# Dissertation Thomas Neusius 2009

inauguraldissertation zur Erlangung der Doktorwürde der Naturwissenschaftlich-Mathematischen Gesamtfakultät der Ruprecht-Karls-Universität Heidelberg Vorgelegt von Diplom-Physiker Thomas Neusius aus Neuwied Tag der mündlichen Prüfung: 24. Juni 2009 ii Thermal Fluctuations of Biomolecules An Approach to Understand Subdiffusion in the Internal Dynamics of Peptides and Proteins Gutachter: Prof. Dr. Jeremy C. Smith Prof. Dr. Heinz Horner iv Zusammenfassung Auf der Basis von Molekulardynamik-Simulationen werden die thermischen Fluktuationen in den inneren Freiheitsgraden von Biomolekülen, wie Oligopeptiden oder einer β-Schleife, untersucht, unter besonderer Beachtung der zeitgemittelten, mittleren quadratischen Verschiebung (MSD). Die Simulationen lassen in einem Bereich von 10−12 bis 10−8 s Subdiﬀusion im thermischen Gleichgewicht erkennen. Mögliche Modelle subdiﬀusiver Fluktuationen werden vorgestellt und diskutiert. Der zeitkontinuierliche Zufallslauf (CTRW), dessen Wartezeitenverteilung einem Potenzgesetz folgt, wird als mögliches Modell untersucht. Während das ensemble-gemittelte MSD eines freien CTRW Subdiﬀusion zeigt, ist dies beim zeitgemittelten MSD nicht der Fall. Hier wird gezeigt, daß der CTRW in einem begrenzten Volumen ein zeitgemitteltes MSD aufweist, das ab einer kritischen Zeit subdiﬀusiv ist. Eine analytische Näherung des zeitgemittelten MSD wird hergeleitet und mit Computersimulationen von CTRW-Prozessen verglichen. Ein Vergleich der Parameter des begrenzten CTRW mit den Ergebnissen der Molekulardynamik-Simulationen zeigt, daß CTRW die Subdiﬀusion der inneren Freiheitgrade nicht erklären kann. Eher muß die Subdiﬀusion als Konsequenz der fraktalartigen Struktur des zugänglichen Volumens im Konﬁgurationsraum betrachtet werden. Mit Hilfe von Übergangsmatrizen kann der Konﬁgurationsraum durch ein Netzwerkmodell angenähert werden. Die Hausdorﬀ-Dimension der Netzwerke liegt im fraktalen Bereich. Die Netzwerkmodelle erlauben eine gute Modellierung der Kinetik auf Zeitskalen ab 100 ps. Abstract Molecular dynamics (MD) simulations are used to analyze the thermal ﬂuctuations in the internal coordinates of biomolecules, such as oligopeptide chains or a β-hairpin, with a focus on the time-averaged mean squared displacement (MSD). The simulations reveal conﬁgurational subdiﬀusion at equilibrium in the range from 10−12 to 10−8 s. Potential models of subdiﬀusive ﬂuctuations are discussed. Continuous time random walks (CTRW) with a power-law distribution of waiting times are examined as a potential model of subdiﬀusion. Whereas the ensemble-averaged MSD of an unbounded CTRW exhibits subdiﬀusion, the time-averaged MSD does not. Here, we demonstrate that in a bounded (conﬁned) CTRW the timeaveraged MSD indeed exhibits subdiﬀusion beyond a critical time. An analytical approximation to the time-averaged MSD is derived and compared to a numerical MSD obtained from simulations of the model. Comparing the parameters of the conﬁned CTRW to the results obtained from the MD trajectories, the CTRW is disqualiﬁed as a model of the subdiﬀusive internal ﬂuctuations. Subdiﬀusion arises rather from the fractal-like structure of the accessible conﬁguration space. Employing transition matrices, the conﬁguration space is represented by a network, the Hausdorﬀ dimensions of which are found to be fractal. The network representation allows the kinetics to be correctly reproduced on time scales of 100 ps and above. Meinen Eltern. viii Few people who are not actually practitioners of a mature science realize how much mop-up work of this sort a paradigm leaves to be done or quite how fascinating such work can prove in the execution. And these points need to be understood. Mopping-up operations are what engage most scientists throughout their careers. They constitute what I am here calling normal science. Thomas S. Kuhn Vorwort Die vorliegende Arbeit – wie sollte es anders sein – konnte nur gelingen Dank der Unterstützung und Hilfe, die ich in den letzten gut vier Jahren erfahren habe. An erster Stelle ist mein Betreuer Prof. Jeremy C. Smith zu nennen, der mir die Möglichkeit gegeben hat, mich mit einem spannenden Thema zu beschäftigen, meine eigenen Ansätze mit großem Freiraum zu entwickeln und immer weiter zu vertiefen. Insbesondere seine Geduld bei bei Erstellung der Manuskripte für die geplanten Veröffentlichungen war bemerkenswert. Weiterhin danke ich Prof. Igor M. Sokolov von der Humboldt-Universität zu Berlin, der mir bedeutende Anregungen geben hat und mich früh bestärkte, das Thema weiterzuverfolgen – eine wichtige Motivation zum richtigen Zeitpunkt. Isabella Daidone hat mir freundlicherweise ihre umfangreichen MD-Simulationen zur Verfügung gestellt und mir bei vielen Detailfragen weitergeholfen. Diskussionen mit ihr, sowie mit Frank Noé und Dieter Krachtus waren wesentlicher Anlaß, das Thema immer wieder von verschiedenen Seiten zu betrachten. Nadia Elgobashi-Meinhardt hat mit ihren Korrekturen des Dissertationsmanuskripts wesentlich dazu beigetragen, daß der Text sich einer lesbaren Form zumindest angenähert hat. Lars Meinhold stand mir immer wieder mit Rat und Tat zur Seite. Ein besonderer Dank geht an Thomas Splettstößer. Seine spontane Quartierwahl im winterlichen Knoxville hat mich vermutlich vor Depressionen bewahrt. Bei allen Mitarbeitern der CMB-Gruppe am Interdisziplinären Zentrum für wissenschaftliches Rechnen möchte ich mich für die gute Zusammenarbeit und die Unterstützung bei Problemen bedanken. Ellen Vogel war eine unverzichtbare Hilfe bei allen formalen Dingen, insbesondere beim Thema Dienstreisen. Bogdan Costescu hat die Rechner der Arbeitsgruppe am Laufen gehalten; ohne ihn wären wir schon lange aufgeschmissen. Ohne die Unterstützung meiner Eltern wäre ein Studium in Heidelberg kaum möglich gewesen. Schließlich möchte ich mich bei meiner Frau Chrisi für ihr Verständnis für meine längeren Abwesenheiten und die moralische Stärkung während der Durststrecken bedanken, die wohl jedes Promotionsprojekt mit sich bringt. Oak Ridge, Tennessee, im März 2009 Thomas Neusius x Contents Zusammenfassung v Abstract vi Vorwort ix 1 Introduction 1.1 The dynamics of biomolecules . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Thermal Fluctuations 2.1 Very brief history of the kinetic theory 2.2 Basic concepts . . . . . . . . . . . . . 2.3 Brownian motion . . . . . . . . . . . . 2.4 From random walks to diﬀusion . . . . 2.5 Anomalous diﬀusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Molecular Dynamics Simulations of Biomolecules 3.1 Numerical integration . . . . . . . . . . . . . . . . 3.2 Force ﬁeld . . . . . . . . . . . . . . . . . . . . . . . 3.3 The energy landscape . . . . . . . . . . . . . . . . 3.4 Normal Mode Analysis (NMA) . . . . . . . . . . . 3.5 Principal Component Analysis (PCA) . . . . . . . 3.6 Convergence . . . . . . . . . . . . . . . . . . . . . . 4 Study of Biomolecular Dynamics 4.1 Setup of the MD simulations . . . . 4.2 Thermal ﬂuctuations in biomolecules 4.3 Thermal kinetics of biomolecules . . 4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3 5 . . . . . 7 7 9 12 17 19 . . . . . . 21 23 25 31 33 34 36 . . . . 39 39 41 50 56 5 Modeling Subdiffusion 59 5.1 Zwanzig’s projection formalism . . . . . . . . . . . . . . . . . . . . . . . 59 5.2 Chain dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 xii CONTENTS 5.3 5.4 5.5 The Continuous Time Random Walk (CTRW) . . . . . . . . . . 5.3.1 Ensemble averages and CTRW . . . . . . . . . . . . . . . 5.3.2 Time averages and CTRW . . . . . . . . . . . . . . . . . . 5.3.3 Conﬁned CTRW . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Application of CTRW to internal biomolecular dynamics . Diﬀusion on networks . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 From the energy landscape to transition networks . . . . 5.4.2 Networks as Markov chain models . . . . . . . . . . . . . 5.4.3 Subdiﬀusion in fractal networks . . . . . . . . . . . . . . . 5.4.4 Transition networks as models of the conﬁguration space . 5.4.5 Eigenvector-decomposition and diﬀusion in networks . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 67 70 75 77 80 80 81 83 88 94 98 6 Concluding Remarks and Outlook 101 6.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 6.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 References 105 Abbreviations and Symbols 113 Index 115 Appendix A Diﬀusion equation solved by Fourier decomposition B Constrained Verlet algorithm – LINCS . . . . . . . C Derivation of the Rouse ACF . . . . . . . . . . . . D Fractional diﬀusion equation . . . . . . . . . . . . . D.1 Gamma function . . . . . . . . . . . . . . . D.2 Fractional derivatives . . . . . . . . . . . . D.3 The fractional diﬀusion equation . . . . . . D.4 The Mittag-Leﬄer function . . . . . . . . . E Exponentials and power laws . . . . . . . . . . . . Zitate in den Kapitelüberschriften . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 119 120 123 124 124 125 126 127 128 130 Jedoch das Gebiet, welches der unbedingten Herrschaft der vollendeten Wissenschaft unterworfen werden kann, ist leider sehr eng, und schon die organische Welt entzieht sich ihm größtenteils. Hermann von Helmholtz Chapter 1 Introduction Some of the most fruitful physical concepts were motivated by biological or medical observations. In particular, thermodynamics owe some of their fundamental insights to the physiological interests of their pioneers. Julius Robert von Mayer was a medical doctor by training and the ﬁrst to realize that heat is nothing else than a form of energy. Also Hermann von Helmholtz’s interest in thermodynamics was closely connected to his physiological studies. The discovery of the motion of pollen grains by the botanist Robert Brown in 1827 [1] and the development of the diﬀusion equation by the physiologist Adolf Fick in 1855 [2] are fundamental landmarks for the understanding of transport phenomena. In the framework of the kinetic theory of heat, Albert Einstein could demonstrate in 1905 [3] that the atomistic structure of matter brings about thermal ﬂuctuations, i.e., the irregular, chaotic trembling of atoms and molecules. The thermal ﬂuctuations give rise to the zig-zag motion, e.g. of pollen grains or other small particles suspended in a ﬂuid. In 1908, Jean Perrin’s skillful experimental observations revealed that Brownian motion is to be understood as a molecular ﬂuctuation phenomenon, as suggested by Einstein [4]. Brownian motion is an extremely fundamental process, and it is still a very active ﬁeld of research [5]. Thermal ﬂuctuations are microscopic eﬀects. However, what is observed on the macroscopic level are thermodynamic quantities like, e.g., temperature, pressure, and heat capacity. Statistical physics, as developed by Gibbs and Boltzmann, based the thermodynamic quantities on microscopic processes and derived the equations of thermodynamics as limiting cases for inﬁnitely many particles. In so doing, Fick’s diﬀusion equation can be obtained from Brownian motion. Thus, diﬀusion processes are powered by thermal energy. The diﬀusion equation turned out to be an approximate description of a fairly large class of microscopic and other processes, in which randomness has a critical inﬂuence, e.g., osmosis, the spontaneous mixing of gases, the charge current in metals, or disease spreading. In particular, diﬀusion is an important biological transport mechanism on the cellular level. Among the microscopic processes giving rise to diﬀusion, one of the most simple scenarios is the random walk model. Due to its simplicity, the random walk proves to 2 Introduction be applicable in a wide range of diﬀerent situations. Whenever microscopic processes can be described reasonably with random walk models, the diﬀusion equation can be employed for the modeling of the collective dynamics of a large number of such processes. Despite the fruitfulness of the diﬀusion equation, deviations from the classical diffusion became apparent as experimental skills advanced. The need to broaden the concept of diﬀusion led to the development of the fractional diffusion equation [6–8] and the corresponding microscopic processes like, e.g., the continuous time random walk (CTRW) [8, 9]. The wider approach paved the way to apply stochastic models to an even larger variety of situations in which randomness is present. A major challenge of contemporary science is the understanding of the behavior of proteins and peptides. Those biomolecules are among the smallest building blocks of life1 . They are employed by the cells for a plethora of functions, e.g. metabolism, signal transduction, transport or mechanical work. Proteins and peptides are chains of amino acids. Proteins usually form a stable conformation, but this does not mean that they are static objects. As proteins (and peptides) are molecular objects they are directly inﬂuenced by the presence of thermal ﬂuctuations; their precise structure undergoes dynamical changes due to strokes and kicks from the surrounding molecules [10]. Hence, any description of the internal dynamics of proteins and peptides must take into account the inﬂuence of thermal agitation, a task that can only be accomplished in statistical terms. The thesis at hand presents one small step towards an understanding of the principles that govern internal dynamics of peptides and proteins. The viewpoint taken up in this work is to look at the most general features of the thermal ﬂuctuations seen in peptides and proteins. More precisely, this work is focused on the long time correlations in the dynamics of biomolecules. In particular, the decelerated time dependence of the mean squared displacement (MSD) is studied in some detail – a phenomenon referred to as subdiﬀusivity. A thorough understanding of the possible stochastic models is required in order to assess their fruitfulness in the context of biomolecular, thermal motion. On that account, this work dwells mainly on the stochastic models, some of which may be of interest not only in biophysics. The theoretical considerations are compared to molecular dynamics (MD) simulations of peptides and proteins. One of the main result of this thesis is that the structure of the accessible volume in conﬁgurational space is responsible for the subdiﬀusive MSD of the biomolecule rather than the distribution of barrier heights or the local minima’s eﬀective depths. Instead of trap models or CTRW, which rely on the statistics of trap depths or waiting times, respectively, and ignore the geometry of the conﬁguration space, we suggest network models of the conﬁguration space. 1 The name protein – πρώτ ειoς (proteios) means in ancient Greek I take the first place“ – em” phasizes its central role for life. The denomination had been introduced by the Swedish chemist Jöns Jakob Berzelius in 1838. 1.1 The dynamics of biomolecules 1.1 3 The dynamics of biomolecules Peptides and proteins are built from a set of twenty-two amino acids (called residues) [11]. All amino acids contain a carboxyl group and an amino group. Two amino acids can form a chemical bond, the peptide bond, by releasing a water molecule and coupling the carboxyl group of one residue to the amino group of an adjacent residue. In that way, chain-like molecules can be built with up to thousands of residues2 . The chain of peptide bonds is referred to as the backbone of the molecule. The cell synthesizes the sequence of a protein (primary structure) which is encoded in the deoxyribonucleic acid (DNA). In aqueous solution proteins spontaneously assume a compact spacial structure; they fold into the natural conformation or native state (secondary structure). The natural conformation is determined by the sequence of the protein and stabilized by the formation of hydrogen bonds. It is an object of current research to discover how the molecule is able to ﬁnd its natural conformation as quickly as it does in nature and in experiment, the problem of protein folding. Peptides are smaller than proteins and usually they do not fold into a ﬁxed three dimensional structure. The number of residues needed to make up a protein varies, but the commonly used terminology speaks of proteins starting from 20 to 30 residues. Experimental techniques that allow the observation of biomolecules on an atomic level started being developed in the 1960s. The ﬁrst protein structures to be resolved by X-ray crystallography were myoglobin [14] and hemoglobin [15]. For their achievement the Nobel Prize was awarded to John Kendrew and Max Perutz in 1962. X-ray crystallography, small-angle X-ray scattering and nuclear magnetic resonance (NMR) techniques are used for structure determination. Dynamical information is provided, e.g., by NMR, infrared spectroscopy, Raman spectroscopy and quasi-elastic neutron scattering. Fluorescence methods give access to single molecule dynamics [10]. Besides experiment and theory a third approach has been established in the ﬁeld of the dynamics of biomolecules. Computer simulation is a widespread tool for exploring the behavior of molecules. Computational methods allow mathematical models to be exploited, the complexity of which does not permit analytical solutions. Computer simulations help to bridge the gap between theory and experiment. In particular, MD simulations are useful with respect to dynamics: they produce a time series of coordinates of the molecule investigated, the so-called trajectory, by a numerical integration of the Newtonian equations of motion. The numerical integration gives access to the detailed time behavior [16–18]. The conformation of proteins is a prerequisite for the understanding of the biological function for which the molecule is designed. However, the protein is not frozen in the natural conformation. Rather the conformation is the ensemble of several, nearly isoenergetic conformational substates [19–22]. Thus, at ambient temperature, the protein maintains a ﬂexibility that is necessary for its biological function [10]. The subconformations can be seen as local minima on the high-dimensional complex energy landscape. 2 Most proteins have some hundreds of residues. The largest known polypeptide is titin, with more than twenty thousand residues [12, 13]. 4 Introduction The substates are separated by energetic barriers, a crossing of which occurs due to thermal activation [23]. There are further classiﬁcations of the substates into a hierarchical organization [10, 20, 22, 24]. The properties of the energy landscape determine the dynamical behavior of the molecule. The concept of an energy landscape was developed in the context of the dynamics of glass formers [25]. Indeed, the dynamics of proteins exhibit some similarities with the dynamics of glasses [20, 26], e.g. non-exponential relaxation patterns [27–29] and non-Arrhenius-like temperature dependence [29, 30]. At a temperature Tg ≈ 200 K a transition of the dynamical behavior is observed for many proteins; below Tg the amplitude of the thermal ﬂuctuations increases proportional to the temperature. Above Tg , a sharp increase of the overall ﬂuctuations is revealed by quasi-elastic neutron scattering [31–34], X-ray scattering [35, 36], and Mößbauer spectroscopy [37–41]. Water plays a dominant role in the dynamical transition, also sometimes referred to as glass transition [29, 34, 42–45]. It is controversial whether the protein ceases to function below Tg [36, 46, 47]. Recently, tetrahertz time domain spectroscopy revealed the dynamical transition of poly-alanine, a peptide without secondary structure [48]. The glass-like behavior of proteins is seen in a variety of diﬀerent molecules, such as myoglobin, lysozyme, tRNA, and polyalanine [31, 33, 34, 48]. The presence of glassy dynamics in very diﬀerent systems indicates that various peptides and proteins, albeit diﬀerent in size and shape, share similar dynamical features [48]. Due to the presence of strong thermal ﬂuctuations in the environment of biomolecules, theoretical attempts to model the dynamics have to include stochastic contributions. The main focus of this work is on subdiffusion in the internal dynamics, i.e., a behavior that is characterized by a MSD that exhibits a time dependence of ∼ tα (where 0 < α < 1) rather than a linear time dependence. This non-linear behavior is seen in a variety of experiments and was also found in MD simulations [49]: • The rebinding of carbon monoxide and myoglobin upon ﬂash dissociation follows a stretched exponential pattern [23]. The non-exponential relaxation indicates fractional dynamics [50]. • In single-molecule experiments, the ﬂuorescence lifetime of a ﬂavin quenched by electron transfer from a tyrosine residue allows the distance ﬂuctuations between the two residues to be measured. The statistics of the distance ﬂuctuations has revealed long-lasting autocorrelations corresponding to subdiﬀusive dynamics [27, 28, 51, 52]. • Spin echo neutron scattering experiments on lysozyme reveal long-lasting correlations attributed to fractional dynamics [53, 54]. • In MD simulations, lysozyme in aqueous solution exhibits subdiﬀusive behavior [53]. There have been diﬀerent approaches to describing subdiﬀusive dynamics and to understanding the mechanism by which it emerges. 1.2 Thesis outline 1.2 5 Thesis outline This thesis is organized as follows. The fundamental concept of diﬀusion theory, its connection to kinetic theory, Brownian motion, and random walk models are reviewed in Chap. 2. The methodology of MD simulations and common analysis tools are assembled in Chap. 3. The sections in Chap. 3 cover the integration algorithm, the terms included in the interaction potential, the energy landscape picture, and the techniques of normal mode analysis and principal component analysis. Sec. 3.6 brieﬂy comments on the problem of convergence. In Chap. 4, we present the MD simulations performed for a β-hairpin molecule folding into a β-turn and oligopeptide chains lacking a speciﬁc native state. After a description of the simulation set up in Sec. 4.1, the thermal ﬂuctuations seen in the MD simulations are analyzed with respect to the principal components. The potentials of mean force are obtained and characterized with respect to delocalization and anharmonicity. Some of the diﬀerent techniques employed for the modeling of the water dynamics in MD simulations, including explicit water, Langevin thermostat, and the generalized Born model, are compared in Sec. 4.2. In particular, the accuracy of the implicit water simulations with respect to the kinetic aspects is assessed by a comparison to the explicit water simulation of the same system. Internal ﬂuctuations of biomolecules have been reported to be subdiﬀusive; the claim is based on experimental and MD simulation results. Therefore, we address the following question: Is subdiffusivity a general feature of internal fluctuations in biomolecules? The kinetic aspects of the MD trajectories are discussed with particular emphasis on the MSD, which is found to be subdiﬀusive. The next chapter, Chap. 5, is dedicated to the central question of the present thesis. What is the mechanism that causes the internal dynamics of biomolecules to be subdiffusive? The chapter is subdivided in ﬁve sections. The ﬁrst section reviews brieﬂy the projection operator approach, as developed by R. Zwanzig and H. Mori in the 1960s. The projection operator approach describes the dynamics with a reduced set of dynamical coordinates, considered to be relevant. It is demonstrated by Zwanzig’s approach that correlations emerge from the ignorance against dynamical coordinates deemed irrelevant. As a consequence, the dynamics appear to exhibit time correlations and to be non-Markovian. Sec. 5.2 deals with the Rouse chain polymer model as an example of a harmonic chain which exhibits subdiﬀusive distance ﬂuctuations. Trap models, prominently the CTRW model, are examined in detail in Sec. 5.3. As the CTRW model has been discussed as a candidate for modeling subdiﬀusive, biomolecular motion, we ask: Can the CTRW model exhibit subdiffusivity in the time-averaged MSD? 6 Introduction We emphasize that subdiﬀusivity in the time-averaged MSD is a prerequisite for the application of CTRWs to MD simulations, as the trajectories provide time-averaged quantities. However, the interest in the question may be not conﬁned to the context of biomolecular dynamics, as CTRWs are used in various, very diﬀerent, physical ﬁelds. The answer to the above question prompts to the ergodicity breaking found in the CTRW with a power-law tailed distribution of waiting times. The non-ergodicity of the model requires a careful treatment of time and ensemble averaging procedures. The inﬂuence of a ﬁnite volume boundary condition is examined. The theoretical results are corroborated by extensive simulations of the CTRW model. An alternative model of subdiﬀusive ﬂuctuations is based on the network representation of the conﬁguration space. The network approach is presented in Sec. 5.4. The network representation allows the time evolution of the molecules to be reproduced by a Markov model. The essential input for the Markov model is the transition matrix, obtained from a suitable discretization of the conﬁguration space trajectory. The features of the conﬁguration space networks are analyzed in terms of the fractal dimensions. The predictions of the network models for the kinetics are compared to the original MD trajectories. Secs. 5.3 and 5.4 contain the principal results obtained in the present thesis. Chap. 6 summarizes the results presented and brieﬂy discusses potential perspectives for future work. Eng ist die Welt, und das Gehirn ist weit. Leicht beieinander wohnen die Gedanken, doch hart im Raume stoßen sich die Sachen. Friedrich Schiller Chapter 2 Thermal Fluctuations In the following chapter, we give a brief overview of the history of the kinetic theory and introduce the fundamental concepts from statistical mechanics, which form the basis of the present thesis. In particular, we present the theory of Brownian motion, the random walk and the phenomenon of anomalous diﬀusion. 2.1 Very brief history of the kinetic theory Joseph Fourier found in 1807 that the conduction of heat is proportional to the temperature gradient. Together with the continuity equation he derived the heat equation for a temperature ﬁeld, T (x, t), ∂2T ∂T =κ 2. (2.1) ∂t ∂x Fourier also developed a method to solve the heat equation, the Fourier decomposition of periodic functions [55]. Fourier’s work was an important contribution to the physics of heat, although, at the time, it was not yet clear what heat actually is. Independently, James Prescott Joule (in 1843) and Robert Mayer (in 1842) identiﬁed heat as a form of energy. On the basis of these works Hermann von Helmholtz established the ﬁrst law of thermodynamics in 1847 [56]. In 1855 the physiologist Adolf Fick found Fourier’s heat equation to govern also the dynamics of diﬀusing concentrations [2]. The second law of thermodynamics was implicitly formulated by Rudolf Clausius in 1850. He also promoted in 1857 the idea of heat emerging from the kinetic energy of small particles, which he called molecules, i.e. small constituents of matter [57]. Later, in 1865, he introduced the concept of entropy to quantify irreversibility [58]. Although the idea of an atomistic structure of matter was suggested earlier – apart from philosophical speculations, Daniel Bernoulli, for example, had had similar speculations in the 18th century – Clausius’ work initiated a new branch in physics, kinetic theory, primarily applied to the thermodynamics of gases. The appeal of kinetic theory arose from its power to unify diﬀerent physical ﬁelds: kinetic theory raised the hope of thermodynamics being based on mechanics [56]. 8 Thermal Fluctuations In the following years several results of thermodynamics could be reproduced with kinetic theory, like the speciﬁc heat of one-atomic gases or the ideal gas law. James Clerk Maxwell derived the ﬁrst statistical law in physics, the velocity distribution of molecules in a gas. One of the great feats of kinetic theory was the H-theorem of Ludwig Boltzmann in 1872 [59]. The atomic picture led him to conclude that the second law of thermodynamics is merely a statistical law. Therefore, a decrease in entropy is unlikely but not a priori impossible. Boltzmann’s result was received with great skepticism1 . As Max Planck remarked2 , Boltzmann did not acknowledge the pivotal role of molecular disorder in his proof of the H-theorem. This inattention may have been the reason why Boltzmann did not attempt to prove kinetic theory on the microscopic level. Boltzmann also considered diﬀerent averaging procedures in his work and argued that averages over an ensemble and averages over time must coincide for most, but not all, mechanical systems [61]. Boltzmann assumed a more far-reaching version, the ergodic hypothesis, which was later disproved by Johann von Neumann and George David Birkhoﬀ, who established the mathematical theory of ergodicity. However, Boltzmann intuitively captured an important property of a large class of physical systems [56]. As a doctoral student at the University of Zürich, Albert Einstein worked on the measurement of molecular weights. In 1905, he published an article about the motion of particles suspended in a resting ﬂuid [3]. Einstein realized that the thermal motion of the molecules in a liquid, as stated by kinetic theory, gives rise to the motion of suspended beads which are large enough to be visible under a microscope. He proposed this eﬀect as an ultimate touchstone of kinetic theory and conjectured it could be identical to Brownian motion, which had been discovered eighty years earlier [1]. At the same time, Marian von Smoluchowski at Lemberg worked on similar questions [62]. The concept of random walks, a term coined by Karl Pearson, was discussed in the journal Nature [63]. In 1908, Paul Langevin developed the idea of stochastic diﬀerential equations [64], a seminal approach to Brownian motion and other problems involving randomness. Einstein’s work laid the basis for Jean Perrin’s observations of Brownian particles in 1909 [4]. Perrin, an experimentalist from Paris, had developed a new type of microscope that allowed him to measure the Brownian motion with a precision previously unattained. His validation of Einstein’s theoretical predictions was among the results which earned him the Nobel Prize in physics in 1926. The discovery of the thermodynamic origin of Brownian motion was the ﬁrst quantitative observation of microscopic thermal ﬂuctuations. 1 The younger Max Planck criticized Boltzmann as Planck did not accept the statistical nature of the second law. Planck doubted the existence of atoms but later changed his mind. Besides Planck’s assistant Ernst Zermelo, who criticized Boltzmann on mathematical arguments, the chemist Wilhelm Ostwald and the physicist-philosopher Ernst Mach were the ones who fiercely denied the existence of atoms. Ostwald claimed to have been convinced otherwise by the results of Perrin’s work. 2 In der Tat fehlte in der Rechnung von Boltzmann die Erwähnung der für die Gültigkeit seines ” Theorems unentbehrlichen Voraussetzung der molekularen Unordnung.“[60] 2.2 Basic concepts 2.2 9 Basic concepts Next, we turn our attention to the basic concept employed in statistical mechanics and diﬀusion theory. Let a system consist of N particles with masses mi . The Cartesian positions of the particles are merged to the 3N position vector, r (t). The coordinates are chosen such that the center of mass is always at the origin. The velocities form the 3N vector, v (t). The 6N dimensional space of the combined position-velocity vector is the phase space, sometimes denoted as Γ and referred to as Γ -space. At each time, t, the point in the phase space, (r (t), v (t)) deﬁnes the state of the system. Let M be the diagonal mass matrix with the components Mij = δij mj . The time evolution of the system is given by Newton’s second principle d2 M 2r = f , (2.2) dt or by equivalent schemes such as Lagrange’s or Hamilton’s formalism. The solution (r (t), v (t)) can be seen as a trajectory in the phase space which is parametrized by the time, t. These phase space trajectories do not intersect. The phase space is ﬁlled by a vector ﬁeld which is analogous to the ﬂow ﬁeld of an incompressible ﬂuid, as seen from the Liouville theorem [65]. 15 15 B 10 10 5 5 velocity v velocity v A 0 0 −5 −5 −10 −10 −15 −5 −4 −3 −2 −1 0 1 position x 2 3 4 5 −15 −5 −4 −3 −2 −1 0 1 position x 2 3 4 5 Figure 2.1: Phase space of the harmonic oscillator. - A: The phase space trajectories of the harmonic oscillator are ellipsoid and form closed orbits due to the periodicity of the process. B: Phase space trajectories of the harmonic oscillator with friction (underdamped case). All trajectories oscillate but approach the equilibrium (x, v) = (0, 0) for t → ∞. Note that adding friction violates the conservation of energy. As an example, the one-dimensional harmonic oscillator with angular frequency ω has the equations of motion mv̇ = −mω 2 r ṙ = v, (2.3) (2.4) where the dots correspond to time derivatives. These equations deﬁne an elliptic ﬁeld in the phase space, cf. Fig. 2.1. If the state of the system at some point is known, i.e., 10 Thermal Fluctuations if the position and velocity are known at some time, the trajectory of the oscillator is deﬁned. Often, however, it is diﬃcult to describe the deﬁnite state of the system, as this would require complete knowledge of all positions and velocities. Rather, the state of the system is described by a probability density, ρ(r , v , t), characterizing the regions of phase space in which the system is likely to be found. The probability density deﬁnes an ensemble. The members of the ensemble are referred to as microstates; the distribution ρ represents a macrostate of the system3 . Statistical mechanics provides the theoretical framework for dealing with phase space probability densities. It bases macroscopic quantities, like temperature and pressure, on the microscopic probability density [65]. For a quantity, φ(r , v ), depending on the microstate of the system, the ensemble average is deﬁned as Z φ(r , v )ρeq (r , v )dΓ, (2.5) hφiens = Γ where the integration over the Γ -space is denoted as dΓ = d3N rd3N v. Imagine a gas of rigid spheres whose only interactions are collisions. Assume the gas is enclosed in a box of ﬁnite volume with elastically reﬂecting walls at constant temperature. The path traveled by an individual particle, its trajectory, is a zig-zag line: when it collides with a second particle it changes its velocity and is scattered in another direction until it bumps into the next particle and so on. The microscopic picture shows “that the real particles in nature continually jiggling and bouncing, turning and twisting around one another,” as Richard Feynman described it [67]. The permanent agitation brings about temporal variations of observable quantities, the thermal fluctuations. The ﬂuctuation of a quantity, φ(r , v ), depending on the positions and velocities, is deﬁned as 2 ∆φ = [φ(r , v ) − hφ(r , v )iens ]2 ens . (2.6) All microscopic degrees of freedom are ﬂuctuating as a consequence of thermal agitation. A key result from kinetic theory is the equipartition theorem. It states that in macroscopic equilibrium every microscopic degree of freedom entering the Hamilton function quadratically, undergoes ﬂuctuations such that the corresponding energy equals kB T /2, where kB is the Boltzmann constant and T the thermodynamic temperature [65, 68, 69]. Hence, a gas or liquid is never at rest on a molecular level4 ; molecules in a gas or liquid keep themselves permanently agitated. The temperature T is a measure of the energy content per degree of freedom. Since the average velocity of an arbitrary particle is zero, applying the equipartition 3 Strictly speaking, the definition of a microstate requires establishing a probability measure in the phase space, Γ . The measure on the phase space emerges from a coarse-graining or a concept of relevance [66]. 4 The case of a temperature lim T → 0 can be treated correctly only in the framework of quantum mechanics – a subject which is not touched in the present thesis. 2.2 Basic concepts 11 theorem, the average kinetic energy per molecule along one direction is 1 2 1 m ∆v = kB T . 2 2 (2.7) Note that the particle has this energy in all three spacial dimensions, so the total kinetic energy per particle is three times as much as in Eq. (2.7). In the equipartition theorem the notion of macroscopic equilibrium deserves some attention. A thermodynamic equilibrium state has the property of being constant in time. The second law states that thermodynamic processes evolve in time towards an equilibrium state. A system that has reached equilibrium will not leave this state unless an external perturbation occurs. The microscopic picture of kinetic theory assumes molecules that are permanently moving. Only the macroscopic properties do not vary in the equilibrium state. The one-dimensional distribution of velocities v of the particles in an ideal gas, each of mass m, at temperature T is given by the Maxwell-Boltzmann distribution r mv 2 m exp − . (2.8) Pmb (v) = 2πkB T 2kB T This distribution is stationary under the random scattering of the gas molecules, as can be demonstrated with the Boltzmann equation. The latter involves the assumption of molecular disorder which allows to neglect the correlations of two particles just having collided (molecular chaos). Besides Eq. (2.8), there is no other distribution that is stationary with respect to the Boltzmann equation. Therefore, Eq. (2.8) gives the equilibrium distribution of velocities of an ideal gas, or of any system that can be described by the Boltzmann equations together with the molecular chaos assumption. Ergodicity Besides the ensemble average, there is a second way to perform an average: time averaging. For a quantity φ(τ ) the time average is deﬁned as 1 hφ(τ )iτ = lim T →∞ T Z T φ(τ )dτ. (2.9) 0 A Hamiltonian system with constraints is restricted to a certain subset or surface5 , S, in the Γ -space. The subset, S, reﬂects the total energy and constraints excluding the “forbidden” points in the phase space. Boltzmann assumed for a Hamiltonian system that the system visits every microstate on the surface S, i.e., every accessible6 microstate, after suﬃcient time. That is, every phase trajectory crosses every phase 5 If the only constraint is the total energy, the dynamics in Γ -space are restricted to a 6N − 1dimensional subset, i.e., a hypersurface in phase space. However, the term surface is used even if more constraints exist. 6 A state is called “accessible” if it is not in conflict with the constraints of the system – e.g., constant total energy – and if it is connected, i.e., if there is a phase space path connecting the state with the initial condition which does not violate the constraints. 12 Thermal Fluctuations space point which is not in conﬂict with the constraints or boundary conditions, and all accessible microstates are equally probable. This is the so-called ergodic hypothesis. In the strict sense, the hypothesis is wrong. However, it can be proven that all phase space trajectories of a Hamiltonian system, except a set of measure zero, come arbitrarily close to every accessible point in phase space (quasi-ergodic hypothesis) and that the time spent in a region of the surface is proportional to the surface area in the limit of inﬁnitely long trajectories. Thus, for a Hamiltonian system, ensemble averages and time averages yield the same results. In general, systems that exhibit the same behavior in time averages as in ensemble averages are called ergodic [65]. For all practical applications, there is a ﬁnite observation time, T and the limit procedure in equation Eq. (2.9) cannot be carried out for a real experiment or for a computer simulation. The following notation will be used to denote ﬁnite time averages Z T −1 hφ(τ )iτ,T = T φ(τ )dτ. (2.10) 0 The validity of equating h·iτ with h·iτ,T holds, of course, only if T is “large enough”. In principle T must be larger than every timescale innate to the system or quantity of interest. Whenever the time T is too short, the system appears to be non-ergodic, irrespective of whether the system is ergodic or not. 2.3 Brownian motion Small, but microscopically observable particles suspended in a liquid undergo a restless, irregular motion, so-called Brownian motion. In 1908, Langevin developed a scheme to model Brownian motion7 which turned out to be an approach of general applicability [64, 69, 70]. He formulated the following stochastic diﬀerential equation for the one-dimensional movement of a microscopic particle of mass m d d2 (2.11) m 2 x(t) = −mγ x(t) + ξ(t). dt dt The ﬁrst term on the right accounts for the friction with the sourrounding particles arising from the collisions with these particles. The friction constant, γ, determines the strength of the interaction between the Brownian particle and its environment, and is linear in velocity v = dx/dt corresponding to Stokes’s friction law. The given expression is just an averaged value; deviations from that value are expressed by the random force ξ. This random force keeps the particle moving. If there were only friction, the particle would come to rest. Eq. (2.11) is called the free Langevin equation, because there is no force ﬁeld present. A term representing the force due to a space dependent potential V (x), e.g., a drift or a conﬁning harmonic potential, can be added to Eq. (2.11). These terms sum up to the total force that acts on the Brownian particle. 7 The term Brownian motion is used in different contexts with slightly different meanings. Here, it is used to describe the microscopic motion of suspended particles, i.e., the original phenomenon observed by R. Brown and originally referred to as “Brown’sche Molecularbewegung” by Einstein, and in a broader sense as the process described by the Langevin equation Eq. (2.11). 2.3 Brownian motion 13 0.5 0 −0.5 −1 2 1 1 0 −1 0 −2 Figure 2.2: Brownian motion. - Three-dimensional realization of Brownian motion. The data are obtained from a numerical integration of the Langevin equation, Eq. (2.11), with the Beeman algorithm [71]. The random force ξ(t) – also referred to as noise – is assumed to be a stationary random variable. The term stationary refers to a quantity or a situation with invariance under time shift. For the random force, ξ(t) ∈ R, this means: ξ is drawn from a distribution P (ξ) that does not depend on the time t. As a consequence, the process itself is stationary. The distribution P (ξ) is assumed to have a zero mean hξi = Z ξP (ξ)dξ = 0, (2.12) that is, there is no drift due to the random force. Furthermore, we assume hξ(t + τ )ξ(τ )i = 2Bδ(t). (2.13) Hence, the variance of P (ξ) is assumed to be ξ 2 = 2B. From Eq. (2.13) it is obvious that, for the Langevin random force ξ, no time correlations exist. The term white noise refers to this lack of time correlations in ξ. Note that Eq. (2.13) does not depend on the time τ , due to the time-independence of P (ξ). Commonly, it is assumed that ξ is a Gaussian random variable, an assumption that can be justiﬁed by the central limit theorem in many applications. Eq. (2.11) is the stochastic analog of Newton’s equation of motion. The averaging procedure in Eqs. (2.12) and (2.13) is to be understood as performed over the random force ξ itself. As the Langevin equation Eq. (2.11) represents a ‘typical’ particle in the ensemble, this averaging procedure corresponds to an ensemble average. More precisely, the deﬁnition of the random force ξ, which is the only random variable of the process, determines a speciﬁc ensemble: the equation of motion, Eq. (2.11) together with the noise, gives rise to a certain population of the phase space. 14 Thermal Fluctuations From Eq. (2.11) the velocity is obtained as −γt v(t) = v0 e −γt +e Z 0 t eγτ ξ(τ ) dτ, m (2.14) in which v0 = v(0) is the initial velocity. After taking the square of Eq. (2.14) and performing the average over the noise, the following expression is obtained B v 2 (t) = v02 e−2γt + , γm2 (2.15) where the cross term vanished due to hξi = 0, Eq. (2.12). The velocity reaches a constant value for long times and from the second law it follows that t → ∞ corresponds to the equilibrium, in which the initial condition is entirely forgotten. Therefore, the time (2γ)−1 can be seen as a relaxation time. In thermal equilibrium, the equipartition theorem can be applied. Under these conditions, the constant B, introduced in Eq. (2.13), can be determined as B = mγkB T . Hence, the correlations of the noise are linear in the temperature and in the friction coeﬃcient. This result is commonly referred to as the fluctuation-dissipation theorem. The friction represents the interaction between the Brownian particle and the environment – but only the dissipative part of it. The thermal agitation also gives rise to the random force, a measure of which is given by the temperature; it keeps the particle moving. Hence, with increasing T , the agitation becomes stronger, which is accounted for by the increased variance of ξ. Therefore, the form of B is a consequence of the conservation of energy. The ﬂuctuation-dissipation theorem assures the conservation of energy: the energy dissipated due to the friction equals the energy transferred to the system by the random force. The ﬂuctuation-dissipation theorem is valid only at equilibrium conditions. In the time interval [t1 , t2 ] the Brownian particle travels a certain distance, called the displacement. If the Langevin process is assumed to have no drift, as e.g., in Eq. (2.12), the motion is symmetric. Hence, the mean displacement equals zero. A quantity of interest is the mean squared displacement (MSD), deﬁned as 2 ∆x (t) = [x(t + τ ) − x(τ )]2 . (2.16) The MSD can be calculated as a time average or as an ensemble average. R The dist placement from the initial position x0 up to time t is x(t) − x0 = ∆x(t) = 0 v(τ )dτ . Performing the integration in Eq. (2.14), the noise averaged MSD reads [70] 2 kB T v2 ∆x2 (t) ξ = 02 1 − e−γt + 2 2γt − 3 + 4e−γt − e−2γt . γ γ m (2.17) After averaging over the initial velocity given by the Maxwell-Boltzmann distribution in Eq. (2.8), i.e. v02 eq = kB T /m, the equilibrium expression is obtained as [69, 70] 2kB T γt − 1 + e−γt . ∆x2 (t) eq = 2 mγ (2.18) 2.3 Brownian motion 15 For short t, the MSD is quadratic in t. The ballistic behavior is a consequence of inertial eﬀects: for short times the particle travels in a straight line with nearly constant velocity. For long t, when inertia has been dissipated, the MSD exhibits a linear time dependence. Langevin equation with harmonic potential The Langevin equation allows an external force ﬁeld to be included in Eq. (2.11). In the following, a one-dimensional Langevin equation with harmonic potential is discussed, m d2 d x(t) = −mγ x(t) − mω 2 x(t) + ξ. 2 dt dt (2.19) The Langevin equation with harmonic potential describes an ergodic system, i.e., for a quantity φ(x) that depends on x, which is itself governed by Eq. (2.19), hφ(x)iτ = hhφ(x)inoise i0 . (2.20) The average h·i0 represents an average over equilibrium initial conditions8 , as is performed to obtain Eq. (2.18). The equilibrium distribution of the velocity is given by the Maxwell-Boltzmann distribution, Eq. (2.8), the coordinate equilibrium distribution obeys Boltzmann statistics, √ ω m mω 2 x2 Peq (x) = √ . (2.21) exp − 2kB T 2πkB T Eq. (2.20) is the ergodic theorem for the Langevin equation. In what follows, we do not diﬀerentiate between time and equilibrium ensemble averages, as is justiﬁed due to the ergodicity of Eq. (2.19). In the case of the harmonic Langevin equation, the equipartition theorem for the kinetic energy is the same as in Eq. (2.7). Additionally, the equipartition theorem can be applied to the vibrational energy, 1 1 mω 2 x2 τ = kB T . 2 2 (2.22) In the expression of the mean squared velocity, Eq. (2.15), derived for the free Langevin equation, the system exhibits a memory of the initial value of the velocity. The Langevin process contains inertial eﬀects due to the present state of the system being inﬂuenced by past states: the dynamics are correlated. This phenomenon is quantiﬁed by the auto-correlation function (ACF), which can be calculated for various 8 Such an average requires a uniquely defined equilibrium state. However, there is no equilibrium distribution of the coordinate in the free, unbounded case in Eq. (2.11). Thus, for quantities depending explicitly on the distribution of initial x values, there is also no equilibrium. Therefore, the ergodic theorem cannot be applied. Furthermore, a time average over any time interval will never converge, as there is no equilibrium to converge to. 16 Thermal Fluctuations quantities. Note that correlations can arise not only from inertia, but also from ignorance against dynamical details. A more general treatment of projection procedures is given in Sec. 5.1. In Eq. (2.13) the ACF of the random force is given. Note that we did not specify an initial condition for ξ and Eq. (2.13) is invariant with time shift, i.e., the right side of the equation does not depend on τ . Other common ACFs are the coordinate ACF9 (CACF), given by hx(t + τ )x(τ )iτ , (2.23) Cx (t) = hx2 (τ )iτ and the velocity auto-correlation function (VACF) Cv (t) = hv(t + τ )v(τ )iτ . hv 2 (τ )iτ (2.24) The two ACFs are normalized; the denominator is given by the equipartition theorem. The ACFs quantify how strongly the present state is inﬂuenced by the history of the system. However, the inﬂuence is a statistical correlation which is not to be confused with a causal relation. If the ACFs decay exponentially, the typical time scale is a relaxation time. Finite time averages must exceed the relaxation times signiﬁcantly for the ergodic theorem to be applicable. A 1 B 0.5 Cv(t) Cx(t) 0.5 1 0 0 γ = 0.1 γ = 0.1 γ = 1.0 −0.5 γ = 1.0 −0.5 γ = 10.0 γ = 10.0 γ = 100.0 −1 0 20 40 60 t [steps] 80 γ = 100.0 100 −1 0 20 40 60 80 100 t [steps] Figure 2.3: Autocorrelation functions of the Langevin process. - Numerical integration of the Langevin equation with a harmonic potential (ω = 1.0), Eq. (2.19), to calculate the CACF and VACF. The integration is done with the Beeman algorithm [71] for different friction values γ = 0.1, 1.0 (underdamped) and γ = 10.0, 100.0 (overdamped). A: CACF, simulation results dotted. The B: VACF, simulation results dotted. analytical CACF, Eq. (2.25), is given as continuous line. The analytical VACF, Eq. (2.26), is given as continuous line. 9 As pointed out above, time averaging requires an equilibrium state to converge. That is why the CACF of the free, unbounded Langevin equation is not defined. There have been cases in which this problem led to confusion. Note that the MSD does not depend on the initial position and is properly defined for the free, unbounded Langevin equation. 2.4 From random walks to diffusion 17 The CACF and VACF of the Langevin process with harmonic potential read, respectively [72, 73] Cx (t) = γ + ̟ −(γ−̟)t/2 −γ + ̟ −(γ+̟)t/2 e + e 2̟ 2̟ (2.25) Cv (t) = −γ + ̟ −(γ−̟)t/2 γ + ̟ −(γ+̟)t/2 e + e , 2̟ 2̟ (2.26) and where the characteristic frequency ̟ = (γ 2 − 4ω 2 )1/2 is used. The VACF of the free Langevin process can be obtained from Eq. (2.26) in the limit ω → 0. Note that the CACF is undeﬁned for the free Langevin particle. There are mainly two diﬀerent cases to discuss, which are illustrated in Fig. 2.3. • Overdamped case γ > 2ω: In this case, ̟ is a real number. Since ω > 0, we have ̟ < γ. As a consequence all terms in Eqs. (2.25) and (2.26) decay exponentially to zero for large t. The leading term in the CACF is exp[−(γ − ̟)t/2]. In the limit γ ≫ 2ω, this term will be constantly equal to one, while the second term in Eq. (2.25) is zero then, due to its vanishing coeﬃcient. Note that the VACF does not show such particular behavior. The dominating term in the VACF is exp[−(γ − ̟)t/2], but its coeﬃcient approaches zero as the friction increases. So for ω → 0, the VACF is equal to its second term, given by e−γt . • Underdamped case γ < 2ω: In this case, the frequency ̟ is an imaginary number. This leads to oscillatory behavior of the CACF and the VACF, which will be bounded by the damping factor exp(−γt/2), giving rise to an exponential decay. The question arises as to how the CACF is connected to the MSD. Given that the CACF exists, one obtains from the deﬁnition of the time-averaged MSD 2 ∆x (t) τ = 2 x2 τ [1 − Cx (t)]. (2.27) Therefore, the CACF and MSD contain essentially the same information. 2.4 From random walks to diffusion “Can any of your readers refer me to a work wherein I should ﬁnd a solution of the following problem, or failing the knowledge of any existing solution provide me with an original one? I should be extremely grateful for aid in the matter.” A man starts in the point O and walks l yards in a straight line; he then turns through any angle whatever and walks another l yards in a second straight line. He repeats this process n times. I require the probability that after these n stretches he is at a distance between r and r + δr from his starting point, O. 18 Thermal Fluctuations This question of the British mathematician Karl Pearson is entitled The problem of the random walk and was published in the Nature magazine in July 1905 [63]. In the following, Pearson’s question is answered for the one-dimensional random walk. If the walker arrives in x at the time t he moves to x + ∆x or x − ∆x after a ﬁxed time interval ∆t. The probability of moving in the positive or negative direction may be 1/2 each. Let W (x, t) be the probability of ﬁnding the walker in the range [x, x + ∆x] at time t. This probability is a distribution of the random variable x, while t is just a parameter. W (x, t) is a probability linked to the ensemble average. Thus, the ensemble average of a x-dependent quantity f (x) reads Z ∞ f (x)W (x, t)dx. (2.28) hf (x)iens = −∞ For the probability W (x, t) the following relation holds 1 1 W (x, t + ∆t) = W (x − ∆x, t) + W (x + ∆x, t). 2 2 (2.29) Expanding the probability in a Taylor series for small ∆t around t and small ∆x around x, respectively, yields W (x, t + ∆t) = W (x, t) + ∆t ∂ W (x, t) + O(|∆t|2 ) ∂t (2.30) and W (x ± ∆x, t) = W (x, t) ± ∆x (∆x)2 ∂ 2 ∂ W (x, t) + W (x, t) + O(|∆x|3 ). ∂x 2 ∂x2 (2.31) Inserting the two Taylor expansions in Eq. (2.29) leads to ∂ (∆x)2 ∂ 2 W (x, t) + O(|∆t|) = W (x, t) + O(|∆x|3 |∆t|−1 ). ∂t 2∆t ∂x2 (2.32) At the limits ∆t → 0 and ∆x → 0, such that D = (∆x)2 /2∆t has a ﬁnite, non-zero value, the diﬀusion equation follows as ∂ ∂2 W (x, t) = D 2 W (x, t). ∂t ∂x (2.33) The constant D is referred to as diffusion constant. The diﬀusion equation is not restricted to the particular setup of the one-dimensional random walk. It can also be obtained as a continuum limit of the Langevin process and a large class of random walks with continuous jump lengths and irregular jump frequency, cf. Eq. (D.58) in Appendix D. Eq. (2.33) is valid when no external force ﬁelds are present. This ﬁeld-free case was implicitly assumed when the probability of moving to the positive and negative direction was uniquely set to 1/2 for either direction. The diﬀusion equation can be extended to processes with external force ﬁelds leading to the Fokker-Planck equation [65, 69]. 2.5 Anomalous diffusion 19 If Eq. (2.33) is applied to all x ∈ R (free, unbounded case), an analytical solution can be obtained. With the initial condition W (x, 0) = δ(x), the solution is given as 1 x2 . (2.34) W (x, t) = √ exp − 4Dt 4πDt The MSD can be obtained from this probability distribution as Z ∞ 2 x2 W (x, t)dx = 2Dt. ∆x (t) ens = (2.35) −∞ Eq. (2.35) corresponds to the long-time, asymptotic behavior of the Langevin MSD Eq. (2.18). The short-time, ballistic behavior is not reproduced by Eq. (2.33). From the comparison with the large-t behavior of Eq. (2.18), it follows D= kB T . mγ (2.36) This identity is known as the Einstein relation, which relates the macroscopic observable, D, with the microscopic quantity mγ [3]. Fourier decomposition allows the diﬀusion equation to be solved, including various types of boundary conditions. Examples are given in Appendix A. 2.5 Anomalous diffusion The consideration of the Langevin process revealed three ranges of the MSD. For short times, the inertia of the Langevin particle causes a ballistic behavior with a quadratic time dependence of the MSD. After the friction has dissipated the inertial contribution, the particle’s MSD exhibits a linear time dependence. If the accessible volume is ﬁnite or, e.g., a harmonic potential is present, the MSD saturates for long times. As the MSD is a continuous function of the time, there are cross over regions between the three ‘phases’ of the MSD. However, the cross over regions extend over short time ranges. In contrast to the MSD of a Langevin particle, the MSD obtained in various experiments exhibits a time dependence diﬀerent from that of all three typical phases. Several experiments from diﬀerent scientiﬁc disciplines reveal a power-law time dependence of the MSD, 2 ∆x (t) ∼ tα , (2.37) with an exponent unequal to one, which cannot be explained as a cross-over eﬀect. α is referred to as MSD exponent. Processes with a power-law MSD, in which α 6= 1, are referred to as anomalous diffusion. If the exponent is larger than one, the MSD is said to be superdiffusive. Likewise, a MSD with an exponent 0 < α < 1 is called subdiffusive. From Eq. (2.27) it can be seen that a power-law behavior applies to both, the MSD and the CACF. Therefore, both the MSD and the CACF serve as indicators of anomalous diﬀusion. 20 Thermal Fluctuations The main interest of the thesis at hand is subdiﬀusive processes. In the following, some hallmark experimental results revealing subdiﬀusive dynamics are presented. The list is by far not exhaustive. Further examples can be found in [8, 74]. • Harvey Scher and Eliott W. Montroll worked in the early 1970s on the charge transport in amorphous thin ﬁlms as used in photocopiers. The transient photocurrent in such media is found to decay as a power-law, indicating persistent time correlations [9]. • The transport of holes through Poly(p-Phenylene Vinylene) LEDs has been observed to be subdiﬀusive with a subdiﬀusive exponent α = 0.45 [75]. The exponent does not depend on the temperature and the subdiﬀusion is hence seen as a consequence of structural disorder. • Financial data, such as the exchange rate between US dollar and Deutsche Mark, can be described in terms of a subdiﬀusive process [76, 77]. • The diﬀusion of macromolecules in cells is aﬀected by crowding, i.e., by the presence of other macromolecules in the cytoplasm such that the MSD is subdiﬀusive [78–82]. The polymeric or actin network in the cell is an obstacle for the diﬀusing molecules, causing their dynamics in the cytoplasm to be subdiﬀusive [83, 84]. • The transport of contaminants in a variety of porous and fractured geological media is subdiﬀusive [85, 86]. • The spines of dendrites act as traps for propagating potentials and lead to subdiﬀusion [87]. • Translocation of DNA through membrane channels exhibits a subdiﬀusive behavior [88]. • Anomalous diﬀusion is seen in the mobility of lipids in phospholipid membranes [89]. The above examples demonstrate the ubiquity of subdiﬀusive behavior [90]. In Chap. 3, the methods applied to perform MD simulations and to analyze the data generated by MD simulations are explained. In Chap. 4 we present MD simulation results demonstrating the presence of subdiﬀusion in the internal dynamics of biomolecules. Various models provide subdiﬀusive dynamics. Those models, which are candidates for the internal, molecular subdiﬀusion are discussed in Chap. 5. Now, we turn to the methods which are used to perform simulations of the time evolution of biomolecules. We give an overview of the established algorithms and introduce some of the standart analysis tools. We also discuss some fundamental issues of simulation, in particular the question of convergence and statistical signiﬁcance. Hence, our trust in Molecular Dynamics simulation as a tool to study the time evolution of many-body systems is based largely on belief. D. Frenkel & B. Smit Chapter 3 Molecular Dynamics Simulations of Biomolecules In this chapter, we present the methods which are commonly used to simulate the time evolution of biomolecular systems with computers. Furthermore, some of the analysis tools established in the ﬁeld are introduced. The dynamics of molecules can be described by the quantum mechanical equations of motion, e.g., the Heisenberg or the Schrödinger equation. However, due to environmental noise, quantum mechanical eﬀects can be approximated in many cases by their classical counterparts [91]. Still, for molecules containing many atoms, the classical equations of motion are too diﬃcult to be treated by analytical means, due to their overall complexity [92, 93]. Only computer simulations enable us to exploit the dynamical information enclosed in the equations of motion. There are further beneﬁts from computer simulations. Experimental techniques cannot resolve the dynamics in all details and it is often complicated to manipulate the systems as one wishes. Computer simulations give access to the full atomic details of a molecule. They allow the parameters of the models to be manipulated, and to test models in regions that are inaccessible in experiments. The concept of a molecular dynamics (MD) simulation is as follows. A number of molecules, e.g., a peptide and sourrounding water molecules, is given in an initial position. The interaction between the atoms form the so-called force field. Essentially, the force ﬁeld and other parameters, like boundary conditions or coupling to a heat bath, build the physical model of the system. The forces are represented by the vector f . If the system consists of N point-like particles, then f has 3N components. In the present thesis, the point-like particles in the MD simulation are the atoms. The force vector is derived from the underlying potential, f =− ∂V (r ) , ∂r (3.1) where the r is the conﬁguration vector which contains the 3N atomic position coordinates, e.g. the Cartesian coordinates of all particles. The equations of motion 22 Molecular Dynamics Simulations of Biomolecules corresponding to the physical model, i.e. M d2 r =f, dt2 (3.2) where the mass 3N × 3N -matrix M containing the masses of all atoms on the diagonal is used, are integrated numerically. The integration is performed for a certain time. The data produced in this initial part of the simulation cannot be used to analyze the system, as the system needs time to equilibrate. The estimation of the time needed for equilibration is a non-trivial problem [94]. After this equilibration period, the data collected can be used for the analysis. Thus, MD simulations generate a time series of coordinates of the molecules in the system, the so-called trajectory. With the initial preparation, the equilibration, and the recording of data, MD simulations are similar to real experiments. Therefore, they are sometimes referred to as computer experiments. In MD simulations, the molecules are assumed to be formed of atoms1 which exhibit classical dynamics, i.e., the time evolution can be characterized by Newton’s second principle or equivalent schemes (Lagrange, Hamilton). The intra- and intermolecular forces are derived from empirical potentials. Non-conservative forces, e.g. friction, are absent on the molecular level2 . The classical mechanics description is in many cases a very good approximation. As long as there are no covalent bonds being broken or formed, and as long as the highest frequencies, ν, in the system are such that hν ≪ kB T – h being Planck’s constant – there is no need to employ quantum mechanical descriptions. To overcome the problems due to the highest frequencies, corrections in the speciﬁc heat and the total internal energy can be included. An alternative is to ﬁx the lengths of covalent bonds. Fixing the bond lengths has the beneﬁt to be more accurate and to allow larger time steps at the same time [96, 97] (see Appendix B). In order to perform an MD simulation some input is required. • The initial positions of the atoms must ﬁt the situation of interest. The topology, i.e., the pairs of atoms that have a covalent bond, must be speciﬁed. For large biomolecules with secondary structure the conformation is obtained from experiments, e.g., from X-ray crystallography. • The initial velocities can be provided as input data. Alternatively, random velocities can be generated from the Maxwell-Boltzmann distribution. 1 The atoms in MD simulation are treated as point particles. In some sense this is a form of the Born-Oppenheimer approximation [95], in which the electrons are assumed to follow instantaneously the motions of the nuclei. The nuclei are three orders of magnitude heavier than the electrons, so electronic motion can be neglected on the time scale of nuclear motion. Due to the fast electron dynamics, it is a valid approach to separate electron and nuclear dynamics. 2 The dissipation of energy by friction can be seen as the transition of mechanical energy to thermal energy, i.e., to a disordered, kinetic energy. For macroscopic objects, the mechanical energy of the macroscopic degrees of freedom is transformed by friction into kinetic energy of microscopic degrees of freedom. In the framework of kinetic theory, the molecular level is the most fundamental one. Therefore, mechanical energy on the atomic level cannot be transformed to “more microscopic” degrees of freedom. Hence, the mechanics of atoms evolves without friction forces. However, sometimes friction forces are introduced to model the interaction with a heat bath, see the paragraph thermostats in Sec. 3.2 3.1 Numerical integration 23 • The force ﬁeld contains the covalent bonds speciﬁed by the topology ﬁle and other pairwise, non-bonded interactions like van der Waals and electrostatic forces. • Boundary conditions determine how to deal with the ﬁnite size of the simulation box in which the molecules are placed. Additionally, the coupling to a heat bath or a constant pressure is often included. In the thesis at hand the GROMACS software package is used [97, 98] with the Gromos96 force ﬁeld [99]. 3.1 Numerical integration In MD simulations, the Verlet algorithm is a common choice for the integration of the equations of motion [18, 100]. The Taylor expansion is the ﬁrst step in deriving the algorithm. Let a system consist of N particles. The 3N -dimensional position vector is r (t), and the forces on the particles are given by the 3N dimensional vector f (t). Using Eq. (3.2), the Taylor expansion is r (t + ∆t) = r (t) + ∆tv (t) + ∆t3 ∂ 3 r (t) ∆t2 −1 M f (t) + + O(|∆t4 |), 2 6 ∂t3 (3.3) where v (t) is the velocity. Likewise, for an earlier time it is r (t − ∆t) = r (t) − ∆tv (t) + ∆t2 −1 ∆t3 ∂ 3 r (t) M f (t) − + O(|∆t4 |). 2 6 ∂t3 (3.4) The sum of the Eqs. (3.3) and (3.4) is rn+1 ≈ 2rn − rn−1 + M−1 fn ∆t2 , (3.5) where the subscript indices replace the time dependence with the similar notation for all time dependent quantities, e.g., r (t) = r (n∆t) = rn . Eq. (3.5) is the Verlet algorithm, with the following properties [18]. • The algorithm is an approximation in the fourth order in ∆t. • It is strictly time symmetric and reversible. • It does not make explicit use of the velocity, nor does it provide the velocity. • It has a moderate energy conservation on time scales of a few high-frequency bond vibrations. • It has a low drift in the total energy on long time scales, i.e., the total energy is well conserved on time scales well above the fastest bond vibrations. 24 Molecular Dynamics Simulations of Biomolecules In principal, the algorithm is designed to integrate the equations of motion. It furnishes a trajectory that is a numerical solution of the diﬀerential equation, Eq. (2.2), characterizing the system. As Eq. (3.5) is an approximation, there is a diﬀerence between the solution obtained from Eq. (3.5) and the exact solution. This error can be decreased by decreasing ∆t. The error that follows from Eq. (3.5) is only one source of the deviation from the “true” dynamics. A system with many degrees of freedom and nonlinear interactions, as most MD simulations in practice are, is extremely sensitive to the initial conditions. This sensitivity lies at the heart of chaotic dynamics. With these problems in mind, the question arises as to the usefulness of MD simulations. First, the aim of MD simulations is usually not to predict precisely how the system evolves starting from a certain initial condition. Instead, statistical results are expected from useful MD simulations. Still, we need to prove that the data generated by MD simulations share the statistics of the underlying equations of motion. There is some evidence that the MD trajectories stay close to some “real” trajectory on sufﬁciently long time scales; the MD trajectories are contained in a shadow orbit around the “true” trajectory [101, 102]. The statement of Frenkel and Smit in the head of this chapter refers to this problem, “Despite this reassuring evidence [...], it should be emphasized that it is just evidence and not proof,” [18]. The approximation in Eq. (3.5) does not break the time symmetry, but is rather a reversible approach. Hence, the Verlet algorithm cannot be used when velocitydependent friction – which would break the time symmetry – is relevant. Conceptually, it is possible to calculate forward as well as backward in time. In practice, this is not feasible as numerical errors will quickly sum up and shift the time evolution to a diﬀerent orbit. The Verlet algorithm does not explicitly calculate the velocities. If needed, the velocity can be obtained by rn+1 − rn−1 + O(|∆t2 |). (3.6) vn = 2∆t This equation is a second order approximation in ∆t of the velocity. Time symmetry is a prerequisite for energy conservation. In classical mechanical systems without friction the total energy is conserved. The approximation of the Verlet algorithm violates energy conservation to some extent. This leads to a moderate energy ﬂuctuation on time scales, which are short and cover only a few integration steps of length ∆t. In contrast, there is only a low energy drift on long time scales, a manifest advantage of the Verlet algorithm. The Leapfrog algorithm An algorithm equivalent to the Verlet scheme is the Leap Frog algorithm which deﬁnes the half integer velocity as rn+1 − rn . (3.7) vn+1/2 = ∆t Hence, the positions are rn+1 = rn + ∆tvn+1/2 . (3.8) 3.2 Force field 25 Eq. (3.5) reads with the velocities vn+1/2 = vn−1/2 + ∆tM−1 fn . (3.9) If the velocities are required at the same time as the positions, they can be calculated as vn+1/2 + vn−1/2 ∆t −1 vn = = vn−1/2 + M fn . (3.10) 2 2 Iterating Eqs. (3.8) and (3.9) is equivalent to the Verlet algorithm. 3.2 Force field The central input of the model used in MD simulation is the force ﬁeld. It consists of two diﬀerent components: (i) the analytical form of the forces, and (ii) the parameters of the forces. The force ﬁeld characterizes the physical model which is exploited by the simulation. In the following, we refer to the Gromos96 force ﬁeld [99], which is the force ﬁeld used in the simulations discussed in the present thesis. The atoms are represented as charged point masses. The covalent bonds are listed and deﬁne the molecules. During the simulation, covalent bonds cannot be formed nor broken. The Gromos96 force ﬁeld usually contains three types of interactions. • Covalent bonds: The forces that emerge from the covalent bonds act between the atoms listed as bonded. The bonded forces include bond stretching, angle bending, proper dihedral bending and improper dihedral bending. • Non-bonded interactions: Coulomb attraction and van der Waals forces lead to interactions between arbitrary pairs of atoms, usually applied only to non-bonded pairs. The Coulomb and van der Waals forces are centro-symmetric and pairadditive. In practice, only interactions within a certain radius are taken into account. The forces are derived from the underlying potential. • Sometimes, additional constraints, such as a ﬁxed end of a polypeptide chain or ﬁxed bond lengths, are included, e.g., to mimic an experimental setup or to increase the time step. The covalent bond has four terms that contribute to the potential energy. The corresponding coordinates are depicted schematically in Fig. 3.1. • If two bonded atoms, i and j, are separated by a distance bij , the bond potential energy is 1 (3.11) Vb = κb (bij − b0 )2 , 2 which accounts for a stretching of the bond length from the equilibrium value, b0 . The stiﬀness of the bond is given by the force constant, κb . 26 Molecular Dynamics Simulations of Biomolecules Figure 3.1: Coordinates used in the force field - Schematic illustration of the coordinates which are used in the bonded energy terms in equations Eqs. (3.11) to (3.14). Figure from L. Meinhold [103]. • The deviation from the equilibrium angle, θ0 , between two neighboring covalent bonds, (i, j) and (j, k), leads to a potential energy of 1 Vθ = κθ (θijk − θ0 )2 , 2 (3.12) where θijk is the angle between the two bonds. In analogy to Eq. (3.11), κθ is the force constant of the harmonic potential (Vθ is harmonic in the angle, not in Cartesian coordinates). The potential Vθ describes a three body interaction. • The proper dihedral angle potential depends on the position of the four atoms i, j, k, and l, the bonds of which form a chain [see Fig. 3.1]. The angle φijkl between the plane deﬁned by the positions of i, j, k and the bond k, l, has an equilibrium value φ0 . The potential energy due to the torsion of φijkl is Vφ = κφ [1 + cos(nφijkl − φ0 )], (3.13) a four body interaction. • The angle between the bond (j, l) and the plane deﬁned by the bonds (i, j) and (j, k) is referred to as improper dihedral, ωijkl , cf. Fig. 3.1. The torsion of ωijkl makes the following contribution to the total potential energy. 1 Vω = κω (ωijkl − ω0 )2 . 2 (3.14) 3.2 Force field 27 The long-range Coulomb force between all pairs of atoms is part of the non-bonded interactions. Its potential has the analytical form X qi qj , (3.15) Ves = rij i,j<i where qi are the electrical charges and rij is the distance between atoms i and j, rij = |r i − r j |. The second non-bonded interaction is the van der Waals force. Its potential is of Lennard-Jones type, i.e., " 6 # X σij 12 σij VvdW = 4ǫij − , (3.16) rij rij i,j<i with the depth of the Lennard-Jones potential, ǫij , and the collision parameter, σij . The calculation of the non-bonded interactions is the most costly part of the numerical integration. Therefore, the potentials are modiﬁed such that they are zero beyond a certain cut-oﬀ value. This can be done by a shift function which preserves the force function to be continuous [97]. In order to eliminate unphysical boundary eﬀects, periodic boundary conditions are imposed. The system is contained in a space ﬁlling box. In the simulations used in the present thesis, the box is a rhombic dodecahedron or a truncated octahedron. The box is surrounded by translated copies of itself, so-called images. As a consequence, boundary eﬀects are avoided. Instead, an unnatural periodicity crops up. The non-bonded, short-range interactions are limited in GROMACS to atoms in the nearest images (minimum image convention). The long-range Coulomb force is treated by methods that are based on the idea of Ewald summation [104], which we now discuss in more detail. Particle Mesh Ewald Method In 1921, Paul Peter Ewald, physicist at Munich, developed a scheme to treat the periodic Coulomb interactions in a crystal [104]. In solid state physics it is a common idea to build a crystal from a unit cell and identical images of the unit cell as illustrated in Fig. 3.2 A. The periodic image cells are identiﬁed by the vector n. Assume there are N point charges qi in the unit cell at the positions ri . The total electrostatic energy including the contributions from the images can be written as N Ves 1X qi φ(ri ), = 2 (3.17) i=1 where φ(ri ) denotes the potential at the position of qi , ri . The potential is given as φ(r ) = X′ 1 qj , 4πǫ |ri − rj + nL| (3.18) n,j where the prime denotes that the sum goes over all image cells n, and j over all particles, save the combination n = 0 and i = j, as this would correspond to a self interaction. 28 Molecular Dynamics Simulations of Biomolecules The sum on the right side of Eq. (3.18) does not generally converge. If it does, it converges slowly. The Ewald summation overcomes the slow convergence of the electrostatic potential. The idea is to split Eq. (3.18) into two parts: a quickly converging short-range term and a long-range contribution which can be treated eﬃciently after it has been transformed into the Fourier k -space. The separation of the two contributions is achieved by an additional potential that screens each of the point charges qi with a Gaussian distribution of charge density of opposite sign. The sum of the point charges and the Gaussians is the short range contribution. The virtual charge distribution must be canceled out by a mirror distribution, i.e., the same distribution with opposite sign, see Fig. 3.2 B. This can be handled more eﬃciently in Fourier space. A third term must be introduced to compensate the self interaction which is due to the energy between the point charge qi and the mirror of its screening Gaussian. ρ + + r c A a − b − B C Figure 3.2: Calculation of electrostatic interactions in a crystal using the Ewald summation. - A: An ideal crystal is obtained by translations {a,b,c} of a unit cell (red). B: In the Ewald summation, each point charge (blue) is surrounded by a neutralizing Gaussian charge distribution (cyan), leading to rapidly convergent series in real space; the corresponding canceling distribution (magenta) is calculated in reciprocal space. C: The irregularly distributed charges (blue) are interpolated onto the vertices (green) of a regular grid. Figure and caption from L. Meinhold [103]. Let us start with the mirror charge density given as ρm (r ) = XX j n qj a 3 2 π exp(−a|r − rj + nL|2 ). (3.19) The constant value a speciﬁes the width of the Gaussian. According to the Poisson equation, the charge density gives rise to a potential − ∆φm (r ) = ρm (r ) , ǫ (3.20) where ∆ is the Laplace operator. The potential corresponding to a point charge, q at the origin is q φpc (r ) = . (3.21) |r | 3.2 Force field 29 It is useful to transform the Poisson equation and the charge density to the Fourier space with the transformation ρm (r ) = 1 X ρm (k )eirk , V (3.22) k where k = 2πl /L is the reciprocal cell vector, l = (lx , ly , lz ) being the lattice vector in Fourier space. The unit cell is assumed to be cubic with side length L and volume V . The Poisson equation reads in the Fourier representation k2 φm (k ) = ρm (k ) , ǫ (3.23) in which |k |2 = k2 . The Fourier transform of the charge density is ρm (k ) = 1 X −ikrj −k2 /4a qj e e . V (3.24) j Combining Eqs. (3.23) and (3.24), the potential in Fourier representation is derived for all k 6= 0. After inverse transformation, the potential reads φm (r ) = 1 X qj ik (r −rj ) −k2 /4a e e . Vǫ k2 (3.25) k 6=0,j P With the deﬁnition ρ(k ) = j qj e−ikrj , the long range contribution to the total electrostatic energy is expressed as Vlr = 2 1 X 1 |ρ(k )|2 e−k /4a . 2V ǫ k2 (3.26) k 6=0 The self-interaction energy, i.e., the energy of qi due to the mirror potential φm , is calculated as follows. Both the charge qi and the Gaussian are shifted to the origin by setting ri = 0. The symmetry allows spherical coordinates to be used. The Gaussian leads to a potential derived from the Poisson equation in spherical coordinates as √ (3.27) φgauss (r ) = qi erf( ar), where the error function erf(·) p is used. The self-interaction potential at ri is given by φsi (ri ) = φgauss (0) = 2qi a/π. The corresponding contribution to the electrostatic energy is r aX 2 1X Vsi = − qi φsi (ri ) = − qi . (3.28) 2 π i i The short range interaction potential is the sum of the potential of the point charge, qi , and the screening Gaussians, φsr (r) = √ qi 1 − erf( ar) , r (3.29) 30 Molecular Dynamics Simulations of Biomolecules where r is the radius in spherical coordinates. The energy Vsr follows from Eq. (3.29) substituting Eq. (3.17). Thus, the electrostatic energy can be rewritten as Ves = Vlr + Vsr + Vsi . (3.30) Instead of evaluating Eq. (3.18), the numerical implementation of Eq. (3.30) is favorable, as the individual terms of the latter converge much quicker. The algorithm contains a numerical inverse Fourier transform of the sum over the wavevectors k in Vlr . This sum scales with the particle number N as O(N 2 ). Solving the Poisson equation with a charge distribution on a grid is much faster than the general case. The employment of a grid can be used to speed up the simulation to a numerical eﬃciency which scales ad N log N [105, 106]. The charge distribution is interpolated with cardinal B-splines – piecewise polynomial functions which are unequal zero on a small interval (i.e. with compact support) –, which allows the Fast Fourier Transform (FFT) to be employed [107] in the evaluation of Vlr , see Fig. 3.2 C. The method is referred to as smooth particle mesh Ewald method [18, 106]. Water models In vivo, biomolecules are always found in aqueous solution. The presence of the surrounding water is crucial for proteins to function [42, 43] and it aﬀects the internal dynamics [108]. Water itself is an intrinsically complex substance and its physical properties are an active ﬁeld of research [109–111]. In the simulations used in the thesis at hand, water is modeled in diﬀerent ways. Most of the simulations are performed with an explicit water model, i.e., in addition to the biomolecule of interest the simulation box is ﬁlled with MD water molecules using the extended simple point charge (eSPC) water model [112]. It describes water as a tetrahedrally shaped molecule with an OH distance of 0.1 nm. At the positions of the oxygen and hydrogen, point charges of –0.8467 e and +0.4238 e are placed, respectively. The eﬀective pair potential between two water molecules is given by a potential of Lennard-Jones type acting on the oxygen positions. The radial distribution that emerges from the eSPC model reproduces the ﬁrst two peaks of the radial distribution function of oxygen-oxygen distances in real water, the diﬀusion constant, and density at 300 K. The dipole moment of 2.35 D is higher than the experimental value of 1.85 D [112]. An alternative is to use implicit water models, in which the dynamics of the water is not modeled in atomic detail. One common approach is the generalized Born/surface area (GB/SA) method [113, 114]. To obtain the correct solvation free energy, Fsol = Fcav + FvdW + Fpol , (3.31) three contributing terms are considered: a solvent-solvent cavity term, Fcav , a solventsolute van der Waals term, FvdW , and a solvent-solute polarization term, Fpol . The ﬁrst two terms are linear in the accessible surface area. The polarization term is obtained 3.3 The energy landscape 31 from a combination of Coulomb’s law with the Born equation, yielding the so-called generalized Born equation. The GB/SA method does not take into account momentum transfer on the solute or friction eﬀects [114]. The model has been compared to explicit water simulations and found to reproduce the dynamics in the case of small peptides [115, 116]. The eﬀect of the solvent can also be included by a Langevin formulation adding a random force and a friction term, cf. Eq. (2.11). Isokinetic thermostat The ensemble corresponding to the equations of motion, Eqs. (3.2) and (B.15), has a ﬁxed volume, V , a ﬁxed number of particles, N , and a ﬁxed total energy E. Experimentally such an ensemble is diﬃcult to establish. Therefore, it is useful to modify the equations of motion and to simulate other ensembles. The isokinetic thermostat [117, 118] allows the (N, V, T ) ensemble to be simulated, where T is the temperature. Let us start with the modiﬁed equations of motion M d2 r = f − βv , dt2 (3.32) in which β is a constant. The β-term allows the temperature to be kept constant. To see this, we calculate the kinetic energy as 3(N − 1)kB T = v T Mv . (3.33) Since the center of mass motion was subtracted, there are 3(N − 1) degrees of freedom. If the temperature is required to be constant, i.e., dT /dt = 0, and using Eq. (3.32) we have d2 r (3.34) 0 = v T M 2 = v T (f − βv ). dt From that it follows β= vT f . vT v (3.35) Inserting Eq. (3.32) into the Leap Frog update for the velocity, Eq. (3.9), leads to vn+1/2 = vn−1/2 + ∆tM−1 (fn − βvn ). (3.36) Iterating Eqs. (3.8) , (3.10), and (3.36) allows the equations of motion to be integrated at constant temperature. The above approach to model the temperature coupling of the system is designated isokinetic thermostat. 3.3 The energy landscape In Sec. 2.2, the idea of ensembles is introduced. Of particular interest are probability densities that correspond to equilibrium macrostates, i.e., macrostates that do not 32 Molecular Dynamics Simulations of Biomolecules Figure 3.3: Energy landscape - Schematic illustration of a rugged energy landscape. The conformation coordinate corresponds to the two-dimensional x-y plane. The potential energy, E, is given on the vertical axis. Figure from Nicolas Calimet. explicitly depend on time3 , ∂ ρeq (r , v , t) = 0 (3.37) ∂t where ρeq is the equilibrium phase space density. Hence, for the equilibrium density we skip the time argument, i.e., ρeq (r , v ). Note that from Eq. (3.37) it does not follow d ρeq (r (t), v (t)) = 0. Nonetheless, averages with respect to the ensemble do that dt not depend on the time, if the ensemble obeys Eq. (3.37). For a molecular system in equilibrium with a constant temperature the distribution of velocities is given by the Maxwell-Boltzmann distribution, Pmb (v), Eq. (2.8), irrespective of the potential energy V (r ). Thus, the conﬁguration r can be considered as statistically independent from the velocity v . The statistical independence is expressed by ρ(r , v ) = ρr (r )Pmb (v ). (3.38) Then, the statistical analysis can be limited to the dynamics in the conﬁguration part of the phase space, the so-called configuration space. The potential energy, V (r ) depends on the conﬁguration of the system, the latter represented by r (t). Formally, V (r ) is a function that maps the conﬁguration space to real numbers. It can be seen as a hypersurface over the conﬁguration space. This hypersurface is rugged, i.e., it has many local minima separated by barriers, ‘peaks’ and ‘ridges’, corresponding to extremely high potential energies, ‘valleys’ and ‘wells’ where the energy is low, see Fig. 3.3. These analogies are the reason for referring to the hypersurface as energy landscape. 3 For a Hamiltonian system, the total time derivative of the density ρ equals zero. This can be proven from the conservation of probability which leads to a continuity equation for ρ [65]. 3.4 Normal Mode Analysis (NMA) 33 M. Goldstein proposed the idea to describe the dynamics of system in terms of a trajectory on the energy landscape in the context of glass forming liquids [25]. Due to the thermally induced decoupling of position and velocity, the dynamics emerges from the properties of the energy landscape. Since the glass-like aspects of protein dynamics were discovered, the concept of the energy landscape has also been applied to biomolecules [20–22, 26, 119]. 3.4 Normal Mode Analysis (NMA) Let a system of N particles be governed by the equations of motion, Eq. (2.2), and the force be conservative, according to Eq. (3.1). The 3N positions are denoted by the 3N dimensional vector r . Often it is convenient to introduce mass-weighted coordinates, i.e., 1 x = M2 r, (3.39) where M is the diagonal matrix with the components Mij = δij mi (mi is the mass of particle i). The equations of motion in the mass-weighted coordinates read d2 x =f. dt2 (3.40) Now assume the potentials of interaction to be purely harmonic, i.e., Vharm (x ) = 1 (x − x0 )T H(x − x0 ), 2 (3.41) where H is a symmetric, real matrix. The position vector, x0 , corresponds to the local minimum of the potential. In the following, it will be assumed that x0 = 0. The coordinates can always be modiﬁed such that the minimum is at the origin, x̃ = x − x0 . The forces can now be calculated as f =− ∂ Vharm (x ) = −Hx . ∂x (3.42) Since H is a symmetric matrix, it can be diagonalized, i.e., there is an orthogonal matrix, U, such that UT HU = Θ, (3.43) where Θ is a diagonal matrix with the eigenvalues, θi , as non-zero components. The orthogonal matrix U corresponds to a transformation of coordinates. In the transformed coordinates g = UT x the equations of motion read d2 g = −Θg . dt2 (3.44) As Θ is diagonal, the 3N equations of motion are completely decoupled when expressed in the coordinates, g . Thus, Eq. (3.44) can be solved for each component gi (t) separately. The solutions of Eq. (3.44), the gi (t), are referred to as normal modes. Once 34 Molecular Dynamics Simulations of Biomolecules the dynamics is given in normal modes, it can be inversely transformed to ordinary coordinates r . Hence, the dynamics can be understood as a linear combination of the contributions of the individual normal modes. The method to change the coordinates of a harmonic potential and using the normal modes is referred to as normal mode analysis (NMA). The decomposition in normal modes is also applicable if the equation of motion, Eq. (3.43), is expanded to include an homogenous4 friction term and a random force, is given as one analogous to the Langevin equation, Eq. (2.19). Each normal mode √ harmonically bound Langevin process with angular frequency ωi = θi [120]. The introduction of noise and friction corresponds to the coupling with a heat bath. The 2 (t) , corresponds to a variance of the normal mode position around the minimum, g i potential energy in a harmonic potential, ωi2 gi2 (t) /2, with an angular frequency ωi ; note that the coordinates are mass-weighted. Due to the equipartition theorem, it is kB T gi2 (t) = 2 . θi (3.45) The above demonstration is based on the assumption of harmonic interaction potentials. Here, harmonicity is assumed for the Cartesian coordinates Eq. (3.41). If the potential is harmonic in any other set of coordinates, NMA can be analogously applied to the equations of motion expressed in the “linear” coordinate set. But normal modes can be a useful tool even for more general potentials [72, 120]; for suﬃciently low temperatures the system will be close to a local potential minimum at position r0 . The potential around this minimum can be approximated by a harmonic expression. The matrix H is obtained as the Hessian of the potential, V (r ), i.e., the components of H are given as ∂ 2 V (r0 ) Hij = , (3.46) ∂ri ∂rj and H characterizes the curvature of the potential at the local minimum in r0 . It must be stressed that this approximation is a local one: each local minimum has an individual Hessian. The accuracy of the approximation depends on the ratio between temperature and the leading correction to the harmonic approximation. In MD simulation the normal modes of a particular local minimum can be obtained from the force ﬁeld at a certain conﬁguration r0 . A crucial prerequisite of the calculation is the position of the minimum, the conﬁguration r0 . The position of the minimum is obtained by an energy minimization with diﬀerent methods, like steepest descent, conjugate gradient, or limited Broyden-Fletcher-Goldfarb-Shanno [97]. 3.5 Principal Component Analysis (PCA) In an MD simulation with N particles, a large amount of high-dimensional data is generated; the trajectory contains 3N coordinates for each time step for which the po4 We here assume that there is a unique friction with all atoms, i.e., the friction is given by a number. A frequency dependent friction is discussed in [120]. 3.5 Principal Component Analysis (PCA) 35 sitions are recorded. Multidimensional data analysis provides schemes that, depending on the quantities of interest, allow the amount of data to be reduced. One such scheme common in MD simulation is principal component analysis 5 (PCA) [122, 124–126]. In NMA, the motion of a system with approximately harmonic interactions is decomposed in diﬀerent modes. In the case of an MD simulation of a large biomolecule at physiological temperature, the interactions are strongly anharmonic. However, there are choices of coordinates that allow diﬀerent kinds of modes to be separated eﬃciently. Assume an MD simulation yields a trajectory, r (t), of length T . The coordinates ri (t) are chosen such that the center of mass has zero velocity. The mass-weighted √ coordinates, x = Mr , have the components xi (t) = mi ri (t). In the analysis of large biomolecules, we consider only the coordinates of the N particles that form the peptide or protein. Sometimes, the set of coordinates is further reduced, e.g., to contain only the positions of the heavier atoms or the Cα -atoms. The surrounding water molecules and ions are assumed to follow considerably faster dynamics. The average value of the ith component of the mass-weighted position vector is denoted by hxi (τ )iτ,T = x̄i , where τ is the variable over which the average is performed and T the length of the time interval in which the average is performed. PCA starts with the covariance matrix, sometimes also called the second moment matrix, C, whose components are given as Cij = h(xi (τ ) − x̄i )(xj (τ ) − x̄j )iτ,T . (3.47) The diagonal elements of the matrix C represent the ﬂuctuations of the molecule. As is known from matrix algebra, the trace of a matrix, i.e., the sum of the diagonal elements, is invariant with orthogonal basis set transformations. The trace of C is the total ﬂuctuation of the molecule, which is independent of the choice of coordinates. The oﬀ-diagonal elements of C give the correlations between the coordinates. In contrast to the ﬂuctuations, the correlations depend on the basis set chosen. The covariance matrix is symmetric by construction. It is diagonalized by the orthogonal matrix W, i.e., WT CW = Λ, (3.48) where Λ has the components Λij = δij λi . The matrix W represents a change from one orthonormal basis set to another. The eigenvalues of the covariance matrix, λi , are the ﬂuctuations along the eigenvectors. We sort the eigenvalues in descending order, λ0 being the largest eigenvalue. The coordinates, q = WT x , in which the covariance matrix is diagonal, are termed principal components (PC). By construction, the PCs are uncorrelated coordinates, but they are not statistically independent. The PCs are collective coordinates, i.e., they involve multiple atoms. As in the case of normal modes, the low PCs are global coordinates involving many particles, while the high PCs are more localized. PCA allows those linear combinations of the original coordinates to be determined, which account for the strongest contributions to the overall internal motion. The time evolution of the component, qi (t), is the PC mode i. 5 The method was invented by the British mathematician Karl Pearson [121], the same who coined the term random walk. PCA has also been referred to as “essential dynamics” [122, 123], “molecule optimal dynamic coordinates” [124], and “quasi-harmonic analysis” [125]. 36 Molecular Dynamics Simulations of Biomolecules The PCs are related to the original coordinates by a linear basis set transformation. The choice of original coordinates determines the class of basis sets which possibly can be determined with PCA. Therefore, it is of crucial importance to chose a proper coordinate set before launching the PCA. Instead of the Cartesian positions of a group of atoms, internal angles are also common starting coordinates for PCA [123, 127, 128]. In Cartesian coordinates, the removal of center of mass motion is obvious, whereas the removal of the molecular rotation is not uniquely deﬁned and leads to some ambiguity [129–131]. 3.6 Convergence MD simulation trajectories do not correspond to real dynamics, but rather imitate real dynamics. As argued in Sec. 3.1, MD simulations attempt to capture the statistical properties of the systems investigated. Provided the trajectory is long enough to ensure a representative sampling of the conﬁguration space, the ergodic hypothesis can be applied, and the ensemble properties can be derived from time averages. In this case, one says the trajectory is converged. MD simulations are not the best way to sample the conﬁguration space eﬃciently. Other methods, such as Monte Carlo simulations or umbrella sampling, are favorable in this respect. However, MD simulations do not only reproduce the equilibrium state but they also contain information about the time evolution of the system. That is, MD simulations are the best classical approximation for the kinetics of the system. One can increase the accuracy of the conﬁguration space sampling by performing multiple simulations instead of a single long trajectory [132]. Real systems have a wide range of intrinsic time scales. Generally, larger systems with larger molecules tend to have longer intrinsic time scales. The simulation has to exceed the longest of these intrinsic time scales to reach the equilibrium state [133]. If the simulation time is shorter than any of the intrinsic time scales, the simulation is likely to provide a poor sampling of parts of the conﬁguration space. The poor sampling may invalidate the statistical properties of the MD trajectory, as the properties cannot be considered representative of the system under investigation. Usually, simulations are considered converged, if, e.g., the ﬂuctuations have reached a plateau value [103]. If the simulations are not converged, it is diﬃcult to draw any conclusions from the trajectories. However, it has been shown repeatedly that the assumption of convergence is not justiﬁed in many cases [133–139]. Assessing the convergence of a given trajectory is a non-trivial task, “because it involves attempting to use what has been measured to deduce whether there is anything of importance which remains unmeasured” [139]. Strictly speaking, the convergence can be assessed only on the basis of an increased amount of data. However, various methods are used to test the convergence with the limited random sample obtained in an individual MD simulation. A common test for convergence is the time dependence of the time-averaged MSD. If the trajectory is converged the time-averaged MSD reaches for long times a constant 3.6 Convergence 37 value. For multiple trajectories, the overlap, i.e. the similarity of the PC coordinates can quantify the convergence [136, 139]. Another measure for convergence is the cluster population in the conﬁguration space. The conﬁgurations are cast into a ﬁnite set of reference structures (clusters). Then the trajectory is cut into pieces. If the various pieces of the trajectory exhibit the same cluster population as the whole trajectory, the simulation can be seen as converged [138]. This technique can also be used to compare diﬀerent simulations [139]. A particular problem is the convergence in the context of PCA. The coordinate set obtained by PCA is unstable for unconverged trajectories [133, 136, 137]. An analytical treatment of Brownian diﬀusion reveals the cosine shape of the PC modes in an unconverged MD simulation. The cosine content allows the contribution of Brownian diﬀusion to the ﬂuctuations to be measured [137]. However, even a zero cosine content does not safeguard convergence of the simulation. After having introduced the methods that allow the time evolution of biomolecules to be simulated and studied, we now turn to examples of such computer simulations. We present in the next chapter results obtained from MD simulations of various biomolecules and discuss their thermodynamics and kinetics in some detail. 38 Molecular Dynamics Simulations of Biomolecules [...] if we were to name the most powerful assumption of all, which leads one on and on in an attempt to understand life, it is that all things are made of atoms, and that everything that living things do can be understood in terms of the jiggling and wiggling of atoms. Richard P. Feynman Chapter 4 Study of Biomolecular Dynamics The results in this chapter have been partially published in T. Neusius, et al., Phys. Rev. Lett. 100, 188103, cf. Ref. [140]. 4.1 Setup of the MD simulations In order to study the thermal ﬂuctuations of biomolecules and the kinetics that underlay the ﬂuctuations, MD simulations of diﬀerent peptides and a β-hairpin protein are performed. The MD simulations presented in the present thesis were performed by Isabella Daidone [141]. Three diﬀerent models are used to simulate the dynamics of solvent water. The comparison between the three water models allows the inﬂuence of water on the kinetics of biomolecules in aqueous solution to be assessed. For the MD simulations, the GROMACS software package [97, 98] and the Gromos96 force ﬁeld [99] are used. In the following, the details of the simulations are listed. Most of the numerical methods are explained in more detail in Chap. 3. The (GS)n W peptides The (GS)n W molecules are polypeptide chains formed of n (here n = 2, 3, 5, 7) repeated GS segments (G = glycine, S = serine) with a ﬁnal tryptophan (W ) residue. The Figure 4.1: (GS)7 W peptide. - 40 Study of Biomolecular Dynamics (GS)n W peptides do not fold into a speciﬁc secondary structure. Simulations of these peptides are performed with the LINCS algorithm [96] with an integration step of 2 fs. The data were saved every picosecond, the simulation length of each simulation is given in Tab. 4.2. The canonical ensemble is used, i.e., the N V T ensemble with constant number of particles, N , constant volume, V , and constant temperature. The constant temperature, T = 293 K, is established by the isokinetic thermostat [117]. Each peptide in extended conformation is placed in a rhombic dodecahedron box. The boundary of the boxes have at least a distance of ≈1.0 nm to the atoms of the peptide enclosed. The aqueous solution is modeled by the eSPC water model [112]. The liquid density is 55.32 mol/l (≈1 g/cm3 ). The real space cut-oﬀ distance is set to 0.9 nm. Periodic boundary conditions are used and long-range interactions are treated by the particle mesh Ewald method [105] with a grid spacing of 0.12 nm and 4th -order B-spline interpolation. n (GS)n W 2 0.8 3 1.0 5 1.9 7 2.5 Figure 4.2: Simulation lengths in units of microseconds (µs). - The β-hairpin MD simulations of the 14-residue amyloidogenic prion protein H1 peptide are performed using a similar setup as for the (GS)n W peptides. The native structure of the β-hairpin is a β-turn. Multiple folding and unfolding events can be observed during the simulation [141]. Thus, unlike the (GS)n W peptides, the β-hairpin has a secondary structure. The data are recorded all two picoseconds, the simulation time is 1 µs. Figure 4.3: β-hairpin molecule. - 14-residue amyloidogenic prion protein H1 peptide. In contrast to the (GS)n W peptides, the β-hairpin simulation with explicit solvent uses a truncated octahedron as simulation box. To check the role of the solvent dynamics, two alternative simulations are performed, besides the simulation in aqueous solution: one with an eﬀective Langevin process imitating the solvent dynamics [97] 4.2 Thermal fluctuations in biomolecules 41 and a simulation of the β-hairpin with implicit solvent using GB/SA [113, 114]. The Born radii are calculated using the fast asymptotic pairwise summation of [142]. The relevant Gromos96 parameters can be found in [143]. To increase the eﬃciency of the surface area calculation, a mimic based on the Born radii is used [144]. Further details of the β-hairpin simulations can be found in [141]. 4.2 Thermal fluctuations in biomolecules Due to the thermal agitation, the conﬁguration, r , of a biomolecule exhibits internal ﬂuctuations in equilibrium. These stationary ﬂuctuations shall be studied in the following section employing PCA. 7 4.5 g g g 4.0 3.5 2 Total á u ñ Multiminima PCA modes Quasi-Harmonic PCA modes 5 Harmonic PCA modes 2 á u ñ (Å ) 3.0 2.5 13 2 4 2.0 7 3 1.5 2 1 1 1.0 3 NMA 5 0.5 0.0 80 100 120 140 160 180 200 220 240 260 280 300 Temperature (K) Figure 4.4: Total position fluctuation of a myoglobin protein as a function ˙ of ¸ temperature. - For low temperatures the position fluctuation, in the figure denoted as u2 , increases linearly with the temperature T . Above 180 K, the fluctuations sharply increase, the so-called dynamical transition. The different colors correspond to different types of PCs, depending on the shape of the free energy profile along that mode (see page 43). The transition is mainly due to the appearance of non-harmonic PCs (red) above the transition temperature, some additional fluctuation is due to the quasi-harmonic modes (green), while the harmonic modes (blue) essentially contribute what is expected from the normal modes, indicated by NMA. The numbers at the curves give the quantity of PCs which belong to the different types of free energy profiles. Figure from A. Tournier [145]. Let the coordinate system of the conﬁguration space be such that the average position vector, r vanishes, hr i = 0. The overall ﬂuctuations of the Cartesian positions1 1 In this section, the same notation as in Chap. 3 is used. However, throughout this section, ordinary mean values, h·i, are to be understood as time averages over finite intervals, h·iτ,T , unless something else is explicitly indicated. 42 Study of Biomolecular Dynamics are X 2 ri . r2 = (4.1) i The thermal agitation is characterized by the temperature, 2 T . Therefore, the overall 2 ﬂuctuation, r , increase with T . The ﬂuctuations, r , of a myoglobin molecule as a function of T is illustrated in Fig. 4.4. The 2 ﬂuctuation exhibits a linear temperature dependence for low T , but above ≈ 180 K, r sharply increases with the temperature, T : the molecule becomes much more ﬂexible. The enhanced ﬂexibility of the molecule is referred to as the “dynamical transition” or “glass transition”. For details see the caption to Fig. 4.4. 2 TIn the case of a harmonic potential, the mass-weighted position ﬂuctuations, x = r Mr , with the mass matrix, M, are proportional to the thermal energy, by virtue of the equipartition theorem. Therefore, in the following, the discussion is focused on the mass-weighted coordinates, x , rather than the Cartesian coordinates, r . 10 ﬂuctuation [u nm2 ] 10 10 10 10 10 10 10 2 B 1 0 −1 −2 −3 −4 10 10 10 10 10 −5 0 10 10 Expl. water Langevin GB/SA ﬂuctuation [u nm2 ] A 50 100 150 mode number 200 250 10 2 (GS)2 W 1 (GS)3 W 0 (GS)5 W −1 (GS)7 W −2 −3 −4 −5 0 50 100 mode number 150 Figure 4.5: PCA eigenvalues. - Fluctuations as measured by PCA eigenvalues. A: PCA eigenvalues of three different simulations of the β-hairpin. All simulation methods employed – explicit water, Langevin thermostat, GB/SA – exhibit the same spectrum of eigenvalues, i.e., the strength of the fluctuations seen in the explicit solvent simulation is reproduced by the implicit solvent methods (Langevin and GB/SA) with high accuracy. B: PCA spectrum of eigenvalues of the (GS)n W peptides (n = 2, 3, 5, and 7): the longer chains are more flexible. In any mass-weighted coordinate system, the total ﬂuctuations are T X r Mr = x 2 = q 2 = λi , (4.2) i where q is the conﬁguration vector expressed in the basis of the PCs and the λi are the eigenvalues, as obtained from PCA. The expression on the right of Eq. (4.2) is the trace of the covariance 2 matrix, Eq. (3.47). As the trace is invariant with orthogonal transformations, x is independent of the basis set chosen. This independence reﬂects the physical nature of the ﬂuctuations, x 2 , which can be measured, e.g., by neutron scattering experiments [31]. The eigenvalues of the individual PC modes are illustrated in Fig. 4.5. The three different water models used for the β-hairpin exhibit a very similar spectrum of eigenvalues 4.2 Thermal fluctuations in biomolecules 43 [Fig. 4.5 A]. The congruence of the three water models with respect to the eigenvalues demonstrates that the equilibrium ﬂuctuations of the explicit water model are correctly reproduced by the implicit water methods. The spectrum of the (GS)n W peptides illustrates the increase of ﬂexibility which comes along with an increasing chain length [Fig. 4.5 B]. The eigenvalues cover a range of ﬁve to six orders of magnitude. The large diﬀerences between the lowest and the highest eigenvalues accounts for the heterogeneity of the PC modes; the lowest PC modes contain much stronger ﬂuctuations than the higher PC modes. Therefore, PCA is a useful tool in identifying low-dimensional subspaces in the conﬁguration space, which contain a signiﬁcant fraction of the overall ﬂuctuations. Potential of mean force The MD simulations used in the present thesis were performed in the canonical ensemble, i.e., the N V T ensemble. The thermodynamics of the canonical ensemble is given by the free energy2 F (T , V, N ) = −kB T log Z, (4.3) R −H/k T B dΓ , follows from the Hamilton function where the partition function, Z = e of the system, H. The integral is over the phase space, Γ . Often, the state of the system is analyzed as a function of an appropriate reaction coordinate, y. Note that y may contain more than one free parameter. The free energy is calculated at a given value of the coordinate, Fy (T , V, N ), that is, the coordinate y acts as a constraint while the coordinates unaﬀected by the ﬁxed value of the reaction coordinate are treated in the canonical ensemble. We stress that for general, curved coordinates the constrained free energy is not uniquely deﬁned, but depends on the coordinates chosen. In the present thesis, only coordinates are used which can be obtained from the massweighted Cartesian coordinates by a linear, orthogonal transformation. Therefore, the constraint y = const. is to be understood as the linear subspace spanned by y. Then, the integration over the remaining, orthogonal coordinates is uniquely deﬁned. In the thermodynamical equilibrium, the probability distribution obeys the Boltzmann statistics. Therefore, the probability distribution of the reaction coordinate, ρ(y), determines the free energy Fy (T , V, N ) = −kB T log ρ(y). (4.4) In the following, we analyze the free energy proﬁles along the PCs, qi . The free energy as a function of the constraint qi is referred to as potential of mean force (PMF) [146]. The equilibrium distribution along the PC mode qi can be obtained from the MD trajectory as a histogram, h(qi ). The free energy is found, up to a constant, as F (qi ) = −kB T log h(qi ), (4.5) in which the arguments T , V , and N are skipped for simplicity. 2 The free energy is sometimes referred to as Helmholtz free energy, in particular in the context of chemistry. 44 Study of Biomolecular Dynamics A 8 B 8 6 2 2 F(q) [kB T ] 4 F(q) [kB T ] 6 0 −10 PC 1 PC 30 4 −5 0 q [nm] 5 10 0 −2 −1 0.1 0.2 0 q [nm] 1 2 8 C F(q) [kB T ] 6 PC 100 4 2 0 −0.2 −0.1 0 q [nm] Figure 4.6: Potential of Mean Force (PMF) of (GS)5 W - The potential of mean force is the free energy profile in which the dynamics of a single PC mode, q, evolves. It is obtained from a histogram, h(q), of the PC mode as F (q) = −kB T log h(q). PC 1 of the (GS)5 W peptide exhibits a multi-minima shape. PC 30 has an anharmonic PMF, but with a single local minimum (quasiharmonic). The PMF of PC 100 is harmonic. The PC modes can be classiﬁed into three groups by virtue of their PMF: (i) PC modes with a harmonic PMF are called harmonic PCs, (ii) PC modes with a single local minimum in the PMF but which are not harmonic are referred to as quasi harmonic PCs, and (iii) those with various local minima, are designated as multi-minima PCs [145]. In Fig. 4.6, three examples of PMFs of the (GS)5 W peptide are illustrated. The lowest PCs exhibit multi-minima shape, while the higher PCs follow in a very good approximation a parabola. The intermediate PCs feature a single minimum but are anharmonic. A similar behavior is found for the β-hairpin, both in explicit water [Fig. 4.8] and implicit water [Fig. 4.9]. Similar free energy proﬁles are found for the GB/SA simulation (not shown). The Langevin simulation and the GB/SA simulation of the β-hairpin reproduce the overall features of the explicit water simulation of the same molecule, as can be seen in Fig. 4.7, although the details of the free energy proﬁles look diﬀerent. Only a small number, usually 10 to 20, of the PC exhibit a multi-minima PMF 4.2 Thermal fluctuations in biomolecules A 45 2 10 B 1 Width of potential σ [nm ] −3 10 2 0 10 2 i 2 i 2 Width of potential σ [nm ] 10 Expl. water Langevin GB/SA −4 10 −1 10 −2 10 (GS) W 2 −3 10 −4 10 (GS)3W (GS)5W (GS)7W −5 0 50 100 PC mode number i 10 150 0 20 40 60 PC mode number i 80 100 Figure 4.7: Fluctuations along PCs - The thermal fluctuations along the PCs of the three β-hairpin simulations (A) and the simulations of the four (GS)n W peptides (B), obtained from the variance of the PC modes. The low PCs exhibit the strongest fluctuations, as is expected from the definition of the PCA. All three simulation methods used for the β-hairpin exhibit a similar pattern of the fluctuations along the PCs. The PCs of the (GS)n W peptides fluctuate more for the longer chains (those with higher n). Note the similarity of the results illustrated in Fig. 4.5. A 8 B 8 6 2 2 F(q) [kB T ] 4 F(q) [kB T ] 6 0 PC 1 PC 30 4 −15 −10 −5 q [nm] 0 5 10 0 −2 0 q [nm] 2 8 C F(q) [kB T ] 6 PC 100 4 2 0 −1 −0.5 0 q [nm] 0.5 Figure 4.8: Potential of Mean Force (PMF) of the β-hairpin simulation with explicit water. - The potential of mean force is the free energy profile in which the dynamics of a single PC mode, q, evolves. It is obtained from a histogram of the PC mode as F (q) = −kB T log h(q). The PMF is shown for PC 1, 30, and 100 of the β-hairpin in explicit solvent. A: Multi-minima mode. B: Quasi-harmonic mode. C: Harmonic mode. 46 Study of Biomolecular Dynamics A 8 B 4 F(q) [kB T ] 6 F(q) [kB T ] 6 8 2 2 0 PC 1 PC 30 4 −10 0 q [nm] 0 10 −2 0 q [nm] 2 8 C F(q) [kB T ] 6 PC 100 4 2 0 −1 −0.5 0 q [nm] 0.5 Figure 4.9: Potential of Mean Force (PMF) of the β-hairpin simulation with Langevin thermostat - The potential of mean force is the free energy profile in which the dynamics of a single PC mode, q, evolves. It is obtained from a histogram of the PC mode as F (q) = −kB T log h(q). The PMF is shown for PC 1, 30, and 100 of the β-hairpin with implicit solvent modeled by a Langevin thermostat. A: Multi-minima mode. B: Quasi-harmonic mode. C: Harmonic mode. at physiological temperatures. However, as found by A. Tournier et al. [145], the overall ﬂuctuations of a biomolecule at ambient temperature are mainly due to these few multi-minima PCs, see Fig. 4.4. The onset of anharmonicity in the lowest PCs has been identiﬁed as the origin of the dynamical transition of proteins at Tg ≈ 200 K [145], illustrated in Fig. 4.4. PCA allows those degrees of freedom to be found, which contribute most to the overall ﬂuctuations of the molecule. The observation of few coordinates which account for the majority of ﬂuctuations, forms the basis for any attempt to reduce the number of degrees of freedom in the modeling of biomolecules, cf. Sec. 5.1. As is illustrated in Fig. 4.7, the lower PCs have a broader PMF. The width of the potential is proportional to the ﬂuctuations along the reaction coordinate. The three simulation methods used for the β-hairpin provide similar widths of the PMFs, Fig. 4.7 A. Thus, the Langevin and GB/SA simulations reproduce the ﬂuctuations as seen in the explicit water simulation. The agreement between the explicit and implicit 4.2 Thermal fluctuations in biomolecules 47 water methods corroborates the ﬁndings in the context of the spectrum of eigenvalues, i.e., the free energy proﬁles and the total ﬂuctuation are well reproduced by all three water simulation techniques. The conﬁgurations of the longer peptides populate a larger volume in the conﬁguration space, Fig. 4.7 B. The larger conﬁguration volume can be understood as an enhanced ﬂexibility of the chain. The decrease of the ﬂuctuations with the mode number appears to be more regular for the (GS)n W peptides, relative to the pattern of the β-hairpin simulation. The kinetics along a reaction coordinate are not fully described by the free energy. This is due to the fact, that the projection to the subspace (or, more generally, the manifold) spanned by the possible values of the reaction coordinate, may lead to time correlations or memory eﬀects, as follows from Zwanzig’s projection operator approach (cf. Sec. 5.1). Only on suﬃciently long time scales, when all memory eﬀects have decayed, the kinetics follow directly from the free energy proﬁle. Therefore, in general, the free energy proﬁles are insuﬃcient to understand the kinetic behavior of a molecule. Participation ratio Both, the PC modes and normal modes, are collective coordinates, i.e., they are linear combinations of the original coordinates, the components of x . As PCA is based on a linear coordinate transformation, the choice of original coordinates constrains the potential PCs to those basis sets, which can be yield by a linear basis transformation of the original coordinates. In the original coordinates, the PC k has the components Wik , i.e., PC k is given as the kth column of the matrix W which diagonalizes the covariance matrix, C, cf. Eq. (3.48). To quantify the delocalization of PC k, the participation ratio is introduced as ηk = 3N −6 X 4 . Wik (4.6) i=1 If the PC k involves just one atom, the PC eigenvector k has the components Wik = δik , leading to ηk = 1. A collective motion that involves all atoms homogeneously √ has the eigenvector components Wik = 1/ 3N − 6, as follows from the normalization of the basis vectors. The participation factor then equals ηk = 1/(3N − 6). Small participation ratios correspond to delocalized PCs, whereas values close to one indicate strong localization. The reciprocal value of ηk corresponds to the average number of degrees of freedom that are involved in the PC, the participation number. The participation number of the β-hairpin simulations is illustrated in Fig. 4.10 A. All three simulation methods exhibit modes that are similarly delocalized, albeit with strong irregularities. The low PCs are spread over the whole molecule, whereas the higher PCs tend to be more localized. The participation number of the (GS)n W peptides has a similar dependence on the mode number, i.e., the lower PCs are strongly delocalized; with increasing mode number the participation number decreases. 48 Study of Biomolecular Dynamics Anharmonicity If the interaction potential of a molecule is purely harmonic, as in Eq. (3.41), the covariance matrix can be expressed in terms of the matrix U, which transforms the original coordinates to the normal modes [Eq. (3.43)], and the normal mode eigenvalues θk X Cij = Uik Ujk θk , (4.7) k where the Uij are the components of the matrix U. From the deﬁnition of PCA it follows that the matrix U solves the PCA eigenvalue equation, Eq. (3.48), for harmonic interaction potentials. Hence, the matrix that transforms the original coordinates to the PCs is identical to the normal mode matrix, i.e. W = U. Therefore, if the interaction potential is purely harmonic, the normal modes, gi (t), collapse with the PC modes, qi (t), and the eigenvalues of the normal modes, θi , are identical to those obtained by PCA, i.e. θi = λi . When describing the ﬂuctuations around a minimum of an anharmonic potential, the diﬀerence between the ﬂuctuations as seen in the simulation and the normal mode contribution – which is an approximation for low temperatures – can quantify, how much is contributed in excess of the harmonic contribution. That is, the fraction of the ﬂuctuations due to the harmonic normal mode approximation allows the anharmonicity to be estimated [24]. In the situation of anharmonic interaction potentials, PCA leads to diﬀerent coordinates than NMA, i.e., U 6= W. The contribution of the normal modes to PC mode i are obtained by projecting the normal mode ﬂuctuations to the one-dimensional subspace spanned by PC mode i. Upon projection, the ﬂuctuation along the principal component i, which would be expected from the normal mode Expl. water Langevin 80 GB/SA 60 40 20 0 0 B100 number of atoms number of atoms A 100 (GS)2 W (GS)3 W 80 (GS)5 W (GS)7 W 60 40 20 20 40 60 80 mode number 100 0 0 20 40 60 80 mode number 100 Figure 4.10: Participation number - A: The participation number of PCs of the different simulations of the β-hairpin (111 atoms). The PCs involve a large number of atoms and are clearly non-local. In particular the low PC modes are non-local. The participation number is not affected by the different simulation methods for the water dynamics. B: The participation number of the (GS)n W (n = 2, 3, 5, and 7) molecules, which consist of 50, 63, 89, and 115 atoms, respectively. The nonlocality is clearly visible and is more pronounced for the lower PCs. 4.2 Thermal fluctuations in biomolecules 49 approximation, is ζi = X (Wji Ujk )2 θk . (4.8) jk The anharmonicity of PC mode i is deﬁned in [24] as s λi ςi = , ζi (4.9) i.e., as the total ﬂuctuations divided by the ﬂuctuations contributed by the normal modes. In a system with purely harmonic interactions, the anharmonicity factor, ςi , equals one, otherwise it is larger than one. It is assumed in the above argument that the mean conﬁguration hr i is given by the conﬁguration at a unique potential minimum. If there are multiple local minima, NMA can be applied to each of the local minima, leaving the deﬁnition of ςi in Eq. (4.9) inconclusive. Therefore, a uniquely deﬁned measure of anharmonicity is required instead of the anharmonicity factor, ςi . −2 −2 10 B Anharmonicity factor ∆i Anharmonicity factor ∆i A Expl. water Langevin GB/SA −3 10 −4 10 (GS)2 W (GS)3 W −3 10 (GS)5 W (GS)7 W −4 10 10 0 20 40 60 80 PC mode number i 100 0 50 100 PC mode number i 150 Figure 4.11: Anharmonicity degree, ∆i for single PCs - For each PC mode the best fit of a Gaussian to the underlying probability distribution function is performed. The anharmonicity degree is calculated using Eq. (4.10). A: Anharmonicity degree ∆i of the β-hairpin. The anharmonicity is similar for the explicit water model, the Langevin simulation, and the GB/SA approach. The PMF of the β-hairpin exhibits a strong anharmonicity for the PC modes above i = 40. B: The anharmonicity of the PMF of the (GS)n W peptides (n = 2, 3, 5, and 7). The PMFs of all peptides exhibit strong anharmonicity for the lower modes. There is a trend of ∆i to decrease with increasing mode number. In order to quantify the deviation of the PMF from a harmonic potential without referring to NMA, one can estimate the diﬀerence between the histogram, h(qi ), and a Gaussian ﬁt, nµ,σ (q), to the histogram in an adequate norm of a function space, e.g., as the L2 -norm3 X (si+1 − si )|h(si ) − nµ,σ (si )|2 . (4.10) ∆i = kh − nµ,σ k2L2 = i 3 The functional space L2 (V ) is defined as the set of functions, f : V → R, which are square R integrable. That is, for all f ∈ L2 (V ) the L2 -norm is finite, i.e., V |f (x)|2 dx < ∞. 50 Study of Biomolecular Dynamics The anharmonicity degree, ∆i , equals zero for a harmonic PMF. The more the PMF deviates from the harmonic one, the higher is ∆i . The anharmonicity degree is illustrated for the β-hairpin and (GS)n W simulations in Fig. 4.11 as a function of the mode number. The lowest modes are strongly anharmonic. With increasing mode number, there is a tendency of ∆i to decrease. Thus, the higher modes are more likely to exhibit a PMF that is approximately harmonic. However, the anharmonicity degree exhibits an irregular shape and scatters irregularly around the trend. All water models employed in the simulations of the β-hairpin lead to similar anharmonicity in the PMF. 4.3 Thermal kinetics of biomolecules The quantities discussed so far can be obtained from the equilibrium distribution in the conﬁguration space. In this sense, quantities like the PMF or the ﬂuctuation along a given mode are stationary and do not represent kinetic aspects of the molecular behavior. In order to describe the kinetic behavior of the ﬂuctuations we analyze the MSD of the PCs. The MSD for the discrete data qn = q(n∆t), with ∆t being the resolution of the trajectory, is obtained as 2 ∆x (t) = ∆x2 (n∆t) = N −n X 1 (qk+n − qk )2 , N −n (4.11) k=1 where N is the total number of frames of the trajectory, the length of the simulation being T = N ∆t. The time averaging procedure in Eq. (4.11) is valid for t ≪ T . For t-values too close to T , the trajectory is statistically not signiﬁcant to perform the time average in Eq. (4.11). Fig. 4.12 illustrates the time-averaged MSD of the individual PC modes of the βhairpin simulations. For the explicit water simulation, Fig. 4.12 A, the PC 1 exhibits a power-law behavior extending from 2 ps up to 100 ns. The simulation length, T = 1µs, is too short to observe a signiﬁcant saturation of the MSD, which is displayed in Fig. 4.12 A only up to t = 0.1 µs, as the MSD can be obtained only for the time scale t ≪ T . The exponent of the power law is in the range of ≈ 0.5, but the value is slightly diﬀerent for the PC 2 and PC 3. The PCs 10, 20, and 30 also exhibit a power-law behavior for the short time scales but they reach more quickly the saturation. A subdiﬀusive MSD is also found in the Langevin simulation, Fig. 4.12 B, and the GB/SA simulation, Fig. 4.12 C. However, the MSD of the implicit water simulations provides power-law exponents ≈ 0.4 and ≈ 0.3, respectively, which are smaller than the explicit water simulation. The diﬀerence between the subdiﬀusion exponents of the explicit water simulation and the GB/SA method prompts to potential inaccuracies of the implicit water simulations when reproducing the kinetics of the β-hairpin. The Langevin simulation mimics the eﬀect of the solvent with an uncorrelated, white noise and a friction term. The GB/SA simply modiﬁes the electrostatic interaction potentials to imitate the presence of water, thereby ignoring friction and exchange of angular momentum. 4.3 Thermal kinetics of biomolecules 10 1 〈∆ x (t)〉 10 2 B 10 PC 1 PC 2 PC 3 PC 10 PC 20 PC 30 10 1 ∼ t0.4 2 2 10 PC 1 Langevin thermostat PC 2 PC 3 PC 10 PC 20 PC 30 explicit water ∼ t1/2 10 2 〈∆ x (t)〉 A 51 0 10 −1 1 10 2 10 10 time [ps] C 3 10 4 10 5 −1 1 10 10 2 10 10 time [ps] 3 10 4 5 10 2 1 〈∆ x (t)〉 10 10 0 PC 1 PC 2 PC 3 PC 10 PC 20 PC 30 GB/SA 2 ∼ t0.3 10 10 0 −1 1 10 2 10 10 time [ps] 3 10 4 5 10 Figure 4.12: MSD of β-hairpin simulations with different water models - The time-averaged MSD of the β-hairpin for various PCs. A: The explicit water simulation reveals a subdiffusive MSD for the lowest PCs. The exponent of the power law is ≈ 0.5. The higher PCs exhibit a quicker saturation. However, for short times, the subdiffusive regime displays MSD exponents similar to those for the low PCs. B: The Langevin dynamics simulation reproduces essentially the subdiffusive behavior found in the MSD of the explicit solvent simulation (A). However, the power law exponent is slightly smaller and equals ≈ 0.4 for PC 1. C: The GB/SA implicit solvent simulation exhibits subdiffusion in the MSD with an exponent ≈ 0.3 in the range of 2 ps to 1 ns. The low PC modes exhibit an increased MSD on a time scale of 1 to 10 ns – a behavior not found for any of the other simulation methods. The removal of the water dynamics can be expressed as a projection to a subspace of the conﬁguration space, which is spanned by the internal coordinates of the β-hairpin. As follows from Zwanzig’s projection formalism, the projection gives rise to memory eﬀects, which are not reproduced by the implicit solvent methods. Zwanzig’s projection approach is discussed in more detail in Sec. 5.1. Surprisingly, the subdiﬀusive behavior, which is in a sense a memory eﬀect, cf. Eq. (2.27), is more pronounced for the implicit solvent simulations. The MSD of the (GS)n W peptide simulations is given in Fig. 4.13. Subdiﬀusive behavior is found in all simulations. The exponents of PC 1 are in the range from 0.65 [(GS)2 W in A and (GS)3 W in B, respectively] to ≈ 0.5 for the (GS)5 W peptide (C). 52 Study of Biomolecular Dynamics A 10 1 B (GS) W 10 (GS) W 2 3 0 10 −1 2 〈∆ x (t)〉 PC 1 PC 2 PC 3 PC 10 PC 20 PC 30 2 〈∆ x (t)〉 10 10 1 C 2 3 10 10 time [ps] 4 10 5 10 PC 1 PC 2 PC 3 PC 10 PC 20 PC 30 0 −1 ∼ t0.65 6 1 10 10 D 5 10 10 2 3 10 10 time [ps] 4 10 5 6 10 2 (GS) W 7 1 〈∆ x (t)〉 10 0 −1 ∼ t0.5 1 10 10 2 3 10 10 time [ps] 4 10 5 1 PC 1 PC 2 PC 3 PC 10 PC 20 PC 30 2 PC 1 PC 2 PC 3 PC 10 PC 20 PC 30 2 〈∆ x (t)〉 10 (GS) W 10 1 −2 10 10 10 10 ∼ t0.65 10 2 10 10 6 10 0 −1 ∼ t0.6 1 10 10 2 3 10 10 time [ps] 4 10 5 6 10 Figure 4.13: MSD of (GS)n W peptides - The time-averaged MSD of the (GS)n W peptides for various PCs. The figures A to D correspond to n = 2, 3, 5, and 7, respectively. For the lowest PCs, all peptides exhibit a subdiffusive MSD over at least three orders of magnitude ˙ ¸in the time domain. An equilibrium is reached for the longest time scales when the MSD, ∆x2 (t) , saturates and does not further increase with increasing time, t. All simulations reach clearly the saturation plateau. The MSD saturates after 103 to 104 ps for the smallest peptide, (GS)2 W , and around 105 ps for the largest peptide, (GS)7 W . Even in the case of (GS)7 W , the MSD of PC 1 reaches a plateau. This gives evidence that the trajectory can be considered as converged with respect to the MSD calculation. A linear least-squares ﬁt to the logarithm of the data is performed in the time range 1 to 10 ps to obtain the exponents of the MSD power-law behavior. The ﬁt is restricted to the shortest time scale to avoid an inﬂuence of the cross-over to the saturation plateau. The exponents can be found in Fig. 4.14 for both the β-hairpin simulations and the (GS)n W peptides. From Fig. 4.14 A it can be concluded, that the subdiﬀusive MSD obtained from the Langevin simulation reproduces the explicit water dynamics reasonably – at least in the range 1 to 10 ps, in which the ﬁt is performed. In contrast, the GB/SA simulation fails to reproduce the kinetic behavior and underestimates the MSD exponents by a factor of two. 4.3 Thermal kinetics of biomolecules A 0.6 B Explicit water Langevin GB/SA 0.8 (GS) W 0.7 (GS) W 2 3 (GS) W MSD exponent α 0.5 MSD exponent α 53 0.4 0.3 5 0.6 (GS) W 7 0.5 0.4 0.3 0.2 0.2 0.1 20 40 60 mode number 80 100 0.1 10 20 30 mode number 40 50 Figure 4.14: MSD exponents - The MSD exponents are obtained with a least-squares fit to the MSD in a double-logarithmic scale. The fit is performed in the time window from 1 ps to 10 ps to avoid an influence of the cross over to the saturation plateau. A: MSD exponents of the different βhairpin simulations. The MSD in the PCs of the β-hairpin, simulated with explicit water, exhibit exponents 6 0.6, indicative of subdiffusion (blue dots). The exponent has a tendency to decrease with the mode number. The subdiffusion exponents are reproduced by the Langevin simulation (red dots). In contrast, the implicit water GB/SA simulation exhibits smaller MSD exponents (green dots), the exponents found in the GB/SA simulation are approximately half as big as those of the explicit water simulation. B: The (GS)n W peptides exhibit a subdiffusive MSD for all n = 2, 3, 5, and 7. The exponents have a tendency to decrease, a behavior more expressed for the smaller peptides. First passage times The ﬁrst passage time of a random walker is the time the walker needs to escape from a given volume. The ﬁrst passage time distribution (FPTD) is a second kinetic quantity analyzed in this section, besides the MSD. The asymptotic behavior of the FPTD is obtained by mapping the dynamics along the principal coordinates, qi , onto a two-state process. For a given PC i, the range of qi values observed during the simulation is partitioned into two parts, qi > a and qi < a. For multi-minima PC modes, the value a is chosen to represent the location of the highest free energy barrier. For the harmonic and quasi-harmonic modes, by symmetry, a is chosen at the position of the minimum of the free energy proﬁle. The results were found to be independent of the precise location of a. The PC mode qi (t) is then mapped to a binary series bi (t) ∈ {0, 1}, such that bi (t) = 0 if qi (t) 6 a and bi (t) = 1 if qi (t) < a. Then, a histogram of the times spent in state 0 and 1 are obtained. As the statistics are poor for the long-time behavior, the FPTD is convoluted piecewise with ﬁlters of diﬀerent sizes. The piecewise ﬁltering is required as the distribution covers four orders of magnitude. The FPTD of the β-hairpin is illustrated in Fig. 4.15 and the FPTD of various (GS)n W peptides is shown in Fig. 4.16. For long times t, the FPTD exhibits a powerlaw behavior, w(t) ∼ t−1−β , again extending to the 10 ns range. The exponent β lies in the range of 0.5 to 0.6 for the lowest modes. A distribution with w(t) ∼ t−1−β and 0 < β < 1 has no mean value. It represents strong, long-lasting correlations in the 54 Study of Biomolecular Dynamics 10 w(t) 10 10 10 10 0 B PC 1 PC 10 PC 30 PC 50 PC 100 PC 150 −2 −4 ∼ t−1.6 10 10 w(t) A −6 10 10 −8 0 10 10 1 2 10 t [ps] C 10 10 w(t) 10 10 10 10 3 10 10 4 0 PC 1 PC 10 PC 30 PC 50 PC 100 PC 150 −2 −4 ∼ t−1.6 −6 −8 0 10 10 1 2 10 t [ps] 10 3 10 4 0 PC 1 PC 10 PC 30 PC 50 PC 100 PC 150 −2 −4 ∼ t−1.7 −6 −8 0 10 10 1 2 10 t [ps] 10 3 10 4 Figure 4.15: First passage time distribution (FPTD), w(t), of the β-hairpin. - The FPTD is obtained by projecting time series of each PC onto a two-state space. For PCs with multi-minima potentials of mean force, the two states are defined by dividing the PC coordinate, q, into the two parts either side of the position of the highest barrier in the potential of mean force. For the higher PCs, which have only a single minimum, by symmetry the space is divided at the position of the minimum. The results are found to be independent of the precise location of the partition. In order to improve statistics at long times the data are piecewise convoluted with filters of different sizes. A: Explicit water simulation, B: Langevin dynamics simulation, C: GB/SA simulation. kinetic behavior. The power-law tail of the FPTD is a consequence of the memory eﬀects that dominate the dynamics. Barrier-crossing processes in a one-dimensional Langevin dynamics lead to an exponential FPTD, as follows from Kramer’s escape theory [69]. The dynamics along the PC i, are not uniquely determined by the free energy proﬁle, i.e. the PMF. On the contrary, the dynamics are strongly inﬂuenced by the dynamics in the orthogonal PCs j 6= i, which contribute correlations to the PC i and give rise to memory eﬀects up to the time scale of ≈ 10 ps. 4.3 Thermal kinetics of biomolecules 55 0 10 B PC 1 PC 10 PC 30 PC 50 PC 100 −2 w(t) 10 −1.6 ∼t −4 10 10 10 w(t) A 10 10 0 PC 1 PC 10 PC 30 PC 50 PC 100 PC 200 −2 −4 ∼ t−1.6 −6 −6 10 10 0 1 10 2 3 10 t [ps] 10 4 10 −8 0 10 10 10 1 2 10 t [ps] 10 3 10 4 0 10 C PC 1 PC 10 PC 30 PC 50 PC 100 PC 200 −2 w(t) 10 −4 10 ∼ t−1.6 −6 10 −8 10 0 10 10 1 2 3 10 10 t [ps] 4 10 Figure 4.16: First passage time distribution (FPTD), w(t), of the (GS)n W peptides. The FPTD is obtained by projecting time series of each PC onto a two-state space. For PCs with multi-minima potentials of mean force, i.e. the lowest, the two states are defined by dividing the PC coordinate, q, into the two parts either side of the position of the highest barrier in the potential of mean force. For the higher PCs, which have only a single minimum, by symmetry the space is divided at the position of the minimum. The results are found to be independent of the precise location of the partition. In order to improve statistics at long times the data are piecewise convoluted with filters of different sizes. A: (GS)2 W , B: (GS)5 W , C: (GS)7 W . Convergence An important issue for the discussion of the MD simulations is the question as to whether the trajectories are long enough to sample the conﬁguration space suﬃciently. That is, has the system reached a state of thermal equilibrium on the time scale of the simulation? As pointed out in Sec. 3.6, this question is diﬃcult to assess: we do not have multiple trajectories of the same system at our disposal to compare with. The MSDs of the PC 1 in Fig. 4.13 A-D exhibit a plateau at least at the time scale t ≈ T /10, i.e. one order of magnitude below the simulation length, T . As the constant saturation plateau is an equilibrium eﬀect, all of the (GS)n W simulations can be considered as converged with respect to the MSD. An analysis of the PMFs aﬃrms the convergence of the (GS)n W simulations. When cutting the trajectory into two pieces, a ﬁrst and a second half, two histograms, from 56 Study of Biomolecular Dynamics 6 7 5 B6 A β-hairpin 4 half B 2 F(q) [k T] 1 half nd 3 2 full trajectory 1st half 2nd half 4 3 2 1 0 (GS)5 W 5 st B F(q) [k T] full trajectory 1 −15 −10 −5 0 q [nm] 5 10 0 −10 −5 0 q [nm] 5 10 Figure 4.17: Free energy profile (PMF) for two halves of simulation - Histograms of the PC mode 1 are obtained from the first and the second half of the simulation, independently. A: The β-hairpin PMF (explicit water simulation) exhibits substantial differences between the first (red) and the second half (magenta) of the simulation. The difference indicates the poor sampling of the configuration space, i.e., the trajectory is unconverged. B: The two halves of the (GS)5 W peptide simulation exhibit the same structure and are in general close to each other. The sampling is much better relative to A, albeit not perfect. the ﬁrst and the second half, are calculated successively, h1 and h2 , respectively. In Fig. 4.17, the free energy proﬁle (i.e. the PMF) for the PC mode 1 is illustrated for the β-hairpin simulation with explicit water (A) and the (GS)5 W peptide. The β-hairpin PMF exhibits strong diﬀerences between the proﬁle obtained from the ﬁrst half and the one obtained from the second half, which is due to poor sampling. In contrast, the two free energy proﬁles of the ﬁrst PC mode of (GS)5 W nearly collapse indicating a rather good convergence. We calculate the L2 -diﬀerence between the two histograms, h1 and h2 , that is X Λi = kh1 − h2 k2L2 = (si+1 − si )|h1 (si ) − h2 (si )|2 . (4.12) i The convergence factor, Λi , is illustrated in Fig. 4.18 for the β-hairpin simulations (A) and the (GS)n W peptides (B). The diﬀerences between the ﬁrst and the second half of the simulations are larger for the lower PCs containing the diﬀusive contribution. In contrast to the (GS)n W peptides, the MSD of the β-hairpin PC 1 in Fig. 4.12 A does not reach a constant plateau. Therefore, the β-hairpin simulation is not fully converged on the time scale of simulation length, T = 1 µs. This is conﬁrmed by the Λi as illustrated in Fig. 4.18. The lowest Λi are approximately a factor ten larger for the β-hairpin in Fig. 4.12 A than the values of (GS)7 W (B). 4.4 Conclusion MD simulations of a β-hairpin molecule have been studied on a time scale of 1 µs. Three diﬀerent methods to imitate the dynamics of solvent water have been used: the 4.4 Conclusion 10 A 57 −2 B (GS)2W 10 Langevin (GS) W GB/SA (GS) W 5 7 i 10 −3 10 10 (GS)3W Λ Λ i Explicit water −3 −4 −4 0 50 100 PC mode number i 150 10 −5 0 50 100 PC mode number i 150 Figure 4.18: Convergence factor. - Histograms of the PC coordinates are obtained from the first and the second half of the simulation, independently. The L2 -norm of the difference between the two histograms is given as the convergence factor, Λi , for every PC i. A: β-hairpin simulation, B: (GS)n W peptides. eSPC explicit water model, a Langevin thermostat and a method using the generalized Born equation. Also, MD simulations of four (GS)n W peptides with n = 2, 3, 5, and 7 have been analyzed on the microsecond time scale. A PCA has been performed for each of the above-mentioned systems. The PMF of the low PCs exhibits strong anharmonicities in all systems. The lowest PCs are delocalized, general motions which involve a large fraction of atoms in the molecule. The PMF and the delocalization of the PCs exhibit the same features irrespective of the water model used. The kinetics of the molecules is characterized in terms of the MSD of the individual PC modes. The MSD of the explicit solvent simulations exhibits a subdiﬀusive pattern in the time range from 1 ps to the range of nanoseconds. The (GS)n W peptides do not fold into a unique secondary structure. The observation of a subdiﬀusive MSD in the PC modes of these peptides proves that a complex, secondary structure as seen by proteins is not a requirement for fractional diﬀusion in the internal coordinates of molecules. Therefore, the subdiﬀusivity appears as a widespread, general feature of biomolecular ﬂuctuations. The MSD found in the β-hairpin for the Langevin thermostat is similar to the subdiﬀusive MSD of the explicit water simulation. The Langevin thermostat mimics the eﬀect of the solvent water dynamics. It does not include, however, any kind of memory eﬀects, potentially arising from the projection of the dynamics to the subspace of the internal coordinates of the molecule, cf. Sec. 5.1. Therefore, the coincidence of the results of the Langevin thermostat and the simulation with the eSPC water model indicates that on the time scale of and above 1 ps, memory eﬀects due to the projection to the internal molecule coordinates are negligible. That is, the detailed water dynamics can be ignored in the analysis of the internal subdiﬀusion, at least on the time scales of 1 ps and above. In contrast, the GB/SA simulation exhibits a subdiﬀusive MSD which is diﬀerent from the behavior found in the explicit water simulation. The exponents of the MSD 58 Study of Biomolecular Dynamics power law are substantially smaller in the GB/SA simulations than the exponents found in the explicit water simulation. The diﬀerence between the simulation with explicit water and the simulation based on the generalized Born equation prompts to inaccuracies of the kinetics, as found with the GB/SA simulations. When projecting the dynamics to a single PC mode, the dynamics is characterized by strong memory eﬀects on time scales up to at least 10 ns. The memory eﬀects are illustrated by the FPTD, which exhibits a power-law decay. The (GS)n W peptide simulations have reached an equilibrium on the timescale of microseconds, whereas the β-hairpin simulation has poorly converged on the time scale of the simulation, T = µs. In any case, the subdiﬀusive MSD found in all of the systems analyzed is not an out-of-equilibrium eﬀect but can be considered to be part of the stationary kinetics at ambient temperature. In the next chapter, we address the question as to which mechanism gives rise to the subdiﬀusive MSD found in the MD simulation presented here. Various models are considered and compared with the MD simulation results obtained. Nihil certi habemus in nostra scientia, nisi nostram mathematicam. Nicolaus von Kues Chapter 5 Modeling Subdiffusion The results in this chapter have been partially published in T. Neusius, et al., Phys. Rev. Lett. 100, 188103, cf. Ref. [140] and submitted for publication as T. Neusius, et al., cf. Ref. [147]. The presence of subdiﬀusive kinetics in the internal coordinates of biomolecules raise the question as to the underlying mechanism. Several models can account for subdiﬀusion, some of which were discussed as candidates for biomolecular ﬂuctuations. In the following chapter, some of these models are reviewed. It is demonstrated how the models compare with the experimental and simulation results. In particular, the continuous time random walk (CTRW) and the diﬀusion on fractal geometries are subjects of the present thesis. First, we discuss brieﬂy the projection approach of R. Zwanzig as described in [69]. Zwanzig’s formalism allows the Hamiltonian equations to be written in a subspace of reduced dimensionality. Correlations and memory eﬀects are the price one has to pay for the reduced number of degrees of freedom. 5.1 Zwanzig’s projection formalism Many systems of physical interest possess so large a number of degrees of freedom that it is impossible to follow simultaneously all of them in an experiment or in theoretical considerations. And even if it is possible, it may be more eﬃcient to omit irrelevant details and describe the system in terms of the relevant aspects, as is done, for example, when Brownian motion is modeled by the Langevin equation. Here, we aim at establishing a framework, in which the system can be approximately described with reasonable eﬀort. Furthermore, it is of high importance to estimate the accuracy of the approximation. Zwanzig’s projection operator approach allows a Hamiltonian system to be described in a subspace of the phase space. The subspace is spanned by the set of relevant coordinates; irrelevant coordinates are neglected. Assume a Hamiltonian system with coordinates qi and momenta pi and the Hamilton function H(q , p). The phase space, Γ , is spanned by the qi and pi . Let A(q , p, t) be a dynamical variable depending on the phase space coordinates, the momenta, and 60 Modeling Subdiffusion time the t. The time evolution of A is given by the Liouville equation ∂ A = LA, ∂t in which the Liouville operator is used, deﬁned as X ∂H ∂ ∂H ∂ − L= . ∂pi ∂qi ∂qi ∂pi (5.1) (5.2) i Note that the Liouville equation is a diﬀerential equation. That is, the time derivative of A at time t∗ depends exclusively on the values of A at that time t∗ . There are no time correlations present, i.e., the values of A in the past t < t∗ or in the future t > t∗ do not inﬂuence the time derivative in t∗ . This property is referred to as the Markov property and a system is Markovian if it complies with that property. All Hamiltonian systems are Markovian, as they all can be described by Eq. (5.1). In what follows, we do assume Eq. (5.1) to be valid, i.e., we assume the system to be Markovian. However, we do not explicitly use the form of the Liouville operator which follows from the Hamilton function. Therefore, the results of this section are valid for systems whose time evolution can be described by an equation of the form Eq. (5.1). The dynamical variable A is a function on Rthe phase space, A : Γ → R. We assume A to be square integrable, i.e., the integral Γ |A|2 dq dp is ﬁnite. The space of all square integrable functions is denoted as L2 (Γ ) and is a Hilbert space with the scalar product Z hA|Bi = ABdq dp. (5.3) Γ The Liouville operator maps the space onto inself, L : L2 (Γ ) → L2 (Γ ). It can be expressed as a matrix with respect to a basis set of L2 (Γ ). Let the set {ψi (q , p)} be a orthonormal basis set of L2 (Γ ). The dynamical variable A can be written as a linear combination of the basis vectors (which are L2 -functions, ψi : Γ → R), X A(q , p, t) = ai (t)ψi (q , p). (5.4) i The scalar product acts as a projection to the basis vectors. Therefore, the coeﬃcients of the expansion of A can be obtained as ai = hψi |Ai. The Liouville equation can be written componentwise as X ∂ ak (t) = Lki ai (t). (5.5) ∂t i The matrix elements of the Liouville operator with respect to the basis {ψi } are obtained as Lki = hψk |Lψi i . (5.6) Note that the Hilbert space L2 (Γ ) is of inﬁnite dimension, irrespective of how many dimensions span Γ . Therefore, the matrices must be compact to ensure the convergence 5.1 Zwanzig’s projection formalism 61 of the inﬁnite sums in the above calculations. Here, we are not dealing with the mathematical diﬃculties of this sort. When describing physical systems with many degrees of freedom one often assumes that not all details of the system are equally important. A lot of the dynamical information can be deemed unimportant or irrelevant for the features of interest. We seek a physical description based on those quantities considered relevant and neglect the irrelevant details. The properties of the system, which we would like to understand, are certainly relevant, but it may turn out that further quantities can not be ignored as they crucially determine the time evolution. There is neither a general rule, how to ﬁnd a set of relevant quantities or coordinates in a given physical system, nor can it be taken for granted that a useful reduced description exists. But the reduced description has proved advantageous in many situations and can sometimes considerably increase the eﬃciency of the physical description. If we chose a set of relevant coordinates, the time evolution can be obtained by a projection of Eq. (5.1) to the subspace spanned by the relevant coordinates. The eﬀort to express the time evolution of dynamical quantities in terms of basis functions in L2 allows projection operators to be written in a simple form. In what follows, we assume a two-dimensional situation: one relevant coordinate and one irrelevant coordinate. The Liouville equation, Eq. (5.1) reads in the two-dimensional representation, ∂ a1 L11 L12 a1 = . (5.7) L21 L22 a2 ∂t a2 The solution of the second component can be written as Z t L22 t eL22 (t−s) L21 a1 (s)ds. a2 (t) = e a2 (0) + (5.8) 0 The solution of a2 (t) can be substituted into the ﬁrst component of the Liouville equation, leading to Z t ∂ eL22 (t−s) L21 a1 (s)ds + L12 eL22 t a2 (0). (5.9) a1 (t) = L11 a1 (t) + L12 ∂t 0 The irrelevant coordinate enters only with its initial condition in the last term, the rest of the equation depends only on a1 , the relevant coordinate. Eq. (5.9) can be read as an equation depending on a1 with a perturbation represented by the last term, which is also referred to as noise term, ξ(t) = L12 eL22 t a2 (0). With the deﬁnitions µ(t) = −eL22 t L21 , Ω = L11 , and γ = L12 the equation for the relevant variable v(t) = a1 (t) reads Z t ∂ µ(t − s)v(s)ds + ξ(t). (5.10) v = Ωv − γ ∂t 0 This is the generalized Langevin equation (GLE). The time evolution of the relevant coordinate v is given by an equation that has some similarity with the Langevin equation, Eq. (2.11). The Langevin equation is the limiting case of Eq. (5.10) for Ω = 0, 62 Modeling Subdiffusion µ(t) = δ(t) and assuming ξ to be white noise. In contrast to the classical Langevin equation, the GLE is not a diﬀerential equation but an integro-diﬀerential equation, which is not local in time, i.e., the derivative of v at time t∗ depends on the values of v in the past, t < t∗ . Therefore, the description of the dynamics of v in the reduced subspace is non-Markovian! The integral with the memory function µ(t) represents correlations in the system. This is the consequence of the projection. In other words: reducing the complexity of a Markovian system by a projection to a subspace gives rise to non-Markovian dynamics and a perturbation term. Eq. (5.10) gives also a justiﬁcation of the noise term in the free Langevin equation, Eq. (2.11): the description in a relevant subspace gives rise to stochasticity. The general case, where the projection leads to a multidimensional subspace, fading out several dimensions, can be treated by a similar approach. In Eq. (5.9), the ai are two parts of the vector A, containing the relevant coordinates for i = 1 and the irrelevant coordinates for i = 2. Reading Eq. (5.9) as a matrix equation, the memory function is a consequence of the oﬀ-diagonal blocks in the Liouville operator, L21 and L12 , which couple the relevant with the irrelevant coordinates. For details, see [69]. The remarkable result of this section is that whenever we ignore parts of a Hamiltonian system, we introduce a stochastic element in the dynamics; and we are confronted with non-Markovian dynamics as a consequence of incomplete dynamical information. The analysis of the MD simulations in Chap. 4 was based on the degrees of freedom of the simulated biomolecule (β-hairpin or (GS)n W peptide). However, in the explicit water simulation the molecule is just a subsystem, the dynamics of which depend on the dynamics of the surrounding water. Therefore, we have to take account of the possibility of memory eﬀects and non-Markovian behavior. The analysis of the diﬀerent water models used in the MD simulation of a β-hairpin molecule indicate that the memory eﬀects due to he water dynamics are not signiﬁcant on the time scale 1 ps, see Sec. 4.3. If external memory eﬀects are to be taken into account in MD simulation, random forces with correlated noise (“colored noise”) can be used, as was recently suggested by Ceriotti et al. [148]. 5.2 Chain dynamics MD simulations are based on a reﬁned, empiric, classical model of the microscopic dynamics. An interpretation of the MD trajectories and their features is impeded by the complexity of the model used. Therefore, simpliﬁed models can help to identify the mechanism that gives rise to a speciﬁc dynamical eﬀect. At best, this allows to check which of the “ingredients” of the model account for which sort of dynamical properties. A common model is the Rouse chain, developed in the context of polymer physics [149, 150]. The Rouse chain consists of N beads, each of mass m. Let zi be the position of bead i. The beads are connected by Hookean springs with the angular frequency ω̃, such that they built a linear chain without branching. In the overdamped 5.2 Chain dynamics 63 approximation, i.e., ignoring inertial eﬀects, the Hamilton function of the chain reads H= N X 1 i=1 2 mω̃ 2 (zi−1 − zi )2 + (zi − zi+1 )2 . (5.11) As the three Cartesian coordinates of each bead decouple, the Rouse chain is treated here as a one-dimensional problem, replacing the vectors zn by zn . The chain dynamics can be described in terms of the normal modes, ζn , which are derived in Appendix C. The eigenfrequencies obtained from NMA are given as ωn = ω̃πn N −1 with n = 0, 1, ..., N − 1. (5.12) The Hamilton function reads in the basis of the normal modes N −1 1 X 2 2 ω ζ (t). H= m 2 n=0 n n (5.13) Note that the normal modes collapse with the PC, as the beads have identical mass m and all interaction potentials are harmonic. To include thermal ﬂuctuations, the chain is coupled to a Langevin heat bath with friction, γ, and a frequency speciﬁc random force, ξn , analogous to Eq. (2.11). In the normal mode coordinates the equations of motion read mγ d ζn = −mω 2 ζn + ξn , dt (5.14) where the inertial terms are ignored as in the Hamilton function (overdamped approximation). The noise ξn is assumed to be white noise [Eqs. (2.12) and (2.13)]. Therefore, we can treat the system as ergodic and do not need to specify the average procedure in what follows in this section. As the ζn (t) contribute quadratically to the Hamilton function, the equipartition theorem can be applied. Hence, the ACF of the normal modes is given as hζn (t + τ )ζk (τ )i = kB T −ωn2 t/γ e δnk . mωn2 (5.15) Using Eq. (2.27), the MSD of the normal modes follows from the ACF as 2 t/γ kB T −ωn . 1 − e [ζn (t + τ ) − ζn (τ )]2 = 2 mωn2 (5.16) Hence, the MSD of the normal modes/PC modes exhibits a linear time dependence for short times and saturates for long times. The typical saturation time of the slowest mode, γ(N − 1)2 , (5.17) τR = π 2 ω̃ 2 64 Modeling Subdiffusion is the so-called Rouse time, which dominates the long time behavior of the model. Finite time averaging, h·iτ,T , needs to exceed the Rouse time, i.e. T ≫ τR , in order to justify the application of the ergodic hypothesis. From Eq. (5.16) it follows that the MSD of the normal modes is not subdiﬀusive in the Rouse model. However, other generalized coordinates can exhibit subdiﬀusive behavior. Consider the distance between two beads, ∆(t) = zi (t) − zj (t). The beads i and j are assumed to be neither neighbored nor close to the chain ends. If t ≪ τR , the ACF of ∆(t) is given as h∆(t + τ )∆(τ )iτ = t−1/2 . (5.18) A detailed derivation is given in Appendix C. The power-law decay of the ACF leads immediately to a subdiﬀusive MSD according to Eq. (2.27). The subdiﬀusive distance ﬂuctuations are a consequence of a superposition of the exponential normal modes. Since the long-time dynamics are given by the slowest mode, the power-law ACF, Eq. (5.18), breaks down at ≈ τR , with an exponential decay for t > τR [151]. The above calculations for the Rouse chain can be extended to include hydrodynamic eﬀects [152] or long-time memory eﬀects using a GLE approach [153, 154]. The model can also be generalized to other bead spring geometries, which include branching, loops and fractal clusters [151, 155, 156]. It can be demonstrated that this allows power laws to occur in the distance ACF with various exponents [155]. The structure of the backbone suggests the application of chain models to the dynamics of peptides [153, 157]. GLE based models have been employed to model long-time dynamics of biomolecules [153, 154]. Cross links and more sophisticated coupling geometries between the beads have been designed for the description of protein ﬂuctuations [151, 155]. Distance ﬂuctuations, as exempliﬁed by the above derivation, are of particular interest: ﬂuorescence quenching by a tryptophan residue can be used to measure the distance ﬂuctuations in single molecule experiments [27, 51, 52, 157]. The observation of a ﬂavin reductase protein complex revealed subdiﬀusive dynamics on the 10−4 to 100 s time scale [27, 51, 52]. However, an estimation of the Rouse time is in the order of a few nanoseconds, i.e., at least four orders of magnitude below the observed subdiﬀusive regime [158]. In the case of the biomolecules presented in Chap. 4, typical normal mode ﬂuctuations are in the range of some squared nanometers, i.e. ∆x2 ≈ 1 nm2 . Approximately, the corresponding frequencies are obtained as ω̃ 2 ≈ 3kB T /m ∆x2 , which are in the range of ω̃ ≈ 1 ps−1 . A typical friction value of a heavy atom at the surface of an biomolecule is γ ≈ 50 ps−1 [159]. Eq. (5.17) allows the Rouse time of the longest biomolecule in Chap. 4 (N = 16, the number of residues) to be estimated as τR ≈ 76 ps. In contrast, the subdiﬀusion found in the (GS)n W peptides and the β-hairpin extends to 10 ns. Therefore, the Rouse chain model – irrespective of potential subdiﬀusive dynamics on short time scales – cannot provide an understanding of the subdiﬀusive dynamics as found in the MD simulations presented in Sec. 4.2. 5.3 The Continuous Time Random Walk (CTRW) 5.3 65 The Continuous Time Random Walk (CTRW) Trap models Non-exponential relaxation patterns, as found in the dynamics of proteins and peptides, are a common property of glassy materials. There are other characteristics shared by glasses and proteins, such as non-Arrhenius temperature dependence [29, 30] and the enhancement of ﬂuctuations above the glass transition [31]. The similarity of biomolecules and glasses is considered as a consequence of various, nearly isoenergetic conformational substates that both, proteins and glasses, can assume [20, 23, 30]. The energy landscape of glasses, and the “glassy” dynamics that follows from it, were successfully described as a hopping between energy traps on a fully connected lattice [160, 161]. Energy traps are understood as local minima of the energy landscape [24]. Kramer’s escape theory [69] allows to ascribe a typical escape time to a potential energy minimum at a given temperature. The typical time of escaping can be identiﬁed with an eﬀective energy depth of the minimum. Then, the distribution of escape times, or, equivalently, the distribution of eﬀective trap depths determines the dynamics of the system [160, 162]. The essential properties of this sort of trap models are equivalently reproduced by the continuous time random walk (CTRW) model [9]. Introduction of the CTRW model Pearson’s random walk, as discussed in Sec. 2.4, has a ﬁxed jump length, ∆x and a ﬁxed time step, ∆t. Here, such random walks with ﬁxed, discrete time step are referred to as classical random walks. The diﬀusion limit corresponds to the behavior on length scales ≫ ∆x and time scales ≫ ∆t. In mathematical terms, this is expressed by the limit (∆x, ∆t) → 0, in which D = ∆x2 /2∆t is constant. The CTRW generalizes the classical random walk, such that both variables, jump length and time step are random variables. The jump length, here denoted as x, is characterized by the jump length distribution (JLD), ϕ(x). Throughout this thesis it is assumed, that the JLD is symmetric around x = 0 and the mean value of the JLD equals zero. The time between two successive jumps is referred to as waiting time, t, and it is taken from the distribution w(t). Assume the JLD has the variance x̄2 and the waiting time distribution (WTD) has the mean value t̄. Then, a derivation analogous to Sec. 2.4 can be applied to derive the diﬀusion equation, Eq. (2.33), in the limit x̄2 → 0 and t̄ → 0 with D = x̄2 /2t̄ constant. Therefore, if both, the variance of the JLD and the mean value of the WTD, are ﬁnite, classical diﬀusion occurs, i.e., Eq. (2.33) is valid and the MSD exhibits a linear time dependence, 2 ∆x (t) = 2Dt, (5.19) as in Eq. (2.35). The situation is diﬀerent, if the distributions do not satisfy the above conditions. If the JLD has a diverging variance, the CTRW is found to be superdiﬀusive, i.e., the 66 Modeling Subdiffusion MSD exhibits a time dependence ∼ tβ , where β > 1 [8]. A subdiﬀusive MSD occurs, if the WTD has an inﬁnite mean value. The present thesis is focused on subdiﬀusion. Therefore, we conﬁne the discussion to the following scenario, as laid out in [8]. We study a one-dimensional1 CTRW with the JLD, ϕ(x) and the WTD, w(t). The JLD is assumed to have mean value zero, to be symmetric, and to have a ﬁnite variance x̄2 . Let the WTD have a power-law tail, i.e., −1−α t , (5.20) w(t) ∼ τ0 with 0 < α < 1 referred to as the WTD exponent. The time τ0 deﬁnes the time unit and is not a relaxation time. As a consequence of the power-law decay of w(t), the distribution has a diverging mean value. The results in this section are independent of the detailed analytical form of w(t) at short times. A ﬁrst jump from the position x = 0 is assumed to occur at t = 0. Note that this are two independent initial conditions: the initial condition for the position is W0 (x) = δ(x), i.e., being at x = 0 at t = 0, whereas the initial condition for the waiting times is having the ﬁrst jump at t = 0. Illustrations of one-dimensional CTRWs with various WTD exponents are given in Fig. 5.1. Application of CTRW The CTRW has been developed in the context of charge currents in semiconductors [9, 90, 163] or, more generally, for transport in amorphous, disordered materials [164]. There is a plethora of applications in very diﬀerent ﬁelds, see [8, 74, 90] for a review. Trapping models have become a powerful model applied to a variety of diﬀerent processes, such as diﬀusion through actin networks [84, 165], diﬀusion crossing a membrane [166], or through porous media [85, 86]. There is a wide variety of cases where, rather than unbounded diﬀusion, distinct boundaries exist and can have critical eﬀects on diﬀusive dynamics [167]. Examples of these are the subdiﬀusive dynamics of macromolecules in the cell nucleus [168], the cytoplasm, which has been shown to emerge from crowding [78, 80–82, 169], in cell and plasma membranes [170–173], and the subdiﬀusion of lipid granules, which is inﬂuenced by the presence of entangled actin networks [79, 83, 84]. The consequences of reactive boundaries [174–176] and diﬀusion through diﬀerent kinds of narrow pores and tubes [177–182] have also been examined. In the context of internal dynamics of biomolecules, CTRW has been suggested as a possible mechanism causing the dynamics to be subdiﬀusive [52, 183], based on the analysis of time-averaged quantities, such as the ACF. However, care must be taken when applying the CTRW model to time averages of measured time series, because the CTRW is an intrinsically non-ergodic process. Recently, the question as to how timeaveraged CTRW quantities behave has attracted theoretical attention [165, 184–188]. In this section, the time averages of CTRW are examined. The time dependence of the 1 Multidimensional CTRWs can be decomposed into a set of one-dimensional CTRWs, if the walk is statistically isotropic. 5.3 The Continuous Time Random Walk (CTRW) 67 time-averaged MSD for unbounded and for bounded diﬀusion are among the results obtained in the present thesis [140, 147]. Simulation of CTRW The theoretical results in this sections are compared to simulations of a CTRW with WTD exponent α. The simulations were performed as follows. The jump lengths are taken from a Gaussian distribution with variance one, implemented as a Box-Muller algorithm [107]. All CTRWs were started at x = 0 with an initial jump at t = 0. A random variable tw with the distribution given in Eq. (5.20) is obtained from the uniformly distributed r ∈ (0, 1) via the transformation tw = r −1/α − 1. With this transformation, the WTD has the following analytical form w(tw ) = α . (1 + tw )1+α (5.21) Here, we set τ0 = 1. Uniformly distributed random numbers, r ∈ (0, 1) were generated with the long-period random number generator of L’Ecuyer with Bays-Durham shuﬄe and added safeguards [107]. 5.3.1 Ensemble averages and CTRW In Sec. 2.4, the diﬀusion equation is derived from the classical random walk. Instead of a microscopic description of the random walk, the diﬀusion equation determines the time evolution of the probability distribution, W (x, t). For a large number, N , of similar particles, the dynamics of which is given by W (x, t), the probability gives rise to a particle density, ρ(x, t) = N W (x, t). Note that the probability W (x, t) corresponds to an ensemble average. In contrast to the individual particle, the density is a macroscopic quantity. With this perspective, the diﬀusion equation is the macroscopic description of a large number of microscopic, classical random walkers. The question arises as to whether a similar equation exists which describes the time evolution of the probability of a CTRW. It has been demonstrated that such an equation can be derived from the CTRW, e.g., by means of Fourier-Laplace transforms [8], which is demonstrated in Appendix D together with some mathematical details. In the case of CTRW, the diﬀusion limit reads (x̄, τ0 ) → 0, such that the generalized diﬀusion constant, deﬁned as Kα = x̄2 /τ0α , has a ﬁnite, non-zero value. The fractional diﬀusion equation (FDE) is [6] ∂ ∂2 W (x, t) = 0Dt1−α Kα 2 W (x, t), ∂t ∂x where the Riemann-Liouville operator is used Z t φ(t′ ) 1 ∂ 1−α dt′ . φ(t) = 0 Dt Γ(α) ∂t 0 (t − t′ )1−α Γ(α) is the Gamma function, see Appendix D.1. (5.22) (5.23) 68 Modeling Subdiffusion position x 20 0 position x −20 0 20 40 60 t [steps] 80 100 100 0 −100 −200 0 position x Class Diff α = 0.9 α =0.75 α =0.5 2000 4000 6000 t [steps] 8000 10000 0 −500 −1000 0 2 4 6 t [steps] 8 10 5 x 10 Figure 5.1: CTRW – individual time series - Time series of the positions, x(t), of individual CTRWs are displayed. The colors refer to various WTD exponents, α = 0.5 (blue), α = 0.75 (red), and α = 0.9 (green). In yellow a classical random walk with the same x̄2 . The three figures illustrate three different time scales, T = 102 , 104 , 106 (top down). The CTRWs exhibit on all time scales a similar pattern of waiting times. The derivation of Eq. (5.22) involves an averaging procedure over the ensemble. The elements of the ensemble are all possible realizations of CTRW respecting the initial condition, in particular, all members of the ensemble have a ﬁrst jump at t = 0. Obviously, there is no other point in time, at which the probability of observing a jump equals one. Therefore, the initial condition breaks the time-shift invariance of the CTRW process. This symmetry breaking is referred to when saying that the CTRW in the ensemble average undergoes aging. Eq. (5.22) is an integro-diﬀerential equation, i.e., it is not a local equation in time: the time derivative at t is inﬂuenced by the values of W (x, t′ ) with t′ 6 t. This accounts for memory eﬀects and makes the process described by Eq. (5.22) non-Markovian. As the memory eﬀects do not decay fast, i.e., the decay follows a power law instead of an exponential decay, the process is non-ergodic. As in the case of the diﬀusion equation, Fourier decomposition allows to give an 5.3 The Continuous Time Random Walk (CTRW) 69 analytical solution of Eq. (5.22) in terms of the Mittag-Leﬄer function2 (MLF) Z ∞ a(k)Eα (−Kα k2 tα )e−ikx dk. (5.25) W (x, t) = 0 The Fourier coeﬃcients a(k) depend on the initial condition of the position and the boundary conditions. If W0 (x) = δ(x), the coeﬃcient function is a(k) = (2π)−1 . The position part of Eq. (5.25) is identical to the classical expression in Eq. (2.34). The diﬀerent behavior is due to the Mittag-Leﬄer decay which replaces the exponential relaxation of the diﬀusion equation [Eq. (2.33)]. The MLF exhibits an asymptotic power-law dependence at long times, Eα (−Kα k2 tα ) ∼ t−α , which is considerably slower than an exponential decay. This causes long-lasting memory eﬀects. In particular, the power-law relaxation pattern does not possess a typical time scale, a consequence of the WTD from Eq. (5.20). Therefore, there is no time scale on which correlations can be ignored. Hence, time-averages over an interval [ts , ts + T ] depend on ts , irrespective of the length of T . The ergodic hypothesis is not applicable3 , the CTRW exhibits weak ergodicity breaking [184–186]. Eq. (5.22) leads to a subdiﬀusive MSD Z ∞ 2 2Kα α t , (5.26) ∆x (t) ens,0 = x2 W (x, t)dx = Γ(1 + α) −∞ which can be derived by means of Fourier-Laplace transformation [8]. The notation h·iens,0 indicates averages over the ensemble with the initial condition of a ﬁrst jump at t = 0 during the time interval [0, t]. Therefore, Eq. (5.26) is the ensemble-averaged MSD from the origin, x = 0, during the time interval [0, t]. The ensemble-averaged MSD is not invariant with time shift. The ensemble-averaged MSD during the time interval [ts , t + ts ] with ts > 0 of a CTRW with initial jump at t = 0, denoted by ∆x2 (t) ens,ts , is diﬀerent from ∆x2 (t) ens,0 . In Fig. 5.2 simulation results with several values of ts illustrate the non-stationarity of the process. Due to the breaking of the time shift symmetry of the CTRW, the process depends on the time of the ﬁrst jump. Loosely speaking, the process ‘looks statistically diﬀerent’ at diﬀerent times; it is not stationary. 2 The Mittag-Leffler function (named after Gösta M.-L. (1846-1927), Swedish mathematician) is defined as ∞ X zn Eα (z) = . (5.24) Γ(1 + nα) n=0 For details see Appendix D. 3 As mentioned for the classical diffusion equation, there is no meaningful stationary solution for the position probability distribution in the absence of a potential or a boundary. Therefore, care must be taken with the ergodic hypothesis when dealing with free, unbounded diffusion. The same argument applies to Eq. (5.22) representing the diffusion limit of the free, unbounded CTRW. When energy potentials are present, the FDE can be generalized to a fractional Fokker-Planck equation, which contains Eq. (5.22) as a limiting case [7]. With appropriate potentials, a stationary solution exists for both, the classical random walk and the subdiffusive CTRW. However, even with a stationary solution, the memory effects embodied by Eq. (5.25) are fundamentally in conflict with the ergodic hypothesis. See Subsec. 5.3.3. 70 Modeling Subdiffusion 4 10 3 〈 ∆ x2(t)〉ens,t s 10 ts = 0 2 ts = 103 10 ts = 104 1 10 ts = 105 ∼ t0.5 0 10 −1 10 2 10 10 4 10 6 10 8 t [time] Figure 5.2: Free, unbounded CTRW – ensemble-averaged MSD - Ensemble-averaged MSD of with WTD exponent α = 0.5. The MSD exhibits a power-law time ˙ ¸ dependence, ˙ an2 CTRW ¸ ∆x (t) ens,0 ∼ tα , as predicted by Eq. (5.26). However, the shifted MSD, ∆x2 (t) ens,t , depends s on ts , i.e., it is not invariant with time shift. The ensemble average is performed over 1 000 individual CTRWs. Assume we observe an CTRW process, starting the observation at a time ts > 0. The time ts is almost always between two jumps. Therefore, a period t1 elapses until we observe the ﬁrst jump. The distribution of the initial waiting times t1 is denoted as w1 (t1 , ts ). The time shift invariance makes the initial WTD, w1 (t1 , ts ) depend on ts . After the initial waiting time has elapsed, a ﬁrst jump occurs at ts + t1 . If a new time coordinate is introduced by t̃ = t − ts − t1 , the ﬁrst jump takes place at t̃ = 0. In this time coordinate the usual CTRW theory can be applied for times t̃ > 0. Hence, the initial WTD is suﬃcient to characterize the situation, in which the observation starts at ts . A CTRW with ts > 0 is referred to as aging CTRW (ACTRW) [189]. 5.3.2 Time averages and CTRW The CTRW is a non-ergodic process. As the mean of the WTD in Eq. (5.20) diverges, there is no typical relaxation time in CTRW. As a consequence, time averages of CTRW quantities are, in general, diﬀerent from the ensemble average of the same quantities. Although many types of experimental measurements provide ensemble averages, certain techniques provide time series, such as, for example, single-particle tracking, single molecule spectroscopy or, in particular, the MD simulations presented in Chap. 3. Time averages are then required to extract statistically signiﬁcant properties from the data. Therefore, the question has arisen as to how the time-averaged properties of CTRW processes behave [140, 165, 184–188]. In the following we focus on the relative probability of being at position x at a time t > ts , provided the walker was at position xs at time ts > 0. This probability can be 5.3 The Continuous Time Random Walk (CTRW) 71 expressed as [190, 191] W (x, t; xs , ts ) = ∞ X Wn (x; xs )χn (t; ts ), (5.27) n=0 in which Wn (x; xs ) is the probability of reaching x from xs in exactly n jumps, and χn (t; ts ) is the probability of making exactly n jumps in the time interval [ts , t]. The variable n is referred to as operational time. Eq. (5.27) decomposes the CTRW into terms depending on its two stochastic ingredients, the WTD, w(t) and the JLD, ϕ(x). Since ϕ(x) has a ﬁnite variance, x̄2 , the probability Wn (x; xs ) is the same as in a random walk, in which the walker jumps with a ﬁxed frequency but the jump length is a random variable with distribution ϕ(x). An approximation of Wn (x; xs ) can be obtained as the solution of the diﬀusion equation, Eq. (2.34), with D = x̄2 /2. The probability Wn (x; xs ) represents an average over all possible series of jump lengths from the JLD ϕ(x). The probability χn (t; ts ) reﬂects the waiting time as a random variable, that is, χn is a probability with respect to an average over all possible series of waiting times from w(t). An outstanding property of the CTRW model is the fact that the time averages of CTRW quantities are random variables, as will be illustrated with the following argument. In this paragraph, let the position coordinate, x, of a CTRW be restricted to the integer values x = 1, 2, ..., K. The random walker jumps from x only to x + 1 or x − 1, with a probability qx or 1 − qx , respectively. At x = 1 and x = K the walker jumps always to the right and left, respectively. The waiting times between two jumps are a random variable characterized by the WTD Eq. (5.20). The probability of being at x after n jumps is denoted by Pn (x). The process is Markovian and in the operational time, it is a classical random walk. Therefore, it can be described with a Master equation [187, 192]. Hence, an equilibrium is reached for large n, i.e. limn→∞ Pn (x) = px . ThePoperational time spent at x is denoted as Nx , the total number of jumps is N = x Nx , and, for large N , px = N Px /N . Let Tx be the time spent at position x, the total observation period is T = x Tx , in the regular time coordinate. An observable φ(x), being a function of x, has the time average over the period [0, T ] X hφiτ,T = wx φ(x), (5.28) x in which wx = Tx /T is the average sojourn time at the position x. It can be demonstrated that the Tx are random variables with a common distribution function that follows from Lévy statistics [187]. As a consequence, the time average in Eq. (5.28) is a random variable [184, 185]. An example is illustrated in Fig. 5.3 A. In the present thesis, the distributions of time-averaged quantities are not examined. Instead, we focus on mean values of time-averaged quantities. Using mean values of time-averaged CTRW quantities allows the application of Eq. (5.27), which involves ensemble averaging. Hence, in what follows the combined ensemble- and time-averaged MSD in the interval [0, T ] is derived. 72 Modeling Subdiffusion We now return to the free, unbounded CTRW with continuous position space. Eq. (5.27) is used to characterize the CTRW when it is observed at time ts . The decomposition of the CTRW into its stochastic ingredients is exploited to obtain the ensemble-averaged MSD between time ts and the time ts + t, which is given as Z ∞ 2 (x − xs )2 W (x, t + ts ; xs , ts )dx. (5.29) [x(ts + t) − x(ts )] ens,0 = −∞ With Eq. (5.27) this can be expressed as [x(ts + t) − x(ts )]2 ens,0 = ∞ X n=0 ∆x2 (n) χn (t; ts ), (5.30) where the MSD of the free, unbounded, classical random walk in operational time with JLD ϕ(x) is used [see Eq. (2.35)], Z ∞ 2 Wn (x; xs )dx = 2Dn. (5.31) ∆x (n) = −∞ This is a ﬁrst step towards the time-averaged MSD, which is obtained by performing an average over ts ∈ [0, T ] on both sides of Eq. (5.30). Thus the time-averaged MSD over the observation period [0, T ] follows as Z T −t 2 1 [x(t + τ ) − x(τ )]2 ens,0 dτ, (5.32) ∆x (t) t = T −t 0 where the combined ensemble-time average is deﬁned as E D . h·it := h·iens,0 ts ,T (5.33) Note that the time average and the ensemble average in Eq. (5.33) can be swapped. The ts -dependence in Eq. (5.27) is due to χn . The ts -dependence of χn is due to the initial WTD, w1 , in Eq. (5.38). Therefore, a time average in the interval [0, T ] can be performed by substituting w1 (t, ts ) using the time-averaged initial WTD Z 1 τ w̄1 (t) = hw1 (t, τ )iτ,T = w1 (t, τ )dτ. (5.34) T 0 The form of the initial WTD w1 (t, ts ) is derived in [189]. It has the long t asymptotic behavior sin πα 1 tαs w1 (t, ts ) ∼ . (5.35) π t + ts tα The time-averaged initial WTD is obtained by integration over Eq. (5.35) in the limit t ≪ T [147] κα (5.36) w̄1 (t) ≈ 1−α t−α , T where the constant κα = sin(πα)/πα is introduced. 5.3 The Continuous Time Random Walk (CTRW) 73 The probability of making exactly n jumps during a time period of length t is the joint probability of making n − 1 jumps in a shorter time t − τ and of ﬁnding a waiting time τ , integrated over all possible values of τ . More precisely, the probability of making n > 2 jumps in the time interval [ts , ts + t] is given by Z t χn−1 (τ + ts ; ts )w(t − τ )dτ. (5.37) χn (t + ts ; ts ) = 0 The probability of making no or one jump in [ts , t + ts ] can be expressed as Z t w1 (τ, ts )w(t − τ )dτ , and χ1 (t + ts ; ts ) = 0 Z t dτ w1 (τ, ts )dτ. χ0 (t + ts ; ts ) = 1 − (5.38) (5.39) 0 Eqs. (5.37) and (5.38) allow an iterative expression to be established, in which χn is expressed as an n − 1-fold convolution integral. In order to evaluate Eqs. (5.37) – (5.39), the Laplace transform t → u is applied. In the Laplace representation, the n − 1-fold convolution for χn corresponds to a product χn (u; ts ) = 1 − w(u) [w(u)]n−1 w1 (u, ts ) u for n > 1. (5.40) For n = 0, χ0 (u, ts ) = [1 − w1 (u, ts )]/u. With Eqs. (5.40) and (5.31) the ensembleaveraged MSD between ts and ts + t in Eq. (5.30) is expressed in the Laplace representation as ∞ h i 1−w X nwn−1 (5.41) L [x(ts + t) − x(ts )]2 ens,0 (u) = 2Dw1 u n=0 w1 = 2D , (5.42) u[1 − w] where the geometric series is used and the functional arguments, u and ts , are skipped. As the ts dependence is entirely due to w1 , the time-averaged initial WTD w̄1 is substituted in Eq. (5.40), i.e., χ̄n (u) = hχn (u; ts )its ,T = 1 − w(u) [w(u)]n−1 w̄1 (u) u for n > 1. (5.43) In the Laplace transform, the long t asymptotic behavior corresponds to small u behavior in the Laplace space. In particular, the value u = 0 corresponds to the integral over the entire t-range. The Laplace transform of the WTD w(t) with power-law decay as in Eq. (5.20) reads w(u) ∼ 1 − Γ(1 − α)uα with u & 0. (5.44) For the time-averaged initial WTD in Eq. (5.36), the Laplace transform has the form w̄1 (u) ∼ (uT )α−1 Γ(1 + α) with u & 0. (5.45) 74 Modeling Subdiffusion A 10 4 B 10 10 2 10 〈〈 ∆ x2(t)〉〉 〈 ∆ x2(t)〉τ,T 10 10 0 10 2 α = 0.5 α = 0.6 α = 0.7 α = 0.8 α = 0.9 ∼t 0 −2 10 10 4 −2 −4 0 10 1 10 2 10 3 10 t [time] 4 5 10 6 10 10 10 −4 10 0 10 1 10 2 3 10 t [time] 10 4 10 5 10 6 Figure ˙5.3: Free, unbounded CTRW – time-averaged MSD. - A: Individual time-averaged ¸ MSD, ∆x2 (t) τ,T , of an CTRW with WTD exponent α = 0.5. The continuous red line is the ˙ ¸ ˙ ¸ average over 100 MSDs, ∆x2 (t) t . The individual MSDs scatter around ∆x2 (t) t , as the timeaveraged CTRW quantities are random variables.˙ B: Time-averaged MSDs of CTRWs with ¸ various WTD exponents α (see legend). The MSD, ∆x2 (t) t , is obtained of individual trajectories as in A. An additional averaging is performed over 100 individual MSDs. Independent of α, all MSDs exhibit a linear time dependence. No subdiffusive behavior is present for the time-averaged MSD of a free, unbounded CTRW, as predicted by Eq. (5.47). Note that a different notation for double average is used in the axis label, hh·ii instead of h·it . With the Laplace transforms of w and w̄1 , the right side of Eq. (5.42), upon averaging over ts , has the following form 2 κα (5.46) ∆x (u) t = 2D 1−α u−2 . T The time-averaged, ensemble-averaged MSD in the interval [0, T ] of the CTRW with WTD exponent α follows upon inverse Laplace transformation of Eq. (5.46) as κα ∆x2 (t) t = 2D 1−α t. T (5.47) In contrast to the ensemble-averaged MSD of Eq. (5.26), the time-averaged MSD is not subdiﬀusive. The linear time dependence of the time-averaged MSD is illustrated 2 in Fig. 5.3 for various WTD exponents α. The MSD, ∆x (t) t depends on the length of the observation of the process. An alternative way to calculate the average ∆x2 (t) t is to exploit the fact that after the initial waiting time, when the ﬁrst jump occurs, the ensemble average can be obtained as the solution of the FDE, i.e. from Eq. (5.25). Z t 2 w̄1 (τ ) ∆x2 (t − τ ) ens,0 dτ. (5.48) ∆x (t) t = 0 The convolution integral in Eq. (5.48) can be evaluated in the Laplace representation 2 as a product. The MSD ∆x (t) ens,0 is obtained from the solution of the fractional diﬀusion equation, Eq. (5.25). 5.3 The Continuous Time Random Walk (CTRW) 5.3.3 75 Confined CTRW The time-averaged MSD of the free, unbounded CTRW was found to not exhibit subdiﬀusive behavior [Eq. (5.47)]. However, the possible applications suggest that, rather than unbounded diﬀusion, ﬁnite volume eﬀects may have a critical inﬂuence on the properties of a CTRW. In this subsection, the confined CTRW is examined, in which the walker is restricted to the interval x ∈ [0, L] by reﬂecting boundaries. 2 A 10 B 10 1 7 T = 10 〈〈 x (t)〉〉 8 1 2 10 2 〈 ∆ x (t)〉τ,T 10 2 T = 10 10 9 T = 10 0 10 T = 10 11 T = 10 10 −1 12 T = 10 0.1 ~t 0 10 2 10 3 10 4 10 5 6 10 10 t [time] 7 10 8 10 10 −2 10 1 10 3 5 7 10 10 t [time] 9 10 11 10 Figure 5.4: Confined CTRW – time-averaged MSD - A: The dotted lines are time-averaged ˙ ¸ MSDs, ∆x2 (t) τ,T , of individual trajectories with WTD exponent, α = 0.9, simulation length, T = 109˙, and L ¸= 20. The continuous red line is the ensemble-average over 1 000 of the individual MSDs, ∆x2 (t) t . The individual MSDs exhibit a common underlying pattern: a linear time 1−α dependence for short times and cross over , for long ˙ ¸ to a subdiffusive time dependence, ∼ t times. However, the individual ∆x2 (t) τ,T do not coincide, as time averages of CTRWs are ˙ ¸ random variables. B: Ensemble-averaged, time-averaged MSD, ∆x2 (t) t for various simulation lengths T (WTD exponent α = 0.9 and boundary width L = 20). In order that each time series should contain the same number of points (107 ), the time resolution of the longer simulations was reduced. Therefore, the short time behavior of the MSDs of larger T is absent. The ˙ ¸ MSDs display a dependency on the simulation length and decrease with increasing T . x2 (t) t is bounded by the value L2 /6. For longer T , the time-averages MSD is shifted to smaller values, but does not reach a constant plateau. Note that a different notation for the double average is used in the axis label, hh·ii instead of h·it . The ensemble-averaged, time-averaged MSD, ∆x2 (t) t, can be derived analogously to the last subsection, if the unbounded MSD in operational time, ∆x2 (n) = 2Dn, is replaced with the corresponding ﬁnite volume MSD4 in Eq. (5.42). The ﬁnite volume 4 Here, the approximation Eq. (A.8) is used rather than Eq. (A.5), for simplicity. It turns out that the approximation does not affect the results inappropriately. However, the mathematical argument can equally be applied to the exact solution Eq. (A.5). 76 Modeling Subdiffusion MSD is given in Eq. (A.8). Then, Eq. (5.42) reads ∞ L2 1 − w X n−1 ∆x2 (u) t = w̄1 w [1 − gn ] 6 uw (5.49) n=0 L2 w̄1 1 − g , = 6 uw 1 − gw (5.50) where g = exp[−π 2 D/L2 ]. Eq. (5.50) can be evaluated for two limiting cases: the u → 0 behavior determines the long t asymptotic form of ∆x2 (t) t, whereas u . 1 corresponds to the short time limit, t & 1. For small u ≪ 1, Eq. (5.50) reads uα−2 L2 1 . ∆x2 (u) t ≈ 6 T 1−α Γ(α + 1) Hence, the long t behavior of the time-averaged MSD is given as 1−α 2 t L2 κα . ∆x (t) t ≈ 6 1−α T (5.51) (5.52) The short-time behavior is obtained from the Laplace representation in the range u . 1 L2 κα 1 − g −2 ∆x2 (u) t ≈ u . 6 T 1−α g (5.53) With (1 − g)/g ≈ 12D/L2 , the time-averaged MSD at short times has the following t dependence 2 κα (5.54) ∆x (t) t ≈ 2D 1−α t. T The time-averaged MSD of the conﬁned CTRW exhibits a two-phasic behavior. In contrast to the free, unbounded CTRW, the conﬁned CTRW has a subdiﬀusive timeaveraged MSD at long t. The subdiﬀusion found has an exponent 1 − α. For short times, the result of the free, unbounded CTRW, Eq. (5.47), is retrieved, as for short times 2 the boundary does not aﬀect the MSD of the random walker. The two phases of ∆x (t) t are separated from each other by a critical time, which is given as tc = L2 12D(1 − α) α1 . (5.55) The critical time, tc , does not depend on the length of the simulation, T . Up to a constant factor, the time tc can be understood as the time, after which the ensembleaveraged MSD of the free, unbounded CTRW equals the plateau value. Therefore, the critical cannot depend on the observation length, T . The subdiﬀusive behavior, 2 time1−α x (t) t ∼ t , found in Eq. (5.52) contrasts with the ∼ tα behavior reported by He et al. [165], which arose from the fact that the time window, in which ∼ tα was ﬁtted to the MSD in [165], was conﬁned to times close to ∼ tc ≈ 27 000. 5.3 The Continuous Time Random Walk (CTRW) 77 8 B 10 data 10 α = 0.7 analyt. fit α = 0.5 6 10 1−α 2 4 1−α 2 4 10 T 2 〈〈 x (t)〉〉 T 10 10 8 α = 0.9 6 〈〈 x (t)〉〉 A 10 2 10 0 0 10 10 1 10 10 3 5 7 10 10 t [time] 9 10 10 11 1 10 10 3 5 10 10 t [time] 7 9 10 10 11 Figure 5.5: Confined CTRW – Simulation ˙ ¸ length scaling and analytical fit - A: Ensembleaveraged, time-averaged MSD, T 1−α ∆x2 (t) t of various T = 108 , 109 , ..., 1014 for each α = 0.5, 0.7 and 0.9 (L = 20). The MSD is multiplied with T 1−α to remove the T -dependence. The coincidence of the data for different simulation˙ lengths, ¸ T , illustrates the correct scaling behavior of Eq. (5.56). B: The T -scaled MSD, T 1−α ∆x2 (t) t , as in A is compared to the analytical expression in Eq. (5.56) for α = 0.3, 0.4, ..., 0.9 (top down). The good fit to the simulation data of the analytical curve demonstrates the validity of the approximations made in the derivation of Eq. (5.56). It also illustrates that the critical time, tc , as given in Eq. (5.55), does not depend on the simulation length, T . The following interpolation between short and long t domains can be performed 2 L2 κα t1−α 1−α α ∆x (t) t = 1 − exp − 2 12Dt . (5.56) 6 (1 − α)T 1−α L 5.3.4 Application of CTRW to internal biomolecular dynamics The CTRW model has been suggested as a possible mechanism for the subdiﬀusive MSD found in the internal dynamics of proteins [52, 183]. Subdiﬀusive behavior is found in single molecule ﬂuorescence spectroscopy [27, 28, 52] and spin echo neutron scattering [54], experimental techniques providing time-averaged ACFs. Thus, the experiments revealed subdiﬀusion in the time-averaged ACF. Likewise, MD simulations indicating subdiﬀusive dynamics provide time-averaged MSDs [53, 140]. However, the subdiﬀusive MSD found in a free, unbounded CTRW is obtained after ensemble averaging, cf. Eq. (5.26), whereas the time-averaged MSD is not subdiﬀusive! Therefore, the free, unbounded CTRW cannot explain the subdiﬀusion in the above mentioned cases. The conﬁned CTRW exhibits a subdiﬀusive, time-averaged MSD for times t > tc . Therefore, the conﬁned CTRW, in contrast to the free, unbounded CTRW, could explain the MD results (and the subdiﬀusion of the time-averaged ACF found in experiment). The fact that the time-averaged MSD obtained from MD simulation reaches a saturation plateau, whereas the time-averaged MSD of the conﬁned CTRW does not, does not necessarily disqualify the conﬁned CTRW as a model for MD simulations. The CTRW can be modiﬁed such that the power-law WTD of Eq. (5.20) holds up to a time 78 Modeling Subdiffusion T1−α〈〈 x2(t)〉〉/L2/α 10 10 2 0 L=20 10 L=40 −2 L=60 L=80 10 −4 −6 10 −5 10 −3 10 10 −1 10 1 3 t/tc 10 5 10 7 10 10 9 Figure˙ 5.6: Confined CTRW, scaling with volume - Ensemble-averaged, time-averaged MSD, ¸ T 1−α ∆x2 (t) t /L2/α of various L = 20, 40, 60, and 80 and various T = 108 , ..., 1013 with α = 0.7. ˙ ¸ The figure illustrates the L-scaling of ∆x2 (t) t as given in Eq. (5.56). It also demonstrates the L-dependence of the critical time, tc , Eq. (5.55). scale tmax (i.e., for t < tmax Eq. (5.20) is valid), but beyond which (i.e. for t > tmax ) the WTD decays faster than t−2 . Equilibrium would be reached on a time scale t > tmax . In this case, for t < tmax the process behaves as a CTRW, whereas for t > tmax classical diﬀusion occurs. Subdiﬀusion is found on time scales tc < t < tmax , (i.e., times shorter than tmax but long enough for the system to explore the accessible volume). For t < tc a linear time dependence occurs. When assuming the conﬁned CTRW model for the MSD as found in the MD simulations of Chap. 4, the diﬀusion constant cannot be obtained: none of the simulations has a time resolution which allows the linear time dependence of the MSD to be observed, but the subdiﬀusive part does not depend on D. To estimate the diﬀusion constant, an MD simulation was performed with an increased time resolution; the coordinates were recorded with 2 fs time resolution, corresponding to the MD integration time step (instead of the time resolution of 1 ps used in the (GS)n W simulations and of 2 ps used for the β-hairpin). The total length of this simulation is 2 ns. The MSD of the simulation is illustrated in Fig. 5.7. For the shortest time scales the MSD grows ballistically, as expected from inertial eﬀects. On the time scale of inertial eﬀects, the CTRW model does not apply; no inertial eﬀects are included in CTRW. The MSD in Fig. 5.7 does not exhibit a linear time dependence, i.e., the MSD crosses over from the ballistic to the subdiﬀusive regime. However, we estimate anupper bound of the diﬀusion constant, D, choosing the smallest D for which ∆x2 (t) 6 2Dt in Fig. 5.7. The approximation of D . 0.35 u nm2 ps−1 is based on the assumption that at the time scale, on which the MSD crosses over from ballistic to subdiﬀusive, i.e. around ≈ 0.3 ps, the MSD exhibits, in principal, a linear time dependence. The diﬀusion constant reﬂects the diﬀusion in 5.3 The Continuous Time Random Walk (CTRW) 2 2 2 −1 〈 ∆ x (t)〉 [u nm ps ] 10 79 10 10 0 (GS) W −2 2 (GS) W 5 0.7*t 10 −4 −2 10 −1 10 0 1 10 10 t [ps] 10 2 3 10 Figure 5.7: MSD of (GS)2 W and (GS)5 W at short time scales. - The MSD grows ballistically at the time scale of the MD integration step ∆t = 2 fs. At around t = 0.3 ps, it crosses over from a ballistic to a subdiffusive time dependence. The tangent to the curve, 2Dt with D = 0.35 u nm2 ps−1 , illustrated as full line colored in magenta, allows the upper bound of the diffusion constant to be estimated. Note that the estimation, independent of the system under investigation, leads to the same diffusion constant; the diffusion in the configuration space is isotropic as long as the diffusion is not affected by the presence of the potential energy landscape. the conﬁguration space for time scales, on which the diﬀusion is not aﬀected by the form of the energy landscape, i.e., time scales on which the conﬁgurational diﬀusion can be considered as free and unbounded. At these short time scales, the diﬀusion is assumed to be isotropic due to the equipartition theorem, which states that all degrees of freedom can be treated in a similar way using the harmonic approximation around a local minimum of the energy landscape. Therefore, the diﬀusion constant is expected to be independent of the system under investigation. This independence is conﬁrmed by Fig. 5.7, as for both, (GS)2 W and (GS)5 W a very similar diﬀusion is observed on short time scales. We assumed classical diﬀusion for estimation of the diﬀusion constant. Why not use Eq. (5.54) for the ﬁt? The CTRW can only be understood as a rough sketch of the dynamics. Obviously, the conﬁguration, i.e. the conﬁguration space coordinate, never is strictly at rest but always ﬂuctuates. Waiting times can only be introduced as periods during which the overall displacement in the conﬁguration space does not exceed a speciﬁed value. During such periods the systems appears to be trapped. Hence, the WTD can be imagined as a consequence of local minima of the energy landscape acting as kinetic traps. However, at short times, the molecule can be assumed to perform unhindered, isotropic diﬀusion, even if it is just inside a single local minimum of the energy landscape. Again, the diﬀusion on short time scales is independent of the system under investigation. We calculate the critical times, tc , for the individual PCs. The time tc of the 80 Modeling Subdiffusion 10 10 tc and tp 10 10 10 10 10 6 B Expl. water Langevin GB/SA 5 10 4 10 3 2 10 10 0 0 10 20 30 mode number 40 50 2 5 (GS) W 3 (GS) W 5 4 (GS) W 7 10 1 6 (GS) W 10 tc and tp A 10 3 2 1 0 0 10 20 30 mode number 40 50 Figure 5.8: Critical time of confined CTRW of (GS)n W peptides. - The crosses, +, illustrate the critical time as function of the PCs and the dots represent the times, tp , on which the PCs saturates to the plateau of saturation. A: The different colors refer to the different water models in the simulation of the β-hairpin, see legend. B: The different colors refer to the (GS) ˙ n W of¸ different n = 2, 3, 5, and 7, see legend. The dots illustrate the times after which the MSD, ∆x2 (t) , reaches the constant plateau. For the lowest PCs the critical time, tc , lies in the order of the saturation time tp . Even for intermediate PCs the critical time tc > 1. Therefore, the subdiffusion seen in the lowest PCs ranging from 1 ps to tp cannot be explained by the confined CTRW model. individual PCs is compared to the time, tp , on which the MSD of the PC mode reaches a constant value. Fig. 5.8 illustrates the critical times of the individual PCs of the (GS)n W as crosses (+), while the saturation times, tp , are given as dots. For the lowest PCs, tp and tc are in the same order of magnitude. The conﬁned CTRW model predicts subdiﬀusive behavior only in the range tc < t < tp . Therefore, the conﬁned CTRW model cannot account for the subdiﬀusion seen in the low PCs of the β-hairpin and the (GS)n W peptides. 5.4 Diffusion on networks In the theory of complex systems, the application of network models has become a common approach [193, 194]. Networks connect the theory of Markov chains to transport processes in fractal geometries and in amorphous or disordered media. As subdiﬀusion is a typical feature of random walks on fractal geometries, networks are a promising approach for ﬁnding the mechanisms that give rise to subdiﬀusive behavior. 5.4.1 From the energy landscape to transition networks Assume a conservative, mechanical system. The conﬁguration of the system is given by the vector x . The interaction potential between diﬀerent parts of the system give rise to a potential energy, which is a function of x . In addition to the dynamics determined by the potential energy, the system is interacting with its environment, modeled as a velocity-dependent friction and a random force, given as stationary white noise. This scenario corresponds to the classical Langevin equation, Eq. (2.11), in multiple 5.4 Diffusion on networks 81 dimensions with an arbitrary underlying potential. We assume that the ﬂuctuationdissipation theorem holds. The friction and the white noise represent the coupling to a heat bath, and the system is assumed to be in thermal equilibrium. For each conﬁguration of the internal coordinates, x , the system has a certain potential energy due to the interaction potential between the systems constituents. The white noise causes the system to populate the potential energy landscape according to Boltzmann statistics. The kinetics of the system arises entirely from the potential energy landscape. If the dynamics of the system is recorded with a time resolution ∆t, for each t = n∆t a vector containing all internal coordinates, x (t) = xn , characterizes the time evolution of the system. If the accessible volume of the conﬁguration space is ﬁnite, the trajectory is long enough, and ∆t suﬃciently small, then the trajectory allows the potential energy landscape to be reconstructed. An approximate representation of the potential energy landscape can be obtained as follows [195]. First, the conﬁguration space is partitioned into discrete states. Then, each possible conﬁguration is attributed to one of the discrete states, e.g., to the most similar discrete state in the conﬁguration space. The similarity follows from the metric used in the conﬁguration space. In doing so, the trajectory, (x1 , x2 , ..., xT ) is mapped onto a sequence of discrete states, (κ1 , κ2 , ..., κT ). If the states are seen as the vertices of a network, the transitions between the states deﬁne the edges of the network. The network represents the possible transitions between the discrete states on a speciﬁed time scale. If a weight factor is assigned to each of the edges, this factor can be chosen such that it characterizes the relative frequency of the corresponding transition. Such a network is referred to as a transition network. The transition network represents the features of the potential energy landscape with arbitrary precision, if the number of discrete states is adequate; in the limit of an inﬁnite number of states, the potential energy landscape is retained from the transition matrix. Using the network representation of the potential energy landscape as input of a Markov model, the complete dynamical behavior can be reproduced. Here, we try to characterize the energy landscape of biomolecules in terms of transition networks. An MD trajectory of a molecule with explicit water can be seen as an example of the above scenario. However, if the dynamics are projected to the internal coordinates of the molecule, the projection may lead to memory eﬀects as demonstrated in Sec. 5.1. The projected equilibrium probability distribution follows from the free energy landscape. In contrast to the potential energy landscape, the free energy landscape is, in general, insuﬃcient to determine the kinetic behavior. From the discussion in Sec. 4.3, it follows that the memory eﬀects due to the water dynamics are small on the time scale of 1 ps. Therefore, a Markov model may still capture the dynamics of the projected trajectory above 1 ps. 5.4.2 Networks as Markov chain models Let us start with a set of states, i.e., regions of the conﬁguration space, N = {n1 , n2 , ... ..., nN }, and a discrete trajectory (κ1 , κ2 , ..., κT ), obtained from the full conﬁguration space dynamics, x (t) = x (n∆t) = xn , as described in the last paragraph. The counting 82 Modeling Subdiffusion matrix, Z(ϑ), contains as components zij the number of transitions from state j to state i on the time scale ϑ = ν∆t, zij = T X n=1 δ[κn − j]δ[κn+ν − i]. (5.57) In general, for a ﬁnite trajectory, the counting matrix is not symmetric. A thermodynamics system tends to an equilibrium state at long times. The equilibrium is characterized by a stationary phase space density. As said above, the dynamics of a Hamiltonian system in contact with a heat bath can be modeled with a multidimensional Langevin equation and the full interaction potential V (x ), d2 d x − Γ x + ξ − ∇x V (x ) = 0. 2 dt dt (5.58) or a suﬃciently strong friction, the overdamped case, the equation reduces to Γ d x + ∇x V (x ) = ξ. dt (5.59) Eq. (5.59) allows the velocity coordinate to be decoupled from the position coordinate. Therefore, we are justiﬁed to assume an equilibrium state in which the probability of being in a volume d3N x at position x is proportional to V (x ) 3N d x. (5.60) exp − kB T The stationary probability distribution of the discretized trajectory is denoted by the vector p s , with the components psi being the equilibrium probability of being in state i. The equilibrium probability density in the conﬁguration space translates to the equilibrium probability of the discretized trajectory as Z V (x ) 3N −1 s exp − pi = Zconf ig d x, (5.61) kB T Vi with the conﬁgurational partition function Z V (x ) 3N d x. Zconf ig = exp − kB T (5.62) In equilibrium, the number of transitions from i to any other state equals the number of transitions from any state to i. Furthermore, if the number of transitions from i to j were unequal to those of the transitions from j to i, zij 6= zji , then the equilibrium dynamics in conﬁguration space would contain loops, i.e., closed paths exhibiting a preferred rotational direction. The presence of such vortices with preferred rotational sense in the conﬁguration space would be in conﬂict with the second law, as such eddies would constitute, in principle, a perpetuum mobile, a perpetual motion machine of the 5.4 Diffusion on networks 83 second kind. Therefore, in the limit of T → ∞, Z(ϑ) is symmetric, zij = zji . The term detailed balance refers to the latter equality. The transition matrix, S(ϑ), contains as components, sij , the relative probability of beingP in state i at time t + ϑ given that at the time t the system was in state j. It follows j sij = 1 for all i. An approximate transition matrix, S̃(ϑ), can be obtained P from the counting matrix as s̃ij = zij / k zkj . It follows that the transition matrix can be approximated on the basis of a ﬁnite discrete trajectory κ(t) = κ(n∆t) = κn as sij ≈ s̃ij = PT n=1 δ[κn − j]δ[κn+ν PT m=1 δ[κm − j] − i] , (5.63) where ϑ = ν∆t. The transition matrix characterizes the kinetics on the time scale ϑ. The matrix S(ϑ) does not contain information about the dynamics on time scales below ϑ. If we assume that the probability of ﬁnding state i at time t∗ + ϑ depends only on the state at time t∗ , but not on the states at the times t < t∗ , then the transition matrix furnishes complete dynamical information with respect to the set of discrete states chosen. If the history of the system does not inﬂuence the present state, i.e., the events at times t < t∗ do not aﬀect the relative probability, the system is Markovian. In Sec. 5.1 it is demonstrated that the Markov property of any system may be lost upon projection to a relevant, low dimensional subspace. Therefore, we cannot expect the dynamics of a solvated peptide or protein to be Markovian in the internal coordinates of the biomolecule, as we neglect the inﬂuence of the solvent dynamics. However, the Markov property may be a beneﬁcial approximation on time scales on which memory eﬀects have suﬃciently decayed. It is demonstrated in Chap. 4 that the Markov property is approximately satisﬁed on the time scale . 1 ps for the systems discussed in the present thesis. Note, however, that the projection to subspaces of fewer dimensions gives rise to further memory eﬀects, which may be relevant on time scales above 1 ps. If the Markov property is a valid approximation on and above a time scale tm , the dynamics for t > ϑ can be represented by the transition matrix, S(ϑ), where ϑ > tm . 5.4.3 Subdiffusion in fractal networks Often, diﬀusion processes occur in disordered materials, e.g., the diﬀusion in porous media [85, 86] or the charge transport in amorphous semiconductors [9]. Fractal lattices proved a beneﬁcial model of disordered media and paved the way to a rigorous mathematical analysis of the dynamics in fractal geometries. The concept of self-similarity – introduced by Benoı̂t Mandelbrot – is the key property of fractal objects which allows to apply scaling arguments. The self-similarity over diﬀerent length scales gives rise to a diﬀusion behavior similar over diﬀerent time scales. Therefore, power-law behavior is a typical feature of fractal behavior. In what follows, we brieﬂy review the concept of fractality. A more detailed discussion of the topic can be found in the review articles [162, 196]. 84 Modeling Subdiffusion Geometrical fractality Geometrical objects can be classiﬁed by their dimension: a line is a one-dimensional object, a plane is two-dimensional and a cube is three-dimensional. There are several concepts in mathematics to introduce the quantity dimension. The topological dimension of a geometrical object is the minimal number of parameters needed to characterize uniquely a single point of the object5 . For example, the position on the earth’s surface can be parametrized by two parameters, e.g., geographical longitude and latitude. Hence, the surface of the earth is a two-dimensional object. The topological dimension has always an integer value. In 1918, the German mathematician Felix Hausdorﬀ introduced a dimension deﬁnition based on measure theory. The Hausdorﬀ dimension coincides for most objects like cubes, planes or curves with the topological dimension, but can have non-integer values for certain, irregular objects. A simpliﬁed approach to the Hausdorﬀ dimension is the following argument. A geometrical object in a d-dimensional space shall be covered by d-dimensional spheres of radius R. Let N (R) be the minimal number of spheres needed to cover the object at a given R. The Hausdorﬀ or fractal dimension is deﬁned as df = − lim R→0 log N (R) . log R (5.64) Fractals are deﬁned as objects for which the fractal dimension is unequal to the topological dimension, df 6= dt . Less rigorously, the fractal dimension can be understood in the following way. Usually the volume of a (massive) object in a d-dimensional space scales with the length as V ∼ Ld . In contrast, if the fractal can be imagined as a massive body, it has a volume scaling as V ∼ Ldf . (5.65) For example, the volume excluded by a fractal in a cube of side length L is typically V ∼ Ldf . The Sierpiński triangles and the Sierpiński gasket – introduced by the Polish mathematician Waclaw Sierpiński in 1915 – are used here as examples of a regular fractal and as toy models for the diﬀusion in disordered media. The ﬁrst order Sierpiński triangle, Γ1 , is an equilateral triangle of side length one Fig. 5.9 A. The second order Sierpiński triangle, Γ2 , is constructed by the combination of three ﬁrst order Sierpiński triangles, the lengths of which are scaled by 1/2, as illustrated in Fig. 5.9. The construction scheme is repeated each time the order is increased by one, cf. Fig. 5.9. The limit Γ∞ = limn→∞ Γn yields the Sierpiński gasket. The Sierpiński gasket has the topological dimension one, dt = 1. The Sierpiński gasket contains three copies of itself, each down-scaled by a factor of one half. Therefore, the fractal dimension can be derived from 2df = 3 to be df = log 3/ log 2 ≈ 1.5850. For 5 The topological dimension, also covering or Lebesgue dimension, of a subset Ξ of a topological space M is strictly defined as the minimal dt ∈ N such that every open cover of Ξ has a refinement, in which every point x ∈ Ξ is included in dt + 1 or fewer open sets. 5.4 Diffusion on networks A 85 B C D Figure 5.9: Construction of the Sierpiński gasket - A: First order Sierpiński triangle, Γ1 . B: Second order Sierpiński triangle, Γ2 . C: Third order Sierpiński triangle, Γ3 . D: Sierpiński triangle of order eight, Γ8 . The Sierpiński gasket is obtained as Γ∞ = limn→∞ Γn . the Sierpiński gasket the Hausdorﬀ dimension exceeds the topological dimension, df > dt , i.e., the Sierpiński gasket is a fractal. Note that the dimension of the embedding space, here the two-dimensional plane, exceeds the fractal dimension. The argument used to derive the fractal dimension of the Sierpiński gasket refers to its self-similarity. By construction, Γ∞ contains identical copies of itself on arbitrary small length scales. Therefore, the Sierpiński gasket has no typical length scale, it is a scale free object – apart from the ﬁnite size of Γ∞ . The regular and deterministic way to construct the Sierpiński gasket allows renormalization techniques to be applied. The non-integer Hausdorﬀ dimension motivated the term fractal coined by Mandelbrot. Although all objects with non-integer Hausdorﬀ dimension are fractals, not all fractals have a non-integer fractal dimension df . An important example is the random walk. Assume a random walker in a two-dimensional plane makes a jump every ∆t = 1 s. The jump length may be a random variable with a Gaussian distribution of variance σ 2 = 1 m2 and all directions are equally likely. Due to the fact that the sum of independent, Gaussian random variables is again a Gaussian random variable, the displacements of the walker on the time scale one second are statistically the same as those on the time scale two √ seconds, provided the variance of the two-second-displacement is scaled by a factor 2. In the diﬀusion limit, i.e., ∆t → 0 and σ 2 → 0 with D = σ 2 /2∆t constant, the fractal dimension of the trajectory of the random walker is given by 2df /2 = 2, i.e., df = 2. Since the topological dimension of the random walkers trajectory is dt = 1, the trajectory, seen as a geometrical object in the embedding space, is a fractal. Note that the fractal dimension equals two, irrespective of the dimension of the embedding space. If the random walk is embedded to a two-dimensional space, i.e. the random walk in the plane, the process is room-ﬁlling. The fractal dimension is reﬂected by the fact that the MSD of the classical, unbounded random walk in space of arbitrary dimension is linear in time [cf. Eqs. (2.18) and (2.35)]. Guigas et al. have conjectured that the subdiﬀusive motion of macromolecules in the cytoplasm increases the fractal dimension of the trajectory, df , such that the trajectories are room ﬁlling in the three-dimensional space [82]. Self-similarity is a surprisingly abundant property in the real world, some of the most impressive examples being the Romanesco broccoli cabbage, fern plants, the shape of clouds and the surface structure of rocks. However, the strict self-similarity on all length scales, as found in mathematics, cannot be observed in nature. Rather, a statistical self-similarity, as in the case of the random walker occurs in nature. Furthermore, 86 Modeling Subdiffusion self-similarity usually is found in a range of length-scales. In that sense, the trajectories of physical Brownian particles are fractal above the length scale deﬁned by the mean free path. So far, the fractal quantity of geometrical objects has been discussed. Now, we proceed to dynamical processes on fractal objects. Dynamical fractality De Gennes raised the question as to how a random walk on a percolation cluster, the path of an “ant in the labyrinth”, would look like [197, 198]. As fractals turned out to be suitable model systems for irregular, amorphous media, and being similar to percolation clusters [199], the random walk on fractal geometries became an active ﬁeld of research in the 1980’s. A review on diﬀusion in amorphous media can be found in [164]. Random walks on fractals can be modeled as random walks on transition networks. Let a transition network have the nodes N = {n1 , n2 , ..., nN } and the edges E ⊂ N ×N . A random walker may jump in unit time from one node, ni , to another node, nj , if ni is connected to nj with an edge, i.e., (ni , nj ) ∈ E. If the edges are of equal weight, the walker, being at ni , chooses with uniform probability one of the neighbores of ni , i.e., one of the nodes which share an edge with ni . If the edges are weighted by a transition matrix, S, then the probability of jumping to node P nj from ni is given as sji . Note that the transition matrix is deﬁned such, that i sji = 1 for all j. The random walk on the network forms a Markov chain. Let the network be embedded to a d-dimensional vector space, V , each node, ni , having a position vi ∈ V . The jump of the random walker on an embedded network from node ni to node nj corresponds to a displacement u = vj − vi . Therefore, the random walk on the network can be mapped to a series of displacements in the embedding P space, u1 , u2 , ..., uT . The displacement from the starting position after t jumps is ti=1 ui . With the vector space metric, the MSD can be calculated as ∆x2 (t) = t X * i=1 ui !2 + = t X i=1 ui2 +2 t t X X i=1 j>i hui uj i . (5.66) For a large network with nodes homogeneously distributed in the embedding space, the mean square of the single displacement is u 2 and the MSD reads t t X X 2 hui uj i . ∆x (t) = u t + 2 2 (5.67) i=1 j>i 2 For 2 networks in which the edges can be considered as uncorrelated, one has ∆x (t) = u t. The linear time dependence is found in the case of regular lattices and corresponds to the MSD of the classical random walk, Eq. (2.35). However, fractal networks usually do not have uncorrelated displacements, i.e. hui uj i = 6 0. Rather than a linear time dependence, the MSD of random walkers on 5.4 Diffusion on networks 10 87 3 n=1 10 n=3 n=4 2 〈∆ x (t)〉 n=2 2 10 n=5 1 n=6 n=7 10 0 n=8 0.86 ∼t 10 0 10 1 10 2 3 10 time 10 4 10 5 10 6 Figure 5.10: MSD of Sierpiński triangles - The MSD of a random walk on the Sierpiński triangle Γn is illustrated for various n = 1, 2, ..., 8 (see legend). The full red line illustrates the power law with the exponent α = 0.86, as predicted from the renormalization argument. fractal networks is found to be subdiﬀusive, i.e. ∆x2 (t) ∼ t2/dw , (5.68) where dw > 2 is the diffusion dimension of the fractal network. Note that the MSD exponent α = 2/dw conveys the same information as the diﬀusion dimension, dw . For all ﬁnite fractal networks, i.e., networks with a ﬁnite number of nodes, the MSD is bounded. Therefore, Eq. (5.64) holds below the time scale on which the MSD saturates to its maximal value. The subdiﬀusion found in fractal networks is due to the geometry of the network and is not a violation of the Markov property. Fractal networks can exhibit subdiﬀusion in equilibrium and as an ergodic phenomenon. An example is illustrated in Fig. 5.10, in which the MSD of random walkers on the Sierpiński triangle, Γn , for various n is illustrated. The diﬀusion dimension of the Sierpiński gasket can be derived with a renormalization argument. In the construction of the Sierpiński gasket going from Γn to Γn+1 corresponds to the replacement of each triangle by the second order Sierpiński triangle, Γ2 , the side length of which is twice as much as for the triangle, Fig. 5.11. We compare the typical transit time in the triangle, τ , to the typical transit time of the second order Sierpiński triangle, τ ′ [196]. The transit time of the triangle, τ , is the time the walker needs upon entering at the top vertex to exit through one of the bottom 0 vertices, Fig. 5.11. In Γ2 , the walker needs a time τ to reach the intermediate nodes, from which it takes him the time A to exit. Hence, τ ′ = τ + A. From the A-vertices, the walker has a 25% chance to leave Γ2 after τ , the chance to go to the top vertex is also 25%, as is the chance jump to the other A-vertex or to the B-vertex. All these jumps take a typical time τ . Therefore, 88 Modeling Subdiffusion τ′ (B) A τ 0 A → (A) 0 B 0 0 Figure 5.11: Transit time on the Sierpiński gasket - Renormalization allows the diffusion dimension of the Sierpiński gasket to be calculated analytically. All vertices are denominated by the time needed for proceeding to the bottom vertices 0 and exiting. (A): The walker enters the triangle at the top vertex and it takes him τ to exit through the bottom vertices. (B): Rescaled triangle, τ → τ ′ . the time to exit from either of them is 1 1 1 1 A = (0 + τ ) + (τ ′ + τ ) + (A + τ ) + (B + τ ). (5.69) 4 4 4 4 With the same reasoning, the time to exit from the B vertex is typically B = (τ + 0)/2 + (τ + A)/2. The following renormalization scheme is found τ′ = τ + A 4A = 4τ + B + A + τ (5.70) ′ 2B = 2τ + A. (5.71) (5.72) The solution of this scheme is τ ′ = 5τ (with A = 4τ and B = 3τ ). The typical transient time is multiplied by 5 each time the side length of the Sierpiński gasket is doubled. The diﬀusion dimension follows as dw = log 5/ log 2 ≈ 2.32, corresponding to an MSD exponent α ≈ 0.86. The exact result for the Sierpiński gasket coincides with the simulation results of (ﬁnite) Sierpiński triangles, as illustrated in Fig. 5.10. The typical distance a random walker is displaced in the embedding space after N steps is R ∼ N 1/dw . The volume in the embedding space in which the walker is to be found, scales as V ∼ Rdf ∼ N df /dw . The probability of the walker being after N steps at the origin is inversely proportional to the volume, i.e., P (N ) ∼ N −df /dw = N −ds /2 . (5.73) Here, the spectral dimension of the network is introduced as ds = 2df /dw [196]. Then, the recurrence probability is P (N ) ∼ N −ds /2 . 5.4.4 Transition networks as models of the configuration space Recently, network representations of the conﬁguration space are a subject of considerable interest [200]. A natural approach to the dynamics induced by the energy landscape 5.4 Diffusion on networks 89 is the location of “basins of attraction” [201], i.e., the regions consisting of the points which lead to the same local minimum if the steepest descent path is followed. Closely connected is the notion of metastable states, i.e., regions of the conﬁguration space which have an extremely long mean escape time. The idea to express the behavior of a dynamical system in terms of the metastable states was ﬁrst discussed in the context of glass-forming liquids [202, 203]. The role of metastable states in the dynamics of biomolecules was analyzed in [204, 205], speciﬁc clustering methods based on the kinetic rather than on the structural similarity were introduced in [206, 207]. Frustration, i.e., the presence of many, nearly isoenergetic local minima, is a typical feature of glassy dynamics [208]. The network of the local minima and transition states has been determined to be a scale free network, i.e., a network whose degree distribution exhibits a power-law tail [209]. Further properties of the networks representing the energy landscape have been discussed, such as clustering, neighbor connectivity, and other topological features [210, 211]. Protein folding is studied in the framework of network theory [195, 212]. Also, network approaches were suggested as a way to ﬁnd adequate reaction coordinates without the restrictions imposed by projection of the trajectory to low-dimensional subspaces [213]. Minimum-cut techniques are applied to the networks to extract low-dimensional representations of the full dynamics [214– 216]. Hierarchical organization of the basins of the energy landscape were reported [206, 209, 213, 217, 218]. ϑ [ps] (GS)2 W (GS)3 W (GS)5 W (GS)7 W β-HP, expl. wat. β-HP, Langevin β-HP, GB/SA 1 6.1 6.2 6.4 6.1 5.1 4.6 5.4 10 6.8 6.8 6.6 6.3 5.3 4.8 5.6 100 7.4 7.6 7.0 6.4 5.7 5.0 5.9 1000 7.9 8.7 7.7 7.3 6.2 5.6 6.5 Figure 5.12: Fractal dimension df - The fractal dimension of the network representing the configuration space, S̃(ϑ). The network is constructed from the MD trajectory projected to a subset of PCs, most often the first ten PCs. Eq. (5.65) allows the fractal dimension to be calculated from the volume scaling of the network. None of the network fills the full ten-dimensional embedding space, all networks exhibit fractal behavior. In the following, the anomalous diﬀusion seen in the kinetics of the β-hairpin and the (GS)n W peptide simulations [see Sec. 4.3] are analyzed using network representations of the conﬁguration space. First, the volume sampled by the MD trajectory in the conﬁguration space is partitioned into 10 000 discrete states. These discrete states are randomly taken from the frames in trajectory. Here, we divide the simulation length, T , into 10 000 pieces of equal length and take the ﬁrst frame of each piece as the center of a discrete state, ri with i = 1, 2, ..., 10 000 being the conﬁguration space coordinates. A frame in the trajectory is now assigned to the discrete state which has the least Euclidean distance in the mass-weighted coordinates. In this way, the entire 90 Modeling Subdiffusion trajectory (x1 , x2 , ..., xT ) is mapped to a discrete time series, (κ1 , κ2 , ..., κT ). From the discrete time series the transition matrix, S̃(ϑ) can be obtained using Eq. (5.63). The network represented by the transition matrix is embedded in the conﬁguration space, the ri being the coordinates of node i. The network representation enables us to study how the energy landscape brings about the subdiﬀusive dynamics of the molecule. As the network analysis is numerically cumbersome in the full conﬁguration space, we perform the analysis on the subspace spanned by the ﬁrst ten PCs. These ten strongly delocalized modes account for more than 50% of the overall ﬂuctuations in the molecule. However, with the considerations in Sec. 5.1, the appearance of memory eﬀects above 1 ps is expected. 2 A: (GS)2 W 2 10 full MSD 〈∆ x2(t)〉 〈∆ x2(t)〉 10 ϑ=1ps 1 10 ϑ=10ps full MSD ϑ=1ps 1 10 ϑ=10ps ϑ=100ps ϑ=100ps ϑ=1ns ϑ=1ns 0 10 0 10 B: (GS)3 W 0 10 1 2 10 3 10 time [ps] 4 10 10 5 10 0 10 6 10 3 10 1 2 10 3 10 time [ps] 10 5 6 10 3 10 10 C: (GS)5 W D: (GS)7 W 2 2 10 10 ϑ=1ps ϑ=10ps 1 ϑ=1ps ϑ=10ps 1 ϑ=100ps 10 full MSD 〈∆ x2(t)〉 〈∆ x2(t)〉 full MSD ϑ=100ps 10 ϑ=1ns ϑ=1ns 0 10 0 10 4 10 0 10 1 2 10 3 10 time [ps] 4 10 10 5 6 10 10 0 10 10 1 2 10 3 10 time [ps] 4 10 10 5 6 10 Figure 5.13: Diffusion on transition networks. - For the (GS)n W peptides with n =2,3,5,7, a random walk is performed on the networks S̃(ϑ) with various time lag ϑ (see legends). The networks are obtained from the projection of the MD trajectories to the subspace of the first ten PCs. The MSD of the random walk on the different networks is compared to the MSD (full red line) found in the first ten PCs of the original MD trajectory. All networks exhibit subdiffusive dynamics to some extent. However, for the time lags ϑ = 1 ps and 10 ps, the network random walk overestimates the MSD. For ϑ = 100 ps the networks reproduce the subdiffusion of the original MSD sufficiently. The networks for ϑ = 1 ns, the MSD is close to the saturation level. Hence, the saturation conceals possible subdiffusive effects. The fractal dimension of the transition network, S̃(ϑ), is obtained using Eq. (5.65). The number, N of edges enclosed in the sphere of radius R centered at ri is computed 5.4 Diffusion on networks 91 for various R. The function N (R) is averaged over 1 000 nodes, and Eq. (5.65) yields the fractal dimension df by a least-squares power-law ﬁt to the data, Tab. 5.12. Networks corresponding to various ϑ are used as Markov models. A random walk is performed on the network with transition matrix S̃(ϑ). The discrete time evolution generated from the transition matrix is now inversely transformed to the conﬁguration space by replacing the discrete state number, i, with the conﬁguration space coordinates of the state, ri . Doing so, a the random walk on the network is mapped to a random walk in the conﬁguration space. The random walk dimension, dw , is obtained from the MSD in the conﬁguration space. It is given together with the MSD exponent α in Tab. 5.14. ϑ [ps] (GS)2 W (GS)3 W (GS)5 W (GS)7 W β-HP, expl. wat. β-HP, Langevin β-HP, GB/SA dw 1 2.6 2.4 2.4 2.3 3.1 2.9 3.5 10 3.0 2.7 2.7 2.5 3.6 3.6 3.8 100 4.3 3.8 3.2 3.0 4.5 4.5 4.4 1000 11.3 9.6 4.4 4.4 5.7 6.4 6.4 α 1 0.8 0.8 0.8 0.9 0.7 0.7 0.6 10 0.7 0.7 0.7 0.8 0.6 0.6 0.5 100 0.5 0.5 0.6 0.7 0.4 0.4 0.5 1000 0.2 0.2 0.5 0.5 0.3 0.3 0.3 Figure 5.14: Diffusion dimension dw and MSD exponent α - A random walk on the networks representing the configuration space, S̃(ϑ), is performed. The network is obtained from the projection of the trajectory to the first ten PCs. The random walk on the network is inversely transformed to the configuration space. The diffusion dimension dw is calculated from the MSD using Eq. (5.64). The MSD exponent, α, used throughout the present thesis to quantify the subdiffusivity, is given in the right panel for comparison with the results of Chap. 4. All networks exhibit subdiffusive dynamics. The network S̃(ϑ) with ϑ = 1 ns is in all cases close to the saturation level. Therefore, the exponents tend to be underestimated as a consequence of finite size effects. For ϑ =100 ps the MSD exponent is close to the values found in the MD simulations, see Fig. 4.14. The networks for ϑ =1 ps and 10 ps tend to overestimate the MSD exponent. This is due to the presence of memory effects arising from the projection to the first ten PCs and the restricted spatial resolution, which becomes important for the dynamics on the shortest time scales. Furthermore, the networks of the β-hairpin simulations may be affected by the poor sampling due to the non-convergence, see Sec. 4.3. The network diﬀusion reproduces the eﬀect of subdiﬀusivity; the MSD exponent α is in the subdiﬀusive regime, Tab. 5.14. However, the networks for ϑ = 1 ps and 10 ps usually exhibit an MSD clearly larger than the original trajectory, given as red line in Fig. 5.13. The reason for the diﬀerence may be memory eﬀects present on the short time scales, which can arise from the projection to the ten-dimensional subspace of the ﬁrst ten PCs. The memory eﬀects cannot be reproduced by the Markovian network dynamics. Therefore, the subdiﬀusivity is underestimated, e.g., the network MSD is larger than the MSD found in the MD simulation data. Also, the limited spatial resolution of a network with 10 000 vertices aﬀects the MSD, and introduces in additional noise, in particular on the short time scales. For ϑ = 100 ps, the network MSD and the original MSD exhibit an acceptable coincidence. The signiﬁcance of 92 Modeling Subdiffusion the networks with ϑ = 1 ns is limited by the fact that the time resolution is close to the time scale of saturation, on which the walker is aﬀected by the ﬁnite, accessible volume. The ﬁnite volume also accounts for the tendency of the MSD exponent α to be underestimated by the random walks on the 1 ns transition network, S̃(ϑ = 1ns). For completeness, the spectral dimension is obtained using Eq. (5.73), see Tab. 5.15. It is most likely a ﬁnite size eﬀect that, in contrast to the argument employed in the discussion of Eq. (5.73), ds 6= 2df /dw . ϑ [ps] (GS)2 W (GS)3 W (GS)5 W (GS)7 W β-HP, expl. wat. β-HP, Langevin β-HP, GB/SA 1 0.7 0.6 1.0 1.1 1.2 1.1 1.2 10 0.8 0.6 0.9 0.9 1.2 1.1 1.1 100 1.2 0.8 0.8 0.9 1.3 1.2 1.1 1000 4.3 2.5 0.9 0.7 1.3 1.4 1.3 Figure 5.15: Spectral dimension ds - The spectral dimension of the networks representing the configuration space, S̃(ϑ). The network is obtained from the MD trajectories projected to the first ten PCs. Eq. (5.73) yields ds using the recurrence probability. The values of ds do not obey to ds = 2df /dw , potentially due to the finite size of the networks with only 10 000 nodes. Influence of projection In the above analysis, the dynamics were projected to the subspace spanned by the ﬁrst ten PCs, i.e., the network is embedded to a d-dimensional embedding space with d = 10. In order to characterize the inﬂuence of the projection, the dynamics of the (GS)5 W peptide is also analyzed for projections to subspaces with d = 1, 3, and 100. The fractal dimension of the networks is given in Tab. 5.17. The networks obtained from the projections to d = 1 and d = 3 dimensions are in the range 1.4–1.7. The values obtained for d = 10 are close to those df values found for d = 100. Hence, the fractal structure of the network S̃(ϑ) is essentially developed in the subspace spanned by the ﬁrst ten PCs. However, the MSD of the network, S̃(ϑ), obtained from the projection to d = 100 dimensions is considerably closer to the original MSD than the one for d = 10. The fact that the d = 10 network exhibits the same fractal exponent as the one for d = 100 but is inferior in the reproduction of the kinetics demonstrates that the higher PC modes (here the PC modes 11 to 100) contain only a small contribution to the overall ﬂuctuations, but are kinetically not insigniﬁcant. The diﬀerences between the network MSD and the MSD obtained from the original MD trajectory has three origins: (i) memory eﬀects arising from the neglected dimensions, i.e., due to the nonMarkovian nature of the trajectory (ii) limited spatial resolution, and (iii) memory eﬀects rooted in the discretization scheme. The above results demonstrate that the conﬁguration space of a molecule can be represented by a network model. A Markov process on the network can reproduce 5.4 Diffusion on networks 10 93 2 10 3 A: d = 1 B: d = 3 1 full MSD 〈∆ x (t)〉 10 2 full MSD 2 ϑ=1ps 2 〈∆ x (t)〉 10 ϑ=10ps 10 ϑ=1ps 1 ϑ=10ps ϑ=100ps 10 ϑ=100ps ϑ=1ns 0 10 10 0 10 1 10 2 3 10 time [ps] 4 10 5 10 10 ϑ=1ns 0 6 10 3 0 10 1 10 2 3 10 time [ps] 4 10 5 10 10 6 3 10 10 C: d = 10 D: d = 100 2 10 〈∆ x (t)〉 〈∆ x (t)〉 full MSD ϑ=10ps 1 2 10 full MSD 2 2 ϑ=1ps ϑ=1ps ϑ=10ps ϑ=100ps 10 ϑ=100ps ϑ=1ns ϑ=1ns 1 10 0 10 0 10 1 10 2 10 3 10 time [ps] 4 10 5 10 6 10 0 10 1 10 2 10 3 10 time [ps] 4 10 5 10 6 10 Figure 5.16: Projection of (GS)5 W to configurational subspaces – network diffusion in various dimensions. - The full MD trajectory was projected to d-dimensional subspaces of the configurational space, with d = 1, 3, 10, and 100. From the projected trajectory a transition matrix, S̃(ϑ), was obtained with lag time ϑ = 1 ps, 10 ps, 100 ps, and 1 ns. A random walk on the network defined by S̃(ϑ), is inversely transformed to the configuration space and the random walker’s MSD calculated. The transition matrices obtained from the projections to low dimensional spaces exhibit a larger difference to the MSD of the original MD trajectory, given as red, full line. For d = 100 the transition networks, S̃(ϑ), with ϑ = 100 ps and 1 ns reproduce the original kinetics with high accuracy, but also for ϑ = 1 ps and 10 ps the difference to the original MSD is considerably smaller than for the networks obtained from projections to subspaces of lower dimension. the kinetics of the molecule. Subdiﬀusive dynamics as seen by the sublinear timedependence of the MSD arise from a fractal-like geometry of the network. The representation of the kinetics is limited by three ﬁndings. Memory eﬀects due to the projection of the dynamics are present and violate the Markov property on short time scales. In order to be numerically tractable and to have suﬃcient statistics, the number of discrete states is limited and restricts the spatial resolution of the network in the conﬁguration space. Finally, the discretization is not uniquely deﬁned. If kinetically diﬀerent structures in the conﬁguration space are mapped on the same network vertex, the transition paths are unnaturally mixed. The mixing disturbs the fractal geometry and makes the network more permeable, i.e., the MSD is larger than the MSD of the 94 Modeling Subdiffusion ϑ [ps] d 1 3 10 100 d 1 (0.5) 3 (0.5) 10 (0.5) 100 (0.4) 1 df 1.6 1.4 6.4 6.3 α 0.9 0.9 0.8 0.7 10 100 1000 1.6 1.4 6.6 6.5 1.7 1.5 7.0 6.8 1.7 1.6 7.7 7.7 1.0 0.9 0.7 0.6 0.8 0.8 0.6 0.5 0.6 0.5 0.5 0.3 Figure 5.17: Projection of (GS)5 W to configurational subspaces – fractal dimension of transition networks. - The full MD trajectory was projected to d-dimensional subspaces of the configurational space, with d = 1, 3, 10, and 100. From the projected trajectory a transition matrix, S̃(ϑ), was obtained with lag time ϑ = 1 ps, 10 ps, 100 ps, and 1 ns. The fractal dimesion of the networks is obtained using Eq. (5.64). The fractal dimension found for d = 10 is close to the values obtained from d = 100, i.e., the fractal shape of the network is mainly contained in the first ten PCs. The MSD exponent, α, is calculated from the random walks illustrated in Fig. 5.16. original trajectory. 5.4.5 Eigenvector-decomposition and diffusion in networks Since random walks on networks can be modeled as Markov processes based on a transition matrix, methods from linear algebra can serve as useful tools in the analysis of diﬀusion problems. The time evolution of a Markov process can be expressed in terms of the eigenvectors of the transition matrix. The diﬀusive contribution of each eigenvector can be quantiﬁed with its transport coeﬃcient. Only a small fraction of eigenvectors dominate the MSD seen for the random walker of a given network. Let S be an irreducible6 transition matrix and p(n) the probability vector after n time steps. Its components pi (n) correspond to the probability of being in state i after n time steps. The initial probability vector is p(0). The time evolution of the probability p on the network is governed by the following equation p(n + 1) = Sp(n). (5.74) The recursive Eq. (5.74) yields the evolution equation p(n) = Sn p(0). (5.75) 6 The graph of a network is irreducible if any vertex can be reached from every other vertex, i.e. the network cannot be decomposed into different, unconnected subgraphs. The unconnected subgraphs of a reducible graph can be treated separately, each subgraph being irreducible. The corresponding transition matrix is called irreducible if it is not similar to a block upper triangular matrix via a permutation. The definitions of irreducibility of the network and the transition matrix are equivalent [192]. 5.4 Diffusion on networks 95 The largest eigenvalue of the matrix S equals one, λ1 = 1, and has multiplicity one, i.e., there is a uniquely deﬁned eigenvector to the eigenvalue λ1 = 1, according to the Ruelle-Perron-Frobenius theorem [192]. The theorem also states that all eigenvalues have an absolute value smaller than one, |λk | < 1 for all k 6= 1. Furthermore, the eigenvector to λ1 = 1 is the only eigenvector with strictly non-negative components. The eigenvector to the largest eigenvalue, λ1 = 1, is the stationary probability vector p s and it is lim p(n) = lim Sn p(0) = p s . (5.76) n→∞ n→∞ P The sum of the components of the stationary eigenvector p s equals one, k psk = 1. From Eq. (5.74) P it follows, that the component sum of any other eigenvector k 6= 1 equals zero, i.e. k pk = 0 if k 6= 1. The probability p(n) can be seen as a diﬀusion process, based on a large number (an ensemble) of random walkers. Eq. (5.74) can be modiﬁed to p(n + 1) − p(n) = (S − 1l)p(n). (5.77) Eq. (5.77) is the discrete analog of the diﬀusion equation, Eq. (2.33). The operator (S − 1l) is the discrete Laplace operator and has the eigenvalues {λi − 1|i = 1, 2, ...}. In the continuous time limit, the µ = − log λ ≈ −(λ − 1) are the (negative) eigenvalues of the Laplace operator. The eigenvectors of the Laplace operator are identical with the eigenvectors of S. From Eq. (5.77) it follows that the eigenfunctions of the Laplace operator decay exponentially with the characteristic time 1/µ, except the eigenfunction with the eigenvalue µ = 0, i.e., corresponding to the equilibrium p s with λ = 1. We introduce the notation ( diag v )ij = δij vj for the components of the matrix diag v , where v is a vector. As pointed out earlier, in general, the transition matrix, S, is not symmetric. From the detailed balance condition it follows sij psj = sij psi for all i, j. (5.78) Therefore, the matrix S̃ = diag (p s )−1/2 S diag (p s )1/2 (5.79) is symmetric, i.e. S̃ = S̃T , where S̃T is the transposed of the matrix S̃. As S̃ is symmetric, it has real eigenvalues {λk } and the eigenvectors, {ṽk }, obey S̃ṽk = λk ṽk , (5.80) and form an orthogonal basis set. The diagonal matrix that is similar to S̃ can be expressed as Ut S̃U = Λ. (5.81) The eigenvectors of S are obtained from the {ṽk } as vk = diag p s ṽk . The transition matrix S has the same eigenvalues, λk , as the matrix S̃. However, the eigenvectors of S do, in general, not form an orthogonal set. Furthermore, the identity S = diag (p s )1/2 S̃ diag (p s )−1/2 (5.82) 96 Modeling Subdiffusion allows together with diag (p s )−1/2 diag (p s )1/2 = 1l (5.83) the powers of the transition matrix to be expressed as Sn = diag (p s )1/2 S̃n diag (p s )−1/2 . (5.84) Using Eq. (5.84), the time evolution Eq. (5.75) can be written as p(n) = diag (ps )1/2 UΛn Ut diag (ps )−1/2 p(0). When using the componentwise notation, Eq. (5.85) reads X pi (n) = (psi )1/2 uik λnk ulk (psl )−1/2 pl (0). (5.85) (5.86) kl The conditional probability of being in state i after n steps starting in state q at time t = 0 is X (5.87) pi (n|q, 0) = (psi )1/2 uik λnk ulk (psl )−1/2 δlq kl = X (psi )1/2 uik λnk uqk (psq )−1/2 . (5.88) k The joint probability of being in state i at step n and in state q at time 0 reads X pi (n; q, 0) = pq (0)pi (n|q, 0) = pq (0) (5.89) (psi )1/2 uik λnk uqk (psq )−1/2 . k Let diq be the distance between state i and state q in the embedding space, diq = kri − rq k. Then, the MSD of the random walk on the network deﬁned by S is given as X X 2 X 2 ∆x (n) = diq pi (n; q, 0) = d2iq pq (0) (5.90) (psi )1/2 uik λnk uqk (psq )−1/2 . iq iq k Note that the above expression is an ensemble average. It can be performed with any given initial probability distribution p(0). In equilibrium, one has to replace p(0) with the stationary distribution p s , so one has X X 2 X 2 diq pi (n; q, 0) = d2iq psq ∆x (n) = (psi )1/2 uik λnk uqk (psq )−1/2 . (5.91) iq iq k Deﬁning the transport coefficients as X d2iq (psq )1/2 (psi )1/2 uik uqk , Rk := (5.92) iq Eq. (5.91) can be expressed as X ∆x2 (n) = Rk λnk . k (5.93) 5.4 Diffusion on networks 97 The long-time limit of the MSD, n → ∞, is given 2 by the largest eigenvalue λ1 = 1 and its transport coeﬃcient R1 , that is, limn→∞ ∆x (n) = R1 . All other eigenvalues have an absolute value smaller than one. Therefore, the contribution of these eigenvalues to the MSD decays for n → ∞, except the coeﬃcient R1 . As the MSD is strictly monotonic increasing, it follows Rk < 0 for k 6= 1. The larger eigenvalues, i.e. the ones close to one, contribute the long-time behavior of Eq. (5.93). If the sum in Eq. (5.93) is performed only over the low-indexed mode numbers, i.e. over the largest 2 eigenvalues, then the short-time behavior will not be reproduced correctly. As ∆x (n = 0) = 0, P P it follows k Rk = 0. Therefore R1 = − k>1 Rk . 3 10 2 10 2 〈 x (t)〉 full MSD 1−3 1−8 1−18 1 10 0 10 0 10 10 1 2 3 10 time 10 4 10 Figure 5.18: MSD of the Sierpinski gasket Γ6 - The red line illustrates the complete MSD, while the other curves correspond to Eq. (5.93) with only a partial sum over the coefficients Rk , over the first three (magenta), over the first eight (cyan) and over the first 18 (green) eigenvalues, see legend. The Rk > 0 of the largest eigenvalues contributes the value of the plateau reached for long t. With decreasing λk , i.e. with decreasing µk and increasing k, the short-time behavior is retained. Only a small number of transport coefficients determine the behavior of the MSD up to high accuracy. The MSD can be expressed as X X ∆x2 (t) = Rk et log λk = Rk e−µk t , k (5.94) k with the rates deﬁned as µk = − log λk > 0 and the continuous time coordinate t instead of the discrete n. The MSD can be decomposed into a sum of exponential contributions with a typical time µ−1 k . The prefactors are the transport coeﬃcients, Rk , with the properties given above. The MSD of a Markov process in Eq. (5.94) gives rise to subdiﬀusion if the transport coeﬃcients, Rk , obey to certain conditions, as brieﬂy outlined in Appendix E. It turns out that only a small number of additive terms suﬃce to provide a power-law MSD 98 Modeling Subdiffusion over several time scales. Therefore, even the Sierpiński triangles, Γn , for n > 3 exhibit a noticeable subdiﬀusivity, as illustrated in Fig. 5.10. As an example, we analyze the eigenvalues of the transition matrix corresponding to the Sierpiński triangles. The eigenvalues of the discrete Laplace operator are illustrated in Fig. 5.19 A. The eigenvalues found for Γn are also found for Γk with k > n, but with a higher multiplicity. The eigenvalues exhibit two series of multiplicities, one with 2, 3, 6, ...(3k + 3)/2 and the other with 1, 4, 13, (3k + 3)/2 − 2 [219, 220]. The transport coeﬃcients, Rk , in Eq. (5.93) determine the MSD seen in the Sierpiński triangle Γn . The ﬁrst exponent, R1 , contributes the saturation plateau asymptotically reached in the limit t → ∞. All other transport coeﬃcients are negative. The transport coeﬃcients Rk (for k 6= 1) are largest for the two eigenvalues k = 2 and 3. It turns out that only isolated transport coeﬃcients give a signiﬁcant contribution to the MSD. The decomposition of the MSD into the diﬀerent exponential contributions, Eq. (5.94), is illustrated in Fig. 5.18. Note the similarity with Fig. E.1. A 1 B 10 eigenvalue λk Γ2 0.5 Γ 3 Γ4 Γ 5 0 Γ6 10 10 10 10 10 10 −0.5 0 100 200 mode number k 300 Γ 1 k 1 transport coefficients |R | Γ 2 10 Γ 1 2 Γ 0 3 Γ −1 4 Γ 5 −2 Γ 6 −3 −4 −5 0 50 100 150 mode number k 200 250 Figure 5.19: Diffusion on the Sierpiński triangle - A: Eigenvalues of Sierpiński triangles with n = 1, 2, 3, 4, 5, and 6. B: Absolute value of transport coefficients, |Rk |, for various Sierpiński triangles. Only a limited number of the Rk has an absolute value significantly larger than zero. The largest |Rk |, apart from R1 , is seen for the eigenvalues 2 and 3, 7 and 8, 17 and 18. 5.5 Conclusion Diﬀerent approaches have been applied to model subdiﬀusion in the internal dynamics of biomolecules. It has been demonstrated with Zwanzig’s projection formalism that projections to a limited set of relevant coordinates gives rise to correlations and memory eﬀects. From the simulations presented in Chap. 4 it follows that the memory eﬀects due to the projection to the internal coordinates of a β-hairpin molecule on the time scale of 2 ps are not signiﬁcant for the kinetic behavior, in particular for the MSD found. The Rouse chain model is brieﬂy reviewed. It is demonstrated how the normal modes of Rouse model determine the time evolution. The distance ﬂuctuations of the Rouse chain are shown to exhibit subdiﬀusion. However, an estimation of the Rouse 5.5 Conclusion 99 time demonstrated that the subdiﬀusive regime does not ﬁt to the conﬁgurational subdiﬀusion seen in MD simulations and in single molecule spectroscopy. The CTRW model has been introduced and was found to not reproduce subdiﬀusivity in the time-averaged MSD. We carefully analyzed the conﬁned CTRW model, i.e., the CTRW in a ﬁnite volume. Although the conﬁned CTRW does exhibit subdiﬀusive dynamics in the time-averaged MSD, it is not an explanation of the subdiﬀusion seen in the conﬁgurational diﬀusion of the MD simulations at hand. The conﬁguration space can be represented by a discrete transition network. It was demonstrated that a Markov process based on the network representation of the conﬁguration space allows the subdiﬀusive MSD of the MD simulation to be reproduced with high accuracy on time scales of and above 100 ps. The fractal-like nature of the transition network was characterized by the fractal dimension and identiﬁed as the essential mechanism that gives rise to subdiﬀusive dynamics. It is sketched that, in principal, the transition matrices allows methods from linear algebra and matrix algebra to be applied to the problem of subdiﬀusion. The transport coeﬃcients arising from the eigenvectors of the transition matrix determine the diﬀusional behavior of the network. However, as being numerical cumbersome, these methods could not yet be applied to the transition networks obtained from the MD trajectories. 100 Modeling Subdiffusion [...] mais n’abandonnez jamais. A la longue votre article sera accepté et publié dans le Physical Review et alors, [...], vous serez devenu un vrai physicien. David Ruelle Chapter 6 Concluding Remarks and Outlook 6.1 Conclusion The present thesis deals with questions from two very active ﬁelds of physical research, the ﬁeld of protein dynamics and the topic of subdiﬀusive transport processes. The cell is the building block of terrestrial life. An understanding of cell activity, in turn, requires the understanding of proteins, the main agents at the sub-cellular level. Therefore, the study of protein dynamics has attracted an enormous, still increasing attention in the last decades. The experimental techniques provide access to details of the molecular processes with unprecedented accuracy. The entanglement of processes on very diﬀerent time and length scales puzzles the theoreticians engaged in the ﬁeld of biophysics. Even on the single-molecule level, the processes span many orders of magnitude. In the case of peptides, the fastest motion are due to covalent bond vibrations, found in the range of femtoseconds (10−15 s) to large domain motions, which occur on the time scale of some hundreds of microseconds (10−1 s). Hence, internal dynamics of peptides span at least 14 orders of magnitude. In order to give a coherent description of the very diﬀerent time scales on which the peptide and protein dynamics occur, the energy landscape picture was successfully employed since the 1970s [22, 195, 221, 222]. Notably the energy landscape idea has been introduced in the context of equilibrium dynamics [23], albeit energy landscape models also proved extremely fruitful for the analysis of protein folding. The energy landscape idea has been extensively studied in the context of glass forming liquids. In contrast to liquids, proteins fold into a relatively well deﬁned conformation, the native state. However, the similarity of proteins and peptides to glassy systems is due to the presence of diﬀerent isoenergetic substates (for biomolecules, found in the same native state), which are separated from each other by energetic barriers. That is, like glass formers, peptides exhibit frustration. The substates themselves may be further structured by even lower internal barriers. Hence, the molecule’s energy landscape is organized by a hierarchy of states allowing a vast range of time and length scales to be covered. 102 Concluding Remarks and Outlook In the thesis at hand, the thermal ﬂuctuations of biomolecules have been studied using MD simulations. The kinetics of four diﬀerent, short peptide chains and a βhairpin molecule has been examined on the basis of MD trajectories extending to the microsecond time scale. In the introduction chapter, three questions are raised. Here, the results obtained in the present thesis shall be summarized in response to these questions. Is subdiffusivity biomolecules? a general feature of internal fluctuations in Subdiﬀusive internal ﬂuctuations have been found in experiments with myoglobin [23, 30, 50], lysozyme [54] and ﬂavin oxireductase [27, 28, 51, 52] and simulations of lysozyme [53], the outer membrane protein f [136], and ﬂavin [183]. Here, the MD simulations of short peptide chains allows the subdiﬀusivity of internal biomolecular motion to be assessed for small and relatively simple systems. The fact that the (GS)n W peptides for all n = 2, 3, 5, and 7 exhibit subdiﬀusive ﬂuctuations corroborates the generality of the eﬀect. Also, the β-hairpin undergoes fractional diﬀusion. A well deﬁned, native state is not a prerequisite of anomalous diﬀusion in the internal coordinates. Therefore, fractional dynamics are likely to be the standard case rather than an exception or the property of speciﬁc, more complex molecules. The comparison between diﬀerent models for the water dynamics has indicated shortcomings of the generalized Born approach (GB/SA) with respect to the kinetic behavior, i.e., while the free energy proﬁle is correctly reproduced the kinetics are not. Although the eﬀect of subdiﬀusion is present in the GB/SA simulation, the MSD exponent is considerably diﬀerent. What is the mechanism that causes the internal dynamics of biomolecules to be subdiffusive? Can the CTRW model exhibit subdiffusivity in the time-averaged MSD? Hitherto, several models have been suggested to describe the experimental and simulation ﬁndings [27, 28, 53, 54, 155, 183]. In this thesis, we were seeking the underlying mechanism of subdiﬀusivity, so we examined only approaches that make explicit statements about the origins of the eﬀect. We did not address ad hoc descriptions employed to derive further molecular properties from fractional diﬀusion. The harmonic chain model, i.e. the Rouse chain, and its extension to more general geometries [151, 155, 156], although exhibiting subdiﬀusive behavior, has been found to fail in the reproduction of the correct time-scales of the fractional regime. The CTRW model has been found to have a subdiﬀusive ensemble average. However, the simulation results and some of the experiments exhibit subdiﬀusion in the time-averaged MSD. Therefore, the time averages of CTRW quantities have been examined carefully. The free, unbounded CTRW does not display a subdiﬀusive, time-averaged MSD. In contrast, the time-averaged MSD of the conﬁned CTRW exhibits a subdiﬀusive behavior. These ﬁndings make the CTRW a candidate for the mechanism responsible for the 6.1 Conclusion 103 anomalous diﬀusion found in biomolecules. However, an estimation of the parameters of the conﬁned CTRW, in particular the critical time which deﬁnes a lower bound for the presence of subdiﬀusion in the time domain, disqualiﬁes CTRW as a model for biomolecular kinetics. As an alternative approach, the diﬀusion on fractal networks has been discussed. A transition network representation of the conﬁguration space arises naturally by dividing the energy landscape in discrete states. The network geometry was analyzed and found to be of fractal nature. The Hausdorﬀ dimension of the transition network was calculated and lies in the range df ≈ 6 to 7. The transition networks obtained from the MD simulation trajectories were also used as an input for Markov models, which allowed random walks on the networks to be performed. The transition network random walks were used to determine the diﬀusion dimension. The dimensions found clearly indicate subdiﬀusive behavior. Also the MSDs, as predicted by the transition networks, were obtained and compared to the results of the original MD trajectories. The kinetics are correctly reproduced at and above the time scale of 100 ps. For shorter times, the kinetics are not completely represented, although subdiﬀusive behavior has still been found. Three potential sources of deviations have been identiﬁed: (i) the dynamics of the projected trajectory is eﬀectively non-Markovian on short time scales; (ii) the spatial resolution of the network is too low; and (iii) the discretization scheme is too simple to resolve the correct kinetics. However, the memory eﬀects accounting for the non-Markovian nature have been found to be insubstantial, as was demonstrated by a comparison of an explicit water simulation to a Langevin thermostat simulation of the β-hairpin. Both these simulations agree in terms of the overall kinetic behavior. It was illustrated that the ﬁrst ten PCs account for most of the internal ﬂuctuations. However, the low amplitude, high-indexed PCs yield important dynamical and kinetic information. Network models taking the higher PCs into account are much more accurate in reproducing the biomolecular dynamics. Brieﬂy, we sketched how algebraic methods may be applied in the context of transition networks to give a better understanding of subdiﬀusive behavior. The fractal geometry of the accessible volume in the conﬁguration space is a consequence of the energy landscape. A high potential energy makes conﬁgurations unfavorable such that they are eﬀectively forbidden. Hence, regions with too high a potential energy value are eﬀectively unaccessible. The conﬁgurations observed in the MD trajectory sample the accessible volume, which is divided by multiple barriers into conﬁgurational substates. The dynamics are further restricted by geometrical constraints. That is, the accessible volume is not compact but contains many unaccessible “islands” which act as obstacles and render the accessible volume fractal-like. Fractal behavior prompts to scale invariance. The power-law behavior of the MSD, ∆x2 (t) ∼ tα , means that the dynamics on the time scale of, say t = 1 ps, are eﬀectively the same as those on the time scale of t′ = 1 ns, provided the distances are scaled as l′ = λα/2 l with λ = t′ /t. Therefore, the subdiﬀusive MSD characterizes an ordering principle of the energy landscape in a speciﬁc range. The kinetics of the peptides studied in the present thesis cover the range from 104 Concluding Remarks and Outlook picoseconds to microseconds. As the molecules studied are relatively small with a simple structure, this time range is suﬃcient to collect enough statistics for a reliable analysis. At least in the case of the (GS)n W peptides, we expect the longest intrinsic time scales to be in the range of 10 ns. For larger systems, much longer time scales will be found. However, we expect similar dynamical behavior, i.e. subdiﬀusivity, to be present on intermediate time scales even for much larger biomolecules. 6.2 Outlook The present work provides a new approach in the analysis of internal ﬂuctuations of biomolecules. Well established network methods may be a key to quantifying the properties of the energy landscape which give rise to subdiﬀusive behavior. However, it is an open question how to discretize the energy landscape in order to reproduce the dynamics on shorter time scales correctly, in particular with respect to subdiﬀusivity. Network models likely allow the organizing principles of the energy landscape to be characterized. The hierarchical organization of the energy landscape, as reported in recent publications [206, 209, 213, 217, 218], could be the basis to establishing a rigorous length-time scaling that determines the MSD exponent α. If function-relevant subgraphs can be identiﬁed, the analysis of network representations of the conﬁguration space can give new insights in the connection between dynamics and biological function [204, 206, 223]. A minimal mechanical model exhibiting subdiﬀusive ﬂuctuations is needed to assess which parts of the interaction potentials used in MD simulations account for the fractional behavior. In the present work, we did not study the temperature dependence of the subdiﬀusive behavior. Recent experiments demonstrated small peptides to exhibit a dynamical transition [48], as earlier found for proteins [31]. An interesting question is whether the subdiﬀusive behavior is lost below the glass transition when the molecule is expected to be trapped in a single potential well. Also the precise time dependence of the MSD at lower temperatures could tell more about the hierarchical nature of the energy landscape. A further challenge is to derive quantities from the theoretical models which can be assessed by experimental techniques. The presence of fractional diﬀusion on the pico- and nanosecond time scale has been shown to give rise to ACFs of Mittag-Leﬄer form [54], regardless of the underlying mechanism. Therefore, it is diﬃcult to put the theoretical models to the test as long as only the ACFs can be measured, such as in spin echo neutron scattering or single molecule ﬂuorescence spectroscopy. Translating a given transition network model into neutron scattering quantities is a promising path to quantify the contributions of the elementary dynamical processes to the dynamical structure factor, the intermediate scattering function, and other measurable quantities. [16] M. Karplus and G.A. Petsko. Molecular dynamics simulations in biology. 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Teubner, Leipzig, 1877. 122 112 REFERENCES Abbreviations and Symbols Abbreviations ACF ACTRW CACF CTRW eSPC FDE FFT FPTD GB/SA GLE JLD LINCS MD MLF MSD NMA NMR PC PCA PMF SPC VACF WTD auto-correlation function aging continuous time random walk coordinate auto-correlation function continuous time random walk extended simple point charge fractional diffusion equation fast Fourier transform first passage time distribution generalized Born/surface area generalized Langevin equation jump length distribution linear constraints solver molecular dynamics Mittag-Leffler function mean squared displacement normal mode analysis nuclear magnetic resonance principle component principle component analysis potential of mean force simple point charge velocity auto-correlation function waiting time distribution Greek symbols α ∆t ∆k Γ dΓ Γ(α) Γ(α, x) Γn Γ∞ MSD exponent, Eq. (2.37) simulation time step or random walk time step anharmonicity degree, Eq. (4.10) phase space integration over phase space Gamma function, Eq. (D.38) incomplete Gamma function Eq. (D.42) Sierpiński triangle of order n Sierpiński gasket γ δij δ(t) ηk ϑ κα Λk µ(t) ξ,ξ χn (t2 , t1 ) ρ τ0 τR ϕ(x) Langevin friction, Eq. (2.11) Kronecker’s delta Dirac’s delta function participation ratio, Eq. (4.6) lag time of transition matrix, Eq. (5.63) prefactor, = sin(πα)/πα convergence factor, Eq. (4.12) memory function, Eq. (5.10) Langevin random force, Eq. (2.11) probability of n jumps in [t1 , t2 ], Eq. (5.27) phase space density typical time scale in WTD Eq. (5.20) Rouse time, Eq. (5.17) JLD Latin symbols AT C Cv (t) Cx (t) D 1−α 0 Dt df ds dt dw Eα (x) E F (N, V, T ) f kB Kα transposed of matrix A covariance matrix, Eq. (3.47) VACF, Eq. (2.26) CACF, Eq. (2.25) diffusion constant, Eq. (2.33) Riemann-Liouville operator, Eq. (5.23) Hausdorff or fractal dimension, Eq. (5.64) spectral dimension, Eq. (5.73) topological dimension diffusion dimension, Eq. (5.68) MLF, Eq. (5.24) set of edges in network free energy, Eq. (4.3) force vector in configuration space, Eq. (3.1) Boltzmann constant generalized diffusion constant, Eq. (5.22) 114 L L L2 (V ) M N N nµ,σ O(kxk) ps q Rk r S S(ϑ) S̃(ϑ) T T Tg tc tp ts V V (r ) v W (x, t) w(t) w1 (t, ts ) x Z Z Abbreviations and Symbols Liouville operator, Eq. (5.2) Laplace transform, Eq. (D.53) space of square integrable functions over V mass matrix number of particles set of vertices in network Gaussian distribution with mean value µ and standart deviation σ Landau symbol stationary probability distribution PC vector transport coefficients, Eq. (5.92) position vector in configuration space transition matrix, Eq. (5.63) transition matrix on time scale ϑ, Eq. (5.63) approximate transition matrix length of observation/length of simulation temperature glass transition temp. critical time in confined CTRW, Eq. (5.55) saturation time start of observation in ACTRW volume energy potential velocity vector in configuration space probability distribution in diffusion equation WTD, Eq. (5.20) initial WTD mass-weighted configuration vector counting matrix, Eq. (5.57) partition function Other symbols hA|Bi h·iτ h·iτ,T h·iens h·iens,0 h·iens,ts h·it L2 -scalar product time average time average over τ ∈ [0, T ] ensemble average CTRW: ensemble average, time relative to first jump CTRW: ensemble average, time relative to ts > 0 CTRW: combined ensemble and time average, also hh·ii ˙ 2 ¸ ∆x (t) MSD Index ACF, 15, 16, 123 anharmonicity degree, 49 anharmonicity factor, 49 anomalous diﬀusion, 19 Bernoulli, Daniel, 7 Berzelius, Jöns Jakob, 2 Birkhoﬀ, George David, 8 Boltzmann, Ludwig, 1, 8, 11 Born-Oppenheimer approximation, 22 Brown, Robert, 1 Brownian motion, 1, 8, 12, 13, 37, 59 CACF, 16, 17 central limit theorem, 13 Clausius, Rudolf, 7 conﬁguration space, 32 convergence, 36, 52, 55 convergence factor, 56 counting matrix, 81 covariance matrix, 35, 42, 48 CTRW, 2, 64 conﬁned, 74 detailed balance, 82 diﬀusion constant, 18, 30 diﬀusion equation, 18, 119 dimension diﬀusion, 86 fractal, 84 spectral, 88 topological, 83 Einstein relation, 19 Einstein, Albert, 1, 8, 12 energy landscape, 3, 31 ensemble average, 10, 68 equipartition theorem, 10 ergodicity, 8, 12, 15, 16, 36, 69, 70, 86 eSPC, 30, 57 Ewald, Paul Peter, 27 FDE, 68, 124, 126 Feynman, Richard, 10 Fick, Adolf, 1, 7 ﬁrst law of thermodynamics, 7 ﬂuctuation-dissipation theorem, 14 force ﬁeld, 21, 25 Fourier decomposition, 7, 19, 68, 119 Fourier transform, 126 Fourier, Joseph, 7 FPTD, 53 fractal dynamics, 85 fractal geometry, 83 fractional derivative, 125 free energy, 43 Gamma function, 124 GB/SA, 30 generalized Born, 30 generalized Langevin equation, 61 Gibbs, Josiah Willard, 1 glass, 4, 33, 64 glass transition, 4, 41, 42, 46 Hausdorﬀ dimension, 84 Hausdorﬀ, Felix, 83 Helmholtz, Hermann von, 1, 7 isokinetic thermostat, 40 JLD, 65 116 Joule, James Prescott, 7 kinetic theory, 7 Kramer’s escape theory, 54, 65 Langevin equation, 12, 15, 59, 61 Langevin thermostat, 31 Langevin, Paul, 8 Laplace transform, 126 Leapfrog algorithm, 24 LINCS algorithm, 40, 120 Liouville equation, 59, 61 Liouville theorem, 9 Mach, Ernst, 8 Mandelbrot, Benoı̂t, 83 Markov property, 60, 61, 86 mass matrix, 9 mass-weighted coordinates, 33, 42 Maxwell, James Clerk, 8 Maxwell-Boltzmann distribution, 11, 14, 15, 22, 32 Mayer, Julius Robert von, 1, 7 MD simulation, 21 mean squared displacement, 14 Mittag-Leﬄer function, 68, 127 molecular disorder, 11 MSD, 14, 15, 19 MSD exponent, 19 INDEX Perrin, Jean, 1, 8 phase space, 9 Planck, Max, 8 PMF, 43 Poisson equation, 28–30 potential of mean force, 43 principal components, 35 protein, 2, 3 protein folding, 3 random force, 12, 13 random walk, 1, 18 reaction coordinate, 43, 47 Riemann-Liouville, 68 Riemann-Liouville operator, 125 Rouse chain, 62, 123 second law of thermodynamics, 7, 8, 11 secondary structure, 3, 4, 22, 40 Sierpiński, Waclaw, 84 Sierpiński gasket, 84 Sierpiński triangle, 84 Smoluchowski, Marian von, 8 subdiﬀusion, 4, 19 superdiﬀusion, 19 Neumann, Johann von, 8 neutron scattering, 3, 4, 42 Newton’s second law, 9 NMA, 33, 34 normal modes, 34 thermostat isokinetic, 31 Langevin, 31 time average, 11, 70 trajectory, 10, 22 transition matrix, 82 transition network, 81 transport coeﬃcient, 96 trap model, 2, 64, 66 operational time, 70 Ostwald, Wilhelm, 8 VACF, 16, 17 Verlet algorithm, 23–25 participation number, 47 participation ratio, 47 particle mesh Ewald method, 27, 30, 40 PCA, 34, 37, 48 Pearson, Karl, 8, 18, 35 peptide, 2, 3 water, 4, 30, 39, 50 explicit, 30, 40 implicit, 30, 40 white noise, 13, 50 WTD, 65–67 WTD exponent, 66 INDEX Zermelo, Ernst, 8 Zwanzig’s projection formalism, 59 117 118 INDEX √ Ainsi il s’en s[u]it, que d1:2 x sera egal à x 2 dx : x. Il y a de l’apperance, qu’on tirera un jour des consequences bien utils de ces paradoxes, car il n’y a gueres des paradoxes sans utilité. G. W. F. Leibniz Appendix A Diffusion equation solved by Fourier decomposition The diﬀusion equation can be solved using Fourier decomposition. The principal idea is the separation of the time variable t and the coordinate x in the probability W (x, t). The eigenfunctions of the Laplace operator in the diﬀusion equation, Eq. (2.33), ∂ ∂2 W (x, t) = D 2 W (x, t) (A.1) ∂t ∂x can be expressed as imaginary exponentials, likewise the time dependence is given by a real exponential. Thus, the probability W (x, t) is obtained as Z ∞ 2 (A.2) a(k)eikx e−Dk t dk. W (x, t) = −∞ The initial condition and the boundary condition determine the Fourier coeﬃcients a(k). In the case of free, unbounded diﬀusion, the initial condition W0 (x) = δ(x) corresponds to a(k) = (2π)−1 . Then, Eq. (A.2) yields the probability distribution W (x, t) of Eq. (2.34), which solves the diﬀusion equation. The MSD is obtained from the solution by integration. The result of the integrations is given in Eq. (2.35). Diffusion with finite volume In the presence of a reﬂecting boundary at x = 0 and at x = L, the probability distribution, W (x, t) is conﬁned to the coordinate interval x ∈ [0, L]. At the boundaries, the Dirichlet condition is satisﬁed ∂ ∂ W (x, t) W (x, t) = = 0. (A.3) ∂x ∂x x=0 x=L The boundary condition is fulﬁlled by the functions cos(πkx/L), with k ∈ N. As a consequence, the integral in Eq. (A.2) is replaced by a discrete sum containing only the above cosine terms. Correspondingly, the time dependence has to be modiﬁed. We impose the initial condition W0 (x) = δ(x − x0 ). The probability reads ∞ 2X 4π 2 k2 πkx 1 exp − 2 Dx . (A.4) ak (x0 ) cos W (x, t; x0 ) = + L L L L k=1 120 Appendix Inserting the probability distribution, W (x, t; x0 ) in Eq. (2.35), the MSD of the ﬁnite volume diﬀusion follows as " # ∞ 2 L2 1 (2k + 1)2 π 2 96 X ∆x (t) = (A.5) exp − Dt 1− 4 6 π (2k + 1)4 L2 k=0 # " ∞ X L2 t = (A.6) 1− θk gk , 6 k=0 where θk = 96/[π(2k + 1)]4 and (2k + 1)2 π 2 gk = exp − D . L2 (A.7) In the main text, the following approximation is used for simplicity L2 ∆x2 (t) = [1 − gt ], 6 (A.8) with g = exp(−12D/L2 ). Eq. (A.8) reproduces the behavior of the full solution for small times t → 0 and it reaches the correct saturation level in the limit t → ∞. However, it fails to reproduce the slowest relaxation time by a factor π 4 /12 ≈ 0.8225, i.e., the relaxation time is underestimated by 18%. Note that all calculations performed in Sec. 5.3 using this approximation can also be analogously carried out with the exact solution to Eq. (5.27). B Constrained Verlet algorithm – LINCS The fastest oscillations in an MD simulation are the bond vibrations which impose an upper bound to the maximal integration time step ∆t. In order to increase the time step, the bond oscillations can be removed from the simulation. It turns out that simulations with ﬁxed bond lengths are a more accurate representation of the physical behavior [96]. To remove bond oscillations, the bond lengths are holonomically constrained to a ﬁxed length. Holonomic constraints can be introduced in the Verlet algorithm with the Linear Constraint Solver (LINCS) algorithm [96], which is brieﬂy reviewed in the following. The positions of a system with N particles are given by the 3N vector r (t), the equations of motion are yield from Newton’s second principle, Eq. (3.2). The forces are assumed to be conservative forces, i.e., they can be expressed as f = −∂V /∂r . Eq. (3.2) is a set of 3N diﬀerential equations. The holonomic, i.e. velocity independent, constraints are given by the equations gi (r ) = 0 i = 1, ..., K, (B.9) B Constrained Verlet algorithm – LINCS 121 where the constraints are assumed to not have an explicit time dependence. Here, to ﬁx the bond lengths, the functions gi (r ) read gi (r ) = |r i1 − r i2 | − di i = 1, ..., K, (B.10) where di is the length of bond i between the atoms i1 and i2 . The vectors with superscript denote the positions of the corresponding atoms. Lagrange’s ﬁrst method allows the constraints to be included; the potential V (r ) is modiﬁed such that it contains the constraints multiplied with the Lagrange multipliers, λi (t), ! X ∂ d2 r (B.11) V − λi gi . −M 2 = dt ∂r i With the notation Bhi = ∂gh /∂ri , a K × 3N matrix B is deﬁned. Using B, Eq. (B.11) can be expressed as matrix equation d2 r = M−1 BT λ + M−1 f . dt2 (B.12) In the next step, the Lagrange multipliers, λi (t), are determined from Eq. (B.12). Solving K of the 3N diﬀerential equations of Eq. (B.12) reduces the problem to 3N − K dimensions, in which the constraints of Eq. (B.9) are always respected. The second time derivative of the constraints, Eq. (B.9), yields −B d2 r dB dr = . dt2 dt dt After multiplying Eq. (B.12) with B and using Eq. (B.13), one has dB dr −1 T −1 BM B λ = − + BM f . dt dt (B.13) (B.14) We deﬁne T = M−1 BT (BM−1 BT )−1 . Multiplying Eq. (B.14) with T and inserting the expression in Eq. (B.12) leads to d2 r dB dr = (1l − TB) M−1 f − T . 2 dt dt dt (B.15) The matrix 1l−TB acts as a projection operator to the 3N −K dimensional subspace to which the dynamics are conﬁned by the constrains. To use Eq. (B.15) as an algorithm in MD simulation, the Taylor expansion, Eq. (3.3), is combined with Eq. (B.15) rn+1 = 2rn − rn−1 + [1l − Tn Bn ](∆t)2 M−1 f − ∆tTn (Bn − Bn−1 )vn−1/2 , (B.16) where the velocity is deﬁned as vn−1/2 = rn − rn−1 ∆t (B.17) 122 Appendix As the constraints, Eq. (B.9), do not depend explicitly on time, the time derivative of the constraints yields 3N X ∂g ∂rj dg 0= = = Bn vn+1/2 , dt ∂rj ∂t (B.18) j=1 which allows Eq. (B.16) to be simpliﬁed to rn+1 = rn + (1l − Tn Bn )[rn − rn−1 + (∆t)2 M−1 f ]. (B.19) The algorithm Eq. (B.19) is analytically correct, but numerically instable. As the constraints, Eq. (B.9), entered the algorithm only in the second time derivative, the algorithm accumulates numerical errors. An additional term −Tn (Bn rn − d ), in which d denotes the vector of bond lengths, can solve this problem partially, rn+1 = (1l − Tn Bn )[2rn − rn−1 + (∆t)2 M−1 f ] + Tn d . (B.20) Additionally the bond lengths must be modiﬁed, as Eq. (B.20) just removes the velocity components along the old bond direction but does not ﬁx the value of the bond lengths. Again, this is a consequence of using the second time derivatives instead of the original constraints. If li is the length of the bond after an update, the projection of the new bond onto the old direction is set to pi = [2d2i − li2 ]1/2 . Then, the position vector is ∗ rn+1 = (1l − Tn Bn )rn+1 + Tn p. (B.21) Eqs. (B.20) and (B.21) built the constrained version of the Verlet algorithm. The original Verlet algorithm, Eq. (3.5), is contained in the square brackets, but the matrix (1l − Tn Bn ) projects it such that it meets the constraints. An analogous Leap Frog version can be obtained [96]. The implementation of the constrained Verlet algorithm Eq. (B.20) involves the inverting of Bn M−1 BTn which is required for the calculation of Tn . A proper rearrangement allows to write (Bn M−1 BTn )−1 = S(1l − An )−1 S, (B.22) with a diagonal matrix S. As An can be shown to be sparse, symmetric, and to have eigenvalues smaller than one, the Neumann series can be applied [224], i.e., −1 (1l − An ) = ∞ X Akn . (B.23) k=0 Truncating the series in Eq. (B.23) after the forth term is a reasonable approximation and makes the inversion much more eﬃcient [96, 97]. C Derivation of the Rouse ACF C 123 Derivation of the Rouse ACF The following treatment of the dynamics of the Rouse chain can be found in [149, 150, 158]. Here, as in the main text, the Rouse chain is treated as a one-dimensional object; the three-dimensional case is obtained by a superposition of the independent, one-dimensional time evolutions. First assume an inﬁnite chain. The modes of the chain can be expressed in terms of the Fourier modes. The solutions of the discrete eigenvalue equation ω̃ 2 [Ψω (i + 1) − 2Ψω (i) + Ψω (i − 1)] = −ω 2 Ψω (i), built an orthonormal basis set, imately given by P i (C.24) Ψω (i)Ψω′ (i) = δωω′ . The Fourier modes are approx- Ψω (i) = cos iω ω̃ . (C.25) The positionsPcan be expressed as a linear combination of the Fourier modes, Ψω , i.e., zi (t) = {ω} ζω (t)Ψω (i). Essentially, this is the normal mode decomposition of the chain; ζω (t) are the normal modes. For a ﬁnite chain, the forces at the ends of the chain equal zero. As a consequence, the set of possible eigenfrequencies, {ωn }, is discrete, ﬁnite, and obeys Eq. (5.12), ωn = ω̃πn N −1 with n = 0, 1, ..., N − 1. (C.26) We use the notation ζn = ζωn for the normal modes. We now derive the autocorrelation of the distance between bead i and j. The distance between bead i and j is ∆(t) = zi (t) − zj (t) X = ζn (t)[Ψn (i) − Ψn (j)] (C.27) (C.28) n ζn (t) cos cos α − cos β = 2 sin = X n iωn ω̃ − cos jωn ω̃ . (C.29) , (C.30) The identity α+β 2 sin β−α 2 allows the deviation, ∆(t), to be expressed as X (i + j)ωn (j − i)ωn ∆(t) = 2 ζn (t) sin sin ω̃ ω̃ n X (j − i)ωn . ≈2 ζn (t) ω̃ n (C.31) (C.32) 124 Appendix The approximation in the last line can be performed if the deviation depends only on the distance of the beads on the chain, i.e., as long as i − j ≪ N and both beads are not at the ends of the chain. Then, the ACF is X ωn ωk h∆(t + τ )∆(τ )iτ = hζn (t + τ )ζk (τ )iτ (j − i) (C.33) ω̃ 2 nk X kB T 2 e−tn /τR (j − i), (C.34) = 2 mω̃ n where the vanishing correlation between diﬀerent normal modes is used, as follows from the equipartition theorem, Eq. (5.15). For long chains, the sum can be replaced by an integral. Hence, Z ∞ kB T −tn2 /τR (j − i)dn (C.35) e h∆(t + τ )∆(τ )iτ = mω̃ 2 0 r Z ∞ kB T τR 2 = (j − i) (C.36) e−y dy. 2 mω̃ t 0 Therefore, the time dependence of the ACF is h∆(t + τ )∆(τ )iτ ∝ t−1/2 . D (C.37) Fractional diffusion equation In Sec. 5.3, the fractional diﬀusion equation (FDE) is employed, serving as the macroscopic equation governing the time evolution of an ensemble of CTRW processes. The equation can be obtained as the continuum limit of the CTRW with a power-law WTD as given in Eq. (5.20). In this section, the derivation of the FDE is reviewed, using Fourier-Laplace transforms, as is done in Ref. [8]. First, some mathematical notations are introduced. D.1 Gamma function The Riemannian Gamma function is deﬁned as Z ∞ tα−1 e−t dt. Γ(α) = (D.38) 0 It obeys the functional equation Γ(α + 1) = αΓ(α). (D.39) The latter can be used as a recurrence relation. As Γ(1) = 1, if follows for n ∈ N Γ(n + 1) = n!. Therefore, the Gamma function is a continuous extension of the factorial. (D.40) D Fractional diffusion equation 125 Further, a useful property of the Gamma function is Γ(1 + α)Γ(1 − α) = πα , sin πα with α ∈ R \ Z. The incomplete Gamma function is deﬁned as Z ∞ tα−1 e−t dt. Γ(α, x) = (D.41) (D.42) x The incomplete Gamma function Γ(−α, u) can be expanded as a chain fraction [107]. It has the following asymptotic behavior for small x Γ(α, x) ≈ Γ(α) − D.2 e−x xα α for x ≪ 1. (D.43) Fractional derivatives Let 0 Jt be the integral operator deﬁned by 0 Jt φ(t) = Z t φ(t′ )dt′ . (D.44) 0 According to Cauchy’s formula for repeated integrals, the n-fold integral, n being integer, can be expressed as n 0 Jt φ(t) 1 = (n − 1)! Z t 0 φ(t′ ) dt′ . (t − t′ )1−n (D.45) A straightforward way to extend Eq. (D.45) from n ∈ N to real α > 0, is to deﬁne the integral operator Z t φ(t′ ) 1 α dt′ . (D.46) J φ(t) = 0 t Γ(α) 0 (t − t′ )1−α Let β = n − α, with 0 < α < 1 and n ∈ N. The fractional derivative of order β ∈ R is deﬁned as dn α β J φ(t). (D.47) 0 Dt φ(t) = dtn 0 t The operator 0 Dtβ is called the Riemann-Liouville operator. The fractional derivative of order α 6= 0, of a power of t is given as 1−α γ t 0 Dt = Γ(1 + γ) γ−1+α t . Γ(γ + α) (D.48) The Laplace transform of the Riemann-Liouville operator reads L[0 Dt−β φ(t)] = u−β L[φ(t)]. (D.49) 126 Appendix D.3 The fractional diffusion equation Let a CTRW have the WTD w(t) ∼ (t/τ0 )−(1+α) , as in Eq. (5.20), with an WTD exponent 0 < α < 1. The JLD, ϕ(x), is assumed to be symmetric around x = 0 with variance x̄2 . The probability of being at position x at time t is denoted as W (x, t). In this subsection, the equation is derived that governs the time evolution of the probability distribution W (x, t). The initial condition is W0 (x) = δ(x), i.e., the CTRW starts at t = 0 at the position x = 0. The probability of having just arrived at position x at time t, denoted by η(x, t), equals the probability of having arrived at x′ at time t′ and of waiting a time t − t′ at x′ and to make a jump of length x − x′ integrated over all possible x′ and t′ . Mathematically this equality is expressed by Z ∞Z t η(x, t) = η(x′ , t′ )w(t − t′ )ϕ(x − x′ )dt′ dx′ + W0 (x)δ(t). (D.50) −∞ 0 The probability W (x, t) can be obtained from η(x, t) as " Z Z ′ t W (x, t) = 0 t−t ′ η(x, t ) 1 − ′′ w(t )dt ′′ 0 # dt′ . (D.51) The convolution equations take a relatively simple form in the Fourier-Laplace representation, in which the variables x and t are transformed to k (Fourier transform1 ) and u (Laplace transform2 ), respectively. Combining the Fourier-Laplace transforms of Eqs. (D.50) and (D.51), the following equation is obtained W (k, u) = W0 (k) 1 − w(u) . u 1 − ϕ(k)w(u) (D.54) The Fourier transform of the JLD is for small k approximately given as ϕ(k) ≈ 1 − x̄2 k2 + O(|x|4 ). (D.55) The Laplace transform, t → u, of the WTD with power-law tail, [Eq. (5.20)], w(u) = αeu uα Γ(−α, u), using the incomplete gamma function. With the asymptotic behavior of Γ(α, u) in Eq. (D.43), the Laplace transform w(u) is derived w(u) ≈ 1 − Γ(1 − α)(τ0 u)α . 1 The Fourier transform of a function, φ(x) is given as Z ∞ φ(k) = φ(x)e−ikxdx. (D.56) (D.52) −∞ 2 The Laplace transform of a function, ψ(t) is given as Z ∞ ψ(u) = ψ(t)e−ut dt. 0 (D.53) D Fractional diffusion equation 127 With the generalized diﬀusion constant, Kα = x̄2 /[2Γ(1 − α)τ0α ], the diﬀusion limit, x̄2 → 0, τ0 → 0, and Kα = const., can be performed. The diﬀusion limit corresponds to the range of large x- and t-values, i.e., the time, t, is large compared to the typical time scale,√t ≫ τ0 , and the typical x is large compared to the mean squared jump length, x ≫ x̄2 . In the Fourier-Laplace space, the diﬀusion limit is performed by the limit (k, u) → 0. Inserting the Eqs. (D.55) and (D.56) in Eq. (D.54), a rearrangement of the terms yields the following equation W (k, u) − W0 (k)/u = −Kα k2 u−α W (k, u). (D.57) In the original coordinates using Eq. (D.49), and with an additional time derivative, the FDE follows3 ∂ ∂2 W (x, t) = 0Dt1−α Kα 2 W (x, t). (D.58) ∂t ∂x D.4 The Mittag-Leffler function The Mittag-Leﬄer function (MLF) is deﬁned as Eα (z) = ∞ X n=0 zn , Γ(1 + nα) (D.59) where α > 0. For α = 1, the MLF equals the exponential function, for α = 1/2, it can 2 be written in terms of the error function, E1/2 (z) = e−z [1 + erf(z)]. Using Eq. (D.49), the Laplace transform of the MLF reads L[Eα (−cz α )] = ∞ X 1 cn Γ(αn + 1) 1 = . αn+1 1−α Γ(1 + nα) u u + cu n=0 (D.60) The MLF, Eα (−[t/tm ]α ), has an asymptotic behavior given as Eα (−[t/tm ]α ) ≈ ( [t/tm ]α exp − Γ(1+α) t ≪ tm tα m tα Γ(1−α) t ≫ tm . (D.61) Therefore, the MLF has a stretched exponential behavior for small t, whereas the large-t behavior is given by a power law. 3 If the Laplace representation of a WTD with finite mean value is substituted in Eq. (D.54), the derivation leads to the classical diffusion equation, Eq. (2.33). As only the first terms in the small-u and small-k expansion determine the diffusion limit, the particular analytical form of WTD and JLD are not significant for the diffusion equation to be valid. The diffusion process is characterized by the variance of the JLD and the mean waiting time (the mean of the WTD). The anomalous diffusion seen for a WTD with power-law tail, Eq. (5.20), appears as a consequence of the diverging mean value of the WTD. 128 Appendix Let 0 < α < 1, and apply the Riemann-Liouville operator to the deﬁnition of the MLF, using Eq. (D.48) 1−α Eα (−ctα ) 0 Dt ∞ X = (−c)n Γ(αn + 1) αn−1+α t Γ(1 + nα) Γ(αn + α) n=0 (D.62) = ∞ X (−c)n−1 (D.63) n=1 Γ(nα) tαn−1 ∞ 1 X (−c)n =− αntαn−1 c n=1 Γ(1 + nα) (D.64) =− (D.65) 1 ∂ Eα (−ctα ). c ∂t Therefore the function ak (t) = Eα −Kα k2 tα is the solution of the equation ∂ ak (t) = −Kα k2 0 Dt1−α ak (t). ∂t (D.66) (D.67) This allows the Fourier decomposition to be applied, analogously to the classical diﬀusion equation. The prominent role of the MLF in the context of fractional diﬀerential equations roots in its correspondence to the exponential function. For the results of classical diﬀusion problems to be extended to fractional diﬀusion, it suﬃces in many cases to replace the exponential time dependence by an MLF time dependence. E Exponentials and power laws The sum of exponentials can give rise to a power-law behavior, as in Eq. (5.94). Here, we present a brief, non-rigorous argument demonstrating under which conditions the sum of negative exponentials yields a power law. Consider the following function f (t) = 1 − R exp(−µt), (E.68) which is characterized by two parameters, R and µ. The function f (t) is illustrated in Fig. E.1 with µ = 1 for various R. The function f (t) has the limiting cases f (0) = 1 − R and f (t → ∞) → 1. In Fig. E.1 it is apparent that f (t) exhibits a powerlaw like transition between the two limiting cases, i.e., in an intermediate range of t, the function f (t) is very similar to a power law. More precisely, in the range t ∈ [− log(0.3)/µ; − log(0.7)/µ], where 0.3 6 exp(−µt) 6 0.7, the function f (t) can be approximated as a power law, f (t) ≈ ctα(R) . Approximately, the following exponents are seen 1 − 0.3R log 0.3 α(R) = log / log . (E.69) 1 − 0.7R log 0.7 E Exponentials and power laws 129 0.9 f(t) 0.8 0.7 R=0.3 0.6 R=0.5 R=0.7 α(R)=0.12 0.5 α(R)=0.22 α(R)=0.36 0.4 −2 10 10 −1 0 10 t 10 1 10 2 Figure E.1: Function f (t) of Eq. (E.68) - The function f (t) of Eq. (E.68) has the value 1 − R for t = 0 and 1 for t → ∞. The intermediate behavior around the time scale µ−1 can be approximated by a power-law curve. f (t) is illustrated for various R, see legend. The approximate power-law exponents are also given in the legend. A ﬁnite series of terms like Eq. (E.68) can lead to a power law over an arbitrary interval [t1 , t2 ], i.e. X 1− Rk exp(−µk t) ∼ tα , for t ∈ [t1 , t2 ] (E.70) k as in Eq. (5.94). The power law at t∗ is dominated by terms with µk ≈ log(2)/t∗ . (E.71) For simplicity, we assume the µk to be so diﬀerent that only one of the µk obeys Eq. (E.71) at a given t∗ , i.e., if µk = log(2)/t∗ then exp(−µj t∗ ) ≈ 1 or exp(−µj t∗ ) ≈ 0 for all j 6= k. Hence, we assume that for any given time scale only one Rk -term must be taken into account. In the general case, various Rk contribute at a given time t∗ , i.e., various eigenvectors can contribute to the same time scale. The present argument treats them as if their respective transport coeﬃcients were summed to one common Rk . At the times tk = log(2)/µk the values of the left side of Eq. (E.70) shall be on the curve ctα . Therefore, it is X Rk α . (E.72) cµ−α Ri + k [log 2] = 2 i<k A further condition can be imposed, R2 = P Rk i<k Ri . (E.73) 130 Appendix The above equation represents the fact that the weight of the slowest term, R2 , and the long-time plateau, R1 , have the same ratio asPthe weight of the term k, dominating at time τk = µ−1 i<k Ri . k , and the slower contributions, The Eqs. (E.72) and (E.73) characterize the conditions under which a sum of exponentials like in Eq. (E.70) gives rise to an power-law behavior on a certain, intermediate interval. Note that Eq. (5.94) has the same form as Eq. (E.70). In Fig. 5.18, it is illustrated how a small number of transport coeﬃcients sum up to the power-law MSD in the case of the Sierpiński triangle. Employing the Laplace representation allows a continuous distribution of transport coeﬃcients equivalent to Eq. (E.72) to be established. Zitate in den Kapitelüberschriften • Thomas S. Kuhn: The Structure of Scientiﬁc Revolutions. – Chicago (IL): The University of Chicago Press 1973, 2nd edition, S. 24 • Hermann von Helmholtz: Antwortrede gehalten beim Empfang der GraefeMedaille zu Heidelberg, am 9. August 1886 in der Aula der Universität zu Heidelberg. In: Vorträge und Reden. – Band 2, Braunschweig: F. Vieweg und Sohn 1903, S. 318 • Friedrich Schiller: Wallensteins Tod, II, 2. • D. Frenkel & B. Smit: [18], S. 73. • Richard P. Feynman: The Feynman lectures, vol. 1, 5th edition, 1970, S. 3-6. • Nicolaus von Kues: Trialogus de possest, ca. 1460, S. 44. Zitiert nach: http://www.hs-augsburg.de/∼harsch/Chronologia/Lspost15/Cusa/cus tria.html. • David Ruelle: Hasard et Chaos. – Editions odile jacob 2000, S. 116. • Gottfried Wilhelm Friedrich Leibniz: Brief an Guillaume François Antoine Marquis de l’Hospital vom 30. September 1695.

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