Dissertation Thomas Neusius 2009

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 . . . . . . . . . .
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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 . . . . . . . . . . . . . .
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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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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65
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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
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119
119
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126
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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.
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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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