a segment follows the mean-square displacement of the center of mass of the coil, (r)2 ∼ t, with the diffusion coefficient proportional to n−1 and the longest relaxation time proportional to n2 . It is easy to see that the Rouse model neglects several basic aspects of the physics: It ignores chain volume and the resulting topological restrictions, i.e., a “phantom” chain approximation, and does not take into account the non-uniform flexibility of copolymers and does not account for hydrodynamic interactions (although some extensions of the Rouse model can do this in a highly approximate way). These approximations are more serious for proteins than for flexible long polymers.

1.2 Simple Lattice Polymers Lattice models of simple liquids, except for providing a clear explanation of the entropy of mixing for an ideal solution, are not so useful. The opposite is the case for polymers. In polymers complex correlations can extend to distances many times larger than the sizes of single monomers. Thus, the local details are of less importance. The two ideal models described in the previous section have close lattice analogs. The freely jointed chain can be represented on a regular lattice, and the asymptotic properties remain unchanged (de Gennes 1979). Since on a lattice the allowed angles between consecutive segments belong to a discrete set, the only differences would be seen for short distances along the chain. The Flory’s type of real chain could be modeled on a lattice in a straightforward and efficient fashion. The simplest possibility is a chain with excluded volume (double occupancy of lattice sites is prohibited) having attractive interactions for non-bonded nearest neighbors. The idea is explained in Fig. 1.3. Such a model enables the detailed Monte Carlo study of polymer collapse transitions for various patterns of interactions and for various topologies of the model polymers (branched chains,

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Fig. 1.3 Square lattice polymers. An ideal chain (left) and a real (with excluded volume) chain (right)

macrocycles, etc.). Interestingly, the critical exponent for an athermal linear chain with excluded volume (the limited case of a high-temperature system) estimated from numerous computational experiments is close (however, clearly not identical) to the 6/5 obtained from Flory’s theory. It also has been proven that the collapse transition for long flexible polymers is continuous and that the observed physics does not depend on the particular type of lattice used. Usually, sampling of the conformational space of a lattice polymer is carried out with the use of various Monte Carlo techniques (Binder et al. 2004; Smith and Lisal 2002; Pakula 2004). A Monte Carlo procedure could be employed to build a large number of completely independent random conformations. Then, such a statistical ensemble can be used for the statistical analysis of the conformational and thermodynamic properties of the model. Alternatively, the ensemble can be constructed in a long iterative process of conformational transformations of a single chain or with a collection of chains. For technical reasons, the second possibility is recommended for studies of multichain systems, where growing all chains in parallel without introducing a statistical bias would be a difficult task (Binder et al. 2004; Smith and Lisal 2002; Frenkel and Smit 2001). Computer simulations of single long flexible polymer provided some important insights into nature of coil–globule collapse transition. It has been shown that with increasing chain length the average size of polymer coil changes faster with temperature and that the collapse transition becomes sharper, although it is always continuous (Kolinski et al. 1987b). Critical exponents describing the chain dimensions at various conditions of solvent quality and temperature have been determined from extensive Monte Carlo simulations for polymers of various topology of the main chain: simple linear, branched (especially star-branched) (Hsu et al., 2004; Sikorski and Romiszowski 1996; Sikorski 1993), and ring polymers. Also, computer modeling of polymers stimulated development of new computational techniques (Grest et al. 1996; Freire 1999; Likos 2006).

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The ideal dynamics of the Rouse chain (see the previous section) also has a lattice analog. Imagine a simple lattice chain, which is ideal (lacking excluded volume), and consequently it is also a “phantom” chain – fragments of a chain can cross its own paths during a random motion. Stochastic dynamics could be simulated as a long sequence of local (involving few bonds) conformational transitions at randomly selected positions along the chain (see Fig. 1.4). It has been shown a long time ago by Verdier and Stockmayer (1962) that the long-time dynamics of such model is equivalent to the Rouse dynamics. Lattice chains could easily be modeled as “real” chains, having excluded volume and topological constraints on their motion. This opens the possibility for computational studies of various complex dynamic phenomena, including the mechanism of polymer collapse (protein folding), diffusion in a restricted space, diffusion in dense solutions. It has been shown that excluded volume of chain molecules (and long-range interactions in general) leads to somewhat stronger dependencies of diffusion coefficient and the longest relaxation time on the chain length, when compared to the corresponding relations for the ideal chains in an infinitely diluted solution. Computer simulations were especially helpful in understanding of mechanism of polymer diffusion in gel, concentrated solutions, and in polymer melts. A system of many mutually entangled polymers is probably one of the most complex (if not the most complex) examples of classical multibody problems. It has been shown that the famous “reptation” theory of de Gennes (1979) describes very well the motion of flexible polymers in gel. The term “reptation” relates to snake-like motion of a polymer chain throughout the net of obstacles superimposed by the crosslinked gel. At the same time, many computer simulations demonstrated that the situation in solutions and melts is more complex (Kolinski et al. 1987a, c, and d) and that the mechanism of diffusion cannot be described in the framework of simple “reptation” theory (Skolnick and Kolinski 1990; Sikorski et al. 1994; Di Cecca and Freire 2002).

Fig. 1.4 Verdier–Stockmayer dynamics of a short simple cubic lattice chain showing typical lattice moves: (a) the corner flip, (b) three-bond permutation, (c) the crankshaft move, and (d) the chain-end move

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1.3 Simple Lattice Polymers with Protein-Like Features The collapse transition of a long flexible polymer chain is continuous. But, relatively short natural polypeptide chains undergo reversible pseudo-first-order cooperative transitions from a random coil denatured state to a structurally organized dense globular state (Anfinsen 1973). Since the single protein–solvent system is rather small, it is probably better to describe this transition as all-or-none. This way one can avoid an implicit reference to the thermodynamic limit. At the same time, the term all-ornone refers to a negligible population of the folding intermediates at the transition temperature. It has been pointed out that polypeptides are relatively stiff polymers. Perhaps, the chain stiffness itself can induce a more cooperative collapse transition. To check this hypothesis, extensive simulations were done almost 25 years ago (Kolinski et al. 1986a). The model chains studied were relatively short and consisted of 50–400 segments restricted to the diamond lattice (see Fig. 1.5). The diamond lattice has been chosen because of its tetrahedral valence angle and the qualitative similarity of its stretched (trans-conformation) segments to the β-strands in globular proteins. Short-range and long-range interactions were modeled in a simple way. Local stiffness was controlled by a potential energy preference for the expanded trans-conformation with respect to the two gauche conformations. Attractive longrange interactions were accounted for with a simple contact potential for the nearest non-bonded neighbors on the lattice. It is probably the simplest possible model of a semiflexible homopolymer in a thermodynamically poor solvent. The degree of stiffness can be controlled by changing the ratio of the stiffness parameter to the contact energy parameter. Monte Carlo simulations revealed interesting behavior for such a simple system. For moderately stiff polymers, the collapse transition was continuous, qualitatively similar to the collapse of a chain with unrestricted flexibility. However, at some critical ratio of the stiffness parameter to the segment attraction parameter, the collapse transition became highly cooperative, with all-or-none thermodynamic characteristics. At the transition temperature, the semiflexible polymers exhibit existence of metastable states, characteristic for first-order phase transitions.

Fig. 1.5 A short fragment of a model chain restricted to the diamond lattice. Semiflexible chains have a preference for the expanded trans-type conformations. The long-range interactions are modeled by a contact potential

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Fig. 1.6 Dimensions of short polymeric chains as a function of temperature. The solid line corresponds to the case of flexible chain and the dashed line describes behavior of semiflexible polymer. Tf indicates the collapse (or folding) temperature

This is illustrated in Fig. 1.6. Two types of structures coexisted at the transition temperature. The swollen random coil state exhibited a low density of contacts between the chain segments and relatively low-average lengths of the fully expanded segments. Upon collapse, the length of the expanded strands increased abruptly, accompanied by an abrupt increase of the number of polymer–polymer contacts. In this way the entropy of the random coil has compensated for the low potential energy of the globule. For relatively short chains, the globule had the structure of a bundle of parallel strands. There was a hypothesis that the collapse transition itself induces formation of secondary structures in proteins. This is true, but only when there is an interplay between the short-range conformational stiffness and the long-range interactions. Such interplay seems to be a fundamental feature of proteins and the major factor responsible for the folding cooperativity. For highly flexible polymers, the local ordering in the globular state was undetectable. Here, however, a word of caution needs to be exercised. The results described were obtained for a model of a homopolymer. Strong, specific sequence patterns of the long-range interactions may actually lead to some ordering of the globular structure. Specific interactions of side chains can also augment folding cooperativity (Pande et al. 1996). The model of a semiflexible polymer has one more striking feature. With increasing chain length (somewhere between 200 and 400 segments, depending on the degree of the local stiffness), the globular structure divides into domains of bundles having different orientations of their axes (Rutkowska and Kolinski 2007). This again resembles globular proteins, where longer polypeptide chains fold into two or more separate domains. Obviously, at the limit of a very long chain, where the persistence length becomes small in respect to the chain length of the model homopolymer, a continuous collapse transition should be recovered. The homopolymeric model of protein collapse has, however, an important non-protein-like feature. The globular structure, although highly ordered, is not unique. The average length of the extended

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strands and the distribution of their sizes may differ quite a bit between particular simulations. Also the topology of connections between the strands is not unique. It is interesting that the cooperative collapse transition of a semiflexible polymer could actually be predicted in the framework of a mean-field-type theory, as it was demonstrated a time ago by Post and Zimm (1979). Nevertheless, the picture emerging from the computer simulations is much deeper, and it is exact in the limits of model simplifications and small statistical errors of the Monte Carlo simulations. Models of homopolymeric semiflexible chains provide the zero-order approximation of the physics of globular protein collapse transition, where the interplay between secondary structure preferences (here the local stiffness) and the long-range interactions leads to the characteristic cooperative behavior (Kolinski et al. 1996).

1.4 Minimal Protein-Like Models The homopolymer model of a semiflexible chain, described in the previous section, has several important protein-like features, except the uniqueness of the globular structure. Real protein chains are heteropolymers, with amino acid units that differ in the strength and the physical nature of their short-range and long-range interactions. In a simplest possible approximation, there are two types of amino acids with respect to their hydrophobicity: polar (P), which tend to be exposed to the solvent on the protein surface, and hydrophobic (H), which tend to be buried inside the globule (Lau and Dill 1989). From the point of view of the chain flexibility, one may distinguish between the three main classes (Skolnick et al. 1989): amino acids (or short sequences of amino acids) that tend to adopt extended (e) conformations, residues that tend to build helical structures (h), and, usually more flexible, residues that prefer coil or turn-type local structures (c). Assuming a beta-barreltype target structure, it is natural to limit these possibilities to the two cases: e and c. Thus, in a very crude approximation, the β-type proteins are built from four types of amino acids: He, Hc, Pe, and Pc. By using the diamond lattice approximation described previously, it is relatively easy to design a number of sequences based on these four types of residues that undergo the all-or-none transition to a unique three-dimensional structure, although the level of the folding cooperativity is rather low. Even a quite complex Greek-key topology (seen in many real proteins) of the globule could be designed and obtained, with high reproducibility, in computer simulations (Kolinski et al. 1986b; Skolnick et al. 1988). This, however, requires somewhat more complex patterns of the chain hydrophobicity and flexibility (Skolnick et al. 1989). Also simple helical motifs could be designed and folded in silico employing these simple rules. Of course, instead of e-type residues, the htype had to be used, for which the right-handed three-bond turns have a preferential energy in the short-range interactions (Grosberg and Khokhlov 1994; Kolinski et al. 1996). Slightly after the studies outlined above, a different approach to design a minimal protein-like model has been proposed by Chan and Dill and pursued by many

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others (Dill et al. 1995; Dinner et al. 1994, 1996). In its classic form, the model assumed the simple cubic lattice representation of the chain conformation and two types of residues: polar (P) and hydrophobic (H). In many applications, the target structure had the form of a 3×3×3 cube consisting of 27 model amino acids. It has been shown that it is possible to design sequences that have a single minimum of the chain conformational energy, which is consistent with the cube. This result was exact, since it was feasible to enumerate all possible compact conformations of such short chains (Dill 1999). The model stresses the hydrophobic collapse as the main feature of protein folding and was used in studies of protein-folding thermodynamics, stability, and folding pathways. Many varieties of modifications to this type of model have been proposed and studied in great detail (Sun et al. 1995; Kolinski and Skolnick 1996; Chen et al. 2004; Abkevich et al. 1994, 1996; Sali et al. 1994; Li et al. 1996; Micheletti et al. 1998). The simple polymer approach to protein folding has proven itself to be very productive – through these the most general features of protein folding become better understood. In spite of many successes, the cubic lattice hydrophobic polar (HP) models (and closely related models) have intrinsically some shortcomings that are difficult to ignore. First of all, the notion of secondary structure, so important in real proteins, is quite unclear in these models. Second, the cube geometry has a peculiar pattern of exposed-buried residues. The 27-mer cube has eight highly exposed corner residues, and obviously, unlike in real proteins, all of them are far apart from each other in space. There is only one completely buried residue in the center of the cube. Again, a typical protein domain has about half of its residues buried inside. These shortcomings can be to some extent improved upon by a different definition of interactions. Recent studies by Kaya and Chan (2002) have shown that in order to reproduce true two-state all-or-none folding transitions, more than two types of amino acids need to be included in the model sequences. Their findings indicated also the high significance of the interplay between short-range and long-range interactions. This is in qualitative agreement with the results from the diamond lattice models described in the previous section. Recently, the problem of the minimal model of protein structure and protein folding has been revisited (Pokarowski et al. 2003, 2005). Chains restricted to the face-centered cubic (fcc) lattice were used to represent protein conformational space. This lattice has a higher coordination number, z = 12, and allows for more flexibility than do other simple lattices. Thus, the lattice anisotropy effects are perhaps less severe. Moreover, this lattice allows for the crude, although not trivial, representation of all basic protein-folding motifs: β-type, α-type, and mixed α/β motifs. Representatives of all these motifs have been designed and studied in computational experiments. It was assumed a priori that a minimal model might require three types of potentials to mimic the complex network of molecular interactions in real proteins. In agreement with previous finding, the short-range (sequence-dependent local conformational stiffness) and the long-range contact interactions (with two types of residues, polar and hydrophobic) were implemented. Additionally, the effect of the main-chain hydrogen bonds has been taken into account in the form of a directional component in the pairwise potentials. In the

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simplest form in the β-type models, it seems to be enough to make the polar-polar (PP) interactions orientation dependent. Indeed, in globular proteins the contacting polar side groups on the surface of a globule are almost always approximately parallel. Without the explicit model of the side groups, their hypothetical orientation can be determined easily from the mutual orientation of the two-segment fragments of the interacting nearest neighbors. A more general definition of the ersatz of the hydrogen bonds could be designed, which has the same meaning and the same effect on the system behavior. Let us focus on the example of the relatively complex Greek-key motif of a two-sheet, six-stranded β-barrel. In order to perform detailed analysis of the system thermodynamics, the Replica Exchange Monte Carlo (REMC) sampling method (Hukushima and Nemoto 1996), using a carefully designed set of local conformational modifications, has been combined with the histogram analysis of the density of states (Ferrenberg and Swendsen 1989). A large number of simulations have been performed spanning a wide range of the relative strength (various scaling factors) of the short-range, pairwise hydrophobic, and orientation-dependent polar group interactions. The low-energy structures, including the putative native-like structure and a set of partly folded structures, some of them near-native, were extracted from the REMC pseudo-trajectories. For each one of these structures, its potential energy could be calculated as a function of the scaling factors for particular types of interactions. The assumption that the native structure potential energy has to be the lowest leads to a set of inequalities, with the interaction scaling parameters as the free variables. Solution of these inequalities determines a set of “good” parameters for the model. An important result of such analysis is that all energy parameters must have non-zero contributions. Otherwise, the native structure would not have the minimum energy. Within the set of allowed interaction parameters, there are many possibilities. This parameter space has been explored in additional long simulations, and the level of folding cooperativity has been estimated for every set of scaling parameters from a relatively dense grid within the allowed subspace. The highest, essentially purely two-state, cooperativity has been observed for the system with relatively strong pairwise interactions (hydrophobic and polar, orientation dependent) and a moderate short-range conformational stiffness. The same has been observed for other structural motifs studied. Thus, it has been proven that the proposed model is a minimal one – the three types of interactions are necessary for the protein-like uniqueness of the native structure and the highly cooperative all-or-none folding transition. Interestingly, all designed motifs exhibited some degeneracy for the native structure. For instance, for the Greekbarrel model, 20 structures have exactly the same topology and exactly the same patterns of interactions for all components of the model force field. The only differences were geometrical details, including mirror-image structures. This is probably physical (except of the mirror-image structures) – there are fluctuations in the native structure of the real proteins, and there are known examples where mobile parts contribute to the entropic stabilization of the native state. The simulations for various motifs have shown that a higher degeneracy of the native state leads to a higher cooperativity of the folding transition. Due to the higher entropy of the globular state, its free energy is lower and consequently the free energy gap between the

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globular state and the manifold of random structures is larger. This leads to a very clear two-state behavior of the model system (Pokarowski et al. 2003, 2005). The minimal protein-like models capture the most general physics of globular protein folding. Nevertheless, they are only generic models, which are of a quite limited use for addressing the more detailed problems of specific folds and protein interactions. Possibly, the fcc model described above, with a larger alphabet for the amino acid sequences, can be used for the crude modeling of specific proteins, although the expected accuracy would be rather low, only an overall topology might be reproduced correctly.

1.5 High-Coordination Lattice Protein Models For many reasons, lattice approaches to polymer and biopolymer modeling are very appealing. Conformational transitions could be rapidly calculated in the discrete space of the lattice. The energy landscape is smoother; many local energy barriers are eliminated due to the simplification of the interaction schemes. Moreover, energy calculations are much faster due to the discrete set of allowed distances and angles. This is especially true for proteins, where the interactions are actually rather complex. On the other hand, a higher geometrical accuracy is needed for the study of specific proteins and for protein structure predictions. For this reason, several lattice models of intermediate-to-high resolution were developed in past, and some of them have proven to be rather effective tools for protein molecular modeling (Kolinski and Skolnick 1996; Kolinski 2004; Kolinski et al. 1995, 1996). In several studies of proteins and protein-like models, the three-dimensional “chess-knight” representation of the alpha-carbon trace was used (Kolinski et al. 1996). The chess-knight lattice is built upon a set of vectors type [2,1,0]. There are 24 such vectors; 6 permutations of the coordinates and 4 permutations of the signs. Due to the restrictions on the values of the planar angles of the alpha-carbon trace in real proteins, the number of possible orientations of a Cα −Cα virtual bond should be smaller and dependent on the orientation of the preceding and following bonds. The chess-knight representation of protein structures is more realistic than chains on simple lattices (compare Fig. 1.7 with Fig. 1.5). However, this model is still an intermediate between the protein-like models and models applicable to real proteins – the effects of the lattice anisotropy remain large. For instance, geometrical fidelity of a projection of short β-strands onto the chess-knight lattice depends on the orientation of the projected fragment with respect to the principal axis of the Cartesian coordinate system. Lattice effects are particularly harmful for the simulated dynamics of lattice systems – the relaxation processes could be significantly distorted. To overcome this problem, a modification of the 210 representation was proposed. The set of Cα -trace vectors has been expanded adding the vectors type [1,1,1] and [2,1,1]. The total number of allowed orientations is thereby increased to 56. The vectors type [2,0,0] were excluded for a technical reason relating to the convenience of coding the excluded volume using additional vertices of the underlying simple cubic lattice. The model becomes significantly more flexible, and the

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Fig. 1.7 A short fragment of the “chess-knight” chain with side groups restricted to the lattice

effect of the lattice anisotropy decreases. In spite of rather non-physical fluctuations of the bond length, the overall accuracy and precision of the protein representation improves a lot. As a result, the model produces a plausible picture of polypeptide chain dynamics and enables the de novo prediction of simple low-resolution protein structures (Godzik et al. 1993). Obviously, the structure prediction algorithm requires a properly designed force field based on statistical potentials derived from known protein structures. Basic principles for the design of the interaction schemes for reduced models will be outlined later for a different lattice representation. Fluctuating bond models proved to be a milestone in lattice modeling of protein structures. Interestingly, in parallel the fluctuating bond concept was extensively employed in studies of generic polymeric systems (Carmesin and Kremer 1988). Several specific representations were developed. It has been proven that the dynamics of the fluctuating bond lattice models reproduces well the Rouse dynamics of the continuous space models. Flexibility and computational efficiency of the fluctuating bond models enabled the detailed study of the thermodynamics and dynamics of long polymers, including the extremely complex dynamics of multichain systems. These findings are important for protein modeling. They provide a justification for applications of the flexible lattice models in studies not only of protein structures but also of protein dynamics and protein-folding mechanisms (Kolinski and Skolnick 2004). Proteins are complex heteropolymers with 20 different side chains that are attached to the main-chain backbone, which is rather generic (with the exception of the proline residues), as in synthetic homopolymers or simple copolymers. Thus a satisfactory model of the main chain is just a starting point for an acceptable protein model. Let us consider a more exact fluctuating bond model than the one described above (Kolinski and Skolnick 1994). In this model the main-chain backbone is also

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reduced to the alpha-carbon trace. The number of backbone vectors is equal to 90. These vectors belong to the following set: {[3,1,1],. . . [3,1,0],. . . [3,0,0],. . . [2,2,1], ...}. The amplitude of the bond fluctuations in this model is relatively small, and the lattice anisotropic effects are essentially negligible. The excluded volume of the main chain can be modeled in a convenient way. It is enough to associate with every lattice position of the alpha carbon the 18 closest points of the underlying simple cubic lattice. These 18 lattice vertices (plus the central one) are excluded to other alpha-carbon units, which are also clusters of 19 lattice points. Such lattice coding of excluded volume simplifies immensely the simulation process – the main chain overlaps could be detected with small computational cost. The main-chain discrete geometry provides a convenient reference frame for the definition of the side-chain positions. For each amino acid, a database of known protein structures could be scanned and a database of the observed side-chain rotamers created, assuming certain level of resolution of the model. During the simulations every update of the main-chain conformation has to be associated with an update of the sidechain positions. It could be efficiently done with a help of “prefabricated” sets of allowed side-chain coordinates. The side chains could be modeled as single or multiple interaction centers. They could be restricted to the underlying lattice or could be off-lattice, however with the lattice-bounded reference frame. In the published applications, of this type of fluctuating bond model, a single off-lattice sphere for rotamers was used. The model enabled reproducible de novo folding of several small proteins, with an accuracy of 2–4 Å with respect to their crystallographic structures after the best superposition with the computational models (Kolinski and Skolnick 1994). The CABS (Cα −Cβ –side group) lattice-based model employs a high-resolution discretization of the polypeptide conformational space (Kolinski 2004). As in previously described models, the framework of a polypeptide chain representation is the alpha-carbon trace (see Fig. 1.8 for explanation of the CABS reduced

Fig. 1.8 Schematic drawing of a short fragment of the CABS model

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representation). The alpha carbons are located on the vertices of a simple cubic lattice with the mesh size equal to 0.61 Å. The virtual bonds connecting the alpha carbons belong to the set of 800 vectors type of v = [i, j, k]. The integer coordinates i, j, k are the all possible triplets for which 29 ≤ |v|2 ≤ 49. This set of vectors reproduces the Cα −Cα distance, equal to 3.78 Å, with fluctuations in the range of ±10%. Protein structures could be approximated with an accuracy of about 0.35 Å cRMSD (coordinate Root-Mean-Square Deviation) after the best superposition of the model Cα -trace with corresponding coordinates of the experimental structure. An example is given in Fig. 1.9. The side groups are modeled by two centers of interactions: beta carbons and the centers of the remaining part of the side chain (where applicable). These are not restricted to the lattice and their positions with respect to the backbone are derived from a proper statistics of the known protein structures. Two, the most probable, rotamers are defined for each residue. The excluded volume is modeled by a set of hard spheres centered on the alpha and beta carbons and in the middle of the Cα −Cα virtual bonds. The side groups are treated as soft spheres. Fig. 1.9 High-resolution lattice representation of the alpha-carbon trace of small globular protein (domain B of protein G) – the CABS model. The average accuracy in respect to the crystallographic coordinates is about 0.35 Å. The most probable coordinates of the side-chain united atoms are calculated basing on the Cα -trace geometry and proper statistics of high-resolution protein structures

The force field of the CABS model consists of several components mimicking the real physical interactions in proteins (Kolinski 2004). The generic short-range conformational biases simulate characteristic protein-like chain stiffness. A set of sequence-specific potentials simulate the short-range conformational propensities. Directional potentials of interactions between alpha carbons and between the centers of Cα −Cα virtual bonds simulate the structure-ordering effect of the main-chain hydrogen bonds. Pairwise interactions between the side-group united atoms are “context-dependent,” i.e., they depend on the mutual orientation of the interacting side groups and on the conformations of the corresponding two-bond segments of the main chain. In an implicit way, the pairwise interactions take an approximate account of the average effects of the surrounding solvent. This force field is knowledge-based – the statistical potentials of mean force are derived from structural regularities seen in known high-resolution protein structures (Skolnick

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et al. 1997a; Kolinski 2004). It is perhaps worth noting that the CABS modeling tool performs certainly no worse (and is computationally more effective) than a similar continuous space-reduced model (Boniecki et al. 2003). This shows that the high-resolution lattice approximations are free of lattice artifacts and, due to their computational efficiency, perfectly suited for large-scale applications. Some practical applications of the CABS model are described in the next section. All the high-coordination lattice models described so far have focused on the design of a convenient representation of the main chain of polypeptides, which subsequently are “decorated” with the side chains, using the main-chain backbone as a reference frame. The SICHO (SIde CHain Only) model is based on a completely different concept (Kolinski et al. 2001). In this approach, the explicit lattice approximation uses the fluctuating bond framework for modeling the virtual chain connecting the positions of the centers of mass of polypeptide side groups. Opposite to the CABS model, in the SICHO model the positions of the main-chain atoms are defined in the reference frame of the pseudo-chain connecting the side groups. The SICHO concept is based on the fact that the packing of the side chains is probably the most sequence-specific property of globular proteins. In principle, the sidechain-based models should be computationally faster than the main-chain-based ones.

1.6 Protein Folding and Structure Prediction with Lattice Models High-resolution SICHO and CABS models (and their clones) have been used in a variety of applications. These include ab initio structure prediction (Ortiz et al. 1999; Skolnick et al. 2001), study of protein dynamics, folding pathways and thermodynamics, prediction of protein structure from sparse experimental data and distant homology, or comparative, structure modeling (Kolinski et al. 1999; Kolinski and Skolnick 2004; Pierri et al. 2008; Skolnick et al. 1997b, 1998, 2003). The ab initio folding with the lattice models is practical only for relatively small (say up to 150 residues) and topologically not too complex proteins. With the increasing size of a query protein the success ratio, as well as the accuracy of the produced structures, decreases. Small proteins (50–75 residues) can be folded to a resolution range approaching 2 Å, while for a 150-residue structure the accuracy would be nearer 3–6 Å, depending on fold complexity. It should be pointed out that the bottleneck of the accuracy for reduced models is not their reduced representation but rather the deficiencies of their force fields. The force fields of the reduced models are being permanently updated and in the future this should be the main factor leading to improvements in the algorithms performance. There is a suggestive result partially confirming this. The CABS model has been used many times for comparative modeling, where in addition to the force field, the folding process has been guided by a set of spatial restraints extracted from structures of homologous (or structurally analogous) proteins. In these circumstances the resulting models could be even as good as the best crystallographic structures with a resolution of

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1.0–1.5 Å, at least for the main-chain atoms (Kolinski and Bujnicki 2005). Every 2 years, the Community Wide Experiment on the Critical Assessment of Techniques for Protein Structure Prediction (CASP) is organized in order to assess the current status of in silico protein structure prediction. The idea of the experiment is simple. Experimentalists provide sequences of amino acids of a number of proteins for which the structures are expected to be solved in the next few months. During this time theoreticians try to predict these structures and deposit them in the CASP databank. A variety of modeling techniques are used by researchers from around the world. Afterward, when the experimental structures become available, a group of experts assesses quality of predictions. Interestingly, the high-resolution latticebased modeling methods systematically perform very well and are among the best methods for protein structure prediction. An example of the CASP6 prediction using the CABS lattice modeling is given in Fig. 1.10. The accuracy of this particular prediction is about 3.0 Å after the best superposition of the predicted structure onto the experimental one (Kolinski and Bujnicki 2005).

Fig. 1.10 Side-by-side view of the crystallographic structure and the predicted structure (the CABS model) for one of the CASP6 targets (target # T0223)

Finally, it should be noted that the reduced structures of SICHO or CABS models can be used as a meaningful starting point for the all-atom reconstruction and structure refinement (Feig et al. 2000; Kolinski and Bujnicki 2005). Such procedures are now pursued in several laboratories, opening the possibility for multi-scale modeling of large biomolecular systems.

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Binder K, Müller M, Baschnagel J (2004) Polymers models on the lattice. In: Kotelyanskii MJ, Theodorou DN (eds) Simulation methods for polymers, M. Dekker, New York, NY Boniecki M, Rotkiewicz P, Skolnick J, Kolinski A (2003) Protein fragment reconstruction using various modeling techniques. J Comp Aid Mol Des 17:725–738 Branden C, Tooze J (1991) Introduction to protein structure. Garland, New York, NY Carmesin I, Kremer K (1988) The bond fluctuation method: a new effective algorithm for the dynamics of polymers in all spatial dimensions. Macromolecules 21:2819–2823 Chen H, Zhou X, Chih YL, Chan GK (2004) Kinetic analysis of protein folding lattice models. Mod Phys Lett B 18:163–172 Creighton TE (1993) Proteins: structures and molecular properties. W. H. Freeman, New York, NY De Gennes PG (1979) Scaling concepts in polymer physics, 1st edn. Cornell University Press, New York, NY Di Cecca A, Freire JJ (2002) Monte Carlo simulation of star polymer systems with the bond fluctuation model. Macromolecules 35:2851–2858 Dill KA (1999) Polymer principles and protein folding. Prot Sci 8:1166–1180 Dill KA, Bromberg S, Yue K, Fiebig KM, Yee DP, Thomas PD, Chan HS (1995) Principles of protein folding – a perspective from simple exact models. Prot Sci 4:561–602 Dinner A, Sali A, Karplus M, Shakhnovich E (1994) Phase diagram of a model protein derived by exhaustive enumeration of the conformations. J Chem Phys 101:1444–1451 Dinner AR, Sali A, Karplus M (1996) The folding mechanism of larger model proteins: role of native structure. Proc Natl Acad Sci USA 93:8356–8361 Feig M, Rotkiewicz P, Kolinski A, Skolnick J, Brooks CL 3rd (2000) Accurate reconstruction of all-atom protein representations from side-chain-based low-resolution models. Proteins 41: 86–97 Ferrenberg AM, Swendsen RH (1989) Optimized Monte Carlo data analysis. Phys Rev Lett 63:1195–1198 Flory PJ (1953) Principles of polymer chemistry. Cornell University Press, New York, NY Flory PJ (1969) Statistical mechanics of chain molecules. Wiley, New York, NY Freire J (1999) Conformational properties of branched polymers: theory and simulations. Branched polymers II. Advances in polymer science, vol 143/1999. Springer, Berlin, pp 35–112 Frenkel D, Smit B (2001) Understanding molecular simulation. From algorithms to applications. Computational science series, vol 1, 2nd edn. Academic, New York, NY Godzik A, Kolinski A, Skolnick J (1993) De novo and inverse folding predictions of protein structure and dynamics. J Comp Aid Mol Des 7:397–438 Grest GS, Fetters LJ, Huang JS, Richter D (1996) Star polymers: experiment, theory, and simulation. Adv Chem Phys 104:67–163 Grosberg AY, Khokhlov AR (1994) Statistical physics of macromolecules. American Institutes of Physics Press, New York, NY Hsu HP, Nadler W, Grassberger P (2004) Scaling of star polymers with 1–80 arms. Macromolecules 37:4658–4663 Hukushima K, Nemoto K (1996) Exchange Monte Carlo method and application to Spin Glass Simulations. J Phys Soc Jpn 65:1604–1608 Kaya H, Chan HS (2002) Origins of chevron rollovers in non-two-state protein folding kinetics. Phys Rev Lett 90:258104 Kolinski A (2004) Protein modeling and structure prediction with a reduced representation. Acta Biochim Pol 51:349–371 Kolinski A, Skolnick J (2004) Reduced models of proteins and their applications. Polymer 45: 511–524 Kolinski A, Betancourt MR, Kihara D, Rotkiewicz P, Skolnick J (2001) Generalized comparative modeling (GENECOMP): a combination of sequence comparison, threading, and lattice modeling for protein structure prediction and refinement. Proteins 44:133–149 Kolinski A, Bujnicki JM (2005) Generalized protein structure prediction based on combination of fold-recognition with de novo folding and evaluation of models. Proteins Suppl 7(61):84–90

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Sali A, Shakhnovich E, Karplus M (1994) How does a protein fold? Nature 369:248–251 Sikorski A (1993) Monte Carlo study of the dynamics of star-branched polymers. Macromol Theory Simul 2:309–318 Sikorski A, Kolinski A, Skolnick J (1994) Dynamics of star branched polymers in a matrix of linear chains: a Monte Carlo study. Macromol Theory Simul 3:715–729 Sikorski A, Romiszowski P (1996) Motion of star-branched vs. linear polymer: A Monte Carlo study. J Chem Phys 104:8703–8712 Skolnick J, Jaroszewski L, Kolinski A, Godzik A (1997a) Derivation and testing of pair potentials for protein folding. When is the quasichemical approximation correct? Prot Sci 6:676–688 Skolnick J, Kolinski A (1990) Dynamics of dense polymer systems: computer simulations and analytic theories. In: Advances in chemical physics, vol 78. Wiley, New York, NY Skolnick J, Kolinski A, Kihara D, Betancourt M, Rotkiewicz P, Boniecki M (2001) Ab initio protein structure prediction via a combination of threading, lattice folding, clustering, and structure refinement. Proteins Suppl 5:149–156 Skolnick J, Kolinski A, Ortiz AR (1997b) MONSSTER: a method for folding globular proteins with a small number of distance restraints. J Mol Biol 265:217–241 Skolnick J, Kolinski A, Yaris R (1988) Monte Carlo simulations of the folding of beta-barrel globular proteins. Proc Natl Acad Sci USA 85:5057–5061 Skolnick J, Kolinski A, Yaris R (1989) Dynamic Monte Carlo study of the folding of a six-stranded Greek key globular protein. Proc Natl Acad Sci USA 86:1229–1233 Skolnick J, Zhang Y, Arakaki AK, Kolinski A, Boniecki M, Szilágyi A, Kihara D (2003) TOUCHSTONE: a unified approach to protein structure prediction. Proteins 53:469–479 Smith WR, Lisal M (2002) Direct Monte Carlo simulation methods for nonreacting and reacting systems at fixed total internal energy or enthalpy. Phys Rev E 66:011104 Sun S, Brem R, Chan HS, Dill KA (1995) Designing amino acid sequences to fold with good hydrophobic cores. Protein Eng 8:1205–1213 Verdier PH, Stockmayer WH (1962) Monte Carlo calculations on the dynamics of polymers in dilute solution. J Chem Phys 36:227–235

Chapter 2

Multiscale Protein and Peptide Docking Mateusz Kurcinski, Michał Jamroz, and Andrzej Kolinski

Abstract The number of functional protein complexes in a cell is larger by an order of magnitude than the number of proteins. The experimentally determined three-dimensional structures exist for only a very small fraction of these complexes. Thus, the methods for theoretical prediction of structures of protein assemblies are extremely important for molecular biology. Association of two (or more proteins) always induces conformational changes of the individual components. In many cases, these induced changes are relatively small and involve mostly the side chains at the association interface. In such cases, the approaches of rigid-body docking of two (or more) structures are quite successful. Quite frequently, however, the docking-induced conformational changes are significant. In such cases, prediction of the resulting structures is extremely challenging. The cases, where experimental structures of some components do not exist, are yet even more difficult. In this chapter, we briefly overview the existing in silico docking methods and describe a multiscale strategy of unrestricted flexible docking of proteins and peptides.

2.1 Introduction In eukaryotic cells, an average protein can participate in several protein–protein (or protein–nucleic acid) complexes. The number of such complexes is larger by an order of magnitude than the number of proteins. Since the number of experimentally solved protein structures (about 60,000) is a small fraction of all proteins, the fraction of structurally annotated protein complexes is very small. Thus, the theoretical, in silico, prediction of molecular structures of multimeric protein assemblies is one

A. Kolinski (B) Faculty of Chemistry, University of Warsaw, Warsaw, Poland e-mail: [email protected]

A. Kolinski (ed.), Multiscale Approaches to Protein Modeling, C Springer Science+Business Media, LLC 2011 DOI 10.1007/978-1-4419-6889-0_2, 

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of the most important task of bioinformatics and computational biology (Wodak and Janin 1978; Valencia and Pazos 2002; Salwinski and Eisenberg 2003; Aloy and Russell 2004). There are relatively dependable computational methods for socalled rigid docking. These methods are applicable, provided that the structures of individual components are known and the conformational changes of these components induced upon docking are small. For proteins of known structures, the last requirement is approximately fulfilled quite frequently (Ritchie 2008). Then the problem reduces to generation of a large number of possible poses, according to the shape complementarity of the components and scoring of binding poses by interaction patterns of the interfaces. The latter task is by no means trivial since at least some of the side chains at the interface certainly change their conformations with respect to the conformations seen in the monomeric state or in different complexes. At the moment, the knowledge-based statistical potentials, either atom-wise or united-atom-wise ones, seem to be most productive in scoring of protein–protein interactions. The ultimate goal of the protein docking could be described as follows: having just a set of sequences, find the structure (structures) of the possible assembles. In general, this may appear to be not feasible, at least at present, but maybe this is not so hopeless. Firstly, with the constant progress in protein structure prediction, mainly via rectification of comparative modeling, now it is possible to predict monomeric structures at least for a half of protein sequences and at least with a moderate resolution. Secondly, provided that bioinformatics methods are developed for identification of structure fragments that may change upon the docking, it should be possible to design methods for semi-flexible docking that accounts for the allowed conformational changes of fragments of the components’ structures. A step toward such a docking methodology is described in this chapter. The method employs a multiscale modeling of proteins and peptides. It is based on CABS modeling software. CABS is a high-resolution, coarse-grained, protein modeling tool (the acronym stands for the united atoms representing a residue in a polypeptide chain: CA alpha carbon of the main chain, CB -beta carbon, and S – the center of side group). The CABS protein structure representation is based on united-atom description of protein structure, where a single residue is represented by several (three or four, depending on the size of side chains) united atoms. The conformational space of CABS polypeptide chains is sampled by means of very efficient Monte Carlo schemes. Details of the CABS design are described in the first chapter of this book and in previous publications (Kolinski 2004). The spatial resolution of CABS allows for quite precise reconstruction of atomic details. This reconstruction process (Gront et al. 2007) for main-chain atoms is very fast and accurate within range of few tens of Angstrom. The reconstruction of side groups is less accurate and depends on the achieved accuracy of the Cα-trace fold. Below, we overview briefly the techniques for rigid docking problem, docking with a highly limited flexibility of some structural elements, and then we outline the more flexible (and fully flexible) docking based on a multiscale approach in which the CABS-based structure assembly is the key step of a molecular complex building procedures.

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2.2 Rigid Docking Procedures Suppose we know the three-dimensional structures of two proteins that form a dimer, although we do not know how these two proteins are posed in the complex, and which residues form the protein–protein interface. Finding the structure of the resulting complex is not a trivial task. Classical rigid docking consists of two or three fundamental steps (Vajda and Kozakov 2009). The first one is the generation of a large number of binary structures. The second one is scoring the structures according to the shape complementarity and interactions at the interface. Finally, one may perform rectification of the best structures by adjustments of conformations of the side chain at the interface. Finding plausible poses in rigid docking is not trivial – this requires a very effective search algorithm. Fast Fourier Transform makes it possible to reduce the six-dimensional problem to a one-dimensional problem. A number of algorithms have been developed for this purpose (Katchalski-Katzir et al. 1992; Vakser and Aflalo 1994; Vakser 1995; Mandell et al. 2001; Del Carpio-Muñoz et al. 2002; Chen et al. 2003; Carter et al. 2005; Kozakov et al. 2006; Sternberg et al. 1998). Alternatively, various geometric hashing procedures could be used (Fischer et al. 1995). The resulting poses, usually several thousands of them, need to be scored in order to produce a small number of plausible structures. Scoring functions span a wide range, from a simple shape complementarity (Chen et al. 2003), through the knowledge-based statistical potentials (Kozakov et al. 2006; Tobi and Bahar 2006; Zhang et al. 2005; Cerutti et al. 2005), physics-based force fields (Koehl 2006; Sheinerman et al. 2000; Jiang et al. 2002) to data-driven docking, supported by available biochemical information (Res and Lichtarge 2005; Res et al. 2005; Lichtarge et al. 1996; Dominguez et al. 2003; Nilges 1995; Anand et al. 2003; van Dijk et al. 2005). It has been also noted that rigid docking could be achieved in a different way. Instead of ab initio computing the assembly structure, sometimes it is more effective to predict binding interfaces of the interacting proteins and then perform docking in a limited conformational space. In some sense, this is yet another variant of data-driven docking (Jones and Thornton 1997; Burgoyne and Jackson 2006). Efficiency of various approaches to protein docking is systematically evaluated within the framework of community-wide experiments of Critical Assessments of PRediction of Interactions (CAPRI) (Carter et al. 2005; Janin et al. 2003).

2.3 Flexible Docking Usually, although not always, protein association induces conformational changes of the components (Bonvin 2006; Camacho and Vajda 2001; May and Zacharias 2005). In many cases, these conformational changes are essentially limited to the interface side chains (Andrusier et al. 2008). RosettaDock algorithm is well suited to deal with such cases (Gray et al. 2003; Daily et al. 2005; Wang et al. 2005). The procedure starts from rigid docking and then the side chains are optimized using

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either a rotamer library or free-space side-chain optimization. Such an approach proven to be very successful in blind predictions within CAPRI (Wang et al. 2007; Schueler-Furman et al. 2005). Recently, Rosetta modeling technology has been applied to fully flexible docking, or rather “folding and docking” (Das et al. 2009), of small homo-oligomeric protein assemblies. The method utilizes available experimental data: nuclear magnetic resonance (NMR) chemical shifts and residual dipolar coupling (RDC). Somewhat similar strategy for semi-flexible docking is adapted in ATTRACT (the name of the algorithm comes from attractive interactions of the interface residues) algorithm (Zacharias 2003; May and Zacharias 2007). The method employs a coarse-grained representation of side chains and the docking procedure consists of two steps: rigid docking and optimization of the resulting poses, allowing for flexibility of the side chains of interface residues. ATTRACT algorithm was also used in docking simulations allowing for backbone flexibility of the loop regions. It has been demonstrated that even limited flexibility improves the docking results for most of the tested cases. Small conformational changes, induced by docking, could be accommodated to some extent by reduced representations or/and coarse-grained potentials describing the interface residues. This is probably one of the reasons for surprisingly good performance of docking procedures based on just shape complementarity and smoothened details of the surface. Recently it has been shown that smoothed lowresolution representation of the surface residues leads to more consistent shape complementarity (Zhang et al. 2009). Another way to increase recognition specificity of the interfaces could be achieved by use of multibody knowledge-based potentials. For instance, four-body statistical pseudo-potentials proven to be useful in protein–peptide docking, allowing for full flexibility of the peptide moieties (Aita et al. 2010). Before the applications in protein docking, four-body potentials proven to be very effective efficient in scoring protein decoys (Krishnamoorthy and Tropsha 2003; Feng et al. 2010). In summary, while small, local conformational changes accompanying protein docking are relatively well handled by a variety of docking algorithms, large deformations of components are more difficult to predict. RosettaDock is one of few exceptions, where de novo prediction of new (compared to known structures of components) structures is sometimes feasible. The multiscale approach described below, based on CABS modeling software, bootstrapped with all atom molecular mechanics, is a step toward fully flexible docking of proteins and peptides.

2.4 Multiscale Flexible Docking with CABS CABS (the acronym stands for the united atoms representing a residue in a polypeptide chain: CA – alpha carbon of the main chain, CB – beta carbon, and the center of side group). Cα trace in CABS is restricted to a high-resolution cubic lattice grid, where the lattice spacing is set to 0.61 Å. Cα−Cα distances in CABS are allowed to fluctuate near the 3.8 Å. An additional pseudo-atom is located in the center of

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the Cα−Cα bond and supports a model of main-chain hydrogen bonds. The accuracy of a projection of high-resolution protein structure onto the lattice is of about half of the lattice spacing. The coordinates of beta carbons and side chains are not restricted to the lattice and are defined in the reference frame defined by the Cα trace. Due to the lattice representation, computations of local conformational transitions of the model chains are extremely fast and, in most cases, they are reduced to straightforward shuffling of integer numbers. Similarly, most of interactions could be computed via simple references to large hashing tables. Such computations with CABS are about two orders of magnitude faster than it would be possible for – otherwise equivalent – continuous space model (Boniecki et al. 2003). It should be pointed out that, due to the fine grid of the lattice representations, the model does not exhibit any lattice artifacts. Actual accuracy of the molecular models generated by CABS is lower than the resolution resulting from the lattice representation. The results of free modeling, when successful, are accurate within a low-resolution range of 2.5–5 Å. Comparative models are more accurate and their accuracy depends on the quality of templates from which the distance restraints between Cα atoms are extracted. The best models have an accuracy of about 1 Å. CABS allows for easy implementation of various restraints, not only Cα−Cα distances from templates but also restraints from sparse experimental data, as chemical shifts, residual dipolar coupling, side chain–side chain contacts from mutagenesis, etc. This opens a convenient framework for the treatment of docking flexibility at various levels. A geometric fidelity of the CABS representation is sufficient for a reasonably accurate all-atom reconstructions (Gront et al. 2007). A good measure of this fidelity is an experiment of projecting the structure onto CABS lattice followed by subsequent reconstruction of atomic details. The reconstruction consists of two stages: the first one is a very fast rebuilding of the main chain and beta carbons, executed by Backbone Building from Quadrilaterals (BBQ) program, the second one could be side-chain fitting via side-chain replacement with rotamer libraries (SCWRL) program by Dunbrack (Canutescu et al. 2003). The accuracy of such a reconstruction cycle is within a range of few tens of angstroms for the main-chain atoms and a range of 1.5–2 Å for the side-chain atoms, depending on the structure. The allatom structures could be generated at any stage of the docking and scored by force fields other than CABS. Details of the CABS knowledge-based force field (Kolinski 2004) and description of combinations of CABS with all-atom molecular mechanics (Kmiecik and Kolinski 2007, 2008; Kmiecik et al. 2007) could be found in earlier publications. Sampling protocols of CABS employ various Monte Carlo-based algorithms. When the folding mechanisms are of interest, simple simulated annealing or isothermal Monte Carlo dynamics could be appropriate. Since CABS conformational updating employs various local rearrangements controlled by a pseudo-random mechanisms, the trajectories from such simulations represent solutions of a certain Master Equation of motions and thereby provide a coarse-grained picture of the system dynamics. In this respect, CABS differs from the most of other structure assembly-reduced models, such as Rosetta (Rohl et al. 2004). There are, however, reduced space models enabling similar studies. The continuous models (like united

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residues (UNRES), (Ołdziej et al. 2005) allow for Molecular Dynamics simulations and, similarly to CABS, for Monte Carlo dynamics. When just structure prediction is a goal, more effective than simulated annealing are various multicopy MC algorithms. Most docking experiments with CABS proceed according to a combination of Replica Exchange Monte Carlo (REMC) (simulated tempering) (Hukushima and Nemoto 1996) with simulated annealing. During a typical simulation, a large number of replicas (50) are subject to slow annealing of the entire stack.

2.4.1 Treating of Flexibility CABS facilitates various levels of docking flexibility. Schematically, different instances of docking could be outlined as follows: A. A semi-flexible docking of two or more proteins of known structures. B. Docking of a fully flexible (unrestrained folding) protein or peptide on a semiflexible scaffold of other proteins. C. De novo assembly (fully flexible, free folding) of protein (peptide) complex. Obviously, the success rate and accuracy of the resulting structures decreases from A to C. The unrestricted de novo assembly of a protein complex (C) is now feasible only for relatively small and structurally not too complex proteins. Various symmetry-resulting (as for homo-dimers, trimers, etc.) restraints could be easily implemented (similarly as it was done before for RosettaDock), increasing the docking accuracy significantly. In semi-flexible docking (A), intra-protein restraints are read from the unbonded structures and the corresponding distances are allowed to fluctuate around their unbonded values. The poses could be generated by the CABS REMC in an unrestricted fashion or the initial poses (structures placed at starting replicas) could be obtained from various fast docking shape-complementarity-based algorithms. To speed up assembly, centers of gravity of the assembled molecules are subject to a weak generic attractive force acting at distances larger than the plausible estimated distance within the complex. In such procedures, trivial flexibility related to different conformations of the interface side chains is approximately accounted for at the stage of the all-atom reconstruction. Flexibility of interacting proteins does need to be treated in a uniform fashion. It is easy to restraint just parts of each molecule, allowing the remaining portions to freely adjust during the docking. Prediction of flexible fragments could be achieved in various ways, including structural comparison of different complexes of proteins of interests, normal modes, or Gaussian Network, analysis of these proteins, etc. A relative strength of restraints could be also included in the input data for the docking. Docking of a fully flexible small protein (or a peptide) to semi-flexible scaffold of other protein (a receptor) is almost always successful, without assuming a priori anything about the pose (except the penalty for large distances between molecules) and internal conformation of the free molecule.

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In principle, the interaction between the interface amino acids does need to be the same as the intra-protein interactions. The interactions between the side chains in CABS model are described by statistical pseudo-potentials derived from regularities seen in known structures. These potentials are context dependent (accounting for mutual orientations of the side chains and local conformations of the main chain). Thus, the potentials account in an implicit way for complex multi-body packing effects. Also the averaged solvent effect is encoded in these potentials. Interestingly, potentials derived separately for the interfaces in known complexes do not differ significantly from the potentials derived for monomeric proteins. In the example docking simulations described in this chapter, generic CABS potentials were used.

2.4.2 Example of Peptide Docking to Receptor Protein Frequently small peptides act as coactivators for larger proteins. Below we describe a typical example of such a docking experiment (Kurcinski and Kolinski 2007). The receptor protein is the vitamin D receptor (or rather the receptor part of the entire protein). The receptor is treated in a semi-flexible fashion. A large number of Cα−Cα intra-molecular distances are extracted from the crystallographic structure of the protein. Additionally, the secondary structure defined according to the define secondary structure of proteins (DSSP) assignment is a part of the simulation input. Assigned secondary structure provides a bias toward the proper short-range geometry and favors the hydrogen-bonding patterns consistent with this secondary structure. The simulation set-up for the receptor is schematically depicted in the left-hand side of the flow chart given in Fig. 2.1. During the simulation, the receptor structure oscillates around its native structure. The initial set of 50 replicas for the REMC simulations are generated by replication of the receptor structure with randomly placed peptide chains near the protein surface. Internal conformations of the peptide and its location in respect to the receptor are both selected in a random fashion. The set-up for the peptide is illustrated in the top right-hand part of the flow chart. The starting replica with the superimposed receptor structure is shown in the top central panel of the flow chart. The main part of the docking simulations is executed by the CABS algorithm. CABS produces a large number of conformations, stored in a pseudo-trajectory read from the lowest-temperature replica. Typically, the CABS output contains some thousands of structures. Single run generates millions of states and requires several hours of a single LINUX computing unit. The structures stored in the pseudo-trajectory are subject to a clustering procedure (hierarchical clustering or K-means clustering). In the case of the example illustrated here, there is only one well-defined cluster of solutions, containing majority of the structures. Remaining structures are scattered in apparently random fashion. The main cluster is very dense with nicely superimposed receptor structures. Only the end and some loop residues deviate a little (0.5–1.0 Å) from the mean structure, which is almost identical with the crystallographic structure. The cloud of the peptide structures is also very well defined (bottom, right-hand panel of Fig. 2.1), with

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Fig. 2.1 Flow chart of multiscale hierarchical peptide–protein docking. See the text for details

the mean-square dispersion below 1 Å. The centroid structure from the main cluster provides a scaffold for the all-atom reconstruction (left-hand panels) of the complex. The reconstructed structure is optimized with all-atom force field and rectified in Molecular Dynamics. The final structures obtained in such test docking are of crystallographic resolution. In several tests of peptide docking to various receptors, the proper pose has always been found to be within the main cluster of solution. For longer peptides (25–30 amino acids), the resulting coordinates of end residues of the peptide were usually of lower accuracy (2–3 Å), although the interface contact maps were always predicted (or rather postdicted) with high accuracy.

2.4.3 Protein–Protein Docking The methodology described in the previous sections could be used for protein– protein docking where one or two (or more) proteins could be treated in a fully

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flexible fashion, without assuming anything about their structures within the complex. Good results of fully flexible, unrestricted docking, could be expected only for relatively small and structurally not too complex proteins, of a size of the Rop homo-dimer that consists of two antiparallel long-helical hairpins or the crambin pseudo-dimer. For larger proteins, properly folded complex structures are not always obtained. At the present status of the CABS methodology, the semi-flexible docking simulations are more dependable, where at least parts of the modeled structures are controlled by weaker or stronger intra-molecular restraints derived from non-bonded structures or from different complexes of the proteins of interest. Two examples of semi-flexible docking results are illustrated in Figs. 2.2 and 2.3. In both cases, the structures of the larger proteins in the complex were strongly restrained to their non-bonded native structures. Docking simulations modified these structures very little (deviation range of 0.3–0.9 Å). Actually, this is very close to the structural differences seen between the individual proteins in the complexes and their unbound structures. The structures of larger protein are shown in gray. The second components of the complexes had higher flexibility, the restraints were much weaker allowing for large fluctuations, ranging within 5–10 Å. In both cases, near-native structures were found in the largest clusters. The resulting poses are qualitatively correct, although the details of the internal structures of these proteins have several errors. For easy comparison, Figs. 2.2 and 2.3 show both experimental (green) and calculated (red) structures. The drawings were done assuming the best superimposition of the larger proteins in the complex. The resulting superimposition of the second proteins illustrates a sum of the errors of pose and the errors of internal coordinates. The sum of these errors (coordinate Root-Mean-Square deviation of the smaller protein after the best superimposition of the larger protein) is 2.6 Å in the first case and 3.8 Å in the second case, respectively. Thus, qualitatively correct poses have been predicted (nothing has been assumed about the mutual orientation

Fig. 2.2 Structure obtained from docking procedure. The “receptor protein” (PDB code 1ppn) shown in gray, “ligand” protein (PDB code 2oct) shown in green in the crystallographic structure (PDB code 1stf), and in red for the final model. See the text for details

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Fig. 2.3 Structure obtained from docking procedure. The “receptor protein” (PDB code 2hnt) shown in gray, “ligand” protein (PDB code 5hir) shown in green in the crystallographic structure (PDB code 4htc), and in red for the final model. See the text for details

of the components) although the structural details have been distorted, especially in the second case. Initial 50 replicas for the REMC simulations were generated by FTdock program.

2.5 Perspectives The problem of in silico flexible docking, especially in cases where the dockinginduced conformational changes are large, is far from being solved. Nevertheless, there are numerous encouraging small steps toward a partial solution of this problem. Multiscale procedures, where flexible docking is performed using various coarse-grained protein models, followed by refinement of the resulting poses by more detailed molecular mechanics seem to be very promising. Here, we described combinations of the CABS-reduced space modeling methodology with all-atom refinements applied to flexible and semi-flexible protein–protein and protein– peptide dockings. The method is now mature enough for large-scale predictions of protein interactomes. In the large-scale applications, it may be necessary to introduce a pre-screening phase employing fast docking procedures based on a shape complementarity. At present, the described method is limited to proteins and peptides. An extension onto nucleic acids will require development of their coarse-grained representation, consistent with the CABS representation of proteins. Also the treatment of small ligands within such multiscale docking procedures requires significant extensions of the knowledge-based force fields, and this is still an open problem.

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Another possibility of applications of the outlined method is related to the assembly mechanisms of protein complexes. CABS sampling techniques and its force field enable meaningful simulations of folding pathways (see Chapter 12). Extension of the method on multimeric assemblies is straightforward, although it will require larger computing resources, due to a higher complexity of the problem. Finally, we would like to note that the coarse-grained protein models are potentially very interesting in the context of yet different class of docking problem, namely, fitting molecular structures into cryo-electron microscopy (EM) (or similar) low-resolution experimental data (Lindert et al. 2009; Orzechowski and Tama 2008; Jolley et al. 2008). This problem, however, is beyond the scope of this chapter.

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Ołdziej S, Czaplewski C, Liwo A, Chinchio M, Nanias M, Vila JA, Khalili M, Arnautova YA, Jagielska A, Makowski M, Schafroth HD, Ka´zmierkiewicz R, Ripoll DR, Pillardy J, Saunders JA, Kang YK, Gibson KD, Scheraga HA (2005) Physics-based protein-structure prediction using a hierarchical protocol based on the UNRES force field: assessment in two blind tests. Proc Natl Acad Sci USA 102:7547–7552 Res I, Lichtarge O (2005) Character and evolution of protein–protein interfaces. Phys Biol 2:S36–S43 Res I, Mihalek I, Lichtarge O (2005) An evolution based classifier for prediction of protein interfaces without using protein structures. Bioinformatics 21:2496–2501 Ritchie DW (2008) Recent progress and future directions in protein–protein docking. Curr Protein Pept Sci 9:1–15 Rohl CA, Strauss CE, Misura KM, Baker D (2004) Protein structure prediction using Rosetta. Methods Enzymol 383:66–93 Salwinski L, Eisenberg D (2003) Computational methods of analysis of protein–protein interactions. Curr Opin Struct Biol 13:377–382 Schueler-Furman O, Wang C, Baker D (2005) Progress in protein–protein docking: atomic resolution predictions in the CAPRI experiment using RosettaDock with an improved treatment of side-chain flexibility. Proteins 60:187–194 Sheinerman FB, Norel R, Honig B (2000) Electrostatic aspects of protein–protein interactions. Curr Opin Struct Biol 10:153–159 Sternberg MJ, Gabb HA, and Jackson RM (1998) Predictive docking of protein–protein and protein–DNA complexes. Current Opinion in Structural Biology 8:250–256 Tobi D, Bahar I (2006) Optimal design of protein docking potentials: efficiency and limitations. Proteins 62:970–981 Vajda S, Kozakov D (2009) Convergence and combination of methods in protein–protein docking. Curr Opin Struct Biol 19:164–170 Vakser IA, Aflalo C (1994) Hydrophobic docking: a proposed enhancement to molecular recognition techniques. Proteins 20:320–329 Vakser IA (1995) Protein docking for low-resolution structures. Protein Eng 8:371–377 Valencia A, Pazos F (2002) Computational methods for the prediction of protein interactions. Curr Opin Struct Biol 12:368–373 van Dijk AD, de Vries SJ, Dominguez C, Chen H, Zhou H, Bonvin AM (2005) Data-driven docking: HADDOCK’s adventures in CAPRI. Proteins 60:232–238 Wang C, Schueler-Furman O, Andre I, London N, Fleishman SJ, Bradley P, Qian B, Baker D (2007) RosettaDock in CAPRI rounds 6–12. Proteins 69:758–763 Wang C, Schueler-Furman O, Baker D (2005) Improved side-chain modeling for protein–protein docking. Prot Sci 14:1328–1339 Wodak SJ, Janin J (1978) Computer analysis of protein–protein interaction. J Mol Biol 124: 323–342 Zacharias M (2005) ATTRACT: protein–protein docking in CAPRI using a reduced protein model. Proteins 60:252–256 Zacharias M (2003) Protein–protein docking with a reduced protein model accounting for sidechain flexibility. Prot Sci 12:1271–1282 Zhang C, Liu S, Zhu Q, Zhou Y (2005) A knowledge-based energy function for protein–ligand, protein–protein, and protein–DNA complexes. J Med Chem 48:2325–2335 Zhang Q, Sanner M, Olson AJ (2009) Shape complementarity of protein–protein complexes at multiple resolutions. Proteins 75:453–467

Chapter 3

Coarse-Grained Models of Proteins: Theory and Applications Cezary Czaplewski, Adam Liwo, Mariusz Makowski, Stanisław Ołdziej, and Harold A. Scheraga

Abstract In this chapter, reduced (coarse-grained) protein models are discussed. Emphasis is given to those models which can be used in simulating the structure, thermodynamics, and dynamics of real proteins and are, at the same time, transferable. The coarse-grained force fields are introduced in a physics-based way as potentials of mean force of polypeptide chains in reduced representations, in which the secondary degrees of freedom have been averaged out. Based on this general formula, three categories of coarse-grained potentials are introduced: (i) statistical potentials derived from structural databases, (ii) potentials obtained by factorization of the parent potential of mean force, which enables us to split the system into smaller subsystems and derive each effective energy contribution independently, and (iii) potentials obtained by the force-matching method. Optimization of the potential function to achieve foldability is discussed. Applications of coarse-grained potentials to predict protein structures and simulate long-time protein dynamics are presented. We conclude that while, with the aid of massively parallel computers, coarse graining enables us to reach millisecond simulation timescales of real-size proteins, and case studies indicate that the results of these simulations are realistic, much work remains to be done to improve the force fields.

3.1 Introduction There are two aspects to the protein-folding problem. These are the determination of the folding pathways and the resulting native structure. Both experimental and theoretical methods are used to solve this problem. This chapter is concerned only

A. Liwo (B) Faculty of Chemistry, University of Gda´nsk, Gda´nsk, Poland; Baker Laboratory of Chemistry and Chemical Biology, Cornell University, Ithaca, NY, USA e-mail: [email protected] This chapter is dedicated to the memory of Urszula Kozłowska, our long-time colleague and coworker. It is unfortunate that her early passing prevented her from participating in writing this chapter.

A. Kolinski (ed.), Multiscale Approaches to Protein Modeling, C Springer Science+Business Media, LLC 2011 DOI 10.1007/978-1-4419-6889-0_3, 

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with the theoretical approach, given the amino acid sequence of the protein. The theoretical approach is based on the thermodynamic hypothesis enunciated by Anfinsen (1973), according to which the native protein adopts the conformation in which the protein plus the surrounding solvent is a system whose free energy is at the global minimum. There are two basic ingredients of the theoretical approach: formulation of an appropriate potential energy function with which to compute the interaction between every pair of atoms in the polypeptide chain and development of an algorithm to identify its global minimum. Originally, the focus was on locating the global minimum of the potential energy, based on a large menu of procedures to search conformational space (Scheraga 1988, 1996; Scheraga et al. 2004), but entropic effects were later introduced (Liwo et al. 2007) to locate the global minimum of the free energy. The initial applications made use of an all-atom potential energy function but, with the computer resources available at that time, the largest structure that could be simulated was the 46-residue protein A, with one of the procedures in the aforementioned menu, namely Electrostatically Driven Monte Carlo (EDMC) (Vila et al. 2003). In order to make further progress, resort was to have a hierarchical procedure (Liwo et al. 1993b; Scheraga et al. 2004) in which the initial search of conformational space was carried out with a coarse-grained United-Residue (UNRES) model of the polypeptide chain to locate the region of the global minimum of the UNRES potential energy. This was followed by conversion of the UNRES model to an allatom one (Ka´zmierkiewicz et al. 2002, 2003) and subsequent optimization of the all-atom model with the EDMC procedure. Coarse-grained representations of proteins have long been of interest in theoretical simulations of protein structure and dynamics (Koli´nski and Skolnick 2004; Tozzini 2005; Colombo and Micheletti 2006; Clementi 2008). The primary reason for this is that they involve much less computational effort than all-atom or united-atom representations of the polypeptide chain; this facilitates speed-up of the simulations of dynamics, folding pathways, and thermodynamics by four orders of magnitude compared to all-atom simulations with explicit solvent (Liwo et al. 2005) and, in turn, simulation of biomolecular processes at the millisecond timescale. Another application of coarse-grained models is prediction of protein structure from amino acid sequence, which becomes increasingly important because of the growing gap between the number of known protein sequences and structures; this gap is not likely to be diminished in the foreseeable future even with the improvement of experimental methods such as X-ray, nuclear magnetic resonance (NMR) spectroscopy, and cryomicroscopy. A number of comprehensive reviews of coarse-grained models applied to biomolecular and soft-material systems (Ayton et al. 2007a; Pincus et al. 2008) and specifically to proteins (Koli´nski and Skolnick 2004; Tozzini 2005; Colombo and Micheletti 2006; Gront et al. 2009) have been published recently. A book on coarsegrained models has been published recently under the editorship of Voth (2008) with contributions from over 30 research groups; this book offers a comprehensive survey of the state-of-the-art of predominantly physics-based coarse-grained models.

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3.2 History of Coarse-Grained Protein Models The history of coarse-grained models of proteins began with the pioneering work of Levitt and Warshel (1975) continued by Levitt (1976). These investigators used one or two centers to represent a side chain and three centers (Cα and two pseudo-atoms for the peptide group). The virtual-bond geometry was derived by averaging protein crystal data, and a large part of the potential energy function was obtained by Boltzmann-averaging the all-atom energy of model systems, while the parameters of hydrophobic/hydrophilic interaction potentials were based on amino acid solubility and partition coefficient data. The search procedure consisted of a series of local minimizations, performed in angular variables, with a pushing potential that prevented a system from returning to the already-found energy minimum, thereby allowing a larger-scale search of conformational space. The method was tried on bovine pancreatic trypsin inhibitor (BPTI) and found to produce protein-like structures when simulated folding was started from an extended chain; however, the native structure of the test protein was not reached. Although the Levitt and Warshel work was not developed further, it was the first attempt to construct a physics-based reduced model of polypeptide chains and laid solid foundations for the development of later physics-based models (Pincus and Scheraga 1977; Gerber 1992; Liwo et al. 1993b, 2008a; Wallqvist and Ullner 1994; Maupetit et al. 2007; Voth 2008; Chebaro et al. 2009), some of which (Liwo et al. 1993b, 2008a; Derreumaux 1997, 1999; Derreumaux and Mousseau 2007; Maupetit et al. 2007; Chebaro et al. 2009), were implemented successfully in energy-based protein structure prediction (Ołdziej et al. 2005) and ab initio protein-folding simulations (Derreumaux 1999; Liwo et al. 2005; Voth 2008). Potentials of this class are discussed in detail in Section 3.5.3. At nearly the same time, Tanaka and Scheraga (1976) introduced the first knowledge-based (statistical) protein energy function. These investigators determined a residue–residue interaction matrix from the database of protein structures known at that time by applying the Boltzmann principle to the frequency of contacts between pairs of residues of given types. Two residues were considered to be in contact when the distance between their Cα atoms was less than 7 Å (Tanaka and Scheraga 1976). Miyazawa and Jernigan (1985) developed a residue–residue contact potential by using a more refined method, in which the random-flight-chain component was removed from the effective contact energies and the quasi-chemical approximation (Fowler and Guggenheim 1949) was used instead of the simple Boltzmann inversion of contact frequency. Later they revised this potential (Miyazawa and Jernigan 1996) by using a larger database of protein structures and more refined approximations (Miyazawa and Jernigan 1999). Burgess and Scheraga (1975) introduced a residue–residue five-state model to examine the conformational states of bovine pancreatic trypsin inhibitor (BPTI). Wang and Wang (1999) proposed the reduction of the 20×20 contact-potential matrix to introduce only five distinct residue types. Structure-based contact potentials were implemented in simulations of protein packing (Gregoret and Cohen 1990), in onlattice-folding simulations (Covell 1992; Pincus et al. 2008) and in fold recognition (Maiorov and Crippen 1992).

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Work on continuous knowledge-based potentials for off-lattice simulations was initiated by Kuntz et al. (1976), who constructed a minimal model with Cβ atoms as interaction sites. The pseudo-energy consisted of a series of quadratic penalty terms accounting for the violation of allowed distances between the Cβ atoms close in sequence, favoring close distances between hydrophobic or oppositely charged residues and penalizing close distances between polar or like-charged as well as between polar or charged and hydrophobic residues. Additionally, native disulfide bond topology and the topology of cysteine residues in Fe–S clusters were enforced, if applicable. The residue–residue interaction penalty terms were scaled by interaction-matrix elements, which were assigned in an arbitrary manner according to the number of carbon atoms and the presence of charged or polar groups. When applied to rubredoxin and BPTI, the approach produced low-resolution structures which resembled native-like topology. Crippen and coworkers (Obatake and Crippen 1981; Crippen and Viswanadhan 1984) implemented the Boltzmann principle to determine continuous residue– residue potentials from protein crystal structures and fitted them to a combination of a Lennard–Jones-like and a Gaussian (Obatake and Crippen 1981) or a Lennard– Jones-like functional form (Crippen and Viswanadhan 1984). However, these potentials could locate near-native structures as stable local minima obtained only when performing local energy minimization starting from the experimental structure of a protein. Yˇcas and coworkers (Yˇcas et al. 1978; Goel and Yˇcas 1979) and Wako and Scheraga (1982a,b) developed approaches in which distances between Cα atoms were restrained to average values determined from the Protein Data Bank (PDB) (Berman et al. 2000), depending on residue types and separation in sequence; therefore, the corresponding target functions could be considered knowledge-based potentials. These approaches had some success reproducing the native-like structure of BPTI, lysozyme, and staphylococcal nuclease. Another successful knowledge-based potential was developed by Sun (1993) who was able to locate the lowest-energy structures of melittin, apamin, and avian pancreatic polypeptide inhibitor by using a genetic algorithm with the experimental radius of gyration of the protein under study as a restraint. Following the above-mentioned works, knowledge-based potentials applicable to fold recognition were developed (Jones and Thornton 1993). By applying the Boltzmann principle and using protein crystal data, Sippl and coworkers (Hendlich et al. 1990; Sippl 1990a,b, 1993; Casari and Sippl 1992) developed continuous potentials of residue–residue interactions dependent on inter-residue distance, residue types, and residue separation in sequence. These potentials were able to recognize the folds of proteins and protein fragments. This approach was later continued by a number of investigators (Reva et al. 1997; Samudrala and Moult 1998) and also applied to protein complexes (Jiang et al. 2002) and protein–ligand complexes (Mitchell et al. 1999). More complex pseudo-energy functions for sequence threading, which include explicit local-interaction and solvation terms (Bryant and Lawrence 1993; Jones and Thornton 1993; Godzik et al. 1993; Miller et al. 1996), with parameters

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optimized by using a set of training proteins (Meller and Elber 2001), were developed later. Buchete et al. (2003, 2004) developed distance-dependent and orientation-dependent statistical potentials for protein-fold recognition; they demonstrated that introducing orientation dependence greatly improves the capability of fold recognition. The orientation dependence was introduced with spherical harmonics. Statistical orientation-dependent side-chain–side-chain interaction potentials, in which each side chain is represented by an ellipsoid, were constructed by Liwo et al. (1997a) and Mukherjee et al. (2005). The statistical potentials based on the Boltzmann principle are discussed in Section 3.5.2. A different approach to derive knowledge-based potentials was developed by Wolynes and coworkers (Sasai and Wolynes 1990; Friedrichs et al. 2001; Goldstein et al. 1992a,b; Hardin et al. 2002; Eastwood et al. 2002, 2003). Instead of using the Boltzmann principle to determine the potentials, these investigators developed energy functions termed associative Hamiltonians, which linked the parameters of the potentials to the structures of a number of proteins from the database, the strength of coupling depending on the homology between the sequence under study and a sequence from the database. This approach can be considered sophisticated comparative modeling, in which sequence homology is incorporated into an energy function. Later versions of the method (Hardin et al. 2002; Eastwood et al. 2002, 2003) contain hydrogen-bonding and long-range side-chain contact energy. Initially a polypeptide chain was represented as a sequence of Cα atoms; in later versions, the Cα , backbone oxygen atoms, and side-chain centers were defined as interaction sites. The approach had some success in protein structure prediction (Hardin et al. 2002; Prentiss et al. 2006, 2008) and in Brownian dynamics simulations of protein folding (Wolynes 2005). Until the early 1990s, both physics-based and knowledge-based coarse-grained potentials were constructed as sums of individual terms. It has, however, become clear that the simplifications inherent in coarse graining make the resulting energy function inaccurate enough to prevent proteins from folding starting from an arbitrary conformation, without adjusting energy-term parameters. Work on this problem was initiated by Crippen and coworkers (Crippen and Viswandhan 1987; Crippen and Snow 1990; Seetharamulu and Crippen 1991) who defined potentialfunction optimization as a linear programing problem in which the difference between the energy of the native structure and the lowest-in-energy non-native structure of a training protein was maximized. They used the energy-embedding method as a global optimization algorithm to search low-energy structures. With this method and with the use of avian pancreatic polypeptide (APP) (Crippen and Snow 1990) and APP and crambin (Seetharamulu and Crippen 1991) as training proteins, the optimized potential energy function was able to locate the native-like structures of apamin and melittin as the lowest in energy. However, later work in which simulated annealing was implemented as a search method (Snow 1992) demonstrated that non-native structures lower in energy were found, showing the critical role of the quality of sampling in potential-function optimization. The physics-based justification of the methodology initiated by Crippen and coworkers was provided

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by Wolynes and coworkers (Bryngelson and Wolynes 1987; Hardin et al. 2002; Goldstein et al. 1992a,b) and Shakhnovich and coworkers (Sali et al. 1994a,b). Force-field optimization is discussed in Section 3.5.5. The first successful application of coarse-grained potentials in ab initio protein folding was made by Koli´nski and Skolnick who developed a high-resolution lattice model of proteins and a statistical potential which included side chain–side chain, local, hydrogen bonding, and mutibody terms accounting for cooperative formation of backbone hydrogen bonds and side-chain contact patterns (Koli´nski and Skolnick 1992, 1994a; Koli´nski et al. 1993). A residue was represented by Cα carbon serving as a center of hydrogen-bonding interactions and a side-chain center. Using a Monte Carlo dynamics algorithm, these investigators performed successful folding simulations of protein A, repressor of primer (ROP) dimer, crambin (Koli´nski and Skolnick 1994b), leucine zipper (Vieth et al. 1994), as well as folding model α-helical (Rey and Skolnick 1993; Olszewski et al. 1996; Sikorski et al. 1998) and β-sheet proteins (Koli´nski et al. 1995) and other folding simulations (Koli´nski et al. 1996, 2003; Koli´nski and Skolnick 1997). Later versions of the Koli´nski–Skolnick model are the side chain-only (SICHO) model (Koli´nski and Skolnick 1998) in which only the side-chain centroids are interaction sites and the CABS (Cα , Cβ , side chain) model (Koli´nski 2004). These models served as a basis of the MONSSTER (Skolnick et al. 1997b) and TOUCHSTONE (Skolnick et al. 2003) approaches to protein structure prediction, which combine multiple sequence alignment, secondary-structure prediction, threading, and coarse-grained simulations. These approaches have been very successful in community-wide experiments on the Critical Assessment of Techniques for Protein Structure Prediction (CASP4– CASP8) (Skolnick et al. 2001, 2003; Zhou et al. 2007a) and have also been applied to structure determination from NMR data (Lee et al. 2006; Latek et al. 2007). The CABS model has recently been applied to study protein-folding pathways (Kmiecik et al. 2006; Kmiecik and Koli´nski 2007, 2008). Following the concept of averaging out the less important degrees of freedom, the United-Residue (UNRES) force field has been developed by Liwo, Scheraga, and coworkers (Liwo et al. 1993a,b; 1997a,b; 1998; 2001; 2004b; 2007; 2008a,b; Ołdziej et al. 2003; Czaplewski et al. 2004b; Kozłowska et al. 2007; Chinchio et al. 2007; Makowski et al. 2007a,b, and c; 2008; Rojas et al. 2007; Kozłowska et al. 2010a,b). In this model, the interaction sites are side-chain centers and peptide groups placed halfway between the consecutive α-carbon atoms, while the Cα atoms assist only in the definition of the geometry and peptide group orientation. The early version of UNRES (Liwo et al. 1993a,b) included only pairwise interactions between sites and virtual-bond-torsional terms. Backbone hydrogen-bonding interactions were described by mean-field-based analytical formulas obtained by Boltzmann-averaging of the energy of peptide-group dipoles with parameters derived based on averaging the all-atom ECEPP/2 (Momany et al. 1975) energy; they correctly reproduced the directionality of backbone-group hydrogen bonding while retaining only one interaction site per peptide group. The interactions between the side chains were described by Lennard–Jones-like potentials with radii taken

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from Levitt and Chothia (1976) and well depths computed from the Miyazawa– Jernigan (1985) interaction energies. Later (Liwo et al. 1997a), the side-chain interaction potentials were revised to include anisotropy in the Gay–Berne model and reparameterized using a database of 197 high-resolution non-homologous protein structures (Liwo et al. 1997a); the local-interaction parameters were also determined from the PDB (Liwo et al. 1997b) and the weights of the energy terms were determined by using a Z-score optimization approach (Liwo et al. 1997b). Subsequently (Liwo et al. 1998, 2001), the force field was defined rigorously as a restricted free-energy (RFE) function of a united peptide chain in which the secondary degrees of freedom were averaged out; this definition can also be used to derive any coarse-grained force field. Using this definition, Liwo et al. (1998, 2001) derived multibody terms, which are necessary for regular secondary structure to form spontaneously in united-residue simulations (Koli´nski et al. 1993). The multibody terms, as well as the other terms of the force field, were gradually parameterized using ab initio energy surfaces of model peptide systems (Ołdziej et al. 2003; Liwo et al. 2004b; Kozłowska et al. 2007, 2010a,b) so that all knowledgebased local-interaction terms have now been replaced with physics-based terms and the knowledge-based side-chain interaction potentials are presently being replaced with potentials of mean force determined from all-atom simulations of models of pairs of side chains in water (Makowski et al. 2007a,b, and c, 2008). A hierarchical method was developed for force-field optimization (Liwo et al. 2002, 2007; Ołdziej et al. 2004) which extends the original concept of Wolynes and coworkers (Bryngelson and Wolynes 1987; Hardin et al. 2002; Goldstein et al. 1992a) and Shakhnovich and coworkers (Sali et al. 1994a,b) to energy-ranking partially native states. The UNRES force field was initially used for energy-based protein structure prediction formulated as a global minimum search and had considerable success in CASP experiments (Liwo et al. 1999; Pillardy et al. 2001a; Ołdziej et al. 2005). Recently (Liwo et al. 2005; Khalili et al. 2005a,b), a molecular dynamics (MD) algorithm was implemented in UNRES which extended the scope of the force field to study protein-folding pathways (Khalili et al. 2006), thermodynamic properties (Nanias et al. 2006; Liwo et al. 2007), and also to reformulate physics-based protein structure prediction as a search for the most probable conformational ensemble at temperatures below the folding temperature (Liwo et al. 2007). UNRES was also extended to simulate multichain proteins (Saunders and Scheraga 2003a,b; Rojas et al. 2007) and dynamic formation and breaking of disulfide bonds during protein folding and unfolding (Czaplewski et al. 2004b; Chinchio et al. 2007). A semi-coarse-grained model, with all-atom backbone and united-residue side chains, was developed by Derreumaux and coworkers (Derreumaux 1997, 1999; Wei and Derreumaux 2002). Side-chain interactions were represented by a 4–8 potential with well depths computed from the Miyazawa–Jernigan (1985) interaction energies and backbone long-range interactions were focused to reproduce hydrogen bonding. Later (Maupetit et al. 2007), the force field was enhanced in hydrogen-bonding correlation terms, which was motivated by the presence of multibody contributions to the free energy of desolvation of groups of hydrogen bonds.

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The force field has been applied to Monte Carlo (Derreumaux 1997, 1999; Wei and Derreumaux 2002) and molecular dynamics (Derreumaux and Mousseau 2007; St-Pierre et al. 2008) folding simulations of peptides and small proteins; recent applications include prediction of protein structure (Maupetit et al. 2007) and a study of protein aggregation (Wei et al. 2007; Mousseau and Derreumaux 2008). The knowledge-based force field of Takada and coworkers (Takada 2001; Chikenji et al. 2001; Fujitsuka et al. 2004) is similar in spirit. Just recently (Ayton et al. 2007b; Zhou et al. 2007b; Noid et al. 2008; Thorpe et al. 2008; Wang et al. 2009), a new general approach to coarse-graining has been developed by Voth and coworkers. This approach is based on matching the forces computed in the coarse-grained model of a given system to the mean forces computed by all-atom MD simulations for the same system and is discussed in more detail in Section 3.5.4. Preliminary applications to a model α-helical and β-hairpin peptide resulted in energy landscapes funneled to the respective native structure. The general purpose coarse-graining scheme and force field of Monticelli et al. (2008) termed MARTINI is similar in spirit, though less rigorously derived but incorporating experimental thermodynamic and structural data in parameterization. When applied to proteins, it needs native secondary structure to work. At the time that the physics-based model of Levitt and Warshel (1975) appeared, G¯o and coworkers (Taketomi et al. 1975; Ueda et al. 1978, Cieplak and Sulkowska, Chapter 8 of this book) laid foundations of the now-popular structure-based models, usually termed G¯o-like models (Das et al. 2005; Schug et al. 2008; Hills and Brooks 2009). These models overemphasize the interactions in the native state (in rigorous G¯o-like models only native contacts result in attractive interactions, all other interactions being repulsive). These models exhibit minimal frustration of the energy landscape. Use of G¯o-like models is based on the assumption that the topology of the native state determines the major features of protein-folding pathways and kinetics. Other structure-based models contain potential energy terms biasing toward the native secondary structure (Brown et al. 2003; Brown and Head-Gordon 2004; Eskow et al. 2004). The elastic network coarse-grained models, developed relatively late (Bahar et al. 1997; Hinsen 1998; Atilgan et al. 2001; Tobi and Bahar 2005; Ahmed and Gohlke 2006; Chu and Voth 2007; Moritsugu and Smith 2008), can be considered another class of structure-based models. In these models, a polypeptide chain is treated as a network of Cα atoms connected by springs of equilibrium length corresponding to the distance in the experimental structure of the protein studied and force constants depending on distance in the experimental structure; in more advanced applications (Chu and Voth 2007), a double-well potential is imposed on each spring. In the simplest model (Bahar et al. 1997), the force constant is zero if the distance exceeds 7 Å and 1 otherwise (thus the spring network is described by the Kirchhoff adjacency matrix); in more sophisticated approaches, the force constants are Gaussians in distance (Hinsen 1998) or are calculated based on molecular dynamics simulations (Chu and Voth 2007; Moritsugu and Smith 2008). The elastic network models are used to study low-frequency motions of proteins, including thermal fluctuations (Bahar et al. 1997; Atilgan et al. 2001), domain motions (Hinsen 1998),

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conformational changes upon folding and unfolding (Chu and Voth 2007; Moritsugu and Smith 2008), and protein–protein binding (Tobi and Bahar 2005). Finally, coarse-grained protein-like models that capture only general features of protein structures, such as the existence of a single native state, are at the other extreme. The oldest are the HP lattice models developed by Dill and coworkers (Chan and Dill 1989, 1990, 1991, 1994; Dill et al. 1995) in which only two types of beads, hydrophobic (H) and polar (P), are present with three contact interaction energy values and then the NPH lattice models developed by Hao and Scheraga (1994) with three types of beads: neutral (N), polar (P), and hydrophobic (H). These models proved invaluable in determining the origin of the general features of protein structures, such as compactness and formation of a hydrophobic core and a hydrophilic exterior, and cooperativity in folding. Protein-like lattice models with more complex interaction patterns were used by Shakhnovich and coworkers (Sali et al. 1994a; Shakhnovich 1997) and Thirumalai and coworkers (Camacho and Thirumalai 1996; Klimov and Thirumalai 1996a,b, 1998) to study foldability criteria.

3.3 Choice of Conformational Space Representation The geometry of united-residue polypeptide chains is represented in continuous or discretized space. For continuous space, the Cartesian coordinates of the interacting sites or the virtual-bond vectors are usually the variables of choice, although curvilinear (angular) coordinates were implemented in the early model of Levitt and Warshel (Levitt and Warshel 1975; Levitt 1976), in the UNRES model (Liwo et al. 1993b, 1997a,b, 1999), by Hoffman and Knapp (1996) (who considered collective coordinates composed of the ϕ and ψ torsional angles of several adjacent peptide groups), and by He and Scheraga (1998). Angular coordinates are used in energy minimization or Monte Carlo search rather than in molecular dynamics simulations; one of a few counter-examples is the work of He and Scheraga (1998) who did Brownian dynamics of model polypeptides using virtual-bond angles and virtual-bond dihedral angles as variables. Use of curvilinear coordinates enables us to reduce the number of degrees of freedom by treating the virtual-bond lengths and, sometimes, the virtual-bond angles as fixed, which reduces the cost of minimization and Monte Carlo algorithms. However, use of curvilinear coordinates in molecular dynamics is not so convenient because it introduces a non-diagonal inertia tensor which depends on conformation (resulting in the necessity of solving a linear-equation system in every MD step to compute accelerations from forces, which requires N3 operations, N being the number of variables) and also results in problems with singularity when mapping the curvilinear to Cartesian coordinates. Regardless of the representation (Cartesian or curvilinear), the obvious advantage of a continuous-space representation is the possibility of applying the algorithms of conformational-space exploration that require an energy gradient (local energy minimization with gradient minimizers and molecular dynamics and its variations).

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Discrete representations of conformational space, in the first place, are identified with lattice models (Koli´nski and Skolnick 2004; Chapters 1 and 12), in which the interaction sites are always located on lattice nodes. Depending on the lattice resolution, the lattice models are divided into low-resolution lattice models, in which the sites connected by site–site virtual bonds are located on neighboring lattice nodes, the intermediate-resolution lattices (e.g., the chess-knight lattice) in which the sites connected by a virtual bond are located on second-neighboring lattice nodes, and high-resolution lattices in which the nodes are spaced by 1.45 Å (cubic lattice) or even by 0.61 Å. Low-resolution lattices can be used only to study protein-like polymers while, with high-coordination lattices, the accuracy of chain representation can be as high as 0.35 Å, which is comparable to the inaccuracy inherent in coarsegrained force fields. Various types of lattice models have been discussed extensively in the excellent review by Koli´nski and Skolnick (2004). The advantage of lattice models over continuous-space models is the possibility of pre-computing and storing the contributions to energy corresponding to chain fragments at certain conformations, which saves CPU time. However, while this advantage was substantial a decade ago when processor speed was small compared to the present, and memory access was relatively cheap, now memory access is the bottleneck [the so-called memory wall (Flynn 1999)]. For example, accessing elements of a 10,000,000 array 100,000,000 times at random takes 11 CPU seconds with an Intel Q9400 2.66 MHz processor, which is an issue to which one has to give some consideration. Moreover, conformational search algorithms that require an energy gradient (local minimization using gradient minimizers, molecular dynamics, and related techniques) are not possible with the lattice approach. Another discretization of the conformational space is accomplished by restricting it to decoys derived from structural databases or fragments derived from the sequences homologous to a target sequence. The first is the so-called threading or fold recognition approach (Bryant and Lawrence 1993; Godzik et al. 1993; Jones and Thornton 1993; Miller et al. 1996; Meller and Elber 2001; Buchete et al. 2003, 2004), while the second one is the fragment approach applied recently by Baker and coworkers (Simons et al. 1997; Rohl et al. 2004). These representations are applied mainly in protein structure prediction, although the fragment approach was also applied in protein folding by Monte Carlo dynamics (Fujitsuka and Takada 2004).

3.4 Interaction Schemes Due to the great diversity of coarse-grained models and force fields, it is very difficult, if not impossible, to provide a unique general formula that covers all of them. If we restrict the discussion to force fields that refer to physical interactions, a general formula for the effective energy, U, could be expressed by Eq. (3.1). U=

 i

ulocal + i

 i

j
uij +

 ijkl...

uijkl...

(3.1)

3

Protein Coarse-Grained Models

45

where ulocal denotes a local-interaction term dependent on a single or a number i of adjacent sites, uij denotes the effective interaction energy between sites i and j, and uijkl. . . denotes a multibody interaction which extends over sites i, j, k, l. . . . The terms of the first two sums resemble those of all-atom force fields, although their physical origin is usually different. The local-interaction terms consist not only of those describing the energetics of virtual-bond stretching, virtual-bond angle bending, and virtual-bond torsional terms but also include more complex terms such as those describing the rotameric states of united side chains and terms that depend on a number of consecutive virtual-bond dihedral angles. Gerber (1992) introduced peptide-geometry restoring terms, which penalize the deviations of virtual-bond length and virtual-bond angles from boundary values resulting from the ideal valence geometry of polypeptide chains; these terms are also local-interaction terms. The uij terms corresponding to interactions between united side chains usually encode the totality of such interactions, including the effect of the surrounding solvent. The side-chain–side-chain interaction terms depend only on the distance (in most force fields) or also on side-chain orientation. Usually backbone hydrogen bonding is accounted for separately through interactions between backbone sites located on Cα atoms, between consecutive Cα atoms, or a number of sites, each representing a peptide group. Unless the peptide groups are represented at atomistic or nearly atomistic detail, the backbone hydrogen-bonding terms include dependence on peptide-group orientation. The presence of multibody terms is the main difference between the interaction scheme of all-atom force fields and those of coarse-grained force fields. Although the multibody terms are also present in some specific and very accurate all-atom force fields, their presence there is not required for a force field to work reasonably well. Conversely, coarse-grained force fields without multibody terms have very restricted application and cannot be used for ab initio folding unless specific biases toward elements of the native structure are introduced. The reason for this becomes clear in Section 3.5.3. One type of multibody terms, which is also present in an all-atom force field with implicit solvent representation, is the solvation term computed from the solvent-accessible surface area or related quantities; another one is the centrosymmetric potential implemented by Koli´nski, Skolnick, and coworkers (Koli´nski et al. 1993; Koli´nski and Skolnick 1994a).

3.5 Derivation of Coarse-Grained Force Fields In this section, three leading methods for developing coarse-grained potentials [statistical potentials (Boltzmann principle), factorization of potentials of mean force, and the force-matching method] are discussed. These three approaches result in potentials that can be used to simulate the structure and dynamics of real proteins and are, in theory, transferable. We omit from this discussion (i) the arbitrary potentials designed to simulate protein-like systems in order to study general properties of protein folding and dynamics (Chan and Dill 1989, 1990, 1991, 1994; Dill et al.

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1995; Camacho and Thirumalai 1996; Cieplak et al. 2002; Chapter 8), (ii) the elastic network potentials (Bahar et al. 1997; Hinsen 1998; Atilgan et al. 2001; Tobi and Bahar 2005; Ahmed and Gohlke 2006; Chu and Voth 2007; Moritsugu and Smith 2008), and (iii) the structure-based potentials (Das et al. 2005; Schug et al. 2008; Hills and Brooks 2009; Brown et al. 2003; Brown and Head-Gordon 2004; Eskow et al. 2004; Chapter 8). The reader is referred to the literature for information regarding these three other categories.

3.5.1 Basic Formulations The coarse-grained energy function can generally be defined as an average of the energy of the corresponding all-atom system, the average being computed over the degrees of freedom that are not present in the coarse-grained representation (the secondary degrees of freedom). An illustration of the correspondence between a united-residue chain and the parent all-atom chain is presented in Fig. 3.1.

Fig. 3.1 Illustration of the correspondence between the all-atom polypeptide chain in water (a) and its coarse-grained (UNRES) representation (b). The side chains in part (b) are represented by ellipsoids of revolution and the peptide groups are represented by small spheres in the middle between consecutive α-carbon atoms. The solvent is implicit in the UNRES model. Reproduced with permission from figure 1 of Czaplewski et al. (2009)

The most physical definition corresponds to the restricted free energy (RFE) or potential of mean force (PMF) obtained by computing the part of the configurational integral of a system corresponding to integrating over the secondary degrees of freedom. If X = (x1 , x2 , . . . , xM ) and Y = (y1 , y2 , . . . , ym ) denote the coarse grained and secondary degrees of freedom (orthogonal to X), respectively, M and m being the dimensions of the space spanned by these coarse-grained and secondary variables, respectively, and E(X;Y) denotes the all-atom energy function, RFE can be expressed by Eq. (3.2) (Liwo et al. 1998, 2001; Izvekov and Voth 2005a,b; Ayton et al. 2007a).

3

Protein Coarse-Grained Models

F(X) = −RT ln

47

⎧ ⎪ ⎨ ⎪ ⎩

Y

⎫ ⎪  ⎬ E(X; Y) dVY + C exp − ⎪ RT ⎭

(3.2)

where Y is the space spanned by Y, R is the universal gas constant, T is the absolute temperature, and C is an additive constant. The choice of C varies from approach to approach; for example, Liwo et al. (1998, 2001) chose C = RT ln VY , which makes F(X) a restricted excess free energy, while Voth and coworkers (Izvekov and Voth 2005a,b; Ayton et al. 2007a,b) chose C = −RT ln ZN /zn , with ZN and zn denoting the configuration integrals computed over the coarse-grained and all-atom conformations, respectively. Equation (3.2) provides the best physical connection between the coarse-grained and the corresponding all-atom system, because exp[−F(X)/RT] is proportional to the probability of a coarse-grained conformation defined by X. Consequently, the ensemble averages computed over F(X) are theoretically equal to those computed over the parent all-atom energy function E(X;Y). Another point is that it is clear from Eq. (3.2) that the effective coarse-grained energy function depends on temperature. This point has recently been addressed by Liwo et al. (2007). Earlier coarse-grained models (Levitt and Warshel 1975; Levitt 1976; Pincus and Scheraga 1977; Liwo et al. 1993b) adopted the Boltzmann-averaged energy as the effective energy function. However, the average energy does not have a direct connection to the probability of a coarse-grained conformation, and it is not straightforward to compute ensemble averages using this quantity. Equation (3.2) defines a multidimensional potential of mean force, the computation or determination of which by direct integration, simulation, or from experimental data for the entire protein is unfeasible. In the next three sections, the determination of this integral by making necessary simplifications is addressed. It should be noted that Sections 3.5.2, 3.5.3, and 3.5.4 of this section do not imply a different physical origin but discuss different approaches to derive the coarse-grained potentials based on Eq. (3.2).

3.5.2 Statistical Potentials (Boltzmann Principle) Knowledge-based potentials, also known as statistical potentials, are commonly used to predict protein structures as well to simulate protein-folding pathways. The basic purpose of this approach (Tanaka and Scheraga 1976; Miyazawa and Jernigan 1985; Sippl 1990a,b, 1993; Covell 1992; Casari and Sippl 1992; Maiorov and Crippen 1992) is to construct an effective energy function [the prototype of which is given by Eq. (3.2)] based on the distributions of inter-residue distances, virtual-bond lengths, bond angles, dihedral angles, geometric parameters characteristic of short-sequence fragments, etc., derived from structures deposited in the Protein Data Bank (PDB) (Berman et al. 2000). The basic equation used in deriving statistical potentials is given by Eq. (3.3)

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W (x; c; s) = −RT ln

N obs (x; c; s) N ref (x; c; s)

(3.3)

where W(x; c; s) is the estimated potential of mean force of a fragment with geometry expressed by the vector x, composition (the kinds of residues involved) expressed by the vector c, and sequence and/or secondary-structure context expressed by the vector s, R is the universal gas constant, T is the absolute temperature, N obs (x; c; s) is the number of counts of fragments of a given composition and sequence context and geometry close to x observed in the database, and N ref (x; c; s) is the reference number of counts (in the absence of any interactions except those imposed by chainconnectivity and excluded-volume constraints). In simple residue–residue contact potentials (Tanaka and Scheraga 1976; Miyazawa and Jernigan 1985; Covell 1992; Maiorov and Crippen 1992; Rooman et al. 1992; Zhou and Zhou 2004), c consists of the kinds of the first and the second residues involved, x is 1 if the distance between the two selected atoms (Cα , Cβ , mass centers) of the residues is less than rcut (usually equal to 7 Å) and 0 otherwise, and the sequence context is ignored. In more refined pair potentials, residues are split into more centers of interaction, and different potentials are developed for the local and non-local interactions or the order of the two residues in the chain; this implies sequence context (Sippl 1990a,b, 1993; Koli´nski and Skolnick 1992; Koli´nski et al. 1993; Godzik et al. 1993). It is clear from Eq. (3.3) that the statistical potentials depend on the database from which the N obs (x; c; s) values were derived. For example, it was shown that statistical potentials derived from the structures of all α-helical proteins are significantly different from those obtained from all-β proteins (Furuichi and Koehl 1998); a similar situation is observed when single-chain and multichain proteins are used for development of potentials (Moont et al. 1999; Lu et al. 2003). Following the work of Miyazawa and Jernigan (1985), non-homologous protein structures with different types of folds are selected for a database. The complete statistical energy function is a sum of terms determined using Eq. (3.3). It should be noted that each of the terms can be identified with the potential of mean force of a protein fragment, and each of them is evaluated independently. Consequently, Eq. (3.3) does not provide a rigorous connection to the PMF of the polypeptide chain under consideration [defined by Eq. (3.2)], because this PMF is not a sum of fragment PMFs, each determined in the context of the whole structures of database proteins. Moreover, there is a possibility that the same contributions will be included in different terms. The statistical potentials have been discussed in detail in the review by Shen and Sali (2006). They can be classified according to the characteristics a, b, c, and d: (a) Protein structure representation: The representations differ by the number, location (e.g., on Cα atoms, Cβ atoms, centers of masses of selected fragments, etc.), and types of sites (point masses, rods, or rigid bodies), and choice of the representation of the conformational space (continuous, lattice, decoys, and fragments).

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(b) Interaction scheme: A number of knowledge-based force fields include only residue–residue interactions that depend only on the distance (Miyazawa and Jernigan 1985; Sippl 1990a,b, 1993; Casari and Sippl 1992; Skolnick et al. 1997a; Samudrala and Moult 1998; Rojnuckarin and Subramaniam 1999; Zhang et al. 2004; Chen and Shakhanovich 2005) or on distance and orientation (Buchete et al. 2003, 2004; Miyazawa and Jernigan 2005). Some of these simple potentials are enhanced by including a solvation term dependent on solventaccessible area (Sippl 1993; Melo and Feytymans 1998). Potentials based on residue–residue interactions are good only for threading but perform poorly in ab initio folding. Potentials containing local-only interactions in the form of virtual-bond dihedral angle terms (Rooman et al. 1991, 1992; Zhou and Zhou 2004; Betancourt 2008) or those containing only hydrogen-bonding interactions (Kortemme et al. 2003) are also known. However, the majority of working potentials contain side-chain–side-chain, hydrogen-bonding, local, and multibody interactions (Koli´nski and Skolnick 1992, 1994a,b; Koli´nski et al. 1993; Sun 1993; Koli´nski 2004). (c) Representation of interactions: The interactions can be represented either in a tabular form (i.e., a PMF value for each different geometry; for contact potentials this is just a single number) (Sippl 1990a,b, 1993; Casari and Sippl 1992; Koli´nski and Skolnick 1992, 1994a,b; Koli´nski et al. 1993; Koli´nski 2004) or by functional forms which are determined by fitting to the PMF values (Obatake and Crippen 1981; Crippen and Viswanadhan 1984, 1987; Sun 1993; Liwo et al. 1997a,b; Buchete et al. 2003, 2004). (d) Reference state definition: Generally the reference state is defined as one in which no specific interactions occur within the fragment under consideration, and the interactions result only from chain-connectivity and excluded-volume constraints. However, the level of sophistication of choosing the reference state varies significantly among the statistical potentials (Sippl 1990a,b, 1993; Casari and Sippl 1992; Jernigan and Bahar 1996; Moult 1997; Skolnick et al. 1997; Samudrala and Moult 1998; Lu and Skolnick 2001; Buchete et al. 2004; Zhou et al. 2006). As an example of the derivation of statistical potentials, the normalized distribution function N obs (r), the reference distribution functions, N ref (r) (r being the distance between the side-chain centers), and their ratio (the correlation function) for the Leu-Leu pair determined from the PDB by Liwo et al. (1997a) are shown in Fig. 3.2a, while the respective PMF together with the fit to a Lennard–Jones (6–12) functional form is shown in Fig. 3.2b. The statistical potentials were the subject of extensive criticism especially related to the fact that they are derived from equilibrium structures and their relationship to potentials of mean force is not clear (Thomas and Dill 1996; Ben Naim 1997). The correctness of reference state definition and the quality of statistical data derived from the databases were also challenged (Thomas and Dill 1996; Ben Naim 1997; Tobi et al. 2000; Summa et al. 2005). Implementators and developers of statistical potentials recognized most of the criticisms related to their methodology and

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Fig. 3.2 Upper graph: sample pair-distribution and pair-correlation functions for the Leu-Leu pair averaged over consecutive 0.5-Å shells used to determine the statistical side-chain–side-chain interaction potentials for this residue pair by Liwo et al. (1997a). (a) Radial normalized pair-correlation function [N obs /N ref in Eq. (3.3)]; (b) the reference normalized pair number of counts [Nref in Eq. (3.3)]; (c) the total normalized number of pair counts [Nref in Eq. (3.3)] determined from the PDB. All curves were normalized to the maximum value of 1.0. Lower graph: the potential of mean force computed from the correlation function [W of Eq. (3.3); dashed line] and the Lennard–Jones fit (solid line). It should be noted that the Lennard–Jones fit does not reproduce the desolvation maximum. The upper graph has been reproduced with permission from figure 3 of Liwo et al. (1997a) and the lower graph was constructed based on the data from Liwo et al. (1997a)

3

Protein Coarse-Grained Models

51

raised further important points. One of the most important extensions of the statistical potentials was the observation that pairwise energies used there could be insufficient to describe internal protein energy correctly and, therefore, introduction of multibody expansion in the potential became necessary (Koli´nski and Skolnick 1992, 1994a,b; Koli´nski et al. 1993; Godzik et al. 1993; Vendruscolo and Domany 1998; Vendruscolo et al. 1999; Chapter 6). On the other hand, this extension led to greater problems with the quality of the statistical data derived from the PDB (Vendruscolo et al. 1999). Despite the conceptual and technical problems, statistical-based potentials enjoyed a significant degree of success in many applications related to the biophysics of proteins. The primary application of the statistical potentials was prediction of the three-dimensional structure of proteins from amino acid sequences (Koli´nski and Skolnick 1992, 1994a,b; Koli´nski et al. 1993; Panchenko et al. 2000; Skolnick et al. 2000; Tobi and Elber 2000; Tobi et al. 2000; Vendruscolo et al. 2000, Koli´nski 2004), protein thermodynamical stability (Gilis and Rooman 1996, 1997), and the structure and stability of protein–protein and protein–ligand complexes (Gohlke and Klebe 2001). However, the statistical potentials based on the Boltzmann principle [Eq. (3.3)] are currently no longer developed or used as extensively as 5–7 years ago (Lazaridis and Karplus 2000; Meller and Elber 2002; Russ and Ranganathan 2002; Buchete et al. 2003, 2004). The reason for this is that current protein structure prediction is based mostly on the template recognition methodology, rather than on a statistical energy function (Kryshtafovych and Fidelis 2009).

3.5.3 Factor Expansion of the PMF The exact RFE of Eq. (3.2) can be evaluated only numerically, and this task requires, at best, as much effort as all-atom simulations with explicit solvent. Therefore, a way around, to which many investigators resort, is to compose the total PMF [Eq. (3.3)] from contributions corresponding to fragments of a system obtained by all-atom simulations of, e.g., models of pairs of interacting side chains in water or as statistical potentials. However, the connection of such composite potentials to the parent PMF defined by Eq. (3.2) is unclear. Liwo et al. (1998, 2001) developed a formal expansion of the PMF, which is based on Kubo’s (1962) cluster-cumulant approach. First, the total all-atom energy of a system composed of n coarse-grained sites is partitioned into the component energies, ε1 , ε2 , . . . , εN , N = n(n + 1)/2, as given by Eq. (3.4). E(X; Y) =

n   k=1 i∈Ik

Eik (X; yk ) +

n  k−1    k=1 l=1 i∈Ik j∈Il

Eik;jl (X; yk ; yl ) =

N 

εi (X; zi )

i=1

(3.4) where the sets {I1 , I2 , . . . , In } contain the indices of all atoms assigned to interaction site 1, 2,. . ., n, respectively, and zi = yi or zi = (yk , yl ) depending on whether

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εi is an intrasite or intersite energy. Each component energy is either a sum of all interatomic interactions within a given extended site or between two extended sites. By inserting  Eq. (3.2) and splitting the RFE into cluster-cumulant

Eq. (3.4) into functions, εi1 , εi2 , ..., εik f , containing increasing numbers of component energies, Eq. (3.2) becomes Eq. (3.5). F(X) =





εi f +

εi εj

i

i<j

 f



εi εj εk

+

 f

+... + ε1 ε2 ...εN f

(3.5)

i<j
where the cluster-cumulant functions are expressed by Eq. (3.6).

εi1 , εi2 , ..., εik

 f

=

k 



 (−1)k−l εim1 , εim2 , ..., εiml

(3.6)

l=1 im1
and ⎧ ⎫   ⎪ ⎪  k ⎨ ⎬  

1 1 exp −β εil (X; zil ) dVyl εi1 , εi2 , ...εik = − ln ⎪ β ⎪ ⎩ VyI ⎭ l=1

(3.7)

I

where β = 1/RT, and Vyl is

the volume of the subspace spanned by the variables yi1 , yi2 , ..., yik . The quantity εi1 , εi2 , ...εik is the RFE containing only a subset of component interactions. Each of the first-order factors εi f is the RFE corresponding to component interaction i. These factors are potentials of mean force of the interactions between isolated sites (which can be identified with, e.g., interactions between the side chains or “hydrogen-bonding” interactions between the peptide groups) and with the potentials of mean force of the local interactions within isolated sites. The PMF-based approaches that construct the effective energy function of coarse-grained systems by analogy to all-atom force fields include mostly the first-order factors which turned out to be an insufficient approximation. Nevertheless, the expansion has to be truncated at some points for tractability and transferability of the resulting force field. The fourth-order expansion appears to be sufficient (Ołdziej et al. 2004). Approximate analytical expressions for the factors can be derived by using Kubo’s (1962) generalized cumulant expansion (Liwo et al. 2001). With this expansion, temperature dependence of the respective terms of the force field can be introduced in a straightforward manner (Liwo

et al. 2007). The second-order factors εi εj f contain RFEs of pairs of component interactions minus the sums of the RFEs of the single-component interactions. They can be regarded as correlation terms pertaining to component interactions i and j, reflecting the coupling between the secondary degrees of freedom pertaining to these interactions. Examples of second-order and third-order correlation terms are virtual-bond torsional and double-torsional potentials, respectively [these terms correspond to the coupling between the local interactions of two or three consecutive amino acid

3

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53

residues, respectively (Liwo et al. 2001; Ołdziej et al. 2003)] and the terms pertaining to the coupling between local and backbone-electrostatic interactions. The third-order factors corresponding to the coupling of the long-range electrostatic (hydrogen-bonding) interactions of a pair of peptide groups and the local interactions of residues adjacent to them, and the fourth-order factors accounting for the coupling of long-range electrostatic interactions of two pairs of consecutive peptide groups and the local interactions of the residues linking these groups, are indispensable for stabilization of regular secondary structures (α-helices and β-sheets), as was also observed earlier by Koli´nski, Skolnick, and coworkers (Koli´nski and Skolnick 1992, 1994a; Koli´nski et al. 1993) by experimenting with interaction schemes in the statistical potentials. The third-order factors mentioned above have a nice physical interpretation as the interaction of two fictitious dipoles located on the interacting peptide groups, with the magnitude and orientation of their dipole moments dependent on the orientation of the peptide-group axes and virtual-bond dihedral angles. An illustration is presented in Fig. 3.3. The correlation terms are also apparent in

(3)

Fig. 3.3 Illustration of the physical meaning of the Uloc-el correlation terms. The arrows represent the components of the dipole moments (which represent the peptide groups between the Cαi and   Cαi+1 Cαj and Cαj+1 atoms, respectively; these peptide groups are represented by light-gray letters and lines) from the cumulant-based expression [Eq. (11) in Liwo et al. (2004b)] for the third-order correlation contribution to the RFE of two interacting peptide groups pi and pj considered together with the local interactions within the adjacent amino acid residues and the vector components of these dipole moments defined by Eq. (11) in Liwo et al. (2004b). The positions of atoms Cαi ,   Cαi+1 , and Cαi+2 Cαj , Cαj+1 , and Cαj+2 are fixed, and so are the vector components μ1i and μ1j   of the dipole moments, whereas the virtual bonds Cαi−1 . . . Cαi Cαj−1 . . . Cαj and the components   of μ2i (μ2j ) can rotate about the Cαi . . . Cαi+1 Cαj . . . Cαj+1 axis by the angle λi (λj ). (a) Parallel orientation of the planes defined by the Cαi , Cαi+1 , Cαi+2 and the Cαj , Cαj+1 , Cαj+2 atoms, as found in parallel β sheets. (b) Opposite orientation of these planes, which are not found in protein structures. The dotted rectangles mark the planes containing the Cαi . . . Cαi+1 Cαj . . . Cαj+1 virtual bonds and the vectors μi (μj ). Reproduced with permission from figure 12 of Liwo et al. (2004b)

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effective interactions between side chains in water as shown by many simulation studies (Czaplewski et al. 2000, 2003; Shimizu and Chan 2001), although the inclusion of such correlation terms does not seem necessary for the coarse-grained force fields to work. The factors can be determined by direct Boltzmann summation over the energy surfaces of the respective systems (Levitt and Warshel 1975; Levitt 1976; Pincus and Scheraga 1977; Liwo et al. 1993b, 2001, 2004b; Ołdziej et al. 2003; Kozłowska et al. 2007, 2010a,b) or by simulations (Makowski et al. 2007b,c, 2008; Monticelli et al. 2008). Subsequently, approximate analytical expressions can be fitted to the factor surfaces defined by Eqs. (3.6) and (3.7). As an example, in Fig. 3.4, we present the PMFs of a pair of valine side chains from Makowski et al. (2007c), which are modeled by ellipsoids of revolution, with an interaction potential approximated by a sum of the Gay-Berne (1981) potential of non-bonded interactions and a Gaussian-overlap model of solvation (Makowski et al. 2007a). The physics-based potentials of side-chain interactions (Makowski et al. 2007a,b, and c, 2008) will replace the respective knowledge-based potentials (Liwo et al. 1997a) used in the present UNRES force field (Liwo et al. 2008a). It should be noted that Eq. (3.5) is not used as is but, instead, the individual factors are implemented as energy terms in the final force field, and each term is

1.5

PMF [kcal/mol]

1

0.5

0

–0.5

–1

–1.5 4

5

6

7 8 9 distance [Å]

10

11

12

Fig. 3.4 The PMF curve for a pair of isobutane molecules (to model a pair of valine side chains). The dashed, dotted, and dot-dashed lines correspond to PMFs determined for the side-to-side, edge-to-edge, and side-to-edge orientation, respectively, obtained by MD simulations. The solid lines correspond to the analytical approximation to the PMFs, with coefficients determined by least-squares fitting (Makowski et al. 2007c) of the analytical expression to the PMF determined by MD simulations. Reproduced with permission from figure 4c of Makowski et al. (2007c)

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55

assigned a weight which is determined by force-field optimization (Liwo et al. 2002, 2007, 2008a,b; Ołdziej et al. 2004). Such a procedure enables us to compensate for the errors inherent in the truncation of the factor expansion, approximations to factors, and the accuracy of the method implemented to compute the free-energy surfaces corresponding to factors.

3.5.4 Force-Matching Method The multiscale-coarse-graining (MS-CG) method of Voth and coworkers (Izvekov and Voth 2005a,b; Ayton et al. 2007a,b; Noid et al. 2007, 2008; Zhou et al. 2007b; Wang et al. 2009) is also based on the RFE function (or many-body PMF) of a given system [Eq. (3.8)]. The CG sites are defined as the centers of masses of groups of atoms, MRI (r), I = 1, 2, ..., N, where r denotes the coordinates of atoms within the coarse-grained group I; this quantity is defined by Eq. (3.8). 

mi ri

i∈{I}

MRI (r) = 

mi

(3.8)

i∈{I}

where {I} is the set of the numbers of the atoms contained within the site I. The determination of the RFE function of a given system (e.g., a polypeptide molecule and the surrounding solvent) is based on all-atom MD simulations of that system and, subsequently, determination of the mean forces acting on the CG sites is obtained by solving the minimization problem given by Eq. (3.9) (Izvekov and Voth 2005a,b, Ayton et al. 2007b, Zhou et al. 2007b).  Nf N  CG  2  1  AA CG  χ = FkI − FkI  3Nf NCG 2

(3.9)

k=1 I=1

where FAA kI is the average force acting on coarse-grained site I at configuration k computed from all-atom MD simulations, FCG kI is the force computed in the CG approximation, Nf is the number of coarse-grained configurations obtained in the MD simulation, NCG is the number of CG sites, and ||o|| denotes the Euclidean norm of a vector. It should be noted that the definition of coarse-grained degrees of freedom as the Cartesian coordinates of the centers of mass of the extended sites [Eq. (3.8)] is less general than separating the variable space into the coarse-grained degrees of freedom X and the secondary degrees of freedom Y, as defined in Section 3.5.1. The only restriction imposed on Y is that they must belong to a space orthogonal to that spanned by X; therefore, the coordinates of X might contain not only the Cartesian coordinates of the extended sites but also those of their orientation. Conversely,

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Eq. (3.8) restricts the treatment to radial-only effective potentials of interaction, although extension to non-radial potentials is possible. Because of the large number of degrees of freedom, long-range forces are expressed in terms of pair contributions, and it is not feasible to determine FCG kI without additional assumptions. In the application to peptide calculations (Zhou et al. 2007b; Thorpe et al. 2008), the pairwise contributions were defined as cubic splines in site–site distances. The spline coefficients were determined by minimization of χ 2 of Eq. (3.9), which is the solution of a linear least-squares problem. Harmonic expressions are used for bonded and 1,3-non-bonded interactions and Fourier expressions for torsional interactions (Zhou et al. 2007b); the respective curves are determined as PMFs from MD simulations and the coefficients of the analytical expressions are determined by least-squares fitting. Because of the pairwise decomposition of the forces, the solvent molecules are treated explicitly. The water molecule is a single spherical site. As pointed out by Noid et al. (2007), the forces determined by minimizing χ 2 of Eq. (3.9), although pairwise, are not equivalent to pairwise mean forces determined by averaging the forces acting within each of the pair, which do not minimize χ 2 of Eq. (3.9). The MS-CG forces incorporate many-body terms implicitly, similar to the Yvon–Born–Green equations in liquid-state theory (Noid et al. 2007). However, explicit multibody terms are not present in the force fields obtained with the present force-matching procedure (Noid et al. 2007). The MS-CG methodology has been applied (Zhou et al. 2007b) to pentadecaalanine and to V5 PGV5 , which fold into an α-helix and a β-hairpin, respectively, with the chemistry at harvard molecular mechanics (CHARMM) force field. Two-bead and four-bead models were tried. The two-bead model consisted of one bead per side chain and one per backbone (−NH−CH−CO−) atoms placed at the sidechain center of mass and on the Cα atom, respectively; in the four-bead model, three backbone-atom centers represent the CO, NH, and CH groups, respectively, and one bead represents the side chain. The force field for each peptide and each coarse-grained representation was derived from all-atom MD simulations with the CHARMM force field and explicit water starting from the folded structure; the corresponding peptide never left the folded basin during the simulation. For pentadecaalanine, both the two-bead and the four-bead model were able to keep the folded (α-helical) structure stable, while only the four-bead model was able to maintain the folded structure of V5 PGV5 . The authors concluded that the failure of the two-bead model for V5 PGV5 was caused by asymmetry of interactions, as opposed to pentadecaalanine and that introduction of the dependence of the forces not only on bead type but also on their positions in the sequence could improve the two-bead model. The four-bead models of both peptides were also used to simulate the folding of these peptides starting from extended structures (Thorpe et al. 2008); both peptides folded to the native structures. This finding is remarkable in view of the fact that only the configurations from the neighborhood of the folded structures were used to derive the force field. The MS-CG approach, though fully general, lacks transferability; a force field derived in this way is, by definition, only good for coarse-grained representation of a

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given system. The transferability has recently been addressed (Wang et al. 2009) by developing the effective force-coarse-graining (EF-CG) method, in which pairwise contributions to forces are parameterized explicitly, which reduces the accuracy of the representation but makes the force field transferable. However, both the MS-CG and EF-CG approaches produce force fields with accuracy no better than that of the parent all-atom force fields which, at least in their present form, are far from having the capacity of ab initio protein folding [this feature arises from the fact that the forces implemented in determining the potentials from Eq. (3.9) are computed from all-atom potentials, and the procedure is aimed at reproducing these forces as well as possible]. Force-field optimization for foldability (Section 3.5.5), which can compensate for the errors inherent in force-matching methods (arising both from approximating the site–site forces with a sum of pairwise central forces and the inaccuracy of the parent all-atom potentials used to parameterize them) by including structural and thermodynamic data of selected training proteins, will probably become necessary for application of the MS-CG and EF-CG methodologies in protein-folding simulations. The failure of the two-bead MS-CG model to keep the stable folded structure of V5 PGV5 indicates two other weaknesses of the MS-CG methodology in its present formulation, namely, the representation of forces as combinations of terms dependent on site–site distance, but not on site orientation, and the absence of explicit multibody terms. The first feature was shown by Buchete and coworkers to be important in side-chain interaction terms (Buchete et al. 2003, 2004) and by Liwo et al. (1993b) in peptide-group interaction terms, while explicit multibody terms are crucial to stabilize regular secondary structure in coarse-grained force fields (Koli´nski and Skolnick 1992; Koli´nski et al. 1993; Liwo et al. 1998, 2001). The absence of these features does not invalidate the MS-CG or the EF-CG approach; however, it precludes extensive coarse graining (e.g., it has forced the introduction of three sites per backbone unit instead of one to obtain good results for the V5 PGV5 peptide) and also forces the introduction of explicit solvent; moreover, the coarse-grained sites must be as spherical as possible (Wang et al. 2009). These deficiencies can probably be eliminated. Spherical harmonic expansion as applied by Buchete et al. (2003, 2004) can be used to introduce orientational dependence. Multibody terms could be introduced by factorization of the mean forces into two-body and many-body terms and use of cumulant-based expansion to derive approximate analytical formulas, as in the derivation of the UNRES force field.

3.5.5 Optimization of an Effective Energy Function Apart from the force-matching method discussed in Section 3.5.4 (which, however, does not lead to transferable potentials), the different terms in an effective coarsegrained energy function have different origin and accuracy (never greater than the accuracy of the parent all-atom energy calculations) and, consequently, are weakly related to each other. Moreover, the factor expansion given by Eqs. (3.5) and (3.7)

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has to be truncated to render the force field usable. Consequently, an effective energy function obtained by direct summation of the different terms might not be good for folding simulations. As mentioned in Section 3.5.2, force-field optimization, originated by Crippen and coworkers (Crippen and Viswandhan 1987; Crippen and Snow 1990; Seetharamulu and Crippen 1991), is used to obtain foldable force fields. Wolynes and coworkers (Bryngelson and Wolynes 1987; Hardin et al. 2002; Goldstein et al. 1992a) formulated the energy-landscape theory, according to which the energy landscape of a foldable protein resembles a funnel with the bottom in the native state. This implied that the glass-transition temperature be significantly lower than the folding-transition temperature and, in turn, that the Z-score value, Z, which is defined by Eq. (3.10), be maximized. Enative  − Enon-native   2 2 Enon-native − Enon-native

Z = 

(3.10)

2  where, Enative , Enon-native , and Enon-native denote the canonical averages of the energy of native-like and non-native conformations and the canonical average of the square of the energy of non-native conformations. By simulation studies of model heteropolymer sequences discretized to a cubic lattice, Sali et al. (1994a,b) found that the native state is located as low in energy as possible relative to any alternative state. Although further work (Camacho and Thirumalai 1996; Klimov and Thirumalai 1996a,b, 1998; Tiana and Broglia 2001; Liwo et al. 2004a) has shown that the energy-gap criterion is not a sufficient one, and the structure of the energy spectrum of the non-native states is important, the energy-gap criterion remains one of the bases of potential-function optimization. The Z-score and energy-gap optimization approaches were implemented by many investigators to optimize potential energy functions (Hao and Scheraga 1996a,b; Liwo et al. 1997b; Lee et al. 2000, 2002; Pillardy et al. 2001b; Eastwood et al. 2003; Fujitsuka et al. 2004). Scheraga, Liwo, and coworkers (Liwo et al. 2002, 2004a; Ołdziej et al. 2004; Liwo et al. 2007, 2008b; He et al. 2009) extended the Z-score or energy-gap optimization concept further by developing a hierarchical optimization method. In this approach the conformational space of a protein is divided into levels. The lowest level (level 0) contains unfolded structures, the highest level contains folded structures, and the intermediate levels contain partially folded structures. Optimization is aimed at obtaining such ordering of the levels that the relative free energy of the folded level is the lowest below the folding temperature and the highest above the folding temperature, while the relative free energy of the lowest (non-native) level is the highest below the folding temperature and the lowest above the folding temperature, as illustrated in Fig. 3.5. The relative free energies of the levels and their composition are obtained from thermodynamic and structural data pertaining to the folding of complete proteins or their fragments. The ordering of the free energies as illustrated in Fig. 3.5 leads to the appearance of a heat-capacity peak at the folding-transition temperature (Liwo et al. 2007, 2008b); an example is shown in Fig. 3.6.

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3

9

2.5

8

2

7

1.5

6

1

5 Tfexp = 325 K Tfcalc = 330 K

0.5

4

0 300

320 340 360 temperature [K]

RMSD [A]

Cv [kcal/(mol×°K)]

Fig. 3.5 Schematic illustration of ordering of the energy levels, which is the goal of the algorithm for optimizing the potential function, using the 1EM7 protein, the structure of which consists of two β-hairpins packed to a middle α-helix. Below the folding-transition temperature (Tf ), the nonnative level (level 0) has the highest free energy, the conformations with only the native C-terminal β-hairpin formed (level 1) have a lower free energy, next are the conformations in which the middle part of the N-terminal β -strand joins the β-hairpin and the middle α-helix starts to form and, finally, the native-like structures with all structural elements formed have the lowest free energy. Above the folding-transition temperature, the free-energy relations are reversed and, at the folding-transition temperature, the free energies should be approximately equal. Reproduced with permission from figure 3 of Liwo et al. (2008a)

380

3

Fig. 3.6 (a) Experimental (filled circles) and calculated with the optimized force field (solid line) free-energy difference between the folded and unfolded states of the 1ENH protein. (b) Calculated (solid line) and experimental (dashed line) heat-capacity curve of 1ENH. Data from Liwo et al. (2008b); reproduced with permission from figures 6a and b of that reference

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3.5.6 “Knowledge-Based” and “Physics-Based” Potentials The coarse-grained potentials are sometimes collectively termed “knowledgebased” and as such counterposed to the “physics-based” all-atom potentials (Shell et al. 2009). However, the definition of coarse-grained potentials as potentials of mean force presented in Section 3.5.1 [Eq. (3.2)] is entirely physical and is strictly related to the thermodynamic probability of a coarse-grained structure to occur at a given temperature; as mentioned in Section 3.5.1, coarse-grained force fields should include dependence on temperature (Liwo et al. 2007). In Eq. (3.2), the secondary degrees of freedom are averaged out as in the case of all-atom potentials the electron degrees of freedom are averaged out to leave only the positions of atomic nuclei. The origin of each averaging is different (the Boltzmann law and the adiabatic approximation, respectively) but the principle is the same. Consequently, it is possible to construct both all-atom and coarse-grained force fields based on solid physical principles. If is often argued that coarse-grained potentials are knowledge-based, because they contain extensive input from experimental data and, moreover, are usually optimized to locate the native structures of training proteins as the lowest-energy or the lowest-free-energy structures. However, unless ab initio molecular quantum mechanics is implemented in protein-folding simulations, no computational approach is free from extensive use of experimental information in parameterization. This is true even for semi-empirical methods of molecular quantum mechanics, which were parameterized based on experimental heats of formation, dipole moments, and ionization potentials of model compounds (Stewart 1990). The all-atom force fields contain a heavy input from crystal data (equilibrium bond lengths and van der Waals radii), infrared (IR) spectroscopy (force constants), far IR, microwave, and NMR measurements (to refine torsional constants), dielectric measurements (to determine polarizabilities in order to parameterize the Lennard– Jones potential), and association enthalpy measurements (to adjust the charges). Ab initio quantum-mechanical calculations are used to derive charges and other parameters. This approach is identical in spirit to those described in Sections 3.5.3, 3.5.4 and, with some statistical approaches described in Section 3.5.1.2. In the factor-expansion approach, the parameters of the factors are derived based on quantum-mechanical calculations (Ołdziej et al. 2003; Liwo et al. 2004b) and on allatom simulations (Makowski et al. 2007b,c, 2008; Monticelli et al. 2008); some are derived from structural and other databases (Liwo et al. 1997a), just as the parameters of all-atom force fields. In the force-matching approach, the whole force field for a given system is derived from all-atom MD simulations (Izvekov and Voth 2005a,b; Ayton et al. 2007a,b; Noid et al. 2007, 2008; Zhou et al. 2007b; Thorpe et al. 2008). On the other hand, fully knowledge-based all-atom potentials, in which the “raw” atom–atom PMFs determined by Boltzmann inversion of the correlation functions [Eq. (3.3)] are used directly without fitting to physics-based functional forms, have also been developed (Melo and Feytmans 1997), The force-field optimization described in Section 3.5.5 is based on energylandscape theory which also has solid physical foundations (Bryngelson and

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Wolynes, 1987; Hardin et al. 2002; Goldstein et al. 1992a; Sali et al. 1994a,b). Given the error inherent in individual energy terms, force-field optimization is a necessary step to take if a force field is meant to have any predictive power. This remark concerns both coarse-grained and all-atom force fields; it should be noted that a growing number of foldability-optimized all-atom force fields have appeared lately (Schug and Wenzel 2006; Yang et al. 2007; Arnautova and Scheraga 2008; Jagielska et al. 2008; Irbäck et al. 2009; Chapter 5). Consequently, if all coarse-grained force fields are knowledge-based because of using experimental information in parameterization, then all-atom force fields must also be considered as knowledge-based. In view of the above discussion, it seems reasonable to distinguish knowledgebased potentials from physics-based ones in the same manner as empirical equations of mechanical engineering for the relationship between stress and rod length or bend, velocity and shear friction, etc., are distinguished from formulas of theoretical mechanics and hydrodynamics. The empirical formulas mentioned above are set up to solve practical problems such as the construction of buildings, cars, bridges, and hydraulic systems and work only within the scope of their application. Consequently, a physics-based potential (coarse-grained or all-atom) must (i) be based on solid foundations that can be traced back to elementary laws of physics (such as the Schrodinger equation in the case of all-atom potentials and the Boltzmann law for coarse-grained potentials), (ii) apply consistent approximations to derive a force field or its terms from this principle, (iii) be meant to work without any ancillary information from structural databases, and (iv) reproduce structure, dynamics, and thermodynamics of folding. The statistical potentials derived fully from structural databases both in form and parameters might meet criterion (i) [although various investigators suggest the contrary (Thomas and Dill 1996; Ben Naim 1997)]; however, they certainly do not meet criterion (ii) and most of them work only with explicit database information. It should be noted, though that, e.g., the CABS statistical potential originally designed to predict protein structures also has the capacity to simulate folding pathways (Kmiecik and Koli´nski 2007,2008).

3.6 Applications in Protein Structure Prediction The physical foundation for prediction of protein structure is the Anfinsen thermodynamic hypothesis (Anfinsen 1973), according to which the native structure of a protein corresponds to the global free-energy minimum of this protein plus the surrounding solvent. Consequently, prediction of protein structure can be formulated as a global optimization problem (Scheraga et al. 2004). Optimization of the potential energy instead of the free energy was carried out until very recently (Liwo et al. 2007). Because of the large number of degrees of freedom, the problem is still intractable at the all-atom level except for relatively small proteins (Vila et al. 2003; Schug and Wenzel 2006, Irbäck et al. 2009; Chapter 5) and, consequently, coarsegrained models have been used extensively, including ROSETTA (Simons et al.

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1997; Rohl et al. 2004), SICHO (Koli´nski and Skolnick 1998), CABS (Koli´nski 2004), C alpha and side chain (CAS) (Zhang et al. 2005), and UNRES (Liwo et al. 2008a). Despite some limitation related to the accuracy of the protein structure description by reduced models, many of them are among the most powerful and successful tools in protein structure prediction. Global optimization of the potential energy function was the method of choice in previous applications of UNRES to protein structure prediction (Lee et al. 1999; Liwo et al. 1999; Pillardy et al. 2001a; Ołdziej et al. 2005). The conformationalspace annealing (CSA) method (Lee et al. 1999; Czaplewski et al. 2004a), which is a combination of energy minimization and a genetic algorithm, was implemented in this early work. As an example, the prediction of CASP3 target T0061 obtained with this approach (the best prediction for this target over all groups participating in CASP3) is shown in Fig. 3.7. Fig. 3.7 Superpositions of the Cα traces of two fragments of the crystal (red) and predicted (yellow) with the UNRES force field and CSA global optimization method structure of HDEA (CASP3 target T0061). (a) Residues D25–I85 (Cα RMSD = 4.2 Å, over 61 residues). (b) Residues W16−K42 (Cα RMSD = 2.9 Å, over 27 residues). Helices 2, 3, 4, and 5 are indicated as H-2, H-3, H-4, and H-5, respectively. Reproduced with permission from figures 1 and 2 of Liwo et al. (1999)

Other search techniques, such as parallel tempering, were also implemented (Skolnick et al. 2001, 2003; Gront et al. 2005); in these approaches an all-atom energy (computed after converting the coarse-grained conformations to all-atom conformations), which is considered as more accurate than a united-residue energy function, was used as a scoring function. However, selecting the lowest-energy conformation or a number of energy-ranked conformations as candidates for the predicted structure ignores conformational entropy, which is a key component of the free energy. This problem was addressed by a number of investigators.

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Rohl et al. (2004) developed the thermodynamic-clustering technique in which lowenergy conformations resulting from a conformational search are accumulated, and a cluster analysis of the resulting set is carried out. The largest cluster is selected as the best prediction; this cluster does not necessarily contain the conformation with the lowest potential energy. A similar approach to ensembles obtained from lattice Monte Carlo dynamics parallel-tempering runs was developed by Skolnick et al. (2001, 2003) and Gront et al. (2005). These approaches are at the other extreme because they assume that only conformational entropy (the number of conformations in a cluster) is important in making a decision about the ranking of candidate predictions, while the potential energy is ignored in scoring and is used only in conformational sampling. Such an approach was motivated by the fact that neither the united-residue nor the all-atom energy functions are accurate yet (Kryshtafovych et al. 2005). Liwo et al. (2007) developed a rigorous thermodynamic-clustering approach, in which the results of a replica-exchange molecular dynamics run are processed by the weighted-histogram analysis method (WHAM) (Kumar et al. 1992, 1995) to calculate the probability of each conformation, then the conformations are ranked according to decreasing probabilities and, finally, those whose probabilities sum up to a cut-off value (usually 0.99) are clustered, and the probability of each cluster is calculated from Eq. (3.11). 

 exp ωi −

i∈{I}

P{I} =   {I} i∈{I}

Ui RT

 exp ωi −



Ui RT



(3.11)

where {I} denotes the set of conformations belonging to the Ith cluster, P{I} is the probability of the Ith cluster, Ui is the energy of the ith conformation, ωi is the logarithm of the weight of this conformation determined by the WHAM method (Liwo et al. 2007), R is the universal gas constant, and T is the absolute temperature. The inaccuracy of energy functions as well as the inefficiency of conformational search techniques for larger proteins seriously limits the accuracy and efficiency of database-free predictions of protein structure (Kryshtafovych et al. 2005). To overcome these limitations, knowledge-based information is often introduced to energy-based prediction methods. One method is to impose geometrical restraints derived from knowledge-based information during a conformational search. Such restraints both reduce the conformational space to be sampled and partially bypass the inaccuracies of the energy function by biasing it toward conformations containing native-like elements. Almost all successful protein structure prediction methods use secondary-structure restraints from secondary-structure prediction. Current secondary-structure prediction algorithms are very fast and very accurate (Wood and Hirst 2005) and, therefore, protein structure prediction methods benefit from using this information (Rohl et al. 2004; Koli´nski and Bujnicki 2005; Ołdziej et al. 2005). Long-range restraints derived from tertiary-contact prediction, which is based on sequence homology, are also included (Skolnick et al. 1997b, 2000, 2001, 2003; Koli´nski and Bujnicki 2005). Geometrical restraints are also introduced in

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scoring the candidate predictions, for example, by checking them against the distances derived from multiple sequence alignment (Paluszewski and Karplus 2009, and references cited therein). Reduction of the conformational space to be sampled is accomplished by sequence threading (fold recognition) and by a fragment approach. In the threading approach, the conformational space is reduced to Cα or Cβ traces derived from the database of known proteins (Bryant and Lawrence 1993; Jones and Thornton 1993; Godzik et al. 1993; Miller et al. 1996; Meller and Elber 2001; Buchete et al. 2003, 2004), which makes the search much easier but restricts predictions to folds already present in the database. Early studies of the fragment technique (Ponnuswamy et al. 1973; Simon et al. 1991) were adopted as a build-up procedure (Vásquez and Scheraga 1985) to compute protein structure. This technique, later used by Baker and coworkers (Simons et al. 1997; Rohl et al. 2004), has so far been the most successful one in protein structure prediction. In the fragment technique used by Ponnuswamy et al. (1973) and by Baker and coworkers (Simons et al. 1997; Rohl et al. 2004), a library of possible conformations of three-, five-, seven-, and nine-residue-long fragments of the target sequence is created first. To construct the library, a multiple sequence alignment of the target sequence to database sequences is carried out, and fragments are selected either by a minimum-energy criterion (Ponnuswamy et al. 1973; Simon et al. 1991), or from the portions of the chains of database protein conformations whose sequences are sufficiently homologous to the respective portions of the target sequences (Simons et al. 1997; Rohl et al. 2004). In the procedure of Baker and coworkers (Simons et al. 1997; Rohl et al. 2004), the conformational search is performed in the conformational space defined by the fragment library. The effective energy function is the ROSETTA coarse-grained statistical potential (Simons et al. 1997; Rohl et al. 2004); however, specific filters (e.g., a check for hydrogen-bonded β-strands) are used during the search. Since 1994, the techniques for protein structure prediction have been evaluated every other year in the Critical Assessment of Techniques for Protein Structure Prediction (CASP) experiments (Moult et al. 2007). Those methods that are based on coarse-grained models of polypeptide chains were assessed to be very successful. The most successful approaches include ROSETTA (Simons et al. 1997; Rohl et al. 2004), threading assembly refinement (TASSER) which used the reduced CAS model of a polypeptide chain (Zhang et al. 2005), and the CABS model (Koli´nski 2004). All methods mentioned above always appear among the best predictors in the new-fold category (with little or no similarity to amino acid sequences for which three-dimensional structure is known) (Jauch et al. 2007).

3.7 Applications to Study Protein Dynamics and Thermodynamics Proteins are dynamic objects. Computing the static native structure from a knowledge of the amino acid sequence is only the first approach to the protein-folding

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problems. The second problem, even more demanding, is to compute structural pathways and rates from the unfolded to the folded form. Protein dynamics plays a fundamental role in biological processes such as enzymatic reactions, signal transduction, immunological processes, cell motility, and also in malignant processes such as cancer and amyloid formation (Dobson 2003). Both experimental and theoretical techniques are used to study protein dynamics and mechanism of protein folding. Experimental studies of protein dynamics are restricted to techniques which provide only indirect and fragmentary information, leaving a wide room for interpretation. Recent advancement of single-molecule studies (Cecconi et al. 2005) facilitates the detailed experimental investigation of the folding pathways of some proteins. However, the development of new methods of simulation and comparison of theoretical and experimental characteristics of protein folding is of crucial importance for advancing our understanding of biological systems. Because of the complexity of the systems, all-atom studies of protein dynamics are restricted mostly to simulations of folded globular proteins around their native states or to unfolding simulations, starting from the experimental structure, except for small proteins (Scheraga et al. 2007). Even for small proteins, only a few trajectories can be run, which does not make it readily possible to compare the results with those of experiments which provide ensemble-averaged properties. To carry out large-scale simulations one has to resort to coarse-grained models of proteins. The method of choice to study protein dynamics is molecular dynamics (MD) (Scheraga et al. 2007). In the MD method, Newton’s equations of motion are solved to obtain the coordinates and momenta of all particles: mi

d2 ri = −∇ri U (ri ) , i = 1, 2, ..., N dt2

(3.12)

where ri is the vector of the Cartesian coordinates and mi is the mass of ith particle, U is the potential energy of the system, −∇ri U (ri ) is the potential force acting on particle i, and N is the number of particles. It should be noted that Eq. (3.12) is valid only if the coordinates describing the system are the Cartesian coordinates of the centers of the sites; a non-diagonal tensor of inertia appears instead of point masses on the left-hand side of the system of the equations of motion if other coordinates such as virtual-bond dihedral angles (He and Scheraga 1998) or virtual-bond vectors (Khalili et al. 2005a) are used. In coarse-grained models, Langevin equations of motion are often used in simulations to mimic the non-conservative forces from the solvent: mi

d2 ri dri + Ri (t) , i = 1, 2, ..., N = −∇ri U (ri ) − mi γi 2 dt dt

(3.13)

where γ i is the friction coefficient of the ith particle, Ri (t) is the vector of random forces arising from the collision of particle i with the molecules of the solvent that are not considered explicitly. Ri (t) has zero mean, and its values taken at different times are δ-correlated:

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    Ri (t) • Ri t = 2mi γi kB Tδ t − t

(3.14)

where T is the absolute temperature of the system, kB is the Boltzmann constant, and δ(x) is the Dirac δ-function. In the overdamped limit, the friction terms prevail over the inertial terms [those on the left-hand side of Eq. (3.13)], and the latter can be neglected, resulting in a system of first-order differential equations often referred to as Brownian dynamics: mi γi

dri = −∇ri U (ri ) + Ri (t) dt

(3.15)

Brownian dynamics is frequently used for coarse-grained models of proteins, but it does not provide control over the kinetic energy because of the neglect of the inertial term, and, therefore, it cannot lead to correct values of thermodynamic properties. Monte Carlo (MC) methods are also used to study protein dynamics, especially in the case of lattice models, for which MD cannot be applied. Lattice representation is most often used in simplistic coarse-grained models, which are not meant to reproduce the detailed features of the protein energy surface, but rather to study general properties of folding. Simulations with simplistic general reduced models have contributed significantly to our understanding of the events that occur in the folding process and of foldability criteria (Veitshans et al. 1996; Shakhnovich 2006). Lattice MC dynamics simulations of the CABS model of polypeptide chains (Koli´nski 2004) were performed by Koli´nski and coworkers (Koli´nski et al. 2003; Kmiecik et al. 2006; Kmiecik and Koli´nski 2007, 2008) to study protein unfolding and folding dynamics. Many coarse-grained protein-folding simulations are carried out with the use of G¯o-like (Ueda et al. 1978; Cieplak et al. 2002; Chapter 8) or other structure-based potentials (Sorenson and Head-Gordon 2002; Brown et al. 2003; Brown and HeadGordon 2004), which are biased toward the native structure, or model potentials (Thirumalai and Klimov 1999) which can reproduce only general features of protein folding. Use of structure-based potentials is based on the assumption that the native-state topology largely determines folding, whereas non-native interactions are of secondary importance. Examples of such applications are studies of the kinetics and sequence of folding events (Hoang and Cieplak 2000; Cieplak et al. 2002; Chapter 8) and the thermal (Cieplak and Sulkowska 2005) and mechanical (Cieplak and Szymczak 2006) unfolding of proteins. Head-Gordon and coworkers have used a model with a bias toward native secondary structure in the backbone potential and interactions between side chains depending on their hydrophobicities to study the folding kinetics of ubiquitin-like sequences (Sorenson and Head-Gordon 2002), as well as protein L and G (Brown and Head-Gordon 2004) by Langevin dynamics. He and Scheraga (1998) used a simpler model with a secondary-structure bias to study the folding of model β-sheet sequences. Small-scale motions are also studied by using elastic network models (Atilgan et al. 2001; Ming and Bruschweiler 2006).

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Many existing coarse-grained potentials perform well; in particular, they can predict the native structure of a protein, only when used in connection with information extracted from protein databases, which is not acceptable when studying protein dynamics. The reason for the deficiency of the coarse-grained potentials lies in their derivation either by analogy to all-atom force fields or from database statistics; neither method offers a first-principle systematic derivation of the force field in a clear way. Recently, we implemented the physics-based united-residue UNRES force field for coarse-grained molecular dynamics (MD) (Liwo et al. 2005; Khalili et al. 2005a,b, 2006). Comparison with all-atom MD revealed that UNRES offers a 4,000fold speed-up relative to all-atom simulations with explicit solvent and more than a 200-fold speed-up compared with simulations with implicit solvent (Khalili et al. 2005b). Consequently, real-time folding simulations with UNRES are readily possible. Initial results from UNRES MD simulations show that we are able to simulate folding events which take place in a microsecond or even a millisecond timescale. An example of a folding trajectory of the 1G6U dimeric protein is presented in Fig. 3.8.

C

C C

C C t=0.07 ns

C t= 0.1 ns

C

C

C

C t= 0.18 ns

t= 0.2 ns

t=0.49 ns

Fig. 3.8 Example of a fast folding trajectory of the two-chain 1G6U obtained with the UNRES force field and Berendsen dynamics. The C terminus of each chain is marked. Reproduced with permission from figure 9 of Rojas et al. (2007)

In our very recent work (Liwo et al. 2010), the capacity of UNRES MD was amplified by parallelizing energy and force calculations. A sample scalability graph obtained with IBM BlueGene is presented in Fig. 3.9. It can be seen that an 80-time speed-up was obtained for about an 800-residue protein with 128 processors, which enables millisecond-scale simulations of proteins of this size to be accomplished in days and, consequently, will enable us to simulate biologically important events such as domain motion during enzymatic catalysis, ligand binding and release, and G-protein binding and dissociation from G-protein coupled receptors. Canonical MD simulation can be used for estimating thermodynamical properties as well as for a global search, but in practice it easily becomes trapped, and thus

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Fig. 3.9 Strong scaling data of the fine-grained UNRES/MD code obtained with IBM BlueGene at Argonnne National Laboratory (intrepid.alcf.anl.gov) for proteins of different size and canonical MD simulations

is not an effective method for searching rough free-energy landscapes of proteins. A single MD trajectory cannot sample the vast thermally accessible configurational space of biomolecules to capture large-scale rearrangements, including reaction pathways, mechanisms, and rates. More efficient conformational sampling algorithms are essential components of methods for studying protein dynamics. One of the most effective sampling methods, the replica-exchange method (RE, also known as exchange MC (Hukushima and Nemoto 1996), or parallel tempering (Hansmann 1997; Chapter 9), was initially developed to improve sampling in glassy systems in statistical physics. However, following Hansmann’s use of the method in simulations of a simple peptide, Met-enkephalin (Hansmann 1997), and Sugita and Okamoto’s formulation of an MD version of the algorithm (Sugita and Okamoto 1999), the RE method has been applied extensively in biomolecular simulations. The replica-exchange MD (REMD) method combines the idea of simulated annealing MD and MC methods and is one of the generalized-ensemble algorithms that perform a random walk in energy space due to a free random walk in temperature space. In the REMD method, n replica systems, each in the canonical ensemble, and each at a different temperature, are simulated. At given intervals, swaps, or exchanges, of the configurational variables between systems are accepted with the Metropolis criterion. This is equivalent to exchanges of temperatures, because the set of the n replica system can be treated as the set of n continuous MD trajectories of

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varying temperatures or the set of n canonical ensembles at particular temperatures with structures from all trajectories sorted by temperature. The replica-exchange method was the subject of a recent review (Earl and Deem 2005) which discussed both the history of the method and its application to various physicochemical simulations. The great potential of the RE method was also recognized in a review of sampling methods for molecular simulation (Lei and Duan 2007). REMD is the method of choice for studies of the thermodynamics of protein folding. Various thermodynamical properties are available as a function of temperature through histogram reweighting techniques (WHAM) (Kumar et al. 1992, 1995). Low free-energy minima are accessible through accelerated relaxation. However, REMD does not provide direct information about kinetics, as opposed to multiple-trajectory canonical MD simulations. We compared three generalized-ensemble algorithms for molecular simulations, namely, a replica-exchange method (RE), a replica-exchange multicanonical method (REMUCA), and a replica-exchange multicanonical method with replica exchange (REMUCAREM) in both MC and MD versions, to determine the thermodynamic characteristics of the UNRES force field for efficient sampling at various temperatures using three-model systems: one peptide (polyalanine of 20 residues length) and two small proteins, an α-helical protein of 46 residues (the B domain of the staphylococal protein A; 1BDD) and an α + β protein of 48 residues (the Escherichia coli Mltd Lysm Domain; 1E0G) (Nanias et al. 2006). For polyalanine and α-helical protein A, all algorithms performed reasonably well, but for a more complicated α + β protein (1E0G), the REMD method, especially in its multiplexed version (MREMD) introduced by Rhee and Pande. (Rhee and Pande 2003), turned out to be the most efficient. Replica exchange was also used to test the coarse-grained OPEP force field on six peptides: decaalanine, (AAQAA)3 , the C-terminal β-hairpin from protein G, the Trp-cage, BBA zinc-finger motif, and a dimer of a coiled coil 7-residue peptide (Chebaro et al. 2009). REMD simulations with OPEP showed the α-helical character of alanine-based peptides and reproduced the experimental structure of the β-hairpin and Trp-cage within 1–2 Å. For all systems studied, the proper thermodynamics was recovered. The experimental melting temperature of the Trp-cage was reproduced within a small deviation of 25 K in contrast to previous all-atom simulations. For the peptide with the BBA fold, a cluster of conformations with the fold close to that determined by NMR (a β-hairpin packed to an α-helix) was obtained as one cluster of conformations; however, this cluster did not have the lowest-free energy. Recently, we compared canonical MD, REMD, and MREMD simulations of protein folding with the UNRES force field in detail, using two- model proteins: an α + β protein of 48 residues (the E. coli Mltd Lysm Domain; 1E0G) and an α-protein of 67 residues (the de novo designed protein; 1LQ7) (Czaplewski et al. 2009). To enhance the sampling in MREMD, the replicas are multiplexed with a number of independent molecular dynamics runs at each temperature (Rhee and Pande 2003). Exchanges of conformations between random replicas of neighboring temperatures are tried as in REMD, but there is a larger number of such pairs in MREMD than in REMD. In MREMD, there are several layers of replicas, each of which has all

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different temperature levels and is equivalent to a single REMD simulation. Exchanges between replicas in different layers are tried as well as exchanges between replicas in the same layer. It should be noted that multiplexing provides better sampling compared to REMD with the same number of replicas (and, consequently, more temperatures). The reason for sampling enhancement in MREMD is that efficient sampling requires diffusion in temperature replica space. Therefore, adding more temperature replicas in REMD (which, at first thought, would seem to increase efficiency) means that the number of swaps grows quadratically and that either longer simulations are needed or exchanges must be attempted more frequently. However, the system needs time to equilibrate with a new temperature distribution after each exchange, which prevents too-frequent exchanges. The MREMD method, on the other hand, takes advantage of both the multiple temperature aspect of REMD, as well as the large number of independent simulations to enhance sampling considerably. Even with a large number of replicas, the simulation time should be long enough so that each trajectory can cover the entire conformational space as well as the entire temperature space (Czaplewski et al. 2009). Moreover, multiplexing effectively extends the intrinsic parallelism of the RE algorithm. Resolution exchange, a variant of the replica-exchange method, combines the efficiency of coarse-grained simulation and the accuracy of all-atom simulation by swapping conformations between replicas simulated at different resolutions (e.g., coarse-grained and all-atom) (Lyman et al. 2006). The crucial step in this method is the reconstruction of the higher resolution system from models at lower resolutions. Many reconstruction algorithms result in biased samples and make the calculation of unbiased thermodynamic averages very difficult. Recently a configurational-bias Monte Carlo procedure was adopted in resolution exchange as a general method to rebuild the missing degrees of freedom (Liu et al. 2008). This approach uses the all-atom Hamiltonian to guide the reconstruction process, and the bias introduced is removed through the acceptance criteria.

3.8 Conclusions and Outlook Returning to the statement in Section 3.1 that describes the two aspects of the protein-folding problem, it is clear that, while there has been considerable progress with coarse-grained models, there is a need for further improvement in force fields and in search procedures. Of the two, optimization of the search procedure seems more advanced at present, given the fine-grain parallel processing (Liwo et al. 2010) that has already led to computations for proteins containing up to 1,000 residues with the currently available computer resources. Molecular dynamics with a coarse-grained model, leading to canonical ensembles with MREMD procedures, has already pushed the simulation timescale into the experimentally accessible one. With anticipated enhancements of computer power in the near future, even longer-time simulations can be expected.

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The situation is not as bright as far as improvement of force fields is concerned. While current coarse-grained force fields seem to be based on reasonable physics with a rigorous definition of the coarse-grained effective energy functions as potentials of mean force [Eq. (3.2)], and adequate procedures were developed to derive and parameterize individual effective energy terms and to determine their relative weights, it is clear that more effort must be expended to improve the physics. At present, the coarse-grained force fields work reliably in protein structure prediction only with a heavy input from knowledge-based information. The better treatment of solvent-mediated side-chain–side-chain interactions (including electrostatic interactions), local interactions, and correlation terms seem to be the primary targets for improvements. The importance of local interactions and, possibly, the associated correlation terms, has been shown by the tremendous success of the ROSETTA approach (Simons et al. 1997; Rohl et al. 2004), which is based on the construction of the backbone from a carefully constructed fragment library (cf. Section 3.6). Nevertheless, improvement of force fields would seem to be the barrier that must be surmounted if we can hope to simulate the dynamics, structure, and thermodynamics involved in folding large proteins in experimentally observed timescales. Finally, it will ultimately be necessary to re-convert our coarse-grained trajectories to all-atom ones in order to be able to specify the interactions on an atomic scale that control the dynamics during protein folding. A possible alternative would involve a hybrid procedure combining aspects of an all-atom model with a coarsegrained one. Acknowledgments We thank Ana Rojas for assistance in preparing Fig. 3.8. This work is supported by grants from the National Institutes of Health (GM-14312), the National Science Foundation (MCB00-03722), and grants N N204 049035 (grant contract number 0490/H03/2008/35) and N N204 152836 from the Polish Ministry of Science and Higher Education. Mariusz Makowski was also supported by a grant from the “Homing” program of the Foundation for Polish Science (FNP) and MF EOG resources. This research is conducted by using the resources of (a) our 800-processor Beowulf cluster at the Baker Laboratory of Chemistry and Chemical Biology, Cornell University, (b) the National Science Foundation Terascale Computing System at the Pittsburgh Supercomputer Center, (c) the John von Neumann Institute for Computing at the Central Institute for Applied Mathematics, Forschungszentrum Juelich, Germany, (d) the Beowulf cluster at the Department of Computer Science, Cornell University, (e) the resources of the Center for Computation and Technology at Louisiana State University, which is supported by funding from the Louisiana legislature, (f) our 45-processor Beowulf cluster at the Faculty of Chemistry, University of Gda´nsk, (g) the Informatics Center of the Metropolitan Academic Network (IC MAN) in Gda´nsk, and (h) the Interdisciplinary Center of Mathematical and Computer Modeling (ICM) at the University of Warsaw.

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Chapter 4

Conformational Sampling in Structure Prediction and Refinement with Atomistic and Coarse-Grained Models Michael Feig, Srinivasa M. Gopal, Kanagasabai Vadivel, and Andrew Stumpff-Kane

Abstract The performance of models at different resolutions is compared in the context of an iterative protein structure refinement protocol. The models considered here consist of an all-atom model described with the CHARMM22 force field in combination with a distance-dependent dielectric implicit solvent approximation, a united-atom model described by the CHARMM19 force field in combination with the effective energy function 1 (EEF1) solvent model, the new intermediate coarsegrained model PRotein Intermediate MOdel (PRIMO), both with Generalized Born and distance-dependent dielectric solvent models, and the lattice-based coarsegrained Side CHain Only (SICHO) model. It is found that the CHARMM19 and SICHO models lead to initially rapid refinement for a set consisting of 11 targets from past critical assessment of protein structure prediction (CASP) competitions, but they are eventually outperformed by the CHARMM22 and PRIMO models in consistently reaching near-native conformations over the course of 100 refinement cycles.

4.1 Introduction The most direct approach to protein structure prediction involves the simulation of folding from an initially unfolded structure to the native state at the global free energy minimum (Osguthorpe 2000; Bonneau and Baker 2001). Such an ab initio protocol is conceptually very appealing because it makes little or no a priori assumptions about the native state. However, the practical relevance of ab initio structure prediction is diminishing because template-based models can be obtained today for the large majority of structure prediction targets (Moult et al. 2009). This is a consequence of the increased number of experimental structures in the Protein Data M. Feig (B) Department of Biochemistry and Molecular Biology, Michigan State University, East Lansing, MI, USA; Department of Chemistry, Michigan State University, East Lansing, MI, USA e-mail: [email protected]

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Bank (PDB) (Berman et al. 2000) and the emergence of new methods that assemble protein structures from structural fragments from a variety of known structures (Rohl et al. 2004; Zhang et al. 2005). Fragment-based methods have been especially successful in predicting structures for “difficult” targets for which sequence homology-based templates that cover the entire sequence are not available (Simons et al. 1999; Zhang et al. 2005). In fact, results from recent rounds of CASP suggest that the structures of virtually any target could have been predicted at least approximately using templates from the PDB and that a lack of good predictions for the most difficult targets is largely a consequence of not being able to identify and assemble suitable templates (Zhang and Skolnick 2005). Protein structures are often used as starting structures for detailed mechanistic studies with computational techniques and as targets for rational drug design. Both applications often require structures at experimental or near-experimental accuracy. Structure predictions at near-experimental accuracy are typically only possible if structures from highly homologous proteins are available as templates. Otherwise, structure refinement is needed to improve initial predictions to near-experimental accuracy. Such refinement would not be able to take advantage of known structures anymore and instead rely on ab initio-like sampling protocols that can find the native state solely based on a given energy function. As a consequence, there has been renewed interest in ab initio prediction protocols. This time there is more of a focus on solving the refinement problem than on folding from extended structures. In both, ab initio folding and refinement, successful methods require a combination of effective conformational sampling methods with realistic underlying energy functions. The sampling method has to be able to reach the native state within available computational resources while the native state should be located at the global free energy minimum of the energy function. Furthermore, the energy should gradually decrease toward the native basin to reflect funnel-shaped energy landscapes of real proteins and guide the conformational sampling toward the native state. One approach to structure refinement is the application of molecular dynamics simulations with atomistic classical force fields to provide a realistic energetic description of solvated proteins in a theoretically rigorous framework. In practice, such simulations are rarely successful in significantly refining structures because extremely long and prohibitively expensive simulations are required to overcome kinetic barriers (Fan and Mark 2004). At the same time, current force fields still exhibit subtle conformational preferences that may in some cases bias the resulting structures toward non-native conformations even if simulations could be carried out for sufficiently long times (Feig et al. 2003; Hornak et al. 2006; Best et al. 2008; Freddolino et al. 2009). A more practical approach involves simplified protein representations in conjunction with empirical energy functions that are sampled via molecular dynamics or Monte Carlo methods (Skolnick et al. 1997; Chinchio et al. 2006; Gopal et al. 2009). Simplified protein models allow comprehensive conformational sampling with moderate computational resources of a wide variety of conformations, including native-like conformations. However, the energy functions used with such models are usually not sufficiently reliable to consistently converge to the native state at the free energy minimum.

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One strategy to overcome difficulties with ab initio methods is to separate conformational sampling and scoring. In this case, a computationally highly efficient method based on an approximate model is used for the sampling stage to quickly and broadly cover conformational space. Representative structures are then extracted from the sampling and evaluated with an accurate scoring function, often at a higher model resolution, to identify the most native-like conformations. Such a protocol is successful if the sampling method is able to generate at least some native-like conformations (not necessarily with the lowest free energy) and if the scoring function can discriminate native-like conformations from other non-native conformations within a given decoy set. A decoupling of sampling and scoring methods then also allows separate assessment and optimization toward the different objectives of the sampling and scoring stages. This chapter will primarily discuss conformational sampling methods and in particular the effect of different model resolutions on sampling efficiency and accuracy. Furthermore, practical relevance dictates that we focus primarily on structure refinement rather than on ab initio prediction from extended structures. In the following, we will first introduce a framework for assessing sampling efficiency within the context of structure refinement and then present concrete examples of different sampling strategies with models at different resolutions.

4.2 Iterative Structure Refinement Framework In order to establish a framework for analyzing conformational sampling for structure refinement application one may consider an iterative structure refinement scheme (Stumpff-Kane et al. 2008). In such a protocol, depicted in Fig. 4.1, an initial structure, typically obtained as a result of template-based modeling, is subjected to conformational sampling with the goal of generating a large number of diverse decoys. A scoring function is then applied to select one conformation from

Fig. 4.1 Iterative structure refinement scheme

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the set of decoys to be used as the starting structure for the next round of sampling. The sampling–scoring cycles are repeated until convergence is reached. Such a protocol would lead to successful refinement if more native-like decoys than the starting structure at a given cycle are generated and selected according to the scoring function at least for most of the cycles. In such a scheme, sampling and scoring become most decoupled when the sampling protocol involves only short simulations so that the underlying energy function cannot significantly bias the distribution of decoys. The degree to which sampling and scoring should be decoupled depends on the quality of the energy function used during the sampling phase. A very approximate energy function may favor the sampling of non-native states during longer simulations while a more accurate energy function may actually help direct sampling toward the native state. A typical protocol may involve short molecular dynamics or Monte Carlo simulations started with different random seeds from the same initial structure at a given cycle. The length of the simulations determines the breadth of the structural distribution of the resulting decoys and the number of refinement cycles that are minimally needed to reach the native state. The breadth of the decoy distribution in turn affects the ability of the scoring function to identify the most native-like conformations in a given decoy set. An ideal sampling method would therefore generate a broad distribution of structures where the fraction of structures that are more native-like than the starting structure at a given cycle is maximized. Sampling methods can be assessed independently from scoring by assuming an idealized refinement protocol where root mean square deviation (RMSD) from the native state (or another suitable metric of nativeness) is used as the scoring function, i.e., the decoy closest to the native state is always selected as the initial structure for the next cycle (see Fig. 4.1). The scheme can be varied by introducing different levels of noise in the RMSD-based scoring function in order to determine what levels of noise can be tolerated to accomplish structure refinement (Stumpff-Kane et al. 2008). An important question that can be answered with such a framework is whether a given sampling method is in principle able to reach the native state. A given sampling method may not be able to reach the native state if the native state does not correspond to favorable free energies or if there are significant kinetic barriers on the path toward the native state that cannot be overcome within the limited amount of sampling during the generation of decoys. Both, the presence of kinetic barriers and the relative energy of the native state depend on the model that is used for the conformational sampling.

4.2.1 Quantitative Measure of Sampling Efficiency In order to introduce a quantitative measure of sampling efficiency it is worthwhile to revisit an earlier analysis by Reva and coworkers of randomly generated protein-like conformations (Reva et al. 1998). It was found that such conformations approximately follow a normal distribution (see Fig. 4.2) so that the probability of

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finding a conformation with a coordinate RMSD from the native of R or less is given according to:

Pr≤R

1 = √ σ 2π

R e

−(r−)2 2σ 2

dr,

(4.1)

0

where the average RMSD and the associated standard deviation σ depend on the specific protein that is being considered. may be estimated from the number of residues as 3.333N1/3 (Maiorov and Crippen 1995; Reva et al. 1998), while a value of 2.0 Å is a reasonable guess for σ for small and medium-sized proteins (Reva et al. 1998). As an example, one may calculate the probability of randomly generating conformations for a 150-residue protein with an RMSD of 15 Å or less as the gray/blue area under the curve which corresponds to a value of about 0.09 (Fig. 4.2). Fig. 4.2 Probability of finding structures at a given RMSD for a protein with 150 residues

The preceding analysis can be extended to analyze the fraction of structures that are more native-like than a given starting structure for the case of random sampling with no specific bias or barriers that would favor or disfavor sampling toward the native state. If one assumes that an ensemble generated from an initial structure with an RMSD R extends maximally to the interval [R−R, R+R] the fraction of improved structures with an RMSD between R−R and R is given according to: R PR−R
A = = A+B

R−R R+R 

e

−(r−)2 2σ 2

e

−(r−)2 2σ 2

dr (4.2) dr

R−R

where A and B are the areas under the curve indicated in Fig. 4.2 PR−R
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In practice, structures generated with a given sampling methods will not be distributed in a symmetric interval [R−R, R+R] around the starting structure even if the underlying energy landscape is completely flat because of a decreasing number of states as the native state, or any specific state for that matter, is approached. However, the symmetric case may be used as a theoretical reference to introduce a sampling efficiency ratio that compares the fraction of structures that are actually better than the starting structure to PR−R
4.3 Protein Models at Different Resolutions A large variety of models have been introduced to describe the interactions and structure of proteins. At the very minimum, such models need to maintain chain connectivity and proper space filling to generate protein-like structures as those used in early models of protein folding. Improved models introduce specific physicochemical characteristics of different amino acid side chains and represent secondary structure elements while the most detailed models used for representing protein structures include full atomistic detail. Atomistic models with physically motivated interaction potentials provide the most accurate description of protein structures that can be afforded today. An even more detailed model based on quantum mechanics remains beyond the realm of what is practical today. The application of classic atomistic models is limited, however, by still significant computational costs and a resulting energy landscape that tends to be very rough and thereby hinders efficient conformational sampling. Coarse-grained models on the other hand have generally much smoother energy landscapes but in those models energetic accuracy may be compromised to the point where the native state does not correspond to a free energy minimum anymore so that sampling of native-like structures is not favorable. In order to contrast the advantages and disadvantages of models at different resolutions in the context of conformational sampling, we will focus here on fully atomistic model based on the chemistry at harvard molecular mechanics (CHARMM) force field and two types of coarse-grained models (see Fig. 4.3). These models will be described briefly in the following.

4.3.1 All-Atom Models of Proteins All-atom models of biomolecules usually represent each atom, including all hydrogens, as spheres with point charges (see Fig. 4.3). The associated interaction potential generally assumes the following classical form (Mackerell 2004):

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Fig. 4.3 Protein models at different resolutions: all-atom resolution (left), PRIMO (center), SICHO (right)

V(r) =

N bonds

Nangles

kbond,i (di − di,0 ) + 2

i=1

+

kangle,i (θi − θi,0 )2

i=1

Ntorsions 

Nimproper    ktorsion,i 1 + cos(ni φi − φi,0 ) + kimproper,i (φi − φi,0 )2

i=1

+



Natoms atoms −1 N i=1

j=i+1

 4εLJ,ij



σLJ,ij rij

12

 −

i=1

σLJ,ij rij

6  +

1 qi qj 4π ε0 rij (4.3)

In Eq. (4.3), bonded terms maintain covalent molecular bonding geometries through harmonic terms for bonds, angles, and improper torsions as well as a Fourier series term for proper torsions. Bonded terms essentially reflect interactions that otherwise would have to be described quantum mechanically and they are either fit empirically to experimental high-resolution structural data and spectroscopic vibrational frequencies or to optimized geometries and normal mode frequencies from ab initio calculations. Non-bonded interactions are described through a combination of electrostatic interactions between point charges qi located at the atomic centers and the well-known Lennard–Jones potential which combines attractive van der Waals dispersion interactions with a highly repulsive term to avoid overlap of atomic spheres. The non-bonded terms are typically also fit to ab initio data (Mackerell 2004). The basic functional form given in Eq. (4.3) is used to describe protein interactions in a number of widely used force fields such as CHARMM (MacKerell et al. 1998), assisted model building with energy refinement (Amber), groningen molecular simulation (GROMOS) (Oostenbrink et al. 2004), and optimized potentials for liquid simulations (OPLS) (Jorgensen et al. 1996) with the main differences in the choice of force constants, torsion terms, partial charges, and Lennard–Jones parameters as well as the balance between non-bonded and bonded terms for 1–4 interactions. The CHARMM force field also introduces a 1–3 distance potential

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(MacKerell et al. 1998) and recently a spline-based torsion cross-correlation term was added to improve the description of protein backbone sampling (MacKerell Jr. et al. 2004a,b). An interaction potential according to Eq. (4.3) primarily relies on basic physical principles instead of empirical knowledge derived from known protein structures. As a result such models are applicable for describing the energetics of a wide variety of systems, including unusual, non-canonical structures, and disordered peptides. In the context of structure refinement all-atom models are furthermore expected to be able to correctly distinguish subtle energetic differences between sub-optimal and optimal packing arrangements. However, the all-atom level of detail also results in highly rugged energy landscapes since very small variations in atomic positions can lead to large changes in energy, e.g., when two contacting atomic spheres increase their degree of overlap or when the distance between two bonded atoms is stretched. The ruggedness of the energy landscape hinders efficient conformational sampling and in particular, the energetic fluctuations during sampling of similar conformations may be large compared to the difference in free energies between mis-folded and correctly folded states. 4.3.1.1 Sampling with All-Atom Force Fields Sampling with all-atom force fields commonly involves molecular dynamics simulations at a given temperature using forces based on Eq. (4.3) (from the gradient of the potential). The calculation of such forces for biological macromolecules is computationally very demanding because the long-range electrostatic interactions need to be calculated for large pairwise distances (Schreiber and Steinhauser 1992; Norberg and Nilsson 2000). Furthermore, simulations are typically required to use 1–2 fs integration time steps in order to adequately sample high-frequency bond vibrations. Conformational sampling can be improved simply by increasing the temperature or with a variety of more advanced enhanced sampling techniques (Cheng et al. 2004; Hamelberg et al. 2004; Micheletti et al. 2004; Okamoto 2004). An alternative strategy involves the use of normal modes to propagate structures along the lowest-frequency modes (Stumpff-Kane et al. 2008). Such modes focus on overall structural changes that involve large parts of a given structure and avoid the high-frequency fluctuations that give rise to the significant energetic fluctuations with atomistic force fields.

4.3.2 Coarse-Grained Models of Proteins Coarse-grained models of proteins reduce the model resolution from fully atomistic models. Such models may range from united-atom models where, e.g., aliphatic carbon–hydrogen pairs are combined into a single particle to models where an interaction site represents a significant part of a given protein structure and includes multiple residues, but the most popular models retain one up to a few interaction sites per amino acid residue. The reduced number of interaction sites greatly reduces the computational cost that is incurred during sampling but at the same compromises

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energetic accuracy. Furthermore, the degree of coarse-graining affects the ability to retain physically motivated interaction potentials. Instead, energy terms of coarsegrained models often have to rely on empirical and/or system-specific terms that limit transferability and applicability for structure refinement applications. While a more general overview over coarse-grained models is given in Chapter 11, we focus here only on contrasting two specific coarse-grained models that have very different characteristics. The first one, PRotein Intermediate MOdel (PRIMO), is a new model that represents proteins at an intermediate level of resolution but is able to retain a largely physics-based interactions potential. The second one, Side CHain Only (SICHO), is a lattice-based residue-level model that relies primarily on empirical and statistical terms. Both are described in more detail in the following sections. 4.3.2.1 PRIMO PRIMO (PRotein Intermediate MOdel) was recently developed as a coarse-grained model that maintains a one-to-one correspondence to fully atomistic models so that physically motivated force field-like interaction potentials can be used that are parameterized by direct comparison with a given all-atom force field (Gopal et al. 2010). Such a model is possible when coarse-grained interaction sites are selected so that the all-atom level of detail can be reconstructed analytically based on standard molecular bonding geometries. As an example, the position of the Cβ atom can always be reconstructed with good accuracy from backbone heavy atoms C, Cα , and N based on the assumption of standard tetrahedral bonding. A coarse-grained model without an explicit Cβ site is therefore still able to maintain close correspondence with a fully atomistic model. The PRIMO model is the result of a systematic reduction of interaction sites to a minimal set that maintains quasi-atomistic resolution (after all-atom reconstruction). The resulting model is shown in Fig. 4.4. It contains three sites per

Fig. 4.4 PRIMO interaction sites

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residue for the backbone and one to five interaction sites for non-glycine side chains. In some cases additional sites were introduced to achieve good space-filling. An example is the explicit Cβ site in alanine that is not needed for maintaining a oneto-one correspondence to the all-atom level but is included to improve packing interactions. All-atom reconstructions based on PRIMO deviate on average by less than 0.1 Å RMSD from structures that were reduced to the PRIMO level and then reconstructed to atomic detail (Gopal et al. 2010). In contrast, all-atom reconstructions from other coarse-grained models at lower resolutions, e.g., with a single site at the side-chain center, rarely achieve a reconstruction accuracy of less than 0.5 Å (Feig et al. 2000). More typical is an accuracy of 1–2 Å due to uncertainties in the common use of rotamer libraries for determining side-chain orientations (Bower et al. 1997). At the same time, the cost for the analytical reconstruction procedure from PRIMO to allatom representations is negligible compared to the cost of iterative reconstruction procedures that are used otherwise. This presents opportunities for the development of efficient multi-scale models. It also allows the use of virtual atomistic interaction sites to improve the description of bonded interactions without an additional cost for calculating non-bonded energy terms that would result if those sites were included explicitly as part of the model. The PRIMO interaction potential follows a force field-like form with additional terms for implicit solvent, an explicit hydrogen bonding interaction potential, and modified bonded terms to maintain correct bond geometries at the coarse-grained level: V(r) =

N bonds

 2 kbond,i di − di,0 +

i=1 Nangles

+



Nbond-splines



 2 kangle,i θi − θi,0 +

i=1 Nangle-splines



i=1 N i torsion N  i=1

  ktorsion,i, j 1 + cos(ni, j φi − φi, j,0 )

j=1

Ntorsion-splines

+



CMAP   N 1−4 d + s1D s2D i i (φi , ψi ) i

i=1

+

+

Natoms −1

N 

i=1

j=i+1

NHBOND 

  di1−3 s1D i

i=1 mult

+

  1−2 d s1D i i

 4εij

i=1

σij rij

12

  N−CO cos α, d s2D i

i=1

+ GGB solvation +

N atom i=1

γi SASAi

−



σij rij

6  +

(4.4) N  N  i=1 j=i+1

qi qj 4π ε0 rij

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Bonded Terms in PRIMO Bonded interactions between PRIMO sites correspond to both real covalent bonds, e.g., between Cα and N backbone sites and virtual bonds, e.g., between N and the combined backbone carbonyl site, CO. For real covalent bonds, standard harmonic terms are used as in standard all-atom force fields to model 1–2 (bonds) and 1–3 (angles) interactions. Otherwise, distance-based spline-interpolated potentials are employed to reproduce non-harmonic functions and in particular the presence of multiple minima (see example in Fig. 4.5). The shape of the effective bond and angle potentials for virtual bond interactions was extracted from the actual sampling in long explicit solvent simulations over 150 ns with the CHARMM22/CMAP force field for each of the amino acid dipeptides. From a strict point of view, the spline-interpolated potentials represent effective potentials that are only applicable for explicit solvent environments. However, because bonded interactions are strong compared to non-bonded interactions with the environment it is assumed here that the dipeptide-derived effective bonded terms can be transferred to condensed protein environments and different non-aqueous solvent environments.

Fig. 4.5 N–CA(Cα )–SC1(Cβ +Cγ ) effective angle interaction in leucine from leucine dipeptide simulation (points) and spline interpolation used in PRIMO (line)

Bonded terms based on PRIMO interaction sites are augmented by bonded terms involving virtual atomic sites that are reconstructed on-the-fly from the PRIMO model. Such terms are necessary to maintain correct bonding geometries at the PRIMO level. For example, in the case of leucine, the N–CA(Cα )–SC1(Cβ +Cγ ) angle term along with other bonded terms involving SC1 is not sufficient to restrict SC1 to a circle around the Cα –Cβ bond as required for sp3 -hybridized molecular

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bonding to Cβ (see Fig. 4.5). This problem is alleviated by reconstructing the Cβ position from the backbone atoms and introducing a harmonic CA–CB∗ –SC1 angle potential where CB∗ is the reconstructed Cβ position. It may appear easier to simply re-introduce CB as an additional site in PRIMO. However, the cost of an additional particle that is propagated along with the other particles and that has to be included in the calculation of non-bonded interactions far outweighs the negligible cost for reconstructing CB on-the-fly as a virtual atom. In all-atom force fields 1–4 interactions are typically modeled as a combination of Fourier-series torsional terms and scaled electrostatic and Lennard–Jones interactions. In PRIMO, 1–4 interactions are modeled as a combination of Fourier series terms, distance-based spline-interpolated terms, and reduced Lennard–Jones interactions to avoid hard-sphere overlap. However, 1–4 electrostatic interactions are omitted because the reduced charged do not provide sufficiently accurate local electrostatic interactions. Instead, the spline-based 1–4 interaction potentials represent the effective interactions in explicit solvent that are extracted from the dipeptide simulations. This approach is therefore also only strictly valid for small peptides in aqueous solvent. In addition to the one-dimensional 1–4 terms, a spline-interpolated twodimensional cross-correlation term based on the CMAP methodology (MacKerell Jr. et al. 2004a) is used in PRIMO to couple the sampling of CO–N–CA–CO and N– CA–CO–N torsions and thereby control the sampling of φ/ψ backbone torsions. Using the CMAP methodology it is possible to nearly exactly reproduce any given target φ/ψ-map (MacKerell Jr. et al. 2004a). The CMAP term in PRIMO is initially parameterized to reproduce the effective φ/ψ distribution in short alanine-based peptides with the CHARMM22/CMAP force field, but can be easily reparameterized to reflect improved φ/ψ distributions that are more consistent with experimental observations such as nuclear magnetic resonance (NMR) J-coupling data (Graf et al. 2007; Best et al. 2008).

Non-bonded Terms in PRIMO Non-bonded terms in PRIMO consist of Lennard–Jones terms, electrostatic interactions, and an explicit hydrogen bonding term. The parameters in the Lennard–Jones term (σ i , εi ), including reduced values for 1–4 interactions, are optimized to reproduce CHARMM all-atom Lennard–Jones interaction energies for a series of peptide test sets. Electrostatic interactions rely on partial charges for each PRIMO site. Because of the coarse-grained nature of PRIMO, the partial charges are generally reduced over all-atom models although formal charges are maintained for acidic and basic amino acids. With the reduced charges (and in the absence of higher-order multipoles) all-atom electrostatic interactions cannot be accounted for completely so that the PRIMO charges cannot be optimized directly by simply fitting to all-atom electrostatic energies. The reduced partial charges are a particular problem for the description of hydrogen bonds which is accomplished implicitly through electrostatics in modern

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all-atom force fields. In PRIMO, an explicit hydrogen potential is therefore reintroduced but with a functional form that differs from earlier implementations. The PRIMO potential relies on a spline-interpolated two-dimensional potential of mean force (PMF) as a function of both hydrogen bonding distance and angle. The PMF was obtained from a statistical analysis of protein structures in the PDB. In order to apply such a PMF in PRIMO, it is necessary (and straightforward) to reconstruct hydrogen atoms involved in hydrogen bonding on-the-fly from the PRIMO sites. The hydrogen bonding potential is the only term in PRIMO that directly encodes information extracted from known protein structures. Partial charges and a scaling factor for the PMF-based hydrogen bonding potential were optimized by fitting total PRIMO internal energies (including bonded and non-bonded terms) to total CHARMM all-atom energies for a series of test peptides. Implicit Solvent Terms in PRIMO Interactions with the environment are represented in PRIMO via implicit solvent. Because PRIMO sites have partial charges the generalized Born (GB) formalism given in Eq. (4.5) can be used to describe the electrostatic contribution to the solvation free energy as an approximation to a continuum dielectric electrostatic model according to Poisson theory (Still et al. 1990; Feig and Brooks 2004): 

GGB

1 1 = −166 − εp εw

 i

j



rij2

qi qj   + αi αj exp − rij2 /Fαi αj

(4.5)

where εp is the solute cavity dielectric constant, εw is the solvent dielectric constant, qi is the atomic charge of the ith PRIMO site, rij is the interatomic distance between the ith and the jth sites, α i is the so-called effective Born radius of the ith site. The key part of any GB implementation is the calculation of the Born radii (Onufriev et al. 2002). In the case of PRIMO, the generalized born with molecular volume (GBMV) formalism (Lee et al. 2003) is used because it results in electrostatic solvation free energies that are based on a molecular surface description and closely match Poisson theory (Feig et al. 2004b). The non-polar contribution to the solvation free energy is modeled simply according to a linear function of the solvent-accessible surface area (Sitkoff et al. 1994). However, instead of using a constant surface tension coefficient for the entire surface, different scaling factors are introduced for different atom types. Such a term is known as an atomic solvation potential (ASP) (Wesson and Eisenberg 1992). The ASP term is included because the electrostatic solvation free energies according Eq. (4.5) does not fully account for the electrostatic solvation energy because it is based on reduced partial charges. The ASP coefficients, which may be positive or negative, were fit by comparing the combined GB and ASP energies to all-atom implicit solvent free energies. It should be noted that the separation of internal energies and implicit solvent terms in PRIMO allows an extension of PRIMO to non-aqueous environments. For

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example, membrane environments can be modeled by replacing the standard GB formalism with an implicit membrane model based on a heterogeneous dielectric environment and ASP parameters that are scaled as a function of z, the distance from the membrane center (Tanizaki and Feig 2005).

Sampling with PRIMO PRIMO is designed as a continuous-space model with a force field-like energy function that can be explored with molecular dynamics sampling techniques although Monte Carlo sampling is also possible. Because of the coarse-grained nature and in particular the lack of explicit hydrogen atoms, a longer integration time step can be employed. In initial tests, PRIMO has been stable with time steps of up to 7 fs. Longer-time steps are likely possible by exchanging the GBMV implicit solvent model with a variant that models a smoother molecular surface based on experiences with all-atom implicit solvent simulations (Chocholousova and Feig 2006).

Transferability For the most part, PRIMO relies on a physically motivated interaction potential that is parameterized by direct comparison with an all-atom energy function. New empirical terms are introduced to model hydrogen bonding and atom-based solvation properties, but in contrast to other similar models no bias toward known secondary structures or other structural constraints is necessary to model a given protein system. In fact, PRIMO is designed to be used like an all-atom energy function with the ability to carry out stable native-state molecular dynamics simulations of any protein and allow peptide folding from extended chains to native-like structures. Initial validation suggests that PRIMO can fulfill that promise, but future more extensive tests of PRIMO will reveal where the coarse-grained nature of PRIMO places limits on its energetic accuracy compared to fully atomistic models.

4.3.2.2 SICHO The SICHO (Side CHain Only) model consists, as the name suggests, of a single interaction site per amino acid residue, located at the side-chain center (Kolinski and Skolnick 1994). Such a model emphasizes side-chain packing over the detailed conformational sampling of backbone conformations. Sampling with the SICHO model is computationally very efficient because of the small number of interaction sites, but it is further accelerated by projecting the SICHO particles onto a cubic lattice with a 1.45 Å grid spacing and allowing only moves between lattice points (see Fig. 4.3) (Kolinski and Skolnick 1994; Skolnick et al. 1997). The SICHO interaction potential consists of a number of statistical and empirical terms. It is summarized in Eq. (4.6) (Kolinski and Skolnick 1994):

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VSICHO = αstiffness

99

stiffness Vi,...,i+6 + αshort

i

+ αpair



short Vi,...,i+8

i

pair Vi, j

i, j

+ αburial



 i

+ α3−body



3−body

Vi, j,k

i, j,k

Viburial + αcontact



+ αmultibody



multibody

Vi, j,k,...

i, j,k,...

Vicontact + αcentrosymmetric Vcentrosymmetric

i

(4.6) The stiffness potential enhances the persistence length along the SICHO chain to result in protein-like conformations with α-helices and β-strands. The statistical short-range potential further favors sampling of specific secondary structure elements. This potential depends on input about known or predicted secondary structure for a given protein. In the context of refinement, this means that the SICHO potential largely prevents the sampling of alternative secondary structures and is best suited for improving the packing of existing secondary structure elements. The statistical pairwise potential implements known preferences for side-chain interactions in soluble proteins. Because of the low-resolution of SICHO, directional dependence of side-chain–side-chain interactions cannot be implemented with just a pairwise term. This deficiency is addressed in part with 3-body and multibody terms. Pairwise side-chain interactions are further modeled with a side-chain contact term that only applies to side chains in direct contact and a side-chain burial term that relies based on the empirical Kyte–Doolittle hydrophobicity scale (Kyte and Doolittle 1982). In addition, excluded volume of SICHO particles is considered implicitly by not allowing SICHO particles to move onto lattice sites within the excluded volume radius of any other site. Finally, based on empirical knowledge about the typical radius of gyration of a folded protein (Crippen and Snow 1990), a centrosymmetric biasing term is introduced to maintain the radius of gyration near that average value. Solvent is not considered explicitly in SICHO since it is included indirectly in the statistical potentials that are derived from soluble proteins. A particular advantage of the lattice projection is that simple lookup tables can be used for all interaction potentials. Many terms of the SICHO potential overlap in representing the underlying physical interactions. As a result, full scaling of all terms would lead to overcounting of certain interactions. Therefore, scaling factors are applied for each potential term. These scaling factors are optimized based on given test sets and to some extent depend on the type of application. For example, ab initio protein-folding applications might benefit from a larger weight for the centrosymmetric potential in order to favor compact, globular structures while such a potential is less helpful in the detailed refinement of a structure that is already near the native state. Sampling with SICHO Sampling with the SICHO model is essentially limited to Monte Carlo sampling since molecular dynamics simulations cannot be used with lattice models (Skolnick

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et al. 1997). In the current implementation of SICHO within the modeling of new structures from secondary and tertiary restraints (MONSSTER) program, the move set consists of two-bond (involving one site) and three-bond (involving two sites) moves as well as infrequent long-range moves involving a larger structural fragment in order to mimic secondary structure rigid body motions. Compared to all-atom models or even the PRIMO model, sampling with SICHO is extremely fast due to the combination of low-model resolution and lattice projection. In the context of structure refinement, it is necessary to convert the low-resolution SICHO models to all-atom detail. The reconstruction of all-atom models from SICHO is possible with an accuracy of about 1 Å through the use of a rotamer library but the optimization step that is part of such reconstruction algorithms adds additional computational costs (Feig et al. 2000).

4.4 Iterative Refinement with Different Protein Models We will now proceed to describe the application of the different models described above within the iterative refinement protocol described in Section 4.2. Test cases consist of predictions from previous CASP competitions and statistics about the starting structures are given in Table 4.1. Table 4.1 Initial structures from template-based predictions for CASP targets used as test set for iterative structure refinement protocol

Target

Number of residues

PDB code of experimental structure

Residues in PDB

Starting Cα RMSD from native in Å

T0132 T0138 T0157 T0167 T0169 T0188 T0190 T0266 T0274 T0275 T0277

154 135 138 185 156 124 114 150 159 137 119

1YLI 1M2F 1NMN 1M3S 1MK4 1O13 2G2N 1WDV 1WGB 1WJG 1WTY

11–152 1–135 2–101; 119–138 1–135; 141–185 1–156 1–107 4–114 3–152 1–159 2–136 2–117

5.10 5.05 6.36 5.52 6.82 3.37 2.96 1.61 4.10 2.88 1.55

4.4.1 Sampling Protocol Starting from each structure an iterative refinement protocol was followed as described above. At each round 1,000 decoys were generated and the structure with the lowest RMSD was used as the starting structure for the next round. This protocol allows the comparison of different sampling methods in the best-case scenario where RMSD is used as the scoring function. A total of 100 refinement cycles were

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completed for each target with the simulation protocols described in more detail in the following sections. 4.4.1.1 All-Atom Molecular Dynamics Simulations At each round, decoys were generated from the final structures of very short molecular dynamics simulations that were started from the given starting structure with different initial velocities. Each simulation was 1 ps long (500 steps with a 2-fs time step). An elevated temperature of 400 K was used to accelerate the exploration of phase space. Because an entire refinement run consisting of 100 rounds × 1,000 decoys × 1 ps corresponds to a total simulation time of 100 ns only computationally inexpensive solvation models were used instead of explicit solvent so that an entire refinement run can be completed in about a day on moderate computational resources. More specifically, the following two force field/solvent model combinations were considered: (1) A full all-atom model with interactions described by the CHARMM22 force field was used in combination with distance-dependent dielectric screening of electrostatic interaction as a crude but very inexpensive approximation of solvation effects (Schaefer et al. 1999). (2) A united-atom model without aliphatic hydrogens as described by the CHARMM19 force field (Neria et al. 1996) was used in conjunction with the EEF1 solvation model (Lazaridis and Karplus 1999). Both sets of simulations were run with the CHARMM program (Brooks et al. 2009), version c36a1.

4.4.1.2 PRIMO Molecular Dynamics Simulations With the PRIMO model, decoys were also generated from short molecular dynamics simulations over 1 ps each (250 steps with a 4-fs time step) at an elevated temperature of 400 K. Two combinations of PRIMO were run: (1) The PRIMO force field was combined with the generalized Born implicit solvent and atomic solvation parameters as described above. (2) The PRIMO force field was also combined with distance-dependent dielectric screening without atomic solvation parameters. All of the PRIMO simulations were run with a locally modified version of CHARMM based on c36a1 that fully implements all of the PRIMO force field features.

4.4.1.3 SICHO Lattice Monte Carlo Sampling Simulations with the SICHO model were run with lattice Monte Carlo simulations using the program MONSSTER (Skolnick et al. 1997) through the MMTSB Tool

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Set (Feig et al. 2004a). These simulations were run at a constant temperature of 1.333 in reduced units of kT (at 300 K), corresponding to 400 K. In each simulation, 100 Monte Carlo moves were attempted. The default potential parameters were employed, except for the centrosymmetric potential which was not applied.

4.4.2 Refinement Toward the Native State The progress toward the native state for each of the 11 test targets with the 5 different sampling protocols is shown in Fig. 4.6. It can be seen that refinement is possible with all sampling methods but there are significant differences in how quickly structures are improved and how close the final structures are to the native structures. SICHO leads to rapid initial refinement in cases where the initial models are at 4 Å or more, but initial structures that are closer to the native are either not significantly refined or actually become worse. Final structures with SICHO are generally within 2.5 and 3.5 Å. In contrast, sampling with the other force fields

Fig. 4.6 Iterative refinement with different sampling methods from targets given in Table 4.1 using RMSD from the native as an idealized scoring function

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requires more cycles to converge but most of the final models are between 1 and 3 Å with CHARMM22/rdie and CHARMM19/EEF1, between 1.5 and 3.5 Å with PRIMO/GB and between 1.5 and 2.5 Å with PRIMO/rdie. The extent to which structures close to the native state (<2 Å) can be reached depends on the model resolution to be able to distinguish 1.5 Å structures from 2 Å structures, on a sufficiently accurate energy function to allow the sampling of near-native states and the absence of significant barriers on the path toward the native state. Overall, PRIMO/rdie achieves the best performance in reaching nearnative states, but the best structures are closest to the respective native states with CHARMM22/rdie and CHARMM19/EEF1. The best structures are slightly further from the native state with PRIMO and remain at a significant distance from the native state for all but one target when the SICHO model is used for sampling. In order to assess the sampling efficiency ratio defined above, we first examined the range of sampling at each cycle with the different sampling methods. The results are shown in Fig. 4.7. It can be seen that CHARMM22/rdie, CHARMM19/EEF1, and both PRIMO models have a relatively narrow range of sampling of 0.04–0.1 Å while the sampling range is much broader for SICHO (0.2–0.5 Å). Broader sampling

Fig. 4.7 Range of sampling (standard deviation) as a function of the average RMSD during refinement of the test targets given in Table 4.2

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generally facilitates the identification of the most native-like structures with a given real scoring function. The number of structures that are better than the initial structure at each cycle is the other determinant for how likely it is that a given real scoring function will be able to select more native-like structures. The average number of decoys that are better than the initial structure after 1, 5, 10, and 20 cycles is given in Table 4.2. It can be seen that the number of structures better than starting structure is initially high but drops quickly to about 1% at 10 cycles and to about 0.1–0.3% after 20 cycles which means that only 1–3 structures out of 1,000 are better than the starting structure. The fraction of structures that are better than the starting structure is highest with CHARMM19/EEF1 and CHARMM22/rdie and lowest with SICHO sampling. While this quantity is an important characteristic of the different sampling methods, it does not reflect how much better the structures are. For example, refinement during the initial cycles is more rapid with SICHO compared to PRIMO. On the other hand, the ability to refine to more native structures with PRIMO over the course of 100 refinement cycles is a result of a small number of structures that are consistently more native-like during later refinement cycles and allow slow convergence to the 1–2 Å range while such structures are not generated with SICHO. Table 4.2 Average fraction of structures better than the starting structure at different refinement cycles

CHARMM22/rdie PRIMO/GB PRIMO/rdie CHARMM19/EEF1 SICHO

1

5

10

20

0.242 0.138 0.166 0.321 0.046

0.047 0.026 0.032 0.059 0.002

0.008 0.013 0.011 0.008 0.002

0.003 0.003 0.001 0.008 0.000

The number of structures that are better than the initial structure can be related to the theoretical probability PR−R
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Fig. 4.8 Sampling efficiency ratio (fraction of structures better than initial model vs. PR−R
CHARMM22/rdie PRIMO/GB PRIMO/rdie CHARMM19/EEF1 SICHO

1

5

10

20

0.56 0.32 0.39 0.71 0.22

0.11 0.06 0.08 0.13 0.01

0.02 0.03 0.03 0.02 0.01

0.01 0.01 0.00 0.02 0.00

function that is used during the sampling to direct sampling toward the native state. Therefore, it is not surprising that the coarse-grained SICHO model performs worst.

4.5 Summary and Outlook Protein structure refinement remains a challenging problem. The idea promoted here is that successful refinement can be accomplished with an iterative framework through a favorable combination of sampling and scoring methods. The focus in

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this chapter is on an assessment of sampling methods with models at different resolutions ranging from all-atom models to the intermediate coarse-grained model PRIMO and the residue-level SICHO model. It is found that the choice of the model affects the ability to generate near-native conformations with CHARMM22/rdie and CHARMM19/EEF1 performing very well compared to the coarse-grained SICHO model. It may be possible to generate even more native-like conformations by combining the CHARMM22 all-atom force field with generalized Born implicit solvent or even explicit solvent. However, the associated costs make such an approach impractical within the iterative protocol tested here. However, interestingly, the PRIMO model also largely matches and to some extent exceeds the performance of the CHAMM22/rdie and CHARMM19/EEF1 models. It does so at a much reduced cost and therefore appears to be the best practical choice among the models tested here. The results shown here rely on an idealized refinement protocol where RMSD is used as the scoring function. In practical applications of structure refinement where the native structure is not known a real scoring function has to be employed instead. The next direction is therefore the exploration of different scoring functions within the protocol described here to examine whether scoring functions are capable of reliably selecting an improved structure at each cycle. This task becomes especially challenging after 10 cycles when less then 1% of the structures are better than the starting model. In reality, the problem might be easier, though, than it seems, because a given scoring function would be expected to guide the conformational sampling along a low-free energy path where RMSD may not always decrease continuously but sampling may be facilitated by avoiding the crossing of large barriers. We hope that the iterative protocol and sampling efficiency metrics introduced here will establish a framework for assessing sampling methods in the context of structure refinement that ultimately will lead to successful structure refinement protocols. Acknowledgments This work was financially supported from NIH grant GM 084953, NSF grant MCB 0447799, an Alfred P. Sloan fellowship (to MF). Access to computer resources at the High-Performance Computer Center at Michigan State University and use of TeraGrid computing facilities under grant number TG-MCB090003 are acknowledged.

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Chapter 5

Effective All-Atom Potentials for Proteins Anders Irbäck and Sandipan Mohanty

Abstract Experimental developments are steadily improving our knowledge of the dynamics and interactions of proteins, knowledge that is important for understanding living systems at the molecular level. Computer simulations provide a complementary tool in this endeavor and offer unique opportunities to elucidate salient features that remain experimentally inaccessible. Simulations are being used to study many different aspects of protein dynamics of varying computational complexity, which makes it necessary to choose models for the calculations with care, depending on the problem at hand. The models in use today range from reduced models with one or a few sites per amino acid to all-atom models with explicit solvent molecules. All-atom simulations have traditionally often been based on quite elaborate potentials. This level of detail may be needed in many applications, like structure refinement, but is not an obvious choice when studying processes like folding and aggregation. Some recent (implicit solvent) models use simpler and computationally more convenient potentials, while retaining an all-atom description of the protein chains. This article discusses some potentials of this type. The usefulness of the approach is illustrated by briefly describing studies, based on one of the potentials, of folding thermodynamics, mechanical unfolding and aggregation.

5.1 Introduction A molecular understanding of living systems requires knowledge of protein dynamics (Smock and Gierasch 2009). The relevant dynamics of a protein may consist of small fluctuations about its native structure or reorientations of its ordered parts relative to each other. In either case, a tiny fraction of the conformational space is explored. For flexible proteins, perhaps with large intrinsically disordered parts

A. Irbäck (B) Computational Biology & Biological Physics, Department of Theoretical Physics, Lund University, Lund, Sweden e-mail: [email protected]

A. Kolinski (ed.), Multiscale Approaches to Protein Modeling, C Springer Science+Business Media, LLC 2011 DOI 10.1007/978-1-4419-6889-0_5, 

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(Dunker et al. 2002; Uversky 2002; Dyson and Wright 2005), the situation is different. When studying such proteins or conformational conversion processes like folding or amyloid aggregation, the competition between different minima on the free-energy landscape inevitably comes into focus. Studying these systems by computer simulation is a challenge, because proper sampling of all relevant free-energy minima must be ensured. This goal is very hard to achieve if explicit solvent molecules are included in the simulations. Many studies are therefore carried out using implicit representations of the solvent. Additional speed can be gained if the degrees of freedom are further reduced, by turning to a coarse-grained protein representation (see Chapters 2, 3, 4, 6, 7, 8, and 12). This step entails a loss of detail in the protein description that may or may not be critical, depending on the question at hand. In this chapter, we will briefly introduce a few intermediate-resolution descriptions, which retain an all-atom representation of the protein chains while treating the solvent implicitly. Incorporating all protein atoms into the computational model has the advantage of simplifying the description of many interactions, like excludedvolume effects. The price to be paid is the increased computational complexity, as simulating protein folding at the atomic level is a notoriously difficult problem. However, the actual computational cost depends not only on the geometric representation, but also very strongly on the form of the force field, for two important reasons. The first involves the cost of a single energy evaluation or a single simulation step, while the second involves the required length, or number of steps, of the simulation. For a protein with Nat atoms, a potential with pairwise additive interactions consists of a sum of ∼Nat2 terms. The number of elementary computational operations required for calculating this sum depends critically on whether or not the interactions can be effectively treated as short-ranged. If so, the use of cell list techniques can reduce the required number of operations from ∼Nat2 to ∼Nat (Frenkel and Smit 1996) and thereby reduce the cost of each step in a simulation. The form of the force field also dictates the shape of the free-energy surface. This strongly affects the number of steps required for proper sampling of the conformational space. The use of a force field with unphysical local free-energy minima, in which the system might get trapped, risks making the simulations unnecessarily slow. The traditional approach to all-atom protein simulations is to use the molecular dynamics (MD) method and force fields like Amber, CHARMM, OPLS, or GROMOS; for a review, see Ponder and Case (2003). These potentials are based on explicit and detailed, although still approximate, representations of the physical forces involved. These details may be important to include, for instance, when modeling ligand binding, in order to accurately describe small perturbations of a folded protein about its native state. However, the number of parameters involved is large, and it is extremely difficult to ascertain whether the parameter values apply for every possible conformation of a protein. This consideration becomes important in the study of large conformational changes, as in protein folding. Consider a peptide with a helical native state. High precision of a force field near this helical state does

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not guarantee a correct ranking of it relative to completely different states, such as a β-sheet state. In simulations of protein folding from a random unfolded state, the properties of the force field over the whole conformation space are relevant. In addition, the volume of the conformation space grows exponentially with the number of residues. This makes it desirable to develop alternative force fields in which the native states are more easily identified as free-energy minima, even at the expense of some details. In this chapter, we discuss some all-atom effective force fields with this goal. They are designed with a fast exploration of the global conformation space in mind, for which they rely on effective descriptions of the forces involved. These effective potentials are furthermore used together with simulation techniques which perform the conformational search faster than does conventional MD. In the following, we briefly describe four recently developed effective potentials for implicit solvent all-atom studies of proteins: the μ-potential of the Shakhnovich group (Hubner et al. 2005), the PFF potential of the Wenzel group (Verma and Wenzel 2009), the discrete molecular dynamics (DMD)-adapted potential of the Dokholyan group (Ding et al. 2008), and a potential developed by us (Irbäck et al. 2009) which we will refer to as the Lund potential. Our purpose is to introduce the reader to some of the prevailing ideas behind this class of potentials rather than to compile an exhaustive list of all potentials that fall in this category. However, it has been demonstrated that each of the above-mentioned set of potentials is capable of folding small proteins with different secondary structure. There is a wide range of possible applications of these methods, and they have indeed been used to study problems other than that of predicting folded structures. To illustrate this, we briefly discuss some studies of folding thermodynamics, mechanical unfolding, and aggregation that we performed using the Lund potential. The four discussed potentials are largely or entirely sequence-based, meaning that their parameters are determined by the amino acid sequence alone. We do not discuss native-centric G¯o-type potentials here (G¯o 1983), which require prior knowledge of the native structure. G¯o-type potentials have been frequently used in recent years, primarily to characterize the folding pathways of natively folded proteins. In a generalized form, the same approach has also been used to study dimerization processes involving domain swapping (Ding et al. 2002; Yang et al. 2004). Using entirely sequence-based potentials instead is much more challenging, but has obvious advantages, for example, when studying proteins with large disordered parts.

5.2 Effective Potentials In this section, we briefly describe the main features of the four above-mentioned implicit solvent all-atom effective force fields. Here, the term “all-atom” indicates that all heavy (non-hydrogen) protein atoms are explicitly represented; to what extent hydrogen atoms are represented individually varies among the four discussed potentials.

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Although we categorize them here in the same class, both in algebraic form and in the procedures followed to derive their parameters, these four potentials are quite different from one another. Furthermore, they have been used together with different simulation algorithms, and for partly different purposes. The μ potential (Hubner et al. 2005) of the Shakhnovich group is composed of two terms, an explicit backbone hydrogen bond potential and a contact potential governing pairwise interactions between hard-sphere atoms. The strength of the contact interaction is determined by the types of the interacting atoms. For this purpose, the backbone N, Cα, C , and carbonyl O atoms are regarded as defining their own types irrespective of their residue. Except in cases of symmetry, each heavy side chain atom is assigned its own type. Hydrogen atoms are not represented individually. This gives a total of 84 atom types and 3570 interaction strength parameters. These parameters are determined through a systematic statistical procedure, using Protein Data Bank (PDB) structures. The backbone hydrogen bond potential also has a simple square well form depending on four pairwise atom–atom distances. The μ-potential has been used together with Monte Carlo (MC) methods. First, it was applied to small helical proteins (Hubner et al. 2005). Recently, a local backbone term was added to the μ-potential (Yang et al. 2007). This term is based on PDB-derived information about conformational preferences of triplets of amino acids and is amino acid specific. With a modulation of the hydrogen bond term based on PSIPRED (McGuffin et al. 2000), this extended potential was found to reproduce well the native structures of a diverse set of 16 polypeptides with 25–77 residues, including α and β as well as mixed α/β proteins (Yang et al. 2007). The PFF potential of the Wenzel group, in its original form (Herges and Wenzel 2004), contains Lennard–Jones interactions, charge–charge interactions, an explicit backbone hydrogen bond potential, and an implicit solvent model based on exposed surface area. Non-polar CHn groups are represented by united atoms. The parameters of the solvent model are optimized to stabilize the native structure of a single protein, the villin headpiece, against competing low-energy decoys. Other proteins are studied using the same parameters. This version of the PFF potential was applied to five small helical proteins (Herges and Wenzel 2004). Recently, a local backbone potential, consisting of a torsion angle potential and residue-specific electrostatic interactions (Avbelj and Moult 1995), was added to the PFF potential (Verma and Wenzel 2009). Using the new potential, native-like minimum energy conformations were observed for 20 polypeptides with 12–56 amino acids and varying secondary structure (Verma and Wenzel 2009). The calculations were performed using basin-hopping techniques for energy minimization (Verma et al. 2006). The potential of the Dokholyan group (Ding et al. 2008) is adapted to the DMD method, which is an accelerated form of MD (Rapaport 1997). The speed up is achieved by approximating the terms in the interaction potential with step functions. Under this approximation, the dynamics is driven by simple collisions rather than by gradients of the potential. The all-atom DMD potential of this group is composed of a local potential and three terms representing Lennard–Jones interactions, solvation effects, and hydrogen bonding. Only polar hydrogen atoms are represented individually. Solvation effects are modeled in terms of effective pairwise interactions, using

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the potential of Lazaridis and Karplus (1999). DMD simulations with this approach were performed for 6 polypeptides with 20–60 residues and different secondary structure (Ding et al. 2008). Folded structures similar to the experimental structures were observed for all six sequences. For the three smallest ones, the folding thermodynamics were studied as well. The Lund potential, like the μ potential, has been developed and parameterized from scratch (Irbäck et al. 2003; Irbäck and Mohanty 2005; Irbäck et al. 2009). The calibration procedure is, however, very different from that followed for the μ potential; the calibration of the Lund potential is based on full-scale thermodynamic simulations of peptides, rather than on statistical analyses of PDB structures. The Lund potential, in which all hydrogens are represented individually, consists of four major terms: a local potential, an excluded-volume term, hydrogen bonds and a residue-specific interaction term between pairs of side chains. The main contribution to the fourth term approximates implicit solvent effects, in particular the hydrophobic effect. The side chain–side chain potential also contains charge– charge interactions, which are assumed to be screened by the (implicit) solvent and therefore effectively short ranged. A full description of the different terms of this potential, on which the calculations discussed below are based, can be found elsewhere (Irbäck et al. 2009). The simulations with the Lund potential have been carried out using PROFASI, which is an open-source MC package for folding and aggregation simulations, written in C++ (Irbäck and Mohanty 2006). The potential was shown to capture both structural and thermodynamic properties of a set of 17 peptides with 10–37 residues and diverse secondary structure. In addition, it was applied to three 49–67-residue systems (a heterodimeric coiled-coil system, a mixed α/β protein, and a three-helix-bundle protein), with very good results (Irbäck et al. 2009). These four potentials share the ability to fold proteins with different secondary structure, using a single parameter set, which is a fundamental and non-trivial requirement (Yoda et al. 2004; Shell et al. 2008). What makes these four potentials successful in this respect? A precise answer to this question is difficult to give, because the terms of a potential are interdependent and dependent on the choice of geometric representation. It is clear, however, that local backbone energetics influence secondary-structure preferences in a very direct manner and therefore play a key role. Indeed, the above-mentioned revisions of the μ- and PFF potentials, which were carried out to be able to describe β-sheet proteins, focused on local interactions. It is also worth noting that all the four potentials contain an explicit hydrogen bond term. Moreover, the hydrogen bond energy has a directional dependence in all four cases, although its precise form varies. By contrast, there are large differences among the potentials in how solvation effects are modeled. Although this might change for larger proteins, it thus seems that the exact form of these effects is not extremely important for small proteins. These computational models have been studied using DMD, MC methods and energy minimization techniques like basin-hopping. MC methods have been applied to systems with torsion angles as their degrees of freedom. A simple but useful MC move for such systems consists of rotating single, randomly selected torsion

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angles individually. For a backbone angle, this move can lead to a highly non-local deformation of the polypeptide. For efficient sampling of compact states, it is useful to also include a local or semi-local backbone move. An example of such a move is Biased Gaussian Steps, which simultaneously updates up to eight consecutive backbone angles in a manner that keeps the ends of the segment approximately fixed (Favrin et al. 2001). The DMD and MC approaches can be much more efficient than conventional MD. To further speed up the exploration of the conformational space, generalizedensemble techniques are currently routinely used (see Chapter 9).

5.3 Applications The ability of the potentials discussed above to reproduce folded structures is an important benchmark. The potentials can also be used to study a range of other problems, where sometimes the experimental information available is very limited. In this section, we briefly describe some studies of folding thermodynamics, mechanical unfolding, and aggregation based on the Lund potential.

5.3.1 Folding Thermodynamics For many proteins, a proper structural characterization requires more than specifying a single conformation. This is often the case for small polypeptides, which tend to be unstable. There are further recent findings indicating that proteins with large unstructured regions are more common than previously thought (Dunker et al. 2002; Uversky 2002; Dyson and Wright 2005). When studying these fully or partially disordered systems, where fluctuations play a central role, thermodynamic equilibrium methods are more relevant than energy minimization techniques. However, to perform proper thermodynamic simulations at the atomic level has long been a challenge even for small peptides. With potentials like those above and with today’s computers, the situation is different. Statistically reliable results can now be easily obtained for peptides with, say, 20 residues, as illustrated by a recent study by us (Irbäck et al. 2009). Here we studied the folding thermodynamics for 17 peptides with 10–37 residues (see Fig. 5.1). For each peptide, a set of 10 independent runs was performed, starting from different random initial conformations. Each run was long enough to contain about a hundred independent folding/unfolding events. To achieve this required  1 day per run on a 2.0-GHz AMD Opteron processor. These calculations showed that our potential not only folds these sequences to structures similar to their experimental structures, but also captures many experimentally observed stability differences among the peptides. As an example, let us consider the β-hairpin GB1p, the second β-hairpin from the B1 domain of protein G (residues 41–56), and two designed variants of GB1p with enhanced stability, called GB1m2 and GB1m3 (Fesinmeyer et al. 2004). The isolated GB1p fragment

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Fig. 5.1 Schematic illustration of native geometries studied using the Lund potential. (a) the Trp-cage (1L2Y), (b) an α-helix, (c) a β-hairpin, (d) a three-stranded β-sheet, (e) an α-helix dimer (1U2U), (f) a three-helix bundle (1LQ7), and (g) a mixed α/β protein (2GJH). Figure adapted from Irbäck et al. (2009)

has been shown to form a marginally stable β-hairpin (Blanco et al. 1994). By circular dichroism (CD) and nuclear magnetic resonance (NMR) experiments, the folded population of GB1p was found to be ∼0.30 at 298 K (Fesinmeyer et al. 2004). At the same temperature, GB1m2 and GB1m3 have estimated folded populations of 0.74 ± 0.05 and 0.86 ± 0.03, respectively (Fesinmeyer et al. 2004). Using tryptophan fluorescence instead of CD and NMR, a somewhat higher apparent folded population was obtained for GB1p (Muñoz et al. 1997), indicating deviations from a simple two-state behavior. We studied the melting properties of these peptides using two different observables, namely, the hydrophobicity energy Ehp and a hydrogen bond-based nativeness measure called qhb (qhb = 1 if at most two native backbone hydrogen bonds are missing and qhb = 0 otherwise). Figure 5.2 shows the calculated temperature dependence of these observables. Two things are worth noting. First, the apparent folded populations based on Ehp are higher than those based on qhb . This probe-dependence is consistent with the experimental findings for GB1p. Second, the calculated stability order among the three peptides is the same independent of which of the two observables we study. The order is GB1p < GB1m2 < GB1m3, which agrees with experimental data. We also applied our potential to three larger systems with 49–67 residues (Irbäck et al. 2009), which were more challenging to simulate. In fact, for a mixed α/β protein with 49 residues, we were unable to obtain statistically accurate thermodynamic results, although the global free-energy minimum could be identified with confidence. On the other hand, the 67-residue three-helix bundle protein GS-α3W (Johansson et al. 1998; Dai et al. 2002) turned out to be quite easy to characterize thermodynamically. Our simulations for these two proteins illustrate that the

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computational complexity depends not only on protein size, but also very strongly on the shape of the folding free-energy landscape. Our GS-α3W simulations took about 10 days of computing time on 64 AMD Opteron processors running at 2.0 GHz and resulted in about 800 independent folding events to the native state. On average, we thus obtained more than one folding event per day of single processor time, which is very good for a protein of this size. A study of GS-α3W using the same geometric representation but the ECEPP/3 force field was recently reported (Meinke and Hansmann 2009). Compared to this study, our simulations found about a hundred times the number of independent folding events, while consuming much smaller computing resources. Figure 5.3a shows how the probabilities for structures with different backbone root mean square deviation (bRMSD) vary with temperature in our GS-α3W

Fig. 5.3 Simulation results for GS-α3W. (a) Variation of histogram of backbone RMSD (bRMSD) with temperature. At high temperatures, there is a broad distribution of bRMSD with values > 10 Å. At lower temperatures there are three clearly separated clusters. Representative structures from these clusters are also shown aligned with the native structure. (b) Temperature dependence of specific heat, Cv , and the ratio hr of the observed helix content and the helix content of the native structure. Figure adapted from Irbäck et al. (2009)

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simulations. Clearly, the protein makes a transition from a rather continuous distribution of bRMSD at high temperatures to a distribution dominated by three well-separated clusters. Analysis of the structures at the lower temperatures shows that all three free-energy minima consist almost exclusively of structures with all three helices of GS-α3W formed. The plot of the ratio of the observed helix content and the helix content of the native state, shown in Fig. 5.3b, further supports this idea. The average value of this ratio approaches 1 as the temperature decreases below 300 K. From CD measurements, the stability of GS-α3W was estimated to be 4.6 kcal/mol at 298 K (Johansson et al. 1998). The specific heat curve, also shown in Fig. 5.3b, indicates that the formation of these structures correlates with the steepest change in energy. The cluster with a center at bRMSD ∼3 Å dominates at the lowest temperatures. The structures contributing to the cluster with ∼8–9 Å bRMSD superficially look like well-folded three-helix bundles. However, the arrangement of the helices is topologically distinct from the native arrangement. The cluster seen at larger bRMSD values is broader and consists of a host of structures in which two of the helices make a helical hairpin, but the third helix is not bound to it. The unbound helix could be at either side of the chain. According to our model therefore, the population at the lowest temperatures consists of ∼80% genuinely native structures, ∼10% three-helix bundles with wrong topology, and ∼10% other structures with as much helix content as the native state. In order to experimentally determine the true folded population of the protein, the experimental probe must be able to distinguish the native fold from the other helix-rich structures described here.

5.3.2 Mechanical Unfolding Single-molecule manipulation by atomic force spectroscopy (AFM) or optical tweezers has become an important and widely used method for examining mechanical properties of proteins (Forman and Clarke 2007). While the monitoring of individual molecules removes ambiguities of bulk experiments, there are still important issues to be addressed in order to extract as much information as possible from these experiments. How does one, for example, extract properties of the equilibrium free-energy landscape at zero force from observed unfolding times at non-zero force? Further, the structural information provided by the experiments is limited, because the chain extension is typically the only conformational variable monitored. We have used the Lund potential to investigate the mechanical unfolding of two proteins, ubiquitin and the fibronectin domain FnIII-10. AFM experiments had shown that both ubiquitin (Schlierf et al. 2004) and FnIII-10 (Li et al. 2005) sometimes unfold via intermediate states when stretched by an external force. The main purpose of our studies was to characterize the mechanical unfolding pathways and intermediates of these proteins. Following the experiments (Schlierf et al. 2004), we studied the unfolding behavior of the 76-residue protein ubiquitin under constant pulling forces in the range of 100–200 pN (Irbäck et al. 2005). As in the experiments, both one-step unfolding and

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unfolding through intermediate states were observed. Moreover, the intermediates seen in our simulations had extensions consistent with experimental data. Having verified this, we characterized the unfolding pathways by examining the order of breaking of secondary-structure elements. We found that there was a statistically preferred unfolding order, which was the same at the different forces studied. Furthermore, the same unfolding order was observed in events both with and without intermediate states. The mechanical unfolding order we found does not agree with thermal unfolding data from experiments at zero force (Cordier and Grzesiek 2002; Chung et al. 2005). Therefore, we also performed a study of the thermal unfolding of ubiquitin (Irbäck and Mitternacht 2006). It turned out that our computational model actually predicts a different thermal unfolding order, consistent with the thermal unfolding experiments. It is also worth noting that several subsequent computational studies of the mechanical unfolding of ubiquitin, based on completely different potentials, have obtained results similar to ours (Li et al. 2007; Kleiner and Shakhnovich 2007; Imparato and Pelizzola 2008). Fibronectin is a giant multimodular cell adhesion protein. The most common type of module is called FnIII and has ∼90 residues. Of particular interest is the tenth FnIII module, FnIII-10, which contains an Arg-Gly-Asp motif that is critical for the cell–fibronectin interaction. The mechanical properties of FnIII-10 have been extensively studied by both experimental and computational methods. The occurrence of unfolding intermediates was predicted by an early simulation study (Paci and Karplus 1999) and verified by AFM experiments (Li et al. 2005). However, computer simulations have also been performed by many other groups (Klimov and Thirumalai 2000; Craig et al. 2004; Sułkowska and Cieplak 2007; Li 2007), and as yet there is no consensus on the precise nature of the unfolding pathways and intermediates. Recently, we therefore performed an extensive study of how the unfolding behavior of FnIII-10 depends on the pulling conditions (Mitternacht et al. 2009). Six constant pulling forces and four constant pulling velocities were studied, and  100 unfolding trajectories were generated in each of these ten cases. We observed both apparent two-state unfolding and several unfolding pathways involving one of three major, mutually exclusive intermediate states. Unlike for ubiquitin, we thus found more than one major unfolding pathway. The relative weights of the different unfolding pathways were further found to depend strongly on the pulling conditions. For weak pulling (low force or velocity), all three major intermediates occurred with a significant frequency. For strong pulling (high force or velocity), one of them dominated over the other two. All our three major intermediates have extensions consistent with existing experimental data. With higher experimental resolution, it should become possible to discriminate between these three intermediates and thereby test the picture emerging from our study. For properties that can be studied experimentally, like the extension of intermediates and the unfolding force, we obtained results in quite good agreement with experimental data for both ubiquitin and FnIII-10. This agreement is encouraging, especially since both studies could be carried out without having to adjust any parameter of the potential.

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5.3.3 Aggregation The ability to aggregate into β-sheet-rich so-called amyloid fibrils is a property shared by many polypeptides, including unstructured polypeptides as well as natively folded proteins from varying structural classes (Selkoe 2003; Chiti and Dobson 2009). The mechanisms of amyloid aggregation are currently being intensely studied, primarily because protein aggregates of this type have been implicated in several diseases, such as Alzheimer’s and Parkinson’s diseases. Of particular interest are the initial stages of aggregation process, because a growing body of evidence indicates that the cytotoxic component in these diseases is small early formed oligomers rather than the full fibrils. Unfortunately, these small aggregates are notoriously hard to characterize. In the quest to do so, computer simulations can play an important role by providing complementary information not accessible experimentally. Simulating protein aggregation is a challenge that has been approached in several different ways (Ma and Nussinov 2006). One approach has been to study the stability of preformed structures or the addition of polypeptides to such a structure. Another simplification that has been made is to use coarse-grained protein descriptions. All-atom studies of spontaneous aggregation, starting from random initial conformations, have typically been limited to systems consisting of a small number of short polypeptides. The Lund potential has been used to study spontaneous aggregation for fibril-forming fragments of the Aβ peptide and protein tau, both associated with Alzheimer’s disease. Aβ and protein tau are the main components of amyloid plaques and neurofibrillary tangles, respectively. One of the best-studied fibril-forming peptides is the 7-residue Aβ(16–22) fragment, which contains the 17–21-residue segment that is commonly referred to as the central hydrophobic core of Aβ. In six-chain simulations for Aβ(16–22), we observed the spontaneous formation of a β-barrel (Irbäck and Mitternacht 2008). In line with previous Aβ(16–22) simulations (Hwang et al. 2004; Klimov et al. 2004; Santini et al. 2004; Röhrig et al. 2006; Gnanakaran et al. 2006; Meinke and Hansmann 2007; Krone et al. 2008), a wide spectrum of meta-stable aggregates was observed in our study. The β-barrel stood out as the by far most long-lived aggregate, indicating that it corresponds to a relatively deep free-energy minimum. The aggregation properties of the hydrophobic Aβ(16–22) fragment were also compared to those of the less hydrophobic Aβ(25–35) fragment, through simulations with up to 20 chains based on the same potential (Cheon et al. 2007). Aβ(16–22) was found to rapidly form disordered oligomers, which were subsequently converted into ordered oligomers through a reorganization process. By contrast, Aβ(25–35) formed ordered oligomers directly, in a one-step process. In particular, this study suggests that the character of the aggregation process critically depends on the balance between hydrophobicity and hydrogen bonding forces. Recently, we studied spontaneous aggregation for relatively large systems of PHF6 peptides (Li et al. 2008), with up to 36 chains. The PHF6 peptide (VQIVYK) is a six-residue fragment of protein tau and is known from experiments to make amyloid fibrils with a parallel β-strand organization (Goux et al. 2004). Based on

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X-ray experiments on microcrystals, it has further been proposed that the β-sheets in PHF6 fibrils, as well as in fibrils of many other peptides, are arranged in tightly packed pairs with a “dry steric zipper” organization at the sheet–sheet interface (Sawaya et al. 2007). In our PHF6 simulations, we saw, as for Aβ(16–22), a multitude of small aggregates. A significant increase in stability was observed for aggregates above a certain critical size, indicating that the formation of stable oligomers occurs through a nucleation process. The oligomers formed in the nucleation step were β-sheet-rich, but they were not necessarily growth competent. Instead further growth required conformational reorganization. Interestingly, growth competence was found to correlate with the alignment of the strands in the β-sheets. Figure 5.4 shows how the fractions of parallel and antiparallel β-sheet structure varied with aggregate size in our simulations. While, as mentioned, PFH6 fibrils have β-sheets with a parallel strand organization, small aggregates from the simulations show a clear statistical preference for the antiparallel organization. However, the fraction of parallel structure increased steadily with aggregate size, and in the simulated aggregates with >18 chains the parallel organization was more common than the antiparallel one. In marked contrast to the plethora of different small aggregates, all larger simulated aggregates with 20 chains shared a common form. Invariably, these large aggregates were composed of two stacked β-sheets, as in Fig. 5.5. This form bears a suggestive resemblance to the dry steric zipper structure proposed for PHF6 fibrils (Sawaya et al. 2007). It would be very interesting to extend these calculations to even larger systems, which we hope to be able to do soon. These studies dealt with short fragments of Aβ and protein tau. We are currently using the same methods to examine monomer and dimer properties of full-length 42-residue Aβ, for the wild-type sequence, and three variants with different aggregation properties.

Fig. 5.4 The fractions of parallel and antiparallel β-sheet structure against aggregate size, as obtained from simulations for 24 PHF6 peptides. Figure reprinted from Li et al. (2008)

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Fig. 5.5 Snapshots of two large aggregates from simulations with 24 PHF6 peptides. V1, I3, and K5 side chains are shown in gray and V4 side chains in dark, whereas Q2 and K6 side chains, for clarity, are omitted. Figure adapted from Li et al. (2008)

5.4 Summary The computer time required to explore the conformational space of a protein depends not only on the geometric representation of the system and the simulation algorithm, but also on the form of the potential. Here, we have discussed some novel effective potentials for implicit solvent all-atom simulations. The use of effective potentials together with efficient algorithms can give a dramatic speed up of the conformational search and thereby make it possible to address questions that otherwise would be computationally out of reach. Also, it is likely that the discussed potentials, which are relatively new, can be refined to further increase their applicability.

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Chapter 6

Statistical Contact Potentials in Protein Coarse-Grained Modeling: From Pair to Multi-body Potentials Sumudu P. Leelananda, Yaping Feng, Pawel Gniewek, Andrzej Kloczkowski, and Robert L. Jernigan

Abstract The basic concepts of coarse-graining protein structures led to the introduction of empirical statistical potentials in protein computations. We review the history of the development of statistical contact potentials in computational biology and discuss the common features and differences between various pair contact potentials. Potentials derived from the statistics of non-bonded contacts in protein structures from the Protein Data Bank (PDB) are compared with potentials developed for threading purposes based on the optimization of the selection of the native structures among decoys. The energy of transfer of amino acids from water to a protein environment is discussed in detail. We suggest that a next generation of statistical contact potentials should include the effects of residue packing in proteins to improve predictions of protein native three-dimensional structures. We review existing multi-body potentials that have been proposed in the literature, including our own recent four-body potentials. We show how these are related to amino acid substitution matrices.

6.1 Introduction Statistical contact potential functions are knowledge-based potentials since they are based on information extracted from sets of protein structures. These knowledgebased potentials are useful to describe in approximate ways the interactions among residues or atoms in proteins. They are useful mainly because it is extremely difficult, if not impossible, to formulate these interactions directly using laws of physics or to determine them experimentally. Statistical potential functions have

R.L. Jernigan (B) Department of Biochemistry, Biophysics, and Molecular Biology, Iowa State University, Ames, IA, USA; L.H.Baker Center for Bioinformatics and Biological Statistics, Iowa State University, Ames, IA, USA e-mail: [email protected]

A. Kolinski (ed.), Multiscale Approaches to Protein Modeling, C Springer Science+Business Media, LLC 2011 DOI 10.1007/978-1-4419-6889-0_6, 

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been derived by analyzing known protein structures in the Protein Data Bank (PDB) and have been used extensively in computational studies of proteins. Usually these are derived for coarse-grained cases, averages for individual amino acids or groups of atoms. In this chapter we discuss the theory and methods used to derive knowledgebased potentials and review the history of the development of statistical contact potentials. Based on the approach used to develop knowledge-based potentials, they can be generally classified into two groups: those derived from statistical analysis of a protein structure database (Tanaka and Scheraga 1976; Miyazawa and Jernigan 1985; Sippl 1990) and potentials based on optimization of recognition of the native structure among a set of misfolded conformations – so-called decoys (Goldstein et al. 1992; Maiorov and Crippen 1992; Thomas and Dill 1996a). We discuss these two classes of knowledge-based potentials, giving emphasis to statistical potentials because these have proven themselves to be important in protein structure prediction, in studies of protein–protein interactions, and in designing protein sequences and are more prominent. There are three types of statistical potentials: distance-independent statistical potentials, distance-dependent ones, and geometric potentials. We will discuss in some detail distance-independent potential functions, with special reference to the Miyazawa–Jernigan contact statistical potentials, which are widely used. We will also review distance-dependent statistical potentials and geometric statistical potentials. Several multi-body potentials that were proposed in the literature (Munson and Singh 1997; Krishnamoorthy and Tropsha 2003), and recently by us (Feng et al. 2007), will be described. We compare and discuss common features and differences among the various pair contact potentials. Potentials derived from the statistics of non-bonded contacts in protein structures in the Protein Data Bank are compared with potentials developed for threading purposes based on the optimization of the selection of the native structures among decoys. The potential server developed recently by Feng et al. (2010) for energy estimation of coarse-grained models of proteins with knowledge-based potentials is also introduced. Further, we briefly discuss the following applications of knowledge-based potentials: (i) discrimination of the native structure from decoys (Park and Levitt 1996; Zhou and Zhou 2002; Gilis 2004), (ii) protein docking (Li and Liang 2005a), (iii) design of new proteins (Hu et al. 2004), and (iv) prediction of protein stability and binding affinity (Gilis and Rooman 1996, 1997; Guerois et al. 2002; Bordner and Abagyan 2004; Hoppe and Schomburg 2005; Zhou and Zhou 2002). And, finally ways to further improve these potentials are discussed. To develop a knowledge-based potential function three main steps are necessary: (i) the description of the sequence and structure of the native protein structure (protein descriptor) in a form suitable for computations; (ii) determination of the functional form of the potential function; (iii) the extraction method to derive potential function parameters.

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6.2 History of Development of Knowledge-Based Potentials Experimental observations led to the remarkable discovery that all information required to specify the three-dimensional structure of a protein is contained in its amino acid sequence (Anfinsen 1973; Anfinsen et al. 1961). These experiments led to the formulation of the thermodynamic hypothesis of proteins, which states that the native state of a protein corresponds to its global free energy minimum under physiological conditions and provides the basis for development of potential functions that are used to calculate effective energy of a protein. Although this is substantially true, we now know that there are exceptions where the protein structure changes in response to local forces or local conditions. Initially atomic potential functions were obtained using principles of physics. Here, thermodynamic properties of a system are measured experimentally and also calculated using quantum mechanics applied to small molecules. By utilizing the appropriate experimental and theoretical values, the parameters in the atomic potential are obtained. These potentials are often referred to as physics-based, physical or semi-empirical effective potential functions, or force fields (Wolynes et al. 1995; Karplus and Petsko 1990; Yang et al. 2006). Although these atomic physics-based potentials have been found promising for protein folding studies, the approach is computationally costly when applied to large proteins because of the large number of atoms in large proteins. As an alternative, knowledge-based potentials have been developed. Here, known protein structures that have been deposited in Protein Data Bank (PDB) are used to generate the parameters in our potential functions (Sippl 1993). These potential functions implicitly take into account many different types of physical interactions such as electrostatic, cation–π, and van der Waals interactions. The number of experimentally determined three-dimensional protein structures in the PDB database is increasing rapidly, and it now becomes possible to derive a variety of improved, more specific potentials. As a result knowledge-based potential functions are becoming extremely popular nowadays in computational studies such as protein structure prediction, analyzing protein–protein interactions, and designing new proteins. The knowledge-based potential functions can usually be divided into two types: atomic-level potentials (Samudrala and Moult 1998; Gatchell et al. 2000; Lu and Skolnick 2001; Zhou and Zhou 2002; McConkey et al. 2003; Hubner et al. 2005; Qiu and Elber 2005; Shen and Sali 2006) and coarse-grained potentials (Tanaka and Scheraga 1976; Miyazawa and Jernigan 1985; Hendlich et al. 1990; Sippl 1990; Hinds and Levitt 1992; Miyazawa and Jernigan 1996; Bahar and Jernigan 1997; Eisenberg et al. 1997; Simons et al. 1999; Tobi and Elber 2000; Zhang et al. 2006; Dehouck et al. 2006; Dong et al. 2006). The latter have been demonstrated to be highly effective for reducing the computational cost in modeling native protein structures, although they are sometimes considered not to be sufficiently rigorous to reflect the entire landscape of a potential energy surface (Thomas and Dill 1996b; Skolnick 2006). The performance and applicability of coarse-grained potential functions are largely modulated by the choice of the coarse-graining scheme. In

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many applications, the ability to accurately calculate the potential energy based solely on Cα positions would certainly be useful. Typical examples are seen in recent studies on modeling protein chain topologies based on low-resolution density maps (Wu et al. 2005a) and on coarse-grained folding simulations based on Cα models (Wu et al. 2005b). Knowledge-based potentials are categorized into two main groups according to how they are extracted from known protein structures. Most important of the two are the so-called statistical or knowledge-based potentials. These potential functions are derived by a statistical analysis of protein structure databases (Tanaka and Scheraga 1976; Miyazawa and Jernigan 1985; Sippl 1990). The other group of knowledgebased potential functions are known as optimized potential functions. This type of potential function is even more empirical. To obtain these potential functions a selected criterion, for example, the maximization of the energy gap between the known native structure and a set of alternative or decoy conformations is used (Dobbs et al. 2002; Hu et al. 2004; Dombkowski and Crippen 2000). Some threebody (Munson and Singh 1997) and four-body contact potentials (Krishnamoorthy and Tropsha 2003; Feng et al. 2007) were introduced that showed improved results.

6.2.1 Inverse Boltzmann Relationship In an ensemble of protein molecules at thermodynamic equilibrium at temperature T, the relative probability P(s) of a microstate s with energy E(s) is given by the Boltzmann distribution P(s) =

exp [−E(s)/kT] Z(q)

(6.1)

where k is the Boltzmann constant and Z(q) is the partition function for the protein sequence q given by, Z(q) ≡



exp[−E(s)/kT].

(6.2)

s

and the sum is over all microstates of the system. By inverting Eq. (6.1), E(s), which is also sometimes termed a potential of mean force, can be obtained directly in the following form E(s) = −kT ln P(s) − kT ln Z(q)

(6.3)

For a given temperature, Z(q) depends only on the protein sequence. Although Z(q) cannot be obtained experimentally, when the sequence q and temperature T are fixed, Z(q) is also fixed. The partition function Z(q) also depends on how the protein is described once its structure and sequence are set, e.g., the level of coarse graining.

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The descriptor s does not affect the probability of occupancy of different energy states. However, obtaining a real energy value from a protein descriptor s depends on the form of the potential function E(s). The general goal in statistical mechanics is to calculate the partition function and probabilities from a given energy function. Then the macroscopic thermodynamic quantities of the system can be derived. This approach in statistical mechanics requires (i) the design of an energy function which models the system reasonably well and (ii) the calculation of the partition function by analytical or numerical techniques. Potential functions can take many forms; but by far the most widely used form for coarse-grained proteins is the weighted linear sum over pairwise contacts (Tanaka and Scheraga 1976; Miyazawa and Jernigan 1985; Tobi et al. 2000; Vendruscolo and Domanyi 1998). In this linear approximation E(s) is given by the dot product of vectors w and s: E(s) =



wi si = w · s

(6.4)

i

where si is the number of occurrences of the descriptor type i and wi is the corresponding weight. The potential function is fully defined when the weight vector w is specified. For optimized knowledge-based linear potential functions, optimization procedures are used to obtain the weight vector w. For statistical knowledge-based potential functions, w is derived using a database of diverse experimentally determined protein structures and computing the frequency distribution of the structural descriptors. If the probability distribution P(s) is measured accurately, the knowledge-based potential function E(s) can be determined by Eq. (6.3). We must specify reference state interaction energies E (s), in order to obtain an effective potential energy function E(s). By subtracting the reference energy from the energy of the mean force we remove all energies corresponding to a starting state, whether it be exposure to water or burial inside a protein. By subtracting E (s), the energy of the reference state (Sippl, 1990), the effective potential energy function can be written as: E(s) = E(s) − E (s) = −kT ln



 Z(q) P(s) − kT ln P (s) Z (q)

(6.5)

where, P (s) and Z (q) are, respectively, the reference state probability of a sequence q corresponding to the descriptor vector s and the partition function of that state. The P’s can be counts also, and an example of how these counts are made is shown in Fig. 6.1. The second term of the above relationship is a constant for a particular temperature because the partition function is constant for a given sequence and does not depend on the descriptor vector s. Following Sippl (1990) we can assume that Z(q) and Z (q) are approximately equal, thus the second term of Eq. (6.5) becomes zero

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X

2 2 2

Fig. 6.1 Counting scheme for pairwise contact potential extraction from structures. The circles represent amino acids and the color the types of amino acids, with the dashed yellow circle representing solvent. Each amino acid is taken as the center of a sphere and the number of pairs with it is counted. In the example a blue circle is the center of the green dashed large sphere, and within this sphere there are contacts between two of each class of interaction as shown on the right side

and the effective or the net potential energy is given by the first term in Eq. (6.5) only as: 

P(s) E(s) = −kT ln P (s)

(6.6)

In order to determine P(s)/P (s), we make further assumptions. By assuming that the probability distribution of each descriptor s is independent, we can write !  P(si ) P(s) = P (s) P (si )

(6.7)

i

If the probability of occurrence is independent of the descriptor we have !  P(si ) i

P (si )

=

!  Pi Si i

(6.8)

P i

where Pi is the probability of occurrence of the ith type structural feature in the native protein state and P i is that in the reference state, and si is the number of microstates having the structural feature of the type i. This leads to the relationship,   P(s) Pi ln = si ln P (s) P i 

(6.9)

i

Eq. (6.6) can now be written as, E(s) = −kT

 i



Pi si ln P i

(6.10)

For a linear potential function, the effective potential energy E(s) of the system is written as a sum of various energetic terms,

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E(s) =



E(si ) =



i

si wi .

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(6.11)

i

In the native protein structures, if si distributions are all linearly independent, then, by comparing Eqs. (6.10) and (6.11) we get  wi = −kT ln

Pi P i

(6.12)

This shows that in native protein structures the probability of each structural feature obeys independently the Boltzmann distribution. In nearly all statistical potential functions this Boltzmann distribution is assumed. Finkelstein et al. (1995) noticed that many structural features observed in proteins, such as distribution of residues between the surface and interior of globular proteins, distribution of dihedral angles, and ion pairs correlate well with the Boltzmann formula. Probability Pi is estimated by finding the frequency of occurrence of the ith structural feature in a database of protein structures. From a database of solved protein structures Pi can be easily determined.

6.2.2 Quasi-chemical Approximation Reference states are critical for the application of empirical potentials, just as in thermodynamics. In the quasi-chemical approximation the random mixing approximation is utilized as reference state: the number of contacts between a particular pair of species is taken to be directly proportional to their relative frequencies. Three reference states are most common. For the first, the preference of an i-type residue for a j-type residue is compared to that of their self-interactions expressed as

i•j+j•j→2i•j

(6.13)

The corresponding effective contact energy eAB , also referred to as self-contact energy, is found from 2 eij (RC ) = 2 Eij (RC ) − [Eii (RC ) + Ejj (RC )]

(6.14)

Another reference state involves the desolvation of residues i and jB prior to their association, as i•0+j•0→i•j+0•0

(6.15)

where “0” indicates solvent molecules. The corresponding solvent-mediated (indicated by superscript 0) effective contact energy is given by

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eij (RC ) = Eij (RC ) + E00 (RC ) − Ei0 (RC ) − Ej0 (RC )

(6.16)

The solvent-residue potentials, Ei0 (RC ) and Ej0 (RC ), are determined from the number of effective solvent “0” molecules coordinating residue types i and j, and the solvent–solvent interaction potential E00 (RC ) contributes a constant amount to each of the contact energies eij 0 (RC ).

6.3 Distant-Independent Potential Functions Miyazawa–Jernigan (Miyazawa and Jernigan 1985, 1996, 1999a,b) contact potentials are historically significant and still widely used today. These potentials take implicitly into account the solvent effects. In the Miyazawa–Jernigan approach, the ith residue is represented by a sphere located at the center of its side chain Ri , except for glycine (Gly), which does not have a side chain, so the position of Gly Cα atom is taken as Ri . A residue pair (i, j) is considered to be in contact if their separation distance is less than a cutoff distance rc = 6.5 Å. The closest residues i and j along the amino acid sequence (|i − j| = 1) are excluded in computations of the statistics of contacts. A contact matrix for all pairs of residues (i, j) is defined as " Cij =

1 if |Ri − Rj | ≤ rc , and|i − j| > 1 0 otherwise

(6.17)

Effective inter-residue contact energies for all pairs of amino acids are evaluated, under the basic assumption that the average characteristics of residue–residue contacts observed in a large number of crystal structures of globular proteins represent the actual intrinsic inter-residue interactions. This empirical energy function includes solvent effects and provides an estimate of the long-range component of conformational energies without atomic details. To obtain this energy function, the quasi-chemical approximation with an approximate treatment of the effects of chain connectivity was employed with the number of residue–residue close contacts computed from a dataset of known protein crystal structures (Miyazawa and Jernigan 1985). The Bethe approximation is a well-known second-order approximation to the mean-field approximation used to describe a system consisting of a mixture of multiple molecular species interacting with each other (Hill 1960). In this approximation, the effects of interactions are taken into account to estimate the average numbers of contacts. Thus, the Bethe approximation is the lowest-order approximation that can provide an estimate of a set of contact energies between species from a given set of the average numbers of contacts between them. Two-body contact energies are estimated using the Bethe approximation along with the basic assumption for residue–residue contacts in protein structures that they are the same as those in mixtures of unconnected amino acids and solvent molecules. Both these approximations are usually used to calculate a partition function for such a system from a given set of interaction energies between molecules. In the mean field approximation, contacts between species are approximated to be random, and a

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partition function of the system is computed. Therefore, if residue–residue contacts in protein structures can be reliably represented to be the same as those in mixtures of unconnected amino acids and solvent molecules, the Bethe approximation will give us a reasonable estimate of actual interaction energies between amino acids. There are many homologous proteins in the Protein Data Bank that may bias the computed statistics. In order to correctly use all structural data, an unbiased sampling of protein structures from the PDB is required in the calculation of contact frequencies. A sampling weight for each protein is devised based upon a sequence homology matrix giving the extent of sequence identity of all pairs of sequences.

6.3.1 Sample Weighing Miyazawa and Jernigan used 1,168 protein structures, whose PDB structures have been determined with a resolution better than 2.5 Å. Then each of the 1,661 sequences in those structures is sampled with a weight determined on the basis of the sequence identity matrix. Average contact energy for the pth residue is given by, < Epc >=

1 p ei nc 2 p

(6.18) p

where, eip is the average contact energy of the ip type of amino acid and nc is the number of contacts with the pth residue. The ordinate corresponds to the sum of the average attractive contact energy and the repulsive packing energy erp . Repulsive packing energies reflect only packing density and do not depend strongly on the type of amino acid at the center. Also, the coordination number of the amino acid does not depend on the type of amino acid. The long-range energy is defined by, Elong =

  Epc + Epr

(6.19)

p

where Epc is the short-range attractive term that becomes effective only when two residues are in close proximity, and Epr is the repulsive term that results from the overlap of residues at high packing densities. Contact energies for the pth residue is Epc (eij ) =

1 eip j ncip j 2

(6.20)

j

where, eip j is the contact energy of the pth residue whose amino acid is i in contact with a type j residue. Here ncip j is the number of i, j contacts with pth residue. Total contact energy for the pth residue is the sum of all the residue pairs contact energies according to the above equation.

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Miyazawa and Jernigan (1985) introduced also two types of energies: eij and e ij , where e ij, is the energy difference detailing the residue interaction specificity accompanying the formation of a contact pair i−j from contact pairs i−i and j−j (that is in buried protein environment, without taking into account the solvent effects), while eij is the energy difference accompanying the formation of contacts between i and j types of amino acids from those amino acids exposed to solvent. A methodology for estimating contact energies eij and e ij was explained in more detail in Miyazawa and Jernigan (1985). Repulsive energies consist of two parts and are estimated for the residue according to,

r Epr = ehc p + ep

(6.21)

The first term on the right-hand side of the above equation, the hard core repulsion term, is considered to be zero for proteins in the PDB and is not included, because known protein structures are usually refined to remove such close contacts. For off-lattice protein models, a repulsive potential is necessary for evaluation of the conformational packing energy of a protein to remove possible steric clashes. Total long-range energy is estimated by summing over contact energies and repulsive energies of all residues in a protein. Assessment of protein stability based on the total number of contacts in a protein can lead to an incorrect result. Thus, in order to measure the stability of any protein structure for a given sequence, it might be better to use the energy that does not include the homogeneous energy for protein collapse but consists only of the remaining energy for aligning residues with the contacts assumed to be present within the protein structure. The alignment energy of residues within a protein structure is therefore defined as

Epc (eij − err ) = Epc (eij ) − Epc (err )

(6.22)

where Epc (err ) is the average collapse energy. It is, however, still inappropriate for determining the stabilities of a given fold for different protein sequences. Miyazawa and Jernigan have addressed this issue using an empirical energy potential with a reference state that not only can be used for protein fold recognition but also can be used for the recognition of sequences (Miyazawa and Jernigan 1999c). Miyazawa and Jernigan (1996) chose the total energy expected for a typical protein with a given amino acid composition and chain length to be the reference energy of the native structure for the given protein sequence. The following difference in energy is considered:

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Elong ≡Elong − (Elong of a typical native structure for a given sequence)  fip ∼ Elong −

(6.23)

p

· (the average number of contacts per residue in a typical native structure) where the second term is the sum of the average contact energies per residue of residue type ip over all positions in the protein. Here the average number of contacts per residue in a typical native structure fi is a function of eij and is estimated by averaging over all proteins. Lastly, a linear dependence on chain length is removed by comparison on a “per residue” basis, which is appropriate for assessing the stability of one protein structure among other folds. The threading of sequences into other folds (Hendlich et al. 1990; Jones et al. 1992) or inverse folding (Bowie et al. 1991) is a good method to evaluate how well a given energy scale can discriminate the native structures as the lowest energy conformations among other folds. A basic assumption underlying the Miyazawa and Jernigan (1996) estimation of contact energies is that, for a large enough sample, the effects of specific amino acid sequences will average out and then the numbers of residue–residue contacts observed in a large number of protein crystals will reflect directly the actual intrinsic inter-residue interaction strengths. The hydrophobic effect is a dominant force in stabilizing the native structure of a globular protein. However, there is a lack of agreement as to its precise magnitude. The hydrophobicity of a small molecule can be measured by transfer energies, for example, of amino acids from a non-polar solvent to water. However, it is not immediately evident that the free energy change accompanying a process in which residues are buried in the folding process of protein is the same as that accompanying the transfer process of amino acids from water to a non-aqueous solvent, because the two processes are obviously different (Lee 1993) and denatured proteins do not fully expose all residues to solvent.

6.4 Distance-Dependent Potential Functions In the Miyazawa–Jernigan potential function a cutoff value of rc = 6.5 Å is used for the distance between a pair of residues to define an occurrence of a contact. This is done by assuming the interactions between amino acids are short range. On the other hand, in distance-dependent potentials the interactions depend on the distance between the amino acids or atoms. Here the distance of interactions is divided

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into a number of small distance intervals (bins) and potential functions are derived by applying the weight equation for each bin. Many distance-dependent potential functions have been developed (Hendlich et al. 1990; Sippl 1990; Jones et al. 1992; Samudrala and Moult 1998; Lu and Skolnick 2001). One of the problems is that these potentials need to approach zero beyond a certain distance. This problem has been discussed and one solution to the problem presented by Bahar and Jernigan (1997). For distance-dependence the effective potential energy for pairwise interactions between two residue types i and j at a distance d, E(i, j; d) can be written in several different forms. the knowledge-based potential (KBP) function developed by Lu and Skolnick (2001) and residue-specific all-atom probability discriminatory function (RAPDF) developed by Samudrala and Moult (1998) are two examples. All of these approaches lead to the same basic formula which can be written as,  E(i, j; d) = − ln

N(i, j;d) N(i, j;d)

 (6.24)

where, N(i, j;d) and N (i, j;d) are, respectively, the observed and expected total number of pairwise interactions for (i, j) pairs within a distance d. The method of obtaining N(i, j;d) is the same for many distance-dependent energy functions. However, N (i, j;d) depends on the way of counting it. Therefore the model of reference state used to compute N (i, j;d) is critical for distance-dependent energy functions. For example, in the model proposed by Sippl (1990) uniform density reference state was assumed to derive the residue-based, distance-dependent potential. Here, for a reference state total number of pairs in a given distance shell is equal to that for folded proteins. This means the distance dependence of the pair probability distribution is the averaged distribution over all atomic or residue pairs. Lu and Skolnick (2001) used this reference state to calculate the expected number of (i, j) interactions at distance d. Samudrala and Moult (1998) used another type of reference state where they took the probability of the distance d between a pair of residues (i, j) to be independent of the contact types (i, j). The problem with the uniform density model is that very different volume distributions of (i, j) pairs in specific regions of the protein may produce the same density if the particular residue pair is along a line. To overcome this problem Zhou and Zhou (2002) developed a new reference state called DFIRE (Distancescaled, Finite Ideal-gas REference state) It is an extension of the ideal gas reference state and residues are modeled as non-interacting points like molecules in an ideal gas. The interacting pair distribution is uniform not only along any line, but also in the whole volume of the protein. Potentials generated using this reference state are termed DFIRE-based potentials. These potentials were tested by using decoys and mutation databases. Their results showed that the DFIRE-based all-atom potential consistently performed better than previous all-atom knowledge-based potentials.

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6.5 Geometric Potential Functions The third class of knowledge-based statistical potentials is based on the shape of the protein molecule. Here various geometric constructs that accurately reflect the shape of the molecules are computed. There are several of these geometric constructs and potentials derived from these constructs have shown significant success. The potential function developed by McConkey et al. (2003) that was based on the Voronoi diagram of the atomic structures of proteins is one of the best-performing atomic-level potential functions used in decoy discrimination. McConkey et al. used the Voronoi tessellation procedure to divide the protein into cells and the cell volume was restricted to a sphere, which defined the atom-solvent accessible area. Atom–atom contacts were calculated from the projection of the cell faces to the surface of the sphere. Their contact-based scoring function was able to distinguish native proteins from corresponding decoy structures with a high degree of accuracy. The Delaunay triangulation (Singh et al. 1996; Zheng et al. 1997; Carter et al. 2001; Krishnamoorthy and Tropsha 2003) is another type of geometric construct. Krishnamoorthy and Tropsha represented proteins by the side-chain centroids of amino acids. Delaunay tessellation of this representation defines all sets of nearest neighbor quadruplets of amino acids. A four-body contact scoring function (log likelihoods of residue quadruplet composition) was derived by analysis of a diverse set of proteins with known structures. A test protein was characterized by the total score calculated as the sum of the individual log likelihoods of amino acid quadruplets. Their results showed that the scoring function distinguishes native from partially unfolded or deliberately misfolded structures. The alpha shape of the protein molecules has also been used to define geometric constructs (Li et al. 2003; Li and Liang 2005a,b).

6.6 Multi-body Potentials All non-geometric statistical potentials discussed so far were two-body potentials and they showed varying levels of success in discriminating native structures from decoys. However, two-body potentials are not capable of recognizing all native folds against large datasets of decoy structures (Vendruscolo et al. 2000). They also cannot properly represent three-dimensional interactions and are unable to represent threedimensional cooperativity properly and can only represent lower-order packing arrangements (Carter et al. 2001). Czaplewski et al. (2000) learned that two-body potentials for methane in water were insufficient for capturing the cooperativity and that it required a three-body term. The Scheraga group has subsequently explored six-body potentials and learned that a four-body term is sufficient to capture the cooperativity present in proteins. The lack of any significant “excess” contributions to the pairwise potentials, which cannot be approximated by one-body components, strongly suggests that more effective structure-specific knowledge-based potentials are required

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(Pokarowski et al. 2005). Betancourt and Thirumalai examined the similarities and differences between the two widely used pairwise potentials proposed by Miyazawa and Jernigan (Miyazawa and Jernigan 1996) and Skolnick et al. (Skolnick et al. 1997) and suggested that pairwise potentials are not sufficient for reliable prediction of protein structures (Betancourt and Thirumalai 1999). Munson et al. (1997) showed small gains in threading by using three-body potentials instead of two-body potentials. Krishnamoorthy and Tropsha (2003) showed that the four-body potentials obtained by using Delaunay tessellation algorithms, which are popular in the study of protein structure, can discern correct sequences or structures and generate better z-scores compared to two-body statistical potentials. However, the four-body contact potentials derived by Delaunay tessellation and most of two-body potentials (Miyazawa and Jernigan 1985, 1996; Vajda et al. 1997; Maiorov and Crippen 1992; Skolnick at el. 1997) neglect the information on the sequence location of interacting residues. Feng et al. (2007) have developed a new scheme for the derivation of four-body potentials, considering in more detail the interactions between backbones and side chains and including some of the sequential information of the protein. They tested this new scheme of four-body potentials by threading against same decoy databases used with Delaunay tessellation four-body potentials (Krishnamoorthy and Tropsha 2003) and concluded that the overall rankings of their potentials are significantly better than the Delaunay tessellation-based potentials.

6.6.1 Four-Body Contact Potentials The four-body contact potentials developed by Feng et al. (2007) give a more cooperative representation of protein interaction energies. They found that these four-body contact potentials can discriminate well between native structures and partially unfolded or deliberately misfolded structures. Four-body contact potentials derived using Delaunay tessellation by Krishnamoorthy and Tropsha (2003) and most of two-body potentials neglect the sequence information of proteins. However, in the four-body contact potentials derived by Feng et al. (2007), the interactions between the backbones and side chains are considered in more detail and some of the sequential information for the protein is included. 6.6.1.1 Construction of Four-Body Contacts Residues are all represented by the geometric centers of the side-chain heavy atoms, except for glycine, where the alpha carbon atom is used (Fig. 6.2). Yellow points are the side-chain geometric centers of four sequential residues i, i + 1, i + 2, and i + 3. The red point is the geometric center of the four yellow points chosen as the center of the interacting group. The six cyan planes, defined by all combinations of pairs of yellow points and the central red point, fully subdivide the space surrounding the red point into four tetrahedra. Blue points represent other residues in close proximity to the red point, the interaction range is defined as being within 8.0 Å of the red point.

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Fig. 6.2 Identification of residue points for use in the four-body contacts Table 6.1 Combination of residue types chosen to reduce the sequential amino acids to eight classes A = {GLU, ASP} (acidic) B = {ARG, LYS, HIS} (basic) C = {CYS} (cysteine) H = {TRP, TYR, PHE, MET, LEU, ILE, VAL} (hydrophobic) N = {GLN, ASN} (amide) O = {SER, THR} (hydroxyl) P = {PRO} (proline) S = {ALA, GLY} (small)

An example of the four contacting bodies for potential is shown by the four residues in boxes. Among these, the three yellow residues form a sequence triplet, whose residue types are reduced to eight classes (Table 6.1) in order to ensure sufficient data for potential extraction. The single blue point in a box within the quadruplet is not close in sequence and is taken to be one of the 20 amino acids. Here the four bodies always have three sequential points and one non-sequential in the quartet of interacting residues. In accumulating the information to construct the potential Feng et al. (2007) have ignored the specific sequence order of the three residues within each backbone triplet, so instead of 83 = 512, they have only 120 different triplets since their specific sequence order is disregarded. Data is collected by including all specific types of residues (20 types) for the fourth point, within a distance of 8 Å from the coordinate center (the red point in Fig. 6.2) and assign them to one of the corresponding four tetrahedra defined by the vectors originating from the red point to the yellow points. This residue is then counted in the specific tetrahedron, and the procedure is repeated for the entire set of proteins and for all quartets defining closely interacting residues. They have thus derived four-body conformational sets composed of the three sequential residue triplets and a single non-sequential nearby residue. Each of the residues can be exposed when it is on the surface of the protein or can be buried when located inside the protein core. These different situations were considered separately. The triplets were separated into three groups by their relative solvent accessible surface area (RSA) in order to remove differences in the chain connectivity effect and in residue packing geometry between surface area and buried

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region. These three groups correspond to buried (with all three residues in the triplet having RSA < 20%, denoted as Bu), exposed (with all three having RSA < 20%, denoted as E), and intermediate (some of the residues in the triplet being exposed, and some being buried, denoted as I). Better results were obtained in discriminating native structures from a large number of decoys by using these four-body potentials categorized by RSA.

6.6.2 Four-Body Contact Potential Energy Function A four-body contact potential energy is calculated according to the inverse Boltzmann principle. First the probabilities P4|X , P3|X , and PA , which are, respectively, the observed frequencies of quadruplets and triplets in each of the sets specified by X = Bu, E, or I and amino acid type singlets (A) in the protein datasets, are calculated using the following equations: P4|X =

number of the specific quadruplets given Bu, E, or I in the data set total number of all types quadruplets given Bu, E, or I in the data set

P3|X =

number of the specific triplets given Bu, E, or I in the data set total number of all triplets given Bu, E, or I in the data set

PA=

(6.25)

number of the specific type of amino acids in the data set total number of all amino acids in the data set

Then, the four-body contact potential energy is obtained using Boltzmann formula,  E4|x = −RT ln

P4|x P3|x PA

 (6.26)

The total free energy for a protein is obtained by summing the four-body contact potential energies over all quadruplets nq . Etotal =



E4|x

(6.27)

nq

This equation is used to estimate the free energy of native structures and their decoys. The results are shown in Fig. 6.3 where the relative values of these four-body contact potentials are shown in color, as the heat map. The figure contains three parts: the left vertical part shows potentials for the buried triplets, the middle section is for exposed, and the third for intermediate burial. The y-axis represents the indices of the 120 types of the possible triplets.

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Fig. 6.3 Relative values of four-body contact potentials showed in color for three types of triplets separately (buried, exposed, and intermediate). Blue is the most favorable energy or most frequently occurring

The abscissa shows the singlet within the sequence-based tetrahedra in contact with the specific triplets indexed on the ordinate. The first 20 characters on the x-axis represent the 20 types of amino acids for triplets in the buried state, the next 20 characters the triplets in the exposed state, and the last 20 characters the triplets in the intermediate state. The values of the potential are colored spectrally from blue to red: negative values representing favorable contacts and positive values unfavorable contacts. Energy values are in units of RT.

6.7 Optimization Method Assumption of Boltzmann statistics and the non-inclusion of chain connectivity in the reference state are among the drawbacks associated with statistical knowledgebased potentials. Knowledge-based potentials derived from optimization have been able to take into account these issues. In the optimization method, the same functional form is used as for statistical potentials. However, the weight coefficients are derived by optimization. Here, the weight coefficients are the parameters of the potential functions for individual types

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of interactions. These parameters are optimized for discriminating the native structure from a large set of decoys. This is in contrast to the case of statistical potentials where the weight coefficients are derived from protein database statistical analysis. In this type of potential function certain inequalities are optimized. For example, from Anfinsen’s thermodynamic hypothesis (Anfinsen et al. 1961; Anfinsen 1973), the native amino acid sequence threaded on the native structure has the best fitness score compared to a set of sequence decoys, which are taken from proteins that are in different folds. Secondly, the native sequence has the highest probability of fitting into the specified native structure (Shakhnovich and Gutin 1993; Deutsch and Kurosky 1996; Li et al. 1996). Thirdly, native protein sequences, when mounted onto their native structure, have the lowest energy compared to a set of possible decoys. As in the case of statistical potentials, choosing a functional form of the optimized potential and the generation of a large set of decoys are starting points. Then a set of parameters are optimized in discriminating the native structure among the decoys. Usually the gap between native energy and the average energy of decoys or the energy gap between the native and decoys with the lowest score, or the z-score of the native protein is maximized (Goldstein et al. 1992; Maiorov and Crippen 1992; Thomas and Dill 1996a; Koretke et al. 1996, 1998; Hao and Scheraga 1996; Mirny and Shakhnovich 1996; Vendruscolo and Domanyi 1998).

6.8 Comparative Analysis of Statistical Protein Contact Potentials to Infer Ideal Amino Acid Interaction Forms Pokarowski et al. (2005) have analyzed 29 different published matrices of protein pairwise contact potentials between amino acids derived from different sets of proteins. Structures are taken either from the Protein Data Bank (PDB) or generated computationally (decoys). Each of the contact potentials is similar to one of the two matrices derived in the work of Miyazawa and Jernigan (1999b). All the known pairwise matrices of contact potentials can be well approximated by simple functions of individual residue properties, such as hydrophobicity and electrostatic properties (Georgescu et al. 2002; Mehler et al. 2002; Sandberg and Edholm 1999; Tollinger et al. 2003; Laurents et al. 2003) for each pair of amino acids. Such an approximation of the contact potential matrices is termed a one-body approximation. The electrostatic properties of an amino acid are represented by its isoelectric point and measured in pH units. The contact potential matrices of hydrophobicity can be approximated by the formula, eij = hi + hj , 1 ≤ i, j ≤ 20

(6.28)

Residue-type-dependent factor h is highly correlated with the frequency of occurrence of a given amino acid-type inside proteins. Electrostatic interactions for the potentials of this class are almost negligible. In the potentials belonging to this class,

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the major contribution to the potentials is the one-body transfer energy of the amino acid from water to the protein environment. Potentials belonging to the second class can be approximated by the formula, eij = c0 − hi hj + qi qj

(6.29)

where c0 is a constant and h is highly correlated with the Kyte–Doolittle hydrophobicity scale. The less dominant, residue-type-dependent factor q correlates with amino acid isoelectric points pI. The electrostatic interactions are quite important for the second class of potentials but are completely negligible for the first class of potentials. Inclusion of electrostatic interactions significantly improves the approximation for this class of potentials. High correlation between potentials of the first class and the hydrophobic transfer energies is well known. It was found that this approximation works well for the second class of potentials as well. The potentials of this class are interpreted as representing energies of contact of amino acid pairs within an average protein environment. Hydrophobicity represents the dominant factor in protein potentials. Energy of demixing of amino acids in a protein environment and electrostatic interactions are less dominant factors. Pokarowski et al. (2005) have shown that the accuracy of the one-body approximation works significantly better for potentials derived from the quasi-chemical principle than for the potentials obtained from the optimization of the prediction of the native structures among decoys. The frequencies of contacts between different amino acids can also be successfully approximated with this method. Therefore

Fig. 6.4 Graphical illustration of correlations among different protein potentials (Reproduced from Pokarowski et al. 2005 with the permission from the “publisher”). The extent of correlation is colored according to the key. The identifications of the individual potentials are given in the Pokarowski et al. (2005)

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hydrophobicity, demixing, and electrostatics are fundamental properties defining potentials from the simple statistics of inter-residue pair contacts for proteins. The one-body approximation can be used to separate contact potentials into two classes. Potentials belonging to the first class are dominated by the one-body energies of transfer of amino acids from water to a protein environment. Potentials belonging to the second class represent energies of contacts of amino acids in a protein environment. Figure 6.4 gives a graphical illustration of correlations among different protein potentials. The clustering of the contact potentials into two groups is readily apparent. The first cluster is centered on MJ3h and SJKG and the second one around MJ3 as shown in Fig. 6.4, corresponding to the two major types of reference states. This scheme makes it possible to analyze and compare various statistical potentials for proteins.

6.9 Statistical Force Fields for Coarse-Grained Protein Models In traditional statistical potentials usually only distances between residues are considered. To improve prediction of protein tertiary structure from amino acid sequence more sophisticated statistical force fields have been recently proposed, where in addition to the distances between residues, several other features of amino acids in protein structures have been incorporated, such as their solvent exposure, orientation of side chains, or their localization in specific secondary structure elements (Zhang and Kim 2000). Examples of such statistical force fields are OPUS-Ca developed by Ma (Wu et al. 2007), CABS force field developed by Kolinski and coworkers (Kolinski 2004), and the United Residue (UNRES) model developed by Scheraga and coworkers (Liwo et al. 1997a,b, 1998, 2001; Pillardy et al. 2001). The OPUS-Ca (Wu et al. 2007) potentials require only alpha carbon positions as input. The contributions from other atomic positions are established from pseudopositions artificially built from a Cα trace for auxiliary purposes. The potential function is formed based on seven major representative molecular interactions in proteins: a distance-dependent pairwise energy with orientational preference, a hydrogen bonding energy, a short-range energy, a packing energy, a tri-peptide packing energy, a three-body energy, and a solvation energy. From the testing of decoy recognition on a number of commonly used decoy sets, it was shown that the new potential function outperforms all known Cα -based potentials and most other coarse-grained ones that require more information than Cα positions. The CABS model is based on lattice representation of proteins’ Cα positions with 800 possible orientations of the virtual Cα −Cα bonds. Only positions of Cα and Cβ atoms and center of the side chain were considered. Knowledge-based potentials of the force field for the CABS model include generic protein-like conformational biases, statistical potentials for the short-range conformational propensities, a model of the main-chain hydrogen bonds and context-dependent statistical potentials describing the side-group interactions. The model is more accurate than the

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previously designed lattice models and in many applications it is complementary and competitive in respect to the all-atom techniques. The UNRES force field of Scheraga and collaborators (Liwo et al. 1997a,b, 1998, 2001; Pillardy et al. 2001) represents proteins as a sequence of Cα atoms linked by virtual bonds attached to united side chains and united peptide groups. Only these united peptide groups and united side chains interact, alpha-carbon atoms serve only for linking purposes. The total energy in UNRES model is a sum of terms representing hydrophobic/hydrophilic and electrostatic interactions, excluded volume, virtual dihedral angle torsions, double torsions, virtual-bond angle bending, energies of side-chain rotamers, and multi-body correlations from a cumulant expansion. Because of this the UNRES force field appears similar to a classical atomic force field used in molecular dynamics simulations.

6.10 Applications of Knowledge-Based Potential Functions Knowledge-based potential functions are used in the study of protein structure prediction, protein design, protein docking, and in protein folding. In protein structure prediction, the conformational space is sampled and the near-native structure is recognized from that ensemble of decoy conformations. The potential function is used to discriminate the near-native structure from the set of decoys. Several decoy sets have been developed to test whether a particular knowledgebased potential function is successful in predicting the native structure from decoys. One example of such decoy sets is, the 4-state-reduced decoy set developed by Park and Levitt (1996). This set contains native and the near-native conformations of 7 sequences along with about 650 decoys for each sequence. Decoys “R”Us provides a number of sets of decoys and acts as a resource for test sets for evaluating the success of potential functions and to obtain improved scoring functions (Samudrala and Levitt 2000). CASP (Critical Assessment of Techniques for Protein Structure Prediction) decoys are also often used in testing knowledge-based potential functions. Studies have been carried out in which performance of different knowledgebased potentials have been compared (Park and Levitt 1996; Zhou and Zhou 2002; Gilis 2004). The way the evaluation is done is by finding the success in ranking the native structure as the lowest energy and also in obtaining a large z-score for the native structure. Our (Feng et al. 2007) four-body sequence-based contact potentials (4CP-seq) have been successful in recognition of the native structure among misfolded decoys datasets from Decoys‘R’Us. The sequential information built into this potential enables better gapless threading results than Delaunay tessellation. However, these potentials fail in recognizing native structures of some proteins like ferredoxin (1fca) (Feng et al. 2007). To improve the performance of four-body potentials we have recently derived a second generation of these potentials. These newer four-body potentials include the spatial orientation information by using four

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non-sequential neighbors as its basis and are named 4CP-non-seq. We further improved our four-body contact potentials by combining 4CP-seq and 4CP-non-seq. Because both 4CP-seq and 4CP-non-seq are long-range potentials, we also add in our short-range potentials (Bahar et al. 1997). The best threading results are obtained by combining these three sets of potentials (see Table 6.2, where the bold column is best). Table 6.2 Threading results – significantly improved performance of our second generation of four-body potentials that combines 4CP-seq (Seq), 4CP-non-seq (nonSeq), and short-range (SR) interactions. Results shown are for (a) the four-state reduced decoy and (b) lattice_ssfit sets from Decoys‘R’Us. Best cases shown in Bold Protein

Rank w/Seq

Ranks w/nonSeq Rank w/SR

Rank w/Seq + nonSeq

Rank w/Seq + 0.5 nonSeq + 0.1 SR

z score

1ctf-a 1r69-a 1sn3-a 2cro-a 3icb-a 4pti-a 4rxn-a 1beo-b 1ctf-b 1dkt-A-b 1fca-b 1nkl-b 1pgb-b 1trl-A-b 4icb-b

6 1 1 1 1 7 7 1 2 13 249 1 19 1 1

1 1 46 26 9 13 9 1 1 47 1 1 3 7 1

2 1 7 1 3 2 3 1 1 14 12 1 1 1 1

2 1 2 1 1 1 1 1 1 1 1 1 1 1 1

−2.5 −3.3 −2.6 −2.6 −2.2 −2.9 −2.5 −5.1 −4.5 −2.6 −2.7 −5.4 −3.3 −3.8 −5.0

25 65 11 56 59 6 8 1 87 1 8 4 3 3 1

Our new combined potentials are able to identify all the native structures in the lattice_ssfit and 5 out of 7 in the 4state_reduced decoys. The other two targets in the 4state_reduced decoy set rank them as nearly best. The successful threading tests have been performed on widely used decoys from Decoys‘R’Us, but we are now generating our own sets of decoys. In some cases such as 4icb from the Decoys‘R’Us lattice_ssfit dataset the decoys are tightly clustered in RMSD from 5–12 Å. Our new decoys are generated to be more challenging by including decoys having root mean square deviations (RMSDs) in the range of 2–5 Å. Our potentials can still identify the native structure even for these more difficult new decoys. Knowledge-based potential functions are not only used to identify the native conformation from a set of decoys, but also they can be used when generating conformations to sample the conformation space (Jernigan and Bahar 1996; Hao and Scheraga 1999). Knowledge-based potentials are also used in protein–protein docking predictions. Li and Liang (2005a) used knowledge-based potentials in predicting protein–protein binding surfaces of several antibody or antibody-related proteins.

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Protein–protein complexes used by them were taken from the 21 CAPRI (Critical Assessment of PRedicted Interactions, Méndez et al. 2005) target proteins. Protein design is another active research area where knowledge-based potentials are used extensively. Protein design means identifying the sequences that are compatible with a given fold but not compatible with other alternative folds (Koehl and Levitt 1999a,b). This is important in designing novel proteins that may not exist in nature but have enhanced or novel biological functions and is especially important in the pharmaceutical industry where development of new drugs is essential. Designing proteins is extremely complex because even for a protein with a small number of residues, there are a huge number of possible sequences. It is difficult to design proteins by searching these large sequence spaces using computational and experimental techniques. Therefore protein design techniques rely heavily on potential functions to execute the search. Hu et al. (2004) used optimized non-linear design potential functions in protein design to distinguish the native sequence that folds to a given protein structure, from a set of 440 decoy sequences. Machine learning algorithms were used in protein design to train the potential function. To test the design potential, and check if the native sequence can be distinguished from the decoy sequences, predicted sequences which gave the lowest scores were compared with the native sequences. All 440 native sequences were successfully distinguished from the decoy sequences. Nonlinear design potential functions are capable of successfully discriminating native sequences from decoy sequences; however, linear potential functions are not. Knowledge-based potential functions are also used in predicting protein stability. It is reasonable to use statistical potentials to estimate protein stability upon mutation because mutations will not change the partition functions of the protein sequences appreciably. In studies done using physics-based empirical potentials and statistical potentials, predicted and experimentally measured stabilities show a good correlation (Miyazawa and Jernigan 1994; Gilis and Rooman 1996, 1997; Guerois et al. 2002; Bordner and Abagyan 2004; Hoppe and Schomburg 2005; Zhou and Zhou 2002). Furthermore statistical potentials have been used to predict quantitative binding free energy of protein interactions, such as protein–protein interactions or protein– ligand interactions (DeWitte and Shakhnovich 1996; Mitchell et al. 1999; Muegge and Martin 1999; Liu et al. 2004; Zhang et al. 2005). Changes of binding free energy upon mutations can also be predicted using knowledge-based potentials (Kortemme and Baker 2002; Kortemme et al. 2004).

6.11 Future Developments Although statistical coarse-grained potentials have been quite successful in the prediction of protein tertiary structures from the amino acid sequence, protein design, mutation stability, or docking studies there is still significant room for further improvements. One aspect in our opinion that is especially important is relating

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inter-residue potentials to the dense packing within protein cores. These dense states are not easy to represent. Polyhedra and lattices represent two of the simplest ways to achieve high packing densities. The face-centered cubic (fcc) lattice is how spheres pack with the highest density. Recently, there has been renewed interest in packing problems in computer science and mathematics, such as a recent paper considering packing of tetrahedra (Haji-Akbari et al. 2009). There are also interesting alternative polyhedra to utilize for representing the clusters of densely packed residues within a protein. Our preliminary data show significant improvements through the use of an icosahedron for fitting the orientational packing of residues in comparison with the fcc lattice (Feng et al. 2008). The fcc lattice and icosahedron are comparable, since both have 12 directions between the central point and its nearest nodes. We have found, however, by analyzing coordination clusters in our dataset that some clusters have even 14 nearest neighbors. Because of this another polyhedron – the tetrakis hexahedron – that has 14 vertices and 24 faces might be also a good model. The tetrakis hexahedron is slightly less regular than an icosahedron. The icosahedron belongs to the category of the most regular Platonic solids, while the tetrakis hexahedron is a non-regular polyhedron belonging to the category of Catalan solids (duals of the Archimedean solids). Because our four-body potentials provide significantly more cooperative representations than pairwise potentials, and also because they have shown important gains in threading calculations, we expect that potentials derived by using icosahedral or tetrakis hexahedron packings might further improve coarse-grained representation of proteins. To compare all different coarse-grained potentials we have recently developed a Potentials‘R’Us database and server for testing and comparison on different statistical potentials and for energy estimations of protein coarse-grained models using a variety of knowledge-based potentials (Feng et al. 2010). Our Potentials‘R’Us database contains two types of four-body potentials, shortrange potentials, and 23 different two-body potentials (Feng et al. 2010). The server is an easily accessible, freely available tool with a web interface that collects all existing and future protein coarse-grained potentials and computes energies of multiple structural models. It allows evaluation of energies of different protein folds and significantly improves the access to a wide variety of knowledge-based potentials. The server accepts multiple structural files in the PDB format (including hundreds or even thousands of decoys) and the results are sent back to users promptly to the supplied e-mail address. The address of the server is http://gor.bb.iastate.edu/potential; 21 two-body potential functions from Potentials‘R’Us webserver (Feng et al. 2010) were tested using as decoys structural models from CASP8 competition to examine how well they recognize the target structures from each corresponding set of decoys (Gniewek et al. 2010, unpublished data). The CASP8 targets were divided into two subsets according to the method used to predict structural models (decoys) for each target. One set comprised targets for which the structural models were predicted using homology modeling (easy targets − 153 cases) and the other set comprised the targets for which the decoy sets were obtained using non-homology (templatefree modeling)-based structure prediction methods (hard targets − 12 cases). All incomplete decoys were removed from the sets.

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z-scores were calculated for decoys. The RMSD between the native structure and the best-fit decoy and the RMSD between the native and the average of the five bestfit decoys for each decoy set were also calculated. The Spearman, Pearson, and the Kendall correlation coefficients were calculated for all the target–decoy pairs using potential energies and the RMSD values from the native structure. The Pearson’s correlation coefficient is a covariance of two variables normalized by their standard deviations: n

− x)(yi − y) n 2 2 i=1 (xi − x) i=1 (yi − y)

ρP =  n

i=1 (xi

(6.30)

The Spearman’s rank correlation coefficient is a measure of statistical dependence between two ranked variables. In the case of existence of tied ranks ρS is computed from the same formula as ρP . In the case of no tied ranks the Spearman correlation coefficient can be computed from a simpler formula:  6 ni=1 di2 ρS = 1 − n(n2 − 1)

(6.31)

with di = xi − yi being the difference between the ranks on the two variables. The Kendal τ coefficient is a measure of rank correlation, i.e., the similarity of ordering of the data when ranked by different quantities, and defined as: τ=

nc − nd 1 2 n(n − 1)

(6.32)

where nc is a number of concordant pairs, nd is the number of discordant pairs, and the denominator is the total number of pairs. The values obtained for each target were next averaged over all the decoy sets in each of the two categories to obtain averaged values for each potential function. The results are shown in Table 6.3a, b. Tested potentials are all knowledge-based coarse-grained potentials, and they usually capture statistics based on coordinates of Cα (sometimes Cβ ) atoms, so they do not take into account atomic details of proteins. We observed that for templatebased targets, BT potential (Betancourt and Thirumalai 1999) perform best (in terms of correlation coefficients, average z-score, and average RMSD). The best RMSD values are in the range of 4–5 Å. There are few other potentials which show similar performances. The targets in the homology set have been modeled with high accuracy; in many cases with an RMSD of about 1–5 Å and the free modeled targets in most cases have higher RMSD values. For free modeled targets, performance (both in terms of correlation coefficients or averaged values of z-score and cRMSD), is worse than that for the homology-based case. Potentials that perform well for homology-based targets also perform well for free modeled targets but without giving as good results as the latter. This is due to the fact that the submitted models usually are more

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Table 6.3 Threading results for two-body potentials for decoys obtained by (a) homology modeling and (b) free modeling Potential

Spearman ρ

(a) Homology modeling Qm 0.3874 Qp 0.4062 HLPL 0.3943 SKOb 0.4307 SKOa 0.4154 SKJG 0.4374 MJPL 0.2998 MJ3h 0.4590 MJ2h 0.3247 TS 0.2751 BT 0.4619 BFKV 0.4546 TD 0.4404 TEl 0.4306 TEs 0.4224 RO 0.2397 MS 0.3837 MJ3 0.3968 GKS 0.3043 VD 0.4081 MSBM 0.0688

Kendall τ

z-score

RMSD (top1)

RMSD (top5)

0.3696 0.3942 0.3795 0.4412 0.4042 0.4309 0.2609 0.4830 0.2980 0.2350 0.4881 0.4799 0.4503 0.4620 0.4509 0.2627 0.3978 0.4013 0.3120 0.4338 0.0453

0.2693 0.2933 0.2846 0.3054 0.2909 0.3068 0.2172 0.3310 0.2349 0.1990 0.3319 0.3269 0.3180 0.3068 0.2988 0.1642 0.2663 0.2756 0.2102 0.2890 0.0458

1.251 1.217 1.178 1.479 1.420 1.410 0.752 1.395 0.812 0.656 1.496 1.453 1.268 1.412 1.388 0.455 1.249 1.291 1.162 1.400 0.019

5.039 6.453 6.678 4.808 5.158 4.614 9.308 4.939 8.049 9.445 4.087 4.919 5.389 4.733 6.109 5.848 5.210 4.612 6.259 4.576 8.578

5.648 5.898 6.013 5.217 5.616 5.091 7.847 5.096 7.571 8.364 4.891 5.264 5.427 4.860 5.267 5.678 5.043 5.168 5.592 5.017 8.286

0.1427 0.0407 0.0346 0.1348 0.1541 0.1550 −0.0184 0.1154 −0.0026 −0.0246 0.1583 0.1260 0.1026 0.1160 0.1093 0.1255 0.1805 0.1786 0.1160 0.1376 0.0555

0.1251 0.1310 0.1342 0.1348 0.1397 0.1327 0.1244 0.1131 0.1302 0.1203 0.1366 0.1332 0.1287 0.1075 0.1025 0.0794 0.1437 0.1468 0.1105 0.1207 0.0654

1.701 1.430 1.317 1.944 2.010 1.880 0.880 2.020 0.996 0.809 2.137 1.977 1.779 1.704 1.585 0.457 1.555 1.661 1.333 1.579 1.052

9.334 9.722 10.309 11.35 10.439 10.783 11.35 8.364 11.797 11.572 7.733 9.249 9.876 10.625 10.881 12.264 10.251 9.621 11.102 10.921 10.754

11.173 10.859 10.676 10.814 10.291 10.486 11.975 9.944 11.863 12.324 10.202 10.454 10.291 10.691 10.775 9.865 10.290 9.984 11.292 11.680 10.693

Pearson ρ

(b) Template-free modeling Qm Qp HLPL SKOb SKOa SKJG MJPL MJ3h MJ2h TS BT BFKV TD TEl TEs RO MS MJ3 GKS VD MSBM

0.1887 0.1586 0.1640 0.1892 0.2127 0.2034 0.1520 0.1463 0.1594 0.1467 0.1861 0.1694 0.1613 0.1501 0.1488 0.1203 0.2173 0.2242 0.1638 0.1627 0.0999

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deviant from the native protein structures for template free modeled cases than for homology-based models. Potentials are not able to recognize proteins’ native structures with 100% accuracy regardless of what is the modeling used to generate the structure. This could be due to the fact that knowledge-based potentials are used along with sophisticated optimization methods to generate models. Therefore in order to obtain better quality assessment it is reasonable to produce decoys using one potential and asses their quality using other scoring functions. Example can be found in McGuffin (2007). Acknowledgments We acknowledge the financial support provided by NIH grants 1R01GM 073095-3, 1R01GM072014-5, and 1R01GM081680-2.

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Chapter 7

Bridging the Atomic and Coarse-Grained Descriptions of Collective Motions in Proteins Vincenzo Carnevale, Cristian Micheletti, Francesco Pontiggia, and Raffaello Potestio

Abstract In proteins and enzymes the necessity that the native state is thermodynamically stable must be appropriately balanced by the capability of the structure to sustain conformational changes and efficiently interconvert among different functionally relevant conformers. This subtle equilibrium reverberates in the complexity of the free-energy landscape which is endowed by a variety of local minima of varying depth and breadth corresponding to the salient structural states of the molecules. In this chapter we will present some concepts and computational algorithms that can be used to characterize the internal dynamics of proteins and relate it to their “functional mechanics.” We will apply these concepts to the analysis of a molecular dynamics simulation of adenylate kinase, a protein for which the structural rearrangement is known to be crucial for the accomplishment of its biological function. We will show that, despite the structural heterogeneity of the explored conformational ensemble, the generalized directions accounting for conformational fluctuations within and across the visited conformational substates are robust and can be described by a limited set of collective coordinates. Finally, as a term of comparison, we will show that in the case of HIV-1 Trans-Activator of Transcription (TAT), a naturally unstructured protein, the lack of any hierarchical organization of the free-energy minima results in a poor consistency of the essential dynamical spaces sampled during the dynamical evolution of the system.

7.1 Introduction Proteins have arguably evolved under the pressure of several concurrent selection mechanisms. Among the key selection factors, the balance between thermodynamic stability and conformational elasticity deserves a particular mention. In fact, on C. Micheletti (B) Scuola Internazionale Superiore di Studi Avanzati, Trieste, Italy; Democritos CNR-IOM and Italian Institute of Technology (SISSA Unit), Trieste, Italy e-mail: [email protected]

A. Kolinski (ed.), Multiscale Approaches to Protein Modeling, C Springer Science+Business Media, LLC 2011 DOI 10.1007/978-1-4419-6889-0_7, 

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one hand the thermodynamic stability of a particular conformational state – the native state – is required to attain a well-defined three-dimensional structure under physiological conditions (temperature, pH, etc.). At the same time, the functionality of proteins and enzymes often relies on their capability to sustain appreciable conformational changes, and this aspect favors conformational heterogeneity. The above-mentioned interplay is best discussed following the development of the key concepts that have progressively shaped the current understanding of enzymatic functionality. The fact that proteins are characterized by a well-defined native state was earlier suggested by H.E. Fischer who, at the end of the Nineteenth century, proposed the lock-and-key paradigm for enzymatic activity. According to this view, both the enzyme and the substrate to be processed are described as rigid molecules that can bind and interact only when having complementary matching shapes. Interestingly, the notion of proteins as static rigid objects was challenged by the very first crystallographic determinations of protein structures (see references in Pontiggia 2008). In particular, the inspection of the X-ray-resolved structures of myoglobin and hemoglobin indicated that (i) the apo-form and holo-form of these heme proteins presented noticeable structural differences and that (ii) the apo conformers were too compact to possibly allow oxygen molecules to diffuse from the solvent toward the heme pocket inside the proteins. Both observations prompted the conclusion that proteins can exist in structurally different substates, corresponding to various minima of the free-energy landscape separated by free-energy barriers of various heights (Frauenfelder et al. 1991). According to the induced-fit theory, the interconversion between the different biologically relevant substates is triggered by the binding/release of the correct ligand(s) which modifies the free-energy landscape favoring the bound/unbound state. The conformational changes involved by such interconversions can range from localized differences, such as dihedral angle isomers, to much larger deformations. The latter may result from the concerted displacement of several secondary elements, possibly also involving changes in the overall tertiary organization. The induced-fit mechanism is extremely valuable for rationalizing many functionally oriented protein structural changes including the selective recognition, binding, and processing of other molecules, the catalysis of biochemical reactions, the mechanical transduction of regulatory signals, and the activation or inactivation of metabolic processes that constitute the life cycle of the cell. Though the scope of applicability of the induced-fit mechanism is fairly large, the advent of a new generation of single-molecule experiments has clarified that the substate interconversion dynamics of several proteins defies this paradigm. For a growing number of enzymes, it is now known that, in thermal equilibrium and in the absence of any ligand, the free biomolecule can appreciably populate not only the inactive (rest) form but also the active one. Thus, the emerging view of catalysis involves the notion of conformational selection: rather than triggering a structural transition, ligand binding occurs only for those molecules already “resembling” the bound state. Hence, ligand binding does not necessarily cause conformational changes, but can shift the equilibrium

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between the populations of the various species already present in solution, favoring the dominant substate. Consequently, a great deal of the proteins’ proficiency relies on their intrinsic flexibility. The unexpected capability of proteins to spontaneously interconvert between different biologically relevant states points at the fact that the internal dynamics of the free enzymes is “innately” predisposed to assist these interconversions (Henzler-Wildman and Kern 2007). Starting from these observations it is appealing to suggest that proteins have evolved not only under the pressure to fold rapidly and reversibly into a welldefined overall shape but also for being capable of harnessing thermal energy toward realizing specific structural fluctuations that are functionally oriented. From the brief account given above it is clear that the phenomenology of functionally oriented structural fluctuations in proteins and enzymes is too rich and complex to be accounted for within a single theoretical paradigm. Indeed, the richness of the observed behavior parallels the complexity of the free-energy landscape which is endowed by a variety of local minima of varying depth and breadth corresponding to the salient structural states of the molecules. Depending on the specific protein of interest, local minima associated with the biologically relevant conformational substates may pre-exist the binding of ligands or can be created by the latter. In either case, it can be argued that a fruitful strategy to study the occurrence of structural fluctuations in proteins is through the characterization of how the potential energy landscape is “navigated” under the effect of thermal fluctuations. An extensive investigation of the near-native free-energy landscape is here presented for adenylate kinase. This enzyme is of high biological interest because it regulates the energy charge of the cell. In recent years it has been the object of an increasing number of studies; the overview presented here is based on the specific theoretical and computational investigations carried out within our group over the past few years. Our analysis will be largely based on atomistic molecular dynamics simulations of the free form of adenylate kinase carried out at constant temperature. A number of methods are used to monitor and characterize the internal dynamics of this enzyme, which is known to be capable of converting spontaneously between two markedly different states. We shall discuss in detail the connection between the small-scale structural changes experienced by the molecule over the subnanosecond time scale, and the much larger collective changes observed over tens of ns (which represents the current limit of computational approaches). The analysis indicates the existence of preferential, “innate,” generalized directions along which both small-scale and large-scale structural rearrangements occur. The robustness of these preferential directions is highlighted by the high degree of consistency found between the essential dynamical spaces obtained by molecular dynamics and those predicted on the basis of coarse-grained elastic network models. The collective character of the innate movements of adenylate kinase is finally established by using a variational scheme to decompose the enzyme in dynamical domains. It is found that virtually all the visited conformational spaces can be described in terms of the relative motion of a limited number of quasi-rigid protein regions. Besides offering a transparent insight into the high level of consistency of the molecule internal dynamics observed over very different time scales, the

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decomposition into quasi-rigid domains provides a clear indication of the feasibility to introduce coarse-grained approaches to probe the breadth of conformational space explored by enzymes having an “innate,” functionally oriented, internal dynamics. As a term of comparison, the same analysis is carried out for extensive molecular dynamics simulations of Trans-Activator of Transcription (TAT), an intrinsically disordered protein. The conformational space visited during the molecular dynamics evolution does not lend easily to a subdivision into a small number of clearly identifiable substates. The lack of a well-defined overall structure is accompanied by a poor consistency of the salient aspects of TAT internal dynamics at various stages of the simulation. The two systems considered here, adenylate kinase and TAT, aptly illustrate the extent to which the free-energy landscape organization can differ across proteins. For TAT, the landscape appears to be sufficiently flat to allow a substantial diffusion in conformational space. The lack of pronounced free-energy minima is not only reflected in the absence of substates with a definite structural organization but also in an inconsistent character of the essential dynamical spaces. By converse, the conformational space of adenylate kinase is well partitioned in structural substates which share a common internal dynamics with a manifest functionaloriented character (as it can bridge the rest and catalytically potent forms of the enzyme).

7.2 Protein Internal Dynamics Observed over Different Timescales: Methods The free-energy landscape of proteins is typically regarded as being highly corrugated and organized in a hierarchy of minima which result from the complex interactions of amino acids in the protein and with the surrounding solvent. At the smallest scale, nearby minima reflect minute structural changes such as distinct rotameric states of sidechains. The protein motion within these minima is well described by a normal mode analysis which can be straightforwardly accomplished by calculating the mass-weighted Hessian in correspondence of the local energy minimum. The barriers separating these energy minima are small enough that, over a typical time scale of ∼10 fs, the system is capable of diffusing to other minima. The effective dimensionality of the available phase space is large enough that the probability of return to a previously visited energy minimum is negligible (Brooks et al. 1995; Janezic and Brooks 1995; Janezic et al. 1995). Yet, over timescales of the order of ∼1 ns, proteins appear to be trapped in specific basins of the landscape, as they mostly fluctuate around a well-defined average conformation. Over this timescale, protein motion is conveniently described as a stochastic diffusion process in an approximately harmonic effective potential (representing the envelope of the above-mentioned local energy minima). Already for time spans of about 1 ns it is possible to detect the presence of largescale concerted displacements of groups of several amino acids. In HIV-1 protease,

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for example, various regions consisting of tens of amino acids can be displaced by several angstroms over 10–20 ns (Janezic et al. 1995).

7.2.1 Low-Energy Collective Excitations Without any assumption on the harmonicity of the potential underlying the dynamical evolution of the system, the modes corresponding to the independent collective protein fluctuations can be obtained through a principal component analysis of the dynamical trajectories (Garcia 1992). Specifically, the modes are aptly identified as the eigenvectors associated with the largest eigenvalues of the covariance matrix, C, whose generic entry is defined as



 

 Cij,μν = ri,μ − ri,μ · rj,ν − rj,ν

(7.1)

where ri,μ indicates the μth Cartesian coordinate of the ith atom and < · > represents the time average over the configurations visited during the simulation. Prior to the calculation of the average structure, the overall roto-translational motion of the protein is removed. Typically, this is achieved by optimally superimposing each “frame” of the trajectory onto a reference one. Initially, the first frame is taken as the reference, and the procedure is next iterated taking the average structure of the previously aligned frames as the new reference, and so on. The principal component analysis can be performed both on a detailed atomistic level, keeping into account all degrees of freedom of the system, and within a coarse-grained description, which is justified in view of the collective character of the structural fluctuations (modes) associated with the principal eigenvectors of the C matrix. This point of view will be adopted hereafter where, as customary, only the Cα atom will be used to represent each amino acid (Amadei et al. 1993; Kitao and Go 1999).

7.2.2 Structural Substates Over time spans of the order of 10 ns, proteins are capable of overcoming the larger barriers (estimated to be of the order of 1 kcal/mol) separating the above-mentioned structurally homogeneous substates. Identifying the substates by analyzing the freeenergy landscape may present difficulties because it is not always straightforward to identify a priori the appropriate order parameters for the system of interest. In these cases, a natural approach is to rely on the intuitive notion that substates should be structurally homogeneous. Accordingly, a suitable measure of structural similarity/dissimilarity, such as the root mean square distance (RMSD), can be used to detect indirectly the presence of local free-energy minima where the system dwells for appreciable time spans. The main substates can, in fact, be detected by a structural clustering of the conformers visited during the trajectory. Notice that as the system should dwell in a substate

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over an uninterrupted time interval (except when substate transitions occur). The time continuity of structures assigned to the same substate should thus be enforced by the clustering procedure. This goal is conveniently formulated following, e.g., the method introduced in Pontiggia et al. (2008), which is based on the K-medoids-partitioning scheme (Kaufman and Rousseeuw 2005). In the following we shall indicate with dij the RMSD after optimal superposition (Kabsch 1978) of two structures visited at times i and j. The sought optimal clustering into K substates is found by identifying the substate-transition times, t1 , t2 , . . ., tK−1 , and the representatives (again labeled by their time index, r1 , r2 , . . . , rK ) which minimize the following dissimilarity cost function:

d=

tk+1 K−1 

di,rk ,

k=0 i=tk

where t0 and tK are, respectively, the initial and final time steps of the simulation. The minimization of the dissimilarity score is accomplished using an iterative scheme (Pontiggia et al. 2008). An initial subdivision in substates is performed by randomly choosing the K – 1 transition times. Each substate representative is identified as the structure with the smallest average distance from the other substate members. A new clustering is next proposed by randomly re-assigning one or more of the K – 1 time subdivisions and ensuring that no two subdivisions coincide (because empty-clusters would result) and the representatives are re-identified. The new subdivision is retained if it leads to a smaller dissimilarity score; otherwise the original subdivision is kept. This procedure is iterated until convergence. Notice that the desired number of clusters, K, is left as input to the user and the choice of the optimal K value requires the inspection of various clustering properties, such as the dissimilarity score as a function of K.

7.2.3 Inter-substate and Intra-substate Fluctuations The conformational rearrangements implied by the inter-substate transitions (hereafter simply termed “jumps”) can be significantly large. On such large scale, both the criteria to be used to remove the roto-translational degrees of freedom and the related concept of average structure, on which the definition of the relaxation modes depend, might leave room for ambiguities. Notwithstanding these ambiguities, the principal component analysis of the trajectory can still be a valuable tool for identifying the salient collective dynamical degrees of freedom. The latter should capture not only the fluctuations within the substates, but also the jumps that bridge them. Following Kitao et al. (1998) the covariance matrix may be decomposed in intraminima and inter-minima contributions

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Cij,μν =

 l

l ωl Cij,μν +



      rj,ν l − rj,ν ωl ri,μ l − ri,μ

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(7.2)

l

where ·l represents an average restricted to the sole configurations in the lth substate and ωl is the fraction of simulation time covered by substate l. The relative importance of intra-substate and inter-substate fluctuations is expected to depend on the number of visited substates. This prompts the interest in examining how the weight of the two terms in Eq. (7.2) depends on the duration of the simulated dynamical trajectory. The connection between these two tiers of conformational fluctuations can also be investigated at a deeper, and more interesting level than motion amplitudes. In fact, a key point is to understand what relationship, if any, exists between the set of lowest-energy modes of an extensive trajectory and those calculated separately for each visited substate. This question is aimed at understanding how the free-energy landscape is actually organized. A high consistency of the intra-substate modes and inter-substate jumps would indicate that the effective dimensionality of free-energy landscape is limited, as a limited number of generalized coordinates would suffice to describe the system fluctuations over a wide temporal range. By converse, the lack of a significant overlap between the modes/jumps would indicate the necessity to use a large number of degrees of freedom to capture the salient features of the free-energy landscape.

7.2.4 Comparison of Structural Fluctuations in Different Substates The degree of overlap of two sets of modes can be established using various quantitative criteria (Amadei et al. 1993; Hess 2000; Pontiggia et al. 2008). Here we shall adopt an intuitive one, based on the calculation of the root mean square inner product (RMSIP): # $  2 $1 → pi · → qj RMSIP = % n

(7.3)

ij

where {p} ≡ {→ pl , . . . → pn } and {q} ≡ {→ ql , . . . → qn } are the two (ortho-normal) sets of modes. The RMSIP value ranges from 0, for complete orthogonality of the {p} and {q} spaces, to 1 in case of their perfect coincidence. Various schemes can be used to establish the statistical significance of an RMSIP value in dependence on the size of the protein and on the number of considered modes, n. For adenylate kinase, which consists of 214 amino acids, the customary choice of n = 10 will be used and RMSIP values equal to 0.7 or more will be considered significant (Amadei et al. 1993).

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7.2.5 Coarse-Grained Description and Modeling of Protein Internal Dynamics As anticipated, a distinctive feature of the structural excitations entailed by the lowenergy modes is their collective character. This observation has several implications. In fact, on one hand, it opens the possibility that the low-energy modes can be captured using simplified coarse-grained approaches which, though inaccurate for an atomistic description of the system, may still appropriately account for the non-local low-energy excitations. The second consideration is that the collective character of the low-energy structural fluctuations suggests the possibility that some groups of amino acids of the protein may move so coherently to be regarded as approximately rigid blocks. Both approaches will be considered here to gain insight into the functional dynamics of adenylate kinase. 7.2.5.1 Elastic Network Models In a seminal study, Tirion (1996) showed that the low-energy excitations of globular proteins calculated with an atomistic force field could be accurately reproduced by a much simpler potential energy function, where all pairs of spatially close heavy atoms were connected by springs penalizing their displacement from the reference positions. It was regarded as highly remarkable the fact that the low-energy system excitations could be well captured by a model with a single parameter (the spring constant, which was equal for all pairs of interacting heavy atoms). The success of the model was explained a posteriori in terms of the large-scale, collective character of the low-energy fluctuations. The salient features of these excitations, which entail structural modulations at scales much larger than a few atoms, can therefore be captured with minimalistic interatomic potentials. In the past decade, this observation provided the motivation to model the internal dynamics of proteins using not only harmonic interatomic potentials (as done by Tirion) but adopting a very coarse-grained structural description where each amino acid is replaced by one or two interaction centers. These models are generally referred to as “elastic network models” because of the quadratic character of the model potential energy which is defined as: F=

1  α αβ β δri Mij δrj 2

(7.4)

ij,αβ

where δriα indicates the αth Cartesian component of the displacement of the ith interaction center (usually taken coinciding with the ith Cα atom), and M is a symmetric interaction matrix which is straightforwardly computed from the reference (native) protein structure (Hinsen 1998; Atilgan et al. 2001; Delarue and Sanejouand 2002; Micheletti et al. 2004). The low-energy modes of the model system are simply given by the eigenvectors of M associated with the smallest non-zero eigenvalues. Despite the quadratic character of the potential energy of Eq. (7.4), the model low-energy

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modes have been shown to be in remarkable accord with the principal directions of motion calculated from extensive molecular dynamics trajectories (Atilgan et al. 2001; Micheletti et al. 2004). We shall accordingly complement the comparison of the intra-substates’ essential dynamical spaces with the low-energy modes predicted using the elastic network model of Micheletti et al. (2004). 7.2.5.2 Identifying Protein Dynamical Domains We finally consider the possibility to detect the existence of approximately rigid domains in proteins based on the analysis of the collective excitations. Rigid subunits, or dynamical domains, correspond to groups of amino acids whose pairwise distances are not significantly changed (in modulus) in the course of thermal fluctuations of the protein. For a given tentative grouping of amino acids into Q domains it is possible to calculate a scoring function which measures the extent to which the protein internal dynamics is distorted after the suppression of the internal structural fluctuations of each domain. The starting ' analysis is the projection of & point of→this the original set of lowest-energy modes, → pl , . . . , pn , on a space which allows only for rigid-body movements (rotations and translations) of the putative rigid subunits. This procedure, which entails standard linear-algebra operations, (see Hinsen and Kneller leads ' to the identification of the rigid-body fits of the original modes &→rb 2000) p rb p l ,...,→ n . The fraction of the molecule’s low-energy fluctuation which is captured by the given decomposition is conveyed via the following quantity, f:  f =

(→rb (2 (p (  l l λl

l λl

(7.5)

with λl indicating the covariance matrix eigenvalue corresponding to the lth mode. The best decomposition of the protein is the one that maximizes f over the possible distinct partitions of amino acids in Q groups. As the space of possible amino acid groupings is too large to allow for a thorough exploration, the optimization of f can be accomplished with approximate stochastic methods (the one discussed here follows the variational scheme of Potestio et al. (2009)).

7.3 Protein Internal Dynamics Observed Over Different Timescales: The Case of Adenylate Kinase Adenylate kinase (AKE) is a small (around 23 kDa) single-chain enzyme whose three-dimensional structure can be subdivided into three main subdomains: a central extremely stable Core is flanked by two mobile domains – the Lid-binding and Ampbinding domains. The primary biological function of this protein is to catalyze the reversible reaction Mg ATP + AMP ↔ Mg ADP + ADP. The binding of the two nucleotides is accompanied by the closure of the domains toward the Core which

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assists and controls the phosphotransfer reaction (Mueller and Schulz 1992; Mueller et al. 1996). The correct movement of the subdomains with respect to the core is, therefore, crucial for function. Because of the large, and rapidly growing, number of experimental and theoretical studies on adenylate kinase (Müller and Schulz 1992; Müller et al. 1996; Sinev et al. 1996; Shapiro et al. 2000, 2002; Han et al. 2002; Wolf-Watz et al. 2004; Kern et al. 2005; Shapiro and Meirovitch 2006; Henzler-Wildman et al. 2007b; Hanson et al. 2007), it appears natural to choose this system to investigate the organization of the near-native free-energy landscape and understand how it reverberates on the protein dynamics observed over various time scales. The conformational rearrangement that takes place upon binding of the substrate can be appreciated in Fig. 7.1 where cartoon representations of the free and bound forms can be compared.

Fig. 7.1 Crystallographic structures of the open free form (a) and of the closed bound form (b) of Escherichia coli adenylate kinase (Mueller and Schulz 1992; Mueller et al. 1996). The flexible Lid (amino acids 114–164) and AMP-binding (amino acids 31–60) domains are colored in black and gray, respectively. The rigid Core domain is colored in white. Reproduced with permission from Pontiggia et al. 2008. Copyright Elsevier Inc

In the closed form the two binding domains are displaced by more than 7 Å with respect to the open structure. The conformational rearrangement of the two domains has been the subject of a large number of experimental and computational studies. Interestingly, nuclear magnetic resonance (NMR) relaxation experiments have identified the structural rearrangement as the limiting step in the reaction turnover (Kern et al. 2005). Moreover, nuclear magnetic resonance (NMR) and fluorescence resonance energy transfer (FRET) experiments have also shown that the closed conformation is appreciably populated even in absence of the ligands (HenzlerWildman et al. 2007b; Hanson et al. 2007). In particular, the binding of the ligands appears to act by shifting the equilibrium between the closed and open populations, stabilizing the closed structure. The correct and efficient functioning of the enzyme relies on the fact that thermal fluctuations are sufficient to excite “intrinsic” functionally oriented movements of the enzymes, where the two mobile subdomains fluctuate between open and closed arrangements over a timescale of about 1 ms. This duration is beyond the reach of present-day atomistic molecular dynamics simulations. Yet, by analyzing the enzyme internal dynamics over time spans of ∼10 ns, it is possible to establish

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several relevant facts about the organization of the free energy of adenylate kinase and their influence on the open/closed interconversion dynamics. The findings discussed hereafter will pertain to a 50-ns long simulation carried out in explicit solvent starting from the open crystal structure. The system, solvated by 18,000 simple point charge (SPC) water molecules in a cubic box, was parameterized with OPLSS-(AA)/L force field (Jorgensen et al. 1996; Kaminski et al. 2001), energy minimized, and gradually heated up to 300 K. The temperature was next adjusted, along with the system density, in a 500-ps long MD simulation at constant temperature (300 K) (Nose’ 1984; Hoover 1985) and pressure (1 bar) (Berendsen et al. 1984). The dynamical evolution of the system in the NVT ensemble, within a cubic simulation box of side l = 8.35 nm, was simulated with the GROMACS software (version 3.3.1) (Van Der Spoel et al. 2005) with an integration time step of 1 fs. Lincs algorithm (Hess et al. 1997) was used to constrain all bond lengths while water internal degrees of freedom were controlled with SETTLE (Mityamoto and Kollman 1992). Long-range electrostatic interaction was treated with the particle mesh Ewald (PME) method (Darden et al. 1993; Essmann et al. 1995). The internal dynamics analysis was first undertaken by identifying the system lowest-energy modes from trajectory excerpts of longer and longer duration, from 500 ps to 50 ns. The plots in Fig. 7.2 present the distribution of the amplitude of the protein motion projected on the first low-energy mode of the entire trajectory. For “converged trajectories” occurring in a quadratic free-energy landscape, the abovementioned distributions are expected to have a Gaussian character. As visible in Fig. 7.2, this property is maintained up to durations of ∼1 ns, though it should be noted that the width of the distribution increases with duration (as pointed out for other systems in Pontiggia et al. 2007). Fig. 7.2 Normalized histograms of Cα displacements along trajectories of increasing duration projected onto the principal direction of the Cα fluctuations

On time intervals of the order of tens of nanoseconds, the system jumps through states that are separated by appreciable barriers of a few kcal/mol and a multimodal character of the projected amplitudes becomes very apparent. The existence of pronounced multiple local minima in the free-energy landscape naturally prompts the comparative investigation of their low-energy modes and their possible influence on how the system can navigate efficiently the pleated landscape. To this purpose, the ensemble of conformations visited during the 50-ns long dynamical evolution was first processed to identify the salient substates (intuitively

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Fig. 7.3 (a) Density matrix of the pairwise root mean square distances of the Cα atoms of configurations sampled in 50-ns MD trajectory of AKE (Pontiggia et al. 2008). The block character of the matrix suggests the presence of substates that can be quantitatively identified with the K-medoids algorithm. The representatives of the eight identified substates are represented in panel b. Reproduced with permission from Pontiggia et al. 2008. Copyright Elsevier Inc

associated with the free-energy minima). The existence of well-populated ensembles of distinct conformers is suggested by the block character of the matrix of root mean square distances (RMSDs) of all pairs of visited structures, see Fig. 7.3. By using the clustering algorithm described in the previous section it was found that the space of visited conformers could be viably partitioned in eight substates, each associated with a dwelling time of about 5–10 ns. The typical RMSD of structures belonging to the same cluster was 2 Å, which is indicative of a good structural homogeneity, while the RMSD between any two representatives of different clusters is about 4.5 Å on average.

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By using Eq. (7.2), the total fluctuation experienced by the protein can be expressed as the sum of two contributions: one pertaining to the intra-substates’ fluctuation and the other from the inter-substates’ jumps. It is of interest to examine the interplay of these two contributions for increasing duration of the trajectory. As shown in Fig. 7.4, the intra-substate contribution is approximately constant in the course of the simulation. This indicates that the various free-energy minima (substates) have approximately the same breadth. On the other hand, the progressive enlargement of the visited phase space is revealed by the rapid increase of the contribution from inter-substate jumps. Besides the comparison of the breadth of phase space occupied by the various substates, it is important to examine what relationship, if any, exists between the essential dynamical spaces which capture the most prominent structural fluctuations of each substate. Because these can be assimilated to the lowest-energy excitation of the corresponding free-energy minima, a further relevant issue is to investigate to what extent they are consistent with the structural deformation bridging the open and closed states of the molecule. For each pair of the eight visited substates the consistency of the top ten essential dynamical substates (obtained from the intra-substate structural covariance analysis) was measured. Such pairwise comparisons resulted in typical RMSIP values of 0.83, which is indicative of very significant correspondences of the low-energy

Fig. 7.4 (a) Time evolution of the total mean square fluctuation of Cα atoms and of two separated contributions due to the intra-substate fluctuations and due to the inter-substate transitions. (b) Evolution of the absolute value of the scalar product between the first intra-substate and inter-substate essential modes and the principal mode of the entire trajectory. Reproduced with permission from Pontiggia et al. 2008. Copyright Elsevier Inc

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mode space. Equally remarkable is the fact that also the difference vectors bridging any pair of substate representatives (not just those visited consecutively in time) are well described by the same low-dimensional space of the above-mentioned lowenergy modes. This is ascertained by calculating what fraction of the difference vector between each pair of substate representatives is captured by the subspace of the ten lowest energy modes of each substate. The distribution of these projections is narrowly peaked around the value 0.92 with a spread of only 0.04. The consistency of the linear spaces describing the low-energy modes within substates and of the difference vectors that connect their representatives is suggestive of the fact that the essential dynamical spaces of adenylate kinase ought to be very consistent across all timescales covered by the atomistic simulation. This fact was confirmed by direct observation, as shown in Fig. 7.4 panel (b). The data reflect the degree of accord of the principal directions of motion obtained by the covariance analysis of trajectory excerpts of increasing duration, from few hundreds of picoseconds to several tens of nanoseconds. The data in the figure indicate a remarkable degree of consistency. This has important implications regarding the scope of molecular dynamics simulations, because the analysis of the essential dynamical spaces over a limited time duration may provide very clear indication of the structural deformations that the system may undergo over much longer timescale. An illustration of this concept is shown by the fact that the top ten modes within the most populated substate encountered during the MD trajectory (lifetime of 9.5 ns) is sufficient to account for as much as 85% of the system fluctuations observed over a 50-ns long time span. In turn, the functional-oriented character of the robust “innate” internal rearrangements of adenylate kinase is apparent a posteriori from the fact that the 96% of the norm of the difference vector bridging the open and closed states is captured by the top ten modes of the complete trajectory. As recently suggested by Henzler-Wildman et al. (2007a) the robustness of the essential dynamical spaces of adenylate kinase appears to be encoded in the overall structural organization of the biomolecule. This point can be aptly illustrated by resorting to coarse-grained description of the biomolecule structure and internal dynamics. In particular, it is instructive to consider the application of an elastic network model to determine the lowest-energy excitation of one particular substate by using, as input, only the Cα trace of the representative. We recall here that elastic network models adopt a minimalistic modeling of the free-energy landscape in the neighborhood of the given reference structure. In particular no sequence-dependent information is directly introduced (the same strength is used for interactions between any pair of amino acids that are in contact in the reference structure). The essential dynamical spaces of the elastic network models, therefore, reflect properties that are readily connected to the neat secondary and tertiary organizations of the biomolecule (Micheletti et al. 2002). By applying the β-Gaussian network model (Micheletti et al. 2004) to the structure of the representative conformer of the most populated substate it was found that the top ten modes have an RMSIP of 0.82 against the essential dynamical space of the entire 50-ns long MD trajectory.

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This striking degree of accord confirms the notion that the salient properties of the lowest-energy modes of adenylate kinase are encoded in the overall conformational organization of the biomolecule. This observation finds a natural interpretative framework in the context of dynamical domains. As will be shown in the subsequent analysis, the consistency of the lowest-energy modes over several timescales appears to reflect the presence of a limited number of amino acids which act as hinge regions for the internal dynamics. In particular, there appear to exist a limited number of protein subdomains which are concertedly displaced as a nearly rigid body, in the course of fluctuations. The quasi-rigid character of these fairly large regions is arguably one of the reasons for the observed consistency of the essential dynamical spaces. We shall hereafter describe the results of the decomposition of AKE in quasirigid units following the application of the optimal subdivision scheme described in the previous section. Subdivisions into Q = 2. . .10 domains were considered. As shown in Fig. 7.5, a very limited number of domains is enough for accounting for most of the protein essential mobility. The subdivision in three quasi-rigid blocks is able to capture 77% of the considered essential dynamics. The three dynamic domains consist of five sequence intervals: one for each of the two mobile domains and three for the nearly fixed core. The correspondence of the three dynamical blocks with the customary tripartite subdivision of AKE into Lid, Core, and AMP-binding subdomains is manifest. It is evident from the analysis presented so far that the dynamical properties of a protein, functionally oriented ones in particular, can be inherent features of the native structure. The interconversion between the open and closed forms of adenylate kinase is not necessarily triggered by the binding to the substrates, but can be induced by the mere thermal agitation. The multiminima character of

Fig. 7.5 Subdivisions of AKE in Q = 2, Q = 3, and Q = 4 rigid subunits. The rigid subdomains, identified by different colors, are shown in panels a, b, and c, respectively. The decomposition was performed taking into account the ten lowest-energy modes. The fraction of essential dynamical motion captured by the subdivision into Q = 2. . .10 rigid domains is shown in panel d. Reproduced with permission from Potestio et al. 2009. Copyright Elsevier Inc

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the free-energy landscape prevents a description of the latter in terms of simple quadratic potentials if a dynamics over the nanoseconds is considered; nevertheless, the basins visited by the system share a striking similarity of the essential directions of motion that are moreover oriented in order to help the biological function. The importance of large-scale conformational changes of the protein structure is made more evident by the dynamical domain decomposition, which shows a clear match between structural and functional modularity. The mobility of the protein is by far concentrated in the hinge-bending motion of the two substrate-binding domains, while a negligible amount of fluctuation is retained by the intra-domain displacements.

7.3.1 Conformational Fluctuations in the Presence of a Nearly Flat Free-Energy Landscape: The Case of TAT The results obtained for adenylate kinase demonstrate how dynamical interconversion among distinct conformational states can be an integral part of protein biological functionality. The observed conformational variability of AKE does not compromise the overall organization of the molecule fold. In fact, the large conformational space of the molecule can be described in terms of the relative movements of a limited number of quasi-rigid domains. Nevertheless, it should be noted that several types of proteins exist for which the intrinsic structural plasticity is so high that it is impossible to introduce a definite notion of overall fold (Dunker et al. 2000, 2002; Tompa 2005). These intrinsically disordered proteins are interesting in that their behavior cannot be possibly rationalized in terms of the lock-and-key paradigm, and they are relevant for the present discussion because they provide a concrete example of biomolecules lacking nearly rigid domains. Defining the essential space of these proteins, i.e. a small set of collective atomic displacements accounting for most of the fluctuations, appears challenging given the absence of well-defined conformational substates and the nearly flat nature of the free-energy surface. Therefore disordered proteins are expected to constitute a limiting case for most of the concepts discussed above and their analysis provides the opportunity to discuss the range of applicability of the methodologies introduced in this chapter. A relevant example of intrinsically disordered protein is the HIV-1 TransActivator of Transcription (TAT) (Shojania and O’Neil 2006). TAT is a viral protein of about 100 amino acids essential for an efficient transcription of the viral genome (Athanassiou et al. 2007). From the analysis of the 15-ns long trajectory of the MAL isolate (Alizon et al. 1986) of TAT is evident a significant conformational heterogeneity (Fig. 7.6, right panel). However, rather than resulting from a dynamic interconversion among distinct well-defined substates, such a broad distribution appears to result from free diffusion across a nearly flat free-energy landscape. Indeed the set of pairwise structural distances (RMSD) does not show any clear structure and no recognizable basins are detected.

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Fig. 7.6 Left panel: The MD trajectories of TAT-MAL isolate and of AKE were subdivided in 1-ns long intervals. The top ten eigenvectors of covariance matrix of each interval were calculated. The histogram shows the distribution of the root mean square inner product (RMSIP) of the essential spaces for any pair of distinct intervals. The RMSIP distributions of TAT-MAL isolate and AKE are shown with dashed and continuous lines, respectively. Right panel: Distribution of the root mean square distances of any pair of configurations sampled by the MD trajectory of the TAT-MAL isolate (dashed line) and of AKE (solid line)

The lack of any hierarchical organization of the free-energy minima is expected to result in a poor consistency of the essential spaces calculated along the trajectory. Indeed the mean square inner product between the top ten eigenvectors of the covariance matrix calculated for pairs of 1-ns long intervals of the trajectory are significantly lower than for adenylate kinase (Fig. 7.6). This observation therefore suggests that conformational fluctuations cannot be easily rationalized as a superposition of simpler motions of groups of residues undergoing rigid-body rotations about hinges and/or axes.

7.4 Concluding Remarks For a large number of enzymes it has been shown that the biological function emerges from a plurality of distinct conformational substates (Henzler-Wildman and Kern 2007). The interconversions among the substates are the ultimate determinants of the effective reaction kinetics of the enzymatic processes. Based on these observations it can be stated that key functional-oriented features are encoded in the free-energy landscape, both in terms of local minima (substates) and in terms of the accessible pathways that connect them. In this chapter we have discussed a number of theoretical concepts and computational algorithms that can be used to characterize the internal dynamics of proteins and enzymes and relate it to their “functional mechanics.” The presented techniques, that are general and transferable, have been applied to the case study of adenylate kinase (AKE). As a term of comparison, the case of a protein showing a very different organization of the free-energy landscape, the HIV-1 Trans-Activator of Transcription (TAT), was also briefly discussed. For adenylate kinase it was shown that the dynamical evolution (followed with constant temperature, explicit solvent molecular dynamics) visits several different conformational substates. Each substate appears structurally homogeneous, though

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the inter-substate conformational variability is substantial. The multi-scale analysis of the system intra-substate and inter-substate fluctuation dynamics reveals that the heterogeneity of the conformational ensemble explored over tens of nanoseconds is accompanied by a striking robustness of the generalized degrees of freedom identifying the lowest-energy excitations of these biomolecules. The latter mostly correspond to the relative motion of a limited number of “dynamical domains” that behave as nearly rigid protein subunits and whose displacement can bridge the inactive (open) and active (closed) forms of the enzyme. The substate-independent location of the hinge regions between these units hints at the fact that the salient, functionally oriented aspects of the internal dynamics of adenylate kinase are encoded in the fundamental structural aspects of the enzyme (such as its secondary and tertiary organizations).

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Henzler-Wildman KA, Lei M, Thai V, Kerns SJ, Karplus M, Kern D (2007a) A hierarchy of timescales in protein dynamics is linked to enzyme catalysis. Nature 450:913–916 Henzler-Wildman KA, Thai V, Lei M, Ott M, Wolf-Watz M, Fenn T, Pozharski E, Wilson MA, Petsko GA, Karplus M, Hubner CG, Kern D (2007b) Intrinsic motions along an enzymatic reaction trajectory. Nature 450:838–844 Hess B (2000) Similarities between principal components of protein dynamics and random diffusion. Phys Rev E 62:8438–8448 Hess B, Bekker H, Berendsen HJC, Fraaije JGEM (1997) A linear constraint solver for molecular simulations. J Comp Chem 18:1463–1472 Hinsen K (1998) Analysis of domain motions by approximate normal mode calculations. Proteins 33:417–429 Hinsen K, Kneller GR (2000) Projection methods for the analysis of complex motions in macromolecules. Mol Simulat 23:275–292 Hoover WG (1985) Canonical dynamics: equilibrium phase-space distributions. Phys Rev A 31:1695–1697 Janezic D, Brooks BR (1995) Harmonic analysis of large systems. II. Comparison of different protein models. J Comp Chem 16:1543–1553 Janezic D, Venable R, Brooks BR (1995) Harmonic analysis of large systems. III. Comparison with molecular dynamics. J Comp Chem 16:1544–1556 Jorgensen W, Maxwell D, Tirado-Rives J (1996) Development and testing of the OPLS all-atom force field on conformational energetics and properties of organic liquids. J Am Chem Soc 118:11225–11236 Kabsch W (1978) A discussion of the solution for the best rotation to relate two sets of vectors. Acta Crystallogr A 34:827–828 Kaminski G, Friesner R, Tirado-Rives J, Jorgensen W (2001) Evaluation and reparametrization of the OPLS-AA force field for proteins via comparison with accurate quantum chemical calculations on peptides. J Phys Chem B 105:6474–6487 Kaufman L, Rousseeuw PJ (2005) Finding groups in data: an introduction to cluster analysis. Wiley. Wiley’s Series in Probability and Statistics, New York, NY. Kern D, Eisenmesser EZ, Wolf-Watz M (2005) Enzyme dynamics during catalysis measured by NMR spectroscopy. Methods Enzymol 394:507–524 Kitao A, Go N (1999) Investigating protein dynamics in collective coordinate space. Curr Opin Struct Biol 9:164–169 Kitao A, Hayward S, Go N (1998) Energy landscape of a native protein: jumping-among-minima model. Proteins 33:496–517 Micheletti C, Carloni P, Maritan A (2004) Accurate and efficient description of protein vibrational dynamics: comparing molecular dynamics and Gaussian models. Proteins 55:635–645 Micheletti C, Lattanzi G, Maritan A (2002) Elastic properties of proteins: insight on the folding process and evolutionary selection of native structures. J Mol Biol 321:909–921 Mityamoto S, Kollman PA (1992) Settle: an analytical version of the shake and rattle algorithm for rigid water models. J Comp Chem 13:952–962 Müller CW, Schlauderer GJ, Reinstein J, Schulz GE (1996) Adenylate kinase motions during catalysis: an energetic counterweight balancing substrate binding. Structure 4: 147–156 Müller CW, Schulz GE (1992) Structure of the complex between adenylate kinase from Escherichia coli and the inhibitor AP5A refined at 1.9 Å resolution. A model for a catalytic transition state. J Mol Biol 224:159–177 Nose’ S (1984) A molecular dynamics method for simulations in the canonical ensemble. Mol Phys 52:255–268 Pontiggia F (2008). Protein structure and functionally-oriented dynamics: from atomistic to coarse-grained models. PhD in Biophysics Scuola Internazionale Superiore di Studi Avanzati (SISSA/ISAS) via Beirut 2–4, 34151 Trieste, Italy Pontiggia F, Colombo G, Micheletti C, Orland H (2007) Anharmonicity and self-similarity of the free energy landscape of protein G. Phys Rev Lett 98:048102–048102

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Chapter 8

Structure-Based Models of Biomolecules: Stretching of Proteins, Dynamics of Knots, Hydrodynamic Effects, and Indentation of Virus Capsids Marek Cieplak and Joanna I. Sułkowska

Abstract Coarse-grained models of biomolecules are developed to provide ways of simulating situations which involve system sizes and time scales that are hard to study by all-atom approaches. These situations usually are associated with occurrence of large conformational transformations. This review discusses a subclass of the coarse-grained descriptions: the structure-based models. These models are defined in a phenomenological way and make use of the knowledge of the native structure that is determined experimentally. We discuss the cases of the DNA molecule and dendrimers but the focus of the review is on proteins. For proteins, the reduction of the number of the degrees of freedom is achieved by representing amino acids by single beads located at the Cα atoms. The beads are tethered together into chains and effective attractive contact interactions are introduced so that the lowest energy state corresponds to the native conformation. There are many variants of such models and each variant comes with its own set of properties. Optimal variants can be selected by making comparisons to experimental data on single-molecule stretching. We discuss the best-performing variants of such models. Among them, there is a model with the Lennard–Jones potential in the native contacts and with the uniform (sequence independent) energy parameter. We apply this model to several problems: theoretical survey of mechanical resistance to stretching of 17,134 proteins comprising not more than 250 amino acids, dynamics of proteins with knots, pulling proteins out of membranes, the role of hydrodynamic interactions, and nanoindentation of virus capsids.

8.1 Introduction Equilibrium dynamical behavior of biomolecules can usually be characterized through studies of fluctuations around an optimal conformation. All-atom simulations are well suited to this task, particularly when the system comprises not more

M. Cieplak (B) Institute of Physics, Polish Academy of Sciences, Warsaw, Poland e-mail: [email protected]

A. Kolinski (ed.), Multiscale Approaches to Protein Modeling, C Springer Science+Business Media, LLC 2011 DOI 10.1007/978-1-4419-6889-0_8, 

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than a thousand or so of amino acids or nucleotides. Many interesting biomolecular phenomena, however, involve large conformational transformations that usually last orders of magnitude longer than a typical 10 ns duration of current advanced all-atom calculations. The list of such phenomena includes protein folding, chemical, thermal, and pressure-induced denaturation, stretching and twisting by various nanotechnological devices such as the atomic force microscope, stretching by fluid flows, refolding in a mechanically constraining force clamp, etc. An even larger level of complexity in the conformational changes arises when considering dynamics of biomolecular assemblies that are functional at the cellular level. Examples of such processes include replication, transcription, translation, protein degradation, organelle transport, cell adhesion, muscle elasticity, cell crawling, membrane fusion, functioning of virus capsids, and many others. A still another category of the dynamic large-scale phenomena involves the effects of molecular crowding – how the behavior of one biomolecule is influenced by the presence of many other biomolecules that are nearby. Even though the all-atom simulation methods have occasionally reached the microsecond time scale (Duan and Kollman 1998; Snow et al. 2002), such studies count more toward breaking records than as established tools allowing for multiparameter and multi-trajectory statistical studies. They are also restricted to small system sizes. Undoubtedly, the growing availability of the peta-flop computing will shift the computational feasibility barriers. Even then, however, it will be still worthwhile to first assess overall properties of a system within a simpler model, as a numerical equivalent of a “back-of-the-envelope calculation,” and only then focus on selected aspects in a more realistic and more detailed way. Another application of the simplified models is making comparison of many biomolecules to select those that are desired for a particular purpose such as that come with large resistance to stretching. Still another is studying assemblies or aggregates of many molecules such as virus capsids. Finally, the simplified models can often be used as toy systems for which some basic questions may get qualitative answers. One example of an enquiry of this kind is what are the relations, if any, between various processes that involve conformational changes. A natural way to develop simplified models starts by coarse graining, i.e., by thinning the number of degrees of freedom. A crucial ingredient of coarse graining is provided by introduction of an overdamped “implicit solvent” that accounts for the immersion in water. The implicit solvent brings in thermostating through random forces acting on the biomolecule. It also introduces damping forces that are proportional to the velocity of the structural units of the biomolecule. Both kinds of these forces are added to the relevant equations of motions for the system. The equations of motion are either of the second or first order. In the former case one implements molecular dynamics of the Newtonian type (Allen and Tildesley 1987) and in the latter of the Brownian dynamics type (Ermak and McCammon 1978). In this chapter, we do not discuss the vast kingdom of lattice models in which the dynamics are intrinsically discretized and defined through a Monte Carlo process. The coarse-graining schemes depend on the biomolecule studied. For a protein, the simplest approach is to represent it by a chain of harmonically tethered beads

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located at the sites of the Cα atoms. One can also consider models with the Cα and Cβ atoms as the building units, as studied, e.g., in Sulkowska and Cieplak (2007). For the DNA, each nucleotide in a strand can be represented by two to five beads, depending on the specific model (Knotts et al. 2007; Niewieczerzal and Cieplak 2009) if the description is to be resolved at the single nucleotide level. Any coarse-graining procedure comes with replacement of microscopic couplings (such as Coulombic interactions between partial charges) by new effective couplings between the decimated degrees of freedom. There is no unique prescription for determining such effective couplings. Here we outline a phenomenological approach that is often found successful, in which the couplings are determined based on the knowledge of the native structure. This philosophy has been proposed by Go and his collaborators (Abe and Go 1981) in the context of folding. Its first molecular dynamics implementation appears to be contained in the paper by Veitshans et al. (1997). When applying structure-based models one makes the assumption that the process studied is largely determined by the native-state topology and non-native interactions are of secondary importance. This assumption should be valid in most stretching processes since unfolding commences in the native state. However, it may or may not be valid in the folding process, depending on the nature of the free energy landscape. Small globular proteins are expected to show little structural frustration (Bryngelson et al. 1995; Goldstein et al. 1992; Clementi et al. 2000) and are believed to be compatible with the assumption. It is possible to include non-native contacts in the model whenever they are deemed essential, as for example in Cieplak (2004) and Wallin et al. (2007). There are many possible molecular dynamics implementations of the structurebased approach, since a definition through structure is not sufficiently unique. In fact, various structure-based models of one protein may lead to a wide range of possible behaviors (Sulkowska and Cieplak 2008). We have enumerated more than 500 Go-like off-lattice models and proposed (Sulkowska and Cieplak 2008) that one may select the optimal classes by considering single-protein stretching (Rief et al. 1997; Fowler et al. 2002; Yang et al. 2000; Carrion-Vazquez et al. 2000, 2009; Galera-Prat et al. 2010) since this particular process starts in the native state, in the vicinity of which the structure-based models are expected to be the most reliable. A comparison of the theoretical results to experimental data on stretching allows us to select several models which, in addition, yield reasonable folding properties. Among the winning models, there is one in which, besides other relevant attributes to be explained later, the effective attractive Cα –Cα interaction in the, so-called, native contact is described by the Lennard–Jones potential  V

6−12

= 4Eij

σij rij

12

 −

σij rij

6  .

(8.1)

Here rij is the distance between the Cα s in amino acids i and j and σ ij is determined pair-by-pair so that the minimum is located at the experimentally established native

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distance rij n . The simplest choice of the energy parameter Eij is to take it to be uniform, Eij = ε, where ε which should be of order 1.5 kcal/mol. It should be noted that, even in the case of the uniform Eij , the structure-based model still carries in itself an information about the sequence, albeit in an implicit way. It comes in two ways. The first is through the structure which affects the parameter σ ij and the structure is formed (in water) for a particular sequence. The second is through the contact map which specifies which interactions correspond to the native contacts. For a given distance between the Cα s in the structure, the native contact may or may not form, depending on the specificity. Note that the potential defined in Eq. (8.1) is very weak for rij exceeding several σ ij which allows for “breaking” the contact for sufficiently large distances as a result of stretching or thermal denaturization. This feature is qualitatively distinct from the situation encountered with the purely confining harmonic potential which is used in the elastic network models (Bahar et al. 1997, 1999; Micheletti et al. 2002). Such elastic models form a special class of the Go-like Hamiltonians. Their application to stretching requires a prior determination of a distorted non-native structure (Sulkowska et al. 2008a). The construction of most structure-based models is based on a distinction between native and non-native contacts in a biomolecule. This leads to a natural characterization of folding or unfolding processes in terms of whether particular contacts are functionally established or broken. For instance, for the proteinic model with the Lennard–Jones native potentials, a contact can be declared broken if the rij distance exceeds 1.5 σ ij . It has turned out (Cieplak and Hoang 2003; Cieplak 2004) that when studying folding, it is interesting to determine when a given contact is established for the first time. If all such contacts are established for the first time simultaneously then one declares folding to be accomplished. On the other hand, when studying unraveling, it is interesting to record the last times at which a native contact still holds (Cieplak et al. 2002, 2004). A convenient way to represent processes with large conformational changes visually is to provide scenario diagrams in which the characteristic times for a particular native contact are displayed against a contact label, such as the sequential distance |j-i|. The advantage of the scenario diagrams over representing processes by the snapshots of conformations is that the scenario diagrams are derived for a statistical ensemble of trajectories whereas the snapshot’s role is more that of providing an illustration. Representing results of all-atom simulations through the scenario diagrams may also prove useful. Even though the all-atom models do not make use of the division into native and nonnative contacts to define the dynamics, monitoring the fate of the native interactions as a function of time may provide important insights. In this review, we outline the phenomenological methods of construction of the structure-based models of proteins, the DNA, and dendrimers. The review is focused on the proteins, however, and the examples of applications discussed here deal with the proteins. These are mechanical stretching of proteins, dynamics of proteins with knots, pulling of proteins out of membranes, the role of hydrodynamic interactions, and nanoindentation of virus capsids.

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8.2 The Structure-Based Models of Proteins In the case of the proteins, the simplest coarse-graining scheme represents amino acids by single beads located at the Cα atoms. The beads are tethered together into chains by harmonic interactions. The minima of these interactions correspond to distances between the consecutive Cα atoms as read off the Protein Data Bank (PDB; Berman et al. 2000) structure file. Usually, these distances cluster around 3.8 Å with a spread of 0.15 Å. However, in cases involving prolines, the distances may vary between 2.8 and 3.85 Å as prolines can accommodate the cis-conformation. The beads generate a soft excluded volume. In our implementation (Cieplak et al. 2002), this volume corresponds to a radius of order 4 Å. This excluded volume prevents interpenetration of the chain in non-native conformations. Further steps in the construction of the model (Cieplak and Hoang 2003; Cieplak et al. 2004) involve introduction of a local backbone stiffness, determination of the contact map, i.e., the list of bead pairs endowed with attractive interactions, and then selection of an appropriate potential for the attraction. The structure-based approach (Abe and Go 1981; Takada 1999) postulates that the potentials used should be minimized for the native structure that is determined experimentally. There are many ways to find such potentials. In Sulkowska and Cieplak (2008) we have analyzed in detail 62 variants of the resulting models. The corresponding Hamiltonians are characterized by attributes described in a short-hand notation by model = {V NAT , S, M, E}

(8.2)

where the first term denotes selection of the potential, the second identifies the way to account for the local backbone stiffness, the third defines the contact map, and the fourth determines the choice of the energy parameters. In addition, one may also take the Cβ atoms into account, but we find that this step usually does not amount to an improvement. Physical properties are found to vary significantly across the variants of the model and one can select the best variants by making comparisons to the experimental data on stretching. The specific parameter used in the comparisons is the maximum value, Fmax , of the force of resistance to stretching at constant speed. Fmax is determined before reaching a fully extended conformation (where the covalent bonds get strained but not ruptured) and it characterizes the elastic structure of the protein. Large values of Fmax are associated with simultaneous shearing of many attractive contacts together. In order to select the optimal variants of the models, we have performed statistical tests of correlations to the experimental values of Fmax (Sulkowska and Cieplak 2008). Out of the 62 variants considered, 15 were found to perform well and their list is provided in table 5 of Sulkowska and Cieplak (2008). When no adjustments for the variations in the experimental pulling speeds are made, the model LJ3 = {6 − 12, C, M3, E0 }

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is singled out as the best performer. Here, 6–12 denotes the Lennard–Jones potential defined in Eq. (8.1) with the uniform, i.e., amino acid-independent, energy parameter. Symbol C signifies that the backbone stiffness is described by a potential term, VC , which favors the native sense of the local chirality (Cieplak et al. 2002; Kwiecinska and Cieplak 2005). The chirality-based stiffness is approximately equivalent to favoring of the native values of the dihedral angles (Sulkowska and Cieplak 2008). An alternative is to have an angular stiffness, denoted as A, with a potential VA , in which also the bond angles tend to adopt the native values as in Veitshans et al. (1997) and Clementi et al. (2000). VA is more constraining than VC , but is less efficient numerically. Our choice of VC is given by Vc =

1  2 κ Ci − Cin , 2

Ci =

(wi−1 × wi ) · wi+1 do3

where Ci n is the chirality of residue i in the native conformation and d0 = |wi | is the distance between subsequent Cα atoms. Here, wi = ri+1 − ri . We take κ equal to 1 as discussed in Kwiecinska and Cieplak (2005). We now discuss the most crucial third descriptor in Eq. (8.2): the contact map. One way to define the contact map is to introduce a cutoff length, e.g., equal to 7.5 Å, and declare all pairs ij of amino acids as making a native contact if rij is less than the cutoff. This way, however, does not take the variable sizes of the amino acids into account and thus introduces spurious contacts while eliminating important longer-ranged contacts (even corresponding to distances of order 12 Å (Cieplak and Hoang 2003)). A more physical way to determine the contact map is through the following procedure. First, amino acids are represented by clusters of spheres associated with the heavy atoms located at their native positions. The heavy atoms are assigned van der Waals radii, as in Tsai et al. (1999) and Settanni et al. (2002), which are then multiplied by 1.24 to account for attraction. If the resulting spheres assigned to one amino acid overlap with the spheres assigned to another in the native state then the corresponding pair of amino acids is declared as generating a native contact. Otherwise, the interactions are declared to be non-native and then they are represented by the soft cores. This procedure is illustrated in Fig. 8.1. The atomic-overlap-based criterion generally agrees with a chemical-based assessment (Sobolev et al. 1999) in which one pays attention to the geometry of a bond. By employing the related CSU software (Sobolev et al. 1999) we find, however, that the overlap criterion often errs when it comes to the sequentially close pairs of the type i,i+2. Such contacts are detected as valid whereas they should be discarded as being usually of the dispersive origin (van der Waals interactions) and thus much weaker than hydrogen bonds or ionic bridges. In the M3 map, such contacts are excluded – as opposed to M2 when they are kept. The choice of the model for the local backbone stiffness has a bearing on the choice of the contact map. Even though the function of VA is similar to that of VC , VA is more encompassing and more sensitive to the local shape of a conformation. Thus when using VA , one does not need to worry about the short-range contacts i,i+3 and deal only with the

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Fig. 8.1 An illustration of a construction of a structure-based model of a protein for the Trp-cage miniprotein, with the PDB code 1l2y, which consists of 20 amino acids. The backbone is shown in its native conformation. The conformation is akin to a β-hairpin in which one of the “arms” is shaped into an α-helix. Heavy atoms are shown for two amino acids. The two large spheres are located on the Cδ atom in leucine-2 and the Cγ atom in proline-19. Their radii are equal to the van der Waals radii enhanced by 1.24. These two atoms yield the biggest overlap between residues 2 and 19. All contacts connected to residue 2 are marked by black lines

M4 map in which one keeps i,i+4 and longer-ranged contacts as done in Clementi et al. (2000). A special kind of contacts is formed by the disulfide bridges. Since these are covalent in nature, they cannot be broken by means of the current nanomanipulation techniques (though they can be dissolved chemically). One can represent them by the Lennard–Jones potential in which the energy parameter is enhanced by an order of magnitude as done in Sulkowska and Cieplak (2007). A more reliable way is to represent them by stiff harmonic bonds as in Sikora et al. (2009). All the potential terms discussed above give rise to forces Fi acting on particular amino acids. Newton’s equations of motion for the ith amino acid involve not only Fi but they also include Langevin terms mimicking the implicit solvent: ••

•

i

i

m r = −γ r +Fi +  i where γ denotes the damping constant and the last term corresponds to ran√ dom forces. The dispersion of the random forces is equal to 2γ kB T, where T denotes temperature and kB the Boltzmann constant. In standard molecular dynamics schemes that pertain to atoms, as opposed to effective atoms in an implicit solvent of our models, the Langevin terms are sometimes used (Grest and Kremer

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1986; Smith et al. 1996; Cieplak et al. 2001) as a minor perturbation to provide thermostating. In such cases the random forces are applied just in one Cartesian direction that is assumed to be irrelevant for the problem under consideration. In the proteinic case, however, the Langevin terms are applied in all directions and affect the dynamics in a major way in order to account for the viscous features of the water solvent. We select γ = 2m/τ as we find that this choice provides sufficient overdamping. Higher values of γ would imitate water better (Veitshans et al. 1997; Klimov and Thirumalai 1997). However, enhancing γ beyond our selected value no longer results in noticeable changes in the force–displacement curves in stretching studies (Cieplak et al. 2004) and makes the folding times start depending on γ in a linear fashion (see figure 15 in Hoang and Cieplak (2000)) that allows for simple scaling estimates. One may consider models in which there is specificity in the parameters associated with individual amino acids. Variations in values of the particle masses do not affect folding times (Cieplak et al. 2004) which are consistent with the overdamped character of the model. Variations in the values of the damping constant, reflecting placement on various rings in the hydrophobicity scale, have been found to affect folding and stretching primarily through the mean of these values (Szymczak and Cieplak 2007a). A similar situation is expected to happen when considering differentiations in the hydrodynamic radii when the model includes hydrodynamic interactions that are discussed in Section 4.4. In standard molecular dynamics simulations (Allen and Tildesley 1987), the characteristic time scale τ is of order 1 ps since the motion of the model atoms is mostly ballistic and thermalization takes place primarily through collisions with other atoms. In the coarse-grained model, however, we deal with effective atoms that represent whole amino acids and their motion is dominated by diffusion. A characteristic time needed to cover typical molecular distance like 5 Å through diffusion is of order 1 ns as argued in Veitshans et al. (1997) and Szymczak and Cieplak (2006). The equations can be solved by various algorithms. Many of our results have been obtained by using a fifth-order Gear predictor–corrector scheme (Gear, 1971) with the time step of 0.005 τ . We have found this algorithm to be very stable and allowing for a relatively large integration step (ten times larger than in more popular schemes). This algorithm yields results similar to those obtained within the Brownian dynamics. The model presented here can also be used for studies of protein stretching by fluid flows (Szymczak and Cieplak 2006) by adding drag forces (due to a hydrodynamic flow) to the damping term. One may consider this as a side advantage of the implicit solvent approach. The models described above can be employed, for instance, to study folding. In such studies, one starts from fully extended conformations and asks when all of the native contacts are established for the first time. The median folding time, determined by studying many trajectories, generally has a U-shaped behavior as a function of T (see, e.g., Cieplak and Hoang 2003). For the LJ3 model the U-shape has a (usually broad) minimum centered around kB T/ε = 0.3. In practice, this optimal temperature for folding plays the role of the room temperature in the model. When studying thermal denaturation (Cieplak and Sulkowska 2005), one starts in the native conformation and then asks when

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certain classes of contacts, for instance, all non-local contacts (|j − i| > 4), get ruptured simultaneously. The simplification of the structure-based models has allowed us for demonstrating that, on average, the order of contact breaking by thermal fluctuations is opposite to the order in which contacts are established during folding – which is implicitly assumed in many all-atom simulations performed in the context of the transition state studies (see, e.g., Li and Daggett, 1994). Pressure-induced denaturation is found (Wojciechowski and Cieplak 2007) to lead to unraveling in a still different way. When studying stretching, we attach two elastic springs to two selected amino acids, typically the two terminal amino acids. One end of one of the springs is anchored to an immovable substrate. One end of the other spring is attached to a point that moves along the direction connecting the selected amino acids. In our constant speed simulations, the speed of pulling, vp , is often chosen to be equal to 0.005 Å/τ as in Sulkowska and Cieplak (2007) and Sikora et al. (2009). If τ is of order 1 ns then this vp corresponds to about 500,000 nm/s, which is three orders of magnitude larger than the typical experimental speeds of 600–700 nm/s. Simulations at near experimental speeds take time but can be accomplished in the structure-based models. On the other hand, the “steered” all-atom molecular dynamics computations (Lu and Schulten 1999; Paci and Karplus 2000) usually incorporate pulling speeds which are seven orders of magnitude bigger. This leads to large values of Fmax that well exceed velocity-dependent logarithmic extrapolations of the experimental data. These values can be reduced by performing stretching in a quasistatic way (Pabon and Amzel 2006). When adjustments accounting for the variations in the experimental speeds are made by a corresponding linear proportioning of the theoretical speeds, the ranking of the variants of the models is changed somewhat (Sulkowska and Cieplak 2008; Cieplak and Sulkowska 2009). In particular, the LJ3 model gets dethroned to the second best, whereas &

10−12, A, M4, EHB,MJ

'

is found to be the true winner. It involves the contact potential of the10–12 form, V 10−12

  n 12  n 10  rij rij = Eij 5 −6 . rij rij

and EHB,MJ assigns the amplitudes in this potential in a non-uniform way. Specifically an energy of ε is assigned to hydrogen-bonded amino acids (the bond between the N and O atoms belonging to the backbone and the bond between the N atom from the backbone the Cβ atom from a side chain) whereas an energy εijMJ proportional to the Miyazawa–Jernigan couplings (Miyazawa and Jernigan, 1996), γijMJ , is assigned to contacts arising through non-hydrogen-bond side chain–side chain interactions. All other contacts get the energy ε. The energy εijMJ is given by εγijMJ / < γijMJ > where the couplings in the denominator are averaged over 210

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different possible pairs of amino acids. This very best model does not appear to have been used in the proteinic literature. However, it is very closely related to the third best-performing model {6−10−12, A, M4, EHB,MJ } that has been introduced by Karanicolas and Brooks (2002). This model uses the potential  V

6−10−12

= 4 Eij



σij 13 rij

12



σij − 18 rij

10



σij +4 rij

6  .

The statistical tests have been run at a fixed temperature (close to the optimal folding) and the differences in performance between the three models are found to be slight (Sulkowska and Cieplak 2008). Therefore, it seems sensible to use the simplest of them all: the LJ3 model with the uniform couplings. This is despite the fact that using Miyazawa–Jernigan-like couplings would enhance the specificity content of a model. Our latest comparisons of the LJ3-derived values of Fmax to the corresponding experimental findings have involved determining Fmax at several values of vp , fitting them to a logarithmic vp -dependence and then extrapolating to the experimental speeds (Sikora et al. 2009). The best slope in the cross-correlation plot yields the effective value of the parameter ε. We find it to be (110 ± 30) pNÅ whereas without the extrapolation we were getting about 71 pNÅ (Sulkowska and Cieplak 2008). The latter value corresponds to 1.6 kcal/mol. There are several other implementations of the Go-like coarse-graining that we did not test. Among them, there is the model of Tozzini et al. (2007) which uses the Morse potentials and couplings determined in the spirit of Miyazawa and Jernigan, i.e., through the frequencies of occurrence of various amino acidic pairs. However, for other sets of couplings that we did test (Sulkowska and Cieplak 2008) the Morse potentials were not found to perform in any distinguished way. It should be noted though that the model of Tozzini et al. contains additional terms in the Hamiltonian.

8.3 The Structure-Based Models of the DNA and Dendrimers Construction of coarse-grained models of the double-stranded DNA (dsDNA) is similar in spirit to that of the proteins. In the absence of stress or torsion, the native structure corresponds to the double helix as exemplified by structure 119D deposited in the Protein Data Bank for the sequence 5 D(CGTAGATCTACGTAGATCTACG)-3 . A pair of nucleotides from two strands can bind either by two or three hydrogen bonds, depending on whether this is the case of the A−T pair or the G−C pair, respectively. Recently, we have constructed several coarse-grained models of the dsDNA (Niewieczerzal and Cieplak 2009) that are resolved at a single nucleotide level. The least coarse grained of them (model I) will be described here in more detail. In this model, the A and T nucleotides are

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Fig. 8.2 A schematic representation of the bead model I of the dsDNA after (Niewieczerzal and Cieplak 2009). An A–T pair of the nucleotides is shown; each of the nucleotides in the pair is represented by four beads. The G and C nucleotides are represented by five beads

represented by four beads (Fig. 8.2) and the G and C nucleotides by five beads. One of the beads represents the phosphate group (bead p), another the sugar group (bead b), and the remaining beads (bead h) participate in formation of the appropriate number of the hydrogen bonds. As in proteins, the hydrogen bonds between the coarse-grained entities are represented by effective Lennard–Jones potentials of strength ε. A further reduced version of this model, denoted as model II, is shown at the top of Fig. 8.3. It comprises no h-beads. Instead, the effective Lennard–Jones couplings of strength of either 2 or 3 ε couple the b-beads directly. In between, there is a similarly defined model III (the bottom panel of Fig. 8.3) which involves three beads. Model III has been proposed and studied by Knotts et al. (2007). In models I and II the effective backbone is meant to connect (harmonically) the consecutive p-beads in each strand. In model III, on the other hand, the backbone zigzags

Fig. 8.3 A schematic representation of models II and III of the dsDNA

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between the s-beads (for sugar) and p-beads to render the true chemical connectivity better. Nevertheless, we find (Niewieczerzal and Cieplak 2009) that the three models perform quite similar at least in the context of mechanical manipulations such as unzipping, shearing, and twisting. They also show sensitivity to the sequence. For instance, an all-CG sequence has a bigger mechanical stability than an all-AT sequence of the same length. We illustrate the coarse-graining process for model I. Each strand is first represented by a chain of p-beads which are 5.8 Å apart on average. The p-bead is placed at the position of the C4∗ atom in the molecule of ribose. The h-beads represent the “head” atoms that may form the hydrogen bonds. In the C nucleotides, the h-beads are located on the O2, N3, and N4 atoms of the bases. In the G nucleotides, on the O6, N1, and N2 atoms. Finally, in the A and T nucleotides on the N6, N1, and N3, O4 atoms, respectively. The h-beads are linked to their “bases,” i.e., to the supporting b-beads. In the native state, the b-beads are located half-way between the p-bead and the center of mass of the h-beads at each nucleotide. The excluded volume diameters of the h-beads, b-beads, and p-beads are taken to be equal to 2.0, 3.4, and 6.0 Å, respectively. The local stiffness of the phosphate chain is represented by harmonic potentials that favor native values of the bond and dihedral angles. All of the interbead interactions within one nucleotide are described by stiff harmonic springs. The stacking interactions between consecutive b-beads in each chain are described by the Lennard–Jones interactions with the minima placed at 4.43 Å apart (on average). As the average value of energy for hydrogen bond interaction in dsDNA in model I we chose 0.6 kcal/mol. In a related single-bead model of the RNA Hyeon and Thirumalai (2005) take ε of about 0.5–0.7 kcal/mol. On average, 2.5 hydrogen bonds are created between the bases in dsDNA. Hence the total average energy of interaction between paired bases in the DNA helix is about 1.5 kcal/mol in our models. This choice is consistent with kB T/ε = 0.4 corresponding to T=300 K. The corresponding unit of the force ε/Å should be then of order 100 pN. The structure-based models should find more extensive applications in the context of many experiments on the DNA that involve manipulation (Cluzel et al. 1996; Bustamante et al. 2003; Koster et al. 2005; Bockelmann et al. 1997, 2002; Bryant et al. 2003; Oroszi et al. 2006; Allemand et al. 2007). As a “warm-up,” we have studied various ways to distort short chains (20 nucleotides in one strand) of the dsDNA within the three models proposed (Niewieczerzal and Cieplak 2009). In the unzipping process, the hydrogen bonds are broken one at a time, by starting from the end at which the unzipping forces are applied. The resulting force, F, ondulates at a single nucleotide resolution as the system unravels. This pattern depends on the nucleotide sequence. Experimentally, the average force, , is of order 14 pN and the amplitude of ondulations is of order 1 pN (Bockelmann et al. 1997, 2002). The force pattern has also been found to depend on the pulling speed weakly (Bockelmann et al. 1997). Our simulations point to a logarithmic dependence on the speed (Niewieczerzal and Cieplak 2009). The force is also a monotonic function of the temperature, since the higher the temperature, the bigger the role of thermal

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fluctuations in unraveling the structure. When one pulls the two strands in opposite ways, then there is a simultaneous shearing of all hydrogen bonds. The resulting force pattern consists of one large force peak, the strength of which depends on the length of the strand (Niewieczerzal and Cieplak 2009). Finally, when one stretches the two strands in the same direction and couples this action with application of a torque one may transform the usual right-handed B form of the dsDNA either to a left-handed form, or to a stretched form, or to a Pauling form. Our coarse-grained simulations of these transitions have been found to be qualitatively consistent with all-atom simulation (Wereszczynski and Andricioaei 2006). The structure-based modeling of molecules is not restricted to biomolecules. Recently, we have developed a coarse-grained model to study interaction of the poly(propylene imine) (PPI) and poly(amidino amine) (PAMAM) dendrimer ink molecules with the self-assembled monolayers of β-cyclodextrins (Thompson 2007; Cieplak and Thompson 2008). Such monolayers are new kinds of molecular printboards used in soft lithography (Schonher et al. 2000; Huskens 2006). An about threefold reduction in the number of the degrees of freedom (as measured by the number of the effective atoms involved) allows for studies of larger areas over which the ink molecules may diffuse and of studies of a set of many molecules. The details of the modeling and further discussion can be found in Cieplak and Thompson (2008). Among the results, we find that anchoring of the ink molecules to the monolayer is of a multi-valent nature and the average valency is found to be controlled by the temperature.

8.4 Examples of Applications of the Structure-Based Models of Proteins 8.4.1 Mechanical Strength of 17,134 Proteins We now illustrate one advantage of the structure-based models: a possibility of performing a rapid comparatory analysis of dynamical properties for thousands of proteins. Specifically, we consider resistance to stretching at constant speed. Only less than a hundred of proteins have been stretched experimentally (see, e.g., Carrion-Vazquez et al. 2009; Galera-Prat et al. 2010) which is just a tiny fraction of structures deposited in the Protein Data Bank. It thus seems worthwhile to explore possible elastic behaviors of all available structures to get an overview and to detect features that may be interesting. In particular, it is useful to predict which of the proteins would be expected to yield large values of Fmax and to explain why they are arising. We have addressed these tasks in several publications. In particular, we have used variant {6 −12, C, M2, E0 } in Sulkowska and Cieplak (2007) to make survey of elastic properties of 7,510 proteins of not more than 150 amino acids (and 239 proteins with larger sizes – up to 851 amino acids). The latest survey (Sikora et al. 2009) involved 17,134 proteins comprising no more than 250 amino acids. It used

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model LJ3 which we find to be more reliable. Structures with gaps in the structure assignment and complexes with the nucleic acids have not been taken into account. The primary objective of the surveys has been to determine theoretical values of Fmax for the terminus-to-terminus stretching. The full rank ordered list of proteins is available at www.ifpan.edu.pl/BSDB/. Another objective has been to correlate the values of Fmax with structural classification schemes such as class, architecture, topology, homology (CATH) (Orengo et al. 1997; Pearl et al. 2005) and structure classification of proteins (SCOP) (Murzin et al. 1995; Andreeva et al. 2008). Even though both schemes are hierarchical in nature they are governed by different principles. CATH uses algorithmic methods and it divides proteins into four classes and then into architectures, topologies, and homologies. SCOP is based on visual inspection and it groups protein species into seven classes (and three quasiclasses) and then into folds, superfamilies, families, and proteins. Both schemes involve sequential information at the lowest stages. We find that large values of Fmax should not arise in α proteins (provided they are single-domained). The architectures that are likely to have member proteins that are particularly resistant to stretching include ribbons, β-barrels, β-sandwiches (like titin), β-rolls (like ubiquitin), and three-layer (aba) sandwiches. The SCOP-based folds which are likely to yield large forces are listed in Sikora et al. (2009). It should be noted that mechanical unfolding is usually unrelated to thermal unfolding (and other kinds of unraveling) and mechanical strength need not go together with thermal stability (Cieplak and Sulkowska 2005). The region of a protein that acts as the most potent source of resistance to pulling is known as a mechanical clamp. It can be identified by inspecting the scenario diagrams that provide information about rupturing events that lead to large force peaks and then by removing sets of corresponding native contacts to see the impact on the value of Fmax (Sulkowska and Cieplak 2007; Sikora et al. 2009). The most common mechanical clamps that are strong involve shearing of many native contacts simultaneously, especially in long parallel β-strands that get additional stabilization from surrounding strands, as illustrated in Fig. 8.4. The top two strongest proteins endowed with this kind of a mechanical clamp have been determined to belong to the PDB codes 1c4p and 1qqr. The corresponding theoretical values of Fmax are 5.1 and 5.0 ε/Å, i.e., they should be within the range of about 550 pN. Among the very strong proteins identified by us there is also scaffoldin c7A (the PDB code 1aoh) with Fmax of 4.3 ε/Å. The experimental value of Fmax for this protein has been recently measured to be 480 pN (Valbuena et al. 2009) which testifies to the predictive value of our approach. In proteins discussed so far, the disulfide bonds, if present, do not play any important dynamical roles. However, we have identified tens of proteins in which such bonds are vital and may even yield Fmax within 1,000 pN range. In fact, the top strongest 13 proteins (Sikora et al. 2009), such as 1bmp (bone morphogenic protein) and 1vpf (vascular endothelial growth factor), all contain a motif corresponding to a cysteine slipknot (Fig. 8.5). One of them forms a knot-loop. This loop is fairly stiff as it is closed by two disulfide bonds. Another loop, the slip-loop, pierces through the knot-loop in the native state. On stretching, the slip-loop may be retracted out

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slip−loop

57

102 knot−loop

57 N C

knot−loop

26 N

beta strands (1c4p)

slip−loop 102

68

61

104 C

cys slipknot (1vpf)

Fig. 8.4 Two kinds of proteinic mechanical clamps that yield large resistance to stretching. The bottom panels indicate the relevant mechanisms in a schematic way. The top panels show the corresponding cartoon representations of proteins in which these mechanisms are operational. The two left panels illustrate the shearing mechanism of two parallel β-strands (shown by two central arrows in the bottom left panel) as exemplified by streptokinase 1c4p. The two right panels illustrate the slipknot mechanism as exemplified by the vascular endothelial growth factor 1vpf

Fig. 8.5 The slipknot conformation in the thymidine kinase 1p6x. The line representation defines the characteristic locations k1 , k2 , and k3

k3 k1 k2

of the knot-loop. We find that this process may often require large forces, of order 10 ε/Å, which are still a factor of 2–4 smaller than those required to rupture covalent bonds (Grandbois et al. 1999). We hope that our theoretical investigation of mechanics of the cysteine slipknot proteins will motivate experimental studies.

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8.4.2 Dynamics of Knots Knot-related topologies, such as cysteine slipknots and cysteine knots, in the native conformations of proteins may be related to the presence of disulfide bonds (Craik et al. 1999, 2001) as discussed in the previous section, but usually they are not. One may refer to such non-cysteinic cases as genuine knots or slipknots. The knots are of the so-called open kind as the mathematically proper definition of a knot requires considering closed lines (loops). In practice, however, there are no difficulties with considering knots formed on open lines when the knots are “deep,” that is, when the entangled region is not close to any of the termini. In fact, this is a situation that is frequently encountered for knotted proteins. Such complex topologies cover only several hundred examples of proteinic structures, which is less than 1% of the deposits contained in the PDB. These structures represent nine distinct folds and come with different biological functions (Mansfield 1994; Taylor, 2000; Taylor and Lin 2003; Virnau et al. 2006; Bolinger et al. 2010). Knots are characterized mathematically by various types of invariants, such as Alexander or Jones polynomials (Livingston 1993). The simplest albeit important example of such an invariant is the crossing number, defined as the minimal number of crossings obtained by smooth transformations of the main chain that do not involve cutting. Most knotted proteins contain the simple trefoil knot for which the crossing number is equal to 3. The most complicated protein knot found so far has the crossing number of 6. Another example of geometrically involved structures is genuine slipknots as discussed by King et al. (2007) and Taylor (2007). A slipknot contains a loop that is partly threaded through another loop. It therefore resembles a knot before it is untied so that upon pulling by both termini the system would eventually acquire a conformation of a straight line. Slipknots are then topologically trivial. Elucidation of the role of knots in the kinetics, thermodynamics, and function of proteins is currently being investigated. Some pioneering experiments include works of Mallam and Jackson (2007), Mallam et al. (2008), Andersson et al. (2009), and Wagner et al. (2005). Nevertheless, the complexity of the structure makes the experimental results difficult to interpret since there are no direct techniques allowing for distinguishing between knotted and unknoted structures. The difficulties compound when trying to asses properties of non-native conformations. These circumstances offer a window of opportunity for theoretical simulations, especially for coarse-grained models since proteins with knots and slipknots often come with large sizes. In order to detect the presence of a knot in a given protein, we apply the KMT algorithm (Koniaris and Muthukumar 1991; Taylor 2000). It involves removing the Cα atoms, one at a time, for as long as the backbone does not intersect a triangle set by the atom under consideration and its two immediate sequential neighbors. If this procedure results in reducing the chain to two nodes, in a series of steps, then the structure is not knotted. Otherwise the structure contains a knot. This algorithm also allows for determination of two end points, k1 and k2 , of the knot along the sequence (Sulkowska et al. 2008b). They are determined by cutting away, one by one, consecutive amino acids from the N terminus and the C terminus, respectively. The

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Fig. 8.6 Knot dynamics in protein 1j85 during stretching. The data are based on the study by Sulkowska et al. (2008). The bottom panel shows the multi-peak dependence of the resistive force on the displacement, d, of the pulling element. The middle panel shows locations of the ends of the trefoil knot. The upper panels illustrate the backbone conformation, or its fragments, at three stages of the stretching process. The stages are indicated by the arrows

sequential position

knot end points correspond to the sequential positions of the amino acids whose removal leads to lack of detection of the knot for the first time. The sequential extension, k2 −k1 , of a knot is a useful characteristic to determine in studies of dynamics of knotted proteins. Describing localization of a slipknot requires considering three sequential positions. One of them, denoted by k3 , is determined by eliminating amino acids consecutively from one terminus until the knotted configuration is reached. The slipknot conformations are not symmetric and k3 can arise either from the N or from the C terminus side. Location k3 is where the slip-loop begins. It ends at k2 . The end of the knot-loop is, simultaneously, a beginning of the slip-loop that pierces through the plane set by the knot-loop. The knot-loop ends at k1 . Locations k1 and k2 are determined by applying the KMT algorithm anew with k3 acting as an effective new terminus in the procedure. Cutting off an amino acid located at k3 reduces a slipknot to a knot. Thus these three points (k1 , k2 , and k3 ) divide the slipknot into two loops: the knot-loop and the slip-loop (Sulkowska et al. 2009b). The KMT algorithm can be applied not only in the native state but also in any other conformation that may arise as a result of mechanical manipulation or during folding. This allows for monitoring the evolution of location of the ends of a knot or of a slipknot during dynamical processes. A natural way of studying the physical properties of the knotted protein is offered by mechanical stretching. Stretching by the protein termini results in knot tightening as studied in Sulkowska et al. (2008b) within our LJ3 model. These studies have been accomplished for 18 proteins including the monomeric methylotransferaze (PDB code 1j85) which is used, as an illustration, in Fig. 8.6. We have found that the

k2 k1

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tightening process results in sudden jumps of the knot ends along the sequence (the middle panel of Fig. 8.6). The jumps take place to selected characteristic sites before arriving at the fully tightened conformation. These sites are typically endowed with a large curvature. These results are in contrast to the well-studied case of knots in homopolymers which tend to diffuse smoothly along the chain (Metzler et al. 2006). In a recent experiment on phytochrome B (Bornschlogl et al. 2009) the tightening process has resulted in reducing final end-to-end distance by 12 amino acids compared to the similar distance that is measured along the backbone. We also note that pulling by some other amino acids may result in untying the knot, as in the experiment described by Alam et al. (2002). Another information about the behavior of knotted proteins can be learned from a simulation in which the protein is first stretched and then released. We found that knots typically do not return to their native locations in such a process (Sulkowska et al. 2008b). Such simulations most likely probe different routes on the free energy landscape compared to chemical or thermal refolding. The force–displacement plot of Fig. 8.7 in itself does not shed light on the microscopic interpretation of the rupture process. An information on what contact is broken at a given displacement is provided by the scenario diagram. The scenario diagram for 1j85, shown in Fig. 8.7, indicates that each force peak comes with an accumulation of rupturing events. The numerical labels used in Fig. 8.7 correspond

knot tightening

Fig. 8.7 The scenario diagram for stretching of 1j85 at constant velocity of 0.005 Å/τ at kB T/ε = 0.3. The y-axis indicates sequential separations of the native contacts. The x-axis indicates pulling distances at which particular contacts get ruptured (rij exceeds 1.5 σ ij for the last time). The data symbols denoted by faint asterisks correspond to contacts which do not involve any secondary structures. The remaining symbols are diversified and they correspond to contacts involving the secondary structures as identified by the numerical labels

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to the secondary structures as counted from the N terminus to the C terminus. Thus 1, 3, and 5 are the β-strands which form a β-sheet. A second β-sheet is formed of strands 7, 8, and 10. This β-sheet forms the core of the knot. The symbols 2, 4, 6, 9, and 11 denote helices. An inspection of the scenario diagram suggests that the unfolding process starts by breaking contacts between the two terminal helices 2 and 11. Afterward, stretching affects contacts in the N-terminal section which involves no knot: contacts 1–8 rupture first, they are followed by contacts 3–5 (without generating a force peak), and then 1–3. These rupturing events are accompanied by breaking down of contacts which do not involve any secondary structures (denoted by the asterisks). The largest force peak is seen to be associated with shearing of the 7–8 and 7–10 contacts which also involves a simultaneous rupture of the hydrogen bonds inside the core of the knot. This rupture makes the knot’s ends jump without affecting the Fmax in a significant manner. We have also studied the influence of knots on other physical properties of proteins (Sulkowska et al. 2009a). Specifically, we considered two proteins with similar structures but different topologies. One protein, 1yh1, contains a knot confined to a location far away from the termini. Another protein, 1c9y, is not knotted. Additionally two other synthetic constructs were made. The first is based on 1yh1 where the knot was removed by changing the way one crossing is made. The second is based on the 1c9y where the cysteine knot was created. This set of structures allows us to show that the presence of the knot affects some kinetic properties. In particular, the knotted structure is found to unravel through a different pathway than the corresponding unknotted structure during a stretching simulation. We have also found that the presence of a knot in our model system enhances their mechanical and thermal stabilities. Furthermore, the unfolding rate for knotted protein has been found to be much lower. In addition, thermal fluctuations may untie the knotted protein. One of the possible ways of unknotting the protein proceeds through the slipknot conformation (Sulkowska et al. 2009a). As generally expected, and demonstrated by Cieplak and Sulkowska (2005) explicitly (see, however, a word of caution by Finkelstein (1997)), the process of thermal unfolding at a high temperature is statistically reverse to folding proceeding under optimal folding conditions. These results motivate us to use the structurebased model to study the folding mechanism of knotted protein. When we consider knots of the simplest type, in principle we can distinguish three possible mechanisms leading to the creation of such a knot. The most straightforward one would require just two steps: creating a loop and threading one end of a protein through it. The second mechanism is more complicated as it involves an intermediate step with a slipknot. The third possibility would involve the creation of an ensemble of loose random knots in the first stage, which may turn into deeper knots as a result of a longer lasting process. In (Sulkowska et al. 2009a) we consider methylotransferaze (PDB code 1od6) and show that knotted protein can fold without involving any attractive non-native contacts that have been considered by Wallin et al. (2007). A similar result has been obtained within a simple and short lattice model with a shallow trefoil knot (Faisca et al. 2010).

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We have found (Sulkowska et al. 2009a) that the topological bottlenecks may be overcome by forming slipknots as necessary intermediate conformations. It is worth noticing that a substantial number of loose randomly knotted structures are observed in our folding studies. Such behavior agrees with the results of simulations by Virnau et al. (2005) and well-known experimental results that flexible polymers or strings (Raymer and Smith 2007) can easily become knotted in a spontaneous way. In most cases, however, random knots observed in folding simulations of proteins do not lead to deep and tighter knots. We should remark that knots in a protein chain at the room temperature do not necessarily behave in the same way as in a polymer chain. For instance, in our stretching simulations (Sulkowska et al. 2008c), the knots wound on homopolymers have been found to diffuse-off the chain usually. Although our findings suggest that the presence of the non-native attractive contacts (Wallin et al. 2007) is not necessary for formation of a knot, these contacts could reduce topological traps such as the wrong sense of making a twist or forming a left-handed conformation instead of the corresponding mirror image. The folding mechanisms of knotted proteins require further experimental and theoretical studies. We note here that these folding processes appear not to involve any external intervention such as provided by, e.g., chaperons (Mallam and Jackson 2007; Andersson et al. 2009).

8.4.3 Proteins in Membranes One needs structure as an input to structure-based modeling. The structure can not only be derived experimentally, but it can also be determined through all-atom simulations. As an example, we consider the case of pulling bacteriorhodopsin out of a membrane (Janovjak et al. 2003). Bacteriorhodopsin is a protein whose crystal structure is known and thus its native contacts can be determined. However, it is also interacting with its phospholipidic environment which holds it in place. The molecular structure of the membrane is not readily available. In order to find effective protein–membrane contacts, we have simulated (Cieplak et al. 2006) the whole systems near its equilibrium to determine the average atomic positions. The system was then coarse grained. In the case of the membrane, the structure was represented by its carbon atoms. We then applied the criterion based on the van der Waals volumes to determine which of these carbon atoms were making contacts with the protein. At this stage, the system is represented by the Cα atoms of the protein and by a cage of frozen C atoms that are within the interaction range from the protein. When one pulls one end of the protein out of the membrane the “walls” would be expected to cave in. Yet this appears to be a dynamically minor effect because we find (Cieplak et al. 2006) the resulting multi-peak force pattern to be remarkably similar to the shape obtained experimentally (though there are problems with the scale of the force). We also can demonstrate the dependence of the pattern on the selection of the terminal amino acids that is being pulled.

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8.4.4 Hydrodynamic Interactions Water solvent is a source of the hydrophobic effect that not only keeps proteins globular but it also mediates a drag force on one amino acid due to the flow induced by motion of another amino acid. The latter effect goes under the name of hydrodynamic interactions (HI) and it enhances cooperative features in the dynamics of proteins, DNA, and simple polymers (see, e.g., Hinczewski et al. 2009). The HI may show in a variety of ways, primarily through emergence of dragging and screening phenomena. The HI are implicit in all-atom simulations involving water molecules, but their role is minuscule in near-equilibrium simulations. The structure-based models come in handy when assessing the role of HI because of the larger time scales available and because implicit solvent models can capture the HI approximately by incorporating the diffusion tensor, Dij , in the description of the system. In the Rotne–Prager form (Rotne and Prager 1969; Yamakawa 1970) of this tensor, the beads of finite hydrodynamic radii are coupled in a way that depends on their separation, r, (as 1/r to the leading order; strictly speaking, the approach is valid at large distances). When dealing with the HI, one can use equations of the Newtonian dynamics (Miller et al. 2008), but it is more straightforward to work within the Brownian dynamics (Ermak and McCammon 1978). The equations of the Brownian dynamics read ri − r0i =

  1  0 0 ∇j · D0ij t + Dij · Fj t + Bi , kB T

where the left-hand side represents displacement of bead i in time step t, the index 0 denotes the values of respective quantities at the beginning of the time step, and Fj is the force exerted on particle j by other particles. B is a random displacement given by distribution with an average value of zero and covariance obeying

a Gaussian Bi Bj = 2D0ij t. We take the diffusion tensor as given by Dii =

kB T I γ

and ⎧    ⎪ 2a2 2a2 ⎪ ⎪ ⎪ 1 + 2 I + 1 − 2 rˆ ij rˆ ij , rij ≥ 2a 3rij rij kB T 3a ⎨ Dij =   ⎪ γ 4rij ⎪ r 8 3rij ij ⎪ ⎪ − I + rij 4aˆrij rˆ ij , rij < 2a ⎩ 2a 3 4a where rij = rj − ri , and a is the hydrodynamic radius. By Stokes’ law, γ = 6π aη, where η is the viscosity of the fluid. The non-diagonal parts of Dij are responsible for the HI effects. The selection of the value of the hydrodynamic radius is not obvious. When one thinks of amino acids as represented by spherical beads

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then one would be inclined to consider a that does not exceed a half of the distance between consecutive Cα s. On the other hand, the residues extend laterally which may give rise to larger drag forces. It has been argued (Garcia de la Torre and Bloomfield 1981; Antosiewicz and Porschke 1989) that a characteristic a of an amino acid could be even as big as 4.2 Å whereas a characteristic van der Waals radius is about 3.0 Å (Zamyatin 1972). The van der Waals volume does not include hydration layers whereas the value of 4.2 Å effectively does so. Such big values would mean existence of an overlap between spheres in a chain, as considered in Banavar et al. (2009), but its usage takes into account the non-spherical properties of the amino acids in an approximate way. However, considering a of 1.5 Å has a numerical advantage because the time unit is governed by the damping constant and is thus proportional to a and hence in most of our HI-related calculations, when we are after qualitative effects, this is the value that we use. It should be noted that characteristic times, like those of folding, scale primarily linearly with a for systems both with and without HI. Therefore comparisons can be made for whatever value of a. We have used this formalism to study several proteins, such as ubiquitin, to establish the following results. The HI lead to (a) reduction in peak unfolding forces when stretching at high constant velocities (Szymczak and Cieplak 2007b; the effect of dragging), (b) reduction in unfolding times when stretching at constant force (Szymczak and Cieplak 2007b; dragging), (c) hindering of unraveling imposed through uniform (Szymczak and Cieplak 2006) and shear (Szymczak and Cieplak 2007c) fluid flows (screening), and (d) acceleration of folding times by a proteindependent factor (Cieplak and Niewieczerzal 2009; dragging) and to faster flap closing dynamics in HIV-1 protease (Cieplak and Niewieczerzal 2009; dragging). Item (d) above is qualitatively consistent with the results obtained for self-attracting homopolymers (Kikuchi et al. 2005) and several proteins (Ryder 2005) within the stochastic rotation dynamics approach (Malevanets and Kapral 1999) that mimics the solvent in the spirit of cellular automata. The acceleration of folding is observed to be confined to the collapse stage of the process and we find it to be by a factor of order 2 and which depends on the protein studied (Cieplak and Niewieczerzal 2009). The value of the factor depends on the nature of the initial unfolded state – in our case it is the full extension – whereas Ryder (Ryder 2005) estimates it to be smaller. Our results on folding times are quite similar to those obtained recently by Frembgen-Kesner and Elcock (2009) on 11 proteins studied within structure-based model developed by Clementi et al. (2000).

8.4.5 Nanoindentation of Virus Capsids Capsids are conglomerates of proteins that protect viral genomes. The RNAenveloping capsids are quite resilient mechanically (though not as much as those which envelope the DNA material) as probed recently through nanoindentation measurements (Michel et al. 2006). One especially well-studied RNA-related capsid belongs to the cowpea chlorotic mottle virus (CCMV). The capsids can be made

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devoid of their genetic content which may remove some of the resiliency. In particular, it has been found (Michel et al. 2006) that an emptied CCMV capsid displays an initial Hookean reversible regime corresponding to the spring constant of 0.14 N/m that persists for up to 20% reduction in the diameter where the maximum force of resistance reaches about 500 pN (Michel et al. 2006; Klug et al. 2006). The elastic regime ends in a collapse that involves buckling. The same CCMV capsid with the RNA in it would reach about 1,000 pN before collapsing. These forces are noticeably larger than those needed for a mechanical unraveling of proteins. As an example, unraveling titin requires a force of about 204 pN (Rief et al. 1997; Carrion-Vazquez et al. 1999). The sturdiness of the capsids comes from the mechanical properties of the individual proteins and from reinforcements provided by the geometry of their assembly in which any single protein is stabilized by its neighbors. CCMV capsids consist of 180 sequentially identical proteins comprising 190 amino acids each in a β-barrel fold. They have the fullerene-like truncated icosahedral symmetry with 12 pentagonal and 20 hexagonal faces. This regular nature of the geometry has suggested to model the capsid as a homogeneous shell and use the finite element method to determine its mechanical properties by studying the resulting curved elastic network (Vliegenthart and Gompper 2006; Buenemann and Lenz 2007; Gibbons and Klug 2008). The plots of the force of resistance versus the degree of nanoindentation obtained in this way agree qualitatively with the experimental findings. It should be noted, however, that the thickness of a capsid is non-uniform since the “faces” themselves actually dissect three-dimensional molecular complexes which are far from being homogeneous. In the CCMV case the Cα atoms extend between ∼97 and 143 Å from the capsid center. Most of the theoretical studies of molecular level aspects of the capsid mechanics have been restricted to the properties of the native state. Such studies include an all-atom-based analysis of the low frequency vibrations in the satellite tobacco necrosis virus (Dykeman and Sankey 2008), elastic network model study of eigenmodes in the HK97 virus (Kim et al. 2003), and a similar model for CCMV (Tama and Brooks 2002, 2004). However, there are few molecular level studies of the nanoindentation processes. One exception is the recent all-atom study by Zink and Grubmueller (Zink and Grubmueller 2009) on the southern bean mosaic virus which, however, used extreme nanoindentation rates and emulated “nanoindentation” by pressing one amino acid into the center of the virus capsid. We have recently performed the structure-based molecular dynamics study of CCMV capsid (Cieplak and Robbins 2010) and demonstrated existence of deviations from the elastic homogeneous shell model. This indicates relevance of the molecular level effects for the mechanostability of the capsid. We have also showed that past the elastic regime, the nanoindentation process is strongly history dependent and hysteretic. The locations of some amino acids have not been determined through the X-ray crystallography (Speir et al. 1995; Carrillo-Tripp et al. 2009) and our model makes use only of the known parts of the structure. Thus the coarse-grained model of the whole capsid involves 28,620 Cα atoms and 62,426 native contacts. (It also involves of order 5 × 108 repulsive non-native contacts which makes it necessary to deal with

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their effects by determining a Verlet list (Allen and Tildesley 1987) periodically). Nanoindentation is implemented by placing the capsid between two parallel plates that generate the h-10 repulsive potentials, where h is the distance away from the plate. In the initial state, the plates do not touch the capsid and then they both move toward each other with a relative speed of (in most cases) 0.005 Å/τ . Figure 8.8 shows example of a nanoindentation trajectory for CCMV capsid. As the separation, s, decreases, the dependence of F on s is basically linear for the first ∼35% of the process as measured by the values of s relative to the separation corresponding to the onset of the force of resistance. In this linear regime the nanoindentation process is nearly reversible: an inversion of the direction of motion of the plates results in a force which is dropping down linearly in a way that is tracking its previous growth. At the end of this Hookean regime the force acquires a value Fm ∼ 5.5 ε/Å, which should be of order 600 pN. The resulting spring constant is close to 0.05 ε/Å2 which should correspond to about 5.5 pN/Å which is softer than the experimental value due to the usage of the Lennard–Jones potential which is fairly broad. At this stage, a collapse follows resulting in the force dropping to

Fig. 8.8 Nanoindentation of CCMV capsid at kB T/ε = 0.3 after (Cieplak and Robbins 2010). The solid line in the main figure shows the force of resistance to indentation against the separation between the compressing planes. The force is averaged over time intervals and between the planes. The dotted line shows the force when the planes are moving back once s reaches a value sc (here sc is 100 Å as an illustration) which is past the Hookean regime. This dotted line is seen not to retrace the compression curve. However, there is a reversibility if the motion back is confined to the linear elastic region (s > 180 Å). The figures at the top show the schematic representation of the shapes of the capsid at the three values of s as indicated. The shapes of the n-gons and the locations of the vertices are obtained through studies of the Cα mass distribution for proteins belonging to the particular n-gons. At s = 284 Å, the capsid is in its native state

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about 3 ε/Å. The collapse results primarily from the breakage of the interproteinic contacts. Finally, F starts to rise indefinitely due to compactification of the hard cores representing the amino acids. After departing the linear regime no process is reversible. Further discussion together with the detailed comparison to the elastic shell model is presented in Cieplak and Robbins (2010). We just note here that the biggest deviations are observed in the strain field, i.e., in the shape of the deformed capsid. In summary, the structure-based models appear to perform well in many relevant applications, especially when one seeks qualitative and comparatory goals instead of chemically precise values. The models have been developed to tackle problems of large conformational changes, large time scales, and large sizes. In this last case, they can also be of interest when studying equilibrium fluctuations. For instance, our results on the single-bead and two-bead fluctuation characteristics of the large human topoisomerase I protein (Szklarczyk et al. 2009) are consistent with all-atom simulations (Chillemi et al. 2007) but have been obtained with a much smaller computational effort. Acknowledgments This work summarizes a decade of research performed with a group of collaborators. In particular with J.R. Banavar, M. Carrion-Vazquez, S. Filipek, T.X. Hoang, H. Janovjak, A. Maritan, P. Marszałek, S. Niewieczerzał, A. Pastore, M.O. Robbins, M. Sikora, K. Staro´n, P. Sułkowski, O. Szklarczyk, P. Szymczak, D. Thompson, and M. Wojciechowski. We appreciate S. Niewieczerzał’s help in preparation of the first three figures displayed here. This work has been supported by the grant N N202 0852 33 from the Ministry of Science and Higher Education in Poland, by the EC FUNMOL project under FP7-NMP-2007-SMALL-1, and by the European Union within European Regional Development Fund, through grant Innovative Economy (POIG.01.01.02-00-008/08).

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Szymczak P, Cieplak M (2007b) Influence of hydrodynamic interactions on mechanical unfolding of proteins. J Phys: Condens Matter 19:285224 Szymczak P, Cieplak M (2007c) Proteins in a shear flow. J Chem Phys 127:155106 Takada S (1999) Go-ing for the prediction of protein folding mechanism. Proc Natl Acad Sci USA 96:11698–11700 Tama F, Brooks III CL (2002) The mechanism and pathway of pH induced swelling in cowpea chlorotic mottle virus. J Mol Biol 318:733–747 Tama F, Brooks III CL (2004) Diversity and identity of mechanical properties of icosahedral viral capsids studied with elastic network normal mode analysis. J Mol Biol 345: 299–314 Taylor WR (2000) A deeply knotted protein structure and how it might fold. Nature 406:916–919 Taylor WR, Lin K (2003) Protein knots – a tangled problem. Nature 421:25–25 Taylor WR (2007) Protein knots and fold complexity: some new twists. Comp Biol and Chem 31:151–162 Thompson D (2007) Free energy balance predicates dendrimer binding multivalency at molecular printboards. Langmuir 23:8441–8451 Tozzini V, Trylska J, Chang C, McCammon JA (2007) Flap opening dynamics in HIV-1 protease explored with a coarse-grained model. J Struct Biol 157:606–615 Tsai J, Taylor R, Chothia, C Gerstein M (1999) The packing density in proteins: standard radii and volumes. J Mol Biol 290:253–266 Valbuena A, Oroz J, Hervas R, Vera AM, Rodriguez D. Mendez M, Sulkowska JI, Cieplak M, Carrion-Vazquez M (2009) On the remarkable mechanostability of scaffoldings and the mechanical clamp motif. Proc Natl Acad Sci USA 106:13791–13796 Veitshans T, Klimov D, Thirumalai D (1997) Protein folding kinetics: time scales, pathways and energy landscapes in terms of sequence-dependent properties. Fold Des 2:1–22 Virnau P, Kantor Y, Kardar M (2005) Knots in globule and Coll phases of a model polyethylene. J Am Chem Soc 127:15102 Virnau P, Mirny LA, Kardar M (2006) Intricate knots in proteins: function and evolution. PLoS Comp Biol 2:1074–1079 Vliegenthart GA, Gompper G (2006) Mechanical deformation of spherical viruses with icosahedral symmetry. Biophys J 91:834–839 Wagner JR, Brunzelle JS, Forest KT, Vierstra RD (2005) A light-sensing knot revealed by the structure of the chromophore-binding domain of phytochrome. Nature 438:325–331 Wallin S, Zeldovich KB, Shakhnovich EI (2007) The folding mechanics of a knotted protein. J Mol Biol 368:884–893 Wereszczynski J, Andricioaei I (2006) On structural transitions, thermodynamic equilibrium, and the phase diagram of DNA and RNA duplexes under torque and tension, Proc Natl Acad Sci USA 103:16200–16205 Wojciechowski M, Cieplak M (2007) Coarse-grained modelling of pressure related effects in staphylococcal nuclease and ubiquitin. J Phys Cond Matter 19:285218 Yamakawa H (1970) Transport properties of polymer chains in dilute solutions. Hydrodynamic interaction. J Chem Phys 53:436–443 Yang G, Cecconi C, Baase, WA, Vetter IR, Breyer WA, Haack JA, Matthews BW, Dahlquist FW, Bustamante C (2000) Solid-state synthesis and mechanical unfolding of polymers of T4 lysozyme. Proc Natl Acad Sci USA 97:139–144 Zamyatin AA (1972) Protein volume in solution. Prog Biophys Mol Biol 24:107–123 Zink M, Grubmueller H (2009) Mechanical properties of the icosahedral shell of southern bean mosaic virus: a molecular dynamics study. Biophys J 96:1350–1363

Chapter 9

Sampling Protein Energy Landscapes – The Quest for Efficient Algorithms Ulrich H. E. Hansmann

Abstract Computer simulations aim to become virtual microscopes that can probe the working of cells on a molecular level. One of the remaining obstacles is still poor sampling. This chapter reviews strategies for faster sampling and discusses their limitations. Recent applications to protein folding document the utility of the described techniques.

9.1 Introduction The goal of computational biophysics is to use molecular dynamics or Monte Carlo simulations as a “virtual microscope” to study processes or molecules in a cell that are not directly accessible in experiments. Already for single, non-interacting globular proteins the numerical difficulties are enormous. This is because the complex form of the intramolecular forces and the interaction with the surrounding environment, containing both repulsive and attractive terms, lead to a rough energy landscape with a huge number of local minima. Simple canonical Monte Carlo or molecular dynamics get trapped in a local minimum and, when relying on an all-atom model, normally do not thermalize within the available CPU time. Even simulations of “mini-proteins” with less than 50 residues are computationally challenging. A common estimate for a typical single-domain protein such as the 153 amino acid long myoglobin is that a single-folding trajectory of ≈10−4 s would take about 3 years on a supercomputer capable of trillions of floating point operations per second (Allen et al. 2001). This is because the computational effort to calculate accurately the physical quantities increases exponentially with the number of residues. Note that the problem is not necessarily solved using coarse-grained models. While current all-atom energy functions may introduce additional local minima U.H.E. Hansmann (B) Department of Physics, Michigan Technological University, Houghton, MI, USA e-mail: [email protected]

A. Kolinski (ed.), Multiscale Approaches to Protein Modeling, C Springer Science+Business Media, LLC 2011 DOI 10.1007/978-1-4419-6889-0_9, 

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in the energy landscape, leading to an additional decrease in sampling speed, roughness is an intrinsic characteristic of protein free-energy landscapes. A valid coarse-grained protein model has to lead to a rough energy landscape, and therefore will suffer still from an exponential increase of computational effort with size of the molecule. The numerical advantage of a coarse-grain model is one of a smaller prefactor (a much faster evaluation of energy), and for a sufficiently large proteins the problem of poor sampling and slow convergence of the simulation will re-appear. Hence, independent on the utilized energy functions and detail of description of interactions within a protein, and between a protein and its surrounding environment, there is a need for techniques that speed up the sampling of low-energy configurations. The task is to find local minima but avoid to get trapped, so that the simulation can continue to explore the protein energy landscape. It is obvious that any such approach will change the dynamics, i.e., in most cases one can no longer follow directly the trajectories by which a protein evolves. However, in many cases it is possible to derive enhanced sampling techniques that guarantee correct thermodynamics. We will focus in the following on techniques where this is the case, i.e., where at least in principle it is possible to calculate correct thermal averages. Because of their correct sampling properties, these methods will also allow to reconstruct the free-energy landscape (or its projection on a set of reaction coordinates) and as a consequence allow for a study of the relevant transitions and mechanism of folding, assembly, and aggregation of a protein. This chapter is organized as follows: we start with a short review of the basic simulation techniques and common global optimization methods. We will then introduce a number of advanced simulation techniques before discussing shortcomings and open problems. A summary and outlook finish this short review.

9.2 Basic Simulation Techniques 9.2.1 Molecular Dynamics In principle, one can follow the time evolution of a protein, its folding, assembly, and interaction with other molecules, through solving numerically Newton’s law for each atom i:

Fi (x1 (t), ...xn (t), t) = mi

d2 xi . dt2

(9.1)

Here, xi (t) describes the coordinates of atom i at time t (i.e., it is a vector). Correspondingly, vi (t) is the associated velocity of atom i at time t. Various schemes exist to integrate numerically these equations of motion, with the most simplest ones being the Verlet algorithm

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xi (t + t) = 2xi (t) − xi (t − t) + vi (t) =

Fi (t) 2 t mi

xi (t + t) − xi (t − t) 2t

211

(9.2) (9.3)

and the leapfrog algorithm xi (t + t) = xi (t) + tvi (t + t/2) vi (t + t/2) = vi (t − t/2) + t

Fi (t) . mi

(9.4) (9.5)

Besides probing directly the kinetics of proteins such molecular dynamics simulations also allow one to calculate equilibrium properties by computing time averages. Solving the equations of motions requires knowledge (i.e., calculation) of the forces Fi at time t. In most cases, these forces will not depend explicitly on time, and the total energy E = Epot (xi ) + Ekin (vi )

with Ekin =

1 mi v2i 2

(9.6)

is conserved and the forces can be written as Fi (x1 (t), ..., xn (t) =

∂ Epot (x1 , ..., xn ). ∂xi

(9.7)

Hence, molecular dynamics requires both knowledge of the potentials and their derivates. As the total energy E = const, a molecular dynamics simulation of the above kind will realize a microcanonical ensemble. However, most experimental settings realize a canonical ensemble (i.e., not the energy E but the temperature T is conserved). This requires coupling of the system to a thermostat, described, for instance, in Frenkel and Smit (2001). Note that temperature and kinetic energy of the system are related by < Ekin (t) >=<

1 N mi v2i (t) >= kB T 2 2

(9.8)

where <> denotes a time average, kB is the Boltzmann constant, N is the number of degrees of freedom, and T is the temperature of the system.

9.2.2 Monte Carlo A canonical ensemble at a temperature T is easily realized in a Monte Carlo simulation than with molecular dynamics. This is because in Monte Carlo, trial moves

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are generated randomly and accepted or rejected according to the Boltzmann weight exp(−E/kB T). For instance, in the Metropolis algorithm one calculates w(c →

c )

" =

1 E−E/kB T

E < 0 otherwise

(9.9)

with E = E(c ) − E(c), and E(c) the energy of the current configuration, and E(c ) that of the trial configuration. The trial configuration replaces the current one if w(c → c ) ≥ R, where the random number R takes values between 0 and 1. As the Metropolis algorithm ensures detailed balance and each configuration can be reached in a finite number of steps (ergodicity), the resulting Markov process will converge to the distribution of protein configurations that corresponds to the canonical ensemble at temperature T. Thermodynamic quantities < O > are now calculated by computing averages over the sampled conformations: < O >=

1  Ok M

(9.10)

where Ok is the value measured for the quantity O in the configuration k, and M the number of measurements. This average approximates the ensemble average  < O >=

dxi dvi O(xi )e−E(xi ,vi )/kB T  . dxi dvi e−E(xi ,vi )/kB T

As E = Epot (xi ) + Ekin (vi ) and Ekin = 1/2 velocities, and  < O >=



(9.11)

mi v2i , it is possible to integrate out the

dxi O(xi )e−E(pot xi )/kB T  . dxi e−Epot (xi )/kB T

(9.12)

As a consequence, for the generation of configurations by way of the Metropolis algorithms (Eq. (9.9)) one needs to calculate only the difference of the potential energies Epot . For this reason, we will write most times simply E when only the potential energy Epot is relevant. Note also that Monte Carlo does not require calculation of derivatives reducing the numerical workload. As the configurations are drawn randomly in Monte Carlo, it is not possible to follow the trajectory of a protein, and therefore Monte Carlo – unlike molecular dynamics – is not suitable for probing the kinetics of folding. On the other hand, Monte Carlo allows one to sample the configurational space much faster through utilizing artificial but fast move sets. These are often necessary because in the canonical ensemble crossing of an energy barrier of height E is suppressed by a factor ∝ exp(−E/kB T). This is the reason for the multiple minima problem and the resulting slowing down of protein simulations discussed in the introduction.

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9.2.3 Optimization Techniques Most proteins are thermodynamically stable at room temperature (Anfinsen 1973). This implies that the biologically active configuration is the global minimum in free energy at T ≈ 270 − 300 K. For many proteins, this state is unique up to oscillations around a fixed structure. For this reason, one can identify the global minimum in free energy with that in potential energy, reducing the prediction of protein structures to a global optimization problem. While deterministic methods (for instance, the αBB algorithm (Androulakis et al. 1997)) have many conceptual advantages, stochastic algorithms are often faster and easier to implement. Take as an example simulated annealing (Kirkpatrick et al. 1983) which is inspired by the crystal growth process and realized by gradually decreasing the temperature in a Monte Carlo or molecular dynamics program. While only a logarithmic annealing schedule will ensure that the simulation finds the global minimum (Geman and Geman 1984), limitations in available computer resources require faster annealing schedules where success is no longer guaranteed. Still, because of its simplicity simulated annealing is often the first choice in protein optimization problems. Genetic algorithms (Holland 1975) and Monte Carlo minimization (Li and Scheraga 1987) are two other stochastic optimization techniques commonly used. As simulated annealing they try to avoid entrapment in local minima and continue to search for further solutions. This is a general characteristic of successful optimization techniques. For instance, in tabu search (Cvijovic and Klinowski 1995) the system is guided away from previously explored areas. This can result in slow convergence as the method does not distinguish between important and unimportant regions of the landscape. A somehow opposite approach (Besold et al. 1999; Wenzel and Hamacher 1999) aims at transforming the original energy landscape in a funnellandscape, where convergence toward the global minimum is fast. However, many landscape-deformation methods are hampered either by the required fine tuning or a priori information, or by difficulties with connecting back to the original landscape. Often, minima on the deformed surface are displaced or merged. The latter problem is avoided in energy landscape paving (ELP) (Hansmann and Wille 2002) which merges ideas from tabu search with energy landscape deformation. In ELP, low-temperature Monte Carlo simulations utilize an effective energy: ) w() E) = e−E/kB T

with ) E = E + f (H(q, t)).

(9.13)

Here, T is a (low) temperature and f (H(q, t)) a function of the histogram H(q, t) in a pre-chosen “order parameter” or “reaction coordinate” q. The weight of a local minimum state decreases with the time the system stays in that state, i.e., ELP deforms the energy landscape locally till the local minimum is no longer favored and the system will explore higher energies. It will then either fall in a new local minimum or walk through this high-energy region till the corresponding histogram entries all have similar frequencies and the system again has a bias toward low

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energies. Since the weight factor is time dependent it follows that ELP violates detailed balance. Hence, the method cannot be used to calculate thermodynamic averages. Note, however, that for f (H(q, t)) = f (H(q)) detailed balance is fulfilled, and ELP reduces to the generalized-ensemble methods (Hansmann and Okamoto 1998) discussed in the following section. We have evaluated the efficiency of ELP in simulations of the 20-residue trp-cage protein whose structure we could “predict” within a root-mean-square deviation (rmsd) of 1 Å (Schug et al. 2005). Energy landscape paving allows also the possibility of zero-temperature simulations (Schug et al. 2005). For T → 0 only moves with ) E ≤ 0 will be accepted. If one chooses: ) E = E + cH(E, t), the acceptance criterion is given by: E + cH(q, t) ≤ 0 ↔ cH(q, t) ≤ −E

(9.14)

where E is the “physical” energy. Hence, energy landscape paving can overcome even at T = 0 any energy barrier. The waiting time for such a move is proportional to the height of the barrier that needs to be crossed. The factor c sets the timescale, and in this sense the T = 0 form of ELP is parameter-free.

9.3 Advanced Simulation Techniques Determining the structure of proteins through global optimization assumes the existence of a cost function whose global minimum describes the native structure. In most cases this is an energy that describes the physical interactions within a protein and between the protein and the surrounding environment, in most cases water. Since neither the available force fields nor the inclusion of solvation effects are perfect, it is not certain that the folded structure (as determined by X-ray or NMR experiments) corresponds to the global minimum conformation. Hence, the accuracy of the force fields sets a limit on any global optimization approach to structure prediction of proteins. Global optimization techniques are also not suitable for investigations of the folding mechanism, the change in shape when interacting with other molecules, or the appearance of mis-folded structures. As with structure prediction, it is necessary to go beyond global optimization techniques and to measure thermodynamic quantities, i.e., to sample a set of configurations from a canonical ensemble and take an average of the chosen quantity over this ensemble. In principle, this is possible with molecular dynamics and Monte Carlo simulations, however, but as argued earlier in this review this requires strategies that lead to a faster sampling of low-energy configurations.

9.3.1 Unfolding Simulations The poor sampling of protein configurations at physiologically relevant temperatures results from their rough energy landscape where barriers of height E are

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suppressed by e−E/kB T . Hence, by increasing the temperature T it becomes easier for a protein to cross energy barriers. This can be used to induce the thermal unfolding of a protein. Such unfolding simulations at high temperature are interpreted sometimes as reversed-in-time folding (Daggett and Fersht 2003; Daggett 2002). This approach has been used in the past with some success (Daggett and Fersht 2003; Daggett 2002), but it is not clear whether in general it is justified in protein simulations. We have recently demonstrated that the C-fragment of Top 7, named by us as CFr, folds by a non-trivial pathway that involves caching of an N-terminal segment in an adjunct helix. Only when all other parts of the proteins are folded and in place, the N-terminal segment unfolds and re-folds to a strand that completes the final structure in a three-stranded sheet. We found that this folding mechanism cannot be interfered from unfolding simulations at high temperatures. In fact, the interpretation of unfolding data in Mohanty and Hansmann (2008) as folding in reversed time would miss the caching mechanism that governs folding of this protein. Likely, such an interpretation is restricted to simple two-state folder and associated with a nucleation mechanism, as observed, for instance, for CI2 (Daggett and Fersht 2003; Daggett 2002).

9.3.2 Advanced Updates A possible strategy to increase sampling of relevant protein configurations are improved updates. Within the context of molecular dynamics these are techniques that either guide the simulation and/or allow for larger time steps in the integrator. In the context of Monte Carlo these are usually collective moves that lead to a larger change in configurations. Examples are the re-bridging scheme (G¯o and Scheraga 1970; Wu and Deem 1999) and the biased Gaussian step method (Favrin et al. 2001). In hybrid Monte Carlo (Duane et al. 1987; Brass et al. 1993) a short molecular dynamics run is used as a collective move to provide a trial configuration, which is then accepted or rejected according to the Metropolis criterion. This allows to follow a trajectory over a long time with a large step size, because the Metropolis step corrects for the discretization errors in the molecular dynamics run. A general problem with all improved updates is that they depend strongly on the chosen model and are often not known a priori. A collective move that avoids this pitfall has been recently proposed by Berg (Berg 2003) under the name Rugged Metropolis (RM). The idea is to bias a Monte Carlo simulation by using informations from a simulation at a higher temperature. Assume a range of temperatures T1 > T2 > . . . > Tr > . . . > Tf −1 > Tf .

(9.15)

The simulation at the highest temperature, T1 , is performed with the usual Metropolis algorithm and the results are used to construct an estimator of the probability density function

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ρ(x1 , . . . , xn ; T1 ). that biases the simulation at T2 . In turn, this simulation provides a bias for the one at T3 and iteratively continued down to Tf . For this purpose, Berg assumes the approximation ρ(x1 , . . . , xn ; Tr ) =

n !

ρ 1i (xi ; Tr ),

(9.16)

i=1

where ρ 1i (xi ; Tr ) are estimators of reduced one-variable probability densities ρi1 (xi ; T) =

 !

dxj ρ(x1 , . . . , xn ; T) .

(9.17)

j=i

Recursively, the estimated probability density function ρ(x ¯ 1 , . . . , xn ; Tr−1 ) is generated as an approximation of ρ(x1 , . . . , xn ; Tr ). The acceptance step in the (biased) Metropolis procedure at temperature Tr is now given by *

PRM

+   exp −β E ρ(x1 , . . . , xn ; Tr−1 ) = min 1, exp (−β E) ρ(x 1 , . . . , x n ; Tr−1 )

(9.18)

Rugged Metropolis has been tested successfully for simulations of small peptides, however, as with other improved updates, by itself the gain in efficiency is not enough to make folding simulations of protein domains (consisting usually of 50–200 residues) feasible. On the other hand, improved updates are very useful when combined with the other techniques that we describe in the following subsections.

9.3.3 Generalized-Ensemble Techniques A very successful approach for improving the sampling of low-energy protein configurations is the generalized-ensemble approach. Its underlying idea is not to sample directly the canonical ensemble but an artificial ensemble tailored to enable efficient search for local minima while at same time avoiding entrapment. These generalized ensembles are defined in such a way that re-weighting techniques allow one to connect back to the canonical (i.e., physical) ensemble and to calculate thermodynamic averages at temperatures of interest (Hansmann 2003). A great number of such ensembles have been proposed, and while not all of them can be discussed in this review, we can classify them in principle according to whether they are generated by a random walk through order parameter space (for instance, energy),

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control parameter space (temperature), or through model space (i.e., different energy functions). 9.3.3.1 Random Walks in Order Parameter Space In generalized ensembles that are defined by random walks in order parameter space, one requires that a Monte Carlo or molecular dynamics simulation leads to a broad distribution of a pre-chosen physical quantity. This allows one to sample both low and high-energy states with sufficient probability. For simplicity we will consider only ensembles that lead to flat distributions in one variable. Extensions to higher dimensional generalized ensembles are straightforward (Kumar et al. 1996). Probably the earliest realization of this idea is umbrella sampling (Torrie and Valleau 1977), but now more common is multicanonical sampling (Berg and Neuhaus 1991). Its first application of these techniques to protein simulations can be found in Hansmann and Okamoto (1993) where a Monte Carlo technique was used. Later, it was also adapted to molecular dynamics (Hansmann et al. 1996). The idea is to assign configurations with energy E a weight w(E) such that the distribution of energies Pmu (E) ∝ n(E)wmu (E) = const,

(9.19)

where n(E) is the spectral density. Since all energies appear with the equal probability, a free random walk in the energy space is enforced: the simulation can overcome any energy barrier and will not get trapped in one of the many local minima. For a wide range of temperatures it is now possible to obtain a canonical distribution by the re-weighting techniques (Ferrenberg and Swendsen 1988): −βE , PB (T, E) ∝ Pmu (E)w−1 mu (E)e

(9.20)

since a large range of energies is sampled. This allows one to calculate the expectation value of any physical quantity O at temperature T by  < O >T =

dEO(E)PB (T, E)  . dEPB (T, E)

(9.21)

The price for the resulting improved sampling is that (unlike in the canonical ensemble) the weights wmu (E) ∝ n−1 (E) are not a priori known (in fact, knowledge of the exact weights is equivalent to obtaining the density of states n(E), i.e., solving the system) and one needs their estimates for a numerical simulation. Calculation of the weights is usually done by an iterative procedure (Berg 2004; Hansmann and Okamoto 1993, 1994). Another efficient recursion is the so-called Wang–Landau sampling (Wang and Landau 2001) where one performs updates with estimators n(E) of the density of states   p(E1 → E2 ) = min n(E1 )/n(E2 ), 1 .

(9.22)

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Each time an energy level is visited, the estimator is updated according to n(E) → n(E) f

(9.23)

where, initially, n(E) = 1 and f = f0 = e1 . Once the desired energy range is covered, the factor f is refined, f1 =

, , f , fn+1 = fn+1 ,

(9.24)

until some small value is reached. In multicanonical simulations the computational effort increases with the number of residues like ≈ N 4 (when measured in Metropolis updates) (Hansmann and Okamoto 1999b). In general, the computational effort in simulations increases with ≈ X 2 where X is the variable in which one wants a flat distribution. This is because generalized-ensemble simulations realize by construction of the ensemble a 1D random walk in the chosen quantity X. In the multicanonical algorithm the reaction coordinate X is the potential energy X = E. Since E ∝ N 2 the above scaling relation for the computational effort ≈ N 4 is recovered. Hence, multicanonical sampling is not always the optimal generalized-ensemble algorithm in protein simulations. A better scaling of the computer time with size of the molecule may be obtained by choosing more appropriate reaction coordinate for our ensemble than the energy. This is the motivation behind the various other realizations of the generalizedensemble approach that exist. All aim at sampling a broad range of energies. In this way the simulation will overcome energy barriers and allow escape from local minima. For instance, in Hansmann and Okamoto (1999a) it was proposed that configurations are updated according to a special choice of the Tsallis generalized mechanics formalism (Curado and Tsallis 1994) (the Tsallis parameter q is chosen as q = 1 + 1/nF ):   β(E − E0 ) −nF w(E) = 1 + . nF

(9.25)

Here E0 is an estimator for the ground-state energy and nF is the number of degrees of freedom of the system. The weight reduces in the low-energy region to the canonical Boltzmann weight exp(−βE). This is because E − E0 → 0 for β → 0 leading to β(E − E0 )/nF << 1. On the other hand, high-energy regions are no longer exponentially suppressed but only according to a power law, which enhances excursions to high-energy regions. In stochastic tunneling (Wenzel and Hamacher 1999)), conformations enter with a weight w(E) = exp(f (E)/kB T). Here, f(E) is a non-linear transformation of the potential energy onto the interval [0,1] and T is a low temperature. The physical idea behind such an approach is to allow the system to “tunnel” through energy barriers in the potential energy surface (Wenzel and Hamacher 1999). Such a transformation can be realized by

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f (E) = e−(E−E0 )/nF ,

219

(9.26)

where E0 is again an estimate for the ground state and nF the number of degrees of freedom of the system. Note that the location of all minima is preserved. Hence, at a given low temperature T, the simulation can pass through energy barriers of arbitrary height, while the low-energy region is still well resolved. The efficiency of this algorithm for protein-folding simulations was demonstrated in Hansmann (1999). In both ensembles a broad range of energies is sampled. Hence, one can use again re-weighting techniques (Ferrenberg and Swendsen 1988) to calculate thermodynamic quantities over a large range of temperatures. In contrast to other generalized-ensemble techniques the weights are explicitly given for both new ensembles. One needs only to find an estimator for the ground-state energy E0 which is easier than the determination of weights for other generalized ensembles. In the context of molecular dynamics the generalized-ensemble idea is utilized in the metadynamics method where Gaussian-shaped repulsive potentials Ubias (s, t) =

 ti

 h exp −

|s − s(ti )|2 2w2

are added iteratively to the energy function. These potentials are centered at updated points s(ti ) of the reaction coordinates in order to discourage the system from revisiting the configurations (Laio and Parrinello 2002). The overall contribution from these auxiliary potentials flattens the underlying curvatures of the free-energy wells, therefore leading to a random walk. The original free-energy potentials are recovered by −Ubias (s, t). Another variant is simulated scaling where one assumes a system with potential U0 = Us + Ue ,. The potential Us represents the energy terms determining local conformations in a region of interest and Ue the remaining environmental energy terms. In order to accelerate sampling, one builds an expanded ensemble with the scaled potential U = λm Us + Ue and re-writes the scaled energy function in the dual-topology hybrid potential form that is usually utilized in free-energy simulations, U = (1 − λm ) UsA (x) + λm UsB (x ) + Ue .

(9.27)

Here, UsA (x) and UsB (x ) represent the unique portions of the energy terms for the two end-point chemical species A and B. When the λm histogram is flattened, the free energy difference between any two λm states is calculated according to  A (λ0 → λ1 ) = −RT [a(λ1 ) − a(λ0 )] = −RTln

f (λ1 ) . f (λ0 )

(9.28)

Here, a(λm ) and f (λm ) represent the weight function and biasing function values.

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Metadynamics-based methods are designed to enhance the crossing of energy barriers by flattening the energy surface. This has the undesired side effect of enlarging the conformation space that needs to be searched in a diffusive motion. As a consequence, the discovery of low-energy configurations becomes a rare event. This diffusion problem decreases sampling efficiency and it increases rapidly with the size of the molecule. This diffusion problem impedes also variants of the generalized-ensemble approach that aim at flat distributions in two or more dimensions. 9.3.3.2 Random Walks in Control Parameter Space The most appropriate generalized ensemble for a protein simulation is not necessarily the one generated by a random walk in a certain order parameter or reaction coordinate. Instead, it is often more efficient to enforce a random walk in a control parameter, for instance, temperature. One often used example is simulated tempering (Lyubartsev et al. 1992; Marinari and Parisi 1992) where the temperature itself becomes a dynamic variable and is sampled uniformly. Temperature and configuration are both updated with a weight: wST (T, E) = e−E/kB T−g(T) .

(9.29)

Here, the function g(T) is chosen so that the probability distribution of temperature is given by  (9.30) PST (T) = dE n(E)e−E/kB T−g(T) = const. Physical quantities have to be sampled for each temperature point separately and expectation values at intermediate temperatures are calculated by re-weighting techniques (Ferrenberg and Swendsen 1988). As common in generalized-ensemble simulations, the weight wST (T, E) is not a priori known (since it requires knowledge of the parameters g(T)) and their estimator has to be calculated. They can be again obtained by an iterative procedure. In the simplest version the improved estimator for g(i) (T) for the ith iteration is cal(i−1) culated from the histogram of temperature distribution HST (T) of the preceding simulation as follows: (i−1)

g(i) (T) = g(i−1) (T) + log HST (T).

(9.31)

In this procedure one uses that the histogram of the ith iteration is given by HST (T) = e−gi−1 (T) Zi (T),

(9.32)

 where Zi (T) = dEn(E) exp(−E/kB T) is an estimate for the canonical partition function at temperature T. Setting exp(gi (T)) = Zi (T) leads to the iterative relationship of Eq. (9.31).

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It is easy to see that the factor g(T) drops out once one considers more than one copies of the system. This is the idea behind replica-exchange method (Hukushima and Nemoto 1996; Geyer and Thompson 1995), also known as parallel tempering, and first applied to protein science (Hansmann 1997). Assume, we have N noninteracting replicas of the molecule, each at a different temperature Ti . In addition to standard Monte Carlo or molecular dynamics moves that effect only one copy, parallel tempering has as an additional update (Hukushima and Nemoto 1996) the exchange of conformations between two copies i and j = i + 1 (i ≥ 1 and j ≤ N). This replica-exchange move is accepted or rejected according to the Metropolis criterion with probability w(Cold → Cnew ) = min(1, exp(−βi E(Cj ) − βj E(Ci ) + βi E(Ci ) + βj E(Cj ))) (9.33) = min(1, exp(βE);

(9.34)

where the factors g(Ti ) and g(Tj ) have been omitted as they do not depend on the replica (configuration) and therefore canceled. This exchange of conformations leads to a faster convergence of the Markov chain than in regular canonical simulations since the resulting random walk in temperatures allows the configurations to move out of local minima and cross energy barriers. Expectation values of a physical quantity A are calculated as usual according to: MES 1  A(Ci (k)), ATi = MES

(9.35)

k

where MES is the number of measurements taken for the ith temperature. Values for intermediate temperatures are calculated using multihistogram re-weighting techniques (Ferrenberg and Swendsen 1988, 1989). The temperature distribution should be chosen such that any relevant energy barrier can be crossed at the highest temperature. Note that parallel tempering does not require Boltzmann weights. The method can be combined easily with generalized-ensemble techniques (Hansmann 1997). Obviously, the method is also not restricted to temperature but can be used with any control parameter, for instance, pH or pressure. 9.3.3.3 Random Walks in Model Space Finally, one can also enhance sampling of low-energy configurations by performing a random walk through an ensemble of systems with slightly altered energy functions. In that way, information is exchanged between varying stages of coarse graining or different local environments. This is the idea behind “model hopping” (Kwak and Hansmann 2005), “Hamilton-exchange method,” and related approaches. Consider, for instance, that the energy function can be separated into two terms: E = EA + aEB . As in parallel tempering, “model hopping” considers N non-interacting copies of the molecule, but adjacent copies are now exchanged with probability

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 &  w(Cold → Cnew ) = min 1, exp − β EA (Cj ) + ai EB (Cj ) + EA (Ci ) + aj EB (Ci )  ' −EA (Ci ) − ai EB (Ci ) − EA (Cj ) − aj EB (Cj )   = min 1, exp{βaEB } . (9.36) Here, a = aj − ai and EB = EB (Cj ) − EB (Ci ). Configurations perform a random walk on a ladder of models with a1 = 1 > a2 > a3 > .... > aN that differ by the relative contributions of EB to the total energy E of the molecule. Take as an example the barriers in the energy landscape of proteins that arise from van der Waals repulsion between atoms that come too close. Assuming that such barriers are a main reason for slow sampling in protein simulations, we have considered a version of “model hopping” where the contributions from the van der Waals energy become successively smaller. While the “physical” system is on one side of the ladder (at a1 = 1), the (non-physical) model on the other end of the ladder (at aN << 1) allows in the extreme atoms to share the same position in space. As the protein “tunnels” through van der Waals energy barriers, sampling of low-energy configurations is enhanced in the “physical” model (at a1 = 1). With this realization of “model hopping” we have “predicted” the structure of a 46-residue protein A in an all-atom simulation within a root mean square deviation (rmsd) of 3.2 Å (Kwak and Hansmann 2005). Model hopping also allows guiding a simulation by information obtained from homologous structures (Gront et al. 2005). Usually, such spatial constraints introduce an additional roughness into the energy landscape which often leads to extremely slow convergence of the simulation. This problem is circumvented in our approach through a random walk in an ensemble of replicas that differ by the strength with that the constraints are coupled to the system. We have demonstrated the usefulness of this approach on some examples of the CASP6 competition (Gront et al. 2005).

9.3.3.4 Optimizing the Efficiency of Generalized-Ensemble Sampling The computational efficiency of generalized-ensemble and replica-exchange techniques is often worse than their theoretical optimum. Bottlenecks and barriers can lead to slow relaxation. In parallel tempering, convergence is given by the relaxation at lowest temperature, which can be gauged by the frequency of statistically independent visits at this temperature. A lower bound for this number is the rate of round trips nrt between the lowest and highest temperature, T1 and TN . We define nup (i) (ndn (i)) as the number of replicas at temperature Ti that came from T1 (TN ). The fraction of replicas moving up

fup (i) =

nup (i) nup (i) + ndn (i)

(9.37)

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describes a stationary distribution of probability flow between temperatures T1 and TN . Maximizing the number of round trips nrt results in a constant transition probability between neighboring nodes and a linear flow distribution among the nodes (Nadler and Hansmann 2007): opt

fup (i) = i/N

(9.38)

Note that in protein simulations relying on explicit solvent the system is dominated by water. In this case the heat capacity C is constant, and the system can be approximated by a D = 2C harmonic oscillator. Within this approximation, one finds then that the optimal temperature distribution is one of √ N opt ≈ 1 + 0.594 C ln(Tmax /Tmin )

(9.39)

replicas, with temperatures distributed according to i−1

opt

Ri

= Tmin

Tmax N−1 . Tmin

(9.40)

Here, Tmax is the highest temperature, Tmin is the lowest temperature, and both quantities have to be chosen in advance (Nadler and Hansmann 2008). If the relaxation at a particular temperature is slower than hopping in temperature, the state space partitions into disjoint free-energy basins connected only via neighboring nodes and forming a tree-like hierarchical network. In this case of broken ergodicity an optimized temperature distribution can be found still iteratively (Trebst et al. 2006) by requiring for temperature Tjk in the kth iteration that 

Tjk

η(opt) (T)dT = j/N ,

(9.41)

T1

where 1 < j < N, the two extremal temperatures T1 and TN remain fixed, and η

(opt)

(T) =

C

1 df , T dT

with the normalization constant C chosen so that  TN η(opt) (T)dT = 1 .

(9.42)

(9.43)

T1

This will again lead to a linear flow distribution, but the acceptance probabilities are not any longer constant. Similarly, one can show that in the case of broken ergodicity weights optimizing the flow through order parameter space (for instance, energy) lead not to a flat distribution (Trebst et al. 2006; Nadler and Hansmann 2007). A direct measurement of the flow distribution is computationally costly: Individual replicas have to cross the full ladder of nodes many times in order to

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ensure sufficient statistics. Such “tunneling” events are especially rare in early stages of the control parameter optimization when round trip times are largest. For this reason, we have proposed to estimate the flow distribution from measurements of mean first passage times allowing to approximate the global flow using also replicas crossing only part of it the ladder. The number of first passage events τ (0 → n) in a simulation decreases with n, while the error increases. At the same time, the error for τ (N → n) increases with decreasing n. Obviously, there exists a node n∗ where the mean first passage times for τ (0 → n), n = 1, . . . , n∗ , and those for τ (N → n), n = n∗ + 1, . . . , N − 1 will be the most reliable ones. Hence, while in the optimization schemes of Trebst et al. (2006) and Nadler and Hansmann (2007) the limiting factor are tunneling events across the full ladder of temperatures, here the statistics is only limited by the number of first passage events from either boundary to n∗ . Defining sums over adjacent inverse transition probabilities h(0 → n) =

n  j=1

1   W βj−1 , βj

n−1  τ (0 → j)

=

j=1

j(j + 1)

+

τ (0 → n) , n

(9.44)

and h(N → n) = =

N 

1   W βj , βj−1 j=n+1 N−n−1  j=1

τ (N → N − j) τ (N → n) + , j(j + 1) N−n

(9.45)

one can derive the following expressions for the flow probabilities:

P(MFPT) (n) = up

⎧ ⎪ ⎪ ⎨1 −

:

n ≤ n∗

⎪ ⎪ ⎩

:

n > n∗

h(0 → n) h(0 → n∗ ) + h(N → n∗ ) h(N → n) h(0 → n∗ ) + h(N → n∗ ) (MFPT)

(9.46)

with a similar relation for Pdown . (MFPT) Starting from a flow distribution Pup reconstructed from mean first passage time analysis, one now can use existing iteration schemes that exhibits fast convergence to the optimal temperature values (Trebst et al. 2006; Nadler and Hansmann 2007). In our simulations, flow distributions derived from mean first passage times lead to temperature sets that are more stable upon iteration than those from flows measured directly (Nadler et al. 2008).

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9.4 Recent Applications An example for applications of the above-described techniques are our recent simulations (Nadler et al. 2008) of GS-α3 W, a three-helix bundle with a single tryptophan buried in the interior of the protein, which was designed to serve as a simple model for the function of redox proteins. With 67 residues the protein is of a size that far exceeded what we could study previously in all-atom simulations. In Fig. 9.1 we show a low-energy structure found in an implicit-solvent replica-exchange Monte Carlo simulation relying on mean first passage times-optimized temperatures. The figure shows the overlap of the minimal structure with the experimentally determined (Protein Data Bank identifier 1LQ7). The root mean square deviation between both structures is only 3.3 Å. A detailed analysis of our data is published in Meinke and Hansmann (2009) and confirms that replica exchange with flowoptimized temperatures allows thermal-folding simulations of such large proteins. Fig. 9.1 Overlap of computational and experimentally determined configuration of a 67-residue protein. Figure taken from Nadler et al. (2008)

Our results for the 49-residue C-terminal CFr of the artificially designed Top7 (Kuhlman et al. 2003; Dantas et al. 2006) may be the most impressive example for the power of generalized-ensemble simulations and the progress made over the last few years. Our simulations of the monomer let to a lowest-energy structure with an α-helix packed against a three-stranded anti-parallel β-sheet, shown in Fig. 9.2, that has an rmsd of only 1.7 Å (Mohanty et al. 2008) to the experimentally determined structure (PDB-code 2GJH). It is also the free-energy minimum at T = 300 K. This result is more remarkable as proteins with end-to-end are particularly challenging. The N-terminal β-strand is synthesized early on, but it cannot bind to the C terminus before the chain is fully synthesized. During this time there is a danger that the β-strand at the N terminus interacts with nearby molecules leading to potentially harmful aggregates of incompletely folded proteins. For slow-folding proteins, one can conjecture a “backtracking” mechanism (Gosavi et al. 2006) where folding succeeds only after breaking of prematurely formed β-contacts.

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Fig. 9.2 Overlay of the free-energy configuration of Cfr with the experimentally determined structure. Figure taken from Mohanty and Hansmann (2008)

However, the Cfr monomer is a fast-folding protein. The risk of mis-folding and aggregation is avoided by another mechanism in our simulations. During much of the folding the N-terminal region (Glu 2-Thr 12) forms an extension of the neighboring helix, residues Lys 13 through Gly 31 (picture 2 in Fig. 9.3). This extended helix forms early but frequently unfolds and refolds. After formation and proper arrangement of the β-hairpin (Tyr 32 through Leu 50) (pictures 3–4 in Fig. 9.3), their arrangement is stabilized through hydrophobic contacts between the two secondary structure elements. At this point, the helix unfolds partially and releases the N-terminal residues (picture 5 in Fig. 9.3) that now attach to the hairpin as the third strand of a β-sheet and complete the native structure (picture 6 in Fig. 9.3).

Fig. 9.3 Six snapshots of the folding pathway. Starting from random initial states (1), the molecule first forms a helix (2) that includes the N-terminal residues. The C-terminal hairpin is formed next (3), aligning with the helix (4). The helix partially unfolds (5) and the released residues complete the native structure as third strand of the β-sheet (6). The figure is taken from Mohanty et al. (2008)

The caching of an N-terminal strand in a helix prevents premature formation of contacts with parts of the molecule having strong β-strand propensities and also with similar parts in other molecules. Thus it acts both as a facilitator of folding and as an inhibitor of aggregation. This caching mechanism may be common in proteins with long-distance β-contacts. It requires that one of the strands exhibits chameleon behavior (Minor and Kim 1996), i.e., either extends an adjacent helix or forms a β-strand when provided with a template for a β-sheet. This is not only important because this mechanism may be common in the large class of proteins with β-sheets

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where the strands have long separation in sequence, but also because it demonstrates that protein design cannot be restricted only on the final fold but needs to include the folding path.

9.5 Conclusion Progress in the development of algorithms over the last two decades has extended dramatically the size of peptides and proteins that are accessible in all-atom simulations. While in the early 1990s such simulations were restricted to pentapeptides such as Met-enkephalin, small proteins with 50–60 residues are now accessible. We can expect that further progress in algorithms and hardware will extend this range to even larger molecules. Research over the last years has also allowed to pinpoint the difficulties that need to be overcome in order to obtain further progress. Probably the most important open problem is that novel and advanced techniques require a tuning of parameters to give optimal sampling and that there are no simple and universal rule for this tuning. Research in the next years will need to focus on the development of procedures that allow for an optimal choice of algorithm and its tuning to the problem at hand. Such research requires the establishment of a set of benchmark systems that enable a fair and decisive comparison of simulation techniques and allow one to measure the scaling of their efficiency with size and complexity of the molecules. Such benchmark systems will also allow one to test and improve the software, in that these algorithms are implemented. As the described techniques can only reduce the sampling difficulties from an exponential scaling to a power law, it is necessary to have software that highly adapted to massively parallel computers and modern architectures such as GPUs and cell processors. Writing and adapting this software will be another major challenge. It is hoped that further advancements in hardware and algorithms will overcome the above-described problems and further establish the use of computer simulations as “microscope” to a point where not only single proteins are studied in atomistic resolution, but also their interactions and finally whole cells can be explored in silico. Acknowledgments Support by the National Science Foundation (research grants CHE-998174, 0313618, 0809002) and the National Institutes of Health (GM62838) are acknowledged.

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Hansmann UHE, Okamoto Y (1998) The generalized-ensemble approach for protein folding simulations. In: Stauffer D (ed) Annual reviews in computational physics, vol. VI. World Scientific, Singapore Hansmann UHE, Okamoto Y (1999a) New Monte Carlo algorithms for protein folding. Curr Opin Struc Biol 9:177–184 Hansmann UHE (1999) Protein folding simulations in a deformed energy landscape. Eur Phys J B 12:607–612 Hansmann UHE, Okamoto Y (1999b) Finite-size scaling of helix–coil transitions in poly-alanine studied by multicanonical simulations. J Chem Phys 110:1267–1276 Hansmann UHE, Wille L (2002) Global optimization by energy landscape paving. Phys Rev Lett 88:068105 Hansmann UHE (2003) Protein folding in silico – an overview. Comput Sci Eng 5:64–69 Holland JH (1975) Adaptation in natural and artificial systems. University of Michigan Press, Ann Arbor, MI Hukushima K, Nemoto K (1996) Exchange Monte Carlo method and applications to spin glass simulations. J Phys Soc (Japan) 65:1604–1608 Kirkpatrick S, Gelatt CP, Vecchi MP (1983) Optimization by simulated annealing. Science 220:671–680 Kuhlman B, Dantas G, Ireton GC, Varani G, Stoddard BL, Baker D (2003) Design of a novel globular protein fold with atomic level accuracy. Science 302:1364–1368 Kumar S, Payne PW, Vásquez M (1996) Method for free-energy calculations using iterative techniques. J Comp Chem 17:1269–1275 Kwak W, Hansmann UHE (2005) Efficient sampling of protein structures by model hopping. Phys Rev Lett 95:138102 Laio A, Parrinello M (2002) Escaping free-energy minima. Proc Natl Acad Sci USA 99: 12562–12566 Li Z, Scheraga HA (1987) Monte Carlo-minimization approach to the multiple-minima problem in protein folding. Proc Natl Acad Sci USA 84:6611–6615 Lyubartsev AP, Martinovski AA, Shevkunov SV, Vorontsov-Velyaminov PN (1992) New approach to Monte Carlo calculations of the free energy: method of expanded ensembles. J Chem Phys 96:1776–1783 Marinari E, Parisi G (1992) Simulated tempering: a new Monte Carlo scheme. Europhys Lett 19:451–458 Meinke JH, Hansmann UHE (2009) Thermodynamics and free-energy driven folding of the 67-residue protein GSα W – A large-scale Monte Carlo study. J Comp Chem 30:1642–1648 Minor DL Jr, Kim PS (1996) Context-dependent secondary structure formation of a designed protein sequence. Nature 380:730–734 Mohanty S, Meinke JH, Zimmermann O, Hansmann UHE (2008) Simulation of Top7-CFr: a transient helix extension guides folding. Proc Natl Acad Sci USA 105:8004–8007 Mohanty S, Hansmann UHE (2008) Caching of a Chameleon segment facilitates folding of a protein with end-to-end β-sheet. J Phys Chem B 112:15134 Nadler W, Hansmann UHE (2007) Generalized ensemble and tempering simulations: a unified view. Phys Rev E 75:026109 Nadler W, Hansmann UHE (2008) Optimized explicit-solvent replica-exchange molecular dynamics from scratch. J Phys Chem B 112:10386 Nadler W, Meinke JA, Hansmann UHE (2008) Folding proteins by first-passage-times optimized replica exchange. Phys Rev E 78:061905 Schug A, Wenzel W, Hansmann UHE (2005) Energy landscape paving simulations of the trp-cage protein. J Chem Phys 122:194711 Trebst S, Troyer M, Hansmann UHE (2006) Optimized parallel tempering simulations of proteins. J Chem Phys 124:174903 Torrie GM, Valleau JP (1977) Nonphysical sampling distributions in Monte Carlo free-energy estimation: umbrella sampling. J Comp Phys 23:187–199

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Chapter 10

Protein Structure Prediction: From Recognition of Matches with Known Structures to Recombination of Fragments Michal J. Gajda, Marcin Pawlowski, and Janusz M. Bujnicki

Abstract The field of protein structure prediction has been revolutionized by the application of “mix-and-match” methods both in template-based homology modeling, as well as in template-free, “de novo” folding. Automated generation of models that are closer to the native structure of the target protein than the structure of its closest homolog is currently possible by recombination of fragments copied from known protein structures or extracted from alternative starting models. It is also the most successful approach in the cases where the target protein exhibits a novel three-dimensional fold. This chapter is an update of a review article published earlier [Bujnicki JM, Chem Bio Chem 7(1):19–27, 2006 Jan 9, Copyright WileyVCH Verlag GmbH & Co. KGaA. Reproduced with permission]. It summarizes the recent developments in both template-based and template-free protein structure modeling and compares the available methods for protein structure prediction by recombination of fragments.

10.1 Introduction The high-resolution three-dimensional structure of a protein is the key to the understanding and manipulation of its biochemical and cellular function. However, the rate of protein structure determination by X-ray crystallography lags behind the rate of determination of new protein sequences. As of January 2010, the National Center for Biotechnology Information’s Non-Redundant RefSeq database (Pruitt et al. 2007) contained 9,662,677 sequences, while the Protein Data Bank (Berman et al. 2000) contained only 36,043 protein structures with non-redundant sequences

J.M. Bujnicki (B) Laboratory of Bioinformatics and Protein Engineering, International Institute of Molecular and Cell Biology, Warsaw, Poland; Laboratory of Bioinformatics, Institute of Molecular Biology and Biotechnology, Faculty of Biology, Adam Mickiewicz University, Poznan, Poland e-mail: [email protected]

A. Kolinski (ed.), Multiscale Approaches to Protein Modeling, C Springer Science+Business Media, LLC 2011 DOI 10.1007/978-1-4419-6889-0_10, 

231

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(50,495 structures total). Currently, the size of the sequence database doubles approximately every 5 years, while the structure database doubles every 8 years. Thus, the gap between the number of known structures and known sequences will continue to widen in the foreseeable future and it is unlikely that it will be ever closed, e.g., the structures will never be solved experimentally for all proteins. Almost 50 years ago Anfinsen demonstrated that all of the information necessary for RNase A to fold to the native structure is contained in its amino acid sequence (Anfinsen et al. 1961). This finding has been generalized to most globular proteins, suggesting that a protein’s structure could be calculated (modeled) based on the knowledge of its sequence and our understanding of the sequence–structure relationships. Thus, the current structural genomics initiative aims to solve experimentally the structures of representative proteins, while the others are hoped to be modeled computationally (Baker and Sali 2001). The theoretical prediction of the native structure of a protein from its amino acid sequence remains, however, one of the most challenging problems in contemporary life sciences.

10.2 Protein Structure Prediction Methods: Classification and Critical Evaluation Efforts to solve the protein-folding problem have been traditionally rooted in two schools of thought (Fig. 10.1). One is based on the principles of physics, e.g., on the thermodynamic hypothesis formulated by Anfinsen, according to which the native structure of a protein corresponds to the global minimum of its free energy (Anfinsen 1973). Accordingly, physics-based methods model the process of protein folding by simulating the conformational changes and searching for the free-energy minimum. The other school of thought is based on the principles of evolution. After experimental determination of the first handful of protein structures it became clear that evolutionarily related (homologous) proteins usually retain the same threedimensional fold (i.e., the arrangement and connectivity of elements of secondary structure) despite the accumulation of divergent mutations (Chothia and Lesk 1986). It was also found that structural divergence is much slower than sequence divergence, although these two features are strongly correlated. Thus, methods have been developed to map the sequence of one protein (a target) to the structure of another protein (a template), model the overall fold of the target based on that of the template, and infer how the target structure will change due to substitutions, insertions, and deletions, as compared with the template (reviews: Cohen-Gonsaud et al. 2004; Krieger et al. 2003). Table 10.1 summarizes the key features of methods discussed in this chapter. Accordingly, methods for protein structure prediction have been divided into two classes: “de novo” modeling, in principal applicable to all types of proteins, including those for which no appropriate templates are available, and “comparative (homology) modeling” (CM), in which the target sequence must be aligned to an evolutionarily related, experimentally solved template structure. In this context it is

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Fig. 10.1 The “evolutionary” and “physical” approaches for protein structure prediction. Given the amino acid sequence, a simulation of either protein evolution or protein folding is carried out, according to quantitative models of either divergence of sequences and structures or physical interactions within the molecule and between the molecule and the solvent. (Bujnicki, 2006, Copyright Wiley-VCH Verlag GmbH & Co. KGaA. Reproduced with permission)

worthwhile to remind that protein structure can be described by a hierarchical system, with levels corresponding to primary sequence (covalent bonding of amino acids), secondary structure (segments of recurring arrangement of amino acids consecutive in the sequence), tertiary structure (mutual arrangement of secondary structures in a protein domain), and quaternary structure (mutual arrangement of domains within a multi-domain protein or different subunits in a multi-protein complex). Knowledge-based methods are usually effective on all levels of this hierarchy, first because evolutionary forces tend to preserve all aspects of protein structure. And if homology is present, it often manifests itself at all levels of hierarchy.

Type

CM FR/CM meta-selector FR/CM meta-selector FR/CM fragment splicer

FR/CM fragment splicer

FR/CM/de novo fragment splicer

CM fragment splicer

CM alignment splicer

De novo fragment splicer

FR/CM/de novo fragment splicer

De novo fragment splicer

De novo fragment splicer

Method

SWISS-MODEL PCONS5 (PROQ) 3D-JURY 3D-SHOTGUN

FRANKENSTEIN3D

GS-KudlatyPred

In silico protein recombination

GENETIC ALGORITHM

ROSETTA

GINZU/ROBETTA

SIMFOLD

PROFESY

Merging of domain models, Monte Carlo-simulated annealing Multi-canonical ensemble Monte Carlo Conformational space annealing

Superposition of templates Superposition of models Superposition of models Superposition and recombination of models Superposition and recombination of models Recombination of supersecondary structure fragments form any kind of initial models Superposition and recombination of models, local realignment Recombination of alignments, local realignment Monte Carlo-simulated annealing

Search strategy

Physical energy function

Physical energy function

Physical energy function with elements of a statistical potential, clustering FR score and statistical potential

Statistical potential

Statistical potential

Statistical meta-potential (MetaMQAP) Statistical meta-potential (MetaMQAPcons)

NA Statistical potential (PROQ) NA NA

Evaluation/selection of models

15 aa fragments from PDB

4–9 aa fragments from PDB

3 and 9 aa fragments from PDB, FR models

alternative target-template alignments 3 and 9 aa fragments from PDB

Comparative models with similar folds

FR/CM/de novo models

FR models

CM templates, loops from PDB FR models FR models FR models

Input and/or fragment library

Table 10.1 Summary of key features of methods discussed in this chapter. This table lists all prediction methods discussed in this chapter, along with their distinguishing features. It is intended as a reference and guide for contrasting approaches

234 M.J. Gajda et al.

De novo fragment splicer

De novo fragment splicer

De novo fragment splicer

FR/CM/de novo fragment splicer FR/CM/de novo fragment splicer

FRAGFOLD

UNDERTAKER

ABLE

TASSER

De novo fragment splicer

FR

SALAMI

Structural descriptors

ZAM

FR/CM/de novo fragment splicer De novo fragment splicer

FRANK/CABS

I-TASSER

Type

Method

Bayesian fragment picking, torsion angle dynamics NA

Monte Carlo-simulated annealing, iterated with restraints from previous rounds Replica Exchange Monte Carlo (on a lattice) Replica Exchange Monte Carlo (on a lattice), iterative enrichment of fragments with templates analogous to ab initio models from previous rounds Replica Exchange Monte Carlo (on a lattice) zipping and assembly from smaller fragments

Simulated annealing or genetic algorithm Genetic algorithm

Search strategy

Table 10.1 (continued)

NA

Physical energy function with elements of a statistical potential Steric clashes, clustering

Statistical potential

Statistical potential, clustering Statistical potential, clustering

Physical energy function with elements of a statistical potential

Statistical potential

Statistical potential

Evaluation/selection of models

Alphabet of 300 fragments each 6 residues long Descriptors (groups of >2 fragments; >3 residues long)

Individual residues

FR models

FR models, SCOP folds most similar to ab initio models

FR models

Supersecondary structures and 3–5 aa fragments Fragments of FR models, and 1–4 aa and 9–12 aa fragments from PDB Individual residues

Input and/or fragment library

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Further, in homology-based modeling errors at different level of structural hierarchy are largely independent from each other (e.g., it is not difficult to correctly predict a protein fold without getting all secondary structures correctly and without any consideration of the quaternary structure). This has been demonstrated in the course of Critical Assessment of Techniques for Protein Structure Prediction (CASP), as many modelers have generated models with correct folds while completely disregarding the quaternary structure, and with significant errors with respect to secondary structure alignment (Moult et al. 2005, 2007, 2009). On the other hand, “de novo” methods typically require accurate prediction at low levels of hierarchy in order to correctly predict higher levels. Their advantage, however, is that they are independent on the modeling of the primary sequence (they do not attempt to model sequence alignment between the target sequence and some other sequence). The “de novo” approach can be further subdivided into “ab initio” methods, i.e., those based exclusively on the physics of the interactions within the polypeptide chain and between the polypeptide and the solvent (Scheraga 1996), and “knowledge-based” methods that utilize statistical potentials based on the analysis of recurrent patterns in the known protein structures and sequences (Kolinski 2004). De novo methods may utilize different representations of the protein chain, frequently employing either coarse-grained models (see the chapter by Liwo et al in this volume) or fragments of experimentally solved structures (from parts of the side chain or the backbone, to individual residues, to groups of residues up to the size of a few elements of secondary structure). The CM approach can also be subdivided into two main trends. One is to model the structure by copying the coordinates of the template (both the backbone and the side chains) in the aligned core regions, which can also include “averaging” over coordinates of multiple templates. The variable regions are modeled by taking fragments with similar sequence from a database of previously observed loops, followed by replacing the mutated side chains with rotamers that satisfy the stereochemical criteria, and (optionally) limited energy optimization, as implemented in SWISS-MODEL (Peitsch 1995). The other possibility is to use the distance and torsion angles and interatomic distances from the aligned regions of the template(s) as modeling restraints, which permits the use of information from multiple, possibly conflicting structures. This approach also requires to idealize the geometry and packing of the entire chain by satisfying stereochemical constraints derived from the database of protein structures, as implemented in MODELLER (Sali and Blundell 1993). The CM approach has been also extended to “fold-recognition” (FR), where one attempts to identify a template with a similar fold that does not need to exhibit significant sequence similarity to the target (e.g., the target and the template may or may not be homologous, but they need to share the common fold) (Godzik et al. 1992; Jones et al. 1992). While the early FR methods relied mostly on the “threading” approach, i.e., evaluation of protein energy as the sum of pairwise residue–residue interactions based on physical or statistical potentials, nearly all contemporary FR methods are based mostly (or exclusively) on sequence comparisons and are tuned to detect distantly related homologs rather than unrelated structural analogs (reviews: Cymerman et al. 2004; Ginalski et al. 2005).

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Another way of subdividing the “comparative” approach is into orthodox (traditional) methods that use entire proteins (or domains) as templates and methods that use different (not necessarily related) structures or their fragments to model different parts of the target sequence (which are discussed in this review). With the decreasing size of fragments, the latter type of comparative methods blends with de novo methods that represent protein structures with fragments of known structures. Recently, a new generation of protein structure prediction methods has been developed that combine comparative modeling with de novo modeling. Typically, an initial model or its significant part is modeled by comparative approach, based on the general prediction of the three-dimensional fold, and then the entire structure or its part is “refined” by de novo methods, often in connection with evaluation of local quality by Model Quality Assessment Methods (MQAPs) (Kryshtafovych and Fidelis 2009). This review describes examples of all the above-mentioned approaches. In order to objectively assess the abilities (and inabilities) of different methods for protein structure prediction, Moult and coworkers organized the biennial Community Wide Experiment on the Critical Assessment of Techniques for Protein Structure Prediction (CASP) to compare computationally predicted protein structures with the “golden standard” of experimentally determined ones. The first assessment experiment (CASP1) was held in 1994 and revealed that computational methods for protein structure prediction perform quite poorly, those based on physics and evolution alike (Moult et al. 1995). Since then, the progress in the field of protein structure prediction has been significant, especially in the “template/knowledge-based” category (e.g., CM and FR), in part due to the improvement in methodology but mostly because of the rapid growth of databases and accumulation of new potential template structures as well as numerous new sequences that can serve as convenient “evolutionary intermediates” in the homology searches. Nonetheless, it appears that in the recent years there has been little progress, if any, in the ability of both comparative and de novo methods (Moult et al. 2009).

10.3 “Meta” Approaches to Template-Based Prediction The series of CASP experiments have shown that the combined use of human expertise and automated methods can often result in successful predictions. This has became especially clear in the cases of very remote homology, where most FR methods return predictions with scores indicating the lack of statistical significance and correct models are “buried” among a number of incorrect models. A group of four human predictors including Daniel Fischer, Leszek Rychlewski, Arne Elofsson, and Janusz M. Bujnicki, pioneered the idea of “meta-prediction” in CASP4, by comparing the models generated by FR servers participating in the satellite experiment CAFASP-21 and submitting manually selected “consensus” predictions as the “CAFASP-CONSENSUS” group. This group performed better than any of the

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individual servers and ranked seventh among all predictors of CASP4 (Fischer et al. 2001). Thus, it was demonstrated that the recurrence of a particular protein fold within the sets of top ten models returned by different servers (and not necessarily at the first position of their ranking) increases the likelihood of a correct prediction and that on the average, no single FR method is better than the combination of a few top methods. Since then, meta-prediction based on FR (Fig. 10.2) has become the most successful approach for template-based modeling and has been applied by a large number of human predictors, including the best performers in CASP5 and CASP6.

Fig. 10.2 The meta-server approach for protein structure prediction. The meta-server is used as a gateway to send the target sequence to various “primary” fold-recognition servers, collect the results (target-template alignments), build the corresponding models, compare them with each other, and either select the most representative structure or construct a hybrid model from the most frequently represented fragments (Bujnicki, 2006, Copyright Wiley-VCH Verlag GmbH & Co. KGaA. Reproduced with permission)

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Following the proven success of manual “meta-predictors,” several groups have implemented fully automated “meta-servers” (Bujnicki et al. 2001b; Douguet and Labesse 2001; Kurowski and Bujnicki 2003; Wu and Zhang 2007a). One of the earliest meta-predictors was the neural network PCONS developed by Elofsson and coworkers (Lundstrom et al. 2001), which collects a set of top models generated by different FR servers and selects the models that were most similar to other models in the set. The second edition of an independent assessment experiment LiveBench (Bujnicki et al. 2001a) organized shortly after CASP4 revealed that PCONS2 (version trained for a few specific “primary” FR servers) exhibited the sensitivity comparable to the most sensitive primary method and the specificity higher than any primary method. The newest version of PCONS is reinforced by methods for protein model evaluation (Wallner and Elofsson 2003), exhibits even higher specificity and is able to use as an input model generated by any set of methods. PCONS is available as a part of various meta-servers, as well as a standalone server Pcons.net (http://pcons.net/) (Wallner et al. 2007). 3D-JURY developed by Rychlewski and coworkers (Ginalski et al. 2003) is another automated meta-predictor that simply selects models from those produced by other servers. It takes as input any set of models, compares all against all, and selects one that appears to contain the largest subset of commonly superimposable coordinates. The most important feature contributing to the success of 3D-JURY and its popularity among the users is its scoring system, which allows confidently identifying the models with correctly predicted fold, even though it does not necessarily recognize the absolutely best model among similar “top solutions.” 3D-JURY is available as an integral part of the Bioinfo.pl metaserver (http://meta.bioinfo.pl/). Based on this algorithm, a very similar method 3D-Consens has been implemented as an optional post-processing tool in the GeneSilico meta-server (https://genesilico.pl/meta2/).

10.4 From Multiple Template-Based Models to Hybrids Apart from meta-predictors that simply select models from the input set, another breed of meta-predictors have been developed that use the unrefined models generated by primary servers as a “structural scrap-yard” from which to obtain “spare parts” to generate new models. 3D-SHOTGUN developed by Fischer (Fischer 2003) was the first fully automated meta-predictor designed to assemble hybrid models from fragments of models obtained from independent FR methods (e.g., from different components of the BIOINBGU server (Fischer 2000)). In the first step, regions of structural similarity are identified for all initial models by pairwise superposition. Subsequently, for each residue in each model, the number of its occurrences in the superimposed regions of other models is counted and a hybrid model is assembled by taking the coordinates of each residue from a model having the highest count. Thus, for each initial model a hybrid model is constructed that contains the most common structural features of all models, often including more residues than any

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of the initial models. In the second step, the assembled models are assigned scores based on the combination of the original scores of their parent models (normalized to a similar scale) and the scores describing the structural similarity of the assembled model to other models, as determined by MAXSUB (Siew et al. 2000). For each cluster of highly similar assembled models only one representative model with the highest score is reported. The rationale of the 3D-SHOTGUN strategy is the same as in the consensus methods (selectors) acting on complete models, namely, that recurring structural features observed in models obtained from different FR methods are more likely to be correct. The initial version of 3D-SHOTGUN generated models that comprised only Cα atoms and commonly exhibited stereochemical problems, such as implausible distances and angles and steric clashes between fragments taken from different parent models. In terms of coverage and root mean square deviation (RMSD) between the model and the native structure, however, the approach of hybrid model construction is superior to selection of one of the stereochemically more acceptable input models, as the hybrids are on the average more complete and superimpose better on the native structure than the initial models. The method is sensitive to initial alignment errors – if none of the initial alignments is correct for a given region, it is unlikely that this region will be modeled correctly in the final structure. A new automated version SHGUM includes a crude refinement step, using MODELLER (Sali and Blundell 1993), to generate full-atom models with idealized stereochemistry and without gaps and collisions and even with a slight improvement in the overall RMSD (Sasson and Fischer 2003). The method is available via the INUB server at http://inub.cse.buffalo.edu/query.html. FRankenstein’s Monster is another approach to meta-prediction by consensus and recombination of fragments, developed in the authors’ laboratory (Kosinski et al. 2003, 2005). It is similar to 3D-SHOTGUN, but goes beyond the identification of geometrical consensus by including evaluation of the models by statistical potentials and features an additional step of local realignment of uncertain regions. This helps not only to overcome the problem of selection of the optimal template, but also to correct initial alignment errors. Briefly, the GeneSilico meta-server (Kurowski and Bujnicki 2003) is used as a gateway to run diverse FR methods and to generate preliminary full-atom models from initial pairwise target-template alignments. The local quality of sequence–structure fit in these models is evaluated by a fitness function of local MetaMQAP score (Pawlowski et al. 2008). The most probable folds are identified by clustering. For each fold, a hybrid model is assembled from fragments that are structurally similar in >40% of all preliminary models, while the remaining non-consensus fragments are selected based on the MetaMQAP local score. The initial hybrid model (the “FRankenstein’s monster”) typically exhibits stereochemical problems similar to those found in models generated by the 3D-SHOTGUN method. However, in the FRankenstein strategy the hybrid model is not directly refined, but instead it is superimposed onto the structures of the templates used, yielding a new target-multiple template sequence alignment, which is used to generate a new, stereochemically acceptable model by an “orthodox” CM procedure. The sequence–structure fit in the new model is re-evaluated with

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MetaMQAP, and regions of low local score are selected for further refinement. For each poorly scored non-consensus region, a set of new alignments is generated by progressively shifting the target sequence with a step of one residue in the direction of either terminus, within the region of overlap between the secondary structure elements found in the template structure and those predicted for the target. All resulting alignments are used to generate a new family of intermediate models, which are again evaluated and recombined to produce a hybrid model. The procedure is iterated until all regions in the protein core obtain acceptable score or if the score cannot be further improved. The FRANKENSTEIN3D method generates models that retain the fragments confidently predicted by consensus (regardless of their fitness according to statistical potentials) and attempts to refine the alignment in the uncertain regions to maximize the fitness score. As demonstrated in CASP5 (Kosinski et al. 2003) and CASP6, where the groups from the author’s laboratory ranked very high in the CM and FR categories (Kolinski and Bujnicki 2005; Kosinski et al. 2005), the application of this approach leads to very accurate target-template alignments, often more accurate than any of the initial alignments, provided that a template with a correct fold is identified by at least one of the FR servers used. The method automatically clusters all templates available from FR servers and allows to automatically build multiple alternative models before alignment optimization. The current version of the method is available as a FRANKENSTEIN3D server at http://genesilico.pl/frankenstein/. Another approach to overcome the problem of template selection and correction of alignment errors by recombination of alternative models was developed by Bates and coworkers (Contreras-Moreira et al. 2003a,b). The in silico Protein Recombination method starts with a population of models built from alternative alignments to one or more templates sharing the same fold and uses a genetic algorithm with two mutually exclusive genetic operators – “recombination” of parent models with crossover points outside the regions of secondary structure and “mutation” by averaging the coordinates of two parent models. The fitness function acting as a selection agent is a free-energy estimate based on protein contact pair-potentials and side-chain solvation energies, estimated from their solvent accessible area. The method was shown to be able to improve alignments by recombining well-aligned regions from the initial models and to produce recombinant models that are comparable to the best initial model. However, the quality of the initial models is the upper limit for the quality of the final model (e.g., unlike the FRANKENSTEIN method, it does not produce new, potentially better alignments). It is also critically dependent on the confident identification of a correct fold. The in silico Protein Recombination method is available as a web server at http://www.bmm.icnet.uk/servers/3djigsaw/recomb/. Another method that implements a genetic algorithm for comparative modeling was developed by Sali and coworkers (John and Sali 2003). It is similar to the FRANKENSTEIN3D approach in that it continuously refers to the target-template alignments, modifies them locally, and assesses the result of these changes by evaluation of the corresponding models, generated by MODELLER (Sali and Blundell 1993). The genetic operators include recombination of the parent alignments (one

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and two-point crossovers) and gap insertions/deletions/shifts that actually generate local changes in the parent alignment. The fitness function is based on a score that combines the evaluation of the model by a statistical potential (Melo et al. 2002), target-template sequence identity, and a measure of structural compactness. The method was shown to increase the average quality of the target-template alignment and the corresponding models, but is dependent on the initial choice of the templates; in addition, the inaccurate statistical potential is generally unable to choose the best model (John and Sali 2003). A recent addition to the repertoire of “recombinators” that act on the level of target-template alignments is the MULTICOM-cluster method (which is also a Model Quality Assessment protocol). It performs greedy merging of the top-scoring FR alignment with other alternative alignments that can fill gaps in the structure, generates models with MODELLER, and clusters the resulting models. In the first iteration, the clustering is used to find references among all single-template models, to obtain a global quality score for each model. In the second iteration, models are compared to references to obtain local quality scores, based on average superposition accuracy to the best reference. The models with best global quality are identified as solutions (Cheng et al. 2009).

10.5 Fragment Assembly: A New Trend in De Novo Protein Structure Prediction Modeling protein structure “de novo” without the template is very difficult, because the conformational space to be searched is so vast that it is practically impossible to simulate the folding of a model that includes all atoms of the polypeptide chain and the surrounding solvent molecules. Methods and resources currently available allow to simulate up to about 1 ms of folding of full-atom representations of only small proteins (<100 residues), while most proteins are larger and fold in the timescale of milliseconds or even seconds. Therefore, the solvent is usually treated implicitly and various simplified models are used that have fewer degrees of freedom and exploit the repetitive nature of protein structure (see the chapter by Liwo and coworkers in this volume). These simplified models typically retain only certain atoms, such as Cα or Cβ or “united atoms” in which several atoms are grouped together, such as the centers of mass of the side chains (Liwo et al. 2005; Sun 1995). The protein structure may be represented using a number of simplified schemes such as lattices or bond angles with discrete values (Geetha and Munson 1996; Kolinski 2004). Despite the considerable reduction of dimensionality of the structure space in simplified models, the polypeptide main chain remains highly flexible and requires many variables per residue to model the protein conformation accurately (Hunter and Subramaniam 2002). Significant progress in the field of “de novo” protein structure prediction has been prompted by the observation that the structure of protein backbone can be represented quite accurately by using short fragments taken from other proteins (Claessens et al. 1989; Jones and Thirup 1986). Fragments up to ten residues long

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provided an efficient method for interpreting electron density maps in protein crystallography and in building protein models from nuclear magnetic resonance (NMR) data (Kraulis and Jones 1987). Classification of protein loops has proven useful in comparative modeling, where the incomplete framework of a protein core has to be amended by “de novo” insertion of polypeptide segments (Donate et al. 1996; Oliva et al. 1997; Tramontano et al. 1989) (see also above). Several groups classified peptide backbone units with fixed or variable lengths into collections of fragments (Bystroff and Baker 1997; Camproux et al. 1999; Kolodny et al. 2002; Micheletti et al. 2000; Unger et al. 1989). Analysis of such recurring fragments identified local sequence–structure correlations in proteins (Bystroff et al. 1996; Han and Baker 1996) and suggested a new method for “de novo” protein structure prediction.

10.5.1 De Novo Modeling by Fragment Assembly (and Subsequent Refinement) ROSETTA developed by Baker and coworkers (Simons et al. 1997) implements a model of folding in which short fragments of the protein chain alternate between different local conformations copied from segments of known, not necessarily homologous, protein structures. The probability of assuming a particular conformation is based on the similarity of the local sequence and predicted secondary structure of the target to sequence and structure from the template library, as in the “traditional” template-based methods for protein modeling. The early version of ROSETTA used the I-SITES library of fragments 7–19 residues in length that corresponded to one of 82 patterns of sequence/structure motifs commonly present in all known structures (Bystroff and Baker 1998). The current version uses a library of fragments 3–9 residues in length extracted from known structures, which are assembled by using a Monte Carlo (MC)-simulated annealing (SA) search strategy, in which fragments are randomly inserted into the protein chain by replacing the backbone torsion angles with those in the fragment. The resulting “decoy” conformation is then evaluated according to a database-derived pseudoenergy function that rewards native protein-like properties. Additionally, a number of heuristic filters can be used to discriminate “protein-like” decoys by virtue of contact order, topology of β-sheets, etc. In the standard protocol, ROSETTA uses a reduced representation with backbone heavy atoms and Cβ atoms explicitly included and the side chains represented by single centroids. ROSETTA is also capable of refinement of models using full-atom representation, special conformation modification operators, and a refined (more “physical”) energy function. During the simulation, a large set of decoys are generated (1,000 to many thousands or even millions, depending on the protein length and computing power used), which are then clustered to identify the largest populations of similar global conformations which correspond to the broadest free-energy minima. Full-atom models with explicit side-chain rotamers can be rebuilt before or after clustering (see the recent review of ROSETTA by Das and Baker (2008)).

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The difference between ROSETTA and most of template-based methods for fragment recombination lies in the stochastic and iterative character of this process and in the utilization of multiple small fragments of different, unrelated proteins (template-based methods use the whole structure of one protein or a few-related templates). Thus, ROSETTA can generate a “de novo” model by allowing the fulllength polypeptide chain to explore conformational space via the fragment insertion search method, even if no homologous or analogous template structure is available. Nonetheless, if a template structure is available, the conserved parts of the target can be built as in traditional CM, while the variable parts are allowed to explore the conformational space with fragments in fashion similar to the “de novo” protocol, but in the context of the template (Rohl 2005). As demonstrated already in CASP5 (Bradley et al. 2003), ROSETTA is capable of generating native-like protein models either de novo (i.e., without any template structure) or by adding long insertions and N- and C-terminal extensions to the template that matches only a part of the modeled protein. Among the “winners” of CASP, ROSETTA continues to be the only “de novo” method that has been made available to the academic community both as a source code of the standalone program and as a web server. It is available in two versions: as a part of the GINZU/ROBETTA meta-server developed by the Baker group (Kim et al. 2004) (http://robetta.bakerlab.org/) that uses a hybrid strategy of template-based, if template is available, and template-free modeling if not, and in conjunction with the alternative fragment library I-SITES and the fragment assignment method HMMSTER (http://www.bioinfo.rpi.edu/~bystrc/hmmstr/), developed and maintained by the Bystroff group (Bystroff and Shao 2002). SIMFOLD is another fragment-assembly method developed by Shoji Takada and coworkers (Chikenji et al. 2003). The original method, which performed quite well in CASP6 as “ROKKO” and “ROKKY”, is similar to ROSETTA. It uses 4–9 residue fragments and its energy function consists of various interactions that are based on physical considerations (Fujitsuka et al. 2004). Yet, SIMFOLD exhibits an important difference: it introduces reversible fragment insertion, e.g., when a new fragment is inserted at a junction between two fragments, the replaced “old” conformation comprising elements of two different fragments is added to the library of fragments, so it can be re-inserted. The latter operation is not possible in ROSETTA, which uses only fragments from the original database. This modification satisfies the detailed balance condition, providing the basis for the application of a multicanonical ensemble Monte Carlo method (MEMC), as used by the human team ROKKO in CASP6. MEMC is more effective in finding low-energy conformations than the conventional SA method (Chikenji et al. 2003). SIMFOLD has been available as a server, but unfortunately ceased to be available publicly. PROFESY developed by Lee and coworkers (Lee et al. 2004) is similar to SIMFOLD in that it attempts to improve the poor sampling efficiency of the traditional SA method and uses a physics-based energy function rather than a statistical potential. The global minimization of the energy function is thus performed by the conformational space annealing (CSA) method (Liwo et al. 1999). The fragment library is constructed using the secondary structure prediction method

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PREDICT and comprises a collection of 15-residue long backbone structures. To our knowledge, this method is not yet publicly available. FRAGFOLD developed by Jones uses supersecondary structural fragments (comprising 2 or 3 sequential secondary structures) from a library of high-resolution protein structures as well as small (3, 4, and 5 residues) fragments (Jones and McGuffin 2003). Possible supersecondary fragments are assigned to the target sequence by a threading procedure similar to that in the GenTHREADER FR algorithm (Jones 1999). The global structure is assembled by a genetic algorithm or a simulated annealing method, in which half of random moves correspond to the insertion of a pre-selected supersecondary structure fragment and the other half involve a completely free choice of one of the small fragments. Conformations that lack steric clashes and pass the checks for protein-like compactness and hydrogen bonding are clustered to identify representatives of the most probable folds. FRAGFOLD is not available as a server, but a standalone version that runs on GPU exists (http://bioinfadmin.cs.ucl.ac.uk/downloads/gpufragfold/). UNDERTAKER is a method developed by Karplus and coworkers that assembles the target structure using fragments of known structures obtained from three sources: a generic library of very short segments (1–4 residues) that must exactly match the target sequence, medium-length segments (9–12 residues) that are assigned by the FRAGFINDER program from the SAM suite, and variable-length segments assigned by FR analysis (Karchin et al. 2003). In addition to fragment replacement, UNDERTAKER implements an alignment replacement operation in which a complete FR match is imported into the model, allowing the replacement of several segments at once in the same orientation as they occur in the template structure. UNDERTAKER uses a genetic algorithm for the stochastic search and includes a crossover operation that allows recombining different conformations. The cost function used to assess the decoys includes many tunable parameters, among which the most important one, as the name of the method implies, is the burial. To our knowledge, UNDERTAKER is not yet publicly available. ABLE developed by Shimizu and coworkers (Ishida et al. 2003) is also based on fragment assembly, but it assigns main-chain dihedral angles individually to each residue. The energy function is similar to that used in ROSETTA. ABLE method has two interesting features that help to avoid problems if the initial distribution of decoys is too broad and no clusters can be identified based on the RMSD as a measure of the distance between the conformations. First, it uses the unit-vector root mean square distance (URMS) (Kedem et al. 1999) as a measure of structural similarity. Second, if not enough clusters with sufficient size and density are obtained, the fragment assembly search is reiterated, but with additional spatial restraints obtained from the consensus substructures in the models generated by the previous minimization procedure. To our knowledge, ABLE is not yet publicly available. The SALAMI method developed by Andrew Torda and coworkers uses a small alphabet of 300 fragment templates derived from clusters of 6-residue fragments in a non-redundant version of the Protein Data Bank (PDB) database (Margraf et al. 2009). Each fragment template includes information about average geometry of residues and distribution of deviations from average values. These distributions

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are used to derive energies that along with steric clash potentials are used to find a lowest-energy conformation. This conformation is a result of simulated annealing in the torsion angle space for each of randomly chosen sequence of fragments. Results for different sequences of fragments are treated as set of decoys and clustered to obtain the final solution. This approach gives more complete coverage of fragment space than many other methods, because it generalizes single, average geometries found in ROSETTA (for example) to fragment template distributions, defined as histograms of torsion angles. SALAMI is available as a web server at http://www.zbh.uni-hamburg.de/salami.

10.5.2 Hybrid Methods Involving Fragment Assembly and Folding Simulations ZAM is an interesting method developed by the group of Kenneth Dill, which uses a unique combination of assembly and ab initio-folding simulation (Ozkan et al. 2007). The target-protein sequence is divided into small pieces, which are folded by a physics-based ab initio method and only then assembled. It will be interesting to see if computational methods such as this one that explore conformation of fragments by ab initio calculations will match the performance of de novo methods that simply derive the fragments from a database of known structures. To our knowledge, the ZAM method is unfortunately not available publicly. An alternative approach to fragment assembly, and one with a long history, is that of the lattice representation, in which residues are restricted to points on a regular three-dimensional lattice (Hinds and Levitt 1992; Skolnick and Kolinski 1991). These methods allow very fast sampling of the conformational space, but their ability to represent the atomic details and to use physics-based energy function is limited. Following the success of fragment-assembly methods, several hybrid methods arose, which combine the strengths of both approaches. TASSER developed by Skolnick and coworkers (Zhang and Skolnick 2004a) starts with an FR analysis based on the PROSPECTOR threading method (Skolnick et al. 2004) that identifies either a single consensus fold or a set of templates with globally distinct folds. Based on the FR alignments, the protein chain is divided into contiguous aligned regions of at least 5 residues (20.7 residues on average, according to the authors’ own benchmark) and gapped unaligned regions. The conformation of aligned regions is copied from the templates and remains unchanged during the assembly, while the unaligned regions are represented on an underlying cubic lattice as in the earlier models developed by Skolnick, Kolinski, and coworkers (Kihara et al. 2001). A series of initial models is generated and submitted for assembly and refinement to parallel hyperbolic Replica Exchange Monte Carlo (REMC) sampling method. Structures generated in the lowest-temperature replicas are subjected to iterative clustering using SPICKER (Zhang and Skolnick 2004b) to identify the final models based on the cluster density. I-TASSER is a newer variant of this method, which substitutes fragments modeled de novo for

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analogous fragments found in known structures in the PDB database and uses them to provide restraints to guide the folding simulation during the following iteration. TASSER and its variants have been very successful in CASP, since CASP6. I-TASSER is available as a web server at http://zhanglab.ccmb.med.umich.edu/ I-TASSER/ (Wu et al. 2007). TASSERLite, a version that allows for refinement of comparative models but not full de novo modeling (Pandit et al. 2006), is available from the Skolnick group website (http://cssb.biology.gatech.edu/skolnick/ webservice/tasserlite/index.html). Another hybrid method, involving the recombination of fragments and latticebased modeling, was developed by the author of this article in cooperation with Andrzej Kolinski, by combining the FRankenstein’s Monster method (Kosinski et al. 2003) (see also above) for generation of initial models, with the reduced lattice model CABS (Kolinski 2004). Briefly, preliminary hybrid models are generated with the template-based recombination method and scored with an MQAP software to identify well-folded fragments, as described earlier. These fragments are not considered directly, but are used as a source of spatial restraints to guide the REMC-folding simulation using the CABS model. The resulting decoys are clustered using the HCPM method (Gront and Kolinski 2005) to identify the final models. This method performed very well in CASP6 evaluation (Kolinski and Bujnicki 2005), but has been never implemented as fully integrated and automated software. One of the authors of this chapter (M.P.) has recently developed a new fragmentbased method, which has participated as “GS-KudlatyPred” in CASP8 (Pawlowski 2009). This method is to some extent related to the “FRankenstein’s Monster approach”, but operates essentially only on the three-dimensional fragments, without toggling between the three-dimensional and alignment representation. As an input this method takes a set of models built with any methods (comparative or de novo) and extracts fragments comprising 1–4 sequentially occurring secondary structures. Each of the initial models is scored by MetaMQAPcons (a derivative of method described in the article by Pawlowski et al. (2008)). Five models with the best global score are selected as the reference models. All possible combinations of supersecondary structure fragments are generated and ranked based on the sum of three components: (1) MetaMQAPcons score of all fragments in the combination; (2) a GDT_TS score describing the fit of each fragment onto the closest “root model”; and (3) degree of structural similarity in the area of overlap between neighboring fragments. In the last step, hybrid models are built for 200 top-scored combinations of fragments, using MODELLER (Fiser and Sali 2003) in a multi-template mode and ranked by the MetaMQAPcons method.

10.5.3 Other Methods Based on Fragment Prediction All the aforementioned methods for protein structure prediction by recombination use contiguous fragments of protein backbone. Another approach to protein

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structure prediction is based on the concept of three-dimensional structural descriptors developed by Kryshtafovych and Fidelis (unpublished analysis cited in Hvidsten et al. (2003)), e.g., substructures that encompass a set of non-contiguous protein backbone fragments residing within a spatial neighborhood of a specific residue. The calculation of descriptors for all known protein structures followed by clustering similar descriptors into groups revealed certain sequence preferences that can be interpreted as propensity of particular residues to be accommodated within particular substructures (Hvidsten et al. 2003), similar to the observation made for single contiguous fragments, e.g., in the I-SITES library (Bystroff et al. 2000). Based on these correlations it is possible to identify descriptors matching the target sequence and to predict a three-dimensional fold that is most compatible with these descriptors, without building an explicit three-dimensional model of the target structure (Hvidsten et al. 2003). In principle, it may be possible to assemble the tertiary structure of the target from the descriptors that contain multiple backbone fragments but to the author’s knowledge such a method has not yet been developed. It remains to be seen if the three-dimensional descriptor approach will ever lead to practically a useful method for tertiary structure prediction that would become available publicly.

10.6 Why Are the Fragments-Assembly Methods So Successful? Template-based methods, especially FR meta-servers, have been found to produce exceptionally good predictions and are now widely used for protein structure prediction. In particular, their relatively low-computational cost makes them very useful for large-scale analyses, e.g., for building models for proteins encoded in whole genomes. However, all template-based methods suffer from the fundamental limitation of being able to recognize only the folds that have been already observed. The results of structural genomics initiatives reveal that the majority of proteins belong to the previously characterized folds, but the percentage of structures with “new” folds or variations of “old” folds that cannot be accurately predicted by FR methods remains significant. On the other hand, physics-based methods for “ab initio” folding are extremely costly in terms of the computing power even if they use reduced representation and do not yet successfully fold large proteins. However, even when a novel fold is discovered, it usually turns out to be composed of common structural motifs, often at the level of supersecondary or even larger structures. Levitt and coworkers (Kolodny et al. 2002) have demonstrated that all proteins in the PDB can be modeled accurately from rigid fragments of unrelated proteins that are concatenated without any degrees of freedom. Skolnick and coworkers (Kihara and Skolnick 2003; Zhang et al. 2005) have shown that most of proteins in the PDB have significant structural alignments to other proteins in a different secondary structure and fold class. Thus, modeling of new folds can be greatly facilitated by assembling them from fragments of known structures identified by “local fold recognition,” rather than attempting to model the whole process of protein folding based on first principles.

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The success of methods based on fragment assembly lies not only in the restriction of the conformational space, but which can be also achieved by other reduced models (e.g., “pure” lattice models) that are less successful. As emphasized by Takada and coworkers (Chikenji et al. 2003), one of the problems of the contemporary energy functions, those based on physics and statistics alike, it the limited ability to capture the subtleties of interactions between the neighboring residues (side-chain/main-chain hydrogen bonding, side-chain configurational entropy loss, etc.), which govern the local torsional propensities. Computing the local interaction energies “ab initio” may lead to accumulation of inaccuracies and greatly decrease the chances of obtaining a globally correct model. Methods that utilize fragments avoid this problem by sampling local conformations that exist in native protein structures, which provides implicit, yet accurate representation of local interactions. Thus, a single-fragment substitution corresponds to instantly transporting the modeled protein from one local energy minimum to another, without the necessity of overcoming local energetic barriers. This enormously speeds up the search for the global energy minimum and allows shifting the focus to the generation of non-local conformational changes and identification of globally native-like structures. The conservation of local structure may have not only physical, but also evolutionary sense. Lupas et al. (2001) proposed a scenario, in which modern proteins evolved from ancient short-peptide ancestors, called antecedent domain segments (ADSs). They suggested that the ancestors of contemporary (sub)domains arose by spontaneous non-covalent association of peptides with native-like and/or tertiarylike structural features, and since such assemblies provided functional advantage (e.g., due to improved stability of the individual fragments or their increased efficient concentration), the fusion of primitive genes encoding these fragments was preferentially selected by evolution. It is noteworthy that attempts to form folded and functional proteins by recombination revealed that successfully recombined fragments called “schemas” often correspond to known supersecondary structural elements (Voigt et al. 2002). This hypothetical “mix-and-join” scenario convincingly explains not only the structure of repetitive proteins, such as propellers, TIM-barrels, and helical bundles, but may also be invoked to explain the origin of more complicated and asymmetrical domains (Soding and Lupas 2003).

10.7 Conclusions and Outlook In the earlier version of this chapter the main author (J.M.B.) predicted that in the “near future” with respect to the year of 2006 (i.e., near past with respect to the year of 2010) we should have seen more integration of the most successful approaches, that is, meta-prediction and assembly of fragments, and further convergence of the evolutionary and physical schools. In opinion of the authors of this chapter this indeed has happened. Essentially all methods that currently score best in CASP rely on some sort of meta-prediction, either with the use of external servers or ones constructed in house. In parallel, emphasis increases on the use of physics-based methods for the refinement of models that are close to the native structure, but could

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be even closer. So what is our prediction of the nearest future, e.g., a few years following the publication of this volume? The results of recent CASP demonstrate that the progress in protein structure prediction is negligible and comes more from the area of “information technology” than “science.” In our opinion this indicates that the field of protein structure prediction has grown old and the only alternative to retirement is a major breakthrough rather than just recombination of what is already available. While we keep our fingers crossed for such a breakthrough on any line of the stalled protein front, we predict that an increased number of researchers will turn away from the field of protein structure prediction to move to other fields. One of the interesting directions is the testing of the applicability of techniques developed for protein three-dimensional structure prediction in the emerging field of RNA threedimensional structure prediction, which on the other hand offers many interesting solutions that could be used to refresh the aging field of protein bioinformatics. The Baker group has already developed a version of “ROSETTA for RNA”, dubbed FARNA (Fragment Assembly of RNA) (Das and Baker 2007). So is the RNA structure the New World for conquistadors from the Protein Continent? It remains to be seen. See you there! Acknowledgments Our recent research in the field of structural bioinformatics has been funded by the Polish Ministry of Scientific Research and Higher Education (grant numbers: POIG.02.03.00-00-003/09, 188/N-DFG/2008/0, N301 106 32/3600, and PBZ-MNiI2/1/2005), by the NIH (grant numbers R01GM081680 and R03TW007163-01), by the European Commission (grant numbers LSHG-CT-2003-503238, LSHG-CT-2005-518238, MRTN-CT-2005019566, 229676, and RIDS 011934), and by HFSP program RGP 55/2006.

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