Advances in Chemical Physics, Advancing Theory for Kinetics and Dynamics of Complex, Many-Dimensional Systems: Clusters and Proteins (Volume 145)

ADVANCING THEORY FOR KINETICS AND DYNAMICS OF COMPLEX, MANY-DIMENSIONAL SYSTEMS ADVANCES IN CHEMICAL PHYSICS, VOLUME 145...

Author: Tamiki Komatsuzaki | R. Stephen Berry | David M. Leitner | Stuart A. Rice | Aaron R. Dinner

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ADVANCING THEORY FOR KINETICS AND DYNAMICS OF COMPLEX, MANY-DIMENSIONAL SYSTEMS ADVANCES IN CHEMICAL PHYSICS, VOLUME 145

EDITORIAL BOARD Moungi G. Bawendi, Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA Kurt Binder, Condensed Matter Theory Group, Institut für Physik, Johannes GutenbergUniversität Mainz, Mainz, Germany William T. Coffey, Department of Electronics and Electrical Engineering, Trinity College, University of Dublin, Dublin, Ireland Karl F. Freed, Department of Chemistry, James Franck Institute, University of Chicago, Chicago, Illinois, USA Daan Frenkel, Department of Chemistry, Trinity College, University of Cambridge, Cambridge, United Kingdom Pierre Gaspard, Center for Nonlinear Phenomena and Complex Systems, Université Libre de Bruxelles, Brussels, Belgium Martin Gruebele, School of Chemical Sciences and Beckman Institute, Director of Center for Biophysics and Computational Biology, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA Jean-Pierre Hansen, Department of Chemistry, University of Cambridge, Cambridge, United Kingdom Gerhard Hummer, Chief, Theoretical Biophysics Section, NIDDK-National Institutes of Health, Bethesda, Maryland, USA Ronnie Kosloff, Department of Physical Chemistry, Institute of Chemistry and Fritz Haber Center for Molecular Dynamics, The Hebrew University of Jerusalem, Israel Ka Yee Lee, Department of Chemistry and The James Franck Institute, The University of Chicago, Chicago, Illinois, USA Todd J. Martinez, Department of Chemistry, Stanford University, Stanford, California, USA Shaul Mukamel, Department of Chemistry, University of California at Irvine, Irvine, California, USA Jose Onuchic, Department of Physics, Co-Director Center for Theoretical Biological Physics, University of California at San Diego, La Jolla, California, USA Steven Quake, Department of Physics, Stanford University, Stanford, California, USA Mark Ratner, Department of Chemistry, Northwestern University, Evanston, Illinois, USA David Reichmann, Department of Chemistry, Columbia University, New York, New York, USA George Schatz, Department of Chemistry, Northwestern University, Evanston, Illinois, USA Norbert Scherer, Department of Chemistry, James Franck Institute, University of Chicago, Chicago, Illinois, USA Steven J. Sibener, Department of Chemistry, James Franck Institute, University of Chicago, Chicago, Illinois, USA Andrei Tokmakoff, Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA Donald G. Truhlar, Department of Chemistry, University of Minnesota, Minneapolis, Minnesota, USA John C. Tully, Department of Chemistry, Yale University, New Haven, Connecticut, USA

ADVANCING THEORY FOR KINETICS AND DYNAMICS OF COMPLEX, MANY-DIMENSIONAL SYSTEMS: CLUSTERS AND PROTEINS ADVANCES IN CHEMICAL PHYSICS, VOLUME 145

Edited by TAMIKI KOMATSUZAKI, R. STEPHEN BERRY, DAVID M. LEITNER Series Editors STUART A. RICE Department of Chemistry and The James Franck Institute, The University of Chicago, Chicago, Illinois

AARON R. DINNER Department of Chemistry and The James Franck Institute, The University of Chicago, Chicago, Illinois

Copyright © 2011 by John Wiley & Sons, Inc. All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com. Library of Congress Cataloging-in-Publication Data: Library of Congress Catalog Number: 58-9935 ISBN: 978-0-470-64371-6 Printed in Singapore oBook ISBN: 9781118087817 ePDF ISBN: 9781118087831 ePub ISBN: 9781118087824 10 9 8 7 6 5 4 3 2 1

CONTRIBUTORS TO VOLUME 145 Akinori Baba, Molecule & Life Nonlinear Sciences Laboratory, Research Institute for Electronic Science, Hokkaido University, Kita 20 Nishi 10, Kita-ku, Sapporo 001-0020, Japan; Core Research for Evolutional Science and Technology (CREST), Japan Science and Technology Agency (JST), Kawaguchi, Saitama 332-0012, Japan R. Stephen Berry, Department of Chemistry, The University of Chicago, 929 East 57th Street, Chicago, IL 60637, USA Sotaro Fuchigami, Department of Supramolecular Biology, Graduate School of Nanobioscience, Yokohama City University, 1-7-29 Suehiro-cho, Tsurumi-ku, Yokohama 230-0045, Japan Hiroshi Fujisaki, Molecular Scale Team, Integrated Simulation of Living Matter Group, Computational Science Research Program, RIKEN, 2-1 Hirosawa, Wako-shi, Saitama 351-0198, Japan, and Department of Physics, Nippon Medical School, 2-297-2 Kosugi-cho, Nakahara, Kawasaki, Kanagawa 211-0063, Japan Shinnosuke Kawai, Molecule & Life Nonlinear Sciences Laboratory, Research Institute for Electronic Science, Hokkaido University, Kita 20 Nishi 10, Kita-ku, Sapporo 001-0020, Japan Akinori Kidera, Department of Supramolecular Biology, Graduate School of Nanobioscience, Yokohama City University, 1-7-29 Suehiro-cho, Tsurumi-ku, Yokohama 230-0045, Japan; Molecular Scale Team, Integrated Simulation of Living Matter Group, Computational Science Research Program, RIKEN, 2-1 Hirosawa, Wako-shi, Saitama 351-0198, Japan Tamiki Komatsuzaki, Molecule & Life Nonlinear Sciences Laboratory, Research Institute for Electronic Science, Hokkaido University, Kita 20 Nishi 10, Kita-ku, Sapporo 001-0020, Japan; Core Research for Evolutional Science and Technology (CREST), Japan Science and Technology Agency (JST), Kawaguchi, Saitama 332-0012, Japan David M. Leitner, Department of Chemistry and Chemical Physics Program, University of Nevada, Reno, NV 89557-0216, USA Chun-Biu Li, Molecule & Life Nonlinear Sciences Laboratory, Research Institute for Electronic Science, Hokkaido University, Kita 20 Nishi 10, Kita-ku, Sapporo 001-0020, Japan v

vi

contributors to volume 145

Yasuhiro Matsunaga, Molecular Scale Team, Integrated Simulation of Living Matter Group, Computational Science Research Program, RIKEN, 2-1 Hirosawa, Wako-shi, Saitama 351-0198, Japan Akira Shojiguchi, Department of Physics, Faculty of Science, Nara Women’s University, Kitauoyahigashimachi, Nara 630-8506, Japan John E. Straub, Department of Chemistry, Boston University, 590 Commonwealth Avenue, SCI 503, Boston, MA 02215, USA Hiroshi Teramoto, Molecule & Life Nonlinear Sciences Laboratory, Research Institute for Electronic Science, Hokkaido University, Kita 20 Nishi 10, Kitaku, Sapporo 001-0020, Japan Mikito Toda, Department of Physics, Faculty of Science, Nara Women’s University, Kitauoyahigashimachi, Nara 630-8506, Japan Yong Zhang, Department of Chemical and Biomolecular Engineering, University of Notre Dame, 182 Fitzpatrick Hall, Notre Dame, IN 46556-5637, USA

INTRODUCTION Few of us can any longer keep up with the flood of scientific literature, even in specialized subfields. Any attempt to do more and be broadly educated with respect to a large domain of science has the appearance of tilting at windmills. Yet the synthesis of ideas drawn from different subjects into new, powerful, general concepts is as valuable as ever, and the desire to remain educated persists in all scientists. This series, Advances in Chemical Physics, is devoted to helping the reader obtain general information about a wide variety of topics in chemical physics, a field that we interpret very broadly. Our intent is to have experts present comprehensive analyses of subjects of interest and to encourage the expression of individual points of view. We hope that this approach to the presentation of an overview of a subject will both stimulate new research and serve as a personalized learning text for beginners in a field. Stuart A. Rice

vii

CONTENTS Preface

xi

Non-Markovian Theory of Vibrational Energy Relaxation and its Applications to Biomolecular Systems

1

By Hiroshi Fujisaki, Yong Zhang, and John E. Straub Protein Functional Motions: Basic Concepts and Computational Methodologies

35

By Sotaro Fuchigami, Hiroshi Fujisaki, Yasuhiro Matsunaga, and Akinori Kidera Non-Brownian Phase Space Dynamics of Molecules, the Nature of Their Vibrational States, and Non-RRKM Kinetics

83

By David M. Leitner, Yasuhiro Matsunaga, Chun-Biu Li, Tamiki Komatsuzaki, Akira Shojiguchi, and Mikito Toda Dynamical Reaction Theory Based on Geometric Structures in Phase Space

123

By Shinnosuke Kawai, Hiroshi Teramoto, Chun-Biu Li, Tamiki Komatsuzaki, and Mikito Toda Ergodic Problems for Real Complex Systems in Chemical Physics

171

By Tamiki Komatsuzaki, Akinori Baba, Shinnosuke Kawai, Mikito Toda, John E. Straub, and R. Stephen Berry Author Index

221

Subject Index

247

ix

PREFACE The simple descriptions of molecular dynamics that we envision for small molecules, and apply to other areas of chemical physics, such as chemical kinetics, are often incomplete or even inappropriate when carried over to large, complex molecules, such as those encountered in biology or nanoscale materials. New tools are needed to sort through the dynamics on the energy landscape that underlie the functional motion of biological molecules and energy transport within them. The aim of this volume is to present some of the theoretical and computational methods that have been developed recently to address this challenge. The following chapters provide a summary of topics presented by the authors at several recent workshops in Japan and the United States. The first two chapters address dynamics and energy flow in biological molecules. Chapter 1 focuses on fast motions and energy transfer in biomolecules, mainly proteins, on the pico- to nanosecond timescale. Besides providing a general introduction to the field, this chapter presents a review of a non-Markovian theory for calculating vibrational energy transfer rates and provides a number of examples. Chapter 2 addresses functional motions of proteins, which can span a wide range of timescales, from nanoseconds to seconds. This chapter provides a review of general concepts and recent computational tools that have been put forth to elucidate functional motions. Chapter 3 addresses dynamics and energy flow within basins on the energy landscape. While developing kinetic models for transitions between such basins is relatively simple if the dynamics within a basin is ergodic, the situation is much more complex when the assumptions of ergodicity break down. This chapter summarizes our understanding of the nature of nonergodic dynamics and the corresponding mixed phase space from a classical perspective, and reviews a quantum mechanical theory for corresponding systems with a mixed vibrational state space. The latter is also used to correct Rice–Ramsperger–Kassel–Marcus (RRKM) theory predictions of the unimolecular reaction rate when dynamics of the reactant is nonergodic. Continuing along these lines, Chapter 4 presents a review of recent work on non-RRKM kinetics from a classical phase space geometrical perspective. Finally, ergodicity in biological systems is further explored in Chapter 5, where local measures of ergodic and chaotic behavior are related to the topography of the energy landscape.

xi

xii

preface

The chapters of this volume summarize important areas in our current understanding of dynamics and configurational changes of biological molecules and other many-dimensional systems. We hope that the material presented here will contribute further to the rapid development in the theory of these complex processes. Tamiki Komatsuzaki R. Stephen Berry David M. Leitner Guest Editors

NON-MARKOVIAN THEORY OF VIBRATIONAL ENERGY RELAXATION AND ITS APPLICATIONS TO BIOMOLECULAR SYSTEMS HIROSHI FUJISAKI,1,2 YONG ZHANG,3 and JOHN E. STRAUB4 1 Molecular

Scale Team, Integrated Simulation of Living Matter Group, Computational Science Research Program, RIKEN, 2-1 Hirosawa, Wako-shi, Saitama 351-0198, Japan 2 Department of Physics, Nippon Medical School 2-297-2 Kosugi-cho, Nakahara, Kawasaki, Kanagawa 211-0063, Japan 3 Department of Chemical and Biomolecular Engineering, University of Notre Dame, 182 Fitzpatrick Hall, Notre Dame, IN 46556-5637, USA 4 Department of Chemistry, Boston University, 590 Commonwealth Avenue, SCI 503, Boston, MA 02215, USA

CONTENTS I. Introduction II. Normal Mode Concepts Applied to Protein Dynamics III. Derivation of non-Markovian VER formulas A. Multidimensional Relaxing Mode Coupled to a Static Bath B. One-Dimensional Relaxing Mode Coupled to a Fluctuating Bath C. Limitations of the VER Formulas and Comments IV. Applications of the VER Formulas to Vibrational Modes in Biomolecules A. N-Methylacetamide (NMA) 1. N-Methylacetamide in Vacuum 2. N-Methylacetamide/Water Cluster 3. N-Methylacetamide in Water Solvent B. Cytochrome c in Water C. Porphyrin V. Summary and Discussion Acknowledgments References

Advancing Theory for Kinetics and Dynamics of Complex, Many-Dimensional Systems: Clusters and Proteins, Advances in Chemical Physics, Volume 145, Edited by Tamiki Komatsuzaki, R. Stephen Berry, and David M. Leitner. © 2011 John Wiley & Sons, Inc. Published 2011 by John Wiley & Sons, Inc.

1

2

hiroshi fujisaki et al. I. INTRODUCTION

Energy transfer (relaxation) phenomena are ubiquitous in nature. At a macroscopic level, the phenomenological theory of heat (Fourier law) successfully describes heat transfer and energy flow. However, its microscopic origin is still under debate. This is because the phenomena can contain many-body, multiscale, nonequilibrium, and even quantum mechanical aspects, which present significant challenges to theories addressing energy transfer phenomena in physics, chemistry, and biology [1]. For example, heat generation and transfer in nanodevices is a critical problem in the design of nanotechnology. In molecular physics, it is well known that vibrational energy relaxation (VER) is an essential aspect of any quantitative description of chemical reactions [2]. In the celebrated RRKM theory of an absolute reaction rate for isolated molecules, it is assumed that the intramolecular vibrational energy relaxation (IVR) is much faster than the reaction itself. Under certain statistical assumptions, the reaction rate can be derived [3]. For chemical reactions in solutions, the transition state theory and its extension such as Kramer’s theory and the Grote–Hynes theory have been developed [4, 5] and applied to a variety of chemical systems including biomolecular systems [6]. However, one cannot always assume separation of timescales. It has been shown that a conformational transition (or reaction) rate can be modulated by the IVR rate [7]. As this brief survey demonstrates, a detailed understanding of IVR or VER is essential to study the chemical reaction and conformation change of molecules. A relatively well-understood class of VER is a single vibrational mode embedded in (vibrational) bath modes. If the coupling between the system and the bath modes is weak (or assumed to be weak), a Fermi’s-golden-rule style formula derived using second-order perturbation theory [8–10] may be used to estimate the VER rate. However, the application of such theories to real molecular systems poses several (technical) challenges, including how to choose force fields, how to separate quantum and classical degrees of freedom, or how to treat the separation of timescales between system and bath modes. Multiple solutions have been proposed to meet those challenges leading to a variety of theoretical approaches to the treatment of VER [11–16]. These works using Fermi’s golden rule are based on quantum mechanics and are suitable for the description of high-frequency modes (more than thermal energy 200 cm−1 ), on which nonlinear spectroscopy has recently focused [17–20]. In this chapter, we summarize our recent work on VER of high-frequency modes in biomolecular systems. In our previous work, we have concentrated on the VER rate and mechanisms for proteins [21]. Here we shall focus on the time course of the VER dynamics. We extend our previous Markovian theory of VER to a non-Markovian theory applicable to a broader range of chemical systems [22, 23]. Recent time-resolved spectroscopy can detect the time course of VER dynamics (with femtosecond resolution), which may not be accurately described by a single

non-markovian theory of vibrational energy relaxation

3

timescale. We derive new formulas for VER dynamics and apply them to several interesting cases, where comparison to experimental data is available. This chapter is organized as follows: In Section II, we briefly summarize the normal mode concepts in protein dynamics simulations, on which we build our non-Markovian VER theory. In Section III, we derive VER formulas under several assumptions and discuss the limitations of our formulas. In Section IV, we apply the VER formulas to several situations: the amide I modes in isolated and solvated N-methylacetamide and cytochrome c, and two in-plane modes (ν4 and ν7 modes) in a porphyrin ligated to imidazole. We employ a number of approximations in describing the potential energy surface (PES) on which the dynamics takes place, including the empirical CHARMM [24] force-field and density functional calculations [25] for the small parts of the system (N-methylacetamide and porphyrin). We compare our theoretical results with experiment when available, and find good agreement. We can deduce the VER mechanism based on our theory for each case. In Section V, we summarize and discuss the further aspects of VER in biomolecules and in nanotechnology (molecular devices). II.

NORMAL MODE CONCEPTS APPLIED TO PROTEIN DYNAMICS

Normal mode provides a powerful tool in exploring molecular vibrational dynamics [26] and may be applied to biomolecules as well [27]. The first normal mode calculations for a protein were performed for BPTI protein [28]. Most biomolecular simulation softwares support the calculation of normal modes [24, 29, 30]. However, the calculation of a mass-weighted Hessian Kij , which requires the second derivatives of the potential energy surface, with elements defined as Kij = √

∂2 V 1 mi mj ∂xi ∂xj

(1)

can be computationally demanding. Here mi is the mass, xi is the coordinate, and V is the potential energy of the system. Efficient methods have been devised including torsional angle normal mode [31], block normal mode [32], and the iterative mixed-basis diagonalization (DIMB) methods [33], among others. An alternative direction for efficient calculation of a Hessian is to use coarse-grained models such as elastic [34] or Gaussian network [35] models. From normal mode analysis (or instantaneous normal mode analysis [36]), the frequencies, the density of states, and the normal mode vectors can be calculated. In particular, the last quantity is important because it is known that the lowest eigenvectors may describe the functionally important motions such as large-scale conformational change, a subject that is the focus of another chapter of this volume [37]. There is no doubt as to the usefulness of normal mode concepts. However, for molecular systems, it is always an approximate model as higher order nonlinear

4

hiroshi fujisaki et al.

coupling and intrinsic anharmonicity become essential. To describe energy transfer (or relaxation) phenomena in a protein, Moritsugu, Miyashita, and Kidera (MMK) introduced a reduced model using normal modes with third- and fourth-order (3) (4) anharmonicity [38], Cklm and Cklmn , respectively, V ({qk }) =

ω2

k 2 qk

k

2

+

1 (3) 1 (4) Cklm qk ql qm + Cklmn qk ql qm qn 3! 4! klm

(2)

klmn

with (3)

Cklm ≡ (4)

Cklmn ≡

∂3 V ∂qk ∂ql ∂qm

(3)

∂4 V ∂qk ∂ql ∂qm ∂qn

(4)

where qk denotes the normal mode calculated by the Hessian Kij and ωk is the normal mode frequency. Classical (and harmonic) Fermi resonance [39] is a key ingredient in the MMK theory of energy transfer derived from observations of all-atom simulations of myoglobin at zero temperature (see Fig. 1). At finite temperature, nonresonant effects become important and clear interpretation of the numerical results becomes difficult within the classical approximation. Nagaoka and coworkers [40] identified essential vibrational modes in vacuum simulations of myoglobin and connected these modes to the mechanism of “heme cooling” explored experimentally by Mizutani and Kitagawa [18]. Contemporaneously, nonequilibrium MD simulations of solvated myoglobin carried out by Sagnella and Straub provided the first detailed and accurate simulation of heme cooling dynamics [41]. That work supported the conjecture that the motion

Figure 1. (a) The excited eigenvector depicted by arrows in myoglobin. (b) Classical simulation of mode-specific energy transfer in myoglobin at zero temperature. (Reproduced with permission from Ref. 38. Copyright 2009 by the American Physical Society.)

non-markovian theory of vibrational energy relaxation

5

Figure 2. Nonequilibrium MD simulation of energy flow from the excited amide I mode in Nmethylacetamide in heavy water. See also Fig. 3. (Reproduced with permission from Ref. 42. Copyright 2009 by the American Institute of Physics.)

similar to those modes identified by Nagaoka plays an important role in energy flow pathways. Nguyen and Stock explored the vibrational dynamics of the small molecule, N-methylacetamide (NMA) often used as a model of the peptide backbone [42]. Using nonequilibrium MD simulations of NMA in heavy water, VER was observed to occur on a picosecond timescale for the amide I vibrational mode (see Fig. 2). They used the instantaneous normal mode concept [36] to interpret their result and noted the essential role of anharmonic coupling. Leitner also used the normal mode concept to describe energy diffusion in a protein and found an interesting link between the anomalous heat diffusion and the geometrical properties of a protein [43]. In terms of vibrational spectroscopy, Gerber and coworkers calculated the anharmonic frequencies in BPTI, within the VSCF level of theory [44], using the reduced model [Eq. (2)]. Yagi, et al. refined this type of anharmonic frequency calculation for large molecular systems with more efficient methods [45], appropriate for applications to biomolecules such as DNA base pair [46]. Based on the reduced model [Eq. (2)] with higher order nonlinear coupling, Leitner also studied quantum mechanical aspects of VER in proteins, by employing the Maradudin–Fein theory based on Fermi’s golden rule [12]. Using the same model, Fujisaki, Zhang, and Straub focused on more detailed aspects of VER in biomolecular systems and calculated the VER rate, mechanisms, or pathways, using their non-Markovian perturbative formulas (described in Section III).

6

hiroshi fujisaki et al.

As this brief survey demonstrates, the normal mode concept is a powerful tool that provides significant insight into mode-specific vibrational dynamics and energy transfer in proteins, when anharmonicity of the potential energy surface is taken into account. III.

DERIVATION OF NON-MARKOVIAN VER FORMULAS

We have derived a VER formula for the simplest situation, a one-dimensional relaxing oscillator coupled to a “static” bath [22]. Here we extend this treatment to two more general directions: (a) multidimensional relaxing modes coupled to a “static” bath and (b) a one-dimensional relaxing mode coupled to a “fluctuating” bath [47]. A.

Multidimensional Relaxing Mode Coupled to a Static Bath

We take the following time-independent Hamiltonian: H = H0S + HB + V 0 =

H0S

(5)

+ VB + HB + V − VB 0

= HS + HB + V

(6) (7)

where HS ≡ H0S + VB

(8)

V ≡ V − VB

(9)

0

In previous work [22], we have considered only a single one-dimensional oscillator as the system. Here we extend that treatment to the case of an NS -dimensional oscillator system. That is, NS p2i ωi2 2 HS = + q + V ({qi }) (10) 2 2 i i=1

HB =

NB 2 p

ω2 + α qα2 2 2 α

α=1

V=−

NS

qi δFi ({qα })

(11)

(12)

i=1

where V ({qi }) is the interaction potential function between NS system modes that can be described by, for example, the reduced model, Eq. (2). The simplest case

non-markovian theory of vibrational energy relaxation

7

V ({qi }) = 0 is trivial as each system mode may be treated separately within the perturbation approximation for V. We assume that |k is a certain state in the Hilbert space spanned by HS . Then the reduced density matrix is iHS t/ ˜ |n (ρS )mn (t) = m|e−iHS t/ Tr B {ρ(t)}e

(13)

where the tilde denotes the interaction picture. Substituting the time-dependent perturbation expansion 1 t ˜ ˜ = ρ(0) + ρ(t) dt [V(t ), ρ(0)] i 0 t t 1 ˜ ), [V(t ˜ ), ρ(0)]] + · · · + dt dt [V(t (14) (i)2 0 0 into the above, we find (1) (2) (ρS )mn (t) (ρS )(0) mn (t) + (ρS )mn (t) + (ρS )mn (t) + · · ·

(15)

where −iHS t/ ρS (0)eiHS t/ |n, (ρS )(0) mn (t) = m|e

= m(−t)|ρS (0)|n(−t) = m|ρS (t)|n (16) t t 1 ˜ ), [V(t ˜ ), ρ(0)]]}eiHS t/ |n (ρS )(2) dt dt m|e−iHS t/ TrB {[V(t mn (t) = (i)2 0 0 t t 1 = dt dt m(−t)|[qi (t )qj (t )ρS (0) (i)2 0 0 i,j

− qj (t )ρS (0)qi (t )]|n(−t)δFi (t )δFj (t )B t t 1 + dt dt m(−t)|[ρS (0)qj (t )qi (t ) 2 (i) 0 0 i,j

− qi (t )ρS (0)qj (t )]|n(−t)δFj (t )δFi (t )B

(17)

Here we have defined |m(t) = e−iHS t/ |m and taken (ρS )(1) mn (t) = 0. Recognizing that we must evaluate expressions of the form Rmn;ij (t; t , t ) = m(−t)|[qi (t )qj (t )ρS (0)|n(−t), −m(−t)|qj (t )ρS (0)qi (t )]|n(−t)

Cij (t , t ) = δFi (t )δFj (t )B

(18) (19)

8

hiroshi fujisaki et al.

and their complex conjugates, R∗nm;ij (t; t , t ), Cij∗ (t , t ), the second-order contribution can be written as t t 1 (ρS )(2) (t) = dt dt [Rmn;ij (t; t , t )Cij (t , t ) mn 2 (i) 0 0 i,j

+R∗nm;ij (t; t , t )Cij∗ (t , t )]

(20)

We can separately treat the two terms. Assuming that we can solve HS |a = Ea |a, we find m|a(qi )ab (qj )bc (ρS )cd d|n Rmn;ij (t; t , t ) = abcd

×e−i(Ea −Ed )t−i(Eb −Ea )t −i(Ec −Eb )t − m|a(qj )ab (ρS )bc (qi )cd d|n abcd

×e−i(Ea −Ed )t−i(Ed −Ec )t −i(Eb −Ea )t

(21)

For the bath-averaged term, we assume the following force due to third-order nonlinear coupling of system mode i to the normal modes, α and β, of the bath [21]: Ciαβ (qα qβ − qα qβ ) (22) δFi ({qα }) = α,β

and we have [21] ++ +− Cij (t , t ) = R−− ij (t , t ) + Rij (t , t ) + Rij (t , t )

(23)

with R−− ij (t , t ) =

2 Dαβ;ij (1 + nα )(1 + nβ )e−i(ωα +ωβ )(t −t ) 2

(24)

α,β

R++ ij (t , t ) =

2 2

2 R+− ij (t , t ) =

Dαβ;ij nα nβ ei(ωα +ωβ )(t −t

)

(25)

α,β

Dαβ;ij (1 + nα )nβ e−i(ωα −ωβ )(t −t

)

(26)

α,β

where Dαβ;ij =

Ciαβ Cjαβ ωα ωβ

and nα is the thermal population of the bath mode α.

(27)

non-markovian theory of vibrational energy relaxation

9

This formula reduces to our previous result for a one-dimensional system oscillator [22] when NS = 1 and all indices (i, j) are suppressed. Importantly, this formula can be applied to situations where it is difficult to define a “good” normal mode to serve as a one-dimensional relaxing mode, as in the case of the CH stretching modes of a methyl group [21]. However, expanding to an NS dimensional system adds the burden of solving the multidimensional Schr¨odinger equation HS |a = Ea |a. To address this challenge, we may employ vibrational self-consistent field (VSCF) theory and its extensions developed by Bowman and coworkers [48] implemented in MULTIMODE program of Carter and Bowman [49] or in the SINDO program of Yagi and coworkers [50]. As in the case of our previous theory of a one-dimensional system mode, we must calculate NS -tiple third-order coupling constants Ciαβ (i = 1, 2, ..., NS ) for all bath modes α and β. B.

One-Dimensional Relaxing Mode Coupled to a Fluctuating Bath

We start from the following time-dependent Hamiltonian: H(t) = H0S (t) + HB (t) + V0 (t)

(28)

= H0S (t) + V(t)B + HB (t) + V0 (t) − V(t)B

(29)

= HS (t) + HB (t) + V(t)

(30)

where HS (t) ≡ H0S (t) + V(t)B

(31)

V(t) ≡ V0 (t) − V(t)B

(32)

with the goal of solving the time-dependent Schr¨odinger equation i

∂| (t) = [HS (t) + HB (t) + V(t)]| (t) = [H0 (t) + V(t)]| (t) ∂t

(33)

By introducing a unitary operator U0 (t) = US (t)UB (t) d U0 (t) = H0 (t)U0 (t) dt d i US (t) = HS (t)US (t) dt d i UB (t) = HB (t)UB (t) dt i

(34) (35) (36)

we can derive an “interaction picture” von Neumann equation i

d ˜ ˜ = [V(t), ˜ ρ(t) ρ(t)] dt

(37)

10

hiroshi fujisaki et al.

where ˜ = U † (t)V(t)U0 (t) V(t) 0 †

˜ = U0 (t)ρ(t)U0 (t) ρ(t)

(38) (39)

We assume the simple form of a harmonic system and bath, but allow fluctuations in the system and bath modes modeled by time-dependent frequencies †

HS (t) = ωS (t)(aS aS + 1/2) HB (t) = ωα (t)(aα† aα + 1/2)

(40) (41)

α

The unitary operators generated by these Hamiltonians are US (t) = e UB (t) = e

−i −i

t 0

t 0

†

dτωS (τ)(aS aS +1/2) dτ

† ω (τ)(aα aα +1/2) α α

(42) (43)

and the time evolution of the annihilation operators is given by †

US (t)aS US (t) = aS e †

UB (t)aα UB (t) = aα e

−i

−i

t 0

t 0

dτωS (τ)

(44)

dτωα (τ)

(45)

To simplify the evaluation of the force autocorrelation function, we assume that the temperature is low or the system mode frequency is high as a justification for the approximation. Substituting the above result into the force autocorrelation function calculated by the force operator, Eq. (22), we find δF(t )δF(t )

CSαβ (t )CSαβ (t ) 2

2 ωα (t )ωβ (t )ωα (t )ωβ (t ) α,β

×e−i[αβ (t )−αβ (t

)]

(46)

where S (t) =

t

dτωS (τ)

(47)

dτ[ωα (τ) + ωβ (τ)]

(48)

0

αβ (t) =

0

t

non-markovian theory of vibrational energy relaxation

11

Substituting this approximation into the perturbation expansion Eqs. (15), (16), (17), we obtain our final result: t t CSαβ (t )CSαβ (t ) (ρS )00 (t) dt dt

2 ωS (t )ωα (t )ωβ (t )ωS (t )ωα (t )ωβ (t ) 0 0 α,β

× cos S (t ) − αβ (t ) − S (t ) + αβ (t )

(49)

which provides a dynamic correction to the previous formula [22]. The timedependent parameters ωS (t), ωα (t), and CSαβ (t) may be computed from a running trajectory using instantaneous normal mode analysis [36]. This result was first derived by Fujisaki and Stock [47], and applied to the VER dynamics of Nmethylacetamide as described below. This correction eliminates the assumption that the bath frequencies are static on the VER timescale. For the case of a static bath, the frequency and coupling parameters are timeindependent and this formula reduces to the previous one-dimensional formula (when the off-resonant terms are neglected) [22]: (ρS )00 (t)

2 CSαβ 1 − cos[(ωS − ωα − ωβ )t] 2ωS ω α ωβ (ωS − ωα − ωβ )2

(50)

α,β

Note that Bakker derived a similar fluctuating Landau–Teller formula in a different manner [51]. It was successfully applied to molecular systems by Sibert and coworkers [52]. However, the above formula differs from Bakker’s as (a) we use the instantaneous normal mode analysis to parameterize our expression and (b) we do not take the Markov limit. Our formula can describe both the time course of the density matrix and the VER rate. Another point is that we use the cumulant-type approximation to calculate the dynamics. When we calculate an excited state probability, we use (ρS )11 (t) = 1 − (ρS )00 (t) exp{−(ρS )00 (t)}

(51)

Of course, this is valid for the initial process ((ρS )00 (t) 1), but, at longer timescales, we take (ρS )11 (t) exp{−(ρS )00 (t)} because the naive formula (ρS )11 (t) = 1 − (ρS )00 (t) can be negative, which is unphysical [47]. C.

Limitations of the VER Formulas and Comments

There are several limitations to the VER formulas derived above. The most obvious is that they are second-order perturbative formulas and rely on a short-time approximation. As far as we know, however, there exists no nonperturbative quantum mechanical treatment of VER applicable to large molecular systems. It is prohibitive to treat the full molecular dynamics quantum mechanically [53] for large molecules. Moreover, while there exist several mixed quantum classical methods [11]

12

hiroshi fujisaki et al.

that may be applied to the study of VER, there is no guarantee that such approximate methods work better than the perturbative treatment [54]. Another important limitation is the adaptation of a normal mode basis set, a natural choice for molecular vibrations. Because of the normal mode analysis, the computation can be burdensome. When we employ instantaneous normal mode analysis [36], there is a concern about the imaginary frequency modes. For the study of high-frequency modes, this may not be significant. However, for the study of low-frequency modes, the divergence of quantum (or classical) dynamics due to the presence of such imaginary frequency modes is a significant concern. For the study of low-frequency modes, it is more satisfactory to use other methods that do not rely on normal mode analysis such as semiclassical methods [55] or path integral methods [56]. We often use “empirical” force fields, with which quantum dynamics is calculated. However, it is well known that the force fields underestimate anharmonicity of molecular vibrations [57]. It is often desirable to use ab initio potential energy surfaces. However, such a rigorous approach is much more demanding. Lower levels of theory can fail to match the accuracy of some empirical potentials. As a compromise, approximate potentials of intermediate accuracy, such as QM/MM potentials [58], may be appropriate. We discuss this issue further in Sections IV.A and IV.C. IV.

APPLICATIONS OF THE VER FORMULAS TO VIBRATIONAL MODES IN BIOMOLECULES

We report our quantum dynamics studies of high-frequency modes in biomolecular systems using a variety of VER formulas described in Section III. The application of a variety of theoretical approaches to VER processes will allow for a relative comparison of theories and the absolute assessment of theoretical predictions compared with experimental observations. In doing so, we address a number of fundamental questions. What are the limitations of the static bath approximation for fast VER in biomolecular systems? Can the relaxation dynamics of a relaxing amide I vibration in a protein be accurately modeled as a one-dimensional system mode coupled to a harmonic bath? Can the “fluctuating bath” model accurately capture the system dynamics when the static picture of normal modes is not “good” on the timescale of the VER process? In Sections IV.A and IV.B, our main focus is the VER of excited amide I modes in peptides or proteins. In Section IV.C, we study some vibrational modes in porphyrin ligated to imidazole, which is a mimic of a heme molecule in heme proteins including myoglobin and hemoglobin. A.

N-Methylacetamide (NMA)

NMA is a well-studied small molecule (CH3 –CO–NH–CH3 ) that serves as a convenient model of a peptide bond structure (–CO–NH–) in theory and experiment.

non-markovian theory of vibrational energy relaxation

13

As in other amino acids, there is an amide I mode, localized on the CO bond stretch, which is a useful “reporter” of peptide structure and dynamics when probed by infrared spectroscopy. Many theoretical and experimental studies on amide I and other vibrational modes (amide II and amide III) have characterized how the mode frequencies depend on the local secondary structure of peptides or proteins [59, 60]. For the accurate description of frequencies and polarizability of these modes, see Refs. 15, 16 and 61–65. The main focus of these works is the frequency sensitivity of amide modes on the molecular configuration and environment. In this case, the amide mode frequencies are treated in a quantum mechanical way, but the configuration is treated classically. With a focus on interpreting mode frequency shifts due to configuration and environment, mode coupling between amide modes and other modes is often neglected. As we are mainly interested in VER or IVR dynamics of these modes, an accurate treatment of the mode coupling is essential. Recent theoretical development of IVR dynamics in small molecules is summarized in Ref. 53. Leitner and Wolynes [7] utilized the concept of local random matrix to clarify the quantum aspects of such dynamics. The usefulness and applications of their approach are summarized both in Ref. 12 and in this volume [13]. However, these studies are focused on isolated molecules, whereas our main interest is in exploring quantum dynamics in a condensed phase. We take a step-by-step hierarchical approach. Starting from the isolated NMA molecule, we add several water molecules to form NMA–water clusters, and finally treat the condensed phase NMA–water system (see Fig. 3). With increasing complexity of our model, the accuracy of our theory, including the quality of the potential energy surface, and the accuracy of the quantum dynamics must diminish. As such, the principal focus of our account is a careful examination and validation of our procedures through comparison with accurate methods or experiments.

Figure 3. Representation of three models employed for the study of VER dynamics in Nmethylacetamide. (a) NMA, (b) NMA with three solvating water, and (c) NMA with first solvation shell derived from simulations in bulk water. ((a and b) Reproduced with permission from Ref. 72. Copyright 2009 by the American Chemical Society. (c) Reproduced with permission from Ref. 47. Copyright 2009 by the American Institute of Physics.)

14

hiroshi fujisaki et al. 1.

N-Methylacetamide in Vacuum

In our studies of isolated NMA [66, 67], we have employed both accurate potential energy surface and accurate quantum dynamics methods to explore the timescale and mechanism of VER. From the anharmonic frequency calculations and comparison to experiment [68], we concluded that B3LYP/6-31G(d) is a method of choice for computation of the electronic ground state potential surface, considering both accuracy and feasibility. For other treatments at differing levels of theory of quantum chemical calculation on NMA, see Refs. 57, 69, 70. After the construction of an accurate potential surface, there are several tractable approaches for treating the quantum dynamics for this system. The most accurate is the vibrational configuration interaction (VCI) method based on vibrational self-consistent field (VSCF) basis sets (see Refs. 48, 49, 66, 67 for details). We employed the Sindo code developed by Yagi [50]. The numerical results for the VCI calculation are shown in Fig. 4 and compared with the prediction based on the perturbative formula [Eq. (50)] and classical calculations as done in Ref. 42. Both approximate methods seem to work well, but there are caveats. The perturbative formula works only at short timescales. There is ambiguity for the classical simulation regarding how the zero point energy correction should be included (see Stock’s papers [71]). The main results for a singly deuterated NMA (NMA-d1 ) are (1) the relaxation time appears to be subpicoseconds, (2) as NMA is a small molecule, there is a recurrent phenomenon, (3) the dominant relaxation pathway involves three bath (a)

(b)

1800

1 Quantum Classical

1600

Full quantum Perturbation

0.8

1200 1000

Pi (t)

Ei (t) (cm–1)

1400

800 600

0.6

0.4

400 200

0.2

0 –200

0

0.5

1

1.5

t (ps)

2

2.5

3

0

0

0.5

1

1.5

2

2.5

3

t (ps)

Figure 4. (a) Time evolution of the energy content of the initially excited amide I mode as well as all the remaining modes of N-methylacetamide. Quantum (solid lines) and classical (broken lines) calculations obtained at the DFT/B3LYP level of theory are compared. (b) Comparison of the VCI calculation (solid lines) with the result of the perturbative calculation (broken lines) for the reduced density matrix. (Reproduced with permission from Ref. 67. Copyright 2009 by Wiley–Interscience.)

non-markovian theory of vibrational energy relaxation

15

Figure 5. The dominant bath vibrational modes coupled to the amide I mode calculated on the B3LYP/6-31G(d) potential energy surface. (Reproduced with permission from Ref. 66. Copyright 2009 by Elsevier.)

modes as shown in Fig. 5, and (4) the dominant pathways can be identified and characterized by the following Fermi resonance parameter [66, 67]:

i| V |f CSkl

(52) ∝ η≡

E (ωS − ωk − ωl ) 2ωS 2ωk 2ωl where i| V |f is the matrix element for the anharmonic coupling interaction and

E = (ωS − ωk − ωl ) is the resonance condition (frequency matching) for the system and two bath modes. Both the resonant condition ( E) and anharmonic coupling elements (CSkl ) play a role, but we found that the former affects the result more significantly. This indicates that, for the description of VER phenomena in molecules, accurate calculation of the harmonic frequencies is more important than the accurate calculation of anharmonic coupling elements. This observation is the basis for the development and application of the multiresolution methods for anharmonic frequency calculations [45, 73]. 2.

N-Methylacetamide/Water Cluster

We next examine a somewhat larger system, NMA in a water cluster [72], an interesting and important model system for exploring the response of amide vibrational modes to “solvation” [62]. The system size allows for an ab initio quantum mechanical treatment of the potential surface at a higher level of theory, B3LYP/augcc-pvdz, relative to the commonly employed B3LYP/6-31G(d). The enhancement in the level of theory significantly improves the quality of the NMA–water interaction, specifically the structure and energetics of hydrogen bonding. Since there are at most three hydrogen bonding sites in NMA, it is natural to configure three water molecules around NMA as a minimal model of “full solvation.”

16

hiroshi fujisaki et al. 1 NMA-d1

NMA-d7

ρ11(t)

0.8 0.6 T1 = 0.93 ± 0.02 ps

0.4 T1 = 0.78 ± 0.02 ps

0.2

Simulation data Exponential fit

1 NMA-d1/(D2O)3

NMA-d7/(D2O)3

ρ11(t)

0.8 0.6 0.4 T1 = 0.48 ± 0.05 ps

0.2

T1 = 0.67 ± 0.07 ps

1 NMA-d1/(D2O)3/PCM

NMA-d7/(D2O)3/PCM

ρ11(t )

0.8 0.6 0.4 T1 = 0.71 ± 0.03 ps

T1 = 0.67 ± 0.03 ps

0.2 0 0

0.2

0.4

0.6

t (ps)

0.8

1

0.2

0.4

0.6

0.8

1

t (ps)

Figure 6. Time evolution of the density matrix for the amide I mode in the NMA–water cluster system after v = 1 excitation. The derived vibrational energy relaxation time constants T1 are also provided. (Reproduced with permission from Ref. 72. Copyright 2009 by the American Chemical Society.)

See Fig. 3(b). NMA–water hydrogen bonding causes the frequency of the amide I mode to redshift. As a result, the anharmonic coupling between the relaxing mode and the other bath modes will change relative to the case of the isolated NMA. Nevertheless, we observe that the VER timescale remains subpicosecond as is the case for isolated NMA (Fig. 6). Though there are intermolecular (NMA– water) contributions to VER, they do not significantly alter the VER timescale. Another important finding is that the energy pathway from the amide I to amide II mode is “open” for the NMA–water cluster system. This result is in agreement with experimental results by Tokmakoff and coworkers [74] and recent theoretical investigation [16]. Comparison between singly (NMA-d1 ) and fully (NMA-d7 ) deuterated cases shows that the VER timescale becomes somewhat longer for the case of NMA-d7 (Fig. 6). We also discuss this phenomenon below in the context of the NMA/solvent water system.

non-markovian theory of vibrational energy relaxation

17

3. N-Methylacetamide in Water Solvent Finally, we consider the condensed phase system of NMA in bulk water [22, 47, 58]. We attempt to include the full dynamic effect of the system by generating many configurations from molecular dynamics simulations and using them to ensemble average the results. Note that in the previous examples of isolated NMA and NMA/water clusters, only one configuration at a local minimum of the potential surface was used. On the other hand, the potential energy function used is not so accurate as in the previous examples as it is not feasible to include many water molecules at a high level of theory. We have used the CHARMM force field to calculate the potential energy and to carry out molecular dynamics simulations. All simulations were performed using the CHARMM simulation program package [24] and the CHARMM22 all-atom force field [75] was employed to model the solute NMA-d1 and the TIP3P water model [76] with doubled hydrogen masses to model the solvent D2 O. We also performed simulations for fully deuterated ˚ 3 containing NMA-d7 . The peptide was placed in a periodic cubic box of (25.5 A) 551 D2 O molecules. All bonds containing hydrogens were constrained using the ˚ cutoff with a switching function for the SHAKE algorithm [77]. We used a 10 A nonbonded interaction calculations. After a standard equilibration protocol, we ran a 100 ps NVT trajectory at 300 K, from which 100 statistically independent configurations were collected. We first employed the simplest VER formula [Eqs.(50) and (51)] [22] as shown in Fig. 7. We truncated the system including only NMA and several water molecules ˚ For reasons of computational around NMA with a cutoff distance, taken to be 10 A.

(a)

(b)

1

1 DFT (0 K) CHARMM (0 K) CHARMM (300 K)

0.9

0.8

0.8

0.7

0.7

ρ11(t)

ρ11(t)

0.9

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

ωc=1.0 cm−1 ωc=10.0 cm−1 ωc=100.0 cm−1

0.2 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t (ps)

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t (ps)

Figure 7. (a) Comparison of the calculation of the ρ11 element of the reduced density matrix at different levels of theory. (b) Calculation of the density matrix with different cutoff frequencies. (Reproduced with permission from Ref. 22. Copyright 2009 by the American Institute of Physics.)

18

hiroshi fujisaki et al.

feasibility, we only calculated the normal modes and anharmonic coupling elements within this subsystem. A number of important conclusions were drawn from these calculations. 1. The inclusion of “many” solvating water molecules induces the irreversible decay of the excess energy as well as the density matrix elements (population). The important observation is that the VER behavior does not severely depend on the cutoff distance (if it is large enough) and the cutoff frequency. The implication is that if we are interested in a localized mode such as the amide I mode in NMA, it is enough to use an NMA/water cluster system to totally describe the initial process of VER. In a subsequent study, Fujisaki and Stock used only 16 water molecules surrounding NMA (hydrated water) and found reasonable results [47]. 2. Comparison of the two isolated NMA calculations suggests that the CHARMM force field works well compared to results based on DFT calculations. This suggests that the use of the empirical force field in exploring VER of the amide I mode may be justified. 3. There is a classical limit of this calculation [22], which predicts a slower VER rate close to Nguyen–Stock’s quasiclassical calculation [42]. This finding was explored further by Stock [78], who derived a novel quantum correction factor based on the reduced model, Eq. (2). In these calculations, many solvating water configurations were generated using MD simulations. As such, information characterizing dynamic fluctuation in the environment is ignored. Fujisaki and Stock further improved the methodology to calculate VER [47] by taking into account the dynamic effects of the environment through the incorporation of time-dependent parameters, such as the normal mode frequencies and anharmonic coupling, derived from the MD simulations as shown in Fig. 8. Their method is described in Section III.B, and was applied to the same NMA/solvent water system. As we are principally concerned with high-frequency modes, and the instantaneous normal mode frequencies can become unphysical, we adopted a partial optimization strategy. We optimized the NMA under the influence of the solvent water at a fixed position. (For a different strategy, see Ref. 79.) The right panels of Fig. 8 show the numerical result of the optimization procedure. Through partial optimization, the fluctuations of the parameters become milder than the previous calculations that employed instantaneous normal modes. The population dynamics calculated by the extended VER formula Eq. (49) are shown in Fig. 9. We see that both partial optimization and dynamical averaging affect the result. The “dynamic” formula, Eq. (49), leads to smaller fluctuations in the results for the density matrix. Apparently, dynamic averaging smoothens the resonant effect, stemming from the frequency difference in the denominator of Eq. (50). For the NMA/solvent water system, the time-averaged value of the

non-markovian theory of vibrational energy relaxation 400

200 100 0

200 100 0

–100

–100

–200

–200

–300

0

200

400

600

(110,110) (108,116) (109,116) (110,116)

300

Δ ωSαβ(t ) (cm–1)

300

Δ ωSαβ(t ) (cm–1)

400

(110,110) (108,116) (109,116) (110,116)

800

–300

1000

0

200

t (fs) 15

600

800

1000

800

1000

(110,110) (108,116) (109,116) (110,116)

10

CSαβ (t) (kcal/mol/Å)

CSαβ (t) (kcal/mol/Å)

15

5 0 –5 –10 –15 –20

400

t (fs)

(110,110) (108,116) (109,116) (110,116)

10

19

5 0 –5 –10 –15

0

200

400

600

t (fs)

800

1000

–20

0

200

400

600

t (fs)

Figure 8. Time evolution of the vibrational dynamics of NMA in D2 O obtained from instantaneous normal mode analysis with (right) and without (left) partial energy minimization. Shown are (upper panels) the frequency mismatch ωSαβ (t) = ωS (t) − ωα (t) − ωβ (t), for several resonant bath mode combinations, and (lower panels) the corresponding third-order anharmonic couplings, CSαβ (t). (Reproduced with permission from Ref. 47. Copyright 2009 by the American Institute of Physics.)

Fermi resonance parameter, Eq. (52), can be utilized to clarify the VER pathways as in the case of isolated NMA [47]. It was shown that the hydrating water (the number of waters is 16) is enough to fully describe the VER process at the initial stage ( 0.5 ps). The predictions of the VER rates for the two deuterated cases, NMA-d1 and NMA-d7 , are in good agreement with experiment and also with the NMA/water cluster calculations [72]. Though the dynamic effect is modest in the case of the NMA/solvent water system, the dynamic formula is recommended when variations in the system parameters due to the fluctuating environment must be taken into account.

20

hiroshi fujisaki et al. (b) 1

0.8

0.8

0.6

0.6

P (t)

P (t)

(a) 1

0.4 0.2 0

0.4 0.2

0

200 400 600 t (fs)

800 1000

0

0

200 400 600 t (fs)

800 1000

Figure 9. VER calculations of amide I mode population P(t) of NMA with use of instantaneous normal mode analysis and partial energy minimization. Shown are results from (a) the inhomogeneous averaging approximation and (b) dynamical averaging. Thick lines represent the ensemble averaged population dynamics, whereas solid lines represent each contribution from a single trajectory. (Reproduced with permission from Ref. 47. Copyright 2009 by the American Institute of Physics.)

B.

Cytochrome c in Water

Cytochrome c, one of heme proteins, has been used in experimental and theoretical studies of VER [21, 80–85]. Importantly, spectroscopy and simulation have been used to explore the timescales and mechanism of VER of CH stretching modes [21, 85]. Here we examine VER of amide I modes in cytochrome c [23]. Distinct from previous studies [21] that employed a static local minimum of the system, we use the dynamical trajectory; in the previous study, the water degrees of freedom were excluded, whereas in this study some hydrating water has been taken into account. We used the trajectory of cytochrome c in water generated by Bu and Straub [85]. To study the local nature of the amide I modes and the correspondence with experiment, we isotopically labeled four specific CO bonds, typically C12 O16 as C14 O18 . In evaluating the potential energy in our instantaneous normal mode analysis, we truncated the system with an amide I mode at the center using a cutoff ˚ including both protein and water. Following INM analysis, we used ( 10 A), Eq. (50) to calculate the time course of the density matrix. The predicted VER is single exponential in character with timescales that are subpicosecond with relatively small variations induced by the different environments of the amide I modes (see Fig. 10 and Table I in Ref. 23 for numerical values of the VER timescales). To identify the principal contributions to the dependence on the environment, we examined the VER pathways and the roles played by protein and water degrees of freedom in VER. Our first conclusion is that, for the amide I modes buried in the protein (α-helical regions), the water contribution is less than that for the amide I modes exposed to water (loop regions). This finding is important because only

non-markovian theory of vibrational energy relaxation

21

Figure 10. (a) 81st and 84th residues of cytochrome c in a loop region. (b) 93rd and 97th residues of cytochrome c in an α-helical region. The cartoon represents the protein using a licorice model to identify the four residues. (The water molecules are excluded for simplicity.) (Reproduced with permission from Ref. 23. Copyright 2009 by the American Chemical Society.)

a total VER timescale is accessible in experiment. With our method, the energy flow pathways into protein or water can be clarified. Focusing on the resonant bath modes, we analyzed the anisotropy of the energy flow, as shown in Fig. 11, where the relative positions of bath modes participating in VER are projected on the spherical polar coordinates (θ, φ) centered on the CO bond involved in the amide I mode, which represents the principal z-axis (see Fig. 1 in Ref. 23). The angle dependence of the energy flow from the amide I mode to water is calculated from the normal mode amplitude average, and not directly related to experimental observables. As expected, energy flow is observed in the direction of solvating water. However, that distribution is not spatially isotropic and indicates preferential directed energy flow. These calculations demonstrate the power of our theoretical analysis in elucidating pathways for spatially directed energy flow of fundamental importance to studies of energy flow and signaling in biomolecules and the optimal design of nanodevices (see summary and discussion for more detail). C.

Porphyrin

Our last example is a modified porphyrin [86]. We have carried out systematic studies of VER in the porphyrin–imidazole complex, a system that mimics the active site of the heme protein, myoglobin. The structure of myoglobin was first

22

hiroshi fujisaki et al. (a)

(b)

φ/π

0.5 0

0.2 0.15 0.1 0.05 0

−0.5 −1 0

0.2

0.4

0.6

0.8

0.4

1

0.35 0.3

0.5

0.25

φ/π

0.45 0.4 0.35 0.3 0.25

1

0

0.2 0.15

−0.5

0.1 0.05

−1

1

0 0

0.2

θ/π

0.4

0.6

0.8

1

θ/π

(c)

(d) 0.7

1

0.35

1

0.6

0.3 0.5

0.5 0.4

0 0.3 0.2

−0.5

φ/π

φ/π

0.5

0.25 0.2

0

0.15 0.1

−0.5

0.1 −1

0 0

0.2

0.4

θ/π

0.6

0.8

1

0.05 −1

0 0

0.2

0.4

0.6

0.8

1

θ/π

Figure 11. Angular excitation functions for the resonant normal modes of water for the (a) 81st, (b) 84th, (c) 93rd, and (d) 97th residues, represented in arbitrary units. (Reproduced with permission from Ref. 23. Copyright 2009 by the American Chemical Society.)

determined in 1958 [87]. Experimental and computational studies exploring the dynamics of myoglobin led to the first detailed picture of how fluctuations in a protein structure among a multitude of “conformational substate” support protein function [88]. Time-resolved spectroscopic studies [17] coupled with computational studies have provided a detailed picture of timescale and mechanism for energy flow in myoglobin and its relation to function. Karplus and coworkers developed the CHARMM force field [24] for heme and for amino acids for the study of myoglobin, and a particular focus on the dissociation and rebinding of ligands such as CO, NO, and O2 [90]. The empirical force field appears to provide an accurate model of heme structure and fluctuations, however, we have less confidence in the accuracy of anharmonicity and mode coupling on the force field. Furthermore, the dependence on spin state is important to the proper identification of the electronic ground-state potential energy surface. We carried out ab initio calculations for a heme-mimicking molecule, iron– porphin ligated to imidazole, abbreviated as FeP-Im. See Fig. 12 for the optimized

non-markovian theory of vibrational energy relaxation

23

Figure 12. Optimized structure of FeP-Im (quintuplet S = 2 spin configuration) at the UB3LYP/631G(d) level of theory. (Reproduced with permission from Ref. 86. Copyright 2009 by the American Institute of Physics.)

structure. We employed the UB3LYP/6-31G(d) level of theory as in the case of the isolated NMA [66, 67], but carefully investigated the spin configurations. We identified the quintuplet (S = 2) as the electronic ground state, in accord with experiment. Our study of VER dynamics on this quintuplet ground-state potential energy surface is summarized here. Additional investigations of the VER dynamics on the PES corresponding to other spin configurations as well as different heme models are described elsewhere [86]. A series of elegant pioneering experimental studies have provided a detailed picture of the dynamics of the ν4 and ν7 modes, in-plane modes of the heme (see Fig. 13), following ligand photodissociation in myoglobin. Using time-resolved resonance Raman spectroscopy, Mizutani and Kitagawa observed mode-specific excitation and relaxation [18, 89]. Interestingly, these modes decay on different timescales. The VER timescales are ∼1.0 ps for the ν4 mode and ∼2.0 ps for the ν7 mode. Using a sub-10 fs pulse, Miller and coworkers extended the range of the coherence spectroscopy up to 3000 cm−1 [91]. The heme ν7 mode was found to be most strongly excited following Q band excitation. By comparing to the deoxy-Mb spectrum, they demonstrated that the signal was derived from

24

hiroshi fujisaki et al.

Figure 13. Time course of the ρ11 element of the reduced density matrix for ν4 and ν7 excitations of v = 1. For the explanation of the insets, refer to Ref. 89. (Reproduced with permission from Ref. 86. Copyright 2009 by the American Institute of Physics.)

the structural transition from the six-coordinate to the five-coordinate heme. Less prominent excitation of the ν4 mode was also observed. The selective excitation of the ν7 mode, following excitation of out-of-place heme doming, led to the intriguing conjecture that there may be directed energy transfer of the heme excitation to low-frequency motions connected to backbone displacement and to protein function. The low-frequency heme modes (<400 cm−1 ) have been studied using femtosecond coherence spectroscopy with a 50 fs pulse [92]. A series of modes at ∼40, ∼80, ∼130, and ∼170 cm−1 were observed for several myoglobin derivatives. The couplings between these modes were suggested. It is a long-term goal of our studies to understand, at the mode-specific level, how the excess energy flow due to ligand dissociation leads to the selective excitation of ν4 and ν7 modes. In our study, we ignore the transition in spin state that occurs upon ligand photodissociation and the associated electron–nuclear coupling that will no doubt be essential to an understanding of the “initial state” of the ν4 and ν7 vibrations following ligand photodissociation. Our focus was on the less ambitious but important question of vibrational energy flow on the ground-state (S = 2) surface following excitation resulting from photodissociation. We employed time-dependent perturbation theory, Eq. (50), to model the mode-specific relaxation dynamics. The initial

non-markovian theory of vibrational energy relaxation

25

decay process of each system mode was fitted by a single-exponential function. The time constant of 1.7 ps was derived for the ν4 mode and 2.9 ps for the ν7 mode. These theoretical predictions, which make no assumptions regarding the VER mechanism, agree well with previous experimental results of Mizutani and Kitagawa for MbCO [18]. Vibrational energy transfer pathways were identified by calculating the thirdorder Fermi resonance parameters, Eq. (52). For the excited ν4 and ν7 modes, the dominant VER pathways involve porphin out-of-plane motions as energy accepting doorway modes. Importantly, no direct energy transfer between the ν4 and ν7 modes was observed. Cooling of the five Fe-oop (Fe-out-of-plane) modes, including the functionally important heme doming motion and Fe-Im stretching motion, takes place on the picosecond timescale. All modes dissipate vibrational energy through couplings, weaker or stronger, with low-frequency out-of-plane modes involving significant imidazole ligand motion. It has been suggested that these couplings trigger the delocalized protein backbone motion, important for protein function, which follows ligand dissociation in Mb. The γ7 mode, a porphin methine wagging motion associated with Fe-oop motion, is believed to be directly excited following ligand photodissociation in MbCO. The coupling of this mode to lower frequency bath modes is predicted to be very weak. However, its overtone is strongly coupled to the ν7 mode, forming an effective energy transfer pathway for relaxation on the electronic ground-state and excited-state surfaces. This strong coupling suggests a possible mechanism of excitation of the ν7 mode through energy transfer from the γ7 mode. That mechanism is distinctly different from direct excitation together with Fe-oop motion of the ν4 mode and supports earlier conjectures of mode-specific energy transfer following ligand dissociation in myoglobin [18, 91].

V.

SUMMARY AND DISCUSSION

This chapter provides an overview of our recent work on the application of the non-Markovian theory of vibrational energy relaxation to a variety of systems of biomolecular interest, including protein backbone mimicking amide I modes in Nmethylacetamide (in vacuum, in water cluster, and in solvent water), amide I modes in solvated cytochrome c, and vibrational modes in a heme-mimicking porphyrin ligated to imidazole. We calculated the VER timescales and mechanisms using Eq. (49), incorporating a fluctuating bath, and Eq. (50), using a static bath approximation, and compared them with experiment when available. The theory is based on the reduced model using normal mode concepts with third- and fourth-order anharmonicity, Eq. (2). Applying the simple time-dependent perturbation theory, and ensemble averaging the resulting density matrix, a non-Markovian theory of VER was obtained. We extended the previous theory due to Fujisaki et al. [22] to

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more general situations (1) where the relaxing “system” has a multimode character and (2) when the system parameters depend on time [47]. We also discussed the limitations of our VER formulas related to the assumptions upon which the earlier theories are based. We are now in a position to discuss the future aspects of our work, and the connection to other biomolecular systems or nanotechnological devices. •

Relation to Enzymatic Reaction. The role of vibrational motions in the mechanism of enzymatic reactions remains controversial [93]. In enzymology, the characterization of the enzymatic reaction rate is essential. Kinetic information is typically derived from substrate–enzyme kinetics experiments. In numerical simulations, the free energy calculation combined with transition state theory is the most powerful and practical way to compute reaction rates. As enzymatically catalyzed reactions typically involve chemical bond breaking and formation, QM/MM-type methods should be employed. Warshel and Parson have examined this issue for several decades and concluded that characterizing the free energy barrier is the most important consideration, noting that the electrostatic influence from the protein (enzyme) plays a key role [94]. However, Hammes-Schiffer and coworkers have identified important situations in which VER might play a role in controlling the rate of enzymatic reactions [93]. Furthermore, Hynes and coworkers applied the Grote–Hynes theory to the enzymatic reactions and investigated the dynamic role of the environment [6]. These recent studies indicate the importance of incorporating vibrational energy flow and dynamics as part of a complete understanding of enzyme kinetics. • Relation to Conformational Change. The relation between vibrational excitation/relaxation and conformational change of molecules is intriguing in part because of the possible relation to the optimal control of molecular conformational change using tailored laser pulses. It is well known that there are dynamic corrections to the RRKM reaction rate, the simplest being k(E)/kRRKM (E) = (1 + νR /kIVR (E))−1

(53)

where νR is the intrinsic frequency of a reaction coordinate, kIVR (E) is a microcanonical IVR rate, and kIVR (E) is the RRKM reaction rate [3–5]. Several modifications to this formula are summarized in Ref. 12. It is obvious that VER affects how a molecule changes its shape. However, this is a “passive” role of VER. Combining RRKM theory and the local random matrix theory [7], Leitner and coworkers theoretically studied the active role of vibrational excitations on conformational change of a peptide-like molecule (called NATMA) [95]. There are two particular modes (NH stretching) in NATMA, and they found that the final product depends on which vibrational

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27

mode is excited [96]. For the same system, Teramoto and Komatsuzaki further refined the calculation by employing ab initio potential energy surface [97]. A possibility to control molecular configurations of peptides or proteins using laser pulses should be pursued and some experimental attempts have already begun [20, 98, 99]. Another interesting attempt should be to address mode-specific energy flow associated with structural change. Recently, Ikeguchi et al. [100] developed a linear response theory for conformational changes of biomolecules, which is summarized in another chapter of this volume [37]. Though the original formulation is based on a static picture of the linear response theory (susceptibility), its nonequilibrium extension may be used to explore the relation between energy flow and conformational change in proteins. In addition, Koyama et al. [101] devised a method based on principal component analysis for individual interaction energies of a peptide (and water) and found an interesting correlation between the principal modes and the direction of conformational change [101]. • Relation to Signal Transduction in Proteins. Though signal transduction in biology mainly denotes the information transfer processes carried out by a series of proteins in a cell, it can be interesting and useful to study the information flow in a single protein, which should be related to vibrational dynamics. Straub and coworkers [41] studied such energy flow pathways in myoglobin and found particular pathways from heme to water, later confirmed experimentally by Kitagawa and coworkers [102] and Champion and coworkers [92]. Ota and Agard [103] devised a novel simulation protocol, anisotropic thermal diffusion, and found a particular energy flow pathway in the PDZ domain protein. Importantly, the pathway they identified is located near the conserved amino acid region in the protein family previously elucidated using information theoretic approach by Lockless and Ranganathan [104]. Sharp and Skinner [105] proposed an alternative method, pump–probe MD, and examined the same PDZ domain protein, identifying alternative energy flow pathways. Using linear response theory describing thermal diffusion, Ishikura and Yamato [106] discussed the energy flow pathways in photoactive yellow protein. This method was recently extended to the frequency domain by Leitner and applied to myoglobin dimer [107]. Though the energy flow mentioned above occurs quite rapidly (∼ps), there are time-resolved spectroscopic methods to detect these pathways in vitro [20]. Comparison between theory and experiment will help clarify the biological role of such energy flow in biomolecular systems. • Exploring the Role of VER in Nanodevice Design. Applications of the methods described in this chapter are not limited to biomolecular systems. As

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hiroshi fujisaki et al. mentioned in Section I, heat generation is always an issue in nanotechnology, and an understanding of VER in molecular devices can potentially play an important role in optimal device design. The estimation of thermal conductivity in such devices is a good starting point recently pursued by Leitner [12]. Nitzan and coworkers studied thermal conduction in a molecular wire using a simplified model [108]. It will be interesting to add more molecular detail to such model calculations. Electronic conduction has been one of the main topics in nanotechnology and mesoscopic physics [109], and heat generation during electronic current flow is an additional related area of importance. • VER in a Confined Environment. We have found evidence for spatially anisotropic vibrational energy flow with specific pathways determined by resonance and coupling conditions. It was shown for amide I modes in cytochrome c that VER may depend on the position of the probing modes [23], making it useful for the study of inhomogeneity of the environment. For example, an experimental study of VER in a reverse micelle environment [110], fullerene, nanotube, membrane, or on atomic or molecular surfaces [111] may all be approached using methods described in this chapter. • Anharmonic Effects in Coarse-Grained Models of Proteins. Recently Togashi and Mikhailov studied the conformational relaxation of elastic network models [112]. Though the model does not explicitly incorporate anharmonicity, small anharmonicity exists, resulting in interesting physical behavior relevant to biological function. Sanejouand and coworkers added explicit anharmonicity to the elastic network models and studied the energy storage [113] through the lens of “discrete breather” ideas from nonlinear science [114]. Surprisingly, they found that the energy storage may occur in the active sites of proteins. It remains to be seen whether their conjecture will hold for all-atom models of the same system. Acknowledgments

The authors gratefully acknowledge fruitful and enjoyable collaborations with Prof. G. Stock, Prof. K. Hirao, and Dr. K. Yagi, the results of which form essential contributions to this chapter. We thank Prof. David M. Leitner, Prof. Akinori Kidera, Prof. Mikito Toda, Dr. Motoyuki Shiga, Dr. Sotaro Fuchigami, and Dr. Hiroshi Teramoto for useful discussions. The authors are grateful for the generous support of this research by the National Science Foundation (CHE-0316551 and CHE-0750309) and Boston University’s Center for Computational Science. This research was supported by Research and Development of the Next-Generation Integrated Simulation of Living Matter, a part of the Development and Use of the Next-Generation Supercomputer Project of the Ministry of Education, Culture, Sports, Science and Technology (MEXT).

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PROTEIN FUNCTIONAL MOTIONS: BASIC CONCEPTS AND COMPUTATIONAL METHODOLOGIES SOTARO FUCHIGAMI,1 HIROSHI FUJISAKI,2,3 YASUHIRO MATSUNAGA,2 and AKINORI KIDERA,1,2 1 Department

of Supramolecular Biology, Graduate School of Nanobioscience, Yokohama City University, 1-7-29 Suehiro-cho, Tsurumi-ku, Yokohama 230-0045, Japan 2 Molecular Scale Team, Integrated Simulation of Living Matter Group, Computational Science Research Program, RIKEN, 2-1 Hirosawa, Wako 351-0198, Japan 3 Department of Physics, Nippon Medical School, 2-297-2 Kosugi-cho, Nakahara, Kawasaki 211-0063, Japan

CONTENTS I. II.

Introduction Experiments on Protein Dynamics A. X-Ray Crystallography B. Nuclear Magnetic Resonance Spectroscopy C. Single-Molecule Spectroscopy III. Equilibrium Dynamics of Proteins A. Basic Concepts Underlying Protein Functional Motions: Introduction B. Harmonic Analysis and Model for Protein Dynamics 1. Normal Mode Analysis 2. Elastic Network Models C. Analysis of Molecular Dynamics Trajectories 1. Quasiharmonic Approximation 2. Principal Component Analysis 3. Beyond Quasiharmonic Approximation IV. Nonequilibrium Dynamics of Proteins A. Models for Protein Functions 1. Lock-and-Key Model Advancing Theory for Kinetics and Dynamics of Complex, Many-Dimensional Systems: Clusters and Proteins, Advances in Chemical Physics, Volume 145, Edited by Tamiki Komatsuzaki, R. Stephen Berry, and David M. Leitner. © 2011 John Wiley & Sons, Inc. Published 2011 by John Wiley & Sons, Inc.

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2. Induced-Fit Model and Allosteric Interaction Model 3. Allosteric Interactions in Hemoglobin 4. General Model of Protein–Ligand Interactions 5. Linear Response Model B. Biased Molecular Dynamics Simulation on Protein Functional Motions C. Path Search and Path Sampling for Biomolecules 1. Minimum Energy Path Search 2. Path Search at Finite Temperature 3. Path Sampling V. Concluding Remarks Acknowledgments References

I.

INTRODUCTION

“Protein function” has diverse implications in biology, for example, catalyzing a chemical reaction in a metabolic pathway, receiving and transmitting a signal in a signal transduction process, and transporting a compound between the inside and the outside of a cell. Indeed, a wide variety of functions appear in the annotation descriptions of proteins in databases [1]. Some of these descriptions represent an activity of a single protein molecule, and some a role in cellular functions. For the purpose of discussing protein “functional” motions as part of this volume, we have to avoid such complications in the biological descriptions of protein functions. Hence, we will start by presenting a definition of “protein function” stated rigorously in physical terms as A series of protein structural changes as a response to an external perturbation such as ligand binding, and accompanying chemical reactions. A protein function starts with an external perturbation imposed on the protein system. The response to the perturbation initiates the relaxation process to a new equilibrium structure, resulting in a structural change in the protein. The structural change sometimes accompanies a chemical reaction in the ligand molecule to convert it into a product. The product molecule or the protein in the new equilibrium structure in turn becomes the next perturbation imposed on another protein in the biological system. This process can occur in multiple stages, and we thus used the term “a series” at the top of the definition of protein function. This definition is a physical counterpart of the definition of systems biology, in which a protein is regarded as a transducer of an input signal converting to an output signal [2]. What is important in these definitions is that a protein molecule does not work in a stand-alone manner, but under the influence of its surroundings, and that protein functions are intrinsically dynamic and nonequilibrium processes. In this chapter, we will review our current understanding of protein dynamics and its relevance to biological functions, mainly focusing on theoretical

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descriptions and computational methodologies to analyze and characterize the functional motions of proteins. There are two aspects in protein dynamics, equilibrium dynamics (Section III) and nonequilibrium dynamics (Section IV). Studies on equilibrium dynamics have provided the fundamental concepts and analytical tools for protein motions, while nonequilibrium dynamics have been investigated to understand the molecular mechanisms underlying protein functions. Let us begin by reviewing the experimental observations of protein dynamics in X-ray crystallography and nuclear magnetic resonance (NMR) spectroscopy, focusing on various experimental techniques to detect different types of information on protein dynamics. In addition to direct information on protein dynamics, we can also obtain information on structural changes in proteins from static structures. Recent advances in structural biology have allowed us to obtain multiple three-dimensional structures of individual proteins corresponding to the initial, final, and intermediate states during the protein function. These data have been accumulated in a database [3] and have become the basis of studies on functional motions. In Section II, we have summarized both equilibrium and nonequilibrium aspects of protein dynamics observed in experiments, including brief discussions on single-molecule spectroscopy. Section III is devoted to theoretical descriptions of the equilibrium dynamics of proteins. The complex hierarchical dynamics of proteins can be attributed to their rugged energy landscape in multidimensional configuration space. However, many experimental studies have indicated that not all degrees of freedom of a protein equally contribute to functional motions, but that functionally relevant motions can be described in a low-dimensional space called the essential subspace. Two major analysis methods used to deduce and characterize the subspace, normal mode analysis (NMA) and principal component analysis (PCA), are explained in this section. Although our goal is to understand the protein functional motions including large conformational changes, the timescale for protein functions to occur is too large for brute-force simulations at atomic resolution despite the recent increase in computing power. To circumvent this annoying situation, a number of studies have been done. First, in Section IV.A, we describe some phenomenological models of protein functions from a historical perspective and introduce the linear response model as a unifying scheme to combine those previous models. The key ingredient of the latter model is the quasiharmonic picture of protein equilibrium dynamics (see Section III.C). Hence, we have a systematic point of view for conformational changes of proteins based on the equilibrium properties of protein dynamics. Next, in Sections IV.B and IV.C, as techniques to overcome the limitations in brute-force simulations, we review two categories of simulation techniques, biased molecular dynamics (MD) simulations and path sampling methods. These techniques make it possible to elucidate the mechanisms of slow transition processes underlying protein functions in atomic detail. In Section V, we conclude this chapter with a brief summary and theoretical future prospects for protein functional motions.

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EXPERIMENTS ON PROTEIN DYNAMICS

Protein dynamics has been extensively explored using a wide variety of experimental techniques to reveal molecular mechanisms underlying protein functions. Each technique has revealed a different picture of protein dynamics. In this section, we briefly summarize the experimental observations on protein dynamics obtained using X-ray crystallography, NMR spectroscopy, and single-molecule spectroscopy. A.

X-Ray Crystallography

X-ray crystallography is the most common experimental method for determining protein structures at an atomic resolution [4] and has given valuable insight into the atomistic mechanisms of protein functions [5]. Recent advances in X-ray crystallography techniques have dramatically increased the number of protein structures deposited in the Protein Data Bank [3, 6] and the accumulated data have demonstrated the diversity of protein architectures [7]. As of May 2011, about 60,000 protein structures were determined using X-ray crystallography and compiled in the Protein Data Bank. X-ray diffraction patterns from a protein crystal provide not only the atomic coordinates but also the information on protein dynamics in the form of the temperature factor [8]. The structure factor F, the square of which is proportional to the diffraction intensity, is described by F (q) =

fj (q) exp iq· rj exp iq·rj

(1)

j

where q is the reciprocal lattice vector, fj is the atomic scattering factor for atom j, rj is the average atomic coordinates, and the term exp(iq·rj ) is called the Debye–Waller factor representing the atomic fluctuations around the average position. Usually, the Debye–Waller factor is reduced to a simple form,

exp iq·rj ∼ exp −q2 Bj /16π2

(2)

with Bj =

8π2 2

rj 3

(3)

based on the isotropic Gaussian approximation. The quantity Bj is called a Bfactor and is related to the mean square fluctuation of atom j, rj2 . There are more elaborated models for the interpretation of the Debye–Waller factor, such

protein functional motions

39

as the anisotropic model [9], the TLS model [8–10], and the normal mode model [10–12]. Particularly, the normal mode model could be successfully used to separate the mean square fluctuations into the contributions from the internal degrees of freedom and the external ones, that is, rj2 = rj2 int + rj2 ext , with the help of the orthogonal properties of the normal modes. The external contributions explain not only the rigid-body displacements of the whole protein molecule in the crystal environment but also the thermal diffuse scattering originating from the acoustic phonon of the crystal [13]. It was found that the external contributions accounted for a significant part of the mean square fluctuations [14]. This indicates that the mean square fluctuations determined by Eq. (3) do not originate solely from the contribution of the internal fluctuations. In fact, the agreement between B-factors and the normal mode analysis became reasonable when the effects of the crystal environment were explicitly taken into account [15]. Atomically detailed features of the temperature factors can potentially provide valuable information on protein dynamics [16]. Since the pioneering work by Frauenfelder et al. [17], the mean square fluctuations from the average coordinates rj2 have often been discussed with respect to their temperature dependence [18– 20]. Rasmussen et al. [21] observed that crystalline ribonuclease A lost its binding affinity to the inhibitor at ∼220 K. The correspondence between the dynamic transition and the onset of the activity leads to a hypothesis that the substrate-binding activity of ribonuclease A requires diffusive anharmonic motions acquired above the transition temperature. This hypothesis is based on the glass transition-like phenomenon in the protein dynamics from harmonic to anharmonic diffusive motions, observed in various spectroscopic measurements such as neutron incoherent scattering [22, 23] and M¨ossbauer spectroscopy [24]. At the same time, Rasmussen et al. [21] claimed that they observed an inflection in the crystallographic temperature factors at the same temperature, ∼220 K, and that this inflection was a result of the glass transition of the protein dynamics. However, because of the coherent nature of X-ray diffraction, the mean square fluctuation determined from crystallographic temperature factors represents static distribution, which has a totally different value from the one determined by incoherent spectroscopic experiments measuring the time correlation. Let us consider an experiment for lowering the temperature of a crystal. Suppose molecules in a crystal are in a double-well potential. When crossing the transition temperature from the anharmonic high-temperature regime to the harmonic low-temperature regime, each molecule starts to be confined in one of the two minima, and in turn affects the incoherent spectrum. However, since numerous molecules in the crystal, above the transition temperature, may be distributed to both potential wells with the population proportional to the Boltzmann factor, this transition to the harmonic regime may fix this population, but may not seriously affect the distribution. Thus, the transition may have little effect on the coherent spectrum. In fact, myoglobin did not show any inflection in the temperature factors at the transition temperature [25]. On the other hand, lysozyme did show an

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inflection in the external part of the temperature factors, but not in the internal part [26]. The cause of the inflection was attributed to the rigid-body motions of the molecule presumably due to the formation of cubic ice in the bulk water domain in the crystal. For ribonuclease A, a similar change in the crystal structure, not the dynamic transition, may be the cause of the inflection in the temperature factors. The crystallographic temperature factor cannot detect the dynamic transition occurring at ∼220 K. Protein dynamics can be observed not only in the temperature factors but also in the coordinates. In the Protein Data Bank, we can find many entries containing more than two sets of coordinates for the same protein, which sometimes show different structures. Those multiple structures constitute important experimental information for protein dynamics. Considering that the structural differences are mainly caused by the difference in intermolecular interactions in the crystal, or typically ligand binding, these structural changes should reflect the motions required in various biological functions, such as catalysis [27–29], regulation [30], and molecular transport [31, 32]. Gerstein and coworkers have compiled the protein structural changes in a database, MolMovDB [33], and classified them hierarchically into several categories based on the size (fragment, domain, and subunit motions) and packing in their domain interface (shear, hinge, and other motions). Hayward and coworkers developed the DynDom database [34, 35], focusing on the domain motions on the basis of the program DynDom [36, 37]. The most important observation in these databases is that many of the structural changes, particularly those for domain motions, occur in the space spanned by large-amplitude/low-frequency modes of the equilibrium fluctuations, which are calculated either using normal mode analysis including its coarse-grained (CG) elastic network model (ENM) or using principal component analysis of molecular dynamics trajectories (for these methods see Sections III.B and III.C) [38, 39]. This means that most protein systems behave linearly in terms of the response to the perturbation of ligand binding, which is explained in Section IV.A.5. Although a pair of protein structures, representing two states in the course of an enzymatic reaction, provides information on the start and end points of the reaction process, they do not have the information either on the kinetics or on the path connecting the two ends. There are attempts to obtain the courses of enzymatic reactions using Laue diffraction [40, 41] or by trapping the intermediate states with the analogues of the reaction intermediates [32, 42–44], which allow production of snapshots along the reaction pathways. For theoretical and computational attempts to extract reaction pathways in simulated biomolecular systems, see Section IV.C. In this aspect of protein dynamics, that is, multiple protein structures, experimental technique is not limited to X-ray crystallography. NMR spectroscopy has also been used to determine multiple structures with different conditions. If we

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extend our scope to lower resolution structures, we have a number of protein structural changes determined using the single-particle analysis of a cryoelectron microscope, producing snapshots of the functional motions [45–49]. Tama et al. combined the data from cryoelectron microscope of ribosome with elastic network model (see Section III.B.2) and clarified the functional role of low-frequency modes in the model system [50]. B.

Nuclear Magnetic Resonance Spectroscopy

Nuclear magnetic resonance spectroscopy is another important method used to determine protein structures [51]. A major difference from X-ray crystallography is that NMR spectroscopy determines the structures in aqueous solution, and thus the NMR structures are free from effects of crystal packing. Moreover, NMR spectroscopy measures the spin relaxation process, and thus is more straightforward for investigating protein dynamics than X-ray crystallography for determining the static distribution. An advantage over the other spectroscopic methods is that NMR can measure the spectrum of each atom or provide complete site-specific information at atomic resolution [52]. Protein molecules move over a wide range of timescales, from picoseconds to milliseconds. To probe such a variety of protein dynamics, various NMR techniques have been developed, which provide a dynamic picture of protein functions [53–56]. The most general method of determining the dynamic structure of a protein is spin relaxation experiments, which are conducted to measure the orientation of heteronuclear pairs with half-spin such as 1 H−15 N. According to Lipari and Szabo [57], the correlation function C(t) of the dipolar relaxation of two spins is decomposed into external and internal degrees of freedom as C(t) = Cext (t)Cint (t)

(4)

Cext (t) ∼ e−t/τc

(5)

with

Cint (t) ∼

P2 (u(0)·u(t)) r(0)3 r(t)3

(6)

where Cext is the correlation function for the external motions allowing a free tumbling motion in a solution with the correlation time τ c , Cint represents the internal relaxation containing the contributions of the distance r of two spins and the orientation of the bond vector u (= r/r) connecting the two nuclei in the laboratory coordinate system in the form of the second Legendre polynomial P2 , and indicates the ensemble average. The internal part Cint is further modeled as a motion in a restricted space determined by the protein structure with the effective

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correlation time τ e :

Cint (t) ∼ r−6 S 2 + (1 − S 2 )e−t/τe

(7)

with 1 Cint (∞) S = −6 = −6 r r 2

P2 (u(0)·u(∞)) r(0)3 r(∞)3

∼

1 r −6

2

2

Y2m (θ, ϕ) 2 r3

m=−2

.

(8) where the relationA(0)A(∞) = A2 and the addition theorem for spherical harmonics Y2m were used, and radial and angular averaging were assumed to be statistically separable. The parameter S2 is called the generalized order parameter ranging from 0 to 1; the lower values of S2 indicate larger amplitude of internal motions. Since the external motion averages out the internal correlation slower than τ c , an experiment in aqueous solutions can be conducted to detect only dynamics that are faster than τ c (∼nanosecond order). This model was further extended to include anisotropic tumbling motions [57], more components in the internal motions [58], and side-chain motions [59, 60]. Due to the equivalence of the timescale between the spin relaxation experiments and the feasible timescale of molecular dynamics simulations, it has become routine to compare the results of two methods [61]. Since the information derived from the relaxation experiments is always localized at a single bond vector, it is important to integrate the piecewise information into a model of cooperative protein dynamics. Vendruscolo and coworkers incorporated the experimental order parameters as the restraints in the structural determination of a protein to obtain the structural ensemble consistent with the dynamical data [62]. The second method to be introduced is residual dipolar coupling (RDC). When dipolar couplings are measured in a slightly anisotropic aqueous medium, where solute molecules are weakly aligned [63], RDC is measured by the splitting D in frequency between two resonances: D ∼

P2 (cos θ) r 3

(9)

where θ is the angle of the internuclear vector to the external field. In the aligning environment, this value is not averaged out by molecular tumbling, but remains nonzero. Therefore, unlike the spin relaxation experiments, the time range of RDC exceeds the tumbling time τ c and extends up to milliseconds. It is also possible to define the generalized order parameter for RDC [64]. The generalized order parameters derived from the spin relaxation experiment extend up to nanoseconds, while those derived from RDC contain information on dynamics up to milliseconds. The use of two kinds of order parameters in the refinement enables us to estimate whether the fluctuations have a subnanosecond or supernanosecond timescale [65]. Lange et al. found in the refinement process that ubiquitin undergoes motions

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above microseconds at the flexible loops, which are supposed to be engaged in the molecular recognition process [65]. The two methods explained above measure dipolar relaxation. On the other hand, transverse relaxation experiment directly detects the chemical exchange between multiple species or a protein molecule with different structures. Protein functions frequently accompany structural changes. However, under the conditions favoring the ground state, the excited state scarcely populates and cannot be detected with a standard approach. The transverse relaxation experiment provides detailed information about the process of structural changes [66]. Suppose that the equilibrium between states A and B is A

k1

B

(10)

k−1

where A is in the ground state and B is in the excited state, k1 k−1 , and the population shifts to the ground state, or pA pB . For such a chemical exchange system, the relaxation dispersion experiment using the Carr–Purcell–Meiboom–Gill (CPMG) spin-echo pulse sequences gives the transverse relaxation rate as [66, 67] 1 pA pB ω2 kex R2 R2 (∞) + (11) 1/2 τcp 2 + p2 ω4 + 144/τ 4 kex cp A Experiments are carried out by measuring the R2 values for various τ cp values (the interval of the consecutive refocusing pulses in the CPMG sequence). By fitting to the series of R2 values thus obtained, we can obtain detailed information about the chemical exchange, that is, the population of the two states, pA and pB , the chemical shift differences between the two states, ω, and their rate of interconversion, kex = k1 + k−1 . This methodology is extremely sensitive to the presence of minor conformers with populations as low as 1%, thus providing invaluable structural insight into intermediate states in protein folding and the catalytic cycle. For example, Wright and coworkers characterized all five higher energy intermediate states of dihydrofolate reductase in its catalytic cycle [68] and determined their populations and the interconversion rates, as well as the structures on the basis of the chemical shift differences. These pieces of experimental information on minor structural components should be one of the important targets of molecular simulations to reveal the molecular mechanism of functionally relevant conformational changes. Paramagnetic relaxation enhancement (PRE) is another method for detecting a minor component in the conformational exchange [69]. This method measures the difference in transverse relaxation rates for the paramagnetic and diamagnetic states of a protein: = R2para − R2dia

(12)

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where R2para and R2dia are transverse relaxation rates for a protein with and without a covalently or tightly bound paramagnetic probe, respectively, such as the unpaired electron of Cu2+ , Mn2+ , or nitroxides. Due to the large paramagnetic moment, the transverse relaxation rate for the dipolar interactions between the unpaired electron and 1 H becomes extremely large and provides information on the minor component in the form of = pA A + pB B for the equilibrium of Eq. (10). Using the distance dependence of the dipolar interaction, r−6 , it is also possible to obtain structural information of the minor component. Using the PRE method, Tang et al. determined the structure of the minor component of maltosebinding protein in the ligand-free form [70]. Maltose-binding protein consists of two domains between which the ligand maltose is bound. According to the model of the population shift (see Section IV.A.4), maltose-binding protein should be in a large fluctuation between the open and closed structures even without maltose. In fact, they found in their PRE experiment that the minor component exhibited a closed structure. Such information on PRE was also used to model the structures of weakly interacting encountered protein–protein complexes with the help of replica exchange simulations using a coarse-grained model [71]. C.

Single-Molecule Spectroscopy

Finally, we briefly touch on recent experimental and theoretical advances in singlemolecule spectroscopy [72–74]. Unlike conventional experimental techniques based on the ensemble average, single-molecule experiments have a possibility of providing information both on the inhomogeneous distribution of fluctuations and on the real-time evolution of protein dynamics, resulting in detailed analysis of the conformational fluctuations of proteins during folding or reactions. Considering that protein dynamics occurs on an extremely rugged potential surface, we can suppose that the trajectory of molecular motion is so complex and heterogeneous that it would be masked by an ensemble average. Recent advances in single-molecule experiments opened the possibility of detecting such complicated protein dynamics. The internal protein motions of a single protein molecule have been observed mainly using the fluorescence (F¨orster) resonance energy transfer (FRET) technique, in which two dyes (a donor and an acceptor of FRET) are attached to two different positions in a protein to estimate the interdye distance. The end-toend distance of a protein has been usually employed to study folding/unfolding processes for various types of proteins, and many insights into the structure and dynamics of unfolded proteins have been obtained [75]. Single-molecule FRET experiments have revealed functionally relevant motions in a number of proteins. For instance, Hanson et al. [76] directly followed domain motions of adenylate kinase on its catalytic timescale and demonstrated the plausibility of the population shift model (see Section IV.A.4). Adenylate kinase is capable of dynamically

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sampling two distinct open and closed conformations, and the substrate binding shifts the conformational equilibrium toward the closed form. Since the FRET measurements provide motional information of small degrees of freedom, such as the distance between two residues, it is indispensable to invoke theoretical and computational techniques to correctly interpret the experimental results. Gaussian fitting is usually employed to decompose the end-to-end distance distribution [75], but such a fitting method needs a long-time average of the FRET data. To deal with real-time data of FRET as a time-evolving distribution, Baba and Komatsuzaki [77] devised a concept, a locally equilibrated state (LES), to analyze the nonequilibrium folding/unfolding dynamics. See the chapter of this volume written by Komatsuzaki et al. for a detailed account of their method. On the other hand, Matsunaga and Kidera proposed to combine the FRET measurement and the particle filter algorithm, which is a kind of the sequential Monte Carlo method, to obtain dynamic and structural information of a protein [78]. III. A.

EQUILIBRIUM DYNAMICS OF PROTEINS

Basic Concepts Underlying Protein Functional Motions: Introduction

Because of strongly heterogeneous atomic interactions, proteins are known to have rugged or frustrated energy landscapes in the first approximation, analogous to the situations with spin glasses [79]. However, due to natural selection, proteins have been evolving in such a way that the frustration has become minimal, leading to a “funneled” energy landscape biased to a native structure [80, 81]. According to such a landscape picture, our theoretical task would be to investigate protein dynamics on a funneled energy landscape, that is, to characterize and calculate conformational transitions between (at least) two basins buried in such a landscape. However, our computing ability is still in infancy as state-of-the-art simulations with huge computer facilities can reach only several hundreds of nanoseconds (10−7 s) for several hundreds of thousands of atom systems, much shorter than the millisecond to second timescales of conformational changes that we will address in this chapter. To circumvent this situation, we introduce simplified strategies based on a harmonic approximation of proteins (Section III.B). This can be verified because a protein often has a rather rigid and closely packed structure such as that in crystals or elastic objects. Normal mode analysis and elastic network models are methods or models for investigating such rigid and closely packed structures, using a Hessian matrix (second derivatives of a potential energy function) as a key ingredient. Through careful analysis of the Hessian matrix, it is demonstrated that protein motions can be decomposed into domain motions, which are experimentally observed in conformational transition databases [33–35]. These domain motions are often relevant to biological functions and can be calculated even at the level of harmonic approximation.

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Quasiharmonic approximation is the next level of approximations, whose basic idea is that the distribution function (of a protein) is approximated by a multivariate Gaussian function with its width calculated by a covariance matrix (Section III.C). If anharmonicity is weak, this analysis should reduce to harmonic analysis. Based on this idea, principal component analysis is used to extract directions (principal modes) with the largest fluctuations by diagonalizing the covariance matrix for the fluctuations of Cartesian coordiantes or dihedral angles etc. It was found and experimentally confirmed that functionally relevant motions of proteins occur in a rather low-dimensional subspace spanned by such principal modes, and the subspace was called the “essential” subspace. This is why principal modes are often employed as reaction coordinates to illustrate the free energy landscape of a protein, and further utilized to enhance sampling in configuration space (essential dynamics sampling). The quasiharmonic approximation is also the basis for the linear response model of conformational changes (Section IV.A.5). We will also discuss some extensions of quasiharmonic approximation in Section III.C.3. B.

Harmonic Analysis and Model for Protein Dynamics

In this and next subsections, we review normal mode analysis and elastic network models, which are frequently used to probe the large-scale internal dynamics of proteins on the harmonic approximation of the potential energy. Though the harmonic approximation is in principle not valid for describing large conformational changes relevant to functional motions, these methods have been found to be powerful enough to study the overall features of the displacement and have provided a great deal of insight into the nature of collective motions in proteins [82, 83]. 1.

Normal Mode Analysis

One of the advantages of normal mode analysis is that it allows the decomposition of high-dimensional molecular motions into independent motions, so that large-amplitude motions corresponding to low-frequency modes can be selectively analyzed. Another advantage is in its analytical form enabling the estimation of thermodynamic quantities, such as conformational entropy and free energy. In addition, the experimental quantities described in the previous section, such as crystallographic B-factors and NMR order parameters, can be easily compared with the fluctuations predicted using NMA. Here, we briefly summarize NMA. The equation of motion for a molecule is described under the harmonic approximation by M¨x = −Hx

(13)

where x is the 3N vector of Cartesian coordinates measured from a minimum energy structure, M is the 3N × 3N diagonal mass matrix, and H is the Hessian matrix (=(∂2 V/∂xi ∂xj ), V being the potential energy). Here, we define a set of

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collective variables or the normal mode coordinates, q, which must satisfy the following equation of motion: q¨ = −q

(14)

where is the diagonal matrix containing the frequency of vibration for each component of q, or = diag(ωi2 ). The Cartesian coordinates x and normal mode coordinates q are linearly transformed into each other as x = Eq

(15)

where E is the transformation matrix. Using q, Eq. (13) is rewritten as MEq = HEq

(16)

and this equation should be identically satisfied for arbitrary q. Hence, the following generalized eigenvalue problem is obtained and has to be solved: HE = ME

(17)

where E and are determined as the eigenvector and eigenvalue matrices of H, respectively, with the normalization condition t EME = I (I being the identity matrix). The reason of the success of NMA in the description of protein dynamics is based on the following conversion under the harmonic approximation from the frequency to the variance–covariance of the coordinates, that is, x t x = kB T H−1

(18)

where denotes the ensemble average, and the zeros of H corresponding to the six external degrees of freedom were ignored. Instead of pursuing the time evolution of the vibrational motions of proteins, the left-hand side of Eq. (18) treats the static distribution. It is well known that the potential surface of proteins is extremely rugged particularly at the low-frequency normal modes. Therefore, in the space of the low-frequency normal modes, the protein dynamics is no more purely vibrational, but the amplitude given by Eq. (18) still gives a good estimation. This follows the philosophy of the quasiharmonic approximation explained in Section III.C. Moreover, the trace of both sides of Eq. (18) gives an appealing result of NMA, 3N

3N−6 xi2 = kB T i=1

j=1

1 μj ωj2

(19)

where μj is the reduced mass for normal mode j. This implies that the total variance is dominated by the low-frequency normal modes, and that only a small number of these modes can describe the distribution resulting from protein dynamics.

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The first applications of NMA were performed in the 1980s to investigate the equilibrium fluctuations of relatively small proteins, such as BPTI [84–86], crambin [86], ribonuclease [86], and lysozyme [86, 87]. These pioneer works showed the dominance of low-frequency normal modes in the total variance, whose eigenvectors are delocalized over the entire protein molecule. Moreover, they showed that the mean square fluctuations of residues obtained from NMA correlate well with the crystallographic B-factors explained in Section II.A. Despite its strong assumption, NMA has also been applied for studying the functional motions of proteins. For example, Ma and Karplus [88, 89] found that the low-frequency normal modes of ras p21 have high correlations with the conformational transition between the GTP- and GDP-bound crystallographic structures. Other applications contain the allosteric mechanism of chaperonin GroEL [90], the cooperative transition of aspartate transcarbamylase (ATCase) [91], the gating mechanism of potassium channel KcsA [92], and the opening and closing dynamics of domain II of type IA topoisomerase [93]. These results converged to the notion of the linear response theory explained in Section IV.A.5. Through a number of applications of NMA, it has been suggested that some proteins can be regarded, in the first approximation, to consist of parts that can be treated as behaving dynamically as rigid bodies in the low-frequency normal modes. This gives a picture of protein dynamics as interdomain motions constructed from rigid-body domains connected by softer interactions. Hayward et al. [94] rigorously formalized this picture in terms of NMA. Suppose that the Hessian matrix can be written, by appropriate rearrangement of the atom order, as a matrix where the values of the elements in the off-diagonal blocks are smaller than the elements in the diagonal blocks. For example,

H= (20) Each diagonal block corresponds to a rigid-body domain of the protein with relatively strong interactions compared to the off-diagonal blocks corresponding to softer interactions between rigid-body domains. If the off-diagonal blocks are small enough, one can consider them as a perturbation onto the system of noninteracting domains, and the unperturbed Hessian matrix is written as

H0 = (21)

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There are six external degrees of freedom (translations and rotations) for each block, and if H0 consists of M blocks, there are 6(M − 1) (excluding the overall translations and rotations of the entire molecule) degrees of freedom describing the motions of the rigid-body domains. In the zeroth-order approximation, it was shown [94] that the low-frequency eigenvectors of the perturbed Hessian matrix H are approximated by these 6(M − 1) modes of the unperturbed Hessian matrix H0 . Then, the displacement of the ith atom (in the kth rigid-body domain) is given by [94] ri = τ k + ψk × ri

(22)

where τ k and ψk are constant vectors determined by the linear combinations of the translations and rotations of the kth rigid body, respectively, and ri is the position vector from the center of mass of the kth domain to the ith atom. Equation (22) means that the rigid-body motion can be described as a screw motion consisting of a combination of the translation and rotation of a screw axis. This notion can be extended to interdomain motions [94], and an interdomain screw axis can be defined to represent an interdomain screw motion. Furthermore, by calculating the curl of the vector field of displacements ∇ × r = 2·ψ, Eq. (22) can also be used to identify screw axes and rigid-body domains from a pair of experimentally determined structures (such as crystallographic structures of open and closed forms) without calculation of the Hessian matrix [36, 37, 94]. This idea that protein dynamics is approximated by a linear combination of the external degrees of freedom of rigid bodies was used to reduce the computational costs of NMA. In the rotations–translations of blocks (RTB) approach, proposed by Sanejouand and coworkers [95], the Hessian matrix is approximated in subset bases defined by the local translations and rotations of M quasi-rigid-body blocks. The projected Hessian matrix onto the subset bases is given by Hb = t PHP

(23)

where P is an orthogonal 3N × 6M matrix built with vectors corresponding to the translations and rotations of each quasi-rigid-body block. These 6M vectors form the new basis of small dimension that corresponds to the projector P. One or a few consecutive residues are usually adopted as quasi-rigid-body blocks. This approach was later improved by Li and Cui [96] by constructing the Hessian matrix elements required in the projection operation on the fly, which dramatically reduces the memory requirements. The RTB approach allows application of the NMA to huge proteins, such as Ca2+ -ATPase [96, 97], myosin-II [97], F1 -ATPase [98], RNA polymerase [99], S-adenosyl-l-homocysteine hydrolase [100], and nicotinic acetylcholine receptor [101].

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Elastic Network Models

The effectiveness of the low-frequency modes of proteins can be utilized in a simpler way. Instead of detailed atomistic force fields, Tirion [102] introduced a new description of the potential energy function that is a form of the Hookean pairwise energy function, V =

C 0 2 (|ri,j | − |ri,j |) 2 0

(24)

ri,j ≤Rc

0 is that of a Here, ri,j ≡ ri − rj denotes the vector connecting atoms i and j, ri,j chosen configuration, and Rc is the cutoff distance. The strength of the potential C is a phenomenological constant assumed to be independent of atom types. Note that this energy function V takes a minimum for a chosen configuration {ri0 }; thus, the prior minimization step is not needed for this energy function. This energy function, called the elastic network model, is based on a concept that a protein can be considered as an elastic object. Generally, an elastic object is represented by a continuum model in which masses are distributed continuously and its behavior is described by a partial differential equation with a stress tensor. In the ENM, the mass density is approximated by discrete point masses and a protein is simply represented as a system composed of point masses connected to each other with the Hookean potential. Tirion showed that the ENM is sufficient for reproducing the low-frequency normal modes obtained by the atomistic force field [102]. Due to its simplicity and low computational cost, the ENM has been applied to a number of (huge) proteins [38, 103–114]. Many successes with the analyses using the ENM suggest that the low-frequency modes of proteins are determined by only the elasticity and the shape of the structure, and the information of atom types is not important for these large-amplitude motions. The perturbation approach discussed in the previous subsection may also be used to explain the reason the subspace spanned by the low-frequency modes of the ENM is similar to that of the detailed semiempirical force fields. When the Hessian matrix with the detailed force field has a block diagonal structure as in Eq. (20), the low-frequency modes are approximately described by the linear combinations of the local translations and rotations of the diagonal blocks of Eq. (21). Since the directions of translations and rotations depend only on the shape of the molecule and are independent of the detailed interactions, the subspace spanned by the low-frequency modes is well approximated by “rough” potential energy like the ENM. In fact, Lu and Ma [115] showed that the low-frequency space of the ENM was found to be robustly similar upon randomization of the Hessian matrix elements as long as the block diagonal structure of the matrix is maintained.

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Another approach to the coarse-grained normal mode model was initiated by Bahar et al. [116]. This approach has its origin in the theory of Gaussian polymer chain of arbitrary covalent geometry [117]. The potential function of the system is described simply by V =

C (ri − rj )2 2

(25)

(i,j)

where ri denotes the coordinates of atom i and the summation is for all pairs of bonded atoms. The partition function of Eq. (25) can be easily integrated as C −V/kB T t Z= e dR= exp − tr(RK R) dR 2kB T 2πkB T 3(n−1)/2 −3/2 = |K| (26) C where R is 3 × N matrix of (r1 , r2 , . . .) and K is the Kirchhoff matrix describing the connectivity of the polymer chain with N −1 for a bonded pair Kij = and Kii = Kik . (27) 0 otherwise k,k = / i Bahar et al. [116] reinterpreted Eq. (26) to represent protein dynamics around the equilibrium structure by regarding ri as the shift from each equilibrium position instead of the coordinates from the common origin, and used the same connectivity as that of Tirion [102]. This is called the Gaussian network model (GNM). According to this model, ri ·rj =

3kB T −1 (K )ij C

(28)

where (K−1 )ij is the (i, j) element of the inverse of K. Since this approach assumes the isotropic Gaussian fluctuation for the bonding interaction, the number of degrees of freedom is only N, the number of Cα atoms, and thus the eigenvectors have only amplitude information. To circumvent this problem, several methods have been devised by modeling the Hessian matrix [118]. Applications of these models [119–128] have become popular for investigating the harmonic motions of (huge) proteins. In particular, they are often used within database frameworks; see Refs. 125 and 128, for example. C.

Analysis of Molecular Dynamics Trajectories

In the previous section, we have discussed several approaches using the harmonic approximation to describe protein dynamics. A major consequence from the

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harmonic approximation and NMA of proteins was that protein dynamics are often characterized as the combination of rigid-body domain motions, which correspond to the space spanned by low-frequency normal modes. In MD simulations, a protein molecule traverses a number of potential minima and yields the distributions that should be different from those predicted by NMA restricted within a single minimum. In this subsection, we review methods to treat such anharmonic nature of the energy landscape especially focusing on how the harmonic approximation can be extended to interpret the observed anharmonicity in MD simulations. 1.

Quasiharmonic Approximation

The quasiharmonic approximation [129] permits us to make an anharmonic correction to a Hessian matrix [130]. The quasiharmonic approximation adopts the following formula of the joint probability function: 1t −1 P(q) ≈ exp − (q − q)C (q − q) 2 (2π)n/2 (det C)1/2 1

(29)

where q = {qi } (i = 1, . . . , n) denotes some internal coordinates of protein and the coefficient C−1 is the inverse of the covariance matrix C = (q − q) t (q − q), evaluated from an MD simulation. Equation (29) was derived from the cumulant expansion of the distribution function up to the second order. The effective energy function V (q) under the quasiharmonic approximation is written by V (q) ≈

kB T t (q − q)C−1 (q − q) 2

(30)

Thus, in the quasiharmonic approximation, kB T C−1 can be interpreted as an effective Hessian matrix, which is evaluated from the distribution function in the form of the covariance matrix C, not related to the vibrational frequency. This approach to use the information of a static distribution is the same as that in NMA. The quasiharmonic approximation is useful for calculating and interpreting thermodynamic properties, such as conformational entropy and free energy [130–132]. This is often performed with the combination of principal component analysis [133], which is explained in the next subsection. 2.

Principal Component Analysis

PCA is a multivariate statistical method for finding an optimal orthogonal linear transformation that captures the variation in a high-dimensional data set, or that best represents the variation in the least-square sense [133]. This method, when applied to MD data, is based on the analysis of the covariance matrix C estimated

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from MD trajectories. Diagonalization of C, t

ECE = ,

t

EE = I

(31)

yields the set of eigenvectors E = {ei } called principal modes and the associated eigenvalues = diag{λ2i }. The eigenvalue λ2i is the variance of the ith principal component Qi defined by the projection of the MD trajectory {q(t)} onto the ith principal mode ei , Qi (t) = t ei [q(t) − q]

(32)

The total variance of the MD trajectory q(t) is written as n i=1

n n

(qi − qi )2 = Q2j = λ2j j=1

(33)

j=1

Thus, principal components Qi having large variances λ2i become a representative set to describe the total variance. Usually, principal modes (and components) are ordered such that the first mode has the largest variance, the second has the next largest, and so on. It is expected that in the low-temperature limit the principal components should converge to normal modes. On the other hand, at higher temperatures, the extent to which the principal modes differ from normal modes depends on how well the anharmonic potential energy surface is described by the quasiharmonic approximation. To elucidate the harmonic and anharmonic aspects in protein dynamics, Hayward et al. [134, 135] compared normal and principal modes of BPTI in vacuum. By measuring the degree of anharmonicity in the principal modes based on the anharmonicity factor [135], they showed that the principal modes naturally divide into anharmonic modes with frequencies below 80 cm−1 and harmonic modes with frequencies above 80 cm−1 . They also showed that the anharmonic modes tend to “soften” the protein giving larger mean square fluctuations than those predicted by NMA. In principle, the validity of the quasiharmonic approximation is limited to cases where the anharmonic nature of the potential energy is rather small [129]. For example, the probability distribution function with double potential minima is never approximated with a single Gaussian distribution function. In these cases, PCA simply provides us the decorrelated “modes” but they statistically depend on each other. Thus, it becomes difficult to single out the important degrees of freedom and to calculate thermodynamic quantities. Nevertheless, PCA has been sometimes unlimitedly used in terms of dimensional reduction or extraction of essential subspace [136]. For example, Kitao et al. [137] compared the principal components of MD trajectories of melittin in vacuum and in water. They found a transition event from one to another local minimum energy region along the first

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and second principal components for the simulation in water, while there was no transition in vacuum, suggesting that the solvent water induces local minima in the potential energy surface. Tournier and Smith [138] studied a dynamical transition [21] at 180–220 K with MD simulations of hydrated myoglobin. They showed that the dynamical transition is well characterized by the appearance of a single double-well principal component that involves a global motion of two groups of α-helices. Although PCA has now become a common tool for investigating largeamplitude motions of proteins [139–148], the results should be carefully interpreted without any overestimations. For example, there are several examples of PCA where the oscillatory behavior of principal components has been interpreted as transitions between local energy minima. By calculating the principal components of a simple random diffusion model with flat potential, Hess [149, 150] showed that the oscillatory behavior of principal components is also derived from this model. He suggested that the oscillatory behavior observed in the PCA may not be attributed to transitions between potential minima but to the sampling deficiency caught by a single energy basin. Another important issue was addressed by Stock and coworkers [151]. To study the conformational dynamics of a protein with PCA using (mass-weighted) Cartesian coordinates, one first needs to eliminate the translation and rotation of the entire molecule to concentrate on the internal motion. Although the translation can be readily separated, the elimination of the rotation (e.g., using the superimposition algorithm [152, 153]) is far from straightforward when fluctuations become large [154]. In this case, it is impossible to unambiguously define a single reference structure, and the resultant correlations are highly dependent on a reference structure [155, 156]. To circumvent this problem, Stock and coworkers [151] proposed to use a transformation of dihedral angles, which maps the angles onto a metric space avoiding the problem that the angles are circular variables. From the analysis of 100 ns MD data of pentaalanine, they showed that their dihedral angle PCA untangles numerous free energy minima with a comparable energy, each of which corresponds to a well-defined structure, whereas the standard Cartesian PCA gives only a single minimum smooth surface due to the mixing of internal and external motions of the molecule. 3.

Beyond Quasiharmonic Approximation

As discussed above, when the assumption of the quasiharmonic approach is broken, it is often difficult to interpret the results of PCA since the obtained principal components are not statistically independent of each other. In particular, this problem becomes critical when one studies conformational transitions of a protein since the intra- and interbasin motions are generally mixed up along the principal components. To circumvent this problem, a number of new methods have been

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proposed, which are based on mutual information [157–160], iterative applications of local PCA (moving normal mode) [161], and information on frequency domain (multivariate frequency-domain analysis) [162]. Kitao et al. [163] investigated the complexity of the energy landscape and barrier-crossing motions in human lysozyme at 300 K using their jumping-amongminima (JAM) model with PCA. The JAM model is based on an idea that a covariance matrix can be decomposed into interbasin (jumping-among-minima) motions and intrabasin fluctuations. When the sampled space is classified into M clusters, the covariance is decomposed as follows: Cij = Cijinter + Cijintra =

M

fk (xi k − xi )(xj k − xj ) +

k=1

M

fk (xi − xi k )(xj − xj k )k

k=1

(34) where fk is the fraction of MD snapshots residing in cluster k. It was shown with the JAM model analysis that the energy landscape of a protein is hierarchical from a small number of large-amplitude anharmonic modes to many small-amplitude harmonic modes. Furthermore, they showed that the intersubstate motions are characterized as a diffusive motion with two typical timescales reflecting the hierarchy of the energy landscape. IV.

NONEQUILIBRIUM DYNAMICS OF PROTEINS A.

Models for Protein Functions

Protein dynamics has usually been characterized in the equilibrium state. However, considering that a protein function occurs as a transition to another equilibrium state induced by a certain external perturbation such as ligand binding, we should extend our scope to nonequilibrium dynamics. In this subsection, we follow the history of the models of protein functions, focusing on their underlying protein dynamics. 1.

Lock-and-Key Model

The very first model of a protein function goes back to the end of the nineteenth century when Emil Fischer introduced the celebrated “lock-and-key” model to explain the specificity of enzymatic reaction to carbohydrates [164]. Of course, Fischer did not have any information about the three-dimensional structure of the ligand-bound form of the enzyme. Remembering that he was the founder of the stereochemistry of carbohydrates, we presume that the enzyme was thought to possess a sort of stereochemistry compatible with the structures of carbohydrates.

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In this sense, the original model presented neither a definite structural image of a protein molecule nor the notion of protein dynamics. However, later the lock-andkey model was used to imply a model representing the shape complementarity between an enzyme and its ligand in molecular recognition, and more specifically rigid-body docking that required no flexibility in the ligand-binding site in an enzyme. Gutteridge and Thornton showed that a number of enzymes exhibited lockand-key-like behavior in the ligand-binding process, or that the conformational ˚ root changes between the ligand-free and ligand-bound forms were small (∼1 A mean square deviation (RMSD)), particularly at catalytic sites [165, 166]. Recently, Koike et al. [29] presented a more rigorous scenario in which the structural changes in enzymes depended on the types of enzymatic reactions. A hydrolase reaction involves the surrounding water molecules and occurs at the protein surface, thus requiring only small structural change (lock-and-key). However, a transferase reaction requires the catalytic groups to be insulated from the water environment, and thus transferases bury the ligand molecule inside the protein by causing large-scale motion closing the cleft (non-lock-and-key). 2.

Induced-Fit Model and Allosteric Interaction Model

A dynamical model of a protein function first appeared in the middle of the last century. At a symposium held in 1957, Koshland presented a paper that proposed the “induced-fit” model [167]. In his original and succeeding papers, Koshland stated that in a certain enzymatic process, ligand binding causes a structural change in the enzyme to form a correct atomic configuration required for the catalytic reaction [168]. Remembering that the first structure of a globular protein, myoglobin, at ˚ resolution was published in 1958 [169], it is remarkable that Koshland could 6A point out the importance of a protein’s flexibility in an enzymatic reaction without referring to the three-dimensional structures of proteins. However, we had to wait until 1966 for the full description of the induced-fit model when Koshland et al. proposed a model of the allosteric interaction of hemoglobin [170]. “Allosteric interaction” was proposed to explain the regulation of enzymatic activity in the metabolic system [171, 172]. One of the classical examples is threonine deaminase [173], which is an enzyme involved in the metabolic pathway of leucine/isoleucine biosynthesis and catalyzes the reaction of threonine to yield 2oxobutanoate. It was also known that this reaction was inhibited by isoleucine, the end product of this metabolic pathway. Changeux [173] proved that this inhibition was not due to competitive binding to a single catalytic site, but there was a distinct binding site for isoleucine, which was distant from the catalytic site for threonine. Therefore, it was called “allosteric” (“allos” in Greek means “other” or “different”). This regulation process has been understood as the binding of isoleucine causing structural change at the distant catalytic site to inhibit the catalytic reaction. This

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kind of long-range dynamical coupling between two loci in a protein molecule has been considered to have originated from a large-scale domain (or the subunit) rearrangement accompanied by effector binding [174]. In this example, we can see that allosteric interactions provide the molecular origin of feedback regulation in a biological system. In the 1960s, we could already find the basis of “systems molecular biology.” However, the first major success of the model of allostery was in studies on cooperative ligand binding in hemoglobin. 3.

Allosteric Interactions in Hemoglobin

We do not intend to go into details of this classical problem, cooperative ligand binding in hemoglobin, but we wish to explain the most fundamental models of the protein function that were established in this problem. For this purpose, we present Eigen’s generalization of allosteric ligand binding in hemoglobin (Fig. 1) [175]. Hemoglobin is an α2 β2 tetramer, each of which possesses a ligand (O2 , CO) binding site on the heme plane. The binding at one site affects the affinity of the other sites allosterically and results in cooperative binding that appears in a sigmoidal oxygen equilibrium curve. Two models have been proposed to explain this cooperative phenomenon, that is, the Monod–Wyman–Changeux (MWC) [176] and Koshland– N´emethy–Filmer (KNF) models [170]. Figure 1 describes the various states of

MWC

O

O

O O

O O

O O O

O

O

O

O O

O O

O O

O O

O O

O O

O O

O

O

O

O

O O

O O

O O

O O

O O

O O

O O

O O

O O

O O

KNF

T-state

R-state

Figure 1. Eigen’s generalization of allosteric ligand binding in hemoglobin. Hemoglobin forms a tetramer whose monomeric unit is described by either a square (T-state) or a circle (R-state). The circles (◦) represent bound ligands. The x-direction is that for the process of conformational changes and the y-direction is that for the process of ligand binding. The areas enclosed by the gray broken lines indicate the two models: the MWC model (the leftmost and the rightmost columns) and the KNF model (the diagonal part).

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hemoglobin, whose varieties appear with two aspects, the number of T/R states in the tetramer (the x-direction; the T-state is a “tense” state with a lower ligand affinity and the R-state is a “relaxed” state with a higher ligand affinity) and the number of ligand molecules bound to the tetramer (the y-direction where the circles (◦) indicate bound ligands). The two models describe two extremes of the kinetic process connecting an all-T-state with no ligand (upper left) and an all-R-state with four ligands (lower right). Here, we have summarized the characteristic features of the two models: •

MWC Model: This assumes an all-or-none equilibrium of conformational transition between an all-T-state and an all-R-state without requiring ligand binding. The equilibrium constant between the two conformations is independent of ligand binding. However, it also assumes that the R-state has a larger affinity to the ligand compared to the T-state. Due to this difference in affinity, the equilibrium shifts to an all-R-state depending on the concentration of the ligand. • KNF Model: This focuses on the sequential events of ligand binding, in which the change from the T-state to the R-state is induced by ligand binding (induced fit). To represent this, we have assumed that ligand binding alters the equilibrium between T and R to favor the R-state. The MWC model stresses the equilibrium fluctuation of the two conformers, while the KNF highlights the intermediate states of ligand binding, or the “encounter complex.” Eigen thought that these two aspects, fluctuations and binding intermediates, were not mutually exclusive, and unified them into the one scheme shown in Figure 1. Note that these original models have been improved over a long time to help explain the experimental evidence accumulated thus far [177]. However, the essence of the two models of the protein function has already been contained in this scheme. 4.

General Model of Protein–Ligand Interactions

After the details on Eigen’s unified model had been published [175], it took almost 30 years for these dynamical models of the protein function to be revived and applied to various protein functions. This was presumably because, during the 1970s and the 1980s, protein crystallography provided a picture of static and inflexible proteins, which totally prevailed over the dynamical one. In the 1990s, protein flexibility and its importance in the protein function became to be accepted mainly due to the gradual accumulation of multiple crystal structures of proteins obtained under different crystallization conditions. In the 1990s, studies on antibody–antigen interactions focused on the flexibility of antibodies as a key factor in antigen recognition [178–180]. As in the MWC model, they observed that antibodies underwent a certain range of conformational

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MWC-like Ligand-free state

Ligand-bound state KNF-like Figure 2. A unified model of protein function based on Bosshard’s model [182]. The triangles, squares, octagons, and circles represent various conformations of a protein molecule. A crescent is a ligand. The size of the pairs of arrows implies the equilibrium constant; different sizes mean a biased population. The squares are the equilibrium form of the ligand-free form. The circles are the ligandbound form. There are two pathways, the MWC-like and the KNF-like. An off-pathway is indicated by the triangles on the left side of the figure.

changes before antigen binding. Independently, Nussinov and coworkers proposed an MWC-like model for the ligand recognition process based on an analogy of the folding funnel concept [181]. Then, Bosshard summarized these findings and presented a unified model for protein–ligand interactions corresponding to Eigen’s generalization or monomer versions of the KNF and MWC models (the former is now simply called the “induced-fit” model and the latter is the “preexisting equilibrium,” “population shift,” or “conformational selection” model) [182]. Figure 2 is an explanatory version of the Bosshard’s unified model. In a similar way to that in Fig. 1, the x-direction is for conformational change and the y-direction represents the ligand-binding process. This describes the ligand-binding process from the unbound state (square) to the ligand-bound state (circle with crescent). There are two extreme pathways, the MWC-like and the KNF-like, whose characteristics are as follows: •

The MWC Path: The ligand binding follows the conformational change of the protein. The ligand-bound form of the protein populates without ligand binding even though its probability is not necessarily large. • The KNF Path: First, the ligand binds to the protein and then the protein starts to change its conformation. The existence of such an encounter complex is the key feature.

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There have been numerous studies on how the experimental system fits either of the two paths [183–191]. Since the ligand-binding process is a kinetic process, each intermediate state is transient and difficult to detect experimentally. Therefore, the MWC and the KNF paths have usually been distinguished simply by the following experimental observations: in the MWC path, the ligand-bound form should exist even without ligands, and in the KNF path, an encounter complex should be observed. However, these two features are not necessarily incompatible with each other in a protein system. The existence of equilibrium fluctuations in a protein molecule has already been established (see experimental observations in Section II.B), and it may be reasonable to imagine that the ligand-binding process goes through transient bindings to multiple binding sites from nonspecific to more specific sites [192]. Therefore, it is natural to imagine stochastic and multiple pathways through various intermediate states, which constitute a variety of protein conformations and ligand-binding sites, although there should be a certain preference depending on each system (Fig. 2). 5.

Linear Response Model

In Section IV.A.4, we have presented a general model of the protein function shown in Fig. 2, which describes the details on the kinetic process occurring in ligand binding. Here, we try to construct a physical framework that satisfies the essence of the model. The model in Fig. 2 tells us about two important features: 1. The protein function occurs as the nonequilibrium process of a response to a specific external perturbation such as ligand binding (the KNF model). 2. The response includes conformational changes in the protein, which is consistent with the equilibrium fluctuations (the MWC model). When focusing only on the conformational changes, these requirements are directly connected to linear response theory described in the classical form: φi (r; t)1 − φi (r)0

dr dt χij (r, r ; t − t )Vj (r ; t )

(35)

j

This describes the instantaneous distribution function of a protein atom i, φi (r; t), at position r and time t, which is perturbed by the external perturbation, Vj (r ; t ), acting on a protein atom j at position r and time t , where 1 is the ensemble average in the perturbed state while 0 is the average in the unperturbed state. The response function is described by χij (r, r ; t) =

1 d δφi (r; t)δφj (r ; 0)0 kB T dt

(36)

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The perturbing potential is the essence of the KNF model and the response function proportional to the ensemble average in the unperturbed state is that of the MWC model. Therefore, Eq. (35) satisfies the two features above. In this formula, the ligand molecule and its trajectory have to be explicitly represented by the perturbing potential of Vj (r; t). As the first step in the application of linear response theory, we have considered only the property at t → ∞ here. Thus, we have 1 dr δφi (r; 0)δφj (r ; 0)0 Vj (r ) (37) φi (r; ∞)1 − φi (r)0 − kB T j

where we have considered the following simple trajectory of the ligand interaction: Vj (r; t) =0

t<0

=Vj (r)

t≥0

(38)

This means that we have limited ourselves to the problem of predicting the structure of the ligand-bound state from the ligand-free state, while ignoring information on kinetic intermediate states. Ikeguchi et al. further simplified Eq. (37) and obtained [193] ri 1 ∼

1 ri rj 0 fj kB T

(39)

j

where the response is described in the expectation of the coordinate shift by the perturbing force of fj , and the response function is now represented by a variancecovariance matrix, which can be easily evaluated by using MD simulation or the elastic network model (see Section III.B.2). The example in Fig. 3 shows the prediction of structural change in the ferric-binding protein when a ferric ion is bound and demonstrates the accuracy of linear response theory in predicting the ligand-bound form. When we can use the elastic network model to evaluate the response function of the variance–covariance matrix, it will be possible to apply Eq. (39) to numerous proteins in the Protein Data Bank. Such a database study found that most structural changes accompanying domain motions could be successfully explained by using Eq. (39) [194]. Recently, the formula for linear response theory has been extended to the one in dihedral angle space [195]. Note that Eq. (39) is not a standard linear response theory that describes influences caused by a weak perturbation. This is because the protein structural changes when ligand binding occurs usually include motions with a large amplitude (say ˚ and a long relaxation time on a rugged potential surface. The from a few to 10 A) reason why Eq. (39) can be used to successfully predict changes in the protein structure lies in the characteristics of protein dynamics; that is, the equilibrium distribution of protein molecule can be well approximated by the quasiharmonic

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Figure 3. Structural change in ferric-binding protein on binding of Fe3+ . (a) Structural changes of Cα atoms, experimental (black) and predicted by Eq. (39) (light gray), are represented by arrows on the backbone trace of the average structure in the simulation (dark gray). The experimental displacement is the difference between the ligand-free form (PDB ID: 1D9V) and the bound form (1MRP). On binding of Fe3+ (gray sphere), the moving domain (residues 1–82, 88–101, 226–276, 308–309; left part) undergoes a closure motion against the fixed domain (residues 83–87, 102–225, 277–307; right part). Model force f applied to the Oε1 of Glu58 (magenta, ball and stick) is depicted by the thick arrow. (b) The magnitudes of the experimental (black) and predicted (light gray) displacements of Cα atoms are in agreement. The scale was adjusted to make the average the same as that in the observation. (Reproduced with permission from Ref. 193. Copyright 2005 by the American Physical Society.)

approximation (Section III.C.1). It can be easily shown that Eq. (39) is directly derived from the quasiharmonic distribution function of Eq. (29). Therefore, the linear response theory of Eq. (39) can be simply understood as a nonequilibrium extension of the quasiharmonic approximation of Eq. (29). B.

Biased Molecular Dynamics Simulation on Protein Functional Motions

Functional motions usually involve structural transitions, which always have a free energy barrier in between the initial and final states. It is thus not feasible for a brute-force MD simulation to surmount the barrier within a practical CPU time except for some limited cases [196, 197]. Deliberate identification of the interactions responsible for the potential barrier sometimes helps to simulate the structural transition [198]. However, such methods cannot be generalized to every protein system. Therefore, to simulate the transition processes within a practical simulation time, various biasing techniques have been developed, including essential dynamics sampling and targeted and steered MD simulations. Essential dynamics sampling [199–201] is one of the biased sampling methods to increase the probability of surmounting the barrier in structural transitions. With this method, an MD step is biased to a preassigned direction, but the biasing operation is limited in the essential subspace defined by a few principal components with the largest amplitude (see Section III.C.2). In many cases, the protein structural changes can be well approximated by domain motions where two rigid domains mutually change their positions, or the structural changes mainly occur in the six external degrees of freedom that determine the special arrangement

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of the domains. These degrees of freedom in the domain motions frequently correspond to large-amplitude principal components. Hence, essential dynamics sampling tries to enhance sampling capabilities only in the subspace connecting the initial and final states of the structural change. Essential dynamics sampling was applied to the structural changes in citrate synthase [202] and adenylate kinase [203]. An alternative method of generating reasonable pathways for conformational changes is targeted MD simulation, in which geometrical constraints are imposed to induce a favorable transition between two structures. The constraining force is usually determined so that it is proportional to the RMSD from a target structure. Due to the simplicity of the algorithm, the targeted MD simulation has been widely used to investigate conformational changes in proteins, such as those in insulin [204–206], ras p21 [89, 207, 208], α -chymotrypsin [209, 210], and other enzymes [211–213]. It has also been applied to the problems of allosteric transitions underlying protein functions, including GroEL [214], lactose repressor protein [215], and chorismate mutase [216]. In addition, channel gating of membrane proteins [217–220] and the movement of tRNA into the ribosome [221] have successfully been simulated using this technique. Steered MD simulation is another biasing method that uses external forces in a certain direction instead of the information of a preassigned target structure [222–224]. This method has been used to simulate single-molecule mechanical measurements such as atomic force microscopy, laser optical tweezers, and biomembrane force probe experiments. Its application covers the problems with mechanical unfolding [225–227], protein–protein interaction [228–230], ligand binding/unbinding [231–242], ligand isomerization [243], and ligand transport across a channel protein [244–249]. Since the steered MD simulation induces a nonequilibrium process due to applied external forces, it has been used to evaluate the equilibrium free energy differences [250, 251] using Jarzynski’s equality [252, 253]. C.

Path Search and Path Sampling for Biomolecules

Now we discuss current theoretical attempts to extract conformational transition pathways in biomolecules. A conformational transition pathway roughly means a plausible path connecting two basins in configuration space. This concept was first realized in chemical reactions of small molecules (∼10 atoms) as exemplified by the intrinsic reaction coordinate (IRC) calculations for molecules [254, 255]. The basic procedure of IRC is as follows. First find a transition state (TS) and then integrate the following equation from the TS: dx(s) ∇V (x) =− |∇V (x)| ds

(40)

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where x are the mass-weighted coordinates, s is the path length, and the right-hand side is the negative gradient (force) of a potential energy function V (x). Stop the calculation when reaching a potential minimum. Though this strategy works for small molecules, several technical difficulties arise when we try to apply this method to large biomolecules (say ∼10,000 atoms). First, it is too difficult to find transition states in a multidimensional rugged energy landscape. There should be several (or many) transition states, and we do not know which one is the most important a priori even if we can find it. Furthermore, in the IRC calculation, it is assumed that there is a single and smooth path connecting two basins, but it is not the case for a multidimensional rugged energy surface. There should be many pathways connecting two basins, in particular, at finite temperatures. One of the problems in the IRC method is that it is based on a local (or differential) view of a potential energy surface, and it lacks a global view, which is useful to characterize a multidimensional rugged energy surface. Path search or path sampling methods that we review in this section are based on such a global view of a potential energy (or free energy) landscape. For further discussions on this topic, the reader is referred to the review papers by Straub [256], Elber [257], and van der Vaart [258]. Related techniques for obtaining a reaction coordinate, such as steered and targeted MD simulations, are summarized in Section IV.B. For simplified algorithms (e.g., morphing methods) for searching paths using coarse-grained models, refer to a recent paper by Weiss and Levitt [259]. 1.

Minimum Energy Path Search

A minimum energy path (MEP) is a fundamental concept in path search for (bio)molecules because it attempts to characterize the topography of a potential energy surface. To our knowledge, Ron Elber paved the way for the path search for biomolecules. Elber and Karplus first introduced a line integral formalism for MEP calculations [260] and then Czerminski and Elber suggested to use the self-penalty walk method for more efficient calculations of MEPs [261]. The self-penalty walk method is explained below because it contains basic elements to calculate reaction paths. Self-Penalty Walk Methods. The self-penalty walk method is based on the assumption that an MEP is obtained by minimizing the following line integral: SV =

1 L

RB

V (R)dl(R)

(41)

RA

where V (R) is a potential energy function, from which one will extract a path, and R is the Cartesian coordinate for a molecule. In this method, we use the following

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discretized form of the target function: S = SV + S C + S R

2 M M−1 1 d ij = V (Ri ) + γ (di,i+1 − d)2 + ρ exp − M λd i=1

(42)

i,j

i=1

where the first term SV is a discretization of the line integral, the second SC is the chain constraint forcing the beads to line up with the same interval d, and the third SR is the repulsion constraint, which tends to avoid the overlapping of beads. One can use any minimization or simulated annealing technique to apply this method to real molecular systems. Czerminski and Elber applied this method to the conformational change of a tetrapeptide [261]. Ohmine and Tanaka used this method to analyze the hydrogen bond network rearrangement of bulk water [262]. Elber and coworkers used this method to make an initial path for an enzymatic reaction of a biomolecule [263]. Though there are successful applications of the self-penalty walk, this method suffers from several deficiencies. First, it is not clear whether this method converges to an exact MEP because of the additional constraints. Another practical issue is how to choose the parameters in the constraint terms. Therefore, it is desirable to derive more parameter-free and constraint-free methods. The nudged elastic band (NEB) method and the string method that we review below are such methods. Nudged Elastic Band Method. The NEB method [264] introduces an interaction between two beads. The total functional one may use is SNEB =

M−1 M k 1 |Ri+1 − Ri |2 + V (Ri ) 2 M i=1

(43)

i=1

Instead of minimizing this functional, the simplest version of the NEB method directly uses the following forces to evolve beads representing a path: FiNEB = FiV ⊥ + FiS (44) ||

with

FiV

⊥

FiS

= FiV − FiV ·τ|| τ|| ||

= FiS ·τ|| τ||

(45) (46)

where FV is the “actual” force calculated from the potential V (R), FS is a “virtual” spring force stemming from two neighboring beads, and τ|| (τ⊥ ) is a parallel

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(perpendicular) direction to the path we calculate. The main point of this method is that we use only the perpendicular component of the actual force (FiV )⊥ along a path. On the other hand, the parallel component of the spring force (FiS )|| is mainly used. Otherwise, the obtained path tends to take a short cut, and the barrier energy along the path cannot be accurately estimated, i.e., a transition state cannot be correctly identified. It is known that this method converges to an MEP very fast with only one parameter (spring constant k). The NEB method has many applications, mainly in the field of solid-state physics [265], and we find several applications for biomolecules as well. Mathews and Case applied this method to the study of the conformational change of a GG noncanonical pair of DNA [266]. Arora and Brooks III studied the conformational change of large proteins (adenylate kinase [267] and dihydrofolate reductase [268]) using this method. Zero-Temperature String Method. The original string method developed by Vanden-Eijnden and coworkers [269] is very similar to the self-penalty walk or NEB method. We review the recent simplified algorithm of the (zero-temperature) string method [270]. The string method starts with an equation similar to the IRC equation, Eq. (40), dRi = −∇V (R)|R=Ri + constraint ds

(47)

and a constraint is essential for this method. The most frequently used constraint is the equidistance constraint between beads (for other constraints, see Refs. 271 and 272). That is, after evolving beads according to Eq. (47), Ri is shifted such that |Ri+1 − Ri | becomes the same for all i except for the end beads. This constraint is physically similar to the spring interaction between beads in the NEB method or the chain constraint in the self-penalty walk method, but the point is that we do not need to introduce additional parameters in the string algorithm (though there is a possibility that the “unphysical” shift of the bead position might lead to energetically unstable situations). Furthermore, the end beads can evolve freely without any constraints. Because of this procedure, in the string calculations, the algorithm automatically finds two minima and a path connecting the minima at the same time. That is, we do not need to prepare two minima before the calculation. Conjugate Peak Refinement. The conjugate peak refinement (CPR) of Fischer and Karplus is based on a different principle of path search [273]. Starting from only two beads located at two minima, the CPR method adds a new bead (peak) between two beads and optimizes the bead position using the conjugate gradient algorithm. This is why the method is called CPR. This method was successfully applied to study the complex conformational switch of ras p21 [274], the mechanism of

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a molecular valve in the halorhodopsin chloride pump [275], and the structural mechanism of the recovery stroke in the myosin molecular motor [276]. 2.

Path Search at Finite Temperature

In some cases, MEPs seem to be enough to clarify the molecular mechanisms of reactions, including protein functions, even at room temperature [274–276]. This is certainly the case when the dynamics is extremely overdamped. However, in general, this is not the case, and the finite-temperature effect on a free energy landscape should be considered. In this section, we discuss such methods for calculating a finite-temperature path. MaxFlux Methods. To include the finite-temperature effect on a path, Huo and Straub extended the self-penalty walk method [261] to implement their MaxFlux method [277]. They replaced SV in Eq. (42) with SMF , defined by SMF =

M 1 exp{βV (Ri )} M

(48)

i=1

where β = 1/(kB T ) is an inverse temperature. This functional was first derived in the work by Berkowitz et al. [278] and the physical meaning of minimizing this functional is to maximize the flux between two basins (MaxFlux). Huo and Straub applied this method to the conformational change of polyalanine [279] and Alzheimer’s amyloid β-peptide [280]. Crehuet and Field suggested combining the NEB and the MaxFlux methods [281, 282], obtaining

FiMF

⊥

κi = FiV ⊥ − τ⊥ β

(49)

where κi is the curvature of the path at bead i and τ⊥ is a perpendicular direction to the path. From this expression, it is clear how to connect an MEP and a finitetemperature path. They applied the modified MaxFlux method to a molecular ratchet to estimate the energy barrier of rotation [281, 282]. Finite-Temperature String Methods. Vanden-Eijnden and coworkers extended the zero-temperature string method to the finite-temperature one by assuming the overdamped Langevin equation for each bead [283], dRi = −∇V (R)|R=Ri + η(s) + constraint ds

(50)

where η(s) is a Gaussian white noise satisfying ηα (s)ηβ (s) = 2kB Tδαβ δ(s − s), and the final path is obtained after taking the statistical average over many

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ensembles. The constraint is usually the same as that for the zero-temperature string method (equidistance constraint). Using this method, we can calculate a reaction tube connecting two basins, the free energy surface along the tube, and the reaction rate between two basins [283]. This method was applied to the conformational change of a coarse-grained model of a protein [284]. Coarse-Grained String Method. The recently introduced coarse-grained string (CGS) method is powerful for extracting a minimum free energy path (MFEP) [285, 286]. This is a modification of the finite-temperature string method, and we use a coarse-grained description of the system using collective variables θα (x). To calculate an MFEP, we solve the following equations simultaneously: γ z˙ α (s, t) =

m

˜ αβ (x(s, t))κ(θβ (y(s, t)) − zβ (s, t)) + constraint M

(51)

β=1

mi x¨ i (s, t) = fi (x(s, t)) − κ

m

(θβ (x(s, t)) − zβ (s, t))

β=1

∂θβ (x(s, t)) + thermostat ∂xi (52)

mi y¨ i (s, t) = fi (y(s, t)) − κ

m

(θβ (y(s, t)) − zβ y(s, t))

β=1

∂θβ (y(s, t)) + thermostat ∂yi (53)

˜ αβ (x) is the metric calculated as where M ˜ αβ (x) = M

N 1 ∂θα (x) ∂θβ (x) mk ∂xk ∂xk

(54)

k=1

x, y are coordinates to calculate the thermodynamic average of the metric and free energy gradient, respectively, s parameterizes a string, and t represents the Langevin dynamics of the string. To implement this method, we run two molecular dynamics simulations such that the collective variable θα (x) is constrained around zα [Eqs. (52) and (53)]. Therefore, this part is just a conventional constrained MD simulation. The difference is that the constraints zα evolve according to Eq. (51). It is mathematically shown that the right-hand side of Eq. (51) becomes a negative gradient of the free energy surface F (zα ) [286] multiplied by the metric, so the constraints zα must be located around an MFEP. Finally, the CGS method gives us trajectories along the MFEP, from which we can calculate thermodynamic properties such as a free energy barrier or a rate constant. This method was successfully applied to alanine dipeptide [285, 286], hydrophobic collapse of a polymer (with ∼100,000 constraints mainly representing the coarse-grained water field) [287, 288], conformational transitions of a

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coarse-grained model of a protein [289], and the all-atom model of adenylate kinase (with 20 constraints using principal components) [290] and Src kinase (with 150 Cartesian position constraints) [291]. 3.

Path Sampling

An assumption in the calculation of an MEP or an MFEP is that the path thus obtained should be the most relevant and unique. However, this is not the case when we consider diffusive processes at finite temperature. Furthermore, we need to evaluate some dynamic properties of the system, where the (local) equilibrium assumption for an MFEP also does not hold. Therefore, a more general concept of “path” that is applicable to nonequilibrium situations is desirable. Transition path sampling (TPS) [292, 293] and action-based methods are applicable to this general situation. Transition Path Sampling. TPS is a powerful and general method for extracting a nonequilibrium path distribution. The basic idea of TPS is that a path x(t) has a statistical weight; that is, we can define a probability for a path P[x(t)]. Often, we discretize a path x1 = x(t1 ), x2 = x(t2 ), . . . , xM = x(tM ) using discretized times ti (i = 1, . . . , M), so the path probability becomes a multidimensional probability distribution function P(x1 , x2 , . . . , xM ). This is conceptually similar to the canonical weight P(x) = e−βE(x) /Z for a configuration x, though the dimension becomes huge for the path probability, and the explicit expression for the probability is not as simple as that of the canonical weight. Pratt was the first to notice the possibility of path sampling using this conceptualization of TPS [294], and Chandler and coworkers devised useful Monte Carlo procedures (shooting and shifting) for TPS [292, 293]. Their methods were successfully applied to many chemical systems as summarized in their review papers [292, 293, 295]. A few biomolecular applications are binding and unbinding pathways of a base pair in a CGC DNA oligomer [296], the folding mechanism of Trp-cage [297], the DNA repair process by polymerase β [298, 299], and flip-flop of lipids in membranes [300]. TPS often relies on standard MD procedures to calculate a trajectory, leading to both advantages and disadvantages. One of the advantages is that it is easy to implement using conventional MD softwares. One of the disadvantages is that we can treat only rare but fast transitions. That is, the transition that we can treat by TPS combined with MD simulation is the one that is easily calculated using the MD simulation (∼100 ns). This can be a problem when we study “very” slow processes in biomolecules such as folding or conformational change of proteins. The action-based methods we describe below are more feasible for dealing with such slow processes. Action-Based Methods. There are at least two categories in the action-based methods of path sampling. One is based on the least action principle of classical

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mechanics, which is used to calculate a dynamic trajectory as accurately as possible [301–307]. The important difference from the conventional MD algorithm (Verlet, Leap-frog, etc.) is that the problem is transformed to a boundary value problem instead of an initial value problem. Due to this transformation, we can take a much larger time step to calculate a trajectory. The burden is proportional to the number of degrees of freedom we treat, which is determined by the number of beads to represent a path. Elber et al. pushed away this burden by carrying out numerous calculations for biomolecular processes, including folding processes of C-peptide [308] and cytochrome c [309], an enzymatic reaction in a protein [263], and the conformational transition of the ligand-binding domain of the mGLuR receptor [310]. For a detailed account of their method, the reader is referred to their review papers [305, 306]. The other category of action-based methods deals with diffusive paths described by Langevin dynamics. We illustrate these types of methods using a onedimensional system with a coordinate x and a friction γ at temperature T . Because we are interested in the long-time dynamics of biomolecules, it is often assumed that the dynamics is well approximated by the overdamped Langevin equation γ x˙ = F (x) +

2γkB T ξ(t)

(55)

where F (x) is a force applied to the system, or equivalently by the forward Fokker– Planck equation for a distribution function P(x, t), ∂2 P(x, t) F (x) ∂P(x, t) ∂P(x, t) =D − ∂t ∂x2 γ ∂x

(56)

where D = kB T/γ, which is imposed by the fluctuation-dissipation theorem. A formal solution for the propagator of the Fokker–Planck equation is given in the path integral form as [311–313] P[ξ(t)] ∝ e−βS[x]

(57)

with S[x] =

γ 4

t

ds[˙x − F (x)/γ]2

(58)

0

where S is the Onsager–Machlup action (or functional) [314]. It is formally easy to extend this formalism to the case of underdamped (usual) Langevin dynamics [315]. This can be regarded as a different representation of TPS tailored for slow processes, and the fact that the path probability looks like a canonical weight

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inspired us to use the replica exchange method to enhance sampling in the path space [316]. To our knowledge, Eastman et al. first applied this method to biomolecular systems [317]. Orland and coworkers introduced another formulation using length instead of time [318–320], which is similar to the stochastic differential equation in length (SDEL) developed by Elber et al. [305, 306]. The Onsager–Machlup action is the basis for dynamic importance sampling developed by Zuckerman and Woolf [321] and was applied to dynamic reweighting of a trajectory by Andricioaei and coworkers [322].

V.

CONCLUDING REMARKS

In this chapter, we have reviewed the current status of computational methods to reveal protein dynamics during conformational changes in atomic detail. Although MD simulations have become much more feasible because of advances in software, hardware, and algorithms during the last few decades, the timescale of conformational transition is still far beyond current computational capabilities. To obtain biological insights into protein functions from numerical simulations by overcoming this timescale problem, a number of theoretical and computational methods have been proposed over the years. In this chapter, we have focused on quasiharmonic picture based on the equilibrium description of protein dynamics and related methods such as normal mode analysis and principal component analysis. These methods are known to be efficient to extract functionally relevant motions: a few lowest-frequency normal modes or a few largest-amplitude principal components are known to correlate well with functionally relevant motions of proteins. Such modes can also be utilized to constrain MD simulations in required directions, leading to the idea of essential dynamics sampling for conformational changes. To proceed to an understanding of the nonequilibrium aspects of conformational changes, various kinds of descriptive models have been proposed to conceptualize the coupling between ligand-binding processes and conformational changes, including the lock-and-key, induced-fit, and conformational selection (population shift) models, as well as the allosteric interaction model. The linear response model has been proposed as a theory bridging these concepts in atomic detail, and its usefulness has been confirmed in a number of protein systems [193-195]. The basic assumption underlying this model is the quasiharmonic approximation, in which a distribution function can be approximated using a multivariate Gaussian function, frequently employed to analyze protein dynamics in simulation studies. To understand the complete transition pathways of conformational changes, however, the quasiharmonic picture is entirely insufficient because it simply predicts the direc-

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tion of conformational change. We have reviewed more sophisticated methods including various biased MD simulations (essential dynamics sampling, targeted, and steered MD simulations) and path sampling methods, which will hopefully lead to a fuller understanding of conformational changes and their biological relevance in individual biomolecular systems. To conclude this chapter, let us mention future prospects in theoretical studies on protein functional dynamics. First, the increasing power of computer hardware that will soon reach over 10 petaflops is indispensable for more efficient MD simulations, as well as the high-speed computers specialized for MD simulations (MD-GRAPE [323] and Anton [324]) and the GPU-based acceleration technologies. Further development of algorithms for MD simulations is also required, especially for the calculation of long-range electrostatic forces because it is the most time-consuming part in MD simulations. For example, the recently introduced Wolf method and its extensions are promising to calculate the electrostatic forces efficiently [325]. In Section IV.C, we discussed path search and path sampling methods in some depth and pointed out the importance of a “path ensemble” instead of a single pathway to interpret nonequilibrium protein dynamics, which is paraphrased as follows. First, a single trajectory connecting the initial and final states is not sufficient, but an ensemble of paths is necessary to fully understand the stochastic nature of protein dynamics. Second, we should go beyond the thermodynamic landscape picture to obtain information on “real” protein functional dynamics in atomic detail. Third, for the purposes of accomplishing first and second, we need more computer resources and better sampling algorithms as well as better force-field parameters. Coarse-grained models have been used and extended to treat the functional motions of much larger biomolecular systems such as F1 -ATPase [326] and myosin (or kinesin) motor proteins [327]. However, the accuracy of CG models is not justified a priori, and the parameters of such CG models should be derived from all-atom simulations [328–330]. Therefore, one possible way of improving the accuracy of CG simulations is to combine the CG model with the all-atom model in simulations. Moritsugu et al. recently proposed a multiscale method of simulations using a CG model together with the all-atom model and succeeded in enhancing the sampling efficiency of all-atom simulations [331]. Finally, we stress the importance of comparative MD simulations for various types of biomolecular systems. Since the biological function is diverse, after the physical and chemical origins of the function of each biomolecular system are understood, a bird’s-eye view of numerous protein systems may be helpful to fill in the gaps between individual biological events and certain general principles. We may have to borrow the methodologies used in bioinformatics to detect such “coherent” pattern in whole protein systems. By combining such information with comparative studies on conformational changes (transition pathways) in various

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biomolecular systems, we can hopefully obtain a systematic understanding of whole biological processes in proteins in atomic detail. Note added in proof Recently, two analysis methods have been proposed to overcome the limitations of PCA and to clarify protein dynamics from MD trajectories [332, 333]. Acknowledgments The authors thank Dr. Ikuo Fukuda, Dr. Tadaomi Furuta, and Dr. Minoru Kubo for useful comments on the manuscript. This research was supported by Research and Development of the Next-Generation Integrated Simulation of Living Matter, a part of the Development and Use of the Next-Generation Supercomputer Project of the Ministry of Education, Culture, Sports, Science and Technology (MEXT).

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NON-BROWNIAN PHASE SPACE DYNAMICS OF MOLECULES, THE NATURE OF THEIR VIBRATIONAL STATES, AND NON-RRKM KINETICS DAVID M. LEITNER1 , YASUHIRO MATSUNAGA2 , CHUN-BIU LI3 , TAMIKI KOMATSUZAKI3,4 , AKIRA SHOJIGUCHI5 , and MIKITO TODA5 1 Department

of Chemistry and Chemical Physics Program, University of Nevada, Reno, NV 89557, USA 2 Molecular Scale Team, Integrated Simulation of Living Matter Group, Computational Science Research Program, RIKEN, 2-1 Hirosawa, Wako-shi, Saitama 351-0198, Japan 3 Molecule & Life Nonlinear Sciences Laboratory, Research Institute for Electronic Science, Hokkaido University, Kita 20 Nishi 10, Kita-ku, Sapporo 001-0020, Japan 4 Core Research for Evolutional Science and Technology (CREST), Japan Science and Technology Agency (JST), Kawaguchi, Saitama 332-0012, Japan 5 Department of Physics, Faculty of Science, Nara Women’s University, Kitauoyahigashimachi, Nara 630-8506, Japan

CONTENTS I. II.

Introduction Fractional Behavior in Classical Systems with Mixed Phase Space A. Minimal Model of Reaction Exhibiting Fractional Behavior B. Fractional Behavior of Reactions C. Wavelet Analysis of Arnold Web D. Dynamical Connection Revealed by Wavelet Analysis E. Nonexistence of Reaction Rate Constants III. Anomaly in Diffusion in Spatiotemporal Multiscale Classical Systems A. Finite-Size Lyapunov Exponent B. Principal Component Analysis Advancing Theory for Kinetics and Dynamics of Complex, Many-Dimensional Systems: Clusters and Proteins, Advances in Chemical Physics, Volume 145, Edited by Tamiki Komatsuzaki, R. Stephen Berry, and David M. Leitner. © 2011 John Wiley & Sons, Inc. Published 2011 by John Wiley & Sons, Inc.

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C. Complexity of Hierarchical Dynamics in Multiscale Nonlinear Systems: A Case Study of Model Protein IV. Energy Flow and Localization in Quantum Systems with Mixed State Space and Reaction Kinetics A. Local Random Matrix Theory B. LRMT Dynamical Corrections to RRKM Theory of Unimolecular Reaction Rates C. Illustrative Example: A Gas-Phase SN 2 Reaction V. Conclusions Acknowledgments References

I.

INTRODUCTION

The evolution of structures of large molecules and clusters depends both on the topography of the energy landscape, in particular barriers that separate minima corresponding to specific conformers, and on the flow of energy within each local minimum visited by the system. A useful starting point for calculating the evolution of structures on an energy landscape is a master equation approach. In this case, incorporation of the energy flow contribution can be finessed by assuming ergodicity and extremely rapid flow within a local minimum on the energy landscape and adopting microcanonical transition state theory, that is, Rice–Ramsperger– Kassel–Marcus (RRKM) theory [1], to calculate transition rates between pairs of conformers. This approach, however, neglects possible restrictions that a given conformer may face in equilibration, that is, the system may be nonergodic, and even if it is ergodic the equilibration time may be long compared to the time for an activated conformer to make the transition to a new structure. Many theoretical [2–7] and experimental studies [8–12] of isomerization kinetics of modest-sized organic molecules highlight the importance of these contributions to the reaction rate. Indeed, recent calculations modeling the conformational dynamics of an IR-excited dipeptide in a molecular beam reveal substantial enhancement of conformational selectivity, consistent with experiment [13], when these non-RRKM theory effects are accounted for, in contrast to the case when simple RRKM theory kinetics are assumed to describe the transition rate between local minima on the energy landscape of the peptide [14]. In this chapter, we review work on the statistical properties of nonergodic systems with a “mixed” phase space (classical system), or “mixed” state space (quantum system), and how account for this property can be incorporated into a correction to the RRKM theory estimate for the transition rate between structures of a molecule or cluster. By mixed we mean nonergodic, where transport through the space is in some regions restricted while in other regions much freer. In reviewing work on this topic, we address the following questions: What are the criteria for the

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establishment of equilibrium in a local minimum of the energy landscape? If the system is nonergodic, so that equilibration cannot be achieved, can we develop a statistical description of energy transport on the energy surface or shell? How can we use such a statistical description to calculate the transition rate between local minima on the energy landscape? It is also important to bear in mind that even if ergodicity holds and the system ultimately can equilibrate, how long this process takes can influence the transition rate between conformers, and corrections to the RRKM theory estimate for the rate are often still required, a point that we also address in this chapter. The classical theory for the influence of a mixed phase space on isomerization kinetics is due to Berne [15]. Berne introduced a correction to the RRKM theory estimate for the reaction rate that accounts for the fraction of phase space that is “irregular,” that is, that part of the energy surface where transport is relatively free. In this case, it is possible for the rate constant to be greater than the RRKM theory estimate for the rate, due to the relatively restricted phase space that is available for reaction compared to that accounted for in the RRKM theory estimate. Bottlenecks to transport that appear in a mixed phase space tend to enhance the rate constant, an effect that has been observed in gas-phase SN 2 reaction kinetic studies [16–18]. Recently, Leitner and Wolynes [19] introduced a quantum mechanical version of the theory that also accounts for an additional quantum bottleneck, namely, the rate of energy flow into and out of the vibrational states that lie in the activated region. Because of the relatively small number of vibrational states that overlap the transition regions when the system is nonergodic, the reaction rate may be greater than the RRKM theory estimate for the rate, as predicted by the classical theory. On the other hand, bottlenecks to energy flow into and out of the transition region can reduce the rate, in either the nonergodic or the ergodic regimes. A recent calculation by Leitner and Gruebele [20] has shown that the latter effect can significantly reduce the effective number of states of the activated complex, that is, reduce the effective number of reaction channels. Numerous classical studies of the mixed phase space region of small molecules over many years have revealed a very complex phase space structure [21]. While the phase space can be rigorously divided into regular and irregular regions for two-degree-of-freedom systems, there is no such simple separation for larger systems. This gives rise to new dynamical effects such as Arnold diffusion for larger systems. In particular, fractional behavior in Hamiltonian chaos is of interest for a variety of systems ranging from billiards, water clusters, and other chemical systems [22–29]. In these studies, unconventional statistical features have been found such as anomalous diffusion, time correlation with an algebraic decay, and 1/f spectra. For systems of two degrees of freedom (2DOF), the dynamical origin of the fractional behavior is well understood based on the hierarchical structures formed by resonant tori [22]. There, tori constitute dynamical barriers for

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transport: The two-dimensional tori are impenetrable boundaries, thereby dividing the three-dimensional equienergy surface into separate regions. On the contrary, for systems of more than 2DOF, the origins of the fractional behavior are not yet well understood: The mechanism for systems of 2DOF is not relevant here since the dimension of tori is not sufficient to block the transport in the phase space. Therefore, understanding the dynamical origins of fractional behavior for systems of more than 2DOF presents a new problem in nonlinear physics. The existence of the fractional behavior in Hamiltonian systems raises a serious question for the foundation of the statistical reaction rate theory. In the conventional theory, reaction processes are supposed to be normal diffusive motions [30]. This idea is based on the assumption that the characteristic timescale for the reaction is much longer than that for trajectories to lose their memories. Then, reaction processes are regarded as being composed of many erratic dynamical motions, leading to the Brownian type of diffusion. However, recent studies on reaction processes cast doubt on this assumption. For example, 1/f spectra are found in simulations of water clusters [24, 25], and power law decays are observed in correlations of vibrational dephasing [31, 32]. These studies indicate that the basic assumption of the statistical reaction theory should be reexamined by taking into account the existence of the fractional behavior in the mixed phase space. We should also note the recent development of the theory of dynamics to include nonstationary processes [33–41]. These studies will offer a basis for understanding fractional behavior in reactions beyond the traditional concept of ergodicity [42]. In the quantum mechanical section, we address limited energy transport in a many-nonlinear-oscillator system, which describes the coupled vibrations of a molecule or cluster. We first present criteria for facile quantum energy flow on the energy shell, a necessary but not sufficient condition for RRKM theory kinetics to hold in unimolecular reactions. This analysis reveals a crisp quantum ergodicity threshold [43–45]. We review one approach to the analysis of the quantum ergodicity threshold and the statistical nature of the vibrational states of a large molecule or cluster called local random matrix theory (LRMT). Below the quantum ergodicity threshold, the vibrational states extend on average over a restricted region of the energy shell. At a given energy, LRMT provides predictions about the extension of vibrational states on the energy shell in this mixed state space, nonergodic region. We shall discuss these predictions, and then review application of LRMT to the calculation of rates of unimolecular reaction, specifically calculation of the transmission coefficient that modifies the RRKM theory estimate for the rate to account for nonergodicity and sluggish energy flow. The specific example of a gasphase SN 2 reaction will be discussed. Interestingly, the above studies cast a new light on the relationship between Maxwell’s demon and molecular bases of functions for biomolecules. In the original argument on Maxwell’s demon, the demon is supposed to be a microscopic

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entity that recognizes information on a molecular level and performs a certain function accordingly [46]. Maxwell pointed out that such an entity would eventually violate the second law of thermodynamics. Later, it is shown that the demon cannot perform its function under fluctuation of equilibrium states [47, 48]. In the present context, the demon can be regarded as a model of biomolecules whose functions are to react to specific molecules, to transfer energy to specific modes, and so on. We can present a new possibility of the demon on the molecular level when nonergodicity in reaction processes is taken into account. This idea is based on recent studies where Maxwell’s demon is considered in terms of fractional behavior in Hamiltonian dynamics [49]. An important aspect of these studies is that Maxwell’s demon is treated from a purely dynamical point of view without assuming the existence of a heat bath. When fractional behavior exists at molecular levels, the demon can utilize long-lasting memories inherent in anomalous fluctuations for performing its function [50]. Thus, nonergodicity provides new resources with which Maxwell’s demon can work. Therefore, studies of nonergodicity in reactions offer a new research arena where the possibility of information processing by molecules is considered beyond a conventional argument based on thermal fluctuation. This issue is also addressed in Chapter 5.

II.

FRACTIONAL BEHAVIOR IN CLASSICAL SYSTEMS WITH MIXED PHASE SPACE

In the conventional theory, reaction processes are supposed to be normal diffusive motions [30]. This idea is based on the assumption that the characteristic timescale for the reaction is much longer than that for trajectories to lose their memories. Then, reaction processes are regarded as being composed of many erratic dynamical motions, leading to the Brownian type of diffusion. For such cases, the distribution P(t) of the reactant is expected to obey the exponential decay following the phenomenological rate equation, and the exponent of the decay is given by the rate constant. However, recent studies on reaction processes cast doubt on this assumption. For example, 1/f spectra are found in simulations of water clusters [24, 25], and power law decay is studied theoretically [28, 29] and is experimentally observed in correlations of vibrational dephasing [31]. Such behavior is also revealed for biomolecules by experimental and numerical studies [51–53]. These studies indicate that the basic assumptions of the statistical reaction theory should be reexamined. Thus, we should ask what are the characteristic features of the reaction processes when ergodicity is not achieved. In order to understand reaction processes beyond the assumption of ergodicity, we have to resort to the dynamical theory of reactions. In this theory, the reaction

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consists of the following three processes: (i) redistribution of vibrational energy in the well, (ii) going over the potential saddle, and (iii) dynamical connection between the saddle and the well, and also the connection among multiple saddles. Redistribution of vibrational energy in the well is determined by the Arnold web, the network of nonlinear resonances in the well. The process of going over the saddle is understood by the normally hyperbolic invariant manifolds (NHIMs) and its stable/unstable manifolds, which is discussed further in Chapter 4. The dynamical connection is studied based on intersection among the stable/unstable manifolds of the NHIMs and the Arnold web (exactly speaking, whiskered tori created by resonances). Here, we focus our attention on the redistribution of vibrational energy in the well and the connection between the NHIM around the saddle and the Arnold web in the well. The statistical features of the reaction depend on the characteristics of the Arnold web, how nonlinear resonances are distributed in the web, and whether their distribution is nonuniform or not. In particular, when resonances are distributed nonuniformly and there exist regions in the web where resonance overlap is sparse, we have nonergodic reactions with fractional behavior. We will also show that the dynamical connection between the NHIM and the Arnold web plays an important role in transient features of the reaction processes. A.

Minimal Model of Reaction Exhibiting Fractional Behavior

In this section, we present numerical manifestation of fractional behavior for a minimal Hamiltonian system of 3DOF describing reaction processes with nonergodic dynamics in the well. The model has a double-well potential where the Arnold web within the well is nonuniform. The processes of going over the potential saddle are regarded as isomerization, where the molecule changes its configuration [54–56], H = H0 + H 1

(1)

3 p2i p21 λ2 λ2 ω 2 q2 − q12 + q14 + + i i + bqi4 2 2 4 2 2 i=2 H1 = exp −(q1 − 1)2 /σ 2 a1 q22 q32 + a2 (q1 − 1)2 (q22 + q32 )

H0 =

Here q1 is a coordinate that is the lowest order approximation of the reaction coordinate, and qi (i = 2, 3) are coordinates that are the lowest order approximation of the bath modes. √ The linear frequencies are ω1 = 1.02, ω2 = 0.94, and ω3 = 1.04, where ω1 = 2λ is the unperturbed frequency at q1 = 1, that is, the bottom of the well. The parameters σ = 0.5, a1 = 0.1, a2 = 0.9, and b = 0.5 are chosen to ensure that the system is trapped in the well by energy exchange between the bath and the reaction coordinates.

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Figure 1. The Arnold web of the primary resonances. The radii of the circles indicate their widths. (Reproduced with permission from Ref. 55. Copyright 2007 by the American Physical Society.)

The system has the primary resonances given by ω¯ 1 = ω¯ 2 , ω¯ 1 = ω¯ 3 , ω¯ 2 = ω¯ 3 , where the nonlinear frequencies are defined by ω¯ i (Ji ) = ∂H0 /∂Ji using the action variables Ji (i = 1, 2, 3) corresponding to (pi , qi ), respectively. In Fig. 1, we show these resonances in the action space by estimating their locations approximately using the unperturbed Hamiltonian. There, we also indicate the widths of the resonances by the radii of the circles. We note that resonance overlap between the primary resonances is sparse in the Arnold web. On the other hand, there exist regions where multiple resonances meet, that is, resonance junctions. Thus, the Arnold web is nonuniform. These features play a crucial role for reaction processes with nonergodicity in the well, giving rise to fractional behavior. B.

Fractional Behavior of Reactions

The fractional behavior of the reaction can be seen in the following three properties: •

The survival probability in the well has the following two ranges as a function of the residence time: One is the range where the probability varies according to the power law and the other where it changes exponentially. • The Fourier spectra of the action variables correspondingly exhibit different characteristics; for trajectories in the range of the power law decay, the Fourier spectrum exhibits 1/f dependence. On the other hand, it shows the Lorentzian feature for those in the range of the exponential decay. • The local diffusion of the action variable corresponding to the reaction coordinate exhibits two types of behavior: anomalous diffusion for the power law decay and normal one for the exponential one. In Fig. 2a and b, the survival probabilities are shown as a function of the residence time tr and as a function of the crossing number n, respectively. In both the figures, the survival probabilities reveal two distinct timescales. For the survival

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−γ

probability P(tr ) as a function of the residence time tr , P(tr ) decays as tr up to about 100 cycles, and for longer timescales, P(tr ) decays as exp(−αtr ). By fitting the function P(tr ), we obtain γ = 0.82 and α = 0.0015. We also see a similar behavior for the survival probability as a function of the crossing number n. In Fig. 3a and b, we display Fourier spectra of the time correlation of the action J1 corresponding to the reaction coordinate, for a trajectory of the power law decay and for a trajectory of the exponential decay, respectively. Fittings by 1/ω

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and a/(ω2 + b) are also shown there, respectively. There, the Fourier spectrum for the power law decay reveals a 1/f behavior, while the Fourier spectrum for the exponential decay follows a Lorentzian behavior. Other trajectories of the power law decay and those of the exponential decay exhibit the same features, respectively. Diffusion in the action space also exhibits the corresponding difference between the trajectories of the power law decay and those of the exponential decay. We define local diffusivity in the action space by σk (t, t0 ) = (Jk (t) − Jk (t0 ))2

(2)

where the square displacement of Jk during the time interval between t0 and t is averaged over an ensemble of the trajectories within a range of the residence time. By changing the time t0 , we can estimate diffusivity locally in time.

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In Fig. 4a, the diffusivity σ1 (t, t0 ) of the action of the reaction coordinate is shown for an ensemble of the trajectories whose residence times tr are in the range of the power law decay, that is, 90 < tr < 100. There, these trajectories exhibit subdiffusion of (t − t0 )p with p = 0.53. On the other hand, in Fig. 4b, the diffusivity σ1 (t, t0 ) is shown for an ensemble of trajectories whose residence times tr are in the range of the exponential decay, that is, 1500 < tr < 2000. Here, we choose t0 = 640, thereby estimating local diffusivity in the middle of wandering

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Figure 5. Dark gray points show the average locations in the action space (a) for the trajectories exhibiting the power law decay and (b) for the trajectories exhibiting the exponential decay. The light gray curves denote typical trajectories in the action space exhibiting (a) the power law decay and (b) the exponential decay. (Reproduced with permission Ref. 55. Copyright 2007 by the American Physical Society.)

within the web. Contrary to Fig. 4a, we can see that the trajectories exhibit nearly normal diffusion, that is, (t − t0 )p with p = 0.93. These differences correspond to whether the trajectories experience fully developed chaos or not. In Fig. 5a, the average locations in the action space (J2 , J3 , J1 ) are plotted for the trajectories exhibiting the power law decay. The location of the trajectory is recorded each time when it crosses a surface of section (q1 = 1 with p1 > 0). After that, all the recorded locations are averaged over its residence time. There, a typical example of the trajectories is also shown. In Fig. 5b, the average locations in the action space (J2 , J3 , J1 ) are plotted for the trajectories exhibiting the exponential decay with a typical example of the trajectories. In Fig. 5, we also indicate the locations of the primary resonances for comparison. There, those orbits showing fractional behavior do not travel around the resonance junctions and those exhibiting the exponential behavior wander around the junctions. Thus, nonuniformity of the Arnold web is crucial for the fractional behavior.

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Wavelet Analysis of Arnold Web

In this section, we use wavelets to investigate the Arnold web. Wavelet analysis has been utilized to investigate the nonlinear resonances in the well. The method enables us to extract resonant features including higher order resonances. Thus, it provides a convenient method to investigate the Arnold web based on time series of trajectories [29, 57–59]. The wavelet is regarded as a windowed Fourier transformation in which the widths of the windows are adjusted according to the frequencies. For a given time series f (t), the wavelet transformation fˆ (ω, t) is defined as fˆ (ω, t) =

√ ∞ ω √ ds f (s) exp −iω(s − t) − ω2 (s − t)2 /δ2 2δ π −∞

(3)

where t is the time and ω plays the role of the frequency in the time region around t. This is a continuous wavelet transformation and is called the Morlet wavelet. The parameter δ determines the width of the windows. If the value of δ is too small, we have difficulty in precisely assigning the frequencies. If it is too large, the information concerning the transient features will be lost. We have found that the value δ = 10 is suitable for extracting the resonance structures. Here, we estimate the spectrogram S(ω, t) = |fˆ (ω, t)|2 , which gives an energy density around the frequency ω at time t. In order to study resonances, we extract the ridges of the spectrogram, that is, local maxima of S(ω, t) as a function of ω at a given value of t [58]. The ridges are regarded as those frequency regions where the energy of the time series concentrates. For given multiple time series fi (t) (i = 1, 2, . . .), we compare the values of the ridges ωi (t) (i = 1, 2, . . .) to / j) are rational numbers. If the see if the ratios of these values ωi (t)/ωj (t) (i = ratios are close to rational numbers with relatively simple denominators, the time series fi (t) and fj (t) are considered to be resonant over the time interval around t. In applying the above procedures to our system, we use the time series of the coordinates qi (t) (i = 1, 2, 3). We estimate the values of ridges from the highest to the nth highest ones. For a time series qi (t), we will denote its kth highest ridge (k) (k) (l) / j) to see by ωi (t) with 1 ≤ k ≤ n. We calculate their ratios ωi (t)/ωj (t) (i = if they satisfy resonance conditions. In order to visualize these resonances, we plot the ratios of the frequencies in (k) (l) (m) (l) the plane where the abscissa is ω2 (t)/ω1 (t) and the ordinate is ω3 (t)/ω1 (t). This plot provides us with another representation of the Arnold web. In the following, we use the highest and the second highest ridges, that is, k = 1, 2, l = 1, 2, and m = 1, 2. Among various combinations of the ra(k) (l) (m) (l) tios (ω2 (t)/ω1 (t), ω3 (t)/ω1 (t)), we expect that the larger the intensities of the ridges, the more they affect the statistical properties. Thus, we order these combinations according to the geometric mean of the intensities of the ridges (l) (k) (m) (k) s1 (t)s2 (t)s3 (t), where si (t) indicates the intensity of the kth ridge for the ith

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degree of freedom. We break them down into two classes, that is, those with larger mean intensities and those with smaller ones, and show those combinations with (l) larger mean intensities. In the following, we simply write ω1 instead of ω1 (t) and so on. In Fig. 6a, we plot the ratios of the frequencies for a trajectory of the power law decay. Other trajectories of the power law decay exhibit similar features. In Fig. 6b, we draw some of the resonance lines identified in Fig. 6a. For example, we can recognize the resonances ω2 /ω1 = a with a approximately equal to 1/3, 1/2, 3/4, 1, 3/2, 2, and 3, the resonances ω3 /ω1 = b with b approximately equal to 3/4 and 3/2, and the resonances ω3 /ω2 = c with c approximately equal to 1/4, 1/2, 3/5, 1, 2, and 3. Moreover, these plots tend to distribute around the intersections of these lines and do not spread along the resonance lines. This reveals the effects of the resonance junctions involving higher order resonances, and these resonance junctions do not overlap with each other. We display in Fig. 6c the trajectory in the action space. It wanders away from the junctions of the primary resonances as in Fig. 5a. In these regions, there exist rich resonance structures created by the junctions of high-order resonances as is shown in Fig. 6b. Thus, we suggest that the resonance junctions involving highorder ones play an important role for understanding the statistical features of the trajectories of the power law decay. D.

Dynamical Connection Revealed by Wavelet Analysis

Here, we consider the dynamical connection between the NHIM and the Arnold web. As for the connection, the following question arises: How does the unstable manifold of the NHIM around the saddle land on the Arnold web? In other words, the problem is whether the landing processes from the saddle to the Arnold web differ between the two ensembles of trajectories: one with the fractional and the other with normal diffusion. In general for nonuniform Arnold webs, three possibilities exist for the landing processes. The first is that the unstable manifold lands on the Arnold web where resonance overlap is sparse. Then, the trajectories in the range of the exponential decay wander around the web and migrate into resonance junctions. The second is that the unstable manifold lands on the Arnold web in the regions near the resonance junctions. Then, the trajectories of the power law wander around the web and migrate into the regions where resonance overlap is sparse. The last is that the trajectories of the power law land on the sparse regions and those of the exponential decay land on the dense regions. This is possible when the unstable manifold spreads enough to cover both the regions. We note that transient features of the statistical properties reveal which cases are realized. In particular, wavelet analysis is convenient to show these transient features by indicating how resonances vary as the trajectory wanders in the Arnold web. At present, only few studies use wavelet to reveal such nonstationary features

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Figure 6. (a) Resonance structures of a trajectory of the power law decay with the residence time tr = 80 [cycle]. (b) Resonance lines for the plots (a) are drawn. (c) Corresponding trajectory in the action space. (Reproduced with permission from Ref. 55. Copyright 2007 by the American Physical Society.)

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Figure 7. Transient resonance structures of a trajectory of the exponential decay with the residence time tr = 800 in unit of the period t1 = 2πω1 . Plots of the ratios of the frequencies are done for the following time intervals: (a) t = 0 to 80, (b) t = 400 to 480, (c) t = 560 to 640, and (d) t = 720 to 800. (Reproduced with permission from Ref. 55. Copyright 2007 by the American Physical Society.)

in the Arnold web [59]. We also show that these transient features give a clue to understand the dynamical connection between the NHIM around the saddle and the Arnold web. In Fig. 7, temporal features of the ratios of the frequencies (ω2 /ω1 , ω3 /ω1 ) are plotted for a trajectory of the exponential decay with the residence time tr = 800. Here, we choose the following four time intervals, (a) t = 0 to 80, (b) t = 400 to 480, (c) t = 560 to 640, and (d) t = 720 to 800; (a) is just after arriving in the web, (b) and (c) are in the middle of wandering the web, and (d) is just before departing from the web. In Fig. 7a and d, rich structures of the junctions of higher order resonances exist as in Fig. 6a. By plotting the trajectory in these time intervals, we can confirm that they wander in regions remote from the junction of the primary resonances. Thus,

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the structures have similar characteristics to that of trajectory of the power law decay shown in Fig. 6a. On the other hand, in Fig. 7b and c, the resonance line ω2 /ω1 = a with a equal to 1 and the resonance line ω3 /ω1 = b with b equal to 1 are especially noticeable. Moreover, some plots spread along the resonance lines ω2 /ω1 = 1 and ω3 /ω1 = 1, that is, the primary resonances involving the reaction coordinate. In Fig. 7b, the plots spread along the line ω2 /ω1 = 1 for 3/2 ≤ ω3 /ω1 ≤ 5/2 and along ω3 /ω1 = 1 near ω2 /ω1 = 1/2, ω2 /ω1 = 1, ω2 /ω1 = 2, and ω2 /ω1 = 3. In Fig. 7c, the plots spread along the line ω2 /ω1 = 1 near ω3 /ω1 = 1/2 and ω3 /ω1 = 1, and along ω3 /ω1 = 1 for 3/2 ≤ ω2 /ω1 ≤ 5/2. These results indicate that the junctions involving higher order resonances overlap with each other along the primary resonances. Based on this, we suggest that the overlapping of the junctions including higher order resonances corresponds to the normal diffusion and the Lorentzian spectra. We also recognize that the transient features of the resonances are similar between Fig. 7b and c. To estimate the similarity of the resonance structures quantitatively, we calculate the distance of the distributions

√ √ of resonances using the Hellinger distance dH [60]: dH (f, g) = ( ( f (x) − g(x))2 dx)1/2 , where f (x) and g(x) are normalized distributions of the N-dimensional quantity x. The results are shown in Fig. 8. We can confirm that the distance between the distributions of the resonances for (b) and (c) is the smallest. This indicates that the trajectory experiences similar resonance structures while it wanders around the junctions of the primary resonances. This is because resonance overlap destroys fine structures of higher order resonances and replaces them by locally ergodic dynamics. On the other hand, the distance between (a) and (d) takes a larger value. This means that a variety of resonance structures exist in the entrance (a) and exit (d) regions since various combinations of higher order resonances are possible. Thus, the pattern recognition and the estimate of the distances between the transient resonance structures offer new approaches to differentiate the statistical properties of trajectories from the dynamical viewpoint. 1.18

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Here, we have shown the temporal variance of the resonant behavior using the wavelet analysis. The wavelet analysis has enabled us to extract resonant behavior including higher order resonances. We suggest that the characteristics of the resonance junctions including higher order resonances play an important role for understanding the statistical features of the trajectories. In particular, nonoverlapping of the junctions of higher order resonances causes the fractional behavior. On the other hand, overlapping of these junctions results in the normal statistical behavior. Thus, the difference of the distributions of the resonance junctions explains the coexistence of the power law and exponential decays. These results strongly suggest that the trajectories of the exponential decay also land on the regions where the web is sparse. After spending some time there, they migrate into the regions where the resonances are dense. When they leave, they migrate to the regions where the web is sparse, and exit the well. Thus, the temporal features of wavelet analysis reveal the dynamical connection. E.

Nonexistence of Reaction Rate Constants

For Hamiltonian systems of two degrees of freedom, fractional behavior is well established [22]. It is shown that a hierarchy of resonant tori exists trapping nearby trajectories for a finite time [61]. On the basis of this property, they are described as “sticky.” Then, the hierarchy of timescales leads to a power law distribution of residential times. However, when the number of degrees of freedom is larger than 2, it is not obvious that the dimensions of the tori are not large enough to work as dynamical barriers. Therefore, fractional behavior in systems of more than two degrees of freedom is a new phenomenon in the study of Hamiltonian chaos. The existence of the fractional behavior indicates that the reaction rate constant cannot be well defined in this system. Then, we need to discuss whether our results are relevant for quantum dynamics. It is known in general that “quantum chaos” is less statistical than classical chaos although the discrepancy is shown to be smaller as the number of the degrees of freedom is larger [62]. Therefore, we expect that fractional behavior of classical chaos manifests itself in the corresponding quantum system. In fact, a recent study shows, for a system of two degrees of freedom, that fractional behavior of classical chaos has corresponding effects in the quantum one [63]. Thus, we think that the fractional behavior is relevant for reaction processes, which are quantum in nature. III.

ANOMALY IN DIFFUSION IN SPATIOTEMPORAL MULTISCALE CLASSICAL SYSTEMS

Biological macromolecules can be regarded as dynamical network systems composed of a large number of diverse oscillators that are nonlinearly coupled to each other. This results in the existence of hierarchical scales in time and space

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inherent to the system. For example, the timescale typically spans from 10−15 to 103 s, corresponding to the vibrational motion of bonds at the short-timescale end, to, for example, protein folding and aggregation. The protein dynamics along a chosen “reaction coordinate” has often been represented in terms of (over- and underdamped) Langevin dynamics. This approach reduces all the other “nonreactive” degrees of freedom into the “bath” characterized by the potential of mean force, friction, and random force. Several theoretical [64–67] and experimental [68] studies have revealed the existence of anomaly in the diffusive process of a chosen physical observable for a wide range of timescales. For example, Garc´ia and Hummer [65] observed the anomalous diffusion along a set of large-amplitude collective modes extracted by principal component analysis (PCA) in cytochrome c. They found that the mean square displacements (MSDs) (x(t) − x(0))2 (· is the ensemble average) of the chosen large-amplitude components x(t) are not simply proportional to t but, depending on the regions where the system may reside, exhibit subdiffusion where the MSDs are proportional to t α with α < 1 and superdiffusion with α > 1 below the temperatures where basin hopping of the system occurs once in nanoseconds. In the study of the autocorrelation function of the fluorescence lifetime fluctuation of flavin oxidoreductase, Yang et al. [68] found that conformational fluctuation of the biomolecule exhibits subdiffusion with characteristic timescales spanning several decades, from milliseconds to seconds. In general, when the autocorrelation function of velocities decays exponentially for large values of t, the system exhibits simple Brownian motions at such timescales. On the other hand, when the autocorrelation function has a long algebraic tail, the system shows anomalous diffusion [69]. Thus, anomalies observed in these studies suggest the existence of nontrivial memory effects from the “nonreactive” degrees of freedom in protein. How can one characterize such complexity observed in biological systems whose dynamics and kinetics span a wide range of time and space scales? In dynamical systems theory, several measures, such as (maximum) Lyapunov exponent (LE) and Kolmogorov–Sinai (KS) entropy, have been well established in quantifying the dynamical instability of trajectories and the randomness of dynamical systems. However, since these measures are defined with respect to the asymptotic evolution of an infinitesimal uncertainty, they do not necessarily capture the complexity of multiscale dynamics in systems. This was first illustrated by Amitrano and Berry in their finite-time Lyapunov exponent analyses of argon clusters [70–72] and the H´enon–Heiles model [71]. It was also shown in a certain class of dynamical systems with different spatiotemporal hierarchies that the mean field dynamics exhibits a low-dimensional regularized motion at the macroscopic level [73]. Boffetta et al. [74] also showed for a number of systems having several different timescales that although the LE is certainly related to the small-scale dynamics, it can be of little relevance to the characterization of the predictability in large scale.

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Our primary interests are the conformational transitions or functionally important motions [75] of biological molecules in different spatiotemporal scales. These motions are usually associated with relatively large amplitudes and low frequencies, occurring under thermal fluctuations. Such “coarse-grained” dynamics cannot be well characterized in terms of the dynamical measures like LE and KS entropy. Recently, several important developments have been devoted to generalize these measures and to bridge between the microscopic and macroscopic scales of complex systems. One is called the finite-size Lyapunov exponent (FSLE) [73, 76–78] (which is a finite-space version of the finite-time Lyapunov exponent [70–72]) and the other is the ( , τ) entropy [79–82]. The FSLE measures the average divergence rate of an ensemble of trajectories nearby a reference trajectory at a certain finite length scale . The ( , τ) entropy measures the average amount of information required to describe the trajectories of a dynamical process as a function of the lengthscale and timescale of observation, and τ. Both measures are expected to converge to the corresponding microscopic counterparts, that is, LE and KS entropy, if they exist, when → 0 and → 0, τ → ∞. Although these procedures are, in principle, applicable to any dynamical system having hierarchical spatiotemporal structures, it is yet nontrivial to identify observable(s) for real multiscale biological systems along which the transitions from strong chaos to low-dimensional dynamics might take place. In this section, we overview our recent development of a technique that combines FSLE and PCA to identify characteristic spatiotemporal scales and to quantify the collectivity buried in the coarse-grained dynamics inherent to multiscale protein systems [83]. We apply the technique to a toy model based on two coupled maps of slow and fast variables, and a model protein of 46 amino beads with three different types of residues [84–88] on an ideal funnel energy landscape. First, through the application of FSLE to the two coupled map system, we demonstrate how the FSLE depends on the definition of distance to measure, and how it can work to detect the multiscale behavior of the model system. Then, for the folding and unfolding transitions of the model protein, we study how motions in different space scales change depending on the choice of degrees of freedom and the state in which the system resides. A.

Finite-Size Lyapunov Exponent

The FSLE was originally introduced in the predictability problem of fully developed turbulence [77] and in systems with several diverse units coupling nonuniformly or uniformly together [73]. The basic idea of the FSLE is to define an average growth rate for different sizes of distance between a reference trajectory and a set of perturbed trajectories in finite length scale (shown in Fig. 9a). As the conventional maximum LE gives information on the average predictability time in the infinitesimal regime, the FSLE λ( ) also gives information on the typical

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Perturbed trajectory

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Scaling

Δ0 Reference trajectory Figure 9. Schematic pictures of the calculation procedure of the FSLE (a) and error-doubling experiments (b).

predictability time for a trajectory with finite uncertainty . For small , the difference between the two trajectories follows a linear dynamics. When can no longer be considered as infinitesimal, all scales with a typical size smaller than experience “anharmonic” or “diffusive” separations and do not contribute to the exponential divergence. In this regime, the behavior of λ( ) is governed by the nonlinear evolution of the perturbed trajectories. In general, λ( ) is smaller than or equal to the maximum LE, and the decrease in λ( ) does follow a systemdependent law. In fact, the behavior of λ as a function of contains important information on the characteristic timescales governing the system [74]. To calculate FSLE, one first defines a distance (or norm) (t) = | x(t)| between the reference and the perturbed trajectories. √ A set of distance thresholds, { n = r n 0 } (where n = 0, 1, . . . , N and r = 2 is chosen here), is then introduced in order to measure the “doubling time” T ( n ) at different thresholds n . The doubling time T ( n ) is the first passage time when a perturbed trajectory passes from the threshold n to the next threshold n+1 . A perturbed trajectory is generated within a certain small deviation min from the reference

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trajectory. min has to be small enough ( min 0 ) to ensure that the deviation at 0 is aligned with the maximally expanding direction. The evolution of the error from the initial value 0 to the largest threshold N gives a set of doubling times {Ti ( n )} at the ith error-doubling experiment. When the largest threshold N has been reached, the perturbed trajectory is rescaled back to the initial distance min with respect to the reference trajectory and one repeats another error-doubling experiment to obtain another set of doubling times {Ti+1 ( n )} (see Fig. 9b). For the ith error-doubling experiment, an effective exponential divergence rate at n is λi ( n ) = ln r/Ti ( n ). Then, after performing N ( 1) error-doubling experiments, the FSLE is defined [77] as λ( n ) =

N

i=1 λi ( n )Ti ( n ) N i=1 Ti ( n )

N ln r = N i=1 Ti ( n )

(4)

It should be noted that the behavior of FSLE crucially depends on the choice of distance and the observable in the noninfinitesimal regime, whereas the maximum LE is independent of the particular definition to measure the distance (except for infinite-dimensional systems [89, 90]). In order to show how the FSLE depends on the distance or the observable, let us consider the case of a slow system S, described by the variable xs , coupled with a fast system F , described by the variable xf . Here we use the following two coupled maps from Ref. 91: xs (i + 1) = (1 − )fs [xs (i)] + g[xs (i), xf (i)] mod 1 xf (i + 1) = (1 − )ff [xf (i)] + g[xf (i), xs (i)] mod 1

(5)

where fs and ff are maps of the unit interval [0, 1] onto itself, fs [xs ] = eλs xs mod 1 ff [xf ] = eλf xf mod 1 gs [xs , xf ] = cos(2π(xs + xf ))

(6)

Slow dynamics of xs and fast one of xf are defined by making λs < λf . A set of perturbed trajectories, {(xs (i), xf (i))}, nearby the reference trajectory (xs (i), xf (i)) are generated from initial condition with sufficiently small deviations from that of the reference (xs (0), xf (0)): xs (0) = xs (0) + xs (0), xf (0) = xf (0) + xf (0)

(7)

where xs (0) and xf (0) were chosen from uniform random numbers in [− min , min ] ( min = 10−10 1).

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λ (Δ)

0.5 0.4

Δ s+f Δs Δf

0.3 0.2 0.1 0 –5 10

10

−4

−3

10 Δ

−2

10

−1

10

Figure 10. The FSLE λ( ) as a function of for the coupled maps of Eq. (5) with the coupling strength = 2 × 10−3 , min = 10−10 , and 0 = 10−5 . The open circle, multisymbol, and plus denote the λ( ) based on s+f (i) = {[xs (i) − xs (i)]2 + [xf (i) − xf (i)]2 }1/2 , s (i) = |xs (i) − xs (i)|, and f (i) = |xf (i) − xf (i)|, respectively. The dotted lines are λf = 0.5 and λs = 0.1.

Figure 10 shows the value of λ( ) versus for this system. Here we introduce three different distances, that is, s+f (i) = {[xs (i) − xs (i)]2 + [xf (i) − xf (i)]2 }1/2 , s (i) = |xs (i) − xs (i)|, and f (i) = |xf (i) − xf (i)|. In the numerical calculation, 1000 error-doubling experiments are performed (i.e., 1000 perturbed trajectories are used) along a single reference trajectory. For small , the dynamical instability of the system is driven by the fast mode, and λ( ) tends toward the (maximum) LE ∼ λf = 0.5. For large values of , while the growth of s+f (i) and f (i) is still governed by fast dynamics, s (i) is governed mainly by the slow dynamics and λ( ) approaches λs = 0.1. This fact is due to the saturation of the error on the fast components of the system, which therefore do not contribute to the exponential growth of the uncertainty at large error levels. In fact, the transition between the two regimes takes place at ∼ = 2 × 10−3 , the strength of the coupling. This demonstrates that the (maximum) LE (= λf ) is not sufficient to capture the predictability of the dynamics of slow mode xs and actual predictability time (with a finite uncertainty > ) is longer than the one implied by the (maximum) LE. s+f and f cannot detect such a regularity because they are mainly “contaminated” by the fast dynamics. Thus, it is quite crucial to “see” the hierarchical dynamics along the slow mode(s), if it exist(s), in order to reveal the essence of the complexity in multiscale dynamics emerging in the noninfinitesimal regime. The question that arises is this: From complex systems like proteins how one can generally extract such slow modes, if they exist, along which regularity emerges at

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some coarse-grained scales? This is the important subject in the next section, and we here adopt the PCA to extract the slow modes of the 46-bead protein model. B.

Principal Component Analysis

The technical idea behind the PCA is to find the orthogonal eigenvectors that best capture the fluctuation in the high-dimensional data set or that best represent it in a least-squares sense [92–94]. The set of PCs is the solution to the eigenvalue problem of the second-moment matrix σ of the bead displacements in the Cartesian coordinate. The diagonalization of σ yields the eigenvectors (or the PCs) {wk }, which represent the directions or the modes of collective motions. The corresponding eigenvalues ak2 (usually they are sorted such that a12 > a22 > · · · ≥ 0) provide the variance of fluctuation along each PC. The motion q(t) along the collective modes is obtained by projecting the data set of original Cartesian coordinates onto the subspace composed of the PCs (e.g., q1 (t) = w1T x(t), where x(t) = {ri (t)} ∈ R3N ; N is the number of beads). PCA has been widely used in studies of high-dimensional motions of proteins, and it has been suggested that PCs well reduce the conformational transitions of proteins [94, 95]. For the details of PCA, see another chapter of this volume [96]. C.

Complexity of Hierarchical Dynamics in Multiscale Nonlinear Systems: A Case Study of Model Protein

We illustrate our method by considering the three-color, 46-bead model protein [84] whose potential and free energy landscapes [85–87], kinetics, and dynamics [88] have been well studied. The model, called the BLN model, which was originally proposed as a lattice model [97] and later on as an off-lattice model [84], is composed of hydrophobic (B), hydrophilic (L), and neutral (N) beads, and the global potential energy minimum for the sequence, B9 N3 (LB)4 N3 B9 N3 (LB)5 L, folds into a ␤-barrel structure with four strands. The potential energy function is described by V = (Kr /2)

bonds

(Ri,i+1 − R0 ) + (Kθ /2) 2

i

+

dihedral

+ 4

angles

(θi − θ0 )2

i

[Ai (1 + cos φi ) + Bi (1 + cos 3φi )]

i nonbonds

Ci

σ/Rij

12

6 − Di σ/Rij

(8)

i<j−3

where Ci = Di = 1 for BB (attractive) interactions, Ci = 2/3 and Di = −1 for LL and LB (repulsive) interactions, and Ci = 1 and Di = 0 for all the other pairs

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involving N, expressing only excluded volume interactions. Kr = 231.2 σ −2 and Kθ = 20 rad−2 , with the equilibrium bond length R0 = σ and the equilibrium bond √ angle θ0 = 1.8326 rad. The units of temperature and time are kB−1 and t = σ M −1 , respectively. For small single-domain proteins, an ideal funneltype energy landscape [98] has been postulated as one of the most fundamental properties, which manifests a two-state transition. In this subsection, in order to shed light on the nature of coarse-grained dynamics of folding inherent to small proteins, we impose the G¯o-type bias [99] toward the global minimum structure of the BLN model and apply our analyses to this biased model [86, 87] (called the G¯o model). For the constant-temperature molecular dynamics simulation, we used Berendsen’s algorithm [100] with a time step 0.0025t in which the system is coupled to an external heat bath with a coupling time of 0.50t. The G¯o model exhibits a sharp two-state-like transition (between folded and unfolded states) at around T ∼ 0.50 kB−1 . Trajectory calculations are conducted over a wide range of temperatures (T = 0.20, 0.40, 0.60, and 2.0 kB−1 ). The FSLE along each PC was evaluated as follows: First, a long reference trajectory was calculated at each temperature. Second, PCA was performed for each obtained reference trajectory and the PCs, {wk }, were determined. Then, perturbed trajectories were generated by introducing perturbations to the reference trajectories and they were evolved to operate the error-doubling experiments. Doubling times were evaluated by projecting both trajectories onto each PC and using the distance defined in the PC subspace. Reference trajectories were recorded after a 250t length equilibration process. Initial perturbation was introduced by adding uniform random numbers in the range of −10−10 σ, 10−10 σ to each position coordinate of the reference trajectory excluding the external motions (translations and rotations) of the entire molecule. The results of 500 error-doubling experiments were used to evaluate each FSLE. FSLEs of the G¯o model along PC1, 7, and 50 are shown at different temperatures in Fig. 11. Regardless of the temperature, λ( ) corresponding to the lower indexed PCs takes lower values than those of the higher indexed PCs as the distance becomes large. This means that the average prediction times for the reference trajectories of the lower indexed PCs are longer than those of the higher indexed ones at coarse-grained scale. To look into more details of the coarse-grained dynamics occurring in PC1 subspace, it is useful to compare the dynamics with ideal stochastic processes. For stationary or nonstationary Gaussian processes, rigorous results are obtained for the so-called entropy by the Kolmogorov formula [79, 101] that is closely related to the FSLE in low-dimensional systems, and we can expect that the same results hold for the FSLE in PC1 subspace [74]. As an example of a prototypical model for self-affine stochastic processes, let us consider the fractional Brownian motion (fBm) [102]. The fBm with H¨older exponent α is defined to be a Gaussian process {XfBm (t)} [103] such that for every t ≥ 0 and h > 0, the increment X(t + h) − X(t)

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Figure 11. FSLEs of the G¯o model in PC1, 7, and 50 subspaces as a function of at different temperatures. The norm is the normalized absolute difference value in R composed of the kth PC wk , that is, qk (t) = |qk (t) − qk (t)|, where qk (t) = wkT x(t) and qk (t) = wkT x (t). x(t) ∈ R3N are the Cartesian coordinates of the reference trajectory at time t and x (t) are that of the perturbed one. The normalization factor is the square of the corresponding eigenvalue ak2 . The solid lines are q1 , broken lines are q7 , and dotted lines are q50 . The markers indicate temperature (circle: T = 0.2; rectangle: T = 0.4; triangle: T = 0.6; cross: T = 2.0). (Reproduced with permission from Ref. 83. Copyright 2007 by the American Physical Society.)

has the normal distribution with mean zero and variance h2α , so that the prob x 1 ability is P(X(t + h) − X(t) ≤ x) = (2π)− 2 h−α −∞ exp(−u2 /2h2α ) du. From 2 the definition, one can deduce X (t) = t 2α and X(t)X(t + h) = (1/2)[t 2α + (t + h)2α − h2α ]. Thus, (X(t) − X(0))(X(t + h) − X(t)) is positive or negative according to whether α > 1/2 or α < 1/2. That is, if α > 1/2, then X(t) − X(0) and X(t + h) − X(t) tend to have the same sign (the motion is persistent or superdiffusive). On the other hand, if α < 1/2, then X(t) − X(0) and X(t + h) − X(t) tend to be of opposite sign (the motion is nonpersistent or subdiffusive). α = 1/2 means a simple Brownian motion. The fBm can be regarded as identical with the small-scale behavior of the Yaglom noise [79], which is the stationary analogue of the fBm, and we can expect that the following relation holds for the FSLE [74, 79]: λ( ) ∝ −1/α

(9)

Figure 12 shows the FSLEs in the PC1 subspace at different temperatures in double logarithmic scale. The definition of the norm is the same as that of Fig. 11. In the figure, the asymptotic behavior of a simple Brownian motion (α = 1/2) from Eq. 9 is indicated by the solid line. At lower temperatures than the folding transition temperature (T ∼ 0.50 kB−1 ), the FSLEs are well fitted to the solid line suggesting that the dynamics in the PC1 subspace should be approximated by

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10

−0.8

Δ

−1

λ (Δ)

10

T = 0.2Δq1 T = 0.4Δq1 T = 0.6Δq1 T = 2.0Δq1

−2

10

−2

Δ

−2.5

Δ

−3

10

−2

−1

10

10 Δq / a 1

1

Figure 12. Double logarithmic plot of the FSLEs of the G¯o model in the PC1 subspace as a function of at different temperatures. The definition of the distance is the same as in Fig. 11. The inclined solid line is the asymptotic behavior of a simple Brownian motion (λ ∝ −2 ). (Reproduced with permission from Ref. 83. Copyright 2007 by the American Physical Society.)

a simple Brownian motion. However, at higher temperatures, amplitudes of the slopes of FSLEs are remarkably smaller than that of a simple Brownian motion. This means that at higher temperatures the motions in the PC1 subspace are persistent or superdiffusive and they have strong positive correlations even at large space scale ( ∼ 0.35a1 ). In order to obtain more insights into the anomalous behavior observed in Fig. 12, we defined the local FSLE, regarded as the extension of local LE [70–72] to finite scales. First, aset of observed times, {Ti ( )}, is decom error-doubling posed into two sets, TiF ( ) and TiU ( ) , depending on the state (folded or unfolded) where the reference trajectory resides when each error-doubling meaNF F surement starts. Then, the local FSLE is defined as λF ( ) = NF Ti ( ) ln r i=1 NU U and λU ( ) = NU Ti ( ) ln r for the folded and unfolded states, respectively. i=1

NF is the total number of error-doubling experiments in the folded state and NU is that of the unfolded state. Since the two states are disjoint and enough to cover the whole configuration space of the system, N = NF + NU and N i=1 Ti = NF F NU U FSLE can be recovered by weighing the i=1 Ti + i=1 Ti hold. The original NU U NF F Ti Ti i=1 λ λU ( ). ( ) + time length spent in each state, λ( ) = i=1 F N N T i=1 i

T i=1 i

non-brownian phase space dynamics of molecules 10

0

−1

λ (Δ)

10

109

10

−2

T = 0.4, Δq1, unfolded T = 0.4, Δq1, folded 10

−2

Δ

−3 –2

10

10

−1

Δq1 / a1 Figure 13. Local FSLEs of the G¯o model in the PC1 subspaces at T = 0.40 kB−1 . The inclined solid line is the asymptotic behavior of a simple Brownian motion (λ ∝ −2 ). (Reproduced with permission from Ref. 83. Copyright 2007 by the American Physical Society.)

The local FSLE tells us how the average divergence rate can change depending on the local morphologies on the configuration space of the system. As a quantity to determine in which state the system resides, the fraction of native contacts, called the Q value, was used. λF ( ) and λU ( ) at T = 0.40 kB−1 are shown in Fig. 13. Whereas λF ( ) approximately follows a Brownian motion, the smaller slope of λU ( ) is ascribed to persistent superdiffusion. This result shows that the dynamics significantly depends on the state where the system resides in, and the system exhibits superdiffusion during the “walking” in the unfolded state. To check the dependency on the coupling strength with heat bath, the same analyses were performed for a heat bath more strongly coupled to the system with coupling times (0.05 and 0.005t). The results are, however, qualitatively similar to the present results. This may indicate that the superdiffusion observed in the PC1 space is less influenced by the fast thermal fluctuation driven by the heat bath, and therefore it should be induced mainly by the interactions with other internal degrees of freedom of the system. What are the implications of the occurrence of superdiffusion in the unfolded state? In the theory of foraging biology [104], it has been shown that superdiffusion outperforms Brownian diffusion as a statistical strategy for finding randomly located objects. In this regard, the superdiffusion could contribute to the efficiency of protein folding where the system searches the native state along the low-indexed PCs on

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the multidimensional configuration space of the system. This conjecture should be carefully investigated through a number of similar analyses for different systems and different thermostats. In particular, an actual protein landscape can be more rugged, as suggested by the all-atom MD simulations of Trp-cage [105] and chignolin [106]. As a consequence of having other deep basins, trajectories going to the global minimum are trapped in those misfolded basins. Under the simplified funneltype energy landscape, the current study just suggests that superdiffusive search would help the first stage of folding process in which the trajectories search deep basins (not necessarily the global minimum). Escaping process from the misfolded states often shows slow diffusion (subdiffusion) and it will bring a different scenario in the current conjecture. The characterization of the escaping process is possible using the same kind of analysis as in the current study, for example, by evaluating the local FSLE in each misfolded state. This subject should be pursued in the future. For modeling the effective description of the observed superdiffusion at the coarse-grained scale, it may be reasonable to adopt the generalized Langevin approach representing the potential of mean force, the memory effect as an additive force, and a frictional kernel. This leads to the well-known fractional Fokker– Planck equation (FFPE) [107]. It would be further noted that if the superdiffusion observed in the current study exhibits strong anomalous diffusion in Ref. 69 as a result of a violation of the hypothesis of the central limit theorem, the probability distribution function does not obey the usual linear equations involving fractional FPE. To assess the validity hypothesis of the central limit theorem, more investigations are needed.

IV.

ENERGY FLOW AND LOCALIZATION IN QUANTUM SYSTEMS WITH MIXED STATE SPACE AND REACTION KINETICS

In this section, we review theoretical work describing the nature of quantum energy flow in molecules at energies where transport is nonergodic. While the manifestations of fractional dynamics in these many-dimensional quantum coupled oscillator systems are only beginning to be explored, the models themselves have classical counterparts that include those described in the previous section, which exhibit fractional dynamics. The theoretical model, local random matrix theory (LRMT), predicts that the vibrational eigenstates, while localized in the nonergodic regime and thus restricting free energy flow over the energy shell, exhibit a wide range of localization lengths at a given energy. Energy flow is restricted on the energy shell, but in different regions may flow among a relatively large or small number of states, the distribution of which is predicted by LRMT. If the molecule is initially excited on the energy shell to a state from which it can reach the transition region, then the molecule can react and the rate constant is well defined. We use LRMT to predict the rate constant in this case. For many states on the energy shell, however, the transition region cannot be accessed and molecules excited to these regions do

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not react. Interestingly, those molecules that do react can do so at a rate that exceeds predictions of RRKM theory. Indeed, rates substantially larger than RRKM theory predictions have been observed for reactions involving organic molecules and clusters, and can be attributed to the influence of nonergodicity during reaction. At the end of this section, we discuss application of LRMT to the calculation of rate constants for unimolecular reactions and provide some examples. We turn first to a brief summary of LRMT, a topic that has been reviewed in the past [108]. Here we focus on those parts of the theory that provide information about vibrational eigenstates at energies where quantum energy flow is restricted and nonergodic, the parts of the theory that we apply to predict dynamical corrections to RRKM theory at these energies. A.

Local Random Matrix Theory

LRMT aims to describe the transition to ergodicity in a quantum mechanical system of many coupled oscillators, which could model, for example, the vibrations of a sizable molecule. At energies corresponding to barriers to conformational change of a few kcal/mol, only a small number of vibrational modes of the molecule are excited, and anharmonic coupling among vibrational modes is generally small. To explore the nature of the vibrational eigenstates and energy flow at these energies, we begin with a quantum coupled oscillator Hamiltonian H = H0 + V

(10)

where H0 =

N

α (nα )

(11)

α=1

V =

m

+

−

m (b† )mα bmα

(12)

α

± † where m = {m± 1 , m2 , ...} and b and b are raising and lowering operators, respectively. If the Hamiltonian is expressed in a basis of normal modes, then m is an anharmonic coefficient. The zero-order Hamiltonian H0 consists of a sum over the energies of the nonlinear oscillators, where each oscillator has frequency, ωα , which depends on the occupation number of that oscillator, nα . The set of zero-order energies, { α }, and coefficients of V , {m }, are treated as random variables with suitable average and variance. The vibrational Hamiltonian defined by Eq. (10) includes direct resonant coupling terms of arbitrary order. In order for coupling of states in the matrix ensemble to be “local,” we assume that the coefficients m decay on average exponentially. The larger, low-order terms in V couple states are close to one another in the vibrational quantum number space, the topology of which can be thought of as an N-dimensional lattice. Each lattice site is coupled locally to

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nearby sites by matrix elements arising, for example, from low-order anharmonic terms in the potential. The zero-order energy for a site in the vibrational state space is determined by the frequencies and nonlinearities of the vibrational modes, which may in principle be known. However, if one such site is coupled to a fairly large number of sites nearby in quantum number space, we can assume differences in the zero-order energies of all these sites to be randomly distributed within order of a vibrational frequency. Seen this way, quantum vibrational energy flow in molecules resembles single-particle transport on a many-dimensional disordered lattice. Exploiting this connection, Logan and Wolynes found a transition for energy to flow globally on the energy shell that occurs at a critical value of the product of the anharmonic coupling and local density of states [44]. Solving self-consistently for the most probable value of the imaginary part of the self-energy, vibrational energy flow is unrestricted when [44, 45] ⎛ ⎞2 T (E) = (2π/3) ⎝ |VQ |ρQ ⎠ ≥ 1 (13) Q

while energy is localized in the vibrational state space at energy E when T (E) is less than 1. Here, ρQ is the local density of states that lie a distance Q away in quantum number space, and |VQ | is the average coupling matrix element to such states. The self-consistent analysis reveals that the more the size of the molecule, the more sensitive is the location of the transition to higher order resonances [45, 109]. We focus in this chapter on the energies where energy flow is nonergodic, and LRMT provides useful information on how the transition is approached when T (E) is less than 1 and energy is localized to a finite number of states on the energy shell. The extent of localization of molecular vibrations can be determined spectroscopically by the dilution factor, or the inverse participation ratio for state n, δn = |cnα |4 (14) α

where cnα are eigenvector components. Equation (3) gives the inverse of the number of vibrational states that overlap a particular zero-order state, n, and is the survival probability of the initially excited state in the infinite time limit. LRMT provides an analytical form for the distribution of the inverse participation ratio, or dilution factor, near a particular energy, which is [110]

Pδ (δ) = γδ−1/2 (1 − δ)3/2 exp − πγ 2 δ/(1 − δ) (15) γ= which holds if T (E) < 1.

3T (E)/2π(1 − T (E))

(16)

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The distribution, which is broad and can appear bimodal, has been confirmed by numerical calculations by Sibert and Gruebele on thiophosgene as the transition to extended states is approached [111]. It also characterizes the range of dilution factors [112] observed by Stewart and McDonald [113] for about 20 organic molecules with energy close to 3000 cm−1 . We note that above the transition, LRMT gives for the probability distribution of |cnα |2 , the Porter–Thomas distribution, as expected for quantum ergodic systems [114]. While not the focus of this chapter, we note that above the transition, T (E) > 1, energy flows over all states of the energy shell. Schofield and Wolynes [115] have argued that energy flow in the vibrational state space both just above and well beyond the IVR transition can be described by a random walk, a picture that has been supported by numerical calculations over a wide range of timescales [116]. The state-to-state energy transfer rate can be estimated by LRMT. Well above the transition we would expect the rate of quantum energy flow between states of q the vibrational state space to be given by kIVR = (2π/) Q |VQ |2 ρQ (E), where the superscript q denotes quantum and will be used to distinguish from collisional contributions to energy transfer rates below. More generally, including the region near the transition, we find the energy transfer rate to be [45] q kIVR = 1 − T −1 (E)(2π/) |VQ |2 ρQ (E) (17) Q

Equation (4.8) goes over to a golden rule-like expression that reveals the locality of energy flow through a crossover region just above the transition, which in practice we find to be quite narrow, particularly when we account for higher order resonances. While the transition itself is increasingly influenced by higher order terms, the larger the molecule, the less pronounced is the influence of high-order anharmonic coupling on vibrational energy transfer rates, though it can be important in large molecules [45, 117]. B.

LRMT Dynamical Corrections to RRKM Theory of Unimolecular Reaction Rates

The microcanonical unimolecular reaction rate for a reaction at energy E over a barrier E0 is given by RRKM theory as [1] kRRKM (E) =

N + (E − E0 ) hρtotal (E)

(18)

where ρtotal (E) is the total density of states of the reactant at energy E and N + (E − E0 ) is the number of states of the transition state with excess energy less than or equal to E − E0 . Dynamics in the phase space is assumed to be ergodic and corrections, or perhaps completely different viewpoints, are needed in the nonergodic regime.

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As discussed in Section I, Berne [15] developed a classical theory for the microcanonical reaction rate when the phase space consists of both irregular (i.e., chaotic and ergodic) and regular regions. The phase space volume, , then has an irregular and a regular part, I and R , respectively, where = I + R . Only trajectories that lie in the irregular region can give rise to an exponential population decay when they cross a reaction barrier, and therefore to the existence of a finite reaction rate. If energy flow within the irregular region is very rapid, then an RRKM-like approach to calculating the rate can be followed for states within the irregular region. The initial excess population, P (0) (t), of a species decays with time, t, as P (0) (t) =

R (E) I (E) −k(E)t + e (E) (E)

(19)

The rate coefficient is k(E) = kNEST (E) =

(E) kRRKM (E) I (E)

(20)

where NEST refers to “nonergodic statistical theory” [15]. Here we summarize a recent quantum mechanical formulation for kNEST , kQunEST (E), a quantum nonergodic statistical theory [19]. As we have seen, LRMT predicts that in the nonergodic regime, where vibrational eigenstates are localized, there is a considerable range of localization lengths, a distribution that can be calculated as described above. Energy can flow from a given state on the vibrational state space to others within a volume of radius corresponding to the localization length. The volume contains the number of zero-order vibrational states that effectively contribute to a vibrational eigenstate, or the participation number, p. We shall refer to the number of states in the “limited ergodic” region of the vibrational state space as Np (E). LRMT then gives the unimolecular reaction rate as kQunEST (E) = κ(E)

κ(E) =

ν(E)N + (E − E0 ) Np (E)

kIVR (E) kIVR (E) + ν(E)

(21)

(22)

The dynamical correction to RRKM theory is given by κ(E) and arises from insufficiently rapid IVR in the limited ergodic region of the state space. The dynamical correction depends on the IVR rate from transition states to states outside the transition region, kIVR , and the barrier crossing rate, ν(E), to go from reactant to product [3]. Equation (12) is just the RRKM rate, Eq. (9), when κ(E) = 1 and when the number of states of the limited ergodic region, Np (E), corresponds to the total number of states on the energy shell.

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Each of the N + (E − E0 ) states of the vibrational state space that are associated with the activated complex, or the transition states, may be coupled to states outside the transition region of the state space. The number of such states is related to the participation number, p. In the D-dimensional quantum number space of the reactant, one of those dimensions corresponds to the reaction coordinate and the other D − 1 to the relevant bath coordinates that comprise the limited ergodic region overlapping the transition states. We can thus estimate, assuming isotropic energy flow, that the number of states in the limited ergodic region that overlap the transition state dividing surface is Np (E) = N + (E − E0 )p(D−1)/D . Inserting this estimate into Eq. (12), we have kQunEST (E) = κ(E)

ν(E) p(D−1)/D

(23)

Since the vibrational eigenstates below the ergodicity threshold are localized, and we are in the “limited ergodic” region, the participation number is smaller than the number of states on the energy shell, N(E), so that Np (E) = N + (E − E0 )p(D−1)/D < N(E). We may therefore have an enhancement of the unimolecular reaction rate compared to the RRKM theory prediction. In the classical theory of Berne [15], κ(E) is implicitly 1, so that there is always enhancement. More generally, the size of the enhancement given by QunEST depends on the transmission coefficient, κ(E). If the energy flow rate, kIVR (E), between transition states and nontransition states within the limited ergodic region is slower than the barrier crossing frequency from reactant to product, ν(E), then q

kQunEST (E) =

kIVR (E) p(D−1)/D

(24)

Even when energy flow within the limited ergodic region is limiting, there can be significant enhancement of the reaction rate. For the limited ergodic region, where q energy can flow among p states, we can use kIVR given by Eq. (8) for the IVR rate in Eq. (15). The participation number, p, can be calculated with LRMT as discussed above, and is obtained from the inverse participation number, δn , the distribution of which is given by Eq. (6). We can calculate a representative value for the participation number, p, as p = 1/δ. We have found [110] that δ =

1

dδPδ (δ) = eπγ

2 /2

0

where Dp is the parabolic cylinder function.

D−2

2πγ 2

(25)

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Illustrative Example: A Gas-Phase SN 2 Reaction

A number of unimolecular reactions have been found to occur with rates faster than predicted by RRKM theory. These have often been shown to lie in the nonergodic regime. We have recently applied QunEST to two reactions, one a gas-phase SN 2 reaction, F− -CH3 Cl to FCH3 -Cl− [19], and the other the isomerization of butanal [20]. Moreover, recent calculations on the isomerization rate of a peptide–water complex, where the water molecule shuttles between two hydrogen bonding sites, also indicate an enhancement of the rate compared to RRKM theory predictions [118]. In this section, we summarize results of the gas-phase SN 2 calculations. We discuss as an example a 12-vibrational mode molecule (N = 12), which we recently presented as a rough caricature of the F− -CH3 Cl complex that isomerizes to FCH3 -Cl− during the F− + CH3 Cl SN 2 reaction. For simplicity, we have taken nine of the frequencies to be evenly distributed between 0 and 1500 cm−1 , and the other three to be 3000 cm−1 , a range similar to that for F–CH3 Cl [18]. Only the nine vibrations in the lower “band” of vibrational frequencies participate in energy flow at the relatively low energies of isomerization, the barrier to which is about 3.5 kcal/mol [18]. This gives as a representative frequency a value of 620 cm−1 . The anharmonic constants used to represent m can be estimated by the “typical” value of the cubic anharmonic constant, 3 , which we took to be 7 cm−1 , and a “decay rate constant” for higher order anharmonicity, σ, which we took to be 10, both within the established limits for organic molecules [112], using as an approximate relation, m = 3 σ 3−m . Details of the calculation on F− -CH3 Cl are provided in Ref. 19. We have calculated T (E) as a function of energy, E, with Eq. (4), and then calculated the participation number, p, with Eq. (16). Results for T (E) and p(E) are shown in Fig. 14. The participation number, p, is calculated to be finite for energies up to the quantum ergodicity threshold, which lies at 1690 cm−1 for the parameters used in the calculation. We note that this value is higher than estimates for the isomerization barrier for the reaction F− -CH3 Cl to FCH3 -Cl− , which are in the neighborhood of 1200 cm−1 [18]. There is thus a sizable region of energy above the barrier for which the vibrations of F− -CH3 Cl are nonergodic. In this example, the rise in p (toward the limit of the entire phase volume) is seen in the figure to be rather sharp, so that there can be an enhancement of the unimolecular reaction rate over a broad range in energy above the reaction barrier. We take the isomerization barrier to be 1200 cm−1 , as noted a reasonable estimate for the isomerization barrier for F− -CH3 Cl to FCH3 -Cl− [18]. We have also taken for simplicity the vibrational frequencies in the transition state to be the same as those for the reactant, and the lowest frequency mode (167 cm−1 ) to be that corresponding to the reaction coordinate with frequency ν. We calculate the total density of vibrational states, ρtotal , by direct count, which we then used to calculate the RRKM rate, kRRKM . We then calculated the quantum nonergodic statistical

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2

200

1.5

150

p

T

100

1

50

0.5

0 1200

1300

1400

1500

1600

1700

Energy (cm–1) Figure 14. The participation number, p(E) (solid curve), and the transition parameter, T (E) (dashed curve), are plotted as a function of total energy for a model of the F− -CH3 Cl complex described in the text. 100

10

Enhancement

Enhancement

100

10

1

0.1

0.01 1200

1300

1400

1500

1700

1600

Energy (cm–1)

1

0.1 2000

3000

4000

5000

6000

7000

Energy (cm–1) Figure 15. Plotted is the QuNEST reaction rate for a model of the F− -CH3 Cl complex with 1200 cm−1 isomerization barrier described in the text relative to the RRKM rate. The QuNEST rate constant has been calculated below the ergodicity threshold, found to be at 1690 cm−1 . The QuNEST rate includes the correction, κ(E), accounting for energy flow. Above the ergodicity threshold, the correction to the RRKM theory rate is calculated as κ(E). The inset shows the quantum nonergodic statistical theory enhancement factor, kQunEST (E)/kRRKM (E), for the model of the F− -CH3 Cl complex at energies below the quantum ergodicity threshold of 1690 cm−1 . The dashed curve is the enhancement when kQunEST is calculated assuming κ(E) = 1. The solid curve is the enhancement calculated with a correction, κ(E), that accounts for the finite rate of energy flow between transition states and nontransition states of the limited ergodic region.

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rate enhancement factor as kQunEST (E)/kRRKM (E). The result of this calculation is displayed in Fig. 15. The bare enhancement of the reaction rate compared to the RRKM theory rate due to reduction of the vibrational state space, which is given by Eq. (12) with κ(E) = 1, is seen to be over an order of magnitude to about 1600 cm−1 , and is almost two orders of magnitude near the threshold energy. Introducing the transmission coefficient κ(E), which corrects for the finite rate of energy flow between transition states and nontransition states within the limited ergodic region, diminishes the enhancement, which is seen in this example to be about a factor of 4 at energies close to the barrier energy. Above the ergodicity threshold of 1690 cm−1 , the ratio of the reaction rate to the RRKM theory estimate for the rate is κ(E). We see in the inset of Fig. 15 that in this model κ(E) will approach values close to 1 only at energies considerably higher than the ergodicity threshold, reaching 0.9 at 7000 cm−1 .

V.

CONCLUSIONS

Detailed experimental and theoretical studies in recent years have begun to clarify the wide variety of unimolecular reactions that exhibit non-RRKM theory kinetics [7]. Perhaps the single most important factor underlying the breakdown of RRKM theory for these reactions is that dynamics at the reactive energies is nonergodic. Here we have reviewed the recent work exploring the nature of dynamics in the nonergodic regime and the recent work on predicting rate constants for unimolecular reactions when dynamics are nonergodic. Undoubtedly, further developments in our understanding of the classical and quantum nature of nonergodic dynamics will lead to refinements in the prediction of unimolecular reaction rates in this regime. We have reviewed here a variety of analyses of phase space dynamics in the nonergodic, mixed phase space regime. In few degree-of-freedom systems, fractional dynamics in phase space characterized by 1/f noise is typically observed, which strongly mediates kinetics of unimolecular reactions. We have summarized how wavelet analysis can be used to characterize phase space dynamics in the nonergodic regime. This chapter has also addressed the non-Brownian, anomalous diffusion exhibited by macromolecules such as proteins, and characterization by finite-size Lyapunov exponents and principal component analysis. We have also reviewed recent theoretical work describing the quantum mechanical vibrational state space of many oscillator systems in the nonergodic regime. Our theoretical approach to this problem, local random matrix theory, can be adopted to introduce corrections to microcanonical transition state theory, or RRKM theory, predictions of unimolecular reaction rates. We refer to this adaptation as quantum nonergodic statistical theory (QuNEST). Here we have reviewed a recent application of QuNEST to a gas-phase SN 2 reaction. As discussed in this chapter, recent

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experiments have revealed a number of reactions where RRKM theory fails to predict the decomposition or isomerization rate, apparently due at least in part to nonergodicity, providing further tests for the present theories and a guide for future developments. Acknowledgments We thank Prof. R. Stephen Berry, Prof. John E. Straub, Prof. Akinori Kidera, Dr. Hiroshi Fujisaki, and Dr. Sotaro Fuchigami for useful discussions. We sincerely thank Prof. R. Stephen Berry for his critical reading of and suggestions on our manuscript. DML gratefully acknowledges support from NSF CHE-0512145, CHE-0910669, and OISE-043716; YM from Research and Development of the Next-Generation Integrated Simulation of Living Matter, a part of the Development and Use of the Next-Generation Supercomputer Project of the Ministry of Education, Culture, Sports, Science and Technology (MEXT); TK from JSPS, JST/CREST, Priority Area “Molecular Theory for Real Systems,” MEXT; and MT from JSPS, Priority Area “Molecular Theory for Real Systems,” MEXT.

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DYNAMICAL REACTION THEORY BASED ON GEOMETRIC STRUCTURES IN PHASE SPACE SHINNOSUKE KAWAI,1 HIROSHI TERAMOTO,1 CHUN-BIU LI,1 TAMIKI KOMATSUZAKI,1,2 and MIKITO TODA3 1 Molecule

& Life Nonlinear Sciences Laboratory, Research Institute for Electronic Science, Hokkaido University, Kita 20 Nishi 10, Kita-ku, Sapporo 001-0020, Japan 2 Core Research for Evolutional Science and Technology (CREST), Japan Science and Technology Agency (JST), Kawaguchi, Saitama 332-0012, Japan 3 Department of Physics, Faculty of Science, Nara Women’s University, Kitauoyahigashi-machi, Nara 630-8506, Japan

CONTENTS I.

Introduction A. Statistical Reaction Theory in a Nutshell B. Limitation of RRKM Theory II. Dynamical Reaction Theory A. Arnold Web B. Normally Hyperbolic Invariant Manifolds C. Dynamical Connections D. Fractional Behavior in Mixed Phase Space III. Remnants of Invariants Buried in Phase Space of Many-Degrees-of-Freedom Systems A. A New Technique to Detect Remnants of Invariants Buried in a Sea of Chaos B. An Illustrative Example, HCN C. Remnants of Invariants Buried in Potential Well of HCN at High-Energy Regime Above Potential Barrier IV. Dimension Reduction by Normal Form Theory A. Dimension Reduction Scheme Based on Partial Normal Form Theory B. Vibrational Energy Transfer in O(1 D) + N2 O → NO + NO V. Bifurcation and Breakdown of NHIM: The Origin of Stochasticity of Passage Through Rank-One Saddle 1. Harmonic Region 2. Nonlinear Quasiperiodic Region Advancing Theory for Kinetics and Dynamics of Complex, Many-Dimensional Systems: Clusters and Proteins, Advances in Chemical Physics, Volume 145, Edited by Tamiki Komatsuzaki, R. Stephen Berry, and David M. Leitner. © 2011 John Wiley & Sons, Inc. Published 2011 by John Wiley & Sons, Inc.

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3. Weak-Chaotic Region 4. Strong-Chaotic (Stochastic) Region A. Bifurcation of No-Return TS: Noncollinear H2 + H Exchange Reaction B. Robust Persistence of No-Return TS and Its Chaotic Breakdown VI. Conclusions References

I.

INTRODUCTION

How and why do systems change state from reactants to products? This has been an intriguing subject in natural science from the days of alchemy. The most crucial breakthrough in chemical reaction theories was to decompose the whole complex dynamical event of a chemical reaction into two separate problems: how does the system “climb” from the potential well to a hypersurface, the so-called transition state (TS), which lies between the reactant and the product states?, and how does the system pass through the hypersurface? For the former, the conventional statistical chemical reaction theories assume the existence of local equilibrium, that is, reacting systems ergodically wander (“thermalize”) in the accessible local phase space region before passing the hypersurface. In other words, this assumption lies in the separation of timescales between the “thermalization (or relaxation into local equilibrium)” within the well and the passage through the TS, and hence, leads to the formulation of reaction rate free from any initial conditions on the coordinates and their conjugate momenta besides the total energy. Experimental characterization of a chemical reaction process can be done by the measurement of the rate constant and the product state distribution [1, 2]. The rate constant may be the most fundamental property of the reaction, while the rovibrational states of the reaction product reflect more detailed dynamics during the course of chemical reaction and also are experimentally more accessible than the intermediate dynamics itself. For the purpose of investigating to what extent the system is thermalized, statistical models have been developed for comparison with experimental and numerical results. These includes RRKM (Rice–Ramsperger– Kassel–Marcus) and TST (transition state theory) rate constants [1, 2] and PST (phase space theory) for the product state distribution [3, 4]. A benchmark study on the reaction of O + CN [5] has established a guiding concept about the dynamics of a reacting system with a well on the potential energy surface (PES). O(3 P) + CN(X2 + ) → CO(X1 + ) + N(4 S)

(1)

→ CO(X1 + ) + N(2 D)

(2)

When the reactions on the two PES (ground and first excited electronic states) were compared, the surface with a deep well resulted in a statistical distribution of the

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product vibrations, while that without a well showed highly excited vibration of CO with an inverted distribution. Thus, a deep potential well results in trapping of the system for sufficiently long time to allow “thermalization” of the system inside the well, whereas the energy is distributed among only some of the rovibrational modes of the products in a short-lifetime intermediate complex. However, quite a few chemical reactions have been found (O(1 D) + HCl [6–8], O(1 D) + H2 O [9, 10], H + NO2 [11–13]) that show nonstatistical product distributions despite a deep well, and also a reaction (O(1 D) + N2 O [14–23]) that shows apparently equilibrated energy distributions despite the absence of a deep well. Even for much larger systems, there exists experimental evidence indicating the persistence of nonstatistical behavior in chemical reactions. Dian and coworkers [24] revealed the existence of mode selectivity in conformational isomerization of N-acetyl-tryptophan methyl amide (NATMA) using infrared–ultraviolet holefilling and IR-induced population transfer spectroscopies. This system consists of 36 atoms with 102 degrees of freedom (dofs) having a much more complicated potential energy landscape than that of systems with just a few atoms [25]. They showed that the destination of the transitions among the three stable conformations on the landscape depended significantly on which conformation was excited and on which of the two NH stretching vibrational fundamentals was excited. There seemed to be no clear timescale separation among these internal dofs because the injected energy of the NH stretching modes whose frequencies rank high in the top 5% of all the vibrational modes of NATMA triggered the isomerization reaction. This implies that an NH stretching mode of relatively high frequency selectively induces a reactive mode of low frequency. Their results show that the vibrational excitation energy initially injected into a specific vibrational mode is not immediately dissipated into all the other dofs but flows to the reactive mode, along a specific pathway, yielding a slow and large-amplitude conformation transition. Otherwise the distribution of destinations would be the same irrespective of which vibrational mode is excited. These are just some of the examples showing the existence of nonstatistical behaviors in chemical reactions, which motivate us to scrutinize reaction processes as dynamical systems beyond the statistical viewpoint. In Section II, we present what we have recently understood about important building blocks in phase spaces of many dimensions, normally hyperbolic invariant manifolds (NHIMs) and their stable/unstable invariant manifolds forming robust boundaries of reaction paths, which are crucial to understanding complex reaction dynamics beyond the scope of statistical reaction theories. In Section III, we present our recent studies elucidating the phase space structure in the regions of potential wells, which reveal a structure that could possibly mediate coherent energy flow. In Section IV, we present our recent studies on dimension reduction by extending normal form theory into “partial” normal form theory. In Section V, we present the hierarchical phase space structure in the region of a rank-one saddle that we

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have elucidated in the past decades, and discuss the possible mechanism for the breakdown of the TS (“point of no-return”) spoiling the one-dimensional nature of the reaction coordinate. Before Section II, let us start by giving a brief overview of the statistical reaction theory and its limitations. A.

Statistical Reaction Theory in a Nutshell

Here, we briefly explain the statistical reaction theory. We start our explanation by discussing the assumptions underlying the theory. The first assumption is that relaxation in the potential well is much faster than the reaction. Then, the reaction takes place while equilibrium is maintained in the well. This assumption leads us to the phenomenological treatment of the reaction described by the rate equation such as dP(t)/dt = −kP(t), where P(t) is the density of the reactant species and k denotes the rate constant. The rate equation shows that P(t) exhibits exponential decay and that the inverse of the rate constant gives the characteristic timescale of the reaction process. The most important task of the statistical reaction theory is to estimate the rate constant k. Here, the concept of TS comes into play. In the conventional theory, a TS is a saddle of the potential in the configuration space where the Hessian matrix has one negative eigenvalue (a saddle of index 1). Suppose that a Hamiltonian of N degrees of freedom is expanded around a saddle located at (q1 , . . . , qN ) = (0, . . . , 0) as follows: 2 N 2 ∞ pj ωj2 qj2 p1 μ21 q12 H= + + Hn (3) − + 2 2 2 2 j=2

n=3

where μ1 and ωj for j ≥ 2 are real, and Hn consists of the terms of nth order in (q, p) = (q1 , . . . , qN , p1 , . . . , pN ). Then, the coordinate q1 lies, locally near the saddle, along the direction of the eigenvector corresponding to the negative eigenvalue of the Hessian matrix at the saddle. This means that, in the conventional theory, (q1 , p1 ) are the reaction coordinate and its canonically conjugate momentum, respectively. The boundary between the reactant and the product is supposed to be given by the hyperplane q1 = 0 in the configuration space. The rest of the degrees of freedom (qb , pb ) = (q2 , . . . , qN , p2 , . . . , pN ) are supposed to be the bath modes. The second assumption of the theory is that orbits never come back once they cross the TS q1 = 0 from the reactant side to the product side. Then, the reaction rate is estimated by counting the number of orbits that cross the TS. Combining the above two assumptions, the rate constant k is given by the ratio between the number NTS of states on the TS and the number Nwell of states in the well, k ∝ NTS /Nwell . This is the formula for the rate constant in the statistical reaction theory, and this theory is called the RRKM theory. However, the statistical reaction theory suffers from several difficulties in both experimental and theoretical aspects. In the following, we discuss these difficulties

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and show how we can overcome them by resorting to dynamical studies of the reaction processes. B.

Limitation of RRKM Theory

Since RRKM theory [26] was proposed, investigations on RRKM have been done, not only theoretically but also experimentally, to examine the validity of the underlying assumption of RRKM, that is, that all the available energy redistributes statistically among the degrees of freedom of the system in the reactant state before the reaction takes place. In this section, we introduce several experimental works related to the issue (Fig. 1). There are two types of experimental investigations related to this issue. One type uses a direct comparison of kinetic rate constants measured experimentally to those calculated by using RRKM [27–37]. The other type of investigations do not resort to direct comparison. For example, examining how the ratio of the final products changes depending on which vibrational modes are excited, which is called mode selectivity, is one possible way to address the issue [24, 38–45]. Suppose there is a molecule that has two vibrational modes with almost the same vibrational frequencies and we observe the ratio of the final products of chemical reactions induced by exciting either of the vibrational modes by one quantum. In this case, the ratio of the final products should not depend on which vibrational mode was excited if the underlying assumption of RRKM is valid because, according to the assumption, the excitation energy redistributes statistically among the degrees of freedom before reaction takes place. Therefore, the existence of mode selectivity provides direct evidence of non-RRKM behaviors. Another way to investigate the issue is the

40 (14)

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Figure 1. Some of the reports on nonstatistical chemical reactions: (1) [27], (2) [29], (3) [30], (4) [28], (5) [31], (6) [32], (7) [33], (8) [46], (9) [47], (10) [34], (11) [35], (12) [36], (13) [48], (14) [24], (15) [39–42], (16) [44], (17) [37], and (18) [45].

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following: Consider molecules that have various sizes but have the same reaction coordinate and the same energy landscape along the coordinate, and observe the kinetic rate constants of their chemical reactions. RRKM demands that the kinetic rate constant decreases by orders of magnitude as the size of the molecule increases by orders of magnitude [46–48]. By using this fact, it is possible to address the issue by examining how the kinetic constant scales with the molecule size. II.

DYNAMICAL REACTION THEORY

In this section, we discuss theoretical difficulties of the statistical reaction theory and show how we can overcome these problems. One of the difficulties is the recrossing problem, that is, the TS defined in the configuration space is not a real boundary between the reactant and the product. It is known that some orbits return to the reactant well immediately after they cross the saddle in the configuration space. Existence of such orbits leads to overestimation of the reaction rate. Moreover, the formula for the rate constant cannot be used for reaction processes where the assumption for the separation of the timescales is questionable. This limits the applicability of the statistical reaction theory. Here, we introduce the key concepts in the dynamical theory of reactions. In this chapter, we consider the following three processes of the reaction: (i) redistribution of energy among vibrational modes in the well, (ii) going over the potential saddle, and (iii) dynamical connections among multiple saddles. For the distribution of vibrational energy, nonlinear resonance among vibrational modes is essential. In the action space, the resonant regions constitute the network called the Arnold web, where exchange of vibrational energy takes place. Therefore, properties of the Arnold web play an important role in our topics. In regard to the description of a reaction process going over the saddle, the normal form theory has been developed recently, which provides mathematically sound definitions for the concepts of TS and reaction coordinate. The theory is based on the phase space structures called normally hyperbolic invariant manifolds. It enables us to define the boundary between the reactant and the product, and to single out the reaction coordinate at least locally in the phase space near a saddle of index 1. For full understanding of the reaction, we need to analyze connections between the NHIMs and the Arnold webs, and connections among multiple NHIMs. This leads us to investigate the phase space structure in the large, especially chaotic itinerancy in the reaction dynamics. A.

Arnold Web

In the traditional theory of reactions, intramolecular vibrational energy redistribution (IVR) in the well is supposed to be statistical. However, this assumption does not hold when ergodicity of the processes in the well is questionable. Then,

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we need to analyze the processes of energy exchange based on the nonlinear resonances among vibrational modes. Thus, analysis of the Arnold web is crucial for understanding IVR. Suppose that a Hamiltonian of N degrees of freedom is expanded near a minimum located at (q1 , . . . , qN ) = (0, . . . , 0) as follows: H=

N 2 pj j=1

2

+

ωj2 qj2 2

+

∞

Hn

(4)

n=3

where ωj for j ≥ 1 are real, and Hn consists of the terms of nth degree in (q, p) = (q1 , . . . , qN , p1 , . . . , pN ). Note the difference from the expansion Eq. (3) around the saddle of index 1. This difference is essential in the following. Suppose that, for the Hamiltonian Eq. (4), we try to eliminate as many coupling terms as possible by transforming the original coordinates and momenta to new ones. If we succeed in eliminating all the coupling terms, the transformed vibrational modes describe independent motions in the well. However, such transformations do not exist in general because of the small-denominator problem [49]. The small-denominator problem occurs when nonlinear resonances take place among the vibrational modes. In general, these resonances constitute a network in the action space called the Arnold web, and resonance overlap results in globally chaotic motions [50]. Thus, characteristics of the Arnold web play an important role in understanding IVR [51, 52]. In particular, whether the Arnold web is uniform or not and how regions of resonance overlap are distributed play a key role for the foundation and limitations of the statistical reaction theory. We will briefly point out this problem in Section II.D. and provide more detail in a separate chapter of this volume. B.

Normally Hyperbolic Invariant Manifolds

The recrossing problem is solved by the dynamical reaction theory, which has been recently developed based on the analysis of phase space structures. The studies started with a pioneering work of Davis and Gray, and the mathematical formulation of the theory was provided by Wiggins et al. The key concept here is NHIMs and their stable/unstable manifolds. We start our explanation of NHIMs with the simplest case, that is, the NHIM around a saddle of index 1. Let us start our discussion by the expansion Eq. (3) of the Hamiltonian near the saddle of index 1. The hyperplane defined by q1 = 0 does not constitute the boundary between the reactant and the product. The true boundary can be constructed by a transformation from (p, q) to a new set of coor¯ = (p¯ 1 , p¯ b , q¯ 1 , q¯ b ). The transformation eliminates all dinates and momenta (¯p, q) the coupling terms between the reaction coordinate (p¯ 1 , q¯ 1 ) and the bath degrees of freedom (¯pb , q¯ b ) = (p¯ 2 , . . . , p¯ N , q¯ 2 , . . . , q¯ N ). Existence of such a transformation

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is guaranteed by the fact that no “resonance” takes place between the mode with the negative eigenvalue of the Hessian matrix and the modes with positive eigenvalues. Then, the boundary is defined by the hyperplane q¯ 1 = 0, and a reaction process going over the saddle is described by the reaction coordinate and momentum (p¯ 1 , q¯ 1 ). This is the essence for the existence of the boundary for a saddle of index 1. The NHIM around the saddle is defined by (p¯ 1 , q¯ 1 ) = (0, 0), that is, a manifold of codimension 2. It is invariant under the dynamics since couplings between the reaction coordinate (p¯ 1 , q¯ 1 ) and the bath modes (¯pb , q¯ b ) do not exist. The dynamics on the NHIM describes the motions of the bath modes at the saddle. If the energy above the saddle is small enough, the bath modes exhibit quasiperiodic oscillations, and the NHIM is foliated by tori. Then, the Lyapunov exponents along the tangential directions of the NHIM are zero. On the other hand, the normal directions of the NHIM have positive or negative Lyapunov exponents. The stable or unstable manifolds of the NHIM consist of those orbits that approach or leave the NHIM, respectively. Note that the dynamics near the NHIM is decomposed into two kinds of motions, that is, the motion of (p¯ 1 , q¯ 1 ) along the stable/unstable manifolds and the tori of (¯pb , q¯ b ). Thus, the NHIM and its stable/unstable manifolds describe the essential aspects of the reaction dynamics near the potential saddle. In general, an NHIM is a manifold where instability (either in a forward or in a backward direction of time) along its normal directions is much stronger than that along its tangential directions [53–55]. Using the Lyapunov exponents, which quantitatively measure instability, we can define an NHIM as a manifold where the absolute values of the Lyapunov exponents along its normal directions are much larger than those along its tangential directions. The stable/unstable manifolds consist of those orbits that approach or leave the NHIM, respectively. For saddles of index 1, these geometric structures enable us to identify the one-dimensional reaction coordinate by the normal directions to the NHIM, and thereby to define the TS as the dividing hypersurface of codimension 1 locally near the saddle. The TS thus defined is free from the problem of recrossing orbits, and can decompose the phase space into the distinct regions of the reactants and the products [56–59] (see also the recent reviews [60–63] and the book [64]). Moreover, the stable/unstable manifolds of the NHIM provide us with the reaction conduit through which all the reactive trajectories pass from the reactant to the product or vice versa. Thus, these manifolds offer a crucial way both to understand controllability of the reaction and to investigate dynamical correlation in reaction processes taking place over multiple saddles. NHIMs are structurally stable under perturbations. The wider the gap of instability is between the normal directions and the tangential ones, the more stable it is. However, as we raise the energy above the saddle, chaos can emerge on the NHIM [65], which is caused by nonlinear resonances among the bath degrees of freedom on the NHIM. It can lead to breakdown of normal hyperbolicity because the Lyapunov exponents tangent to the NHIM have a possibility of being

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comparable to those normal to the NHIM [61]. Then, we can no longer define a dividing hypersurface even locally near the saddle. Thus, breakdown of normal hyperbolicity raises serious questions concerning applicability of the concept of TS for reactions taking place high above the saddle. We will discuss this problem in Section V. C.

Dynamical Connections

In order to fully understand the reaction dynamics beyond the statistical theory, global aspects of the dynamics should be taken into account, that is, dynamical connection among multiple NHIMs. This problem was first met when Ezra and his coworker studied the isomerization of a system of more than two degrees of freedom. Here, we present the essence of the problem of why global aspects of the reaction matter. We present the arguments based on simple dimensional counting of various manifolds in the phase space: NHIMs, their stable/unstable manifolds, and intersections between stable and unstable manifolds. The crucial point of the argument is that, for a system of more than two degrees of freedom, one NHIM can have dynamical connections with multiple NHIMs through intersections between their stable and unstable manifolds. For a system of n degrees of freedom, let us take two NHIMs with 2r1 and 2r2 normal directions, respectively. Then, their dimensions are 2n − 2ri (i = 1, 2) in the phase space, and the dimensions of the stable and unstable manifolds are 2n − ri (i = 1, 2), respectively. If their stable and unstable manifolds intersect, that is, homoclinic/heteroclinic intersection exists, the dimension of their intersection in the phase space is 2n − r1 − r2 . When we consider the intersection on the equienergy surface, its dimension on the surface is 2n − r1 − r2 − 1. Thus, the dimension d of the intersection on the Poincar´e section is d = 2n − r1 − r2 − 2. For example, let us consider two saddles of index 1 and intersection between their stable and unstable manifolds, that is, r ≡ r1 = r2 = 1. This is the case the traditional reaction theory treats. For n = 2 and r = 1, d equals 0; that is, the intersections on the Poincar´e section are just points. On the other hand, for n = 3 and r = 1, d equals 2. This means that we can continuously change initial conditions on the unstable manifold. Then, we think of the possibility that tangency takes place on the Poincar´e surface. This is the phenomenon that Ezra and his coworker found for a system of more than two degrees of freedom. Moreover, it is indicated that tangency on the Poincar´e surface indicates that the NHIM has dynamical connection with multiple NHIMs. Therefore, the problem of how multiple NHIMs and the Arnold webs are connected through heteroclinic/homoclinic intersections is an important problem [61]. Moreover, the network that consists of multiple NHIMs and the intersections among them gives rise to chaotic itinerancy. (See Ref. 66 for a recent review

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of chaotic itinerancy.) This is a new feature of dynamical reaction theory, which is to be explored in future studies. D.

Fractional Behavior in Mixed Phase Space

In the conventional reaction theory, existence of the rate constant is presumed. This is based on the assumption that the processes in the well take place statistically. Then, the distribution P(t) of the reactant obeys an exponential decay following the phenomenological rate equation. However, this assumption does not necessarily hold when ergodicity in the well is not achieved. Then, we should ask what are the characteristic features of the reaction processes for such reactions. We can show that the following fractional behavior characterizes such reactions: •

The survival probability behaves differently as a function of the residence time: the probability varies according to a power law in a range of the residence time, and it changes exponentially in another range. • The Fourier spectra of the action variable of the reaction coordinate correspondingly exhibit different characteristics. For trajectories corresponding to power law decay, the spectrum exhibits 1/f dependence. On the other hand, for trajectories corresponding to exponential decay, the spectrum exhibits Lorentzian features. • The local diffusion of the action variable of the reaction coordinate also exhibits two types of behavior; anomalous diffusion for power law decay and normal diffusion for exponential decay. These characteristic differences correspond to whether the trajectories experience fully developed chaos or not. Those orbits that do not travel around fully developed chaos show fractional behavior, and those that do exhibit the exponential behavior. Thus, nonuniformity of the Arnold web is essential for the existence of fractional behavior. For more detailed discussion, we refer the reader to a separate chapter and the original papers of our study [67–69]. Note again that the following difference is important between Hamiltonian systems of two degrees of freedom and those of more than two. For Hamiltonian systems of two degrees of freedom, fractional behavior is well established [70]. It is shown that a hierarchy of resonant tori exists trapping nearby trajectories for a finite time [49]. On the basis of this property, they are described as “sticky.” Then, the hierarchy of timescales leads to a power law distribution of residential times. However, when the number of degrees of freedom is larger than 2, it is not obvious that the same argument holds because the dimension of tori is not large enough to work as dynamical barriers. Therefore, fractional behavior in systems of more than two degrees of freedom is a new issue in the study of Hamiltonian chaos. The existence of the fractional behavior indicates that the concept of the reaction rate constant is no longer valid. This is a new research area in the fields of reaction

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dynamics. In order to fully appreciate the importance of the problem, quantum systems should be studied to see whether fractional behavior exists or not. Note that, in general, “quantum chaos” is less statistical compared to the corresponding classical chaos, although the discrepancy is shown to be smaller as the number of the degrees of freedom becomes larger [71]. Thus, we expect that fractional behavior of classical chaos has corresponding effects in quantum systems. In fact, a recent study shows that, for a system of two degrees of freedom, fractional behavior of classical chaos has corresponding effects in the quantum system [72]. Thus, we think that the fractional behavior is relevant for reaction processes, which are quantum in nature. III.

REMNANTS OF INVARIANTS BURIED IN PHASE SPACE OF MANY-DEGREES-OF-FREEDOM SYSTEMS

In this section, we briefly review the concept of “remnant of invariant” buried in chaos in realistic chemical reaction systems and its manifestation in threedimensional molecular systems. The concept remnant of invariant dates back to Shirts and Reinhardt [73]. They found in the H´enon–Heiles system of two degrees of freedom (dofs) H=

2 i=1

p2i q2 + i 2 2

1 + q12 q2 − q23 3

(5)

that, even in highly chaotic phase space regions where no tori survive, the system is trapped on a low-dimensional surface in the phase space for a certain time duration much longer than the characteristic vibrational timescale of the system and suddenly hops onto another surface in an intermittent manner. We show one typical example of this behavior in Fig. 2. Shirts and Reinhardt [73] called the surfaces remnants of destroyed invariant tori because they are isosurfaces of the formal third integral constructed by using Pad´e resummation of the Birkhoff– Gustavson normal form expansion [74] that provides a true integral of motion if the system is integrable. Note that their findings also ensure that the phase space volume occupied by remnants of invariants is finite and, thus, the dimensionality of the volume must be equal to that of the ambient space, because otherwise it would be impossible to detect remnants of invariants numerically. The H´enon– Heiles system is a system of two dofs that does not have several common features possessed by high-dimensional systems, such as Arnold diffusion [75, 76] due to energy exchange through resonance. However, most realistic chemical reaction systems have many dofs and have much more complicated, nonlinear potential energy landscapes than that of the H´enon–Heiles system. As far as we know, the manifestation of remnant of invariants in higher dimensional systems has not been

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fully understood yet. In order to address the problem, we revisit the concept of remnants of invariants, taking an isomerization reaction of HCN as an example [77, 78]. There are mainly two different approaches to extract the geometrical structure of phase space in terms of Lie canonical perturbation theory (LCPT)1 (see also the details of the method in Section IV). The first and most prevailing approach is to transform H such as Eqs.(24 and 39) and truncate it at (m + 1)th order. If one can confirm that a solution of the truncated Hamiltonian is close to the solution of the original Hamiltonian H (see, for example, Ref. 79), one can assign the 1

Our usage of LCPT and normal form (NF) theory in this chapter is the same. The naming of the former comes from the methodology based on Lie transform, but that of the latter from the name of the resultant representation after the nonlinear transformation of variables (this is, in principle, irrespective of which methodology would be applied to a dynamical system for the transformation). The latter has been used not only for Hamiltonian systems but also for dissipative dynamical systems.

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geometrical structures associated with H in terms of the truncated Hamiltonian. However, in general, it is very difficult to prove that the two solutions are equivalent unless they both are sufficiently close to the stationary point. The other approach is the following: One traces the new coordinates and momenta p¯ (m) , q¯ (m) in H¯ as functions of (p, q) obeying the original Hamiltonian H: p¯ i (p, q) = e−Fm e−Fm−1 · · · e−F1 pi (m)

q¯ i (p, q) = e−Fm e−Fm−1 · · · e−F1 qi (m)

(6)

where we denote m by the order of perturbation and Fν (ν = 1, . . . , m) is the Poisson bracket with a function fν (ν = 1, . . . , m) such as Fν = {·, fν }

(7)

For example, if the system is transformed into a set of isolated oscil (m) lators by (p, q) → p¯ (m) , q¯ (m) , the associated action integrals I¯i (p, q) ≡ (m) (m) 1 ¯ i (p, q))2 + (¯qi (p, q))2 } should, in principle, be constants of motion dur2 {(p ing the dynamical evolution obeying H(p, q). Considering the former approach, one can neither extract the underlying remnants of invariant manifolds that might behave as a bottleneck during transport in phase space nor reveal any intermittent hopping motion among such remnants of invariants. On the other hand, the latter approach using Eq. (6) to trace the dynamical evolution obeying the original Hamiltonian should reveal how the system enters and escapes from remnants of invariants by means of the evolution (m) of the action integrals I¯i . Furthermore, one can also grasp the order of resonance through which the energy exchange takes place by monitoring the ratios between ¯ I¯i(m) . the frequencies ∂H/∂ In common LCPT calculations, the transformed variables in Eq. (6) are usually written as an expansion series truncated at a certain finite order, for example, (m + 1)th order, (m)

p¯ i

≈ pi − {pi , f1 } + · · · + (terms of mth order)

(m) q¯ i

≈ qi − {qi , f1 } + · · · + (terms of mth order)

(8)

However, as shown later in an HCN isomerization reaction, the truncation in the coordinate transformation of LCPT procedure can result in apparent abrupt fluctuation of the new action integrals when one traces them along the original, true Hamiltonian. This partially originates from the fact that Eq. (8) gives rise to the loss of the symplecticness property of p¯ (m) , q¯ (m) whereas the original transformation [Eq. (6)] is symplectic (see, for example, Refs. 80 and 81). Note that such an apparent fluctuation of the action integrals has not been studied carefully

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in the literature and it is a quite difficult task to identify the origin of fluctuation of the action integrals, namely, whether due to the truncation or due to intrinsic chaos, especially, in the potential wells at high energies. A.

A New Technique to Detect Remnants of Invariants Buried in a Sea of Chaos

We present a new technique to preserve the symplecticness property of the transformed variables based on the formula derived by Hori [80] and later by Deprit [81]. The technique can detect remnants of invariants buried in the potential well even at energies higher than the potential barrier. Our idea is very simple and general: provided that a generating function f is calculated analytically in advance, the canonical transformation associated with f , (P, Q) = e{·,f } (p, q) (9) (p,q)=(p0 ,q0 )

can be calculated by solving the ordinary differential equations dp () ∂f =− , d ∂q

dq () ∂f = d ∂p

(10)

with the initial condition (p0 , q0 ) = (p (0) , q (0)) if the canonical transformation [Eq. (9)] exists. The new canonical variables (P, Q) are identified as (p (1) , q (1)). The canonical transformation in Eq. (9) can be easily generalized for function ¯ I¯i , thanks A (p, q) other than (p, q) such as action integrals and frequencies ∂H/∂ to the relation of ¯ (P, Q) e−{·,f } A (p, q) = A(e{·,f } p, e{·,f } q) = A

(11)

This relation means that value of A (P, Q) is equal to that of A the functional evaluated at the point e{·,f } p, e{·,f } q [80–82]. We utilize this idea consecutively by replacing f by −fν in calculating Eq (6) without any truncation like Eq. (8). The schematic picture of our procedure is presented in Fig. 3. First, we solve the ordinal differential equation ∂f1 dp = , d ∂q

dq ∂f1 =− d ∂p

(12)

starting from the original point p(0) , q(0) until becomes 1. The resultant (p (1) , q (1)) is regarded as the transformed canonical variables at the first order, that is, p¯ (1) , q¯ (1) , which corresponds to the first operation e−F1 pi and e−F1 qi in Eq. (6). Next, starting from the point p¯ (1) , q¯ (1) , we further propagate the

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137

- (m)) (p- (m),q - (1)) (p- (1),q - (2)) (p- (2),q

(p(0),q(0))

dp/dτ = -{p,f2} dq/dτ = -{q,f2}

Figure 3. A schematic of multiple “-time” evolutions in terms of the generating functions. In terms of the numerical propagation using a set of the generating functions, one can exactly calculate the canonical transformation of Eq. (6) up to the desired order.

“system” in terms of the second-order generating function f2 in the same manner: dp ∂f2 = , d ∂q

dq ∂f2 =− (13) d ∂p for the unit time duration, resulting in p¯ (2) , q¯ (2) . By repeating this procedure, one precisely evaluate the new canonical variables up to a chosen order m (m)can (m) p¯ , q¯ . Note that Eq. (6) itself preserves the symplecticness property if (p, q) is symplectic [80, 81]. Therefore, our procedure does not suffer from the loss of symplecticness property and the validity range of the new canonical coordinates is expected to be much wider than for those truncated at finite order in Eq. (8). B.

An Illustrative Example, HCN

As an illustrative example, we investigate an isomerization reaction of HCN molecule. This molecule consists of three atoms: H, C, and N. Restricting the system to zero total angular momentum, the Hamiltonian can be described by the following three dofs, r (distance between C and N atom), R (distance between H and the center of mass of C and N), and γ (angle between H and C as seen from the center of mass of C and N) in Jacobi coordinates. The corresponding Hamiltonian is 1 2 1 2 1 1 1 H= p2γ + V (r, R, γ) pr + + (14) pR + 2μ 2m 2 μr 2 mR2 where μ = (mC mN )/(mC + mN ) is the reduced mass of the CN diatom, m = (mH (mC + mN ))/((mH + mC + mN )) is the reduced mass of the full system, and the potential V (r, R, γ) is taken from Ref. 83. This molecule has two minima that have collinear configurations, one is called HCN and the other is CNH. The potential barrier height between the two wells is −0.444 kcal/mol. The HCN well and the CNH well and the saddle point that lies between the two wells correspond to γ ≈ 0, r ≈ π, and γ ≈ ±1.168 rad, respectively. In addition, the depth of the

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CNH well is more shallow than that of the HCN well and, inside the CNH well, there are two very shallow wells that correspond to γ ≈ ±1.99. Therefore, the CNH well has a more rugged potential energy landscape than the HCN well. In what follows, we set the total energy of this system to E = −0.420 kcal/mol, which is high enough for trajectories to cross over the HCN–CNH isomerization barrier so that we can observe isomerization events during our simulation. We use adaptive step-size control Runge–Kutta integrator of fourth order to integrate the Hamiltonian equation of the system. The Hamiltonian is expanded as a power series with respect to normal coordinates (p, q) around a potential minimum of either HCN or CNH as presented by Eq. (4). 1 2 ωi pi + qi2 + O (3) 2 3

H=

(ω1 > ω2 > ω3 )

(15)

i=1

Using the LCPT procedure, one can evaluate the new canonical coordinates and the ¯ the associated action integrals I¯i(m) , and the frequencies conjugate momenta (¯p, q), ¯ I¯i(m) up to a certain order m using both the virtual ε-time evolution obeying ∂H/∂ the set of the generating functions Fk and the conventional truncation scheme of Eq. (8). Here, I¯1 and I¯2 roughly correspond to the stretching motions of r and R coordinate, respectively, and I¯3 to the bending motion associated with γ in both the wells. C.

Remnants of Invariants Buried in Potential Well of HCN at High-Energy Regime Above Potential Barrier

Figure 4 illustrates how the three action integrals evolve along a typical reactive trajectory passing through the potential well in the isomerization process. In Fig. 4a, we present a typical reactive trajectory running through the potential well. One can see that γ takes value around 0 in t = 60–225 fs, which means that the system wanders in the HCN well during this time regime. Figure 4b–d shows plots of the three action integrals along the reactive trajectory, obtained by Eq. (8). Note that, if the actions are well defined in the potential well, they are expected to evolve slowly along trajectories inside the well. One can see in Fig. 4b–d that all the actions change rapidly in the potential well and the increase in perturbation order from fourth to seventh does not improve this situation. The Pad´e resummation technique has often been performed to extrapolate them to higher order in terms of a set of the truncated calculations [73]. It was found, however, that the Pad´e resummation technique does not work because the Pad´e coefficients could not be well determined for these abruptly fluctuating actions. One might interpret that such abrupt fluctuation of action integrals is caused by nonintegrability of the system

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(b)

(a) Inside HCN well 2 1.8 1.6 1.4 1.2 1 0.8 0

2π

γ (radian)

r, R (A)

6 5 4 O - (2,5) 3 I1 2 1 0 0 50 100 150 200 250 300 60 80 100 120 140 160 180 200 220 Time (fs) Time (fs) (d) (c) 30 6 25 5 - (2,5)20 - (2,5)4 I3 15 I2 3 10 2 5 1 0 0 60 80 100 120 140 160 180 200 220 60 80 100 120 140 160 180 200 220 Time (fs) Time (fs) Figure 4. (a) A representative reactive trajectory of HCN isomerization. r (thin solid line), R (2) (5) (dotted line), and γ (thick solid line). (b–d) The action integrals I¯i (thick solid line) and I¯i (thin solid line) obtained by Eq. (8) inside the HCN well (t = 60 − 230 fs).

resulting from the high nonlinearity in the potential well at energies above the potential barrier. However, there exists yet another possible source of the abrupt fluctuation of the actions, that is, the truncation in the coordinate transformation of the LCPT procedure. It can result in apparent abrupt fluctuation of the new action integrals when one traces them along the original, true Hamiltonian. This is due fact (m)to the that Eq. (8) gives rise to the loss of the symplecticness property of p¯ , q¯ (m) . In Fig. 5, we show the three actions calculated to fifth order by the -time evolution technique using fν (ν = 1–5) along the same trajectory as shown in Fig. 4a. One can see that, compared to Fig. 4b–d, the abrupt fluctuation observed in the actions in Fig. 4b–d is much more suppressed with fewer spurious peaks while the same generating functions are used. Note that large fluctuations still survive at certain times duration, for example, peaks at around 60 and 80 fs. To further look into the origin of the appearance of the peaks, first we check the convergence properties with respect to the order of the perturbation in the -time evolution procedure. In Fig. 6a and b, we show the two consecutive “trajectories” of the ordinal differential equation with “Hamiltonian” −Fk (from k = 3 to 7) initiated from phase space points (p, q) taken from two different time regions. One is the time region where the actions fluctuate abruptly (t = 81.7581 fs) and the other is one where the actions vary slowly (t = 115.138 fs). Here, the “trajectories”

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20 - (5) I1

5

- (5) I2 15

4

- (5) I3

- (5) I 1,2 3

- (5) 10 I 3

2 5 1 0 60

100 120 140 160 180 200 220 Time (fs) 115.138 (fs) 81.7581(fs) 80

0

Figure 5. The new actions calculated by the -time evolutions using fν (ν = 1, 2, 3, 4, and 5) for each dof along the same trajectory as in Fig. 4a inside the HCN well (t = 60 ∼ 230 fs).

2 1.5

(a) Time = 81.7581 fs Second order First order

1 p

1

Fourth order

0.5

Third order Fifth order

0 –0.5 –1 –2

1.3

–1

0

1

q

2

3

4

5

1

(b) Time = 115.138 fs

1.2 First order

1

Fourth order

p

1

1.1

0.9 0.8 0.7

Fifth order Second order Third order

0.6 –1.4 –1.2

–1

–0.8 –0.6 –0.4 –0.2 q1

0

0.2

Figure 6. The projection of the consecutive, -time evolutions obeying “Hamiltonian” −fν (from ν = 1 to 5) on (p1 , q1 ), where the initial phase space point (p, q) is taken at (a) t = 81.7581 fs and (b) t = 115.138 fs.

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are projected onto the (p1 , q1 ) plane. (Note that distance in this plane does not necessarily correspond to the actual distance on the underlying six-dimensional phase space, but we observed almost the same tendency when we projected the “trajectories” onto the (p3 , q3 ) plane [77].) For the “trajectory” initiated from t = 81.7581 fs, where the large abrupt fluctuation exists, one cannot see any tendency to converge with respect to the order k of the generating functions, but for the “trajectory” from t = 115.138 fs where no significant fluctuation survives, one can expect the existence of a tendency to converge. A more systematic test for the convergence is shown in Ref. 78. In addition, we also found that the first local Lyapunov index also takes relatively large value when the actions turn abruptly, which also indicates that the abrupt change of the actions is due to the intrinsic chaos in phase space [77]. The most striking consequence of our technique is the discovery of the slowly evolving actions from t = 90 to 230 fs, even beyond the isomerization threshold of HCN. The apparent “irregular” dynamics of HCN can be understood as a process of slow energy exchange among these actions buried in the phase space. (Note again that the abruptly fluctuating actions obtained by the conventional truncation procedure cannot capture the underlying mechanism of energy transfer.) To demonstrate the underlying motions in the phase space during the process, the trajectory from t = 100 to 220 fs in Fig. 5 is projected onto six different planes in Fig. 7, that is, the normal coordinates and the conjugate momenta of the second and third modes (pi , qi ) (i = 2, 3) (Fig. 7a and b), the corresponding normal (5) (5) form coordinates and the conjugate momenta (p¯ i , q¯ i ) evaluated by the conventional truncation procedure (Fig. 7c and d), and those by our -time evolution (5) (5) technique (Fig. 7e and f). One can immediately see that (p¯ i , q¯ i ) evaluated by the truncation procedure cannot provide us with any insight into the underlying mechanism of energy exchange among modes in the phase space. The projections of the trajectory onto the normal mode planes (pi , qi ) exhibit more subdued oscil(5) (5) lations than the projections onto (p¯ i , q¯ i ) evaluated by the truncation procedure, but still look complicated and chaotic. However, in Fig. 7e and f (and Fig. 7g and (5) (5) h with the color grade), (p¯ i , q¯ i ) evaluated by our -time evolution technique provides a clear picture of the underlying motions associated with energy exchange among the modes of slowly varying actions: while the action of the third mode (reactive mode) gradually decreases from t = 90 to 150 fs (see Fig. 5), implying the release of the energy into the other bath modes, the oscillation motion persists (5) (5) with slightly decreasing (increasing) amplitude of the oscillation in the (p¯ 3 , q¯ 3 ) (5) (5) ((p¯ 2 , q¯ 2 )) plane. Then, the pattern of the amplitude change of the two normal form coordinates becomes opposite, reflecting that the action of the reactive mode turns to gradually increase (i.e., the gain of the energy from the bath modes). As mentioned earlier, the origin of fluctuation observed in action integrals has not been carefully paid attention, and it was difficult to identify whether the

(a)

(b) Time = 220 (fs)

3 4

2

Time = 100 (fs)

2

p2

1 p3

0

0

–1 –2

Time = 220 (fs)

–2 –3

–4 –3

–2

–1

(c)

0 q2

1

2

–4

–2

0 q3

2

4

–4

–2

0 q3

2

4

2

4

(d)

3

4

2

Time =100 (fs)

2

1

p2

Time = 100 (fs)

3

p3

0

–1

0

–2

–2

–4

–3 –3

–2

–1

(b) 3 2

0

q2

1

2

3

(f) 4

Time = 220 (fs)

Time = 100 (fs)

2

Time = 220 (fs)

p3

p2

1 0

–1

0 Time = 100 (fs)

–2

–2 –4

–3 –3 –2 –1

0 q2

100 (fs) 3

3

–4

Time = 220 (fs)

–2

0 q3

100 (fs)

(h)

220 (fs)

4

Time= 100 (fs)

Time= 220 (fs)

3

2

0

p

p

2

1

2

220 (fs)

(g) 2

1

–1

0

Time = 100 (fs)

–2

–2

–4

–3 –3 –2 –1

0 q

2

1

2

3

–4

–2

0 q3

2

4

Figure 7. The projection of the trajectory in Fig. 4a from t = 100 to 220 fs onto the normal coordinates and momenta of the second and third modes (pi , qi ) (i = 2, 3) (a and b), the corresponding (5) (5) normal form coordinates and momenta (p¯ i , q¯ i ) evaluated by the truncation procedure (c and d), and those by the -time evolution technique (e and f) ((g and h) the color grade indicates the history along the time evolution).

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origin of fluctuation is due to the truncation or intrinsic chaos in the potential wells. Our new technique has great potential not only to identify the origin of fluctuation in the action integral but also to reveal the underlying mechanism of energy transfer among the modes such as energy “dissipation/accumulation” from/to reactive to/from nonreactive modes in terms of nonlinear resonance among the modes constructed by the technique. IV.

DIMENSION REDUCTION BY NORMAL FORM THEORY

In this section, we present the formulation of normal form (NF) that has been used in several parts of the preceding sections in this chapter. NF theory is one of the most powerful tools to tackle the problem of multiple degrees of freedom coupled to each other. To observe the difficulty introduced by mode coupling, suppose a Hamiltonian of an N-degrees-of-freedom system given by 1 2 H= p + V (q) 2 N

=1

=

N

=1

1 2 κ 2 p + q

2 2

+

ajk q1 j1 · · · qN jN p1 k1 · · · pN kN

(16)

j,k

where V (q) is a potential energy. In the second line, the potential energy is expanded in Taylor series with the expansion coefficients κ for the quadratic part and ajk for higher order terms. We have κ > 0 for all if the equilibrium point is a minimum point of the potential energy surface, or κ < 0 and κ > 0 for ≥ 2 if it is a rank-one saddle point. The equation of motion given by Eq. (16) is, for mode 1 as an example, d q 1 = p1 + ajk j1 q1 j1 −1 · · · qN jN p1 k1 · · · pN kN dt j,k

d ajk k1 q1 j1 · · · qN jN p1 k1 −1 · · · pN kN p1 = −κ1 q1 − dt

(17)

j,k

The first terms in the above equations correspond to the conventional normal mode picture and include only mode 1. The second terms show that the motion along the mode 1 depends on the value of other coordinates q2 , . . . , qN , p2 , . . . , qN . This dependence of the motion on the other modes is called “coupling.” The coupling introduces the major complication of the dynamics in that the motion along each mode cannot be analyzed independently of others.

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The idea of NF is to introduce a new set of coordinates by a coordinate ¯ with which the number of coupling terms is transformation (q, p) → (¯q, p), as small as possible. The scheme is based on the canonical perturbation theory proposed by Deprit [81]. It is recently introduced in the field of molecular dynamics in the context of transition state theory [59, 62, 64, 84–87], and also reaction dynamics [88]. In this section, we describe the mathematical formulation of normal form (Section IV.A) and its application to the reaction dynamics of O(1 D) + N2 O → NO + NO (Section IV.B). A.

Dimension Reduction Scheme Based on Partial Normal Form Theory

The NF theory is formulated here for a rank-one saddle point. Extension to a minimum point or higher rank saddles is straightforward. The Hamiltonian of the system is decomposed into a series expansion with a formal parameter ε of perturbation, which we set ε = 1 after all the following calculation is done: H=

∞

εν Hν(0)

(18)

ν=0 (0)

The zeroth-order H0 is an integrable Hamiltonian and the higher order terms (ν ≥ 1) will be regarded as “perturbation” to the system. One way to introduce the formal parameter ε is to scale the coordinates by (q, p) → (εq, εp) and H → ε−2 H, although it is not always the case [89]. In the former case, the zeroth order corresponds to the harmonic part 1

1 2 p1 − λ 2 q 1 2 + p 2 + ω 2 q 2 2 2 N

(0)

H0 =

(19)

=2

We have assigned number 1 to the reactive mode, with the imaginary frequency iλ. The other modes are vibrational modes with harmonic frequencies ω ( = 2, . . . , N). The higher order terms become homogeneous polynomials: ajk q1 j1 · · · qN jN p1 k1 · · · pN kN (20) Hν(0) = |j|+|k|=ν+2

N

with |j| = =1 j . It is convenient in the later calculations to introduce complex-valued normal mode coordinates by x1 =

λq1 + p1 , (2λ)1/2

x =

ω q − ip

, (2ω )1/2

ξ1 =

p1 − λq1 , (2λ)1/2

ξ =

p − iω q

(2ω )1/2

( = 2, . . . , N)

(21)

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145

then the harmonic part becomes (0)

H0 =λx1 ξ1 +

N

iω x ξ

(22)

=2 def

The action variables for the harmonic approximation are defined as I1 = x1 ξ1 and def

I = ix ξ . Note the relation x = −iξ ∗ , where the star denotes complex conjugate. The linear transformation [Eq. (21)] is a canonical transformation, so the Poisson bracket for two functions F and G is given by N ∂F ∂G ∂G ∂F {F, G} = − ∂q ∂p

∂q ∂p

=1

=

N ∂F ∂G ∂G ∂F − ∂x ∂ξ

∂x ∂ξ

(23)

=1

¯ which We introduce a coordinate transformation to a new set of variables (¯x, ξ), are called NF coordinates. The purpose is to reduce the number of the coupling ¯ expressed in the new coordinates. In the Hamil¯ x, ξ) terms in the Hamiltonian H(¯ tonian, the off-diagonal terms, that is, terms with different powers of x¯ and ξ¯ , denote couplings among the modes. This can be seen as follows: If the transformed Hamiltonian H¯ takes the following form: ¯ = λ¯x1 ξ¯ 1 + ¯ x, ξ) H(¯

N

=2

iω x¯ ξ¯ +

j j j a¯ j x¯ 1 ξ¯ 1 1 x¯ 2 ξ¯ 2 2 · · · x¯ N ξ¯ N N

|j|≥2

(24) where a¯ j ’s are the coefficients of the polynomial, then all of the new action variables I¯1 = x¯ 1 ξ¯ 1 , and I¯ = i¯x ξ¯ ( = 2, . . . , N) are constants of motion and the system is fully integrable. The Hamiltonian H¯ of the form (24) is called a full normal form. ¯ by Lie canonical perturWe construct the NF transformation from (x, ξ) to (¯x, ξ) bation theory [81]. Following the formulation by Dragt and Finn [90], we perform successive operations of Lie transformations: x¯ = exp(−εF1 ) exp(−ε2 F2 ) · · · exp(−εn Fn )x

(25)

ξ¯ = exp(−εF1 ) exp(−ε2 F2 ) · · · exp(−εn Fn )ξ

(26)

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where n is the order of perturbation and Fν (ν = 1, . . . , n) is an operation of Poisson bracket with a function fν : Fν = {·, fν }

(27)

¯ ε) is then given by ¯ x, ξ, The transformed Hamiltonian H(x, ξ, ε) → H(¯ H¯ = exp(εn Fn ) · · · exp(ε2 F2 ) exp(εF1 )H

(28)

¯ ε) and H¯ (μ) ¯ by If we define H¯ (μ) (¯x, ξ, x, ξ) ν (¯ H¯ (μ) = exp(εμ Fμ )H¯ (μ−1) = exp(εμ Fμ ) · · · exp(ε2 F2 ) exp(εF1 )H H¯ (μ) =

∞

εν H¯ (μ) ν

(29) (30)

ν=0 (μ) we obtain the following recursion formulas for H¯ ν :

¯ (μ−1) ν < μ : H¯ (μ) ν = Hν

(31)

¯ (μ−1) + Fμ H¯ (0) ν = μ : H¯ (μ) μ = Hμ 0 s ∞ Fμ (μ−1) ¯ (μ−1) + ν > μ : H¯ (μ) H¯ ν−sμ ν = Hν s!

(32) (33)

s=1

In the final Hamiltonian H¯ = H¯ (n) =

∞

ν=0 ε

νH ¯ (n) ν ,

the terms of order ν are

¯ (n−1) = · · · = H¯ (ν) ¯ (ν−1) + Fν H¯ (0) H¯ (n) ν = Hν ν =H ν 0

(34)

(ν−1)

is given from the because of Eqs. (31) and (32). In the above equation, H¯ ν (n) calculations in lower orders. Then we render the final Hamiltonian H¯ ν into a “desirable” form by setting Fν “appropriately.” The meaning of the desirable form and appropriate will be made more concrete in the following. (ν−1) is expressed in the form of a polynomial: In the present case, H¯ ν (ν) (x, ξ) = αjk x1 j1 x2 j2 · · · xN jN ξ1 k1 ξ2 k2 · · · ξN kN (35) H¯ (ν−1) ν |j|+|k|=ν+2 (ν)

where αjk is the coefficient of the polynomial. Since the definition of Fν in Eq. (27) (0) (0) (0) (0) gives Fν H¯ = {H¯ , fν } and H¯ = H has the form of Eq. (22), we can 0

0

0

0

dynamical reaction theory

147

¯ eliminate the terms with certain values of (j, k) from the final Hamiltonian H, by setting fν =

α(ν) jk (j,k)

γjk

x1 j1 x2 j2 · · · xN jN ξ1 k1 ξ2 k2 · · · ξN kN

(36)

where the denominator γjk is given by def

γjk = λ(j1 − k1 ) + i

N

ω (j − k )

(37)

=2

To obtain a well-defined transformation, the range of the summation in Eq. (36) is determined so that the polynomial series converges. For example, if we have γjk ≈ 0 for certain combinations of (j, k), then the corresponding coefficients of Eq. (36) take large values, resulting in divergence of the series. This is called the problem of small denominators [91]. This situation arises when the bath mode frequencies ω2 , . . . , ωn are nearly in the ratio of integers and called the “resonance” effect. Such values of (j, k) cannot be included in the summation of ¯ If all the coupling terms could be eliminated Eq. (36) and have to be kept in H. ¯ from H, we then have the full NF [Eq. (24)], with which all the action variables are conserved quantities. However, we often have to keep some coupling terms ¯ in H: H¯ =λ¯x1 ξ¯ 1 +

n

iω x¯ ξ¯ +

=2

+

a¯ j (¯x1 ξ¯ 1 )j1 (¯x2 ξ¯ 2 )j2 · · · (¯xN ξ¯ N )jN

|j|≥2 j j b¯ jk x¯ 11 x¯ 22

j kN · · · x¯ NN ξ¯ 1k1 ξ¯ 2k2 · · · ξ¯ N

(38)

d (j−k)∝d

which includes only those combination of the exponents (j, k) satisfying (j − k) ∝ d, where the integer vector d = (d1 , d2 , . . . , dn ) covers all the terms that cannot be eliminated. We denote the coefficients of polynomial in the coupling terms by b¯ jk . Note that, even with the coupling terms, the form [Eq. (38)] is simpler than the original form, because the number of terms in Eq. (38) is restricted by the condition (j − k) ∝ d. The form like Eq. (38) is sometimes called a “partial NF” (PNF). Which terms should be included in H¯ must be determined based on convergent property of the transformation as a function of order n. The (asymptotic) convergence can be checked by monitoring the energy error [86] E(n) = |H¯ (n) − H|, where H¯ (n) is the NF Hamiltonian truncated at the nth order and H is the true Hamiltonian. The detection of the nonlinear terms that must be kept in H¯ can be

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done by “bisection” method. We set a certain threshold c for the denominator γjk and include only terms with γjk > c in the summation of Eq. (36), to avoid small denominators. If we put very large value for c, there is no term in fν . The series is then ¯ = (x, ξ). trivially convergent, since the transformation is then the identity (¯x, ξ) (0) On the other hand, c = 0 means full NF, which is often divergent. Set clower = 0 (0) (m) (m) and cupper sufficiently large. If the intermediate value c = (clower + cupper )/2 (m+1) (m) (m) (m = 0, 1, 2, . . .) leads to convergence, set cupper = (clower + cupper )/2 with (m+1) (m) (m+1) (m) (m) (m+1) (m) clower = clower , otherwise set clower = (clower + cupper )/2 with cupper = cupper . (m+1) (m+1) Check the convergence with a new intermediate value c = (clower + cupper )/2 (m) and repeat this procedure. By doing so, cupper always leads to convergence and (m) (m) (m) clower to divergence, while their difference |cupper − clower | decreases as 2−m with increasing m. This procedure is repeated until there is only one combination of (m) (m) (j, k) between clower and cupper . (Note that if the system is well approximated by (m) (m) the full NF at the finite order n one investigates, clower and cupper become zero (m) (m) and there exists no term between clower and cupper . However, whenever there exist (m) (m) resonances (small denominators), one finds some values of clower and cupper between which one combination of (j, k) exists to make the series diverge.) Then we obtain one d = j − k that should be included in H¯ [Eq. (38)] in order to (m) have convergence. If we include all terms with γjk > clower in the summation of Eq. (36) excepting j − k = d, the series now becomes convergent. To simplify (“slim up”) the new Hamiltonian as much as possible, restart the procedure with (0) (m) ¯ c lower = 0 and c (0) upper = clower , to find the next term to be included in H. This procedure can be repeated until we have detected all the terms that must be kept ¯ in H. With the reduced number of the coupling terms, we can examine the value of each term along the trajectories. Depending on the system, it can happen that some of the terms have negligibly small effects. Then, we can further restrict the range of d in Eq. (38) by ignoring those small terms. If the range of the summation covers only those terms with d1 = d2 = · · · = dm = 0 for some m < N, the resulting Hamiltonian takes the following form: H¯ =λ¯x1 ξ¯ 1 + +

N

iω x¯ ξ¯ +

=2

b¯ jk x¯ 1 ξ¯ 1

j1

a¯ j (¯x1 ξ¯ 1 )j1 (¯x2 ξ¯ 2 )j2 · · · (¯xN ξ¯ N )jN

|j|≥2

· · · x¯ m ξ¯ m

jm

jm+1 j km+1 kN x¯ m+1 · · · x¯ NN ξ¯ m+1 · · · ξ¯ N .

(39)

j,k

With this Hamiltonian, the actions for the modes 1, 2, . . . , m are constants of motion. Thus, we can separate these modes and there remain (N − m) dofs to be

dynamical reaction theory

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investigated. The effective Hamiltonian for these (N − m) modes is given by H¯ eff (¯xm+1 , . . . , x¯ N , ξ¯ m+1 , . . . , ξ¯ N ; I¯1 , . . . , I¯m ) =λI¯1 +

m

ω I¯ +

=1

+

N

iω x¯ ξ¯

=m+1

j j j jm a¯ j I¯11 · · · I¯m x¯ m+1 ξ¯ m+1 m+1 · · · x¯ N ξ¯ N N

|k|+|j|≥2

+

j

j

j

k

kN m+1 jm m+1 a¯ jk I¯11 · · · I¯m x¯ m+1 · · · x¯ NN ξ¯ m+1 · · · ξ¯ N

(40)

j,k

with I¯1 = x¯ 1 ξ¯ 1 and I¯ = i¯x ξ¯ ( = 2, . . . , m) appearing as constant parameters. This completes our dimension reduction from an N-dof system to an effective (N − m)-dof system. If desired, one can introduce “real-valued NF coordinates” in parallel to Eq. (21):

B.

q¯ 1 =

x¯ 1 − ξ¯ 1 , (2λ)1/2

p¯ 1 =

ξ¯ 1 + x¯ 1 (2/λ)1/2

q¯ =

x¯ + iξ¯

, (2ω )1/2

p¯ =

ξ¯ + i¯x

(2/ω )1/2

( = 2, . . . , N)

(41)

Vibrational Energy Transfer in O(1 D) + N2 O → NO + NO

In this section, we present an analysis [88] on the mechanism of the fast energy transfer in the latter reaction: O(1 D) + NN O

→ NO + N O

(42)

where the prime symbol has been introduced to distinguish the two NO products. The normal form analysis starts with diagonalization of the quadratic part to give normal modes [Eq. (19)]. We have taken a linear-shaped equilibrium point of the O(1 D) + N2 O reaction that resulted in the most equilibrated vibrational distribution of two NO molecules [92]. The normal mode calculation yields three stretching modes and two bending modes [88, 92]. One of the bending modes is an unstable mode. One of the stretching modes can be assigned as NN stretching, while the other two are symmetric and antisymmetric stretching of two NO bonds. By the method presented in the previous section, we have reduced the number of coupling into 12 types in H¯ from the 1001 terms in the original Hamiltonian H. In the present system, all the terms in H¯ had j1 = k1 in Eq. (38), where the number 1 is assigned to the unstable bending mode. Therefore, I¯1 = x¯ 1 ξ¯ 1 is a local constant

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q−2

r2

q−3

q−3

q−4

q−4 r1

r1 rNN

rNN

Figure 8. Relations between the normal form coordinates (¯q2 , q¯ 3 , q¯ 4 ) and three nuclear distances, viewed from two different directions.

of motion. This means that the motion along mode 1 is decoupled from the others. Similar situations had been found in former studies [59, 62, 65, 82, 93–100]. It is attributed to the fact that there can be no resonance between imaginary and real frequencies [101]. Furthermore, the NN stretching and the other bending modes are only weakly coupled to the two NO stretching modes. This allows analysis as a 2-dof system consisting of the two NO stretching as a first approximation. To provide an idea for the normal form coordinates, we draw in Fig. 8 the normal mode coordinates in the three-dimensional space spanned by the nuclear distance of N–O, N –O , and N –N denoted by r1 , r2 , and rNN , respectively. We can see that the coordinate transformation is approximately a linear transformation near the origin, corresponding to the normal mode transformation, while the axes show curved nature in the outer region due to nonlinearity. Note that the true transformation is performed in 10-dimensional phase space. In the figure, we only show the threedimensional section by fixing the other coordinates (¯q1 , q¯ 5 , p¯ 1 , . . . , p¯ 5 ) to zero. The coordinates q¯ 4 and q¯ 3 correspond roughly to the symmetric and antisymmetric stretching of the two NO bonds. Note, however, they also have projection on rNN and the dependence on the latter has a curved shape. After introducing these curved coordinates, the system reduced to a 2-dof system, enabling analysis by Poincar´e surface of section. The surface revealed a normal mode-type structure with a slight distortion. The energy transfer between the two NO bonds was thus explained by beating mechanism. The details of beating mechanism are provided in Ref. 88. V. BIFURCATION AND BREAKDOWN OF NHIM: THE ORIGIN OF STOCHASTICITY OF PASSAGE THROUGH RANK-ONE SADDLE In Sections III and IV, we have described our recent developments that enable us to analyze the phase space structure of many-body systems in terms of remnant of invariants and dimensional reduction in order to address the question how the

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systems “climb” from the potential wells to a hypersurface, the so-called transition state, which lies between the reactant and the product states. In this section, we will focus on the question how one can extract the TS as a no-return dividing hypersurface from the multidimensional phase space by presenting our recent understandings on the conditions for its existence. In particular, we will discuss the possible fates of the no-return TS as the system energy increases above a rank-one saddle. These include bifurcation and possible mechanisms for the breakdown of TS implying the spoiling of the one-dimensional nature of the reaction coordinate. In the low-energy regime above the saddle point energy where the passage dynamics is completely regular (i.e., integrable) and the normal hyperbolic invariant manifold (NHIM) simply consists of unstable quasiperiodic orbits, the versatility of the no-return TSs and the reaction paths in the phase space has been well examined in evaluating the accurate reaction rates in isomerization reactions of six-atom cluster [93, 96] and of the isomerization of the HCN/CNH [65, 99], ionization of a hydrogen atom in crossed electric and magnetic fields [59], and also the escape of asteroids from Mars [102]. However, reactions in nature can generally take place in a wide range of energies above the saddle in which the dynamics changes from harmonic and integrable to fully chaotic and nonintegrable. A schematic picture of the dynamics hierarchy [87, 93, 103] above the saddle are shown as follows (see Fig. 9a). Suppose we consider the Hamiltonian of an n-dof system near a rank-one saddle: H=

n 1

i=1

2

p2i + ωi2 qi2 +

ji +

ajk q1 j1 · · · qn jn p1 k1 · · · pn kn

(43)

ki >2

where (q1 , p1 ) are the reactive normal modes with imaginary frequency ω1 = −i|ω1 | and (qi , pi ) , i ≥ 2, are the bath normal modes with real frequencies ωi . The ajk ’s denote the expansion coefficients of the anharmonic terms. 1. Harmonic Region At sufficiently small energy above the saddle point energy, the Hamiltonian, Eq. (43), is well approximated by the normal mode (harmonic) Hamiltonian H

n 1

i=1

2

p2i + ωi2 qi2

(44)

The simple normal mode dynamics ensures that the reaction mode q1 is completely separable from the other coordinates, supporting the conventional idea that the surface defined by q1 = 0 provides the no-return TS.

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(a)

(c)

p'3

p'2

p'1

Strong– chaotic (stochastic)

Weak -chaotic Nonlinear quasiperiodic and bifurcation Harmonic (normal mode)

q'1

Product Reactant

es

th Ba

q'1

Reactive mode

od

...

Effective potential

Esad

q'3

q'2

m

(b)

q1

Saddle

(d)

p3

p2

p1

q2

Center

q3

Center

...

...

p"3

p"2

p"1

q"1

Saddle

q"2

q"3

...

Chaotic

Figure 9. (a) A schematic showing the hierarchy of transition dynamics above a rank-one saddle. The potential profile along the reactive and bath modes is shown by the solid and dashed lines, respectively. For energy sufficiently close to Esad , the dynamics is well approximated by the normal mode Hamiltonian Eq. (44), and the no-return TS can be simply defined by q1 = 0. As the energy increases to the nonlinear quasiperiodic region, the harmonic picture breaks down and q1 = 0 can no longer serve as a no-return TS. Nevertheless, one can still perform a sequence of Lie canonical transformation to obtain the fully normal form Hamiltonian Eq. (45) that provides us with the simple geometrical structure as shown in (b) in the saddle region. The NHIM defined by p¯ 1 = q¯ 1 = 0 and its stable/unstable manifolds are denoted by the dark black dot and the dashed lines, respectively. This simple hyperbolic geometry on the (p¯ 1 , q¯ 1 ) plane allows us to extract the no-return TS q¯ 1 = 0 (shown as thick black lines) easily. Nonlinear couplings between the reactive and bath dofs in the quasiperiodic region can also introduce bifurcation of the NHIM. Part (c) shows an example of pitchfork bifurcation of the NHIM and the corresponding “effective potential.” In (c), the no-return TS, which originates from the two NHIMs shown by dark black dots on the (p1 , q1 ) plane, can “shift” away from the saddle. In the weak-chaotic region, nonlinear resonances among the bath modes can lead to the onset of chaos in the bath dofs as shown in (d). In this case, one can study the effect of chaotic motions in the bath modes on the robust existence of reactive invariant of motion and no-return TS in terms of the partial normal form Eq. (7).

2.

Nonlinear Quasiperiodic Region

As energy increases, the anharmonic terms in Eq. (43), which couple different dofs, become nonnegligible leading to the breakdown of the harmonic picture. Nevertheless, there exists a certain energy range above the saddle energy where almost all the dofs of the system locally maintain their action variables approximately constants of motion in the saddle region. More precisely, the Hamiltonian Eq. (43) in this energy regime can be transformed locally into the full normal form

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H¯ as ¯ q(p, q), p(p, ¯ H(¯ q)) =

n

ωi I¯i +

i=1

j j a¯ j I¯11 I¯22 · · · I¯njn

(45)

ji ≥2

where the transformed action variables are given by I¯1 = i(p¯ 21 /|ω1 | − |ω1 |¯q12 )/2

(46)

for the “new” reactive modes (¯q1 , p¯ 1 ) and I¯i = (p¯ 2i /ωi + ωi q¯ i2 )/2

(47)

with i ≥ 2 for the bath modes (¯qi , p¯ i ). Also the a¯ j ’s denote the coefficients of the anharmonic terms. Since the Hamiltonian Eq. (45) is independent of the angle variables (i.e., cyclic), all of the transformed action variables I¯i are constant of motion and therefore the system is fully integrable locally in the saddle region. The crux that leads to the full normal form is to generate a sequence of nonlinear Lie canonical transformations presented in the region of a rank-one saddle [59, 82]. This construction provides us with a new phase space geometrical structure, the NHIM, that can be regarded as a generalization of “saddle” in many-dof phase space. This building block enables us to define a robust no-return TS as a dividing hypersurface to decompose the multidimensional phase space into the distinct regions of reactant and product. Moreover, the orbits that asymptotically approach to (leave from) the NHIM construct the stable (unstable) invariant manifolds, which form the boundary of the reaction paths in the phase space through which all reactive trajectories necessarily follow. The geometrical picture in the saddle region implied from the full normal form Hamiltonian Eq. (45) is summarized in Fig. 9b. On the other hand, it is presumed in the setting of the normal form Hamiltonian Eq. (45) (i.e., with the local constant of motion I¯1 = i(p¯ 21 /|ω1 | − |ω1 |¯q12 )/2) that the geometrical structure associated with the reactive modes (¯q1 , p¯ 1 ) is simply hyperbolic as shown in Fig. 9b. This means the existence of only a single hyperbolic fixed point on the (¯q1 , p¯ 1 ) plane in the saddle region. However, Pechukas and his coworker [104, 105] first showed in the late 1970s that the no-return TS, termed periodic orbit dividing surface (PODS) at that time, for the collinear triatomic exchange reaction HX + H (with X = H, Cl, F) (a 2-dof system) undergoes a bifurcation starting at certain energy above the saddle in which the stability of the PODS changes from unstable to stable, leading to a serious overestimation of the reaction rate. They also found numerically that new PODSs emerge away from the saddle that can serve as new “reaction bottleneck” that controls the rate of reaction. Moreover, several experimental evidences indicate the existence of a topological

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change in the phase space geometry in the saddle region in many-dof system, such as in the decomposition of vibrationally excited ketene [106]. Recently, we presented a new method [103] to study the topological changes in the reactive dof and the corresponding bifurcation mechanisms of NHIM and noreturn TSs for many-dof systems. The basic idea is to generalize the sequence of Lie canonical transformation to obtain a different normal form Hamiltonian from Eq. (45) without the restriction of simple hyperbolic geometry for the reactive modes (see the schematic picture in Fig. 9c). It is found that the control parameters for the bifurcation are composed of the total energy and the transformed action variables of the bath dofs defined in the phase space. In particular, as long as the transformed action variables of the bath dofs are approximately constants of motion in the saddle region, the method enables us to predict in what circumstance the bifurcation of TS happens and what kind of topological changes occur. We note here that the bifurcation of NHIM, or in other words, the bifurcation of unstable/stable periodic orbits, can take place without the presence of chaos when the system is completely integrable. The topological changes associated with the reactive modes are due to the nonlinear coupling with the bath dofs. We also expect that a bifurcation cascade of NHIM in the saddle region should occur as the energy, being one of the control parameters, keeps increasing above the saddle, which can lead to the transition to chaos as in the case of most nonlinear systems [107, 108]. In Section V.A, we will demonstrate our method of predicting how and when the NHIM and the no-return TS bifurcate in terms of the noncollinear H2 + H exchange reaction with zero total linear and angular momenta (a 3-dof system). 3.

Weak-Chaotic Region

The discussion above so far concerns the identification of the no-return TS in the phase space for energies at which the reaction system remains completely integrable locally in the saddle region. However, the full integrability of the system in the saddle region, that is, the existence of n local constants of motion for an n-dof system [cf. Eq. (45)], can be spoiled by the nonlinear resonances as energy increases. In this case, can the no-return TS still be defined when the original Hamiltonian can no longer be transformed to the fully integrable normal form Eq. (45) and what are the effects of chaotic motion on the definability of the noreturn TS that enables us to calculate the correct transition rate and predict the fate of saddle crossing? To make the following discussion simple, we assume that no bifurcations of NHIM or topological changes (as discussed above) take place in the reactive dofs. The breakdown of integrability of the system can be understood in terms of the “small-denominator problem” appearing in the perturbation series of the Lie canonical transformation that brings the original Hamiltonian Eq. (43) to the full normal form Eq. (45). In our case of reaction dynamics across a rank-one saddle

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[cf. the Hamiltonian Eq. (43)], the resonance conditions are in general of the form m1 ω 1 +

n

mi ωi ≈ 0

(48)

i=2

with the integers mi (i = 1, . . . , n) not all equal to zero. Since the frequencies ω1 = −i|ω1 | associated with the reactive mode are purely imaginary and ωi (i ≥ 2) are real for the bath dofs, it is then expected (see, for example, Hernandez and Miller in their semiclassical TST studies [85]), at least for not so high energies, that the resonance condition cannot be satisfied among both the reactive and bath dofs, and therefore validates the canonical perturbation theory in the quasiperiodic region discussed above. However, as energy increases, resonances among the real frequencies ωi (i ≥ 2) of the bath dofs may start to take place (i.e., the resonance condition ni=2 mi ωi ≈ 0 can be satisfied), which result in the onset of chaos in the bath dofs and the destruction of constants of motion I¯i with i ≥ 2 (see Fig. 9d). On the other hand, we note from Fig. 9b that the identification of no-return TS relies only on the existence of constant of motion associated with the reactive mode, but not on those from the bath dofs. Therefore, one can generalize the full normalization scheme (i.e., the sequence of Lie canonical transformation that casts the original Hamiltonian into the fully integrable normal form) to the “partial” normalization scheme in which solely the reactive dof is normalized such that it can be separated from the bath dofs. In contrast to the full normal form Eq. (45), the partial normal form Hamiltonian (in double-prime notation to distinguish it from the full normal form) is given by n p2 ω2 qi2 i H (I1 , pb , qb ) = ω1 I1 + + 2 2 i=2

+f1 (I1 , pb , qb ) + f2 (pb , qb )

(49)

2 where I1 = i(p2 1 /|ω1 | − |ω1 |q1 )/2 is the transformed action variables of the re action dof and (pb , qb ) denote the transformed bath modes (pi , qi ) with i ≥ 2. The function f1 denotes all the anharmonic terms that couple the reactive action I1 with the transformed bath dofs (pb , qb ) such that f1 = 0 when I1 = 0, whereas the function f2 denotes the anharmonic terms involving the transformed bath dofs (pb , qb ) only. The anharmonic terms f1 and f2 can in general contain nonlinear resonances that cause the onset of chaos and nonintegrability in the bath dofs (see Fig. 9d). Since the partial normal form Hamiltonian Eq. (49) is independent of the angle variable θ1 , only the reactive action variable I1 is conserved in the saddle region. This ensures the existence of no-return TS provided that the original Hamiltonian can actually be transformed into Eq. (49), that is, if I1 (p (p(t), q(t)), q (p(t), q(t))) is invariant under the time evolution (p(t), q(t))

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governed by the original Hamiltonian in the saddle region. The partial normal form Hamiltonian also allows us to scrutinize the effects of chaotic motion in the bath dofs on the existence of constant of motion in the reactive modes, or in other words, on the definability of NHIM and no-return TS. Moreover, since chaos in the bath space can only occur if the dimensionality of the bath modes is greater than 1, this means that the definability/breakdown of no-return TS by chaos in the bath dofs is a new phenomenon inherent to many-dof system (i.e., system with (n ≥ 3) dofs). In Section V.B, we will demonstrate using a three-dof model system of saddlecrossing dynamics the robust existence of no-return TS as the motion of the bath dofs changes from regular to chaotic. In particular, it is found that the transformed action variable I1 strongly persists as invariant in the saddle region even when isolated stochastic layers emerge in the phase space (pb , qb ) of the bath dofs. The robust persistence of the reactive action variable with the corresponding hyperbolic structures (i.e., NHIM and its stable/unstable) at moderately high energies above the saddle has also been reported from the numerical studies of the isomerization of six-atom clusters [93, 96]. 4.

Strong-Chaotic (Stochastic) Region

As the energy further increases, the nonlinear resonances become strong enough to create global chaos in the bath space. This can destroy almost all the constants of motion (i.e., invariant torus) in the bath modes. Moreover, the robust invariant action variable associated with the reactive mode starts to break down and the one-dimensional nature of the reaction coordinate, which is separable from the bath dofs even in the weak-chaotic region, does not hold when global chaos arises in the bath space. It was also evident from the numerical study of six-atom cluster isomerization [93, 96] that the no-return TS in the phase space breaks down in the high-energy region. Moreover, the experiment on ketene decomposition [109, 110] demonstrated that the monotonic increase in the rates ceases at the high energy indicating the destruction of the invariants of motion associated with the reactive modes. In the strongly chaotic region, the partial normal form Eq. (49) introduced above can still provide us with the physical insights to understand the breakdown mechanism of the no-return TS and the corresponding reactive invariant of motion [87]. In Section V.B, we will show that the transformed action variable I1 does not lose its invariance completely in the saddle region when global chaos emerges. Instead, the invariance of I1 breaks down only locally where the instability of the bath motions, which can be quantified by the local Lyapunov exponents (LLEs), becomes comparable to those of the reactive mode. In other words, the breakdown of the no-return TS is originated from the breakdown of normal hyperbolicity. Therefore, the concept of “vague” TS may still be defined for the regions of the phase space where the instability of the bath dofs is smaller than those of the reactive direction. The complete breakdown of the concept of no-return TS is

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expected for extremely high energy above the saddle when the unstable periodic orbits are dense in the bath space with instabilities comparable to those of the reactive mode almost everywhere. A.

Bifurcation of No-Return TS: Noncollinear H2 + H Exchange Reaction

We first present our recent method [103] to predict the topological change along the reactive dof and to scrutinize the definability of no-return TS under the occurrence of the bifurcations. The idea is to relax the restriction of simple hyperbolic geometry for the reactive mode by transforming the original Hamiltonian Eq. (43) into a particular “partial” normal form H in which only the bath dofs are normalized (here the single-prime notation is used): H (p1 , q1 , Jb ) =

n 1 2 2 + p1 − |ω1 |2 q1 ωi Ji 2 i=2

(50)

+ g1 (p1 , q1 , Jb ) + g2 (Jb ) 2 where the subscript b means the bath modes and Ii = (p2 i /ωi + ωi qi )/2 with i ≥ 2 are the transformed bath action as in the case of full normal form (see Eq. (47)). The two functions g1 and g2 contain the anharmonic terms of the “partial” normal form variables (p , q ), and g1 is defined so that g1 = 0 when p1 = q1 = 0. Although Eqs. (50) and (49) are both termed partial normal form Hamiltonian in this section, we emphasize here that they correspond to partial normalization procedures applying to different sets of dof, namely, the bath dofs are normalized in Eq. (50) for the investigation of bifurcation in the reactive mode, whereas solely the reaction dof is normalized in Eq. (49) for the study of the effects of chaotic motions in the bath space. As in the case of the full normal form Eq. (45), H in Eq. (50) is fully integrable in the saddle region with n constants of motion, E and Ji with i = 2, . . . , n. However, the reactive mode in H is not restricted to be simple hyperbolic in contrast to the full normal form [cf. Eq. (46)]. In particular, g1 containing the higher power terms in (p1 , q1 ) with coefficients depending on Jb that determines the phase space topological changes in the reactive mode. With the bath actions Jb being invariants of motion in the saddle region, H in Eq. (50) can be regarded as a one-dimensional Hamiltonian of the reaction mode (p1 , q1 ) with n − 1 control (or external) parameters of bifurcation composed of E, and any (n − 2) independent variables from the set of n − 1 bath actions Jb . In particular for 2-dof systems, the total energy E is the only control parameter of bifurcation. The phase space portrait on the (p1 , q1 ) plane, such as the number of stable/unstable fixed points, can be changed from one set of control parameters to another. Moreover, we note that the validity of the transformation to the partial normal form Eq. (50) relies on the fact that the Jb actually preserve as approximate invariants along the time evolution of the original Hamiltonian Eq. (43).

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We apply the above partial normal form procedure to the 3-dof noncollinear H2 +H exchange reaction with zero total angular and linear momenta. The Hamiltonian for the noncollinear case is given by p2r1 + p2r2 pr pr cos γ pγ sin γ pr1 pr2 + 1 2 H= − + mH mH r2 r1 mH 2 pγ 1 1 cos γ + V (r1 , r2 , γ) + + 2− 2 mH r1 r1 r2 r2

(51)

where r1 and r2 are the two H–H distances, γ the bending angle between two H–H bonds, and mH the hydrogen mass. In this subsection, the units for energy, action, length, mass, and time are set to be eV, 5.4 × 10−2 eV fs, Bohr radius, amu, and 5.4 fs, respectively. The potential V (r1 , r2 , γ) is taken from Ref. 111, which has a rank-one saddle with Esad = 0.396 located at the saddle point r1 = r2 = 1.701 and γ = π. The Hamiltonian Eq. (51) is first Taylor expanded around the saddle point and then transformed to the partial normal form [Eq. (50)] up to the 12th order in the power of (p , q ). In order to properly elucidate the phase space geometry in the remote regions from the saddle, we found that it is crucial to employ the Pad´e approximants [87] to the power series expansions in obtaining the partial normal form. Moreover, we choose E and J3 (the transformed action for the bending motion of the two H–H bonds) as the control parameters of bifurcation. The changes in the phase space portrait on the (p1 , q1 ) plane for different values of control parameters are shown in Fig. 10a for the range of E and J3 in which the transformation to the partial normal form Eq. (50) is valid; that is, the transformed actions J2 and J3 are actually conserved approximately along the time evolution of the original Hamiltonian Eq. (51). Also shown in Fig. 10a are the effective potentials corresponding to the phase space portrait of each type. Four different topological structures are identified and labeled as type I–IV. In type I, the system follows a simple hyperbolic geometry in (p1 , q1 ) in which there is only a single unstable fixed point, that is, a single NHIM, and therefore the no-return TS is simply given by q1 = 0 (shown as the thick dark line). In the type II case, a set of two new unstable/stable fixed points emerge in q1 at q1 > 0 and q1 < 0, giving rise to two metastable states in the saddle region. This topological change, however, does not affect the location of the no-return TS defined at q1 = 0. For the type IV portrait, the hypersurface defined by q1 = 0 can no longer serve as a no-return TS due to the seriously “local” recrossing by the trajectories belonging to the intermediate state at q1 ≈ 0. Instead, a pair of new no-return TSs in the remote regime emerge that are in general originated from the unstable fixed points on the (p1 , q1 ) plane with the highest value in the effective potential. The type III portrait exists in between types II and IV along the “smooth” topological change of the

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(b) q'1

IV (a)

p'1

I

I

III

1

I

q'

2 J3' 0 10

II (c) q'1 p'1

IV

q'1

1

J3'

1

q'

3

p'1

II

I

Effective potential

Figure 10. (a) Four different phase space portraits emerge for the 3-dof noncollinear H2 + H exchange reaction. On the (p1 , q1 ) plane, the unstable/stable fixed points, the separatrices, the noreturn TSs, and some representative trajectories are represented by dark/gray dots, dashed lines, thick dark lines, and arrowed solid lines, respectively. Also shown in (a) are the 1D effective potential corresponding to each type of phase space portrait. Bifurcation diagrams and the corresponding phase space portrait as a function of J3 at energy (b) E = 0.55 and (c) 0.63. The unit of J3 is 10−2 eV s. In the bifurcation diagrams, the thin dark and gray lines denote the location of the unstable and stable fixed points in q1 , respectively. The thick dark lines represent the locations of the unstable fixed points with highest effective potential that give rise to the no-return TS (reaction bottleneck) that is expected to dominate the reaction rate. At E = 0.55, the simple hyperbolic geometry persists until an inverted saddle node bifurcation occurs at J3 ≈ 17. At E = 0.63, there is no topological bifurcation for the values of J3 considered, but along the transition from type II to IV, the no-return TS starts to migrate away from q1 = 0.

phase space in which each of the three unstable fixed points with the same effective potential can give rise to a no-return TS. Figure 10b and c presents the bifurcation diagram and the corresponding phase space portrait on the (p1 , q1 ) plane as functions of the partial normal form action of bending motion J3 at two different energies E = 0.55 and 0.63. The location of the unstable/stable fixed points and the no-return TS in q1 with the classification of topological types I–IV are shown by thin dark/gray and thick dark lines, respectively. At E = 0.55, we found that the transformed action J3 remains to be invariant through the course of reaction governed by the original Hamiltonian Eq. (51) except for the moments when the system just enters/leaves the saddle region. Therefore, the partial normal form Eq. (50) enables us to predict in what circumstance which type of topological bifurcation takes place at this energy regime.

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On the other hand, the situation at E = 0.63 starts to change. While for most of the trajectories in the saddle region the nonlinear bending actions J3 behave fairly well as approximate invariants, the J3 demonstrates chaotic fluctuations transiently for a small set of trajectories (figures not shown here). Although it is known that, for the cases where no bifurcation takes place, the no-return TS still robustly persists even when the bath actions cease to behave as approximate invariants [59, 93, 96, 101], one can no longer identify a priori the unpredictable changes of position and bifurcation of the no-return TSs for this small set of trajectories due to the chaotic time evolution of the control parameter J3 (t). We note that this phenomenon is inherent only to (n ≥ 3)-dof systems whose number of the control parameters of bifurcation is more than one. B.

Robust Persistence of No-Return TS and Its Chaotic Breakdown

In this subsection, we investigate the definability of the no-return TS under the influences of the chaotic motion in the bath space. In order to clearly distinguish the effect of chaos from the bifurcation phenomenon described in Section V.A, we consider the following 3D model Hamiltonian as a prototype of isomerization reaction whose parameters are chosen such that no topological changes and bifurcation occur in the energy range of interest: H=

1 2 1 (p1 + p22 + p23 ) + a1 q12 + a2 q14 + (ω22 q22 + ω32 q32 ) + g(q1 , q2 , q3 ) (52) 2 2

where the anharmonic coupling between the reactive mode q1 and the bath dofs q2 and q3 is given by g(q1 , q2 , q3 ) = e−α

sad (q −qsad )2 1 1

βsad q22 q32 + γ sad q12 q22 + q32

well1 (q −qwell1 )2 1 1

+ e−α

well2 (q −qwell2 )2 1 1

+ e−α

βwell1 q22 q32 + γ well1 q12 q22 + q32 (53)

βwell2 q22 q32 + γ well2 q12 q22 + q32

with parameters a1 = − αsad =

35 , 75

1 , 16

a2 =

2 , 1875

q1sad = 2,

αwell1 = αwell2 = 1,

ω2 = 1,

βsad = 8,

ω3 = 0.809

γ sad = 0.75

q1well1 = −q1well2 = 14.1421

βwell1 = βwell2 = γ well1 = γ well2 = 1

(54)

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The ratio between the normal mode frequencies of the bath dofs, ω2 and ω3 , is chosen approximately equal to the golden mean in order to avoid linear resonance. The imaginary frequency associated with q1 in the vicinity of the saddle is estimated as ω1 −0.924i. The nonzero value of q1sad aims at avoiding specific symmetry of the potential energy function in q1 . The Hamiltonian Eq. (52) is transformed to the partial normal form Eq. (49), in which only the reactive mode is normalized, up to the 15th order in the saddle region. In terms of the partial normal form Eq. (49), we can look at the change of dynamics inside the NHIM (defined by p = q = 0 where the no-return TS is originated from) as the energy E increases above the saddle. The dynamics of the bath dofs inside the NHIM is described by the 2D Hamiltonian H (I1 = 0, p2 , p3 , q2 , q3 ) = E. It is well known [108] that the phase space dynamics of 2D systems can be visualized by the Poincar´e surface of section (PSOS). In Fig. 11a–c, we construct the PSOS projected on the (q3 , p3 ) plane with the conditions q2 = 0 and p2 > 0 for E = 0.025, 0.05, and 0.15 above the saddle point, respectively. For both E = 0.05 and 0.15, it is expected that the fully integrable normal form picture Eq. (45) breaks down due to the appearance of stochastic layers (weak chaos) in Fig. 11b and global chaotic regions (strong chaos) in Fig. 11c. This implies that the dynamics of the bath modes become nonintegrable and the constants of motion J¯ 2 and J¯ 3 [cf. Eq. (45)] that exist at the quasiperiodic region are destroyed at high energies. Let us first look at the weakly chaotic case at E = 0.05 where only isolated hyperbolic periodic orbits (HPOs) and small stochastic region exist. In Fig. 11c, a few reactive trajectories with different initial conditions in the reactive mode and approximately the same amount of energy distributed into each bath dof are projected on the (q1 , p1 ) plane, where (q1 , p1 ) are the reactive momenta and coordinate defined in Eq. (52). These trajectories are evolved numerically using the original Hamiltonian Eq. (52). The inset of Fig. 11d shows that local recrossing occurs near q1 = p1 = 0 such that the hypersurface q1 = 0 cannot serve as a noreturn TS. However, after transforming to the partial normal form coordinates and momentum defined in Eq. (49), these reactive trajectories follow simple hyperbolic dynamics in the saddle region and the recrossing is “rotated” away as shown in Fig. 11e. The invariance of the normalized action I1 is shown in Fig. 11f, which implies the robust existence of no-return TS q1 = 0 even though the dynamics of the bath modes is weakly nonintegrable and chaotic. Intuitively, the strong persistence of the invariant of motion I1 in the weakly chaotic case can be understood from the fact that the HPOs in the bath space are still isolated and are not dense enough to create instabilities that are comparable to those of the reactive direction. This means that the condition of normal hyperbolicity still holds and therefore nonlinear resonances between the reaction and the bath space are not yet strong enough to destroy the reactive invariant of motion. However, as energy keeps increasing with the onset of global chaos, one can expect that

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p"3

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p"3 0.4

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0.05

q1

–0.05

0.05

q"1

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(e)

0

0.05

q"1

(f)

Figure 11. The PSOS on the (q3 , p3 ) plane for the dynamics of the bath dofs inside NHIM (defined by p1 = q1 = 0 and I1 = 0) in terms of the partial normal form Hamiltonian Eq. (49) with (a) E = 0.025, (b) E = 0.05, and (c) E = 0.15 demonstrating regular dynamics, weak chaos, and strong chaos, respectively. A few reactive trajectories in the weakly chaotic case E = 0.05 are projected on the normal mode variables (p1 , q1 ) defined in Eq. (52) (d) and on the partial normal form variables (p1 , q1 ) defined in Eq. 49 (e). The insets of (d) and (e) show that the recrossing in the saddle region is “rotated’ away in the partial normal form picture. (f) The robust persistence of the invariant motion of the reactive mode I1 and the validity of the partial normal form Eq. (49) against the weakly chaotic motion in the bath space.

the dense distribution of HPOs in the bath space should lead to the breakdown of normal hyperbolicity and definability of no-return TS. In order to look closer into the breakdown mechanism in the high-energy regime, it is again useful to extrapolate the canonical perturbation series in obtaining the partial normal form Eq. (49) in terms of the Pad´e approximant. This allows us to understand how and where the partial normalization scheme diverges in the multidimensional phase space. For any given perturbation expansion of a physical quantity A (e.g., A can be q1 , I1 , p3 , etc.) in the partial normal form picture as functions of the normal mode momentum and coordinates (p, q) up to the nth order: A (p, q) = 2 A0 (p, q) + · · · + n An−2 (p, q)

(55)

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(b) –iI"1 0.001

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15

t

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(c) 0.005

–0.005

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Re(λ2)/Re(λ1)

–0.005

0.8 0.5 0.4

t

Figure 12. Part (a) shows the projection of a few reactive trajectories at E = 0.15 onto the partial normal form (q1 , p1 ) plane with Pad´e approximation. The inset of (a) magnifies the center region to show the loss of invariance for the stable/unstable manifold and the no-return TS (see the text). The dashed lines correspond to the stable/unstable invariant manifolds predicted from the partial normal form Eq. (49). Part (b) shows the fluctuations and singularities in the time evolution of the transformed action I1 for the trajectory shown in the inset of (a). (c) The appearance of the strong singularities in I1 , and therefore the undefinability of the no-return TS, is closely related to the breakdown of normal hyperbolicity, shown by the shaded regions, which is characterized by the moments when the instability of the bath space Re(λ2 ) and the reactive mode Re(λ1 ) are comparable.

with i (i ≥ 2) keeping track of the power in (p, q). The Pad´e approximant A[N,M] with N + M + 1 ≤ n − 2 is a rational approximation to A [112] defined by A[N,M] ≡ PN ()/PM ()

(56)

where PN () and PM (), whose coefficients are functions of (p, q), are polynomials of order N and M in , respectively. Previous studies on the nonlinear resonances in vibrational dynamics (i.e., resonances among real frequencies inside a potential well) show that the poles of the Pad´e approximant have the physical significance corresponding to the nonanalyticity of A (p, q) and to the resonance (chaotic) regions of the dynamical system [112]. In the following, all physical quantities in the partial normal form picture will be extrapolated by the [N = 6, M = 6] Pad´e approximant. We now consider the high-energy regime at E = 0.15 where the bath dofs are strongly chaotic as shown in Fig. 11c. We plot in Fig. 12a a few reactive trajectories (p1 (p(t), q(t)), q1 (p(t), q(t))) at this energy evolving according to the original Hamiltonian Eq. (52) with evenly distributed bath mode energies. One can see that the Pad´e approximant of p1 and q1 picks up several singularities of the canonical transformation “locally” in the phase space. There are more singularities for the trajectory that has the smaller amount of energy distributed to the reactive

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mode (i.e., trajectory closer to the stable/unstable manifold). It is because when more energy is distributed to the bath modes, the dynamics of the bath space becomes more chaotic leading to a higher probability of resonance between the bath and the reactive dofs. Moreover, trajectories away from the stable/unstable manifolds cross the saddle relatively “faster” and reduce the chance to resonate. The inset of Fig. 12a shows that not only the stable/unstable manifolds (denoted by the dashed lines) predicted by the partial normal form Eq. (49) cease to be invariant but also the hypersurface defined by q1 = 0 stops to serve as a no-return TS. The breakdown of the simple hyperbolic geometry in the reactive mode (p1 , q1 ) can also be seen from the fact that the transformed action I1 is no longer a constant of motion in the saddle region. We show in Fig. 12b the fluctuation and singularities of I1 (p(t), q(t)) for the trajectory shown in the inset of Fig. 12a. To establish the relation between the breakdown of the no-return TS and the chaotic properties of the bath space, we employ the local Lyapunov exponent (LLE) analysis to the reactive trajectory closest to the NHIM shown in the inset of Fig. 12a using the Jacobian method developed by Hinde and Berry [113]. The real part of the LLE measures the instability of the dynamics locally in time and tells us how hyperbolic the system is along the trajectory. Figure 12c shows the ratio of the real part of the two largest LLEs, Re(λ2 )/Re(λ1 ), in which λ1 and λ2 represent the instability along the reactive and bath modes, respectively. Apart from the time of entering and exiting the saddle region, one can see that there is a close correspondence between the strong singular peaks in the action I1 and the moments when Re(λ2 ) and Re(λ2 ) become comparable, that is, the breakdown of normal hyperbolicity due to the appearance of hyperbolic dynamics in the bath space (shown by the shaded regions in Fig. 12c). Furthermore, the value of I1 changes after passing the shaded regions indicating that energy can flow from the bath dofs to the reactive mode and vice versa. There are also some weak singular peaks that are outside of the shaded regions at which the condition of normal hyperbolicity still holds. These small singularities may simply be the spurious poles in the Pad´e approximant that has no physical significance since the action value I1 remains the same before and after these peaks. Our results suggest that the destruction of one-dimensional nature of the reactive mode and the no-return TS (p1 , q1 ) could be caused by the resonance between the unstable reactive mode and the hyperbolic structures emerged in the bath spaces. The appearance of hyperbolic structures (or imaginary frequencies) in the bath dofs with real unperturbed frequencies ωi (i ≥ 2) are higher order effects that cannot be discussed by the linear resonance condition [Eq. (48)]. A detailed study of the nonlinear resonance between unstable modes with imaginary frequencies and its effect to reaction dynamics (e.g., that can also occur between the unstable directions of saddles with higher rank) remains to be an open challenge and will be discussed elsewhere. In this section, we have scrutinized the origin of stochasticity in transition dynamics and its consequences on the definability of a locally no-return TS. When the local recrossing events are solely caused by anharmonic but integrable effects

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in the Hamiltonian, the full normal form Hamiltonian Eq. (45) is able to “rotate” away the local recrossings and extract the no-return TS in the phase space. At higher energy where stochasticity and recrossing are caused by chaotic and nonintegrable effects in the bath space, the full normal form fails to resolve the stochasticity and recrossing, while the partial normal form Hamiltonian Eq. (49) is introduced to extract the correct “reactive” invariant of motion and no-return TS. We have shown that the corresponding no-return TS extracted from the partial normal form Eq. (49) persists robustly even though the bath modes become weakly chaotic. However, the definability of no-return TS started to be ruined in the region where the breakdown of normal hyperbolicity occurs, and the systems become completely nonintegrable, that is, with energy being the only constant of motion. However, the initial breakdown of normal hyperbolicity is shown to take place only locally around the NHIM and the stable/unstable manifolds. Therefore, a “vague” TS can still be defined in the region of the phase space where normal hyperbolicity holds.

VI.

CONCLUSIONS

Statistical theories are based on a uniform distribution in the phase space. The motion is assumed to be chaotic enough to generate stochastic nature. Given an initial condition, the system starts to wander around all the places in the phase space and, after sufficient time, every phase space point can be visited by the system with equal probability. There is no structure to distinguish one region of the phase space from another. Geometrical structure is an opposing concept to this assumption. Structures like invariant manifolds prohibit the system from penetrating from one region to another. If an initial condition is on one side of the invariant manifold, the system will be confined in that region. Then we can make “prediction” about the fate of the system at least in a coarse-grained level of narrowing the region of the phase space that can be visited by the system. In this sense, the conventional TST make two seemingly contradictory assumptions. On one hand, the system must be thermalized in the well and all the initial conditions are distributed uniformly. On the other hand, the system should possess a clear structure that divides the space into the reactant and the product sides. Once the system crosses the boundary surface from the reactant to the product side, it is “predicted” to remain there and never cross the surface again. Analysis by partial NF has solved this apparent contradiction. Even if the motion in the well or in the bath space around the saddle point is chaotic, the imaginary frequency along the reactive mode at the saddle point is off-resonant with the other modes. This enables the separation of the dynamics of the reactive mode from the others, leading to the regularity in the reactive mode. Even for high enough energy to make the vibrational modes chaotic, the regularity along the reactive mode is maintained up to a certain energy. In addition to giving mathematical ground for the existence of a no-recrossing hypersurface (TS), the NF analyses went further

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to reveal the bifurcation and the chaotic breakdown of TS for much higher energy. The convergence properties of different types of NF (partial or full) elucidated a hierarchical structure of the dynamics crossing over the reaction barrier. The transition from one level of the hierarchy to another is understood in terms of the dynamical concept of “resonance” appearing in the NF procedure. The NF analyses also revealed the existence of dynamical structures in well regions, which had been considered to be simply uniform in the traditional rate theory. The improvement in convergence by the -time evolution method discovered hidden constants of motion, whose level sets make invariant manifolds, in a seemingly chaotic region in a deep potential well. This may lead to nonuniform distribution in the well region, and therefore to non-RRKM behavior of the reaction rate, which can be analyzed by the methods presented here. In conclusion, the NF analyses and the accompanying dynamical concepts reviewed in this chapter offer great tools for understanding the nonuniformity of the phase space of a chemical reaction going beyond the statistical theories. The NF method, however, is not a panacea. The series expansion often suffers from divergence, especially when the energy of the system becomes high and the nonlinearity grows. There is no ensuring that the dimension reduction procedure by NF always reduces the system dimension to 2. The dimension reduction may stop after reducing to a three- or more dimensional system, whose dynamics cannot be easily visualized by Poincar´e surfaces. The failure of NF is usually associated with an emergence of highly chaotic motions, but it must be taken carefully. Since the NF is based on the perturbational expansion with the zeroth order being usually a harmonic approximation, its failure implies only that the dynamics of the system is “far from harmonic,” which may be chaotic but not necessarily so. It does not exclude the possibility of the appearance of new structures that are very different from harmonic oscillators. What we can learn from the divergence of the NF is yet an open question. Moreover, since the NF utilizes Taylor expansions at a stationary point, its validity is limited in some convergence region around the stationary point. A chemical reaction often involves multiple stationary points (saddles and minima) in the intermediate region, as well as the reactant and the product regions as the dissociation limits of the reaction intermediate. We can make expansions around each of the stationary points and obtain understandings of what happens in the vicinity of each stationary point. The connections among them as well as to the reactant and the product limits are then the next issue. Methodologies and concepts other than those reviewed in this chapter may be desired for understanding these issues. References 1. J. I. Steinfeld, J. S. Francisco, and W. L. Hase, Chemical Kinetics and Dynamics, Prentice-Hall, New Jersey, 1989. 2. R. D. Levine, Molecular Reaction Dynamics, Cambridge University Press, Cambridge, 2005.

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ERGODIC PROBLEMS FOR REAL COMPLEX SYSTEMS IN CHEMICAL PHYSICS TAMIKI KOMATSUZAKI1,2 , AKINORI BABA1,2 , SHINNOSUKE KAWAI1 , MIKITO TODA3 , JOHN E. STRAUB4 , and R. STEPHEN BERRY5 1 Molecule

& Life Nonlinear Sciences Laboratory, Research Institute for Electronic Science, Hokkaido University, Kita 20 Nishi 10, Kita-ku, Sapporo 001-0020, Japan 2 Core Research for Evolutional Science and Technology (CREST), Japan Science and Technology Agency (JST), Kawaguchi, Saitama 332-0012, Japan 3 Department of Physics, Faculty of Science, Nara Women’s University, Kitauoyahigashimachi, Nara 630-8506, Japan 4 Department of Chemistry, Boston University, 590 Commonwealth Avenue, SCI 503, Boston, MA 02215, USA 5 Department of Chemistry, The University of Chicago, 929 East 57th Street, Chicago, IL 60637, USA

CONTENTS I.

Introduction A. Ergodicity B. Mixing C. Multiplicity of Ergodicity in Complex Systems D. The Ergodic Problem in Real Systems II. Origin of Statistical Reaction Theory Revisited A. Traditional Ideas of the Dynamical Origin of Statistical Physics 1. Birkhoff’s Individual Ergodicity Theorem 2. Requirement of Ergodicity B. Issues on Openness and/or Inhomogeneity C. New Developments in Dynamical System Theory D. Biomolecules as Maxwell’s Demon III. Ergodicity in Isomerization of Small Clusters IV. Exploring how proteins wander in state space using the ergodic measure and its application A. The Kinetic Energy Metric as a Probe of Equipartitioning and Quasiequilibrium Advancing Theory for Kinetics and Dynamics of Complex, Many-Dimensional Systems: Clusters and Proteins, Advances in Chemical Physics, Volume 145, Edited by Tamiki Komatsuzaki, R. Stephen Berry, and David M. Leitner. © 2011 John Wiley & Sons, Inc. Published 2011 by John Wiley & Sons, Inc.

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B. The Kinetic Energy Metric as a Probe of Internal Friction C. The Force Metric as a Probe of the Curvature in the Energy Landscape D. Extensions of the Ergodic Measure to Internal Energy Self-Averaging E. Probing the Heterogeneity of Energy Flow Pathways in Proteins V. Extracting the Local Equilibrium State (LES) and Free Energy Landscape from SingleMolecule Time Series A. Extracting LES from Single-Molecule Time Series B. Revisiting the Concept of Free Energy Landscape C. Extracted LES of a Minimalistic Protein Model at Different Temperatures D. Outlook VI. Future Perspectives Acknowledgments References

I.

INTRODUCTION

How many variables or parameters are required to reveal the process of the evolution or the changes of the states in complex systems such as complex chemical networks or proteins? Consider a system of N degrees of freedom interacting with the surrounding environment of M degrees of freedom. If M is zero, the system is regarded as being isolated and usually described in a microcanonical ensemble of constant energy E. On the contrary, the case of M being infinity corresponds to condensed phase dynamics with dissipation and fluctuation arising from the surrounding environment, which is often characterized by constant temperature T and a distribution of atomic friction coefficients. First, let us briefly review isolated reacting systems, that is, M = 0. The dynamic evolution takes place in the phase space of 2N dimensions at a constant energy E. In principle, one should be required to use 2N − 1 independent variables to describe the events. However, as described in Chapter 4, in the well-known statistical reaction theories such as transition state theories, the rate of reaction can be formulated in terms of a substantially smaller number of parameters. For the case of condensed phase systems, the relative ratio of the number of parameters required to describe the reaction rate per the actual dimension of the system is far smaller in condensed phase than for the isolated system. For instance, Kramers theory characterizes the effects exerted by the environment through the temperature T , potential of mean force, random force, and friction. The rate of reaction can be again formulated in terms of a substantially smaller number of parameters such as barrier height, friction, and temperature with a chosen “reaction coordinate.” What is the fundamental assumption that enables us to substantially reduce the actual dimension of the system to represent the rate of complex chemical reactions? The key concept is (local) ergodicity and the resulting separation of timescales; that is, the characteristic timescale to attain ergodicity just within the reactant states is significantly shorter than the timescale of the reaction from the reactant to the

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product states. As a result, the rate of reaction is found to be independent of the initial condition at the reactant state under constant temperature or energy. This is crucially relevant to the fundamental question of how complex, chemical and biological systems evolve or change their states in time. A.

Ergodicity

The best known definition of ergodicity often used in statistical mechanics is the property that the time average of a characteristic of an ergodic system is indistinguishable from the ensemble average for the distribution over all accessible points in the system’s phase space. More precisely, the time average of an arbitrary function f , which is complex-valued in general, defined in a space (mathematically a smooth manifold M) in the phase space of the system, is indistinguishable almost everywhere on the M from the ensemble average over all accessible points on the M. Here, the propagation in time t obeys the behavior of a dynamical system, denoted here by Ut , which maps a point on the M uniquely (one-to-one) to a point on the M while preserving the measure on the M through Ut . Expressed as an equation, it is described as

1 t→∞ t

f (x)dP(x) = lim M

t

f (Ut x(0))dt

(1)

0

where P(x) denotes a measure defined on the M that can be normalized (i.e., probability measure M dP(x) = 1), x are continuous variables defined on the M, and Ut x(0) is the time propagation of x(0) from 0 to t, corresponding to x(t ). Let us exemplify this concept in terms of several systems: A two-dimensional dynamical system well studied in the context of ergodicity may be the “stadium billiard," almost a circle but with the two semicircles separated by parallel straight lines. Almost all trajectories in this enclosure pass through the entire interior of the enclosure; the exceptions are trajectories perpendicular to the straight sections and the one trajectory that passes between the centers of the two hemicircles. The introduction of probability measure plays an essential role to establish the concept of ergodicity. It is because the integration in terms of the probability measure singles out such events of measure zero whenever the probability measure exists. For Hamiltonian systems, the probability measure is given by the phase space volume suitably normalized because the measure-preserving condition is guaranteed by the Liouville theorem. We discuss the case in which there is no such a measure to be normalized in Section II. The second example is a two-dimensional torus where the ratio of the two frequencies ω1 and ω2 is irrational, that is, ω1 /ω2 = / n/m (n, m: arbitrary positive integers). If the system satisfies this irrational condition, no trajectory can be closed on the torus (one calls such motions quasiregular) and every trajectory densely covers all the surface of the torus. Hence, we recognize the motion as being ergodic

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on the torus. It should be noted that the question of ergodicity depends on which space M one considers. In the case of a torus defined by two invariants of motion, there exists no other independent invariant of motion on the torus to decompose the M into disjoint sets. This implies that the invariants of motion on the torus are regarded as global (and trivial) invariants of motion on that space because no invariant of motion exists to divide that space into disjoint sets. Note, however, that if one considers the M as the whole phase space of constant energy, the invariants of action surely prevent the system from wandering through accessible phase space almost everywhere. The system does not behave as ergodic in the whole phase space of constant energy. The third example is a one-dimensional harmonic oscillator. One can regard this as a system composed of a particle with a finite angular velocity confined to a circle where the particle moves along a diameter of the circle, bouncing elastically each time it reaches the circular boundary. This integrable system also satisfies the condition of ergodicity; that is, irrespective of the initial condition on the circle, the system can cover all the accessible points and hence the time average of any function defined on the circle is equivalent to the ensemble average. Note that this is different from the case of two-dimensional torus whose ratio of the two frequencies is rational. In the latter case, depending on the initial condition on the torus, trajectories cover different regimes on the torus because of the difference in phase. What type of property must dynamical systems possess in order to be ergodic? It has been proved that at least for a time evolution Ut defined on the phase space X that is one-to-one and preserves the probability measure P (e.g., Hamiltonian systems), if either P(M) = 1 (more in general P(M\X) = 0) or P(M) = 0 holds for any subset M in X with M = U(−t) (M) (namely, if a set M is invariant under Ut in X and such an invariant set exists solely as global or empty), the system cannot have (nontrivial) invariants of motion to decompose the M into disjoint sets but can have only (trivial) global invariant of motion almost everywhere through the M. (Here, M\X denotes the subset of X that contains all the elements that do not belong to M.) Followed by several theorems, the resultant dynamics is known to satisfy the ergodic condition by Eq. (1) on the M [1]. The most important consequence here is that ergodicity implies neither the existence of chaos nor the loss of correlation in dynamics. (The more detailed discussions are given in Section II.) The notation U(−t) (M) above should be understood in general as an inverse image of Ut , that is, U(−t) (M) ≡ Ut−1 M = {x|Ut x ∈ M}

(2)

If the time evolution of the system is invertible, that is, U(−t) x can be defined uniquely for each x ∈ X, it coincides with the backward propagation of M: U(−t) (M) = {U(−t) x|x ∈ M}

(3)

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However, if, for example, two different points x and y are mapped into the same point z = Ut x = Ut y, the image U(−t) z of the point z cannot be well defined while the inverse image of {z} can be defined as a set U(−t) {z} = {x, y}. In either case, U(−t) (M) is a measurable set for any measurable set M if the time evolution Ut is given by a measurable map. B.

Mixing

Most systems that interest us in the fields of chemistry and biology are more or less related to the property called “mixing,” which results from the existence of chaos. The explanation of mixing is rather simple: suppose coffee in a cup; add milk to the coffee in a ratio of, for example, 70% coffee and 30% milk by volume. After stirring the (mixed) solution many times enough to mix them up, whenever one takes any arbitrary fraction of the solution, the ratio of the coffee and the milk one will find is 70% versus 30%. This situation is represented in general as follows: Suppose two arbitrary subsets A and B in the phase space X and the inverse image U(−t) B of the subset B. The mixing condition is formulated mathematically as lim P(A ∩ U(−t) B) = P(A)P(B)

t→∞

(4)

This equation means that the probability that an arbitrary point x in A will end up in B after t iterations (t is considered to be an effectively infinite number) [the left-hand side of Eq. (5)] is just the same as that of finding the B in the whole X and independent of the position of A and B in X [the right-hand side of Eq. (5)]: limt→∞ P(A ∩ U(−t) B) = P(B) P(A)

(5)

A less restrictive condition is called weak mixing, which states that the longtime average of the difference between P(A ∩ U(−t) B) and P(A)P(B) vanishes: 1 t lim P(A ∩ U(−t ) B) − P(A)P(B) dt = 0 (6) t→∞ t 0 If a system is strong mixing, it satisfies the condition of weak mixing. The converse is not true. Intuitive interpretation of the weak mixing is that the mixing condition P(A ∩ U(−t) B) ≈ P(A)P(B) for very large t is satisfied for “most of the time” [1] since exceptional instances are wiped away by the process of averaging. Any weak mixing transformation [satisfying Eq. (6)] directly results in ergodicity [2]. Assume that B is an invariant set with respect to the time propagation Ut , that is, U(−t) B = B. Take A to be a complement to B so that P(U(−t) B ∩ A) = 0. We substitute this in the left-hand side of Eq. (6), and obtain 1 t |P(A)P(B)| dt = P(B)P(A) = (1 − P(A))P(A) (7) 0 = lim t→∞ t 0

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Thus, P(A) must satisfy either P(A) = 1 or P(A) = 0. This means that weak mixing implies ergodicity (one can prove easily that strong mixing also does so). It should be noted that ergodicity does not necessarily imply mixing, or even weak mixing. As just introduced above, the typical system is an integrable Hamiltonian system such as a two-dimensional torus whose ratio of the two frequencies is irrational, or just the one-dimensional pendulum. The timescale for approaching the mixing state (“equilibrium state”) or that of losing memory of the initial condition or correlation in dynamics in the phase space is approximately regarded as the inverse of Kolmogorov–Sinai entropy [3]. In the literature on chemical physics, relatively little attention has been paid to the conceptual difference between mixing and ergodicity. It may be because most systems of chemical and biological interest are expected to be inherently subject, to some extent, to both chaotic and mixing properties of nonlinear systems. C.

Multiplicity of Ergodicity in Complex Systems

Ergodicity is a property that can be verified only if one can examine both time and (phase) space averages. However, an interesting challenge arises if the system of interest has a rough, complicated potential surface. The reason is that the system may explore local regions thoroughly on short timescales yet require much longer times to escape from one, such local region, and move to another. If the potential surface has two or more relatively deep local minima that are separated by high or very narrow saddles, even if the system can, in principle, pass over those saddles, such passages can be relatively rare events, compared to the frequency of exploring all the places in the region of one of those local minima. Consequently, it is not unusual to find that a complex system can display two or more degrees of ergodicity. On a fairly short timescale, the system may exhibit only local ergodicity, but on a sufficiently long timescale, the system can explore its entire accessible space and be fully ergodic. If the landscape is sufficiently complex, there may be more than two or even more identifiable stages to the evolution of ergodicity. An illustration of this behavior appears in small atomic clusters, particularly in the range of temperatures and pressures within which the cluster may exist as a “solid” or a “liquid,” with the two phases in dynamic equilibrium, like two isomers. Under these conditions, one can see each phase-like form for some well-defined time interval, easily long enough for the internal vibrational modes to equilibrate, yet the system passes from one form to the other in some random fashion. If one tests for ergodicity using an ensemble and a single dynamical system on a long trajectory, one can probe for this property on a short or long timescale. If one looks on a very long timescale, one sees a single kind of behavior that involves exploring the entire accessible phase space, including the solid and liquid regions. If, on the other hand, one looks at a relatively short timescale with the probe (which we shall discuss shortly), then one sees two distinct kinds of behavior. One is ergodic

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but only in the liquid region, and the other is ergodic in the solid-like region; the timescales on which one sees this kind of behavior are too brief for the system to be able to pass between solid and liquid. Presumably, the same kind of timescale separation holds for structural isomers that correspond to structures accessible to a molecule but only on a relatively long timescale. The demonstration of this behavior appears in the distributions of sample values of Lyapunov exponents, the values of the exponential rates at which neighboring trajectories diverge. If these are obtained from long trajectories, then the distributions are unimodal, centered around the single most probable value. However, if the distributions are taken from shorter trajectories, then they are bimodal, with one maximum for the clusters in the liquid region and another for those in the solid region [13, 14] (see also Section III). The other illustrative example may be free energy landscape. In Section V, it is indicated that the morphological feature of the landscape depends on a timescale of observation. The longer the timescale, the more the number of detectable metastable states decreases, and the smoother the landscape implied by the observation. D.

The Ergodic Problem in Real Systems

The traditional concept of ergodic behavior is derived from mathematical analyses that, in turn, treat infinitely long pathways and arbitrarily large ensembles. Physical systems are finite and many of those of great interest now are small, and the timescales on which we may wish to observe them can be very brief indeed. Hence, it is appropriate to introduce heuristic analogues of the rigorous properties of ergodicity and chaos, based on the system in question satisfying some chosen criterion based on a finite, perhaps very long, time interval. If the system satisfies the chosen criterion, we may safely treat it as if it were truly chaotic or ergodic, within time intervals shorter than that of the criterion. (Sometimes this behavior has been called “cryptoergodicity” or “cryptochaos.”) In chemistry, ergodicity has been one common central property that one assumes in establishing several theories such as reaction rate theory, free energy landscape, and so on. One has also known many cases, for example, non-RRKM kinetics and the intramolecular vibrational energy redistribution (IVR) problem, that do not satisfy ergodicity. However, one has paid little attention to validating the concept of ergodicity in systems in which it is probably valid, and furthermore little has been done to explore new insights concerning the system’s dynamics in terms of the concept. What we want to address here is what the appropriate tests or criteria should be, which enable us to use the concept of ergodicity, or, more precisely, to avoid invalidating the application of the concept of ergodicity in the problems of real systems in chemical physics. In that spirit, we are really asking for tests of cryptoergodicity, in the sense that we want to know when we can suppose

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a system appears, in whatever ways are significant for our investigation, to be ergodic. We are concerned not with the rigorous mathematical property but with the observable behavior of the system. It is expected that plausible tests of this property, cryptoergodicity, depend on the size of system or the condition on our knowledge about the system (e.g., whether we can know the equation of motion of the system or we can solely monitor a physical quantity of the system). The finiteness of timescale is even essential and rather inherent to a lot of phenomena of chemical interest. A typical illustrative example is a bimolecular reaction: Two molecules (reactants) collide with each other to form a metastable intermediate complex. After some time, the complex dissociates into a different set of two molecules (products). While the motion in the complex can be chaotic involving most of all degrees of freedom and subject to the issue of ergodicity, in the products and the reactants limits there are two separate molecules. Since the two molecules far apart cannot interact with each other, the system cannot be ergodic in all its dimensions before the formation and after the dissociation of the complex. The intermediate complex is the only form to be subject to ergodicity through the full dimension but the lifetime of the intermediate complex is finite. The finiteness and the value of lifetime in this case are determined by the system (not the problem of observation). Another example is a system that exhibits transitions among multiple well regions. The degree of chaos and ergodicity can be different for different wells. They are subject to the competition between the strength of chaos and the residence time of each well, and depend on the extent that the system can attain ergodicity (or rather cryptoergodicity) in that well. This is essential for heterogeneity to emerge in establishing cryptoergodicity (we will discuss this aspect in more detail for proteins in Section IV). The more important question to be addressed is what we actually learn from the concept of ergodicity about complexity of systems such as the question of what the system actually feels under a thermally fluctuating environment. Here, it must be noted that the introduction of the term “cryptoergodicity” is not only due to the limitation of observation but also inherent to the problems themselves whenever they invoke the change in their states. We will also come back to this issue from the viewpoint of open phase space in Section II.B. This chapter addresses the ergodic problems relevant to real complex systems from small-body systems such as atomic clusters to proteins. Here, we start with an overview, the historical background of the concept of ergodicity, and the implication of the concept in the sense of statistical mechanics in Section II. Then in Section III, we show how the local Lyapunov exponent distribution can unveil the ergodic property of inert gas clusters. This system may be regarded as representative of small-body systems, in contrast to systems with complex internal constraints, for example, proteins, but cluster dynamics is rich enough to start to discuss because clusters exhibit phase transition-like behavior even with small, finite number of degrees of

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freedom. However, when the system of interest becomes much larger than those, although one can still compute the local Lyapunov exponent distribution, the device of the local Lyapunov exponent distribution becomes almost impossible to use. In Sections IV, we turn to the so-called ergodic measure developed for elucidating the rate of self-averaging of physical observables and characterizing the timescale of quasithermalization, and show the existence of heterogeneous multiple timescales to attain ergodicity, depending on the moiety of a protein. In Section V, we review our recent studies on the other measure to evaluate attainability and multiplicity of ergodicity in complex protein systems when one cannot access the underlying equation of motion of the system but just a time series of certain physical variables of the system such as interdye distance. We present our recent progress in deepening our understanding of the free energy landscape at single-molecule level.

II.

ORIGIN OF STATISTICAL REACTION THEORY REVISITED

The most fundamental assumption of the statistical reaction theory is the separation of timescales; that is, the characteristic timescale for establishing equilibrium in the potential well is assumed to be much shorter than that for the reaction to take place. The chemical reaction proceeds while local equilibrium is maintained in the potential well. This makes it possible to apply the methods of the equilibrium statistical physics to chemical reactions. However, the recent development of theoretical and experimental studies on reaction processes reveals the necessity of going beyond the conventional statistical reaction theory [21, 22]. We consider the foundation and limitations of the statistical physics, especially its relevance for understanding reaction processes involving biomolecules. In the context of reactions, the following two features become crucial. First, reaction processes take place in open phase space regions in the sense that trajectories flow into and out of them, while the phase space is closed in the conventional statistical physics. Second, the system is inhomogeneous for reaction processes involving biomolecules, while it consists of identical particles in the traditional statistical physics. We will explain why these two features present serious issues concerning the foundation of statistical reaction theory. In this section, we start our discussion with a brief review of the traditional ideas on the dynamical origin of the statistical physics. Then, we go on to argue why the above two features of the reaction processes necessitate serious reconsideration on the foundation of statistical physics. Finally, we discuss recent development of the dynamical theory concerning the statistical physics such as Sinai–Ruelle–Bowen (SRB) measure and infinite ergodic theory, and present possibility of these new ideas in the study of reaction processes.

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Traditional Ideas of the Dynamical Origin of Statistical Physics

In the study on the mechanism of approaching equilibrium, Boltzmann introduced the model, now called the Boltzmann equation [24], using the oneparticle distribution P(p, q) defined on the phase space (p, q), where p and q are the momentum and the coordinate of the one particle, respectively. Under the assumption of molecular chaos, that is, the motions of molecules are supposedto be completely uncorrelated, he showed H-theorem, that is, the quantity H ≡ P(p, q) log P(p, q)dp dq monotonically decreases in time, indicating irreversible approach to equilibrium. His derivation of the H-theorem met the objection from Loschmidt, who asserted that the H-theorem contradicts the time-reversal symmetry of Newton’s equation of motion. In order to defend his derivation of H-theorem, Boltzmann introduced the ergodic hypothesis implying that H-theorem is relevant for a dominant part of the phase space. Their argument triggered the development of the theory of ergodicity, which is now well established in the sense of mathematics. Here, we give a brief explanation of the theory of ergodicity. The following discussion is not limited to the Hamiltonian systems, that is, the subjects of the traditional studies of the statistical physics. It is also applicable to dissipative systems since dissipative systems can have invariant measures, which are not the phase space volume. Thus, the argument can be applied to reactions involving biomolecules surrounded by an environment, in addition to unimolecular reactions of isolated systems. We follow the traditional argument for the foundation of statistical physics. Several good references exist both for mathematicians [35, 36] and for nonmathematicians [26, 27]. In statistical physics, the idea of ergodicity plays the role that corresponds to that of the law of large numbers in the probability theory [31]. In traditional statistical physics, observed values of the physical quantity are generally assumed to be equivalent to time averages over the infinite time interval. In order to apply the equilibrium statistical methods, these time averages should be independent of initial conditions. In order to justify the above idea of ergodicity in statistical physics from the standpoint of the dynamical system theory, the first thing to ask is whether time averages over infinite time interval exist or not. To approach this, we state Birkhoff’s individual ergodicity theorem. The theorem guarantees existence of time averages over the infinite time interval for physical quantities of a certain class.

1.

Birkhoff’s Individual Ergodicity Theorem

Suppose that the time evolution Ut with the time t is defined on the phase space X such that Ut preserves the probability measure P defined on X, that is, for any subset A of X, P(A) = P(U(−t) (A)) holds. Let us consider a physical quantity f (x)

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defined for x ∈ X, where f (x) belongs to the set L1 (P) of functions that satisfy the following condition: |f (x)| dP(x) < ∞ (8) X

This condition requires that the integral of the quantity f (x) in the region X+ ≡ {x ∈ X|f (x) > 0} and that in the region X− ≡ {x ∈ X|f (x) < 0} converge, respectively. Let x(t) ≡ Ut x denote the trajectory with an initial condition x(0) = x ∈ X. Then the following quantity exists: 1 fˆ(x) ≡ lim t→∞ t

t

f (x(t ))dt

(9)

0

for initial conditions almost everywhere concerning the probability measure P. Moreover, the function fˆ(x) is invariant under the time evolution Ut , that is, fˆ(x) = fˆ(Ut x), and f (x)dP(x) (10) fˆ(x)dP(x) = X

X

holds. According to the theorem, the time average fˆ(x) of the quantity f (x) exists for the trajectory with the initial condition x. Moreover, the invariance of the function fˆ(x) means that the time average fˆ(x) is constant for any initial conditions over each individual trajectory, hence the theorem is called the individual ergodic theorem. However, the time average fˆ(x) can take different values for different trajectories. Therefore, an additional requirement is needed to guarantee that time averages do not depend on initial conditions. This is the requirement of ergodicity. (Some references call it metrical transitivity, see, for example, Ref. 27.) 2.

Requirement of Ergodicity

Suppose that the time evolution Ut is defined on the phase space X such that Ut preserves the probability P defined on X. The evolution Ut is called ergodic if either P(A) = 0 or P(X\A) = 0 holds for any subset A of X with the property A = U(−t) (A), that is, A is invariant under Ut . We denote X\A the complement of A, that is, the subset of X that contains all the elements not belonging to A. For a time evolution Ut that satisfies the requirement of ergodicity, Birkhoff’s individual ergodicity theorem indicates that the time average of the physical quantity

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f ∈ L1 (P) equals its ensemble average for almost every initial condition x ∈ X, that is, fˆ(x) =

f (x )dP(x )

(11)

X

almost everywhere on X. This can be proved as follows. Denote A(a) ≡ x ∈ X|fˆ(x) ≥ a for an arbitrary value a. Both A(a) and X\A(a) are invariant subsets because of Birkhoff’s individual ergodicity theorem. Then, either P(A(a)) = 0 or P(X\A(a)) = 0 holds based on the requirement of ergodicity. Thus, for an arbitrary value a, either P(A(a)) = 1 or P(A(a)) = 0 holds. This means that P(A(a)) is discontinuous at some value aˆ , indicating that fˆ(x) takes the constant value aˆ almost everywhere. Moreover, this constant equals the ensemble average X f (x)dP(x). In the traditional argument of statistical physics, we consider the time evolution Ut under the Hamiltonian H. The measure-preserving condition is guaranteed by the Liouville theorem, that is, the probability measure P is given by the phase space volume suitably normalized, as long as Ut is defined on a certain compact subset X of the phase space. Assuming the requirement of ergodicity, Birkhoff’s individual ergodicity theorem indicates that the time averages of the physical quantities exist and do not depend on initial conditions, that is, the idea of ergodicity in the sense of statistical physics is justified. The following is a historical comment [27]. In the original idea, ergodicity meant that every point in the phase space was visited by a trajectory. However, it is impossible for a one-dimensional trajectory to cover the whole phase space of multiple dimensionality. Something one-dimensional cannot occupy all the points in a space of higher dimension. Therefore, the concept of ergodicity must be relaxed. Now, ergodicity is understood to mean that a trajectory covers the phase space densely, that is, it comes arbitrarily close to every point in the phase space. In Birkhoff’s individual ergodicity theorem, the condition f ∈ L1 (P) for the physical quantity f (x) is crucial. When physical quantities do not belong to this set, we can have a different situation. Also note that the situation differs completely for the cases with unnormalizable measures. These issues will be discussed later in Section II.C. In the mathematical formulation of ergodicity, the time averages are defined over the infinite time interval. For physical situations, however, the time averages must be taken over finite time intervals. We are thus led to the question, “To what extent is ergodicity attained in the physical sense?” This issue will be discussed in the next section. In physical problems, the correlation f (0)f (t) ≡ X f (x)f (Ut x)dP(x) is also of interest for the physical quantity f (x) in the set L2 (P) of functions that satisfy

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the following condition: |f (x)|2 dP(x) < ∞

(12)

X

Suppose that the correlation decays exponentially with the characteristic timescale tc . Then, the two values f (x) and f (Ut x) of the physical quantity f ∈ L2 (P) can be considered as independent as long as the time difference t is larger than tc . This enables us to obtain the central limit theorem for the physical quantity f ∈ L2 (P) [31]. The above argument leads us to another important property of the dynamical systems, that is, mixing. We call the time evolution Ut mixing, when the crosscorrelations f (0)g(t) ≡ X f (x)g(Ut x)dP(x) decay to zero for any physical quantities f, g ∈ L2 (P) with their ensemble averages equal to zero. We have presented the definition of mixing in terms of measure theory in Section I.B. To see the equivalence of this definition with the measure-based definition of Eq. (4), put f = χA and g = χB , the characteristic functions of the sets A and B: χA (x) =

1 0

(x ∈ A) (x ∈ / A)

(13)

Then, the correlation χA (0)χB (t) ≡ X χA (x)χB (Ut x)dP(x) is equal to the probability P(A ∩ U(−t) B), since the integrand χA (x)χB (Ut x) equals 1 when both x ∈ A and Ut x ∈ B hold, that is, x ∈ A ∩ U(−t) B, otherwise χA (x)χB (Ut x) = 0. Subtracting the averages χi ≡ X χi (x)dP(x) = P(i) (i = A, B), respectively, we obtain (χA (0) − χA ) (χB (t) − χB ) = χA (0)χB (t) − χA χB that approaches zero as the time t goes to infinity, indicating that χA (0)χB (t) approaches

χA χB . Thus, we obtain P(A ∩ U(−t) B) that goes to P(A)P(B) as t goes to infinity. B.

Issues on Openness and/or Inhomogeneity

Here, we consider the foundation of the statistical reaction theory especially for those reactions involving biomolecules. The following two features become important: an openness and/or inhomogeneity. The first issue is statistical properties within open phase space regions. In the traditional idea, the phase space is supposed to be compact, that is, closed and of finite volume. Moreover, trajectories do not flow into the phase space region and never leave it, thereby staying there for infinite time from the past to the future. The Liouville theorem guarantees that the measure-preserving property holds for the phase space volume, that is, the Lebesgue measure, and the probability measure is normalizable. In this sense, the phase space is closed in traditional statistical physics. On the other hand, in reaction processes, trajectories flow into and out of

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the phase space region or regions that correspond to the potential well or wells. In this sense, the phase space is open in the chemical reactions. In the chemical reactions, trajectories stay within the phase space region of a well only for a finite time interval. After entering the phase space region and staying there for some time, trajectories leave the region by going over a saddle and enter a new region, leading to chemical change. Thus, ergodicity in the statistical reaction theory concerns the question of the extent statistical statements are valid within finite time intervals. In the traditional theory of reactions, it is supposed that the trajectory visits almost everywhere in the phase space region in the well. If ergodicity in this local sense is satisfied, reaction processes become statistical and independent of specific initial conditions. A closely related question was presented recently as a criticism of the traditional understanding of ergodicity [26]. In the traditional understanding, it is supposed that the trajectory visits the phase space region densely. However, Gallavotti pointed out that for systems of many degrees of freedom, it takes too long in the physical sense for the trajectory to cover the whole phase space densely. In other words, for macroscopic systems, the traditional understanding of ergodicity is irrelevant as the foundation of statistical physics. Both the above arguments concern the necessity of introducing a criterion and a characteristic timescale so that we can estimate if ergodicity holds effectively in the physical sense. Such a criterion was proposed by Thirumalai and Straub [55, 56] called the ergodic measure. The quantity concerns fluctuation of time averages over finite timescales. If the fluctuation behaves consistently with the asymptotic behavior predicted by the law of large numbers, we can conclude that the statistical limit is effectively attained in the physical sense within finite timescales. Note that, in introducing such criteria, we do not need to require that each trajectory covers densely the whole phase space. Rather, we need to estimate whether the asymptotic limit in the sense of the law of large numbers is attained or not. The reason why we focus our attention on this point is the following. In the traditional discussion of ergodicity, we treat homogeneous systems consisting of large numbers of identical particles. In these systems, a trajectory does not need to cover the whole phase space densely to exhibit statistical properties predicted based on ergodicity. It only suffices to cover a representative region of the phase space. Because of the permutation symmetry in systems consisting identical particles, time averages over such a representative region can be almost the same as the time average over the whole phase space. Moreover, such a representative region can be much smaller than the whole phase space. The characteristic timescale for ergodicity to hold in the physical sense can be much shorter than the timescale to cover densely the whole phase space. The above argument leads us to the second issue that is, ergodicity for inhomogeneous systems. For biomolecules such as proteins, the above argument on a representative region is not readily applicable since these molecules tend to be

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heterogeneous in their amino acid sequences. Moreover, in reaction processes involving biomolecules, we consider statistical aspects not necessarily in the macroscopic scale but in mesoscopic scales. For example, Thirumalai and Straub have shown, using the ergodic measure, that the degree of attaining ergodicity differs depending on the parts of the protein [56]. Their results indicate the possibility that some parts of the protein still remain out of thermal equilibrium while other parts recover equilibrium. Thus, ergodicity in parts of the biomolecule is of interest as a possible tool to see nonequilibrium within a single molecule. Such nonequilibrium situations can play an important role in the functional behavior of biomolecules as we point out later in Section II.D. These arguments show that the ergodicity problem in the physical sense becomes even more important as we pay attention to biomolecules in mesoscopic scales. Then, openness and/or inhomogeneity become two key issues. C.

New Developments in Dynamical System Theory

Recently, new developments in the dynamical system theory offers some clues to investigate the issues related to ergodicity discussed in the previous sections. Here, we address two recent results, that is, the Sinai–Ruelle–Bowen (SRB) measure and an extension of the Birkhoff’s individual ergodicity theorem. First, we discuss the SRB and related measures [29–32, 34, 50]. In the traditional understanding of statistical physics, it is supposed that the phase space volume (exactly speaking, the Lebesgue measure) is the only relevant measure for statistical physics. However, in chaotic scattering processes, for example, fractal exists in the scattering events, which is singular with respect to the Lebesgue measure. In chaotic dissipative systems, a consideration of fractals becomes important due to the presence of strange attractors. These phenomena lead us to ask what the relevant physical measure is, in the sense that it corresponds to observation in experiments and numerical simulation. The SRB measure is the measure that is smooth along the unstable invariant manifold, while it is singular along the stable invariant manifold. For compact uniformly hyperbolic systems, it is proved that the SRB measure exists [49]. Its existence can be intuitively understood as follows. Suppose a typical distribution of initial conditions on the phase space in the sense that its Lebesgue measure is positive. Through the time evolution, the distribution is stretched repeatedly along the unstable manifold. Under these processes, nonuniformity of the distribution becomes less and less pronounced leading eventually to a smooth distribution. Along the stable manifold, to the contrary, folding processes make nonuniformity of the distribution more and more steep, eventually giving rise to a singular distribution. Suppose that we have arbitrary initial distributions that is typical in the sense that its Lebesgue measure is positive. The distribution approaches the SRB measures

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under the time evolution, that is, the SRB measure is the natural invariant measure, the measure that a typical distribution of initial conditions approaches under the time evolution. Moreover, it is conjectured that the SRB measure is structurally stable, that is, it is not sensitive to random noise or a change of the parameters of the system. In this sense, it is considered as the physical measure, that is, the measure based on time averages obtained by physical observation [31, 50]. The SRB measure is expected to give a clue to understand nonequilibrium phenomena such as turbulence [53]. The theory of SRB measure has also revived the argument between Boltzmann and Loschmidt, leading to the fluctuation theorem. The fluctuation theorem states that universal behavior exists in the ratio between the probability of increasing entropy and that of decreasing entropy [26, 33]. Thus, the theory of SRB measure opens a new research area in nonequilibrium physics from the viewpoint of the dynamical systems. The above discussion leads us to extend further the SRB measure to even wider situations. We should note that, in the requirement of ergodicity, whether ergodicity holds or not depends on which measure you use. Moreover, the phase space volume is not necessarily an appropriate invariant measure in chaotic scattering and systems with dissipation, as we have explained. Thus, we need to think of the question which measure we should use. The clue to answer this question is given by the existence of variational principles. The SRB measure can be characterized by the variational principle [29, 30, 34, 50]. This corresponds to the fact that equilibrium distributions are characterized through a variational principle as attaining the maximum of entropy or the minimum of the free energy. In this sense, the SRB measure enables us to extend the concepts of equilibrium distribution to nonequilibrium situations. Based on this similarity, a measure that can be characterized by the variational principle in general is called the Gibbs measure. The variational principle is formulated using Lagrange multipliers. The canonical distribution in equilibrium statistical physics is obtained by the variational principle under the constraint that the energy is given. We think further of the variational principle where the values of any physical quantities (not necessarily energy) are given by observation. This generalization introduces a new concept of measures, that is, the Gibbs measures. For open hyperbolic systems, Gaspard and Dorfman [52] introduced a measure that is characterized by the variational principle, that is, the Gibbs measure. This measure is concentrated on the saddles of the chaotic scattering, that is, the repeller in the phase space. Given an arbitrary typical distribution of initial conditions, the closer those trajectories approach the repellers, the longer they remain in the scattering region. In the asymptotic limit of an infinite timescale, the invariant measure is thus defined on the repellers in the phase space. The measure has a finite value only for scattered trajectories since only the scattered trajectories are counted. Chaotic scattering introduces the singular measure that is concentrated

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on the repellers. In this sense, the variational principle here means that observation of scattering trajectories uniquely singles out the relevant physical measure, which is not the Lebesgue measure. They show that the measure plays an important role in quantifying statistical properties of stationary events for open systems such as scattering and reaction processes where fractal structure becomes manifest in the invariant distributions. An interesting question arises if we can extend further the concept of the Gibbs measure to normally hyperbolic invariant manifolds (NHIMs). The NHIMs are manifolds where hyperbolicity on the normal directions is stronger than that on the tangential directions. Thus, the definition of NHIMs corresponds to extensions of repellers to multidimensional dynamical systems. Therefore, normal directions to the NHIM play the role of the reaction coordinate. On the other hand, the tangential directions to the NHIMs consist of vibrational modes, which can be coupled with each other and be chaotic, as long as their hyperbolicity is weaker than hyperbolicity along the normal directions. Any typical distribution of initial conditions in the initial state will approach the NHIM located near the saddle as these trajectories leave the well leading to the reaction. The nearer they approach the NHIM, the longer they take to leave the well. Thus, we can construct the measure on the NHIM similarly to that on the repellers. Second, we discuss an extension of the Birkhoff’s individual ergodicity theorem [28, 38–44, 47]. and its relation to nonstationary processes in reactions [45]. Recently, the Birkhoff’s individual ergodic theorem has been extended in the following two directions: (i) those cases in which the physical quantity f (x) does not belong to L1 (P) with normalizable probability measures P and (ii) those cases in which the invariant measure is not normalizable, that is, the cases that can be treated by what is known as the infinite ergodic theory [28]. For these cases, the concept of time averages is extended, and a new formulation of the law of large numbers is introduced. Then, an interesting new feature is that the asymptotic limit of time averages itself exhibits random fluctuation. Moreover, its distribution reveals a certain universal behavior. For example, Aizawa and his group have shown these universal characteristics for a class of one-dimensional maps and certain billiard systems [39–44]. The existence of universal fluctuation suggests that the statistical reaction theory can be extended to those reactions in which the traditional concept of ergodicity does not hold. Such cases can include the reaction processes in the mixed phase space where the reaction rate constant does not exist because of the fractional behavior such as power law in the distribution of the residence times, anomalous diffusion, and 1/f spectra. See the chapter 3 and Ref. [62–64]. The infinite ergodic theory can be important in those phenomena where extreme events play a crucial role in reaction processes. These days, extreme events in natural and social science receive an intense attention [25, 37] since these extreme events play a decisive role in phenomena such as earthquakes and great depressions,

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although they are rare. In particular, when the probability of extreme events is larger than that predicted by the Gaussian distribution, the predictions based on the Gaussian can lead to catastrophic disasters in the society. In considering extreme events in reactions involving biomolecules, existence of the gap is important between the characteristic timescale for reactions (as rapid as picoseconds for ligand binding or local conformational change) and that for biological functions (as slow as milliseconds to seconds or hours for protein folding, signaling, or transport). This wide gap in the characteristic timescales implies that even extremely rare events in terms of microscopic reactions can be considered frequent in the timescales of biological functions. This phenomenon is similar to the geological events, in which earthquakes are rare events in the characteristic timescale of individual human being while they are frequent on the timescale of geological events. Inspired by such a similarity, the term “protein quake” was coined for describing behavior of the protein [51]. These authors also noticed a hierarchical structure of substates and an associated distribution of bottlenecks that give rise to “broken ergodicity” and nonergodic behavior of the protein on a given finite dynamical timescale. Existence of common features in the protein and geological events suggests that the study of extreme events from the viewpoint of the infinite ergodic theory can lead to finding new universal aspects in nonequilibrium phenomena. In order to analyze reaction processes from the viewpoint of extreme events and their universality, we need to extend the study of Aizawa’s group to multidimensional dynamical systems. For example, Shojiguchi et al. have shown that nonstationary and power law behavior exists in systems where resonance overlap in the Arnold web is nonuniform and sparse in the well [62–64]. There is a possibility that the asymptotic distributions of physical quantities in such nonstationary systems exhibit the universal distribution. D.

Biomolecules as Maxwell’s Demon

In order for biomolecules to play a role in information processing, they must be under nonequilibrium conditions as the celebrated argument of Maxwell’s demon indicates [23, 57, 58]. Maxwell’s demon is a tiny existence of a molecular size, which can differentiate molecules, one from another, on the basis of a property such as energy. Maxwell showed that its existence would lead to violation of the second law of thermodynamics [23]. Now the commonly accepted view is that the fluctuation of equilibrium conditions invalidates the original argument of Maxwell [57, 58]. However, there is still a possibility that nonequilibrium conditions enable the demon to work its task of differentiating molecules, that is, a kind of information processing [57, 59]. In particular, the demon is studied based on the fractional behavior of dynamical systems although their studies are limited to systems of two degrees of freedom [60, 61].

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The question then arises how nonequilibrium conditions are maintained at the molecular level, and whether a dynamical mechanism exists that contributes to maintain nonequilibrium conditions. In order to investigate these questions, the theory of reactions should go beyond the traditional concept of ergodicity. This study will reveal an intrinsic dynamical mechanism of biomolecules so that the molecule is capable of exhibiting the ability to process information.

III.

ERGODICITY IN ISOMERIZATION OF SMALL CLUSTERS

Small clusters of atoms have emerged as very useful tools to help us understand how ergodic and chaotic behavior enter in the kinetics and dynamics, not only of their own motions but also of much more complex systems. This is partly because analyzing the behavior of a system of 3, 4, ...,10, ..., even to 50 or 100 particles is now a reasonable task with modern computing tools and partly because the complexity of the multidimensional configurational and phase spaces in which the particles move grows extremely rapidly with the dimensionality of the space, that is, with the number of degrees of freedom of the multiparticle system. Some of the aspects of ergodicity that have emerged from the study of clusters are as follows: the importance of the differences in behavior in different local regions of the multidimensional potential surface, the utility of local probes such as local Lyapunov exponents, and the time evolution of ergodicity, from local to global character. We can learn how to identify and characterize the specific directions in phase space that are responsible for the magnitude and direction of the local Lyapunov exponents, the components that are the primary local propagators of ergodicity. The Lyapunov exponents, particularly their local analogues (which we simply call “local Lyapunov exponents,” based on finite trajectories of some desired length), reveal the directions and extent to which a trajectory tends to carry a system away from its locality and hence the extent to which a trajectory moves to explore some different region of configuration and phase space. We remind the reader that Lyapunov exponents are the measures of how neighboring trajectories diverge or converge locally from one another, and that for Hamiltonian (conservative) systems, these appear in positive and negative pairs. The traditional concept of Lyapunov exponent is based on the average behavior over the full, accessible phase space. We begin this discussion with a short review of how we learn the different kinds of behavior in different regions of the potential surface. The first indication of this came from the observation that the positive Lyapunov exponents of the threeparticle triangular Lennard–Jones cluster, LJ3 , and the sum of those exponents, the Kolmogorov entropy, increase with the energy of the system, up to the range in which the system can just pass over the energy saddle of the linear configuration. In that energy range, the system behaves in a more ordered fashion than at slightly

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lower energies [4]. Another measure studied in that investigation was the effective Hausdorff dimension, the dimension of the space in which the three atoms move on a timescale consistent with observations, for example, nanoseconds, but brief compared to the time for mode coupling in nearly harmonic molecules, for example, milliseconds [5–7]. Simulations at low energies, corresponding to about 2–10 K, show Hausdorff dimensions of 3.1–3.5, as one would expect from the three normal modes of vibration of a triangular molecule such as LJ3 when its motions are essentially harmonic. The deviation from precisely 3 is a measure of the degree of mode coupling at those energies. However, at an energy corresponding to 18.2 K, the Hausdorff dimension is 5.9; the maximum possible is the number of degrees of freedom in phase space that are not individually conserved, which, for n is 6n − 10 or 8. Hence, the Hausdorff dimension tells us that this three-body system is already quite nonrigid at this energy, although it doesn’t quite have full freedom in its phase space. Likewise, the Kolmogorov entropy (K-entropy) or sum of Lyapunov exponents increases steadily at an accelerating rate from energies corresponding to about 1 K up to a maximum at an energy equivalent to 28 K, drops to a local minimum around 30 K, and then increases again. The drop occurs just at the energy that allows passage over the saddle at the linear configuration of the molecule [8]. In the fully chaotic liquid range, the n positive Lyapunov exponents λn increase according to a power law λn = αnβ . The slope α increases rapidly with increasing temperature or energy; the exponent β is essentially unity at all energies or temperatures [8]. This analysis also examined the way the K-entropy depends on the range of interaction between atoms; this range can be varied systematically if, as used in this work, one represents the interaction between pairs of atoms with a Morse potential, V (r) = exp[−2ρ(r − r0 )] − 2 exp[−ρ(r − r0 )]. A value of ρ of 3 corresponds to the longest range of pairwise potential known between two atoms in a diatomic molecule; a value of about 7 corresponds, likewise, to the shortest range exhibited by pairs of atoms in diatomic molecules. Short-range interactions give rise to very rough energy landscapes with extensive parts of the topography at high energies; long-range interactions give rise to smoother landscapes with deep, well-defined minima [9]. The study by Hinde et al. [8] showed that the Kentropy of three-particle clusters with Morse interactions between particles has an energy dependence that clearly distinguishes the systems with very long-range interactions from others with shorter ranges of interaction. Those with ρ of 3 have K-entropies that rise monotonically with energy and flatten at high energies; those with ρ of 5 or more have maxima in their K-entropies, as they move to regions of high potential energy and hence lower kinetic energy on their potential surfaces, as Fig. 1 shows. Other, closely related systems have revealed similar behavior. Linear triatomic clusters have larger maximum Lyapunov exponents than triangular clusters at the lowest energies at which the linear form can exist; but at higher energies, the

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Figure 1. K-entropy for three-body systems with Morse potentials of various ranges. The shortest range here is that with ρ of 7. The span from 3 to 7 is approximately that of the known diatomic molecules, when they are represented by Morse potential interactions. (Reproduced with permission from Ref. 8. Copyright 1992, by the American Institute of Physics.)

two are very similar [10]. If some of the energy of the system is in rotational motion, then the system tends to be less chaotic, as indicated by a lower maximum Lyapunov exponent than for the case of pure vibration. However, varying the energy in rotation can reveal periodic transitions between regular and chaotic motion [11]. This point was explored in more detail to reveal that the volume of phase space occupied by regular trajectories is a nonmonotonic function of the angular momentum and depends on the coupling between kinetic and potential energy [12]. The second way that atomic clusters have opened an approach to the study of ergodicity and chaos has been in the area of finding timescales for the establishment of ergodic behavior in ever larger regions of the energy landscape [13, 14]. The probe to find the range of exploration in this approach is the distribution of effective Lyapunov exponents for brief, moderate, and long time intervals, but always just of finite-time-based Lyapunov exponents, not extrapolated to infinite time as one would determine traditional Lyapunov exponents. Clusters are particularly useful for this because, given their small sizes, they can exhibit dynamic coexistence of different phase-like forms in equilibrium over ranges of temperature and pressure, whether solid and liquid or different solid forms. Typically, under such conditions

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Figure 2. Distributions of sample values of Lyapunov exponents for three-atom clusters (“Ar3 ” with Lennard-Jones potentials), taken from finite-path samples, at two temperatures. (a) 28.44 K; (b) 30.65 K; at 28.44 K, the system is below the linear saddle; at 30.65 K, it can pass over the saddle. The lowest distributions are based on 8192 time steps, and the successively higher sets are based on half the number of steps of the distribution below, so the second lowest are based on 4086 steps, and the highest on the shortest number, only 256 time steps. (Reproduced with permission from Ref. 14. Copyright 1993 by the American Physical Society.)

of coexistence, the residence time in one phase-like form is long relative to the time of vibrational periods or of thermal equilibration of the vibrational degrees of freedom [15, 16]. Figure 2 shows two sets of distributions of the sample values of the largest Lyapunov exponent for Ar3 from molecular dynamics simulations at two temperatures, 28.44 and 30.65 K. The lowest “curves” are based on 8192 time steps of 10−14 s; the next higher, on 4086 steps, and so on, to the highest, which is based on only 256 time steps. The crucial point is that for short times, and a suitable temperature, even the argon trimer shows a bimodal distribution of Lyapunov exponents. This is more vivid with Ar7 , for which Fig. 3 shows the distributions of sample values of the largest Lyapunov exponents for short trajectories, of only 256 steps, as functions of both the value of the exponent and the kinetic energy at which that value occurred. The essential point of these figures is the passage from a narrow, unimodal distribution at low energies, through a region of bimodal distribution, to a high-energy region where the distribution is again unimodal but broad. Figures 2 and 3 show how, for brief times, systems explore only local regions; but for longer times, they visit their entire accessible phase space. Moreover, with probes such as the one used here, we can determine the timescale for passage from localized behavior to global. Global studies reveal some of the characteristics of larger clusters, notably their phase behavior, but the information in the distributions of local Lyapunov exponents gives additional insight into coexistence of phases even for clusters of over a thousand atoms [17]. One recognizes intuitively, and Hessian matrices demonstrate, that different directions of motion play different roles in the multidimensional configuration space of a several- or many-particle system. One very recent development has

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Figure 3. Distributions of sample values of finite sample Lyapunov exponents for a seven-particle cluster with Lennard–Jones interactions (“Ar7 ”). The distributions are based on 256 time steps; they are expressed as functions of kinetic energy Ek and λ, the local value of the Lyapunov exponent. The distributions correspond to total energies of −0.355, −0.341, −0.328 and −0.300 × 10−13 erg. The units of λ are bits per 10−14 s and of Ek are 10−15 erg. (Reproduced with permission from Ref. 14. Copyright 1993 by the American Physical Society.)

explored this issue, with the goal of identifying the coordinates that play the most important roles in carrying a system from one local region to another [18]. This study uses a simple, Lennard–Jones cluster of three atoms as its model, in order to explore the distributions and participation ratio spectra of both traditional and local Lyapunov exponents. With even this very simple system, one can see that ergodicity develops on different timescales for different regions of phase space. Naturally, the regions most susceptible to unstable trajectories are those nearest to saddles. This particular study uses Gram–Schmidt vectors rather than the actual Lyapunov vectors, but the former are very close approximations to the latter, especially for very local investigations. This three-body system is a convenient device to begin to explore the kinds of information that one can extract from traditional and local Lyapunov exponents and the distributions of the latter. This is, in some

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ways, a consequence of the fact that this system, with nine degrees of freedom and seven constants of motion (three components of momentum and angular momentum, and energy), has only two pairs of nonzero Lyapunov exponents, which, of course, come in matching positive and negative pairs. We refer to the larger as λ1 and its most negative counterpart as λ18 , and the other two as λ2 and λ17 . The investigation evaluated not only the Lyapunov exponents themselves but also the inverse participation ratios [19, 20], which measure the number of degrees of freedom that participate in the direction associated with each Lyapunov exponent. The distributions of the local Lyapunov exponents narrow steadily, as the length or duration of the trajectory extends. The distributions are quite narrow for trajectories of 2000 or more time steps, but very broad for only 100 or 200 time steps. Some bimodality of the sort observed by Amitrano and Berry was also seen in this work. This behavior is clear in the distributions in Figs. 4 and 5, for the larger and smaller positive, finite-interval Lyapunov exponents and the corresponding inverse participation ratios. Low values of the latter indicate many of the modes are participating in the direction corresponding to that Lyapunov exponent. One can

Figure 4. Distributions of the larger Lyapunov exponent λ1 for ranges of sample time intervals l, in (a) and (b), and of the corresponding participation ratios Y1 . The participation ratio is a measure of the number of degrees of freedom that contribute to the direction of motion of each Lyapunov eigenvector. For (a) and (c), an amount of energy E = −1.58 was put initially into the symmetric stretching mode, and for (b) and (d), the same energy was put initially into the asymmetric bending mode. The shortest interval sampled was 100 time steps, indicated by the thin curve without any dot, the lowest in (a) and (b), and the longest, 4000 time steps, is the most peaked in all four panels. (Reproduced with permission from Ref. 18.)

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Figure 5. Distributions of the smaller Lyapunov exponent λ2 for the same ranges of sample time intervals as in Fig. 4. All the notations are the same as in that figure. The most significant difference here is the bimodality of the two lowest curves in (a), corresponding to the system being in either of the two regions for such short intervals. (Reproduced with permission from Ref. 18.)

also see that the asymmetric bending mode of this triangular system plays an earlier role in inducing chaotic behavior than does the symmetric bending mode, in the sense that the asymmetric mode couples with the symmetric stretch at lower energies than does the symmetric bend. A result that emerges from these calculations is a coupling, perhaps surprising, of the excited symmetric stretching mode and symmetric bending mode with the asymmetric bending mode. Symmetry strictly forbids this, but tiny round-off errors in computation are sufficient to create small perturbations that break the symmetry and enable the coupling of asymmetric and symmetric modes. Hence we can recognize the utility of local Lyapunov exponents as devices to help elucidate local dynamics, beyond the global features revealed by the traditional Lyapunov exponents. IV. EXPLORING HOW PROTEINS WANDER IN STATE SPACE USING THE ERGODIC MEASURE AND ITS APPLICATION The “complexity” of the energy landscape of proteins is responsible for the rich behavior observed in the dynamics of proteins [65–67]. The rugged energy surface arises from the presence of many energy scales in proteins due to the intrinsically

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heterogeneous nature of the systems [68]. The equilibrium and dynamical properties of proteins are thought to be determined by a temperature-independent multidimensional potential hypersurface consisting of many minima (conformational substates), maxima, and saddle points. That general view of solids and liquids has a long history, dating back to Eyring. However, the ambitious project to provide a more quantitative assessment of the character of the underlying hypersurface using computational simulations has established in the “inherent structure” theory of Stillinger and Weber [69] and the “conformational substates” view of Frauenfelder [70]. In this picture, the distribution of energies for the minima, the volume of the basins, and the distribution of barrier heights separating these substates determine the thermodynamics and dynamics of the system. This point has been confirmed by the disorder seen in X-ray crystallographic studies and in the wide distribution of timescales for protein motion seen in the ligand photodissociation/rebinding experiments of Frauenfelder and coworkers on heme proteins [71]. Beginning with the pioneering study of Czerminski and Elber [68], computational studies have provided an increasingly quantitative description of the distribution of minimum energy conformations, the rate of exploration of these conformations, and relation to observable properties such as free energies and relaxation for small peptides [72, 73], model proteins [74, 75], and atomistic models of larger peptides [76] and proteins [77]. Recently, there has also been a focus on the application of sophisticated measures of phase space structures, typically restricted in applications to small molecules of relatively few degrees of freedom, to larger molecules and peptides. A focus of particular interest is the identification of local modes in proteins that may couple selectively to a few specific protein modes but relatively weakly to the larger density of states of the surrounding protein and solvent. Leitner and coworkers have pioneered the application of a number of methods, originally developed for the study of energy transfer in solids, to vibrational energy and heat flow in proteins [78] (see Chapter 3). Those methods have been applied and extended by Straub and coworkers to identify mode-specific energy transfer pathways of amide I vibrations in small peptide-like molecules [79, 80], globular proteins [81], and porphyrin and heme groups [82, 83] (see Chapter 1). While applications to the study of energy flow in proteins have focused on dynamics in a constant temperature ensemble, there have also been significant experimental [84] and theoretical studies [85] focused on Hamiltonian (constant energy) flow in peptide-like molecules and small peptides. Significant developments enabling the experimental and theoretical study of biomolecules in the gas phase coupled with dramatic enhancements in computational power have led to the application of sophisticated methods for the study of phase space structures, previously restricted to the study of a few degrees of freedom systems and small molecules, to biomolecules [86]. A beautiful example can be found in the work of

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Farantos who applied methods for computation of periodic orbits to examine the phase space structure of the alanine dipeptide [87]. An extension of that approach has recently been applied to interpret vibrational spectra in proteins [88]. These applications demonstrate the significant potential for the future study of the phase space structure of biomolecular systems. A.

The Kinetic Energy Metric as a Probe of Equipartitioning and Quasiequilibrium

One approach to exploring the nature of the rugged energy landscape and the rate at which observable properties are sampled is through measuring the convergence of averages over dynamics trajectories using replica molecular dynamics – the generalized ergodic measure originally introduced by Thirumalai and Mountain [89–91] and applied to a wide range of systems including proteins [92, 93]. This technique to examine the rate of sampling kinetic energy and atomic force has been shown to be a useful analytical tool for investigating timescales for energy equipartitioning and conformational space sampling [92, 94]. Interestingly, similar measures have been developed in other fields with issues of broken ergodicity where a state of quasiequilibrium is established other than the canonical thermal distributions, including self-gravitating systems [95]. In this section, we review the theory of the ergodic measure applied to estimate the rate of self-averaging of physical observables and characterize the dynamics in phase space using replica molecular dynamics. To provide insight into the behavior of the ergodic measure, the kinetic energy metric is evaluated analytically for the Langevin model and the force metric is evaluated analytically for a system of normal modes. In each case, the rate of convergence is shown to provide a measure of fundamental properties of the system dynamics on an underlying energy landscape. Suppose we have an observable F that can be written as a function of time Fi (t) for the ith atom of a system of N atoms, such as the kinetic energy Fi (t) = mi v2i (t)/2. Writing the time average of Fi (t) as fi (t) and the average of fi (t) over all N atoms of the system as f¯ (t), we define the mean square difference of the individual fi (t) ’s from the average f¯ (t) as (t) =

N 1 [fi (t) − f¯ (t)]2 N

(14)

i=1

This is known as the fluctuation metric [94]. It can be shown that for an ergodic system after a short time the function (t) decays to zero as 1/t as (0)/ (t) Dt (see Fig. 6) [90]. The power law decay of (t) to zero at long times implies that the system is “self-averaging” and the slope is proportional to a diffusion constant for the exploration of the range of values (space) accessible to the variable F (t). This is a necessary, but not sufficient, condition for the system dynamics to be ergodic.

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U(r)

r

Ω(0)/Ω(t)

Ergodic

Broken ergodicity t Figure 6. Two trajectories depicted on the background of a rugged energy landscape (top) and the corresponding reciprocal ergodic measure (0)/ (t) for an ergodic system and a system demonstrating “broken ergodicity” (bottom).

The slope of (t) is proportional to the generalized diffusion constant D for the observable F that can be written D (0) = l2 /τ where (0) is the mean square fluctuation of the property F and τ is the timescale for taking a “step” of generalized mean square length l2 = (0) in sampling the fluctuations of the property F in phase space. In this way, the ergodic measure may be used to explore the rate of exploration of phase space in complex systems characterized by a rugged free energy landscape. Imagine that phase space is divided in two regions A and B by an impassable barrier. Given enough time any trajectory will explore all of the allowed phase space. For a set of trajectories started in region A, (t) will decay to zero, and the property F (t) will appear to be self-averaging. However, unless we have started one of our trajectories in region B we cannot know that the partition exists and the system in not ergodic. Therefore, the decay of (t) to zero is a necessary but not sufficient condition for ergodicity. The ergodic measure is readily calculable while alternative measures of ergodicity (or stochasticity) such as Lyapunov exponents [96] are considerably more involved and not as obviously relevant to the convergence of thermodynamic

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properties as the ergodic measure. While there are strong connections between the convergence of the ergodic measure and the rate of spectral entropy production [89], the ergodic measure has been shown to provide significantly greater insight into the underlying protein dynamics. Moreover, the ergodic measure is readily computed for systems of arbitrarily large dimension, a great advantage in studies of protein dynamics. It is possible to derive the diffusion constant for the kinetic energy metric by assuming that the velocity is a Gaussian random variable. The kinetic energy (or local temperature) metric KE (t) can be expressed in terms of the fluctuations of the kinetic energy δfi (t) = (mv2i (t) − 3kB T )/2 as t 1 t KE (t) = 2 ds1 ds2 Ci (s1 − s2 ) (15) t 0 0 where in the limit of large N we identify Ci (t) as the equilibrium time correlation function of the fluctuations in the kinetic energy about its equilibrium average value. Berne and Harp noted that the velocity may be modeled as a Gaussian random variable if the information entropy corresponding to the probability of having the velocity at time t and the velocity at time 0 is maximized [97]. Through that approximation the autocorrelation function for any higher moments of the velocity may be calculated in terms of the normalized velocity autocorrelation function ψi (t) for the ith atom. The autocorrelation function for the fluctuation of the kinetic energy about its equilibrium average value may then be written Ci (t) = (3/2)(kB T )2 ψi2 (t) and the diffusion constant for the kinetic energy metric is DKE

−1 1 1 ∞ 2 = dtψi (t) 2 N 0

(16)

i

providing a means to determine the slope of KE (0)/ KE (t) DKE t for a particular model of the system dynamics characterizing ψi (t). B.

The Kinetic Energy Metric as a Probe of Internal Friction

The Langevin description of the motion of atoms in proteins is often used to interpret kinetic experiments. The dynamics captured by the Langevin model is the foundation of modern reaction rate theory. In particular, the Langevin model in combination with a normal mode description of the protein has been used to interpret inelastic neutron scattering data for proteins [98]. Starting from the normal mode description and assuming the friction tensor to be diagonal, the velocity autocorrelation function for each of the 3N normal coordinates is of the form ψi (t) = exp(−βi t/2)[cos(ai t) + (βi /2ai ) sin(ai t)] with ai2 = ωi2 − βi2 /4 and ωi2 = κi /mi , where ωi , κi , mi , and βi are the normal mode frequency, the effective

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harmonic force constant and effective mass, and friction of the ith normal mode of the system, respectively. Attempts to determine the friction acting on atoms in a protein, whose motion is bounded, have relied on fits to approximate forms of the correlation functions for the position and velocity [99, 100]. For the kinetic energy metric, using Eq. (16) we find that the generalized diffusion constant assumes the form DKE =

−1 −1 1 1 1 1 = 2 N 2βi β

(17)

i

when κi ≥ 0 ∀ i. This is a useful result. It shows that through a straightforward determination of the diffusion constant for the kinetic energy metric, the friction acting on the motion of a particle may be characterized. Note that the value of DKE is dominated by the smallest value of the friction. The asymptotic convergence of the ergodic measure provides information on those degrees of freedom that most slowly approach equilibrium. This reflects the heterogeneous nature of phase space that has been associated with interesting properties in chemical kinetics, particularly in the low friction energy diffusion regime [101]. As has been shown by Baba and Komatsuzaki in their recent work focusing on the interpretation of single-molecule spectroscopy [122], a detailed understanding of the timescale for the establishment of local equilibrium between states is essential to the interpretation of single-molecule dynamics in terms of dynamics on a free energy landscape. Presumably, the rate of attaining local equilibrium, observed in that work, is intimately connected to the distribution of rates for establishing local self-averaging, discussed here in the context of the ergodic measure. Their definition of local equilibrium states (LES) and its use in the decomposition and construction of a free energy landscape is explored in Section V. Average values of the relaxation rate cannot be used to characterize the rates of relaxation for specific protein modes. The method presented here provides a straightforward means of estimating the friction for all atoms of the protein in a way that allows for a global analysis of the dominant pathways for kinetic energy relaxation. More detailed normal mode-based approaches have been developed to provide insight into the mode-specific nature of the energy transfer pathways defining the system’s exploration of phase space [78]. C.

The Force Metric as a Probe of the Curvature in the Energy Landscape

The fluctuation metric KE (t) provides a measure of the timescale for selfaveraging a given observable, such as the kinetic energy, over a single trajectory. An alternative is to calculate averages over independent trajectories and measure the rate at which these independent averages converge to the same value as they must

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for an ergodic system. This idea has been applied to assess the rate of convergence for the various contributions to the atomic force (bonds and angles, dihedrals, and nonbonded van der Waals and Coulombic) in peptides and proteins [94]. To explore this idea, we define the force metric and evaluate it for the special case of a harmonic system. The time average of the force on the ith atom, fai (t), is defined as for the fluctuation metric where a indicates that the average is calculated over the ath trajectory. Given two trajectories a and b, with independent starting configurations, we define the difference between the averages calculated over each trajectory as the metric dF (t) =

N 1 a |fj (t) − fjb (t)|2 N

(18)

j=1

In an ergodic system, at long times self-averaging is achieved and the metric decays to zero as dF (t)/dF (0) 1/DF t, where DF means the corresponding diffusion constant. Conversely, if dF (t) does not decay as a power law we infer that there trajectories a and b must sample distinct free energy basins. When the system dynamics consists of small excursions about a well-defined average structure, it may be a good approximation to model the system dynamics using a quenched normal mode approximation. The potential is expanded in a Taylor series of the 3N coupled coordinates about a mechanically stable equi↔ librium position where κ is the 3N × 3N dimensional force constant matrix or matrix of second derivatives about the equilibrium position. The normal mode ↔ transformation diagonalizes κ by a unitary transformation defined by the matrix ↔ a resulting in 3N normal mode coordinates and corresponding frequencies ωi . The force metric may then be written as dF (t) =

3N 4kB T

mi [1 − cos(ωi t)] NF 2

(19)

i

where defined the “average mass of the ith normal mode” as mi =

N we have N 2 using the normalization condition that 2 m |a | j ji j j |aji | = 1. In this context, F is a vector observable of interest, such as the force on a given atom or internal coordinate, and F 2 is its mean-square value. For proteins the average mass is fairly independent of the mode number and we ¯ where M¯ is the average atomic mass for the peptide or can approximate mi = M, protein atoms [93]. The initial value of the force metric can be expressed in terms of ¯ B T ω2 , where the second moment of the vibrational density of states dF (0) = 6Mk

3N 2 2

ω = (1/3N) i ωi , leading to a remarkably simple form for the asymptotic limit of the force metric: dF (0)/dF (t)

1 2 2 t ω 2

(20)

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2 /5 providing a Application to the Debye model leads to the result ω2 = 3ωD means of calculating the approximate Debye frequency for the system from the curvature of the total force metric. For the underdamped normal modes, inertial motion dominates and the metric shows superconvergence as 1/t 2 . However, in the case that the modes are coupled through nonlinear forces, phase space may be divided by vague tori [102] (see also the generalization to multidimensional systems in Section III in Chapter 4) or as an Arnold web [103] (see also Section II in Chapter 3), slowing convergence (see Fig. 6). In such cases, the ergodic measure may be used to assess the extent of broken ergodicity, as a convenient alternative to the more demanding computation of Lyapunov exponents [89]. The use of the force metric to assess the average curvature of the underlying free energy landscape has a variety of potential applications. For example, Stillinger and LaViolette applied the inherent structure theory to examine the validity of the Lindemann melting criterion that solids become unstable when the mean-square fluctuations of the atoms approach 10% of the lattice spacing, for simple solids [104]. That idea was extended to examine the nature of the folding transition in proteins [105] leading to the interesting conclusion that in the native state the interior of proteins may be considered “solid-like” while the surface behavior is more “liquid-like.” Deeper insight into the nature of those phases could presumably be gained through computation of the distribution of Lyapunov exponents, as demonstrated by Berry and coworkers in their studies of the dynamics of atomic clusters and discussed earlier in this chapter. More recently, a connection has been established between the curvature of the underlying free energy landscape and the stability of proteins [106]. In that work, the mean square gradient of the potential, related to the underlying curvature of the free energy landscape, was shown to be intimately connected to the statistical temperature, T (E) = (∂S(E)/∂E)−1 , and the depth of the free energy basin containing the native state of the protein.

D.

Extensions of the Ergodic Measure to Internal Energy Self-Averaging

Using the fluctuation metric, we have shown that the kinetic energy is equipartitioned on a timescale of picoseconds. Evidence suggests that longer time relaxation associated with conformational transitions in the peptide is best explored using the metric of the nonbonded (Coulombic and van der Waals) potential energy. The equations for the energy metric are found by substituting the scalar energy eαi (t) for the ith atom in the αth trajectory into expressions where the corresponding force vector leads to N 1 a dE (t) = [ei (t) − ebi (t)]2 N

(21)

i=1

In a rugged energy landscape, the nonbonded energy metric shows rapid initial convergence followed by a slow, long-time decay. The initial convergence

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is significantly greater at higher temperatures (see Fig. 6). In many proteins, the plateau in the reciprocal metric is reached within 3 ps for T < 240 K [93]. This indicates that at lower temperatures the peptide motion is confined to fluctuations about a single local free energy basin without significant conformational transitions on a 75 ps timescale. At 300 K, there is a significant region of linear convergence followed by a plateau beyond 15 ps. This behavior clearly indicates the presence of a wide distribution of timescales for the protein motion. Moreover, within the short timescale several free energy basins are sampled as indicated by the change in the slope of dE (t). The longer time relaxation is related to infrequent barrier crossing, largely in the form of dihedral angle transitions (discussed below) and the diffusive motion of subdomains of the protein that may shift in relative orientation [107]. E.

Probing the Heterogeneity of Energy Flow Pathways in Proteins

In the study of heme protein dynamics, the concept of rapid intramolecular vibrational relaxation within the heme has been a significant focus of experimental and theoretical study for decades [108, 109]. In the protein carboxymyoglobin, the heme group and its host protein share only one covalent bond—that between the proximal histidine and the iron atom. The heme is otherwise kept in place by roughly 90 van der Waals contacts with surrounding protein atoms, much like a molecule in solution. The extent of this isolation can be brought to light using a modified form of the ergodic measure defined by an average over all atoms of the reference system that may, for example, be the heme alone, the protein alone, or the protein and solvent bath. The results are shown in Fig. 7 and summarized in Table I [110].

1

ΩKE(t)

400 ΩKE(0) / ΩKE(t )

Total Protein Heme Solvent

Total Protein Heme Solvent

0.5

300 0 0

200

10

20 Time (ps)

30

100 0

0

10

20

30

Time (ps) Figure 7. Convergence of the reciprocal ergodic kinetic energy metric KE (0)/ KE (t) computed from the molecular dynamics trajectory of the protein carboxymyoglobin in aqueous solution at 300 K for subsets of atoms in the protein, solvent, and heme.

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tamiki komatsuzaki et al. TABLE 1 Summary of the Inhomogeneous Character of the Average Static Friction Computed for Specific Elements of the Solvated Heme Protein System in Carboxymyoglobin at 300 K Region

γ0 (ps−1 )

System Total Protein Heme Solvent

2.12 1.87 6.76 7.49

Backbone Oxygens Nitrogens ␣-Carbons Carbonyl carbons Hydrogens

9.56 9.45 6.92 3.78 0.91

Residues Charged Aliphatic Aromatic

2.39 1.69 1.09

Energy redistribution within the heme itself occurs on a faster timescale than the thermalization of the heme with its environment. This indicates that the ratelimiting step in heme relaxation following photolysis is the “doorway” to energy transfer between the heme and its surroundings. The location and effectiveness of a given IVR pathway can depend sensitively on the nonlinear dynamics of the multimode system and the details of coupling between local modes that determine the character of dynamics in phase space. Overall, application of the ergodic measure to explore protein dynamics has deepened our appreciation for the inhomogeneous character of the distribution of rates of phase space sampling that correlates with the details of the protein structure. V.

EXTRACTING THE LOCAL EQUILIBRIUM STATE (LES) AND FREE ENERGY LANDSCAPE FROM SINGLE-MOLECULE TIME SERIES

In this section, we review our recent studies on the extraction of LES, in which the system exhibits ergodicity in the metastable state, from single-molecule time series such as interdye distance in single-molecule measurements. We also revisit the concept of the free energy landscape of proteins and discuss what kinds of energy landscape proteins actually experience during the course of their time evolution.

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Recent experimental developments in single-molecule spectroscopy have provided us with several new insights into not only the distribution of the observable but also the dynamical information at single-molecule level [111–115]. Fluorescence resonance energy transfer (FRET) experiments monitor the fluorescence intensity from donor (D)/acceptor (A) molecules attached in a single protein. The observed quantities are expected to trace the time evolution and the distribution of the D–A distance at the single-molecule level. For example, some experimental studies have indicated the existence of abnormal diffusion depending on the timescale at which one observes the dynamical events [114], heterogeneous pathways for protein folding in adenylate kinase [113], and different timescales of relaxation in an intermediate state and in the unfolded state of iso-1-cytochrome c [116]. Yang et al. [114] showed in single-molecule time series of flavin reductase with a bound flavin that abnormal diffusion emerges for a timescale less than 10−1 s in the fluorescence lifetime fluctuation while it turns to normal Brownian diffusion for longer timescales than 10−1 s. Rhoades et al. [113] observed a broad distribution in the timescale of the folding transitions and an importance of non-Markovian conformational dynamics especially for slow transitions (>1 s) by trapping adenylate kinase within surface-tethered lipid vesicles. Kinoshita et al. [116] found using a new single-molecule detection technique employing a capillary flow system that iso-1-cytochrome c (known as having a collapsed intermediate state) exhibits relatively slower conformational dynamics in the unfolded state, compared to that in the intermediate state. As argued by Talaga et al. [117], for large conformation changes of proteins, there exist some spurious interactions caused by immobilization from the surface linkage when using direct surface-linking techniques. It should be noted that the above experimental systems [113, 116] were specially designed so as to be free from the artifacts with the surface linkage technique and to detect large conformational change for longer time duration compared to the confocal microscopy experiment. However, the most fundamental question of what type of energy landscape single molecules actually see during the course of such dynamical evolution remains unresolved. To address this question, an essential goal must be the development of a device or means to extract the relevant information concerning the local equilibrium states and their network from scalar single-molecule time series. A.

Extracting LES from Single-Molecule Time Series

There exist several problems in the single-molecule measurements [118–121] in addition to the problem discussed by Talaga et al. [117]. One of the most cumbersome obstacles is the so-called “degeneracy problem” due to the dimensionality of the observable: even when the system resides in different physical states, the value of the observable (scalar time series) is not necessarily different and may be degenerate due to the finite resolution of the observation and the nature of

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the projection of the underlying multidimensional information onto the chosen observable. Baba and Komatsuzaki have recently developed a new method for extracting LES from a given scalar time series as free as possible from the degeneracy problem and constructing an effective free energy landscape [122]. In short, the crux is to evaluate states not by the value of the time series at each time but by the short-time distributions in the neighborhood of each time with a time window τ. The shorttime distributions reflect not only the value at each time but also the higher order moments, that is, variance, skewness, kurtosis, and so on in the vicinity of the time. Thus, the short-time distribution can differentiate the states that are degenerate in the value itself as much as possible under the limitation of the available information (i.e., solely the scalar time series). The secondary crux is to present the criteria of assigning if an obtained state can be regarded as LES or not within the timescale τ by checking the timescale separation between the τ and the escape time from the state candidate. This procedure naturally revisits the concept of the free energy landscape and provides us with a fundamental question of what type of energy landscape the system actually follows in a chosen timescale. In the following, we present their procedure briefly. Figure 8 shows their scheme to construct a set of state candidates from time series s(t). Suppose that s(t) is recorded with an equal interval from t1 to tn . First, they extract “short segments” in a time window (tm − τ/2, tm + τ/2] in the (τ) (s) vicinity of tm and construct the short-time probability density function gm (m = k + 1, k + 2, . . . , n − k, where k is larger than the size corresponding to τ/2) (see Fig. 8a). Second, they quantify the degree of proximity of two probability density functions by using Kantorovich metric [123] defined by dK (pi pj ) =

∞

−∞

ds

s

−∞

ds pi (s ) − pj (s )

(22)

where pi (s) and pj (s) are two arbitrary probability density functions with respect to s. It was found [122] that the dK is much more appropriate than the most commonly used measures, for example, Kullback–Leibler divergence (relative entropy) [124] and Hellinger distance [125], in differentiating the distance between two probability density functions. This is due to the fact that Kullback–Leibler divergence and √ Hellinger distance give rise to a single value (the former is ∞ and the latter is 2) when pi (s) and pj (s) have no overlap in the variable s. Figure 8b (τ) (s) by projecting onto illustrates the metric relationship (regarding dK ) among gm a two-dimensional plane so as to maintain the metric relationship among them [126] (note that in the actual procedure they do not need any projection of the full (τ) (s) at dimension into such a lower dimension). Each node corresponds to each gm (τ) a different time tm . Third, Baba and Komatsuzaki partition the set of gm (s) into a union of “clusters” on the full-dimensional metric space as illustrated by clusters

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τ

(a)

s(t)

4 3 2 1

t

(b)

10

Normalized freq. dist

(c)

8 6 4 2

1.0 1.5

2.5 3.0 3.5

2.0

s Figure 8. (a) A(τ)time series s(t). For every mth time step tm , the short-time probability density (τ) function gm (s) ( gm (s)ds = 1) is evaluated for a time window (tm − τ/2, tm + τ/2]. (b) A two(τ)

(τ)

dimensional projection of a set of gm (s) so as to maintain the “metric” relationship among the gm (s) (τ) [determined by Eq. (22)]. Each node or circle corresponds to each gm (s) at different tm (for the (τ) visual clarity, not all but every other 10,000 sampled points of gm (s) are plotted from the time series (τ) in (a)). The set covered by the dotted line indicates the full set of gm (s) corresponding to the full s(t). Different subsets (covered by solid lines) of different nodes correspond to the different state (τ) “candidate” where the composite gm (s) are classified as the same group on this metric space in the full dimension. (c) The corresponding frequency distributions of the four major state “candidates” with respect to s. (Reproduced with permission from Ref. 122. Copyright 2007 by the National Academy of Sciences.)

(τ) (s), all of surrounded by solid lines in Figure 8b. Each cluster composed of gm whose shapes are almost the same, naturally provides a candidate of state. Figure 8c shows the corresponding frequency distributions of the four major state candidates. Note here that this procedure does not need to assume any shape of the distribution function associated with the underlying states (although one has often assumed it as Gaussian in the conventional analysis with fixing the total number of states). The most important step is to incorporate the concept of local equilibrium states into the “candidates of states.” They considered the following criteria in the timescale for assigning the candidate of state as an LES:

τeq (i) < τ < τesc (i)

(23)

where τeq (i) denotes the characteristic timescale of the system to be locally equilibrated within the state i and τesc (i) that of the system to escape from the state i

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to the other. That is, they considered a “candidate of state” (= a grouped subset (τ) (s)) should satisfy the condition, if it can be regarded as an LES, that a of gm timescale for which the system is locally equilibrated inside the “state candidate i” is shorter than the timescale of observation τ and the τ is longer than a timescale at which the system escapes from it. One can easily check the latter condition τ < τesc (i) by identifying which “candidate of state” the system resides in, enters to, and leaves from along the time series by checking to which cluster the short-time distribution at each time belongs. However, the equilibration timescale of the state i, τeq (i), in the former condition (τ) (s). Theoretically, the state classified as an LES should is inaccessible by using gm possess a unique local distribution of the observable whenever the system revisits the same state along the course of time evolution. It is because, by definition, the system cannot escape any LES before locally equilibrated, resulting in a certain unique distribution function of any physical quantity. Therefore, their method can implicitly take into account the former condition in the procedure of grouping (τ) (s) into a set of approximately unique distribution functions. Their method, gm hence, assigns a state candidate as an LES if τ < τesc (i) is satisfied, otherwise as a non-LES at the given time window τ. B.

Revisiting the Concept of Free Energy Landscape

In order to validate the existence of barriers or “transition states” on the free energy landscape, the other condition is required, that is, (local) detailed balance. One can assess the local detailed balance as follows: The above procedure enables us to evaluate residential probability Pi of the ith LES, that is, how frequently the system (re)visits the ith LES along the time series. In addition, one can evaluate the rate constants ki→j (and kj→i ) from the i(j)th LES to the j(i)th LES, that is, the averaged, inverse of life time for which the system resides in the i(j)th LES before leaving to the j(i)th LES. The local detailed balance condition is represented by ki→j Pi kj→i Pj

(24)

Suppose that we can utilize canonical transition state theory (TST) of the reaction rate, that is, ‡

ki→j = kB T/ h exp(−Fi→j /kB T )

(25)

‡

where Fi→j , kB , h, and T denote the free energy barrier from the ith LES to the jth LES, Boltzmann constant, Planck constant, and absolute temperature, respectively. Then one can evaluate the free energy barrier F ‡ on the free energy landscape that links the free energy minima Fi and Fj of ith LES and jth LES (see Fig. 9): Fi = −kB T ln Pi

(26)

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Figure 9. Free energy surface.

and ‡

F = Fi − kB T ln

h ki→j kB T

= Fj − kB T ln

h kj→i kB T

(27)

where Fi and F ‡ , respectively, denote the relative free energy of the ith LES and the relative free energy barrier linking the ith and jth LES. Note here that unless the local detailed balance is satisfied, the second equality in Eq. (27) does not hold. This implies that one can neither identify the relative free energy of the barrier F ‡ nor construct the landscape of free energy unless the detailed balance holds. (Kramers theory [127] and Grote–Hynes theory [128] tell us that the canonical TST provides an upper bound of the rate constant. The free energy barrier derived from Eq. (27) can be affected by the existence of viscosity from the environment [129, 130]. An appropriate correction may be required for better estimation of the free energy barrier [112].) Let us revisit the concept of free energy landscape. The free energy landscape F (Q) is usually defined as a function of m-dimensional progress variables Q: E(p, q) m Z(Q) = . . . dq dpδ (Q(q) − Q) exp − (28) kB T F (Q) = −kB T log Z(Q)

(29)

where E(p, q) denote the total energy of the system (e.g., protein) as a function of its momenta p and coordinates q coupled with a heat bath of temperature T . δm , Q(q), and Z(Q) denote multidimensional Dirac’s delta function defined by δm (z) = δ(z1 )δ(z2 ) · · · δ(zm ), progress variables by which free energy landscape is depicted (usually a certain function of q), and the partition function with respect to Q, respectively. As for the definition, one can always compute this landscape as a function of the chosen Q by assuming that all the degrees of freedom distribute according to the Boltzmann distribution except a set of the “frozen” Q. Quite often one has argued the dynamics in Q on the landscape of F (Q). The question to be

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addressed here is this: on what type of energy landscape a single protein actually feels along the evolution in the Q space? In the other words, what should be the hidden assumption to validate the picture of dynamics on free energy landscape F (Q)? The key condition is that the system must attain ergodicity to yield the Boltzmann distribution within the space complement to the Q space in the sense of statistical mechanics (see Section II.). That is, in general, the characteristic timescale with respect to the Q motions is much longer than those of the other degrees of freedom so that the system can move about the complementary space at each “constant” Q. Furthermore, such timescale separation between Q and the other degrees of freedom should maintain through the whole regime where F (Q) is constructed. Some of the other degrees of freedom complement to Q may become slow compared to the timescale in Q depending on the regime of the whole space. However, such a possibility of changing the relationship in timescale between each variable Q and the rest should not be invoked in the context of the free energy landscape. One may notice the essential difference between the free energy profile in Fig. 9 composed of the LES and the free energy barrier and free energy landscape defined by Eqs. (28) and (29). The former is defined by LESs and free energy barrier based on the local ergodicity and local detailed balance that should be considered in the space or coordinates to describe the landscape (e.g., some Q). That is, the former definition refers to some requisite conditions in the Q dynamics. On the other hand, the latter definition of free energy landscape [Eqs. (28) and (29)] never requires any a priori condition in the Q dynamics themselves except the timescale separation. The “free energy landscape” defined by Eqs. (28) and (29) is more appropriately referred to as “the potential of mean force landscape” because it invokes no constraint on the Q dynamics. C.

Extracted LES of a Minimalistic Protein Model at Different Temperatures

Suppose that we have a set of single-molecule time series of interdye distance d(t) that could be converted from the FRET intensity as a function of temperature such as that in Fig. 10. How can one retrieve the underlying local equilibrium states and their free energy landscape from such limited information? Baba and Komatsuzaki illustrated the potential of their method by using the scalar time series of the end-to-end distance of an off-lattice 3-color, 46-bead model protein [131, 132] at several temperatures [122]. This model (called the BLN model) is composed of hydrophobic (B), hydrophilic (L), and neutral (N) beads. The global potential energy minimum for the sequence, B9 N3 (LB)4 N3 B9 N3 (LB)5 L, folds into a ␤-barrel structure with four strands (see Fig. 11). Two peaks are seen in the heat capacity: one corresponds to the collapse temperature (∼ 0.65 [134]) at which the BLN model transitions from the

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Figure 10. An “interdye distance” d(t) of a protein at different temperatures. The d(t) plotted here is the end-to-end distance of an off-lattice 3-color, 46-bead model protein [131, 132]. Temperature T , from the top to the bottom column, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, and 2.0/kB ( is the unit of energy of the model protein and kB the Boltzmann constant). The isothermal MD simulation was performed by Berendsen’s thermostat [133]. After a long simulation for equilibration, the value of d was recorded every 100 steps along the 13 million-step trajectory. Here the MD step, t, corresponds to ∼1/180 of the timescale of one vibration of the bond. The coupling constant γ with the thermostat was chosen as ∼1/(200t) so that the canonical distribution can be quickly attained during the course of MD simulation [122].

extended to the compact, collapsed states, and the other to the folding temperature (0.27 [135]−0.35 [134]) where it folds into the global potential energy minimum [136, 137]. Figure 12 shows the result of the analysis of time series d(t), that is, the normalized frequency distributions f (d) of the assigned LES and non-LES with a chosen time window. The f (d) is defined by, in the absence of any broadening effects of signal in the measurement, ∞ f (d) = δ(d − d(tm ))/ dsδ(s − d(tm )) (30) m∈cluster i

m∈all clusters −∞

The larger the f (d), the more the system resides in the LES/non-LES. Here the time window τ was set to be 10,000t, which corresponds to ∼55 oscillations of the bond vibration and 50 times longer than the timescale of the coupling ∼ 1/γ.

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Figure 11. The global minimum structure of BLN protein model. The black, white, and gray color beads correspond to hydrophobic (B), hydrophilic (L),

and neutral (N) beads,

respectively. The potential energy function is described by V = (Kr /2) i (ri − r0i )2 + (Kθ /2) i (θi − θ0i )2 +

[A(1 + cos i ) + B(1 + cos 3i )] + 4 i<j−3 S1 [(σ/Rij )12 − S2 (σ/Rij )6 ], where S1 = S2 = 1 i for BB (attractive) interactions, S1 = 2/3 and S2 = −1 for LL and LB (repulsive) interactions, and S1 = 1 and S2 = 0 for all the other pairs involving N, expressing only excluded volume interactions. Kr = 231.2σ −2 and Kθ = 20/rad2 , with the equilibrium bond length r0i = σ and the equilibrium bond angle θ0i = 1.8326 rad.

At 0.3 and 0.4 /kB , four and three large LESs are identified. As T increases to 0.5/kB , one can see three large LESs into which some of the LES/non-LES at 0.4/kB are unified. The three large LES observed at 0.5/kB are interpreted as unified into one large LES at 0.6/kB . This implies that the system quite often traverses back and forth the three unified LES at 0.6/kB , crossing over the barriers that link the three LES observed at 0.5/kB within the chosen τ. Note also that some “delocalized” distributions start to emerge (with low probabilities) besides this large unified state at 0.6/kB . At 0.7/kB , although the “compact” large LES ceases to exist, delocalized distributions become more significant with higher probabilities. Note that from 0.6 to 0.7/kB , the “center” of the LES migrates from the short to the long end-to-end distance regions, which corresponds to the transition from the collapsed state to the unfolded state. This manifests the existence of Tc between 0.6 and 0.7/kB , which is consistent with the previous assignment in terms of heat capacity ∼0.65/kB [134]. At 0.8/kB above Tc , two distributions are classified as LES while the other distributions violate Eq. (22) in the τ. The most striking messages of this illustrative example are twofold: First, at single-molecule level with a finite timescale, LES and non-LES can generally coexist as a function of temperature and timescale of the observation. Second, one can see that none of the distributions is classified as LES at 0.7/kB within the chosen τ. This indicates that, on the timescale τ, in neither the compact states nor more delocalized, denatured states, the system can be well equilibrated (i.e., the residential times in them are shorter than the τ) although either at higher or lower

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0.7ε 1 0 0.8ε 1 0 2.0ε

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4 6 8 10 End-to-end distance, σ

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14

Figure 12. f (d) of LES/non-LES constructed from the d(t) at different temperatures, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, and 2.0/kB . In the insets of 0.3–0.5, the magnified graphs in the horizontal axis are also given. The unit of vertical axis is 10−2 . The f (d) indicated by the dotted lines are assigned as non-LESs. (Reproduced with permission from Ref. 122. Copyright 2007 by the National Academy of Sciences.)

temperatures the system can more likely equilibrate at least in some of the states at the same chosen timescale τ. We believe that this property of the attainability of local ergodicity depending on the “temperature” is generic irrespective of the system when the system has several potential basins [138].

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Normalized frequency distributions of LESs

4

(a)

2 0

(b)

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(c)

4 2 0

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End-to-end distance, σ Figure 13. The dependence of LES/non-LES on the time window τ at T = 0.4. The number of the sample points to evaluate the local distributions (nS ) corresponds to (a) 400, (b) 500, and (c) 2000, respectively. The unit of vertical axis is the same as in Fig. 12. The solid and dashed lines indicate the LES and non-LES distributions, respectively. The gray arrows indicate the merger of a non-LES into an LES as nS = 400 → 500. The black arrows indicate the change from LES to non-LES as nS = 400 → 500 and the merger of a non-LES into an LES as nS = 500 → 2000. (Reproduced with permission from Ref. 122. Copyright 2007 by the National Academy of Sciences.)

It was indicated in Fig. 13 that the topography of the landscape depends on the timescale of observation; two metastable states are unified as one if the timescale of observation is longer than the escape timescale for which the system can visit mutually these two states [122, 138]. The different time windows naturally lead to different coarse-grained LESs and different coarse-grained free energy landscapes, if they exist, the system should find at the different timescales. This corresponds to a manifestation of transition from a set of two “ergodic” states (i.e., locally ergodic) to a single “ergodic” state, as observed in small atomic clusters in Section III, that is, a bimodal to unimodal peak in the local Lyapunov exponent distribution for a “phase transition” in a small finite system. D.

Outlook

In this section, we have reviewed a new method to extract effective free energy landscapes from single-molecule time series. If and only if the local equilibrium and the local detailed balance are satisfied in a chosen timescale of observation τ, one can construct the effective free energy landscape for the regions where the system wanders frequently. In the procedure, the equilibration time for which

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the system can be locally equilibrated inside the chosen state(s) was not assessed directly but rather implicitly taken into account by checking invariance of the probability density function of the observable. One should note the emphasis of “local” in the phrase “local detailed balance.” At the single-molecule level, in principle, detailed balance does not hold globally unless the length of time goes almost to infinity and the ergodicity in the sense of statistical physics is known to be satisfied in advance. However, one can still expect the existence of the detailed balance in a certain local regime with a certain timescale τ where free energy basins and barriers are well defined.

VI.

FUTURE PERSPECTIVES

In this chapter, we have addressed the fundamental background and practical approaches for addressing ergodic problems in complex systems in chemical physics. Real complex systems have provided us with challenging subjects in ergodicity: multiplicity of ergodicity, heterogeneity of timescales to attain ergodicity for each moiety of the system, the openness of the phase space, and so on. Multiplicity of ergodicity is closely related to the issue of the time-dependent nature of the free energy landscape whose morphological features depend on the timescale of observation. The openness of the phase space demands that the observer incorporate the finiteness of the timescale of the observation when considering the ergodic properties of a system. In exploring the ergodic character of real systems, one must carefully choose criteria or measures of ergodicity and self-averaging. In practice, it is impossible to assure that the system satisfies the mathematical condition of ergodicity. Local Lyapunov exponent distributions provide insight into the distinction between chaotic and less chaotic regions, dimensionality of the exploration of the system through the full phase space, and a timescale of merging two or more ergodic regions. When one can compute the local Lyapunov exponent distribution, this measure provides substantial insight into the detailed system dynamics. However, when the size of the system becomes large, say, more than a hundred degrees of freedom or so, it is impossible to compute such an elaborate distribution. For those cases, the ergodic measure provides a practical measure of the rate of self-averaging that may be readily applied to systems of any dimension if one can know the equation of motion. Furthermore, the more apparent distinction between small-body systems and larger many-body systems such as proteins is the existence of multiple time- and space scales. As presented in Section III of Chapter 3, the divergence of nearby trajectories at infinitesimally small deviation in the phase space is expected to become almost irrelevant to protein dynamics in the large. For many-body systems such as proteins, many outstanding problems to be addressed now are rooted in the existence of a strongly inhomogeneous distribution of timescales that must be sampled in order to attain ergodic sampling. Moreover,

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the exact timescale will depend on the moiety of the system, as shown in Section IV, and its relationship with protein function in the space of nonequilibrium degrees of freedom. When one cannot know the equation of motion or cannot compute dynamical information because of the limitation of computational power, a possible measure to address the concept of ergodicity should be based on the limited dynamical information obtained by experiments such as single-molecule time series. The method of LES analysis developed by Baba and Komatsuzaki can be a strong tool to retrieve the ergodic property from single-molecule measurements. Recently, their approach was applied to investigate which interdye distances (which pair of amino acid residues) are suitable to retrieve the underlying free energy landscape of protein G [139]. The much more challenging question is this: How is the concept of ergodicity/ nonergodicity relevant to biological functions in more complex systems such as signal transduction in a cell or protein–protein interaction and other recognition/signaling problems? To look into this challenging problem, it is crucial to establish a device or means to quantify the robustness of the concept in terms of time series by linking with single-molecule biophysics. The argument of Maxwell’s demon in Section II and the existence of heterogeneous distribution of timescales to achieve ergodicity at each moiety and hierarchy in Sections III and IV may suggest us that the system has not to “feel” full ergodicity or full equilibrium at timescales to characterize or be relevant to biological functions even though most degrees of freedom are more likely expected to be thermalized. That study provides a glimpse of the insights that may be achieved through the application of methods for probing ergodicity to complex biological systems. Acknowledgments T.K. and S.K. thank Dr. Hiroshi Teramoto for his fruitful discussions and comments. J.E.S. gratefully acknowledges the generous support of the National Science Foundation (CHE-0750309) and Boston University’s Center for Computational Science. T.K. acknowledges financial support from JSPS, JST/CREST, Grant-in-Aid for Research on Priority Areas “Systems Genomics” and “Real Molecular Theory,” MEXT. M.T. acknowledges financial support from JSPS, Grant-in-Aid for Research on Priority Area “Real Molecular Theory,” MEXT, and Nara Women’s University Intramural Grant for Project Research. S.K. acknowledges the support by research fellowships of the Japan Society for the Promotion of Science for Young Scientists. RSB wishes to acknowledge the hospitality of the Aspen Center for Physics, where much of his contribution was developed.

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AUTHOR INDEX Numbers in parentheses are reference numbers and indicate that the author’s work is referred to although his name is not mentioned in the text. Numbers in italic show the page on which the complete references are listed. Aaronson, J., 86(33), 187(28), 120, 217 Abel, M., 101(82), 121 Abramavicius, D., 13(64), 32 Ackermann, F., 58(179), 78 Ackley, S. F., 16(74), 32 Adachi, S., 99(62), 133(71), 120, 168 Adamo, C., 3(25), 30 Adib, A. B., 70(313), 82 Agard, D. A., 27(103), 33 Agarwal, P. K., 26(93), 33 Agbo, J. K., 26(95), 84(14), 116(118), 196(85), 33, 119, 122, 219 Agrawal, R. K., 41(48), 74 Ahn, D. S., 127(45), 168 Aizawa, Y., 86(34–38, 40), 187(38–39, 41–44), 120, 217, 218 Akagi, H., 125(18, 19), 167 Akimoto, T., 86(37, 39, 40), 187(39–41), 120, 217, 218 Akutsu, H., 41(51), 74 Albeverio, S., 187(37), 217 Alexandrov, V., 40(39), 50(113), 74, 76 Al-Laham, M. A., 3(25), 30

Allegrini, P., 101(81), 121 Almo, S. C., 40(40), 74 Amadei, A., 53(136), 54(139, 141–142), 62(199–201), 105(93), 77, 79, 121 Amara, S. G., 63(249), 80 Amemiya, T., 40(29), 56(29), 61(194), 71(194), 74, 78 Amitrano, C., 100–101(70–71), 108(70–71), 177(13, 14), 191(13, 14), 192–193(14), 121, 217 Anderson, S., 40(41), 74 Andricioaei, I., 71(322), 82 Anfinrud, P. A., 40(41), 74 Ankeny, D. J., 127(39), 167 Ansari, A., 101(75), 188(51), 195(66, 67), 121, 218 Apostolov, R., 72(323), 82 Araki, M., 36(2), 73 Arango, J., 42(59), 75 Archontis, G., 3(24), 17(24), 22(24), 30 Armstrong, M. R., 25(91), 33 Arnold, E., 63(235), 80 Arnold, V. I., 133(75), 147(91), 175(2), 168, 169, 216

Advancing Theory for Kinetics and Dynamics of Complex, Many-Dimensional Systems: Clusters and Proteins, Advances in Chemical Physics, Volume 145, Edited by Tamiki Komatsuzaki, R. Stephen Berry, and David M. Leitner. © 2011 John Wiley & Sons, Inc. Published 2011 by John Wiley & Sons, Inc.

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222

author index

Arora, K., 66(267, 268), 81 Asplund, M. C., 2(17), 22(17), 29 Astumian, R. D., 70(312), 82 Ataka, S., 14(68), 32 Atilgan, A. R., 51(116, 118–120), 76 Aubry, S., 28(114), 33 Aumann, K. D., 39(19), 74 Aurell, E., 101(77, 78), 121 Austin, A. J., 3(25), 30 Autieri, E., 71(320), 82 Avez, A., 175(2), 216 Ayala, P. Y., 3(25), 30 Baba, A., 45(77), 85(24), 87(24), 100(72), 101(72, 87), 105(87), 106(87), 108(72), 200(122), 202(103), 206(122), 207(122), 210(122), 213–214(122, 138), 75, 119, 121, 219, 220 Baboul, A. G., 3(25), 30 Backus, E., 5(42), 14(42), 18(42), 31 Bader, J. S., 12(54), 31 Baeck, K.-K., 127(45), 168 Baek, S. J., 127(45), 168 Baer, T., 84(1, 11), 127(36), 119, 167 Bahar, I., 3(27, 35), 51(116, 118–126), 63(249), 30, 76, 77, 80 Bai, D., 70(307), 82 Bairoch, A., 36(1), 73 Baker, N. A., 54(148), 77 Bakken, V., 3(25), 30 Bakker, H. J., 11(51), 31 Baldisseri, D. M., 42(59), 75 Balk, M. W., 84(12), 127(28), 119, 167 Bandekar, J., 13(59), 31 Bar, I., 127(44), 168 Barkai, E., 86(41, 42), 187(45, 47), 205(115), 120, 218, 220 Baronavski, A. P., 28(110), 33 Barone, V., 3(25), 30

Barrantes, F. J., 63(241), 80 Barre, J., 85(26), 120 Barrick, D., 24(92), 27(92), 33 Bartels, C., 3(24), 17(24), 22(24), 30 Barth, A., 13(59), 31 Bartunik, H., 39(19), 74 Bashford, D., 17(75), 32 Baskin, J. S., 127(29, 30, 46), 128(46), 167, 168 Batson, B., 72(324), 82 Baudry, J., 63(223), 79 Bax, A., 42(58, 59, 63), 75 Bayas, M. V., 63(228), 79 Beard, W., 63(212), 79 Bechtel, H. A., 127(39), 167 Beck, C., 185(29), 186(30), 217 Beck, R. D., 127(42), 168 Beck, T. L., 190(4), 216 Becker, O. M., 22(90), 46(82), 196(72), 32, 75, 218 Becker, S., 42(65), 43(65), 75 Belega, E. D., 191(12), 217 Bell, A. T., 66(271, 272), 81 Bellott, M., 17(75), 32 Benkovic, S. J., 26(93), 33 Berendsen, H. J. C., 3(29), 17(77), 40(36), 48(94), 49(94), 52(132), 53(136), 54(139–143), 62(199–201), 105(93), 106(100), 211(133), 30, 32, 74, 76, 77, 79, 121, 122, 220 Berendzen, J., 101(75), 188(51), 195(66), 195(67), 121, 218 Berger, C., 58(179, 180), 78 Berghuis, A. M., 48(93), 76 Berkowitz, M., 67(278), 81 Berman, H., 37(3), 38(3), 73 Berman, H. M., 38(6), 73 Berne, B. J., 2(5), 12(54, 56), 26(5), 65(264), 85(15), 114(15), 115(15), 199(97), 200(101), 29, 31, 80, 119, 219

author index Berry, R. S., 2(13), 100(70–72), 101(70–72, 85), 105(85), 108(70–72), 130(63, 64), 136(82), 144(64), 150(82, 93–98), 151(93, 96), 153(82), 156(93, 96), 160(93, 96), 164(113), 177(13, 14), 190(4, 8), 191(8, 13, 14), 192(14–16), 193(14, 18), 194(18), 195(18), 210(132), 211(132), 121, 168, 169, 216, 217, 220 Best, R. B., 42(62), 75 Bettati, S., 58(177), 78 Bhat, T. N., 38(6), 73 Bhattacharyya, S., 44(76), 75 Biferale, L., 101(82), 121 Biggin, P. C., 51(127), 77 Bigwood, R., 113(112), 116(112), 122 Billeter, S. R., 26(93), 33 Billing, G. D., 2(4), 26(4), 29 Bizzarri, A. R., 100(66), 121 Björkman, A. J., 40(32), 74 Bloem, R., 2(15, 16), 13(15, 16), 16(16), 29 Blumenfeld, R., 100(64), 120 Boczko, E. M., 211(137), 220 Bodo, G., 22(87), 56(169), 32, 78 Boehr, D. D., 41(56), 75 Boffetta, G., 100(74), 101(77, 78), 102(74), 103(91), 106(74), 121 Bolhuis, P. G., 69(292, 293, 295, 297), 81 Boltzmann, L., 180(24), 217 Bondar, A. N., 67(275), 81 Booth, C., 63(215), 79 Borchardt, R. T., 49(100), 76 Boresch, S., 3(24), 17(24), 22(24), 30 Borkovec, M., 2(5), 26(5), 200(101), 29, 219 Bornhauser, S., 58(179), 78 Bosshard, H. R., 58(179, 180, 182), 78 Bostock-Smith, C. E., 60(183), 78 Botan, V., 5(42), 14(42), 18(42), 31

223

Bouchet, F., 85(26, 27), 120 Bounouar, M., 14(69), 32 Bourgeois, D., 40(43), 74 Bourne, P. E., 38(6), 73 Bowen, R., 185(30, 49), 186(30), 217, 218 Bowers, K. J., 72(324), 82 Bowers, M. T., 127(31, 33), 167 Bowman, J. M., 9(48, 49), 14(48, 49, 70), 31, 32 Bowne, S. F., 101(75), 188(51), 195(66), 121, 218 Bradley, M. J., 54(148), 77 Brändén, C.-I., 38(5), 73 Brandes, U., 206(126), 220 Brauman, J. I., 85(17), 127(32), 119, 167 Braunstein, D., 195(67), 218 Bredenbeck, J., 2(20), 27(20), 29 Briggs, M. E., 101(80), 121 Brillouin, L., 87(47), 188(57), 120, 218 Brody, T. A., 113(114), 122 Brokaw, J., 44(76), 75 Brooks, B., 3(28), 48(85, 87), 50(108), 30, 75, 76 Brooks, B. R., 3(24), 17(24), 22(24), 50(109, 110), 30, 76 Brooks, C. L., III, 3(24), 17(24), 22(24), 41(50), 52(129), 53(129), 66(267, 268), 210(134), 211(134, 137), 212(134), 30, 74, 77, 81, 220 Brouard, M., 125(15, 16), 167 Brown, G. G., 127(37), 167 Broyde, S., 63(212), 79 Bruccoleri, R. E., 3(24), 17(24), 22(24), 30 Brumfeld, V., 60(185), 78 Bruschweiler, R., 42(64), 54(156), 75, 77 Bryngelson, J. D., 106(98), 121

224

author index

Bu, L., 2(21), 4(41), 9(21), 20(21, 80, 85), 27(41), 29, 31, 32 Bujnicki, J. M., 50(114), 76 Buldyrev, S. V., 109(104), 122 Burant, J. C., 3(25), 30 Burbanks, A., 150(99), 151(99), 169 Burioni, R., 51(128), 77 Burk, D. L., 48(93), 76 Caflisch, A., 3(24), 17(24), 22(24), 60(191), 216(139), 30, 78, 220 Cain, S. M., 41(45), 74 Calabrese, R. V., 101(80), 121 Calvo, F., 192(17), 217 Camden, J. P., 127(39), 167 Cammi, R., 3(25), 30 Campbell, D. M., 84(9), 119 Cannistraro, S., 100(66), 121 Cao, J, 205(120), 220 Cao, W., 20(83), 24(92), 27(92), 32, 33 Cárdenas, A., 70(305, 306), 71(305, 306), 82 Cárdenas, A. E., 70(309), 82 Carlini, P., 100(66), 121 Carpenter, B. K., 85(16), 119 Carragher, B. O., 41(45), 74 Carrion-Vazquez, M., 63(227), 79 Carter, S., 9(48, 49), 14(48, 49), 137(83), 31, 169 Casassa, M. P., 125(10), 167 Casati, G., 99(63), 133(72), 120, 168 Case, D. A., 3(30), 42(61), 66(266), 30, 75, 81 Cassi, D., 51(128), 77 Castiglione, P., 100(69), 110(69), 121 Cavanagh, J 41(52), 75 Caves, L., 3(24), 17(24), 22(24), 30 Ceccarelli, M., 70(304), 82 Cecconi, F., 51(128), 77 Cencini, M., 100(74), 101(82), 102(74), 106(74), 121

Chaban, G. M., 12(57), 14(57), 31 Chakraborty, A., 66(271), 81 Chakraborty, A. K., 69(296), 81 Challacombe, M., 3(25), 30 Champion, P. M., 20(82, 83), 24(92), 27(92), 32, 33 Chandler, D., 68(287), 69(293, 296), 84(2), 81, 119 Chandrasekhar, J 17(76), 32 Chandre, C., 94(58), 120 Changeux, J. P., 56(172, 173), 57(174, 176), 78 Chao, J. C., 72(324), 82 Cheatham, T. E., III, 3(30), 30 Cheeseman, J. R., 3(25), 30 Chen, K., 63(235–237, 248), 80 Chen, L. Q., 41(45), 74 Chen, W., 3(25), 30 Chen, W.-K., 127(48), 128(48), 168 Chen, X., 39(12), 73 Cheng, F., 63(235), 80 Cheng, N., 41(49), 74 Cheng, P.-Y., 127(48), 128(48), 168 Cheng, X., 49(101), 54(147), 76, 77 Cherayil, B. J., 87(53), 120 Child, M. S., 153(104), 169 Chirikov, B. V., 129(50), 168 Chivers, P. T., 54(148), 77 Cho, M., 3(36), 5(36), 11(36), 12(36), 13(63), 30, 31 Choi, J.-K., 3(36), 5(36), 11(36), 12(36), 30 Choi, J.-M., 127(45), 168 Chong, S. H., 39(25), 74 Chu, J.W., 44(76), 75 Chung, H. S., 16(74), 32 Chung, Y.-C., 127(48), 128(48), 168 Church, B. W., 196(73, 74), 218 Cianetti, S., 20(84), 32 Ciccotti, G., 17(77), 65(264), 68(285), 32, 80, 81 Cioslowski, J 3(25), 30

author index Clapés, P., 26(93), 33 Clarkson, J. R., 84(13), 119 Cleveland, C. B., 125(9), 167 Clifford, S., 3(25), 30 Clore, G. M., 42(58), 43(69), 44(70, 71), 75 Coker, D. F., 65(264), 80 Collet, P., 180(31), 185(31), 186(31), 217 Compoint, M., 63(218), 79 Conway, J. F., 41(49), 74 Corcelli, S. A., 13(62), 15(62), 31 Cossi, M., 3(25), 30 Cova, S., 100(68), 205(114), 121, 219 Covell, D. G., 51(120), 76 Cover, T. M., 206(124), 220 Cowan, M. L., 25(91), 33 Crehuet, R., 26(93), 67(281, 282), 33, 81 Cremeens, M., 2(21), 9(21), 20(21), 29 Crim, F. F., 127(40), 167 Crisanti, A., 101(77, 78), 103(91), 121 Crisma, M., 5(42), 14(42), 18(42), 31 Cross, J. B., 3(25), 30 Cross, P. C., 3(26), 30 Csajka, F. S., 69(300), 81 Cuendet, M. A., 63(230), 79 Cui, Q., 3(24, 25, 27), 17(24), 22(24), 49(96–99), 60(187), 30, 76, 78 Culik, S. J., 9(48), 14(48), 31 Cuniberti, G., 28(109), 33 Cusack, S., 39(22), 74 Czerminski, R., 64(261), 65(261), 196(68), 80, 218 da Luz, M. G. E., 109(104), 122 Dagdigian, P. J., 125(11), 167 Daidone, I., 63(202), 79 Dang, T. T., 127(42), 168 Daniels, A. D., 3(25), 30 Dannenberg, J. J, 3(25), 30 Dapprich, S., 3(25), 30

225

Darden, T., 3(30), 30 Daskalakis, V., 197(88), 219 Dauxois, T., 85(26, 27), 120 Davis, M. J., 153(105), 169 Davis, M.J., 129(51), 168 Dawson, P. E., 2(21), 9(21), 20(21), 29 Decius, J. C., 3(26), 30 DeFlores, L. P., 16(74), 32 de Groot, B. L., 42(65), 43(65), 54(140–143), 62(199–201), 75, 77, 79 De Leon, N., 130(57, 58), 168 De Los Rios, P., 28(113), 33 DeGrado, W. F., 205(117), 220 Delarue, M., 40(43), 50(103, 107), 74, 76 Dellago, C., 69(292, 293, 295), 81 del Valle, M., 28(109), 33 Demidov, A. A., 20(83), 24(92), 27(92), 32, 33 Demirel, M. C., 51(118, 119), 76 Deneroff, M. M., 72(324), 82 Depristo, M. A., 42(62), 75 Deprit, A., 136(81), 137(81), 169 Derreumaux, P., 5(42), 14(42), 18(42), 31 Dewan, J. C., 39(20), 74 Diamond, R., 39(10), 73 Dian, B. C., 27(96), 125(24, 25), 127(24, 37, 38), 196(84), 33, 167, 219 Dianoux, A. J., 39(23), 74 Diau, E. W. G., 127(47), 128(47), 168 Diaz, J. F., 63(207–209), 79 Dijkhuizen, L., 65(263), 70(263), 80 Dijkstra, A. G., 2(15, 16), 13(15, 16), 16(16), 29 Dijkstra, B. W., 65(263), 70(263), 80 Ding, J., 63(235), 80 Dinner, A. R., 3(24), 17(24), 22(24), 69(296), 30, 81

226

author index

DiNola, A., 106(100), 211(133), 122, 220 Dintzis, H. M., 22(87), 56(169), 32, 78 Dlott, D. D., 2(19), 29 Dobson, C. M., 42(62), 75 Doniach, S., 50(106), 71(317), 76, 82 Doreleijers, J. F., 41(51), 74 Dorfman, J. R., 101(80), 185(32), 186(52), 121, 217, 218 Doruker, P., 51(122), 76 Doster, W., 39(22), 74 Douglas, J. F., 101(81), 121 Douglass, K. O., 127(37), 167 Dragt, A. J., 145(90), 169 Drenth, J., 38(4), 73 Driscoll, P. C., 42(58), 75 Dror, R. O., 72(324), 82 Duan, W., 63(237), 80 Duan, Y., 54(146), 77 Duda, R. L., 41(49), 74 Duderstadt, K., 44(76), 75 Dunbrack, R. L., 17(75), 32 Durell, S. R., 51(118), 76 Duvaud, S., 36(1), 73 Duxon, S. P., 125(15, 16), 167 Dyson, H. J., 41(56), 43(68), 75 Eastman, P., 71(317), 82 Eastwood, M. P., 72(324), 82 Eaton, E. A., 205(112), 209(112), 219 Eaton, W. A., 44(75), 45(75), 58(177), 203(109), 75, 78, 219 Echols, N., 40(33, 39), 45(33), 50(113), 74, 76 Eckart, C., 54(154), 77 Eckmann, J.-P., 180(31), 185(31, 50), 186(31, 50), 217, 218 Edelman, M., 87(49), 188(60), 120, 218 Edelstein, S. J., 57(174), 78 Edholm, O., 52(132), 77 Edman, L., 205(119), 220

Eggenberger, J., 58(180), 78 Eguchi, J., 84(8), 119 Ehara, M., 3(25), 30 Eichinger, B. E., 51(117), 76 Eigen, M., 57(175), 58(175), 78 Eijsink, V. G. H., 54(139), 77 Ekonomiuk, D., 60(191), 78 Elbaum, M., 60(185), 78 Elber, R., 5(43), 5(44), 22(90), 64(257, 260, 261), 65(261, 263), 70(263, 302, 305, 306, 308–310), 71(305, 306), 196(68), 203(107), 31, 32, 80, 82, 218, 219 Elmaci, N., 101(85), 105(85), 191(10), 210(132), 211(132), 121, 217, 220 El-Mashtoly, S. F., 27(102), 33 Endo, S., 3(31), 30 Endres, N. F., 41(46), 74 Engelborghs, Y., 63(207–210), 79 Engelman, D., 50(113), 76 Engels, M., 63(204, 205), 79 Enriquez, P. A., 125(15, 16), 167 Erman, B., 3(35), 51(116, 119, 120), 30, 76 Etchebest, C., 50(105), 76, Evans, D. A., 26(95), 84(14), 125(25), 196(85), 33, 167, 119, 219 Evans, D. J., 186(33), 217 Evanseck, J. D., 17(75), 32 Ezra, G.S., 129(51), 168 Faccioli, P., 71(318–320), 82 Falcioni, M., 100(74), 101(82), 102(74), 106(74), 121 Falconer, K., 106(103), 122 Fang, Y., 2(19), 29 Farantos, S. C., 197(87, 88), 219 Farès, C., 42(65), 43(65), 75 Farkas, O., 3(25), 30 Farr, G. W., 41(47), 74 Farrelly, D., 151(102), 169 Fayer, M. D., 2(8), 29

author index Feig, M., 3(24), 17(24), 22(24), 30 Felker, P. M., 127(27), 167 Feng, Z., 38(6), 73 Feng, Z.-P., 50(111), 76 Fenichel, N., 130(53–55), 168 Fenton, W. A., 41(47), 74 Fernandez, J. M., 63(227), 79 Ferrand, M., 39(23), 74 Field, M. J., 17(75), 48(91), 63(211), 67(281), 32, 76, 79, 81 Filmer, D., 56(170), 57(170), 78 Findlay, J. B. C., 54(141), 77 Finn, J. M., 145(90), 169 Fioravanti, E., 40(43), 74 Fischer, A., 68(285), 81 Fischer, E., 55(164), 78 Fischer, K. H., 45(79), 75 Fischer, S., 3(24), 17(24), 22(24), 17(75), 67(275), 66(273, 274), 67(274, 276), 30, 32, 81 Fleming, G. R., 3(36), 5(36), 11(36), 12(36), 84(12), 127(28), 30, 119, 167 Flomenbom, O., 205(121), 220 Flores, J., 113(114), 122 Flores, S., 40(33), 45(33), 74 Flynn, T. C., 63(215), 79 Foote, J., 58(178), 78 Foresman, J. B., 3(25), 30 Fox, D. J., 3(25), 30 Francis, M. K., 101(80), 121 Francisco, J. S., 2(3), 26(3), 124(1), 29, 166 Frank, J., 41(48, 50), 74 Frauenfelder, H., 22(88), 39(16–18), 45(81), 101(75), 188(51), 195(65, 66), 196(70, 71), 32, 73, 75, 121, 218 French, J. B., 113(114), 122 Frisch, M. J., 3(25), 30 Fuchigami, S., 3(37), 27(37, 100), 55(162), 61(194, 195), 71(194,

227

195), 73(333), 105(96), 33, 78, 79, 82, 121 Fujimura, Y., 125(14, 18–20, 23), 144(88), 149(88, 92), 167, 169 Fujisaki, H., 2(21–23), 3(37), 4(41), 6(2, 472), 9(21, 22), 11(22, 47), 12(58), 13(47, 72), 14(66, 67), 15(66, 67, 72), 16(72), 17(22, 47, 58), 18(22, 47), 19(47, 72), 20(21, 23, 47), 21(23, 86), 22(23), 23(66, 67, 86), 25(22), 26(47), 27(37, 41), 28(23), 69(290), 71(316), 105(96), 196(79–83), 29–32, 81, 82, 121, 219 Fukuda, I., 72(325), 82 Fukuda, R., 3(25), 30 Fukui, K., 63(255), 80 Furuta, T., 69(290), 81 Gabdoulline, R. R., 60(192), 78 Gadea, F.-X., 3(32), 49(95), 30, 76 Gagliardo, J., 72(324), 82 Gallavotti, G., 180(26), 184(26), 186(26), 217 Gammon, R.W., 101(80), 121 Gan, W., 69(291), 81 Ganim, Z., 16(74), 32 Gao, J., 3(24), 17(24, 75), 22(24), 30, 32 Gao, M., 63(222), 79 Gao, Y., 27(102), 33 Garberoglio, G., 71(319), 82 García, A. E., 100(64, 65), 120 Gaspard, P., 101(79, 80), 106(79), 107(79), 185(34), 186(34, 52), 121, 217, 218 Gassmann, A., 39(25), 74 Gasteiger, E., 36(1), 73 Geissler, P. L., 69(292, 293), 81 Gerber, R. B., 5(43, 44), 12(57), 14(57), 31

228

author index

Gerstein, M., 40(33, 39), 45(33), 50(113), 60(184), 74, 76, 78 Geva, E., 12(55), 31 Ghosh, A., 70(305, 306), 71(305, 306), 82 Gibbs, A. L., 98(60), 120 Gibson, Q. H., 22(90), 32 Gill, P. M.W., 3(25), 30 Gillilan, R. E., 70(301), 81 Gilliland, G., 38(6), 73 Girardet, C., 63(218), 79 Glotzer, S. C., 101(81), 121 Go, N., 3(28, 31), 39(11, 14, 25, 26), 45(80), 48(84), 53(134, 135, 137), 55(163), 105(94), 106(99), 30, 73–75, 77, 78, 121, 122 Goh, C. S., 60(184), 78 Gohlke, H., 3(30), 30 Golan, A., 127(44), 168 Gomperts, R., 3(25), 30 Gong, J., 130(60), 168 Gonzalez, C., 3(25), 30 Goodrow, A., 66(272), 81 Gorbunov, R. D., 5(42), 13(65), 14(42), 18(42), 31, 32 Gordon, H. L., 211(135), 220 Goto, S., 36(2), 73 Goto, Y., 205(116), 220 Gouda, N., 197(95), 219 Gowen, B., 41(47), 74 Grant, B., 49(101), 76 Grassberger, P., 103(89), 190(5–7), 121, 217 Graul, S. T., 127(31, 33), 167 Green, J. R., 193–195(18), 217 Gregurick, S. K., 12(57), 14(57), 31 Griesinger, C., 42(64, 65), 43(65), 75 Groenhof, G., 3(29), 30 Gronbech-Jensen, N., 71(317), 82 Gronenborn, A. M., 42(58), 75 Grossman, J. P., 72(324), 82

Grubmüller, H., 42(65), 43(65), 55(157, 158), 75, 77 Gruebele, M., 2(12), 11(53), 13(12, 53), 28(12), 85(20), 86(31, 32), 87(31), 113(111, 112), 116(20, 112), 29, 31, 119, 120, 122 Gruia, A. D., 67(275), 81 Gruia, F., 24(92), 27(92), 33 Gu, Y., 63(249), 80 Guarneri, I., 99(63), 133(72), 120, 168 Guevara, J., 67(280), 81 Gullingsrud, J., 63(223, 240), 79, 80 Gulmen, T. S., 11(52), 31 Gumble, E. J., 187(25), 217 Guo, H., 17(75), 32 Guo, J., 127(46), 128(46), 168 Guo, W., 63(220), 79 Guo, Z., 210(134), 211(134), 211(137), 212(134), 220 Guo, Z. Y., 211(136), 220 Gussakovsky, E., 205(113), 219 Gustavson, F., 133(74), 168 Gutierrez-Laliga, R., 28(109), 33 Gutteridge, A., 56(165, 166), 78 Ha, S., 17(75), 32 Haak, J. R., 106(100), 211(133), 122, 220 Hada, M., 3(25), 30 Hagan, M. F., 69(296), 81 Hahn, S., 13(63), 31 Haliloglu, T., 3(35), 30 Hall, B. A., 51(127), 77 Halonen, L. O., 137(83), 169 Ham, S., 13(63), 31 Hamm, P., 2(17, 20), 5(42), 14(42), 18(42), 22(17), 27(20), 29, 31 Hammes-Schiffer, S., 26(93), 33 Hancock, G., 125(22), 167 Handy, N. C., 9(49), 14(49), 31 Hanes, J., 58(180), 78 Hänggi, P., 28(108), 33

author index Hanson. J. A., 44(76), 75 Haouz, A., 40(43), 74 Hara, Y., 4(40), 30 Harada, J., 39(13), 73 Harada, K., 27(102), 33 Haran, G., 205(113), 219 Harano, Y., 41(51), 74 Harp, G. D., 199(97), 219 Harris, S. A., 60(183), 78 Hartmann, H., 39(18, 19), 73, 74 Hase, W. L., 2(3), 26(3), 84(1), 84(7), 85(18), 116(18), 118(7), 124(1), 29, 119, 166 Hasegawa, J., 3(25), 30 Hasha, D. L., 84(8), 119 Hasson, T., 41(45), 74 Hattori, M., 36(2), 73 Haustein, E., 44(72), 75 Havenith, M., 2(12), 13(12), 28(12), 29 Haverd, V., 125(22), 167 Havlin, S., 109(104), 122 Hayashi, S., 63(243), 80 Hayashi, T., 13(64), 27(102), 32, 33 Hayward, S., 40(34–37), 45(34, 35), 46(82), 48(94), 49(94), 53(134, 135), 54(140, 142, 144), 55(163), 62(198), 63(202, 203), 105(94), 74, 75, 76, 77, 78, 79, 121 Head-Gordon, M., 66(272), 81 Helbing, J., 2(20), 27(20), 29 Hendrix, R. W., 41(49), 74 Henrick, K., 37(3), 38(3), 73 Henry, E. R., 58(177), 203(109), 78, 219 Henzler-Wildman, K., 41(53), 75 Herbert, J. M., 84(13), 119 Herek, J. L., 127(47), 128(47), 168 Hernandez, R., 144(85), 155(85), 169 Hertz, J.45(79), 75 Hespenheide, B., 40(33), 45(33), 74

229

Hess, B., 3(29), 54(149, 150), 30, 77 Heyden, A., 66(271), 81 Higo, J., 54(145), 77 Himeno, R., 72(323), 82 Hinde, R. J., 164(113), 190(8), 191(8), 169, 217 Hinsen, K., 48(91), 50(104), 76 Hinterdorfer, P., 60(185), 78 Hirakawa, M., 36(2), 73 Hirao, K., 5(45, 46), 14(66), 15(45, 66), 23(66), 31, 32 Hirata, F., 53(137), 77 Hirata, S., 5(45, 46), 15(45), 31 Hirata, Y., 85(24), 87(24), 119 Ho, C. R., 72(324), 82 Hochstrasser, R. M., 2(17), 13(63), 22(17), 203(109), 205(117), 29, 31, 219, 220 Hodoscek, M., 3(24), 17(24), 22(24), 30 Hofrichter, J., 58(177), 78 Holiday, R. J., 127(40), 167 Holmes, K. C., 67(276), 81 Holyoak, T., 60(188), 78 Honda, Y., 3(25), 30 Honeycutt, J. D., 101(84), 210(131), 211(131), 121, 220 Honma, K., 125(14), 167 Horak, D., 67(276), 81 Hori, G., 136(80), 137(80), 169 Horwich, A. L., 41(47), 74 Hoshino, K., 196(75), 218 Houston, P. L., 125(21), 167 Hratchian, H. P., 3(25), 30 Huber, R., 40(27), 74 Hudspeth, E., 84(10), 127(35), 119, 167 Hummer, G., 44(71), 63(250, 251), 68(288), 100(64, 65), 75, 80, 81, 120 Hünenberger, P. H., 54(155), 77 Huo, S., 67(277, 279, 280), 81

230

author index

Hynes, J. T., 2(6, 7, 9), 26(6, 7), 209(128), 29, 220 Iben, I. E. T., 101(75), 188(51), 195(66), 121, 218 Ichiye, T., 105(92), 200(100), 121, 219 Ierardi, D. J., 72(324), 82 Ihee, H., 40(41), 74 Ikeda, K., 54(145), 99(62), 133(71), 77, 120, 168 Ikeguchi, M., 27(100), 61(193–195), 62(193), 71(193–195), 105(95), 110(105), 33, 78, 79, 121, 122 Ille, F., 66(274), 67(274), 81 Im, W., 3(24), 17(24), 22(24), 30 Impey, R.W., 17(76), 32 Inaka, K., 39(14), 73 Inoue, G., 125(14), 167 Ioannidis, Y. E., 41(51), 74 Ionascu, D., 24(92), 27(92), 33 Irvine, A. M. L., 125(12, 13), 167 Ishida, M., 3(25), 30 Ishikura, T., 27(106), 33 Ishima, R., 43(67), 75 Isin, B., 51(122), 76 Isralewitz, B., 63(222, 223, 225, 226, 231), 79, 80 Itoh, M., 36(2), 73 Ivanov, I., 54(147), 77 Iwahara, J., 43(69), 75 Iwasaki, K., 50(112), 76 Iyengar, S. S., 3(25), 30 Izrailev, S., 63(231, 232), 80 Jacob, F., 56(171, 172), 78 Jacobsen, K. W., 65(264), 80 Jacoby, E., 63(204, 206), 79 Jaffé, C., 130(59, 62), 144(59, 62, 86), 147(86), 150(59, 62), 151(59, 102), 153(59), 160(59), 168, 169 Jager, M., 44(74), 75

Jain, E., 36(1), 73 Jansen, T. l. C., 13(64), 32 Jaramillo, J., 3(25), 30 Jarzynski, C., 63(252, 253), 80 Jelesarov, I., 58(179), 78 Jellinek, J., 192(15), 193(18), 194(18), 195(18), 217 Jensen, F., 63(254), 80 Jensen, M. O., 63(244, 247), 80 Jentsch, V., 187(37), 217 Jernigan, R. L., 50(114), 51(118, 120–121), 76 Jia, Y., 205(117), 220 Jiang, H., 63(235–237, 241, 248), 80 Jiménez, A., 26(93), 67(282), 33, 81 Johns, J. E., 127(37), 167 Johnson, B., 3(25), 30 Johnson, J. B., 195(67), 218 Johnson, J. E., 41(49), 74 Johnston, R. L., 101(87), 105(87), 106(87), 196(75), 121, 218 Jolliffe, I. T., 52(133), 77 Jonas, J., 84(8, 9), 119 Jónsson, H., 65(264), 80 Jordon, K. D., 116(118), 122 Jorgensen, W. L., 17(76), 32 Joseph, S., 63(221), 79 Joseph-McCarthy, D., 17(75), 32 Joti, Y., 39(25), 40(26), 73(332), 74, 82 Joyeux, M., 144(89), 169 Juanico, B., 28(113), 33 Jung, Y., 205(115), 220 Juraszek, J., 69(297), 81 Just, W., 154(107), 169 Juurlink, L. B. F., 127(41), 167 Ka, B. J., 12(55), 31 Kabsch, W., 40(40), 54(152, 153), 74, 77 Kajimoto, O., 125(14, 18–20, 23), 144(88), 149(88, 92), 167, 169

author index Kaledin, A. L., 14(70), 32 Kamagata, K., 205(116), 220 Kamata, T., 50(112), 76 Kamiya, N., 72(323), 82 Kanada, R., 72(327), 82 Kandt, C., 62(197), 79 Kanehisa, M., 36(2), 73 Kaneko, K., 100(73), 101(73), 131(66), 121, 168 Kantz, H., 187(37), 217 Karasawa, H., 5(46), 31 Karnchanaphanurach, P., 100(68), 205(114), 121, 219 Karplus, M., 3(24, 28), 17(24, 75), 22(24, 90), 48(85, 88, 90), 48(87, 89), 49(98), 60(187), 52(129–131), 53(129), 63(89, 214, 216), 64(260), 66(273), 105(92), 158(111), 196(72), 200(99, 100), 202(105), 203(107), 30, 32, 75, 76–81, 121, 169, 218, 219 Katayama, T., 36(2), 73 Kawai, S., 125(23), 130(62), 144(62, 86, 88), 147(86), 149(88, 92), 150(62), 167, 168, 169 Kawashima, S., 36(2), 73 Kay, L. E., 41(55), 42(58–60), 75 Kaye, S. L., 51(127), 77 Keating, K., 40(33), 45(33), 74 Keith, T., 3(25), 30 Kendrew, J. C., 22(87), 56(169), 32, 78 Kenis, P., 206(126), 220 Kenkre, V. M., 2(8), 29 Kent Wenger, R., 41(51), 74 Kern, D., 41(53), 75 Keshavamurthy, S., 85(28, 29), 87(28, 29), 94(29), 120 Keskin, O., 51(118), 76 Keyes, T., 3(36), 5(36), 11–12(36), 202(106), 30, 219 Khinchin, A. I., 180(35), 217

231

Kidera, A., 3(37), 4(38), 27(37, 100), 39(11, 14, 25), 40(26, 29), 45(78), 55(161, 162), 56(29), 61(193–195), 62(193), 69(290), 71(193–195, 316), 72(331), 105(95, 96), 110(105), 30, 73–75, 78, 79, 81, 82, 121, 122 Kikugawa, G., 72(323), 82 Killelea, D. R., 127(41), 167 Kim, J., 202(106), 219 Kim, S. K., 127(45, 46), 127(45, 46), 128(46), 154(106), 156(109), 168, 169 Kim, T., 63(229), 79 Kim, Y. C., 44(71), 75 Kim, Y. S., 13(63), 31 Kim, Z. H., 127(47), 128(47), 168 King, D. S., 125(10), 167 Kinoshita, M., 205(116), 220 Kitagawa, T., 2(18), 4(18), 23(18, 89), 24(89), 25(18), 27(102), 203(108), 29, 32, 33, 219 Kitao, A., 48(94), 49(94), 53(134, 135, 137), 55(163), 62(198), 73(332), 105(94), 76–79, 82, 121 Kitao, O., 3(25), 30 Klafter, J., 110(107), 205(121), 122, 220 Klein, M. L., 17(76), 32 Klene, M., 3(25), 30 Klepeis, J. L., 72(324), 82 Klimov, D. K., 209(130), 220 Kloczkowski, A., 50(114), 76 Kloster, M., 50(114), 76 Knapp, E. W., 39(24), 74 Knoester, J., 2(15, 16), 13(15, 16), 16(16), 29 Knox, J. E., 3(25), 30 Kobayashi, C., 63(239), 80 Kobayashi, T. J., 27(101), 33 Kobus, M., 13(65), 32 Koga, N., 72(326), 88

232

author index

Koike, R., 40(29), 56(29), 61(194), 71(194), 74, 78 Kolinski, A., 50(114), 105(97), 76, 121 Kolmogorov, A. N., 106(101), 122 Kolossváry, I., 72(324), 82 Komaromi, I., 3(25), 30 Komatsuzaki, T., 2(13), 27(97), 45(77), 88(54, 56), 88–93(55), 96–98(55), 94(59), 97(59), 100(72), 101(72, 83, 87–88), 105(87), 107(83), 108(72, 83), 109(83), 130(63–65), 132(67–69), 134(77, 78), 136(82), 141(77), 144(64, 87, 88), 149(88), 150–151(65), 150(82, 93–98, 100), 151(87, 93, 96, 103), 153(82), 154(103), 156(87, 93, 96), 160(93, 96), 187(62–64), 188(62–64), 196(75, 86), 200(122), 205(116), 206(122), 207(122), 210(122), 213(122, 138), 214(122, 138), 29, 33, 75, 120, 121, 168, 169, 218, 219, 220 Kong, F., 125(17), 167 Kong, Y., 48(92), 63(215–217), 76, 79 Konishi, T., 130(64), 144(64), 197(95), 168, 219 Kopidakis, G., 28(114), 33 Korabel, N., 86(41), 187(47), 120, 218 Koshland, D. E., Jr., 56(167, 168, 170), 78 Kostov, K. S., 101(88), 121 Kosztin, D., 63(223, 232), 79 Kou, S. C., 87(53), 120 Koyama, M., 27(102), 33 Koyama, Y. M., 27(101), 33 Kozlov, V. V., 147(91), 169 Kramers, H. A., 209(127), 220 Krammer, A., 63(225, 226), 79 Kraut, J., 40(44), 74 Krebs, W. G., 40(39), 74 Krilov, G., 12(56), 31

Krimm, S., 13(59), 31 Kroenke, C. D., 43(66), 75 Krüger, P., 63(204–206, 210), 79 Kruglik, S. G., 20(84), 32 Krumhansl, J. A., 100(64), 120 Kruus, E. J., 125(7), 167 Krzanowski, W. J., 206(125), 220 Kubo, M., 24(92), 27(92), 33 Kubo, R., 4(39), 180(27), 181(27), 30, 217 Kucheida, D., 39(24), 74 Kuchnir, L., 17(75), 32 Kuczera, K., 3(24), 17(24, 75), 22(24), 49(100), 30, 32, 76 Kudin, K. N., 3(25), 30 Kuharski, R. A., 84(2), 119 Kumar, A. T. N., 24(92), 27(92), 33 Kumar, S., 59(181), 78 Kundu, S., 39(15), 73 Kuppens, S., 63(208), 79 Kushick, J., 52(131), 77 Kushick, J. N., 52(130), 77 Kuskin, J. S., 72(324), 82 Kuzmin, M., 86(43), 120 la Cour Jansen, T., 2(15, 16), 13(15, 16), 16(16), 29 Lacapˇcre, J. J., 50(104, 105), 76 Lakomek, N.-A., 42(65), 43(65), 75 Lange, O. F., 42(65), 43(65), 55(157, 158), 75, 77 Lansing, J. C., 43(68), 75 Larson, R. H., 72(324), 82 Lau, F. T. K., 17(75), 32 Lau, W. L., 205(117), 220 Laughton, C. A., 60(183), 78 LaViolette, R. A., 202(104), 219 Law, R. J., 49(101), 63(219), 76, 79 Lawrence, C. P., 2(10), 13(62), 15(62), 29, 31 Layman, T., 72(324), 82 Lazaridis, T., 3(24), 17(24), 22(24), 30

author index Leckband, D., 63(228), 79 Leder, L., 58(179), 78 Lee, C., 13(63), 31 Lee, I-R., 127(48), 128(48), 168 Lee, J., 127(45), 168 Lee, J. P., 67(280), 81 Lee, K., 127(45), 168 Lee, K.-S., 127(45), 168 Lee, R., 40(35), 45(35), 74 Lee, R. A., 40(34, 37), 45(34), 74 Leff, H. S., 87(48), 188(58), 120, 218 Lehnert, U., 50(113), 76 Lei, H., 54(146), 77 Leitner, D., 111(108), 122 Leitner, D. M., 2(1, 7, 12, 13), 5(43), 13(12), 26(7, 95), 27(107), 28(12), 84(3, 4, 6, 14), 85(19, 20), 86(45), 87(52), 100(67), 112(45, 109, 110), 113(45, 112, 117), 114(3, 19), 116(19, 20, 112, 118), 190(4), 196(78, 85), 200(78), 28, 29, 31, 33, 119–122, 216, 219 Lentfer, A., 40(40), 74 Lesk, A. M., 38(7), 73 Letokhov, V. S., 86(43), 120 Levine, B., 84(6), 119 Levine, R. D., 86(30), 87(30), 124(2), 120, 166 Levitt, M., 48(86), 64(259), 76, 80 Levy, R. M., 52(131), 77 Li, C. B., 2(13), 88–93(55), 88(54, 56), 94(59), 96–98(55), 97(59), 101(83, 87), 105(87), 106(87), 107–109(83), 130(65), 132(67–69), 144(87, 88), 149(88), 150(65, 100), 151(65, 87, 103), 154(103), 156(87), 187–188(62–64), 202(103), 29, 120, 121, 168, 169, 218, 219 Li, G., 49(96–99), 76 Li, H., 63(227), 125(17), 79, 167

233

Li, P., 20(82), 32 Li, S., 13(62), 15(62), 31 Li, W., 72(330), 82 Li, X., 3(25), 30 Liashenko, A., 3(25), 30 Lichetenberg, A. J., 176(3), 198(96), 216, 219 Lichtenberg, A.J., 99(61), 129(49), 132(49), 120, 168 Lidar, D. A., 5(43), 31 Lieberman, M. A., 99(61), 129(49), 132(49), 176(3), 198(96), 120, 168, 216, 219 Light, J. C., 124(3, 4), 167 Lightstone, F. C., 63(219), 79 Lim, M. H., 2(17), 22(17), 29 Lin, A. W., 41(45), 74 Lin, J., 41(51), 74 Lindahl, E., 3(29), 50(107), 30, 76 Lindorff-Larsen, K., 42(62), 75 Linssen, A. B. M., 53(136), 54(139), 62(199), 105(93), 77, 79, 121 Lipari, G., 41(57), 42(57), 75 Lipman, E. A., 205(112), 209(112), 219 Lipscomb, W. N., 63(216), 79 Liu, G., 3(25), 30 Liu, H., 48(93), 54(146), 63(225–227), 76, 77, 79 Liu, M. S., 50(111), 76 Liu, X., 63(241), 80 Liu, Z., 63(246), 80 Livny, M., 41(51), 74 Lo, M. W., 151(102), 169 Lockless, S. W., 27(104), 33 Logan, D. E., 86(44), 112(44), 120 Lohr, L. L., 191(12), 217 Lonascu, D., 24(92), 27(92), 33 Longarte, A., 27(96), 125(24), 127(24, 38), 196(84), 33, 167, 219 Loria, J. P., 43(66), 75

234

author index

Louie, T. M., 100(68), 205(114), 121, 219 Lounnas, V., 60(192), 63(234), 78, 80 Lourderaj, U., 84(7), 118(7), 119 Lovejoy, E. R., 154(106, 109), 156(110), 169 Lu, B., 49(101), 63(238), 76, 80 Lu, C., 24(92), 27(92), 33 Lu, H., 63(227), 79 Lu, H. P., 44(73), 75 Lu, J., 40(33), 45(33), 74 Lu, M., 39(12), 50(115), 73, 76 Ludemann, S. K., 60(192), 63(234), 78, 80 Luntz, A. C., 125(6), 167 Luo, G., 87(53), 100(68), 205(114), 120, 121, 219 Luo, R., 3(30), 30 Luo, X., 48(93), 63(235–237, 248), 76, 80 Ma, B., 59(181), 78 Ma, J., 3(24), 17(24), 22(24), 39(12), 48(88–90, 92), 49(98), 50(115), 63(89, 214–217), 30, 73, 76, 79 Ma, S., 63(220), 79 MacFadyen, J., 71(322), 82 Machlup, S., 70(314, 315), 82 MacKerell, A. D., Jr., 3(24), 17(24, 75), 22(24), 30, 32 Mackowiak, M., 84(9), 119 Mading, S., 41(51), 74 Madura, J., 17(76), 32 Maeda, M., 205(116), 220 Maisuradze, G. G., 100(67), 121 Májek, P., 70(310), 82 Malick, D. K., 3(25), 30 Mandelbrot, B. B., 106(102), 122 Maragliano, L., 68(285, 286), 81 Marcel Dekker, 46(82), 75 Marcus, R. A., 127(26), 167

Margolin, G., 86(42), 187(45), 120, 218 Mark, A. E., 3(29), 54(144, 155), 30, 77 Markley, J. L., 41(51), 74 Maroni, P., 127(42), 168 Marques, O., 3(32), 49(95), 30, 76 Marsden, J., 151(102), 169 Marston, C.C., 129(51), 130(57), 168 Marszalek, P. E., 63(227), 79 Martí, J., 69(300), 81 Martin, J., 20(84), 32 Martin, M. J., 36(1), 73 Martin, M. R., 127(39), 167 Martin, R. L., 3(25), 30 Martinez, T. J., 84(6), 119 Maspero, G., 99(63), 133(72), 120, 168 Mathews, D. H., 66(266), 81 Mátrai, J., 63(210), 79 Matsuda, S., 2(21), 9(21), 20(21), 29 Matsumoto, A., 50(112), 76 Matsunaga, Y., 2(13), 3(37), 27(37), 45(78), 55(162), 69(290), 101(83, 87, 88), 105–106(87), 105(96), 107–109(83), 130(65), 150(65), 151(65), 196(75, 86), 29, 30, 75, 78, 81, 121, 168, 218, 219 Matsushima, M., 39(14), 73 Matthews, K. S., 63(215), 79 Mattos, C., 17(75), 32 Maxwell, J. C., 87(46), 188(23), 120, 217 Maziuk, D., 41(51), 74 Mazzino, A., 100(69), 110(69), 121 McCammon, J. A., 49(101), 54(147), 60(190), 63(213, 238, 240), 67(278), 200(99), 76, 77, 78, 79, 80, 81, 219 McDonald, J. D., 113(113), 122 McElheny, D., 43(68), 75

author index McGarvey, P., 36(1), 73 McLeavey, C., 72(324), 82 McWhorter, D. A., 84(10), 127(34, 35), 119, 167 Meador, W. E., 40(30), 74 Means, A. R., 40(30), 74 Mehta, M.A., 130(57, 58), 168 Meiler, J., 42(64, 65), 43(65), 70(308), 75, 82 Mello, P. A., 113(114), 122 Melton, J. S., 39(15), 73 Mennucci, B., 3(25), 30 Merz, K. M., Jr., 3(30), 30 Mesters, J., 38(4), 73 Metzler, R., 72(324), 110(107), 82, 122 Meuwly, M., 22(90), 32 Michalet, X., 44(74), 75 Michielin, O., 63(230), 79 Michnick, S., 17(75), 32 Mikami, T., 2(11), 11(11), 29 Mikhailov, A. S., 28(112), 33 Mikkelsen, K. V., 2(4), 26(4), 29 Milburn, D., 40(33), 45(33), 50(113), 60(184), 74, 76, 78 Millam, J. M., 3(25), 30 Miller, M. A., 101(86), 105(86), 106(86), 121 Miller, R. J. D., 25(91), 27(98), 33 Miller, T. F., III, 68(287), 81 Miller, W. H., 144(84, 85), 155(85), 169 Miller, Z., 41(51), 74 Millet, O., 42(60), 75 Milligan, R. A., 41(45, 46), 74 Millonas, M. M., 87(50), 188(59), 120, 218 Mills, G., 65(264), 80 Milstein, C., 58(178), 78 Min, W., 87(53), 120 Mitomo, D., 54(145), 77 Mittermaier, A., 41(55), 75

235

Miyaguchi, T., 86(38), 187(44), 120, 218 Miyashita, O., 4(38), 30 Mizutani, Y., 2(18), 4(18), 23(18), 23–24(89), 25(18), 27(102), 203(108), 29, 32, 33, 219 Moffat, K., 40(41), 74 Moliner, V., 2(6), 26(6), 29 Moller, K. B., 2(9), 29 Monod, J., 56(171, 172), 57(176), 78 Montgomery, J. A., Jr., 3(25), 30 Moore, C. B., 154(106), 156(109, 110), 169 Moraes, M. A., 72(324), 82 Moretto, A., 5(42), 14(42), 18(42), 31 Morgan, J. D., 67(278), 81 Moritsugu, K., 4(38), 55(161), 69(290), 72(329, 331), 30, 78, 81, 82 Morokuma, K., 3(25), 30 Morriss, G., 186(33), 217 Moser, J., 150(101), 160(101), 169 Mouawad, L., 3(33), 30 Mountain, R. D., 197(90, 91), 219 Mowbray, S. L., 40(32), 74 Mozzarelli, A., 58(177), 78 Mu, Y., 54(151), 77 Muhandiram, D. R., 42(60), 75 Mukamel, S., 13(64), 32 Muller, C. W., 40(28), 74 Müller, U., 14(71), 32 Munier-Lehmann, H., 40(43), 74 Munson, K., 63(219), 79 Muratore-Ginanneschi, P., 100(69), 110(69), 121 Murdock, J., 134(79), 169 Murrel, J. N., 137(83), 169 Myshakin, E. M., 116(118), 122 Nagaoka, M., 4(40), 30 Nagayama, K., 3(31), 87(51), 30, 120 Nagy, A., 27(98), 33

236

author index

Nagy, A. M., 25(91), 33 Nair, P., 127(37), 167 Nakai, H., 3(25), 30 Nakajima, T., 3(25), 30 Nakamura, H., 37(3), 38(3), 72(323, 325), 73, 82 Nakamura, H. K., 54(145), 77 Nakamura, S., 110(106), 122 Nakano, A., 66(265), 81 Nakasako, M., 40(26), 74 Nakatani, E., 41(51), 74 Nakatsuji, H., 3(25), 30 Nanayakkara, A., 3(25), 30 Naritomi, Y., 73(333), 82 Navrotskaya, I., 12(55), 31 Negrerie, M., 20(84), 32 Neishtadt, A. I., 147(91), 169 Némethy, G., 56(170), 57(170), 78 Nevo, R., 60(185), 78 Neya, S., 27(102), 33 Ngo, T., 17(75), 32 Nguyen, D. T., 17(75), 32 Nguyen, P. H., 5(42), 13(65), 14(42), 18(42), 54(151), 55(159, 160), 31, 32, 77, 78 Nicholson, L. K., 42(59), 75 Niefer, B. I., 125(7), 167 Nilsson, L., 3(24), 17(24), 22(24), 30 Nilsson, L. M., 63(242), 80 Nishikawa, T., 3(28), 48(84), 30, 75 Nitzan, A., 2(2), 28(108), 29, 33 Niu, C., 63(237), 80 Noé, F., 66(274), 67(274), 81 Noguti, T., 3(28), 48(84), 30, 75 Nordholm, S., 84(5), 119 Northrup, S. H., 2(7), 26(7), 67(278), 29, 81 Norton, R. S., 50(111), 76

Nussinov, R., 59(181), 78 Oberhauser, A. F., 63(227), 79 Ochterski, J. W., 3(25), 30 Ogilvie, J. P., 25(91), 33 Ohmine, I., 3(36), 5(36), 11(36), 12(36), 65(262), 85(24), 87(24), 30, 80, 119 Okazaki, I., 4(40), 30 Okazaki, K., 60(186), 72(327), 78, 82 Okazaki, S., 2(11), 11(11), 29 Okuda, S., 36(2), 73 Okumura, H., 4(40), 30 Olafson, B. D., 3(24), 17(24), 22(24), 30 Olender, R., 70(302, 308), 81, 82 Olmstead, W. M., 85(17), 119 Omori, S., 27(100), 61(195), 71(195), 33, 79 Onsager, L., 70(314, 315), 82 Onuchic, J. N., 106(98), 209(129), 121, 220 Onufriev, A., 3(30), 30 Orland, H., 71(318–320), 82 Ormos, P., 195(67), 218 Ortiz, J. V., 3(25), 30 Osterheld, T. H., 127(32), 167 Ostermann, A., 39(25), 74 Ota, M., 40(29), 56(29), 110(105), 74, 122 Ota, N., 27(103), 33 Ovchinnikov, V., 3(24), 17(24), 22(24), 30 Owrutsky, J. C., 28(110), 33 Oxtoby, D. W., 2(8), 29 Ozorio De Almeida, A.M., 130(57), 168 Paci, E., 3(24), 17(24), 22(24), 30 Pahl, R., 40(41), 74 Pai, E. F., 40(40), 74 Pakoulev, A., 2(19), 29

author index Palacián, J., 130(59, 62), 144(59, 62), 150(59, 62), 151(59), 153(59), 160(59), 168 Paladin, G., 101(76–78), 121 Palmer, A. G., III, 41(54), 43(66), 75 Pan, A. C., 69(289), 81 Pan, P. W., 211(135), 220 Pandey, A., 113(114), 122 Pang, A., 51(127), 77 Pang, Y., 2(19), 29 Papageorgopoulos, D. C., 127(42), 168 Paparella, F., 103(91), 121 Parak, F., 39(18, 19, 24, 25), 196(71), 73, 74, 218 Park, C. R., 125(8), 167 Park, S., 63(244), 80 Parrinello, M., 70(303, 304), 81, 82 Parrish, R. G., 22(87), 56(169), 32, 78 Parson, W. W., 26(94), 33 Passerone, D., 70(303, 304), 81, 82 Pastor, R. W., 3(24), 17(24), 22(24), 30 Pate, B. H., 84(10), 127(34, 35, 37), 119, 167 Pechukas, P., 124(3), 153(104), 167, 169 Pederiva, F., 71(318–320), 82 Peng, C. Y., 3(25), 30 Perahia, D., 3(33), 48(91), 30, 76, 77 Perera, R., 51(127), 77 Peters, B., 66(271), 81 Petersson, G. A., 3(25), 30 Peti, W., 42(64), 75 Petratos, K., 40(40), 74 Petrella, R. J., 3(24), 17(24), 22(24), 30 Petry, W., 39(22, 23), 74 Petsko, G. A., 38(8), 39(8, 17, 18, 20, 21), 40(40), 54(21), 73, 74 Pettitt, B. M., 52(129), 53(129), 77 Pfister, R., 5(42), 14(42), 18(42), 31

237

Phan, I., 36(1), 73 Phillips, D. C., 22(87), 56(169), 32, 78 Phillips, G. N., Jr., 39(15), 73 Piazza, F., 28(113), 33 Picaud, F., 63(218), 79 Pikovsky, A., 194(19), 217 Piryatinski, A., 2(10), 29 Pisano, P. J., 125(21), 167 Piskorz, P., 3(25), 30 Pluckthun, A., 58(180), 78 Politi, A., 194(19), 217 Pollak, E., 153(104), 169 Pomelli, C., 3(25), 30 Ponzi, D. R., 39(18), 73 Poole, R. K., 24(92), 27(92), 33 Poon, B. K., 39(12), 73 Pople, J. A., 3(25), 30 Porter, R. N., 158(111), 169 Post, C. B., 3(24), 17(24), 22(24), 30 Postma, J. P. M., 106(100), 211(133), 122, 220 Potts, A. R., 84(11), 127(36), 119, 167 Pouthier, V., 2(14), 29 Pratt, L. R., 69(294), 81 Priest, E. C., 72(324), 82 Procaccia, I., 190(5–7), 217 Prodhom, B., 17(75), 32 Prokhorenko, V., 27(98), 33 Prompers, J. J., 42(64), 54(156), 75, 77 Provenzale, A., 103(91), 121 Pryor, A. W., 39(9), 73 Pu, J. Z., 3(24), 17(24), 22(24), 30 Qi, G., 40(35), 45(35), 63(203), 74, 79 Quenneville, J., 84(6), 119 Quiocho, F. A., 39(12), 40(30, 31), 73, 74 Raab, J., 206(126), 220 Rabuck, A. D., 3(25), 30 Radhakrishnan, R., 69(298, 299), 81 Radons, G., 194(20), 217

238

author index

Raghavachari, K., 3(25), 30 Raicu, V., 25(91), 33 Rajagopal, S., 40(41), 74 Rajagopalan, P. R., 26(93), 33 Ramesh, S. G., 11(52), 31 Ramseyer, C., 63(218), 79 Ranganathan, R., 27(104), 33 Ranson, N. A., 41(47), 74 Raposo, E. P., 109(104), 122 Rapp, G., 40(40), 74 Rashkin, A., 197(93), 201(93), 219 Rashkin, A. B., 184(55), 218 Rasmussen, B. F., 39(21), 54(21), 74 Ratner, M. A., 5(44), 31 Rauhut, G., 15(73), 32 Razaz, M., 40(34), 45(34), 74 Rech, I., 100(68), 205(114), 121, 219 Redaschi, N., 36(1), 73 Rees, F. S., 127(37), 167 Rega, N., 3(25), 30 Reich, Z., 60(185), 78 Reichl, L. E., 154(108), 169 Reiher, W. E., 17(75), 32 Reinhardt, W. P., 133(73), 138(73), 163(112), 202(102), 168, 169, 219 Reinstein, J., 40(28), 74 Remington, S., 40(27), 74 Ren, W., 66(269, 270), 67(283), 68(283), 81 Rennekamp, G., 39(19), 74 Reuscher, H., 39(19), 74 Reuter, N., 50(104), 76 Rex, A. F., 87(48), 188(58), 120, 218 Rey, R., 2(9), 11(52), 29, 31 Rhee, A., 63(229), 79 Rhoades, E., 205(113), 219 Rice, S. A., 130(60), 130(64), 144(64), 168 Rigler, R., 205(119), 220 Ringe, D., 38(8), 39(8, 21), 54(21), 73, 74

Rizzo, T. R., 127(42), 168 Robb, M. A., 3(25), 30 Robertson, K. M., 63(245), 80 Roca, M., 2(6), 26(6), 29 Roccatano, D., 54(144), 63(202), 77, 79 Roche, O., 63(211), 79 Roder, H., 205(117), 220 Rodseth, L. E., 40(31), 74 Roitberg, A., 5(44), 31 Romesberg, F. E., 2(21), 9(21), 20(21), 29 Roper, D., 63(206), 79 Rosca, F., 20(83), 24(92), 27(92), 32, 33 Rose, J. P., 101(85), 105(85), 210(132), 211(132), 121, 220 Roseman, A. M., 41(47), 74 Rosenwaks, S., 127(44), 168 Ross, S. D., 151(102), 169 Rothstein, S. M., 211(135), 220 Roux, B., 3(24), 17(24, 75), 22(24), 63(212), 69(289, 291), 32, 79, 81 Ruelle, D., 185(49, 50), 186(50, 53), 218 Ruffo, S., 85(26), 120 Ruiz-Pernia, J. J., 2(6), 26(6), 29 Ryckaert, J.-P., 17(77), 32 Rylance, G. J., 101(87), 105(87), 106(87), 196(75), 121, 218 Saalfrank, P., 28(111), 33 Saam, J., 63(243), 80 Sacchi, M., 127(42), 168 Sachs, G., 63(219), 79 Sadus, R. J., 50(111), 76 Safer, D., 41(45), 74 Sage, J. T., 20(82), 32 Sagle, L. B., 2(21), 9(21), 20(21), 29 Sagnella, D. E., 4(41), 27(41), 184(56), 203(110), 31, 218, 219

author index Saibil, H. R., 41(47), 74 Saito, N., 4(39), 180(27), 181(27), 30, 217 Saito, S., 3(36), 5(36), 11(36), 12(36), 85(25), 85(24), 86(25), 87(24, 25), 30, 119, 120 Sakata, M., 39(13), 73 Sakuraba, S., 73(332), 82 Salmon, J. K., 72(324), 82 Salvador, P., 3(25), 30 Sanbonmatsu, K. Y., 63(221), 79 Sander, C., 48(86), 76 Sanejouand, Y.-H., 3(32), 28(113), 40(38), 49(95), 50(38, 103, 105), 30, 33, 74, 76 Sato, M., 2(11), 11(11), 27(100), 61(193), 62(193), 71(193), 105(95), 29, 33, 78, 121 Satoh, D., 110(106), 122 Sauder, D. G., 125(10, 11), 167 Sauke, T. B., 101(75), 188(51), 195(66), 121, 218 Sauke, T. S., 195(67), 218 Sawaya, M. R., 40(44), 74 Sayos, R., 125(15), 167 Scalmani, G., 3(25), 30 Schaefer, M., 3(24), 17(24), 22(24), 30 Scheek, R. M., 62(201), 79 Scheurer, Ch., 14(69), 32 Schlauderer, G. J., 40(28, 42), 74 Schlegel, H. B., 3(25), 30 Schlenkrich, M., 17(75), 32 Schleyer, P. v. R., 3(24), 17(24), 22(24), 30 Schlichting, I., 40(40), 74 Schlick, T., 63(212), 69(298, 299), 79, 81 Schlitter, J., 63(204–207, 209), 79 Schlögl, F., 185(29), 186(30), 217 Schmidt, J. R., 13(62), 15(62), 31 Schmidt, M., 40(41), 74 Schneider, V., 206(126), 220

239

Schnell, J. R., 43(68), 75 Schofield, S. A., 113(115, 116), 129(52), 122, 168 Scholl, R., 195(67), 218 Schotte, F., 40(41), 74 Schowen, R. L., 49(100), 76 Schreiber, T., 103(89), 121 Schröder, G. F., 42(65), 43(65), 75 Schuetz, P., 216(139), 220 Schuler, B., 44–45(75), 205(112), 209(112), 216(139), 75, 219, 220 Schulte, A., 195(67), 218 Schulte, C. F., 41(51), 74 Schulten, K., 63(222–228, 231–233, 243, 244, 247), 79, 80 Schulz, G. E., 40(28, 42), 74 Schuster, H. G., 154(107), 169 Schwieters, C. D., 44(70), 75 Schwille, P., 44(72), 75 Scuseria, G. E., 3(25), 30 Searle, M. A., 60(183), 78 Sega, M., 71(318–320), 82 Segal, D., 28(108), 33 Semmes, D. H., 127(29, 30), 167 Semparithi, A., 85(29), 87(29), 94(29), 120 Sen, T. Z., 50(114), 76 Sengers, J. V., 101(80), 121 Seong, N. H., 2(19), 29 Serva, M., 101(76), 121 Sezer, D., 69(289), 81 Shalloway, D., 196(73, 74), 218 Shan, Y., 72(324), 82 Sharff, A. J., 40(31), 74 Sharp, K., 27, (105), 33 Shaw, D. E., 72(324), 82 Sheeran, M., 20(83), 32 Shen, J., 48(93), 63(235–237, 248), 76, 80 Shen, L., 63(235), 80 Shen, Y., 48(92), 63(217), 76, 79 Shi, Q., 12(55), 31

240

author index

Shibata, T., 100(73), 101(73), 121 Shiga, M., 2(11), 11(11), 12(58), 17(58), 71(316), 29, 31, 82 Shigeto, S., 2(19), 29 Shimizu, A., 28(109), 33 Shimizu, K., 110(106), 122 Shindyalov, I. N., 38(6), 73 Shinkai, S., 86(35, 36), 187(42, 43), 120, 218 Shirikov, B. V., 133(76), 168 Shirts, R. B., 133(73), 138(73), 163(112), 202(102), 168, 169, 219 Shojiguchi, A., 88–93(54–56), 94(59), 96–98(55), 97(59), 132(67–69), 144(87), 150(100), 151(87), 156(87), 187–188(62–64), 202(103), 120, 168, 169, 218, 219 Shrivastava, I. H., 51(126), 63(249), 77, 80 Shudo, A., 85(25), 86(25), 87(25), 120 Shyamsunder, E., 101(75), 188(51), 195(66), 121, 218 Sibert, E. L., 11(52), 113(111), 31, 122 Sigler, P. B., 63(214), 79 Silbey, R., 205(115), 220 Silman, I., 63(236, 237), 80 Sim, E., 12(56), 31 Simmerling, C., 3(30), 30 Simons, J. P., 125(15, 16), 167 Sine, S. M., 54(147), 77 Singer, S. J., 84(2), 119 Sjodin, T., 20(83), 24(92), 27(92), 32, 33 Skinner, J. J., 27(105), 33 Skinner, J. L., 2(10), 13(62), 15(62), 29, 31 Skolnick, J., 105(97), 121 Skrynnikov, N. R., 42(60), 75 Sligar, S. G., 22(88), 24(92), 27(92), 45(81), 196(70), 32, 33, 75, 218

Sloan, J. J., 125(7), 167 Smith, I. W. M., 125(12, 13), 167 Smith, J. C., 199(98), 17(75), 54(138), 66(274), 67(274, 275, 276), 72(329), 77, 81, 82, 32, 219 Smith, R. R., 127(41), 167 Snow, C., 63(203), 79 Socci, N. D., 106(98), 209(129), 121, 220 Sokurenko, E. V., 63(242), 80 Sorensen, D. C., 39(15), 73 Sotomayor, M., 63(224), 79 Spengler, J., 72(324), 82 Spurlino, J. C., 40(31), 74 Srajer, V., 40(41), 74 Stanley, H. E., 109(104), 122 States, D. J., 3(24), 17(24), 22(24), 30 Stefanov, B. B., 3(25), 30 Steigemann, W., 39(18, 19), 73, 74 Steinfeld, J. I., 2(3), 26(3), 124(1), 29, 166 Stephenson, J. C., 125(10), 167 Stern, H. A., 70(306), 71(306), 82 Stern, P. S., 48(86), 76 Steven, A. C., 41(49), 74 Stewart, G. M., 113(113), 122 Stillinger, F. H., 196(69), 202(104), 218, 219 Stock, A. M., 39(21), 54(21), 74 Stock, G., 5(42), 6(47), 11(47), 13(47, 65), 14(42, 67, 71), 15(67), 17(47), 18(42, 47, 78), 19(47), 20(47), 23(67), 26(47), 54(151), 196(79), 31, 32, 77, 219 Stockner, T., 62(196), 79 Stote, R., 17(75), 22(90), 32 Strain, M. C., 3(25), 30 Stratmann, R. E., 3(25), 30 Stratt, R. M., 3(36), 5(36), 11(36), 12(36), 30 Straub, J. E., 2(5, 21, 22, 23, 36), 3(36), 4(41), 6(22), 9(21, 22),

author index 11(22, 36), 12(36), 13(72), 14(66, 67), 15(66, 67, 72), 16(72), 17(22, 75), 18(22), 19(72), 21(86), 22(90), 23(66, 67, 86), 25(22), 26(5), 27(41), 20(21, 23, 80, 81, 85), 21(23), 22(23), 28(23), 64(256), 67(277, 279, 280), 100(72), 101(72), 108(72), 127(37), 184(55, 56), 196(76, 77, 79–83), 197(92–94), 200(101), 201(93, 94), 202(106), 203(110), 29, 30, 31, 32, 80, 81, 167, 29, 121, 218, 219 Stuchebruckhov, A. A., 86(43), 120 Su, F. E., 98(60), 120 Suenram, R. D., 127(37), 167 Sugny, D., 144(89), 169 Sullivan, S. M., 60(188), 78 Sussman, J. L., 63(236, 237), 80 Suzek, B. E., 36(1), 73 Swaminathan, S., 3(24), 17(24), 22(24), 200(100), 30, 219 Sweeney, H. L., 41(45), 74 Swift, R. V., 60(190), 63(213), 78, 79 Swint-Kruse, L., 63(215), 79 Szabo, A., 41(57), 42(57, 58), 63(250, 251), 205(121), 75, 80, 220 Tachikawa, M., 12(58), 17(58), 31 Taiji, M., 72(323), 82 Tajkhorshid, E., 63(243, 244, 247), 80 Takada, S., 60(186), 63(239), 72(326, 327, 330), 78, 80, 82, 88 Takagi, J., 50(112), 76 Takahashi, S., 205(116), 220 Takahashi, T., 87(51), 125(23), 120, 167 Takano, M., 87(51), 120 Takayanagi, M., 4(40), 30 Takeuchi, H., 14(68), 32 Talaga, D. S., 205(117), 220 Tama, F., 3(32), 40(38), 41(50), 46(83), 49(95), 50(38), 30, 74–76

241

Tanaka, H., 65(262), 80 Tang, C., 43(69), 44(70, 71), 75 Tang, J., 205(117), 220 Tang, P., 63(246), 80 Tarus, B., 196(76, 77), 218 Tasumi, M., 13(60, 61), 14(68), 31, 32 Tejedor, C., 28(109), 33 Temiz, N. A., 51(123), 77 Terada, T., 69(290), 72(331), 110(106), 81, 82, 122 Teramoto, H., 27(97), 134(77, 78), 141(77), 33, 168 Terashima, T., 2(11), 11(11), 29 Theobald, M., 72(324), 82 Thirumalai, D., 5(43), 12(56), 50(110), 101(84), 184(55, 56), 196(76, 77), 197(89–94), 199(89), 201(93, 94), 202(89), 203(110), 209(130), 210(131), 211(131, 136), 31, 76, 121, 218–220 Thomas, A., 48(91), 76 Thomas, J. A., 206(124), 220 Thomas, W. E., 63(242), 80 Thornton, J., 56(165, 166), 78 Thorpe, M., 40(33), 45(33), 74 Tidor, B., 3(24), 17(24), 22(24), 30 Tieleman, D. P., 62(196, 197), 63(245), 79, 80 Tilton, R. F., Jr., 39(20), 74 Tirion, M. M., 3(34), 50(102), 51(102), 30, 76 Tobi, D., 51(124), 77 Toda, M., 2(13), 4(39), 88–93(55), 88(54, 56), 94(59), 96–98(55), 97(59), 99(62), 100(72), 101(72), 108(72), 130(61, 64, 65), 131(61), 132(67–69), 133(71), 144(64, 87, 88), 149(88), 150(65, 100), 151(65, 87, 103), 154(103), 156(87), 179(21, 22), 180(27), 181(27), 187(62–64), 188(62–64), 29, 30, 120, 121, 168, 169, 217, 218

242

author index

Toda, M., 88(54), 120 Todd, B. D., 50(111), 76 Togashi, Y., 28(112), 33 Tokimatsu, T., 36(2), 73 Tokmakoff, A., 2(8), 16(74), 29, 32 Tolmie, D. E., 41(51), 74 Tomasi, J., 3(25), 30 Tomoda, S., 27(101), 33 Toniolo, C., 5(42), 14(42), 18(42), 31 Tooze, J., 38(5), 73 Topper, R. Q., 130(58), 168 Torchia, D. A., 42(59), 43(67), 75 Torii, H., 13(60, 61), 31 Tournier, A. L., 54(138), 77 Towles, B., 72(324), 82 Toyota, K., 3(25), 30 Trautman, J. K., 205(111), 219 Trubnikov, D. N., 191(12), 217 Trucks, G. W., 3(25), 30 Tsai, C. J., 59(181), 78 Tsernoglou, D., 39(17), 73 Tsironis, G. P., 28(114), 33 Tsuchiya, T., 197(95), 219 Tsuda, I., 131(66), 168 Tsurumaki, H., 125(20), 167 Tsybin, Y. O., 2(14), 29 Tuckett, R. P., 125(12, 13), 167 Tung, C. S., 63(221), 79 Tunon, I., 2(6), 26(6), 29 Ueda, H. R., 27(101), 33 Ueno, J., 27(100), 61(193), 62(193), 71(193), 105(95), 33, 78, 121 Uitdehaag, J. C., 65(263), 70(263), 80 Ulrich, E. L., 41(51), 74 Ursby, T., 40(43), 74 Utz, A. L., 127(41), 167 Uzer, T., 85(21), 94(58), 130(59, 62), 144(59, 62, 86), 147(86), 150(59, 62), 151(59, 102), 153(59), 160(59), 119, 120, 168, 169

Valadié, H., 50(105), 76, Vale, R. D., 41(46), 74 Valle, M., 41(50), 74 van Aalten, D. M. F., 54(139, 141, 142), 62(199, 200), 77, 79 van der Spoel, D., 3(29), 54(140), 30, 77 van der Vaart, A., 64(258), 80 van der Veen, B. A., 65(263), 70(263), 80 van Gunsteren, W. F., 54(155), 106(100), 200(100), 211(133), 77, 122, 219, 220 Van Ness, J. W., 106(102), 122 van Nuland, N. A. J. 62(201), 79 Van Wynsberghe, A., 49(99), 76 Vanden-Eijnden, E., 66(269, 270), 67(283), 68(283–287), 81 Varotsis, C., 197(88), 219 Vekhter, B., 101(85), 105(85), 210(132), 211(132), 121, 220 Vela-Arevalo, L. V., 94(57), 120 Venable, R. M., 3(24), 17(24), 22(24), 30 Vendruscolo, M., 42(62), 75 Venturoli, M., 68(284), 81 Vergni, D., 101(82), 121 Verheyden, G., 63(210), 79 Vershik, A., 206(123), 220 Viappiani, C., 58(177), 78 Viswanathan, G. M., 109(104), 122 Vitkup, D., 202(105), 219 Vodopyanov, K., 127(39), 167 Vogel, H. J., 54(140), 62(196), 77, 79 Vogel, V., 63(225, 226), 63(242), 79, 80 von Deuster, C., 60(189), 78 Vonrhein, C., 40(42), 74 Vos, M. H., 20(84), 23(89), 24(89), 32 Voth, G. A., 3(25), 70(310), 72(328), 30, 82 Vreven, T., 3(25), 30

author index Vriend, G., 54(139, 143), 77 Vulpiani, A., 51(128), 100(69, 74), 101(76–78, 82), 102(74), 103(91), 106(74), 110(69), 77, 121 Vyas, N. K., 39(12), 73 Waalkens, H., 150(99), 151(99), 169 Wade, R. C., 60(192), 63(234), 78, 80 Wagner, D., 206(126), 220 Wako, H., 3(31), 30 Wales, D. J., 26(95), 84(14), 101(86), 105(86), 106(86), 125(25), 190(8, 9), 191(8), 196(75, 85), 33, 119, 121, 167, 217–219 Walter, K. F., 42(65), 43(65), 75 Walters, P., 174(1), 180(36), 216, 217 Wang, H., 54(147), 85(18), 116(18), 77, 119 Wang, M., 49(100), 76 Wang, Q., 39(12), 73 Wang, S. C., 72(324), 82 Wang, W., 20(83), 32 Wang, X., 63(241), 125(17), 80, 167 Wang, X.-J., 101(79), 106(79), 107(79), 121 Wang, Z., 2(19), 29 Warshel, A., 26(94), 33 Warth, T. E., 63(217), 79 Watanabe, D., 18(79), 32 Watanabe, M., 17(75), 32 Watkins, L. P., 44(76), 205(118), 75, 220 Weber, T. A., 196(69), 218 Weber-Bornhauser, S., 58(180), 78 Weikl, T. R., 60(189), 78 Weinan, E., 66(269, 270), 67(283), 68(283), 81 Weinstein, H., 70(310), 82 Weiss, D. R., 64(259), 80 Weiss, S., 44(74), 75 Weissig, H., 38(6), 73 Wells, A. L., 41(45), 74

243

Wells, S., 40(33), 45(33), 74 Wendt, H., 58(179), 78 Wereszczynski, J., 71(322), 82 Westbrook, J., 38(6), 73 Westley, M. S., 125(21), 167 Wharton, D., 24(92), 27(92), 33 Wiegand, G., 40(27), 74 Wiegel, F. W., 70(311), 82 Wiesenfeld, J. R., 125(8, 9), 167 Wiggins, S., 94(57, 58), 130(56, 59), 144(59), 150(59, 99), 151(59, 99), 153(59), 160(59), 120, 168, 169 Wikoff, W. R., 41(49), 74 Willis, B. T. M., 39(9), 73 Wilson, K. R., 70(301), 81 Wilson, C. A., 40(39), 74 Wilson, E. B., Jr., 3(26), 30 Wilson, K., 40(40), 74 Wilson, K. R., 70(301), 81 Wilson, S., 63(212), 79 Windshugel, B., 67(276), 81 Winter, P. R., 27(96), 127(38), 196(84), 33, 167, 219 Wiorkiewicz-Kuczera, J., 17(75), 32 Witkoskie, J. B., 205(120), 220 Wittinghofer, A., 40(40), 74 Wolfram, S., 103(90), 121 Wolfrum, J., 124(5), 167 Wollmer, A., 63(204, 206), 79 Wolynes, P. G., 2(7), 26(7), 11(53), 13(53), 22(88), 45(81), 84(3, 6), 85(19), 86(44, 45), 106(98), 112(44, 45, 109, 110), 113(45, 112, 115–117), 114(3), 116(19, 112), 129(52), 195(65), 196(70), 200(99), 209(129), 29, 31, 32, 75, 119–122, 168, 218–221 Won, Y., 3(24), 17(24), 22(24), 30 Wong, C. F., 63(238), 80 Wong, M. W., 3(25), 30 Wong, S. S. M., 113(114), 122 Wong, V., 86(31), 87(31), 120

244

author index

Woodcock, H. L., 3(24), 17(24), 22(24), 30 Woods, R., 3(30), 30 Woolf, T. B., 71(321), 82 Wriggers, W., 63(233), 80 Wright, P. E., 41(56), 43(68), 75 Wroblowski, B., 63(207, 209), 79 Wu, C., 54(146), 77 Wu, X., 3(24), 17(24), 22(24), 30 Wulff, M., 40(41), 74 Wuttke, R., 216(139), 220 Wyatt, R. E., 113(115, 116), 122 Wyckoff, H., 22(87), 56(169), 32, 78 Wyman, J., 57(176), 78 Xie, X. S., 87(53), 100(68), 205(111, 114), 120, 121, 219 Xiong, B., 48(93), 76 Xu, C., 51(124), 77 Xu, Y., 63(235–237, 241, 246), 80 Xu, Z., 62(197), 63(214), 79 Xun, L., 100(68), 205(114), 121, 219 Yagi, K., 5(45), 9(50), 14(66, 67), 15(45, 66, 67), 18(79), 23(66, 67), 196(79), 31, 32, 219 Yamagishi, A., 54(145), 77 Yamaguchi, Y. Y., 85(23, 26, 27), 119, 120 Yamanishi, Y., 36(2), 73 Yamashita, T., 144(88), 149(88, 92), 169 Yamato, T., 27(106), 33 Yang, H., 44(76), 100(68), 205(114, 118), 75, 121, 219, 220 Yang, H.-L., 194(20), 217 Yang, L., 63(212), 79 Yang, L.-W., 51(125), 77 Yang, S., 69(291), 81 Yang, W., 3(24), 17(24), 22(24), 30 Yang, X.-F., 125(12), 167

Yanguas, P., 130(59, 62), 144(59, 62), 150(59, 62), 151(59), 153(59), 160(59), 168 Yao, H., 41(51), 74 Yao, S., 50(111), 76 Yaris, R., 105(97), 121 Yazyev, O., 3(25), 30 Ye, X., 20(83), 24(92), 27(92), 32, 33 Yeh, S., 24(92), 27(92), 33 Yin, D., 17(75), 32 Yin, Y., 63(247), 80 Yip, C. M., 63(229), 79 Yonetani, T., 24(92), 27(92), 33 Yonezawa, Y., 72(323, 325), 82 Yoon, A., 127(40), 167 York, D. M., 3(24), 17(24), 22(24), 30 Yoshioka, C., 41(46), 74 Young, C., 72(324), 82 Young, P. E., 42(59), 75 Young, R. D., 101(75), 188(51), 195(66), 196(71), 121, 218 Yu, E. Z., 40(33), 45(33), 74 Yu, H., 40(39), 125(17), 74, 167 Yu, X., 2(12), 5(43), 13(12), 28(12), 87(52), 29, 31, 120 Yuge, T., 28(109), 33 Yura, K., 50, (112), 76 Yurtsever, E., 191(10, 11), 217 Zaccai, G., 39(23), 74 Zakrzewski, V. G., 3(25), 30 Zanni, M. T., 2(17), 22(17), 29 Zare, R. N., 127(39), 167 Zaslavsky, G. M., 85(22), 87(49), 99(22), 132(70), 188(60, 61), 119, 120, 168, 218 Zerbe, O., 5(42), 14(42), 18(42), 31 Zewail, A. H., 127(47), 128(47), 168 Zhang, D., 63(240), 80

author index Zhang, J. Z. H., 113(115), 122 Zhang, Y., 2(21, 22), 4(41), 6(22), 9(21, 22), 11(22), 13(72), 15(72), 16(72), 17(22), 18(22), 19(72), 20(21, 81), 21(86), 23(86), 25(22), 27(41), 196(80, 82, 83), 29, 31, 32, 219 Zhao, M., 130(60), 168 Zheng, M., 63(248), 80 Zheng, W., 50(106, 108–110), 76 Zhong, Q., 28(110), 33 Zhong, W., 63(220), 79 Zhou, Y., 202(105), 219

245

Zhu, Q., 125(17), 167 Zhu, W., 63(237, 248), 80 Zhuang, W., 13(64), 32 Zimmermann, J., 2(21), 9(21), 20(21), 29 Zou, H., 63(248), 80 Zscherp, C., 13(59), 31 Zuckerman, D. M., 71(321), 82 Zweil, A. H., 127(27, 29, 30, 46), 128(46), 167, 168 Zwier, T. S., 27(96, 99), 84(13), 33, 125(24, 25), 127(24, 38, 43), 196(84), 119, 167, 168, 219

SUBJECT INDEX Ab initio calculations, for heme-mimicking molecule, 22 N-Acetyl-tryptophan methyl amide (NATMA), 125 Action space, 93 average locations, 93 diffusivity in, 91, 92 resonant regions, 89, 128, 129 trajectory in, 95, 96 Adenylate kinase, 205 Allosteric interaction, 56 Angular excitation functions, 22 Antibody–antigen interactions, 58 Argon clusters, 100 Arnold web, 88, 89, 95, 97, 128–129, 131 nonuniformity of, 93, 132, 188 of primary resonances, 89 wavelet analysis, 94–95 Biased molecular dynamics (MD) simulations, 37 Birkhoff ’s individual ergodicity theorem, 180–181 Brownian motion, 100, 107–109

Cryptoergodicity, 177, 178 Cytochrome c in water, VER dynamics, 20–21 Decay rate constant, 116 Degree-of-freedom systems, 118 Diffusion, 86 abnormal, 205 in action space, 91 anisotropic thermal, 27 anomaly in, 99–101 Arnold, 85, 133 Brownian type of, 86, 87, 205 constant, 197–200 normal, 93, 95, 98, 132 in protein, 5 random, 54 Dynamical reaction theory Arnold web, 128–129 dynamical connections, 131–132 mixed phase space, fractional behavior in, 132–133 normally hyperbolic invariant manifolds, 129–131 Energy transfer (relaxation) phenomena, 2, 4 reduced model, using normal modes, 4

Central limit theorem, 110, 183 Confocal microscopy, 205 Cryptochaos, 177

Advancing Theory for Kinetics and Dynamics of Complex, Many-Dimensional Systems: Clusters and Proteins, Advances in Chemical Physics, Volume 145, Edited by Tamiki Komatsuzaki, R. Stephen Berry, and David M. Leitner. © 2011 John Wiley & Sons, Inc. Published 2011 by John Wiley & Sons, Inc.

247

248

subject index

Equilibrium dynamics of proteins, 45–46. See also Protein dynamics Ergodicity, 86, 87, 173–175 exploring proteins in state space, 195–197 force metric as probe, 200 of curvature in energy landscape, 200–202 in isomerization of small clusters, 189–195 effective Hausdorff dimension, 190 Gram-Schmidt vectors, 193 Hessian matrices, 192 K-entropy, 191 Lyapunov exponents, 189–195 kinetic energy metric as probe, 197–199 of equipartitioning and quasiequilibrium, 197–199 of internal friction, 199–200 multiplicity, in complex systems, 176–177 probing heterogeneity of energy flow pathways, 203–204 problem in real systems, 177–179 property, dynamical systems possess, 174–175 Error-doubling experiments, 103, 106 Escaping process, 110 Exponential decay, 87, 89–93, 95, 97, 99, 126, 132 FeP-Im, optimized structure, 23 time-dependent perturbation theory, 24–25 vibrational energy transfer pathways identification, 25 Fermi resonance, 4 parameter, 15, 19, 25

Fermi’s-golden-rule, 2 Ferric-binding protein, structural change, 62 Finite-size Lyapunov exponent (FSLE), 101–105 Finite-time Lyapunov exponent, 100 Flavin reductase, 205 Fluorescence resonance energy transfer (FRET), 44, 45, 205, 210 Fokker–Planck equation, 70 Fourier law, 2 Fourier spectra, 89 Fourier transformation, 94 Fractional Brownian motion (fBm), 106 Fractional Fokker–Planck equation (FFPE), 110 Free energy landscape extraction of, 204–205 utilizing canonical transition state theory (TST), 208–209 revisiting the concept, 208–210 Functional motions of proteins, 37 Gas-phase SN 2 reaction, 85, 86 Gaussian network model (GNM), 51 Grote–Hynes theory, 2 Hamiltonian dynamics, 87 Hamiltonian systems, 86, 88, 99, 132, 134, 173, 174, 176, 180 Hellinger distance, 98 Heme cooling, 4 Hemoglobin. See also Nonequilibrium dynamics, of proteins allosteric interactions in, 57 Eigen’s generalization of allosteric ligand binding in, 57 MWC and KNF model, explaining cooperative phenomenon, 58 Hénon–Heiles system, 100, 133

subject index

249

Hilbert space, 7 Hölder exponent, 106

Lyapunov exponent (LE), 100 Lyapunov index, 141

Intramolecular vibrational energy relaxation (IVR), 2 Intrinsic reaction coordinate (IRC) calculations, 63, 64. See also Reaction coordinate Iso-1-cytochrome c, 205 Iterative mixed-basis diagonalization (DIMB) methods, 3

Maradudin–Fein theory, 5 Markovian theory, 2 Maxwell’s demon, 86–87 Mean square displacements (MSDs), 100 N-Methylacetamide VER/IVR dynamics, 11, 13 in vacuum, 14–15 in water cluster, 15–16 in water solvent, 17–20 Microcanonical unimolecular reaction rate, 113 Mixed phase space, 187 fractional behavior in, 132–133 (See also Non-Brownian phase space dynamics) on isomerization kinetics, 85 Mixing, 175 condition, mathematical representation, 175–176 MMK theory of energy transfer, 4 Modest-sized organic molecules isomerization kinetics, 84 Molecular dynamics trajectories, analysis, 51–52. See also Protein dynamics beyond quasiharmonic approximation, 54–55 principal component analysis, 52–54 quasiharmonic approximation, 52 Multiscale nonlinear systems, hierarchical dynamics case study of model protein, 105–110 characterization of escaping process, 110 coarse-grained dynamics, 106 error-doubling measurement, 108

Jumping-among-minima (JAM) model with PCA, 55 Kirchhoff matrix, 51 Kolmogorov formula, 106 Kolmogorov-Sinai (KS) entropy, 100, 176 Kramer’s theory, 2 Landau–Teller formula, 11 Langevin dynamics, 70 Lie canonical perturbation theory (LCPT), 134 Liouville theorem, 173 Local equilibrium state. See also Free energy landscape dependence of LES/non-LES on time window, 214 extraction of, 204–205 checking conditions, 208 constructing set of state candidates from time series s(t), 206–207 from single-molecule time series, 205–208 using Kantorovich metric, 206 Local random matrix theory (LRMT), 86, 110, 111–113 applications, 111 Lorentzian spectra, 98

250

subject index

Multiscale nonlinear systems, hierarchical dynamics (Continued ) FSLE along each PC, evaluation, 106, 108, 109 generalized Langevin approach, 110 occurrence of superdiffusion, 109 NMR spectroscopy, 38 Non-Brownian phase space dynamics classical systems with mixed phase space, fractional behavior, 87–99 Arnold web, wavelet analysis, 94–95 dynamical connection revealed by wavelet analysis, 95–99 fractional behavior of reactions, 89–93 reaction rate constants, nonexistence of, 99 quantum systems, energy flow and localization with, 110–118 gas-phase SN 2 reaction, 116–118 local random matrix theory, 111–113 RRKM theory, LRMT dynamical corrections to, 113–115 spatiotemporal multiscale classical systems, anomaly in diffusion, 99–110 finite-size Lyapunov exponent, 101–105 hierarchical dynamics complexity, model protein case study, 105–110 principal component analysis, 105 Nonequilibrium dynamics, of proteins, 55. See also Protein dynamics

biased molecular dynamics simulation, 62–63 models for protein functions, 55 allosteric interaction model, 56–57 allosteric interactions, in hemoglobin, 57–58 induced-fit model, 56 linear response model, 60–62 lock-and-key model, 55–56 protein–ligand interactions, general model, 58–60 Nonequilibrium MD simulation of energy flow from excited amide, 5 of solvated myoglobin, 4 Nonergodic statistical theory (NEST), 114 Nonergodic systems statistical properties, 84 Nonlinear spectroscopy, 2 Non-Markovian conformational dynamics, 205 Non-Markovian perturbative formulas, 5 Non-Markovian theory, 2, 3 Non-Markovian VER formulas, derivation, 6. See also VER formulas relaxing mode multidimensional, coupled to static bath, 6–9 one-dimensional, coupled to a fluctuating bath, 9–11 Nonstatistical behaviors, in chemical reactions, 125 No-return transition state (TS) bifurcation of, 157–160 robust persistence and chaotic breakdown, 160–165 Normal form theory

subject index dimension reduction scheme, 143–149 vibrational energy transfer, 149–150 Normally hyperbolic invariant manifolds (NHIMs), 88, 95, 97, 125 bifurcation and breakdown, 150–151 (See also No-return transition state (TS)) harmonic region, 151–152 nonlinear quasiperiodic region, 152–154 strong-chaotic (stochastic) region, 156–157 weak-chaotic region, 154–156 Normal mode analysis (NMA), 37 Normal mode calculations, for protein, 3 Nuclear magnetic resonance (NMR) spectroscopy, 37, 41–44 Participation number, 117 Path sampling, for biomolecules, 37, 69 action-based methods, 69–71 transition path sampling, 69 Path search, for biomolecules, 63–64 minimum energy path search conjugate peak refinement (CPR), 66–67 nudged elastic band method, 65–66 self-penaltywalk methods, 64–65 zero-temperature string method, 66 path search at finite temperature coarse-grained string method, 68–69 finite-temperature string methods, 67–68 MaxFlux methods, 67

251

PCA. See Principal component analysis (PCA) Porter–Thomas distribution, 113 Power law decay, 95 Fourier spectrum for, 91 trajectory, resonance structures, 96 Principal component analysis (PCA), 37, 52–55, 100, 101, 105, 106 Protein dynamics, 37, 38, 100 elastic network models, 50–51 Hookean pairwise energy function, 50 Kirchhoff matrix describing connectivity, 51 perturbation approach, 50 potential function of system, 51 experiments on, 38 NMR spectroscopy, 41–44 single-molecule spectroscopy, 44–45 X-ray crystallography, 38–41 normal mode analysis, 46–49 applications, 48–49 Protein equilibrium dynamics, 37 Quantum chaos, 99 Quantum coupled oscillator Hamiltonian, 111 Quantum ergodic systems, 113 Quantum nonergodic statistical theory (QuNEST), 117, 118 Quantum number space, 112 Quasiharmonic approximation, 46, 52, 53, 61, 62, 71 QuNEST reaction rate, F− -CH3 Cl complex, 117 Rate constant, 87 Reaction coordinate, 26, 46, 64, 88, 89, 98, 100, 115, 116, 126, 128–130, 132, 151, 156, 172, 187

252

subject index

Remnants of invariants buried in phase space of many-degrees-of-freedom systems, 133–136 isomerization reaction, HCN molecule, 137–138 technique to detect, in sea of chaos, 136–137 buried in potential well of HCN, 138–143 Rice–Ramsperger–Kassel–Marcus (RRKM) theory, 84, 85, 111, 119, 126 condition for, 86 limitation of, 127–128 LRMT dynamical corrections, 113–115 Schrödinger equation, 9 Self-consistent analysis, 112 Sinai-Ruelle-Bowen (SRB) measure, 179, 185 Single-molecule spectroscopy, 38, 44–45 Spectrogram, 94 Statistical reaction theory, 126, 179 biomolecules as Maxwell’s demon, 188–189 dynamical origin, traditional ideas, 180 Birkhoff ’s individual ergodicity theorem, 180–181 requirement of ergodicity, 181–183 dynamical system theory, new developments, 185–188 concept of Gibbs measure to NHIMs, 187

SRB measure, 185–186 openness and/or inhomogeneity, 183–185 SRB measure and infinite ergodic theory, 179 Stochastic differential equation in length (SDEL), 71 Survival probability, 89, 90 Transient resonance structures, 98 Transmission coefficient, 115, 118 Vibrational energy relaxation (VER), 2 applications to vibrational modes in biomolecules, 12 cytochrome c in water, 20, 21 N-methylacetamide, 12–20 porphyrin, 21–25 future perspective, 26–28 limitations, 11, 12 Vibrational Hamiltonian, 111 Vibrational self-consistent field (VSCF) theory, 9 Vibrational spectroscopy, 5 Von Neumann equation, 9 Wavelet analysis, 95 of Arnold web, 94–95 dynamical connection revealed by, 95–99 X-ray crystallography, 37, 38–41 Yaglom noise, 107 Zero-order energy, 112 Zero-order vibrational states, 114