Brownian dynamics at boundaries and interfaces in physics chemistry and biology

Applied Mathematical Sciences

Zeev Schuss

Brownian
Dynamics at
Boundaries and
Interfaces
In Physics, Chemistry, and Biology


Applied Mathematical Sciences
Volume 186
Founding Editors
Fritz John, Joseph Laselle and Lawrence Sirovich
Editors
S.S. Antman

P.J. Holmes

K.R. Sreenivasan


Advisors
L. Greengard
J. Keener
R.V. Kohn
B. Matkowsky
R. Pego
C. Peskin
A. Singer

A. Stevens
A. Stuart

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Zeev Schuss

Brownian Dynamics
at Boundaries and Interfaces
In Physics, Chemistry, and Biology

123


Zeev Schuss
School of Mathematical Sciences
Tel Aviv University
Tel Aviv, Israel

ISSN 0066-5452
ISBN 978-1-4614-7686-3
ISBN 978-1-4614-7687-0 (eBook)
DOI 10.1007/978-1-4614-7687-0
Springer New York Heidelberg Dordrecht London
Library of Congress Control Number: 2013944682
Mathematics Subject Classification (2010): 60-Hxx, 60H30, 62P10, 65Cxx, 82C3, 92C05, 92C37,
92C40, 35-XX, 35-B25, 35Q92
© Author 2013

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Preface
Brownian dynamics serve as mathematical models for the diffusive motion of microscopic particles of various shapes in gaseous, liquid, or solid environments. The
renewed interest in Brownian dynamics is due primarily to their key role in molecular and cellular biophysics: diffusion of ions and molecules is the driver of all
life. Brownian dynamics simulations are the numerical realizations of stochastic
differential equations (SDEs) that model the functions of biological microdevices
such as protein ionic channels of biological membranes, cardiac myocytes, neuronal
synapses, and many more. SDEs are ubiquitous models in computational physics,
chemistry, biophysics, computer science, communications theory, mathematical finance theory, and many other disciplines. Brownian dynamics simulations of the
random motion of particles, be it molecules or stock prices, give rise to mathematical problems that neither the kinetic theory of Maxwell and Boltzmann nor
Einstein’s and Langevin’s theories of Brownian motion could predict.

Kinetic theory, which assigns probabilities to configurations of ensembles of
particles in phase space, assumes that the ensembles are in thermodynamic equilibrium, which means that no net current is flowing through the system. Thus it is
not applicable to the description of nonequilibrium situations such as conduction
of ions through protein channels, nervous signaling, calcium dynamics in cardiac
myocytes, the process of viral infection, and countless other situations in molecular
biophysics.
The motion of individual particles in the ensemble is not described in sufficient
detail to permit computer simulations of the atomic or molecular individual motions in a way that reproduces all macroscopic phenomena. The Einstein statistical
characterization of the motion of a heavy particle undergoing collisions with the
much smaller particles of the surrounding medium lays the foundation for computer
simulations of the Brownian motion. However, pushing Einstein’s description beyond its range of validity leads to artifacts that baffle the simulators: particles move
without velocity, so there is no telling when they enter or leave a given domain.
Theoretically, they cross and recross interfaces an infinite number of times in any
finite time interval. Thus the simulation of Brownian particles in a small domain
surrounded by a continuum becomes problematic. The Langevin description, which
includes velocity, partially remedies the problem. There is, however, a price to pay:
the dimension, and therefore the computational complexity, is doubled.
v


vi

Preface

Computer simulations of diffusion with reflection or partial reflection at the
boundary of a domain, such as at the cellular membrane, are unexpectedly complicated. Both the discrete reflection and partial reflection laws of the simulated
trajectories are not very intuitive in their peculiar dependence on the geometry of
the boundary and on the local anisotropy of the diffusion tensor. The latter is the
hallmark of the diffusion of shaped objects. A case in point is the diffusion of a
stiff rod, whose diffusion tensor is clearly anisotropic (see Sect. 7.7). It is not a

priori clear what should be the reflection law of the rod when one of its ends hits
the impermeable boundary of the confining domain. This issue has been a thorn in
the side of simulators for a long time, which may be explained by the unexpected
mathematical complexity of the problem. It is resolved in Sects. 2.5 and 2.6.
The behavior of random trajectories near boundaries of the simulation imposes a
variety of boundary conditions on the probability density of the random trajectories
and its functionals. The quite intricate connection between the boundary behavior
of random trajectories and the boundary conditions for the partial differential equations is treated here with special care. The analysis of the mathematical issues that
arise in Brownian dynamics simulations relies on Wiener’s discrete path integral
representation of the transition probability density of the random trajectories that
are created by the discrete simulation. As the simulation is refined, the Wiener integral representation leads to initial and boundary value problems for partial differential equations of elliptic and parabolic types that describe important probabilistic
quantities. These include probability density functions (pdfs), mean first passage
times, density of the mean time spent at a point, survival probability, probability
flux density, and so on. Green’s function and its functionals play a central role in
expressing these quantities analytically and in determining their interrelationships.
The analysis provides the means for determining the relationship between the time
step in a simulation and the boundary concentrations.
Key mathematical problems in running Brownian or Langevin simulations include the following questions: What is the “correct” boundary behavior of the random trajectories? What is the effect of their boundary behavior on statistics, e.g.,
on the pdf? What boundary behavior should be chosen to produce a given boundary
behavior of the pdf? How can the higher-dimensional Langevin dynamics be adequately approximated by coarser Brownian dynamics? How should one choose the
time step in a simulation? Another curse of computer simulations of random motion
is the ubiquitous phenomenon of rare events. It is particularly acute in molecular
biophysics, where the simulated particles have to hit small targets or to squeeze
through narrow passages. This is the case, for example, in simulating ionic flux
through protein channels of biological membranes. Finding a small target is an important problem in Brownian dynamics simulations. Can the computational effort
be reduced by providing analytical information about the process? While numerical
analysis gives error estimates for given simulation schemes on finite time intervals,
simulations are often required to produce estimates of unlimited random quantities
such as first passage times or their moments. Thus we need to know how much
computational effort is needed for an estimate of the random escape time from an

attractor or a confining domain.


Preface

vii

In this book, we address these and additional mathematical problems of computer simulation of Itô-type SDEs. The book is not concerned with numerical
analysis, that is, with the design of simulation schemes and the analysis of their
convergence, but rather with the more fundamental questions mentioned above. The
analysis presented in this book not only is applicable to the Euler scheme, but can
also be applied to many other simulation schemes. While the singular perturbation methods for the analysis of rare events that are due to small noise relative
to large drift were thoroughly discussed in Schuss (2010b, 2011), the analysis of
rare events due to the geometry of the confining domain requires new mathematical methods. The “narrow escape problem” in diffusion theory, which goes back to
Lord Rayleigh, is to calculate the mean first passage time of a diffusion process to a
small absorbing target on an otherwise reflecting boundary of a bounded domain. It
includes also the problem of diffusing from one compartment to another through a
narrow passage, a situation that is often encountered in molecular and cellular biophysics and frustrates numerical simulations. The new mathematical methods for
resolving this problem are presented here in great analytical detail.
The exposition in this book is kept at an intermediate level of mathematical rigor.
Experience shows that mathematical rigor and applications can hardly coexist in
the same course; excessive rigor leaves no room for in-depth development of analytical methods and tends to turn off students interested in scientific applications.
Therefore, the book contains only the minimal mathematical rigor required for understanding the mathematical concepts and for enabling the students to use their
own judgment of what is correct and what requires further theoretical study. All
topics require a basic knowledge of SDEs and of asymptotic methods in the theory
of partial differential equations, as presented, for example, in Schuss (2010b). The
introductory review of stochastic processes in Chap. 1 should not be mistaken for
an expository text on the subject. Its role it to establish terminology and to serve
as a refresher on SDEs. The role of the exercises is give the reader an opportunity to examine his/her mastery of the subject. Other texts on stochastic dynamics
include, among other titles, (Arnold 1998; Friedman 2007; Gihman and Skorohod

1972; McKean 1969; Øksendal 1998; Protter 1992). Texts on numerical analysis
of stochastic differential equations include (Allen and Tildesley 1991; Kloeden and
Platen 1992; Milstein 1995; Risken 1996; Robert and Casella 1999; Doucet et al.
2001; Kloeden 2002; Milstein and Tretyakov 2004; Honerkamp 1994). A solid
training in partial differential equations of mathematical physics and in the asymptotic methods of applied mathematics can be derived from the study of classical texts
such as (Zauderer 1989; O’Malley 1974; Kevorkian and Cole 1985) or (Bender and
Orszag 1978). Many of the applications and examples in this book concern molecular and cellular biophysics, especially in the context of neurophysiology. Basic
facts on these subjects should not be acquired from mathematicians or physicists,
but rather from professional elementary texts on the subjects, such as (Alberts et al.
1994; Hille 2001; Koch 1999; Koch and Segev 2001; Sheng et al. 2012; Cowan et
al. 2003; Yuste 2010; Baylog 2009). Wikipedia should be consulted for clarifying
biochemical and physiological terminology.


viii

Preface

This book is aimed at applied mathematicians, physicists, theoretical chemists,
and physiologists who are interested in modeling, analysis, and simulation of microdevices of microbiology. A special topics course from this book requires good
preparation in the theory of SDEs, such as can be found in Schuss (2010b).
Alternatively, some of the topics discussed in this book can be interspersed between
the topics of a more general course as applications and illustrations of the general
theory.
The book contains exercises and worked-out examples. Hands-on training in
stochastic processes, as my long teaching experience shows, consists in solving the
exercises, without which understanding is only illusory.
Acknowledgments Much of the material presented in this book is based on my
collaboration with D. Holcman, A. Singer, B. Nadler, R.S. Eisenberg, and many
other scientists and students, whose names are listed next to mine in the author

index.
Tel Aviv, Israel

Zeev Schuss


List of Figures
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9

Reflected trajectories . . . . . . . . . . . . . . . . . . . . . .
Oblique and normal reflections . . . . . . . . . . . . . . . . .
Marginal density of x(T ) with oblique reflection . . . . . . .
Marginal density of y(T ) with oblique reflection . . . . . . .
Numerical solution the FPE with oblique reflection . . . . . .
Another numerical solution of the FPE with oblique reflection
The reflection law of Xt in Ω . . . . . . . . . . . . . . . . . .
Marginal density of x(T ) with normal and oblique reflections .
Marginal density of y(T ) with normal and oblique reflections .

3.1
3.2
3.3

Typical baths separated by membrane with channel . . . . . . .
Simulation in (0, 1) with normal initial distribution . . . . . . .
Simulation in (0, 1) with initial residual of the normal
distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Concentration profiles with time-step-independent injection rate
Concentration profile with time-step-dependent injection rate . .
Concentration vs. displacement of a Langevin dynamics
simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.4
3.5
3.6

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61
69
78
79
79
80
81
82
83

. . 96
. . 102
. . 102
. . 103
. . 103
. . 106

4.1

The domain D and its complement in the sphere DR . . . . . . . . 130

5.1
5.2
5.3

Variance of fluctuations in the fraction of bound sites . . . . . . . . 143
Schematic drawing of a synapse between two neurons . . . . . . . . 145
Model of a dendritic spine . . . . . . . . . . . . . . . . . . . . . . 146

6.1
6.2
6.3
6.4

Double-well potential surface . . . . . . . . .
Contours and trajectories . . . . . . . . . . .
A potential well with a single metastable state
Dumbbell-shaped domain . . . . . . . . . . .

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166
167
167
168

7.1
7.2

7.3
7.4

Escaping Brownian trajectory . . . . . . . . . .
Composite domains . . . . . . . . . . . . . . .
Receptor movement on the neuronal membrane
An idealized model of the synaptic cleft . . . .

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200
200
201
201
ix


x

List of Figures
7.5
7.6
7.7
7.8
7.9
7.10
7.11
7.12
7.13
7.14
7.15
7.16
7.17
7.18
7.19
7.20

Leak trajectory . . . . . . . . . . . . . . . . .
Escape through a funnel . . . . . . . . . . . . .
Funnel formed by a partial block . . . . . . . .
A small opening near a corner of angle α . . .
Narrow escape from an annulus . . . . . . . . .
Escape near a cusp . . . . . . . . . . . . . . .
Escape to the north pole . . . . . . . . . . . . .
A surface of revolution with a funnel . . . . . .
Conformal image of a funnel . . . . . . . . . .
Drift of a projected Brownian motion . . . . . .
Narrow straits formed by a cone-shaped funnel
Rod in strip . . . . . . . . . . . . . . . . . . .
Conformal image of a rod in a strip . . . . . . .
Boundary layers . . . . . . . . . . . . . . . . .
NET from a domain . . . . . . . . . . . . . . .
Organization of the neuronal membrane . . . .

8.1

Probability to exit through a single pump on the neck
membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
Exit probability in a synaptic cleft . . . . . . . . . . . . . . . . . . 280
Exit probability in a synaptic cleft, 20 AMPAR channels
in PSD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

8.2
8.3

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212
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226
227
235
240
241
244
246


List of Symbols
We use interchangeably · and E(·) to denote expectation (average) of a random
variable, but E(· | ·) and Pr{· | ·} to denote conditional expectation and conditional
probability, respectively.

A, B . . .

Matrices—bold uppercase italic letters

AT

The transpose of A

A−1

The inverse of A

δ(x)

Dirac’s delta function (functional)

det(A)

The determinant of the matrix A

E(x), x

The expected value (expectation) of x

Δ, Δx

Greek uppercase delta, the Laplace operator (with respect to x):
∂2
∂2
∂2
+ 2 + ···+ 2
2

∂x1
∂x2
∂xd

e(t)

ˆ (t) − x(t)
The estimation error process: x

F

The sample space of Brownian events

J[x(·)]

Functional of the trajectory x(·)

L2 [a, b]

Square integrable functions on the interval [a, b]

Mn,m

Space of n × m real matrices

m1 ∧ m2

The minimum min{m1 , m2 }
xi

xii
n ∼ N (μ, σ 2 )

List of Symbols
The random variable n is normally distributed with mean
μ and variance σ 2

N ∼ N (μ, Σ)

The random vector N is normally distributed with mean
μ and covariance matrix Σ

∇, ∇x

Greek nabla, the gradient operator (with respect to x):
∂
∂
∂
,
,...,
∂x1 ∂x2
∂xd

∇·J

T

The divergence operator
∂J1 (x) ∂J2 (x)

∂Jd (x)
+
+ ···+
∂x1
∂x2
∂xd

Pr {event}

The probability of event

pX (x)

The probability density function of the vector X

Q

The rational numbers

R, Rd

The real line, the d-dimensional Euclidean space

Vx

The partial derivative of V with respect to x : ∂V /∂x

tr(A)

Trace of the matrix A

Var(x)

The variance of x

w(t), v(t)

Vectors of independent Brownian motions

x, f (x)

Scalars—lowercase letters

x, f (x)

Vectors—bold lowercase letters

xi

The ith element of the vector x

x(·)

Trajectory or function in function space


List of Symbols

xiii

|x|2 = xT x

L2 norm of x

x·y

Dot (scalar) product of the vectors x and y:
x · y = x1 y1 + x2 + y2 + · · · + xd yd

˙
x(t)

Time derivative: dx(t)/dt



List of Acronyms
BKE

Backward Kolmogorov equation

CKE

Chapman–Kolmogorov equation

epdf

Equilibrium probability density function

FPE

Fokker–Planck equation

FPT

First passage time

i.i.d.

Independent identically distributed

i.o.

Infinitely often

ODE

Ordinary differential equation

OU

Ornstein–Uhlenbeck process

PAV

Pontryagin–Andronov–Vitt

pdf

Probability density function

PDE

Partial differential equation

PDF

Probability distribution function

SDE

Stochastic differential equation

TSR

Transition state region

TST

Transition state theory

(G)TS

Generalized transition state

(G)TST

Generalized transition state theory
xv

Contents
1

Mathematical Brownian Motion
1.1 Definition of Mathematical Brownian Motion . . . . . . . . . .
1.1.1 Mathematical Brownian Motion in Rd . . . . . . . . . .
1.1.2 Construction of Mathematical Brownian Motions . . .
1.1.3 Analytical and Statistical Properties of Brownian Paths .
1.2 Integration with Respect to MBM. The Itô Integral . . . . . . .
1.2.1 Stochastic Differentials . . . . . . . . . . . . . . . . . .
1.2.2 The Chain Rule and Itô’s Formula . . . . . . . . . . . .
1.3 Stochastic Differential Equations . . . . . . . . . . . . . . . . .
1.3.1 The Langevin Equation . . . . . . . . . . . . . . . . . .
1.3.2 Itô Stochastic Differential Equations . . . . . . . . . . .
1.3.3 SDEs of Itô Type . . . . . . . . . . . . . . . . . . . . .
1.3.4 Diffusion Processes . . . . . . . . . . . . . . . . . . . .
1.4 SDEs and PDEs . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.1 The Kolmogorov Representation . . . . . . . . . . . . .
1.4.2 The Feynman–Kac Representation and Terminating
Trajectories . . . . . . . . . . . . . . . . . . . . . . . .
1.4.3 The Pontryagin–Andronov–Vitt Equation for the MFPT
1.4.4 The Exit Distribution . . . . . . . . . . . . . . . . . . .
1.4.5 The PDF of the FPT . . . . . . . . . . . . . . . . . . .
1.5 The Fokker–Planck Equation . . . . . . . . . . . . . . . . . . .
1.5.1 The Backward Kolmogorov Equation . . . . . . . . . .
1.5.2 The Survival Probability and the PDF of the FPT . . . .

2 Euler’s Scheme and Wiener’s Measure

2.1 Euler’s Scheme for Itô SDEs and Its Convergence . . . . . .
2.2 The pdf of Euler’s Scheme in R and the FPE . . . . . . . . .
2.2.1 Euler’s Scheme in Rd . . . . . . . . . . . . . . . . .
2.2.2 The Convergence of the pdf in Euler’s Scheme in Rd
2.2.3 Unidirectional and Net Probability Flux . . . . . . .
2.3 Brownian Dynamics at Boundaries . . . . . . . . . . . . . .

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35
35
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45
xvii


xviii
2.4
2.5
2.6

2.7
2.8

Contents
Absorbing Boundaries . . . . . . . . . . . . . . . . . . . . .
2.4.1 Unidirectional Flux and the Survival Probability . . .
Reflecting and Partially Reflecting Boundaries . . . . . . . . .
2.5.1 Reflection and Partial Reflection in One Dimension . .
Partially Reflected Diffusion in Rd . . . . . . . . . . . . . . .
2.6.1 Partial Reflection in a Half-Space: Constant Diffusion
Matrix . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.2 State-Dependent Diffusion and Partial Oblique
Reflection . . . . . . . . . . . . . . . . . . . . . . . .
2.6.3 Curved Boundary . . . . . . . . . . . . . . . . . . . .
Boundary Conditions for the Backward Equation . . . . . . .
Discussion and Annotations . . . . . . . . . . . . . . . . . .

3 Brownian Simulation of Langevin’s
3.1 Diffusion Limit of Physical Brownian Motion . . . . . . . . .
3.1.1 The Overdamped Langevin Equation . . . . . . . . .
3.1.2 Diffusion Approximation to the Fokker–Planck
Equation . . . . . . . . . . . . . . . . . . . . . . . .
3.1.3 The Unidirectional Current in the Smoluchowski
Equation . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Trajectories Between Fixed Concentrations . . . . . . . . . .
3.2.1 Trajectories, Fluxes, and Boundary Concentrations . .
3.3 Connecting a Simulation to the Continuum . . . . . . . . . . .
3.3.1 The Interface Between Simulation and the Continuum
3.3.2 Brownian Dynamics Simulations . . . . . . . . . . .
3.3.3 Application to Channel Simulation . . . . . . . . . . .

3.4 Annotation . . . . . . . . . . . . . . . . . . . . . . . . . . .

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60

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67
75
82
85

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89
90
90

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92

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4 The First Passage Time to a Boundary

4.1 The FPT and Escape from a Domain . . . . . . . . . . . . . . .
4.2 The PDF of the FPT and the Density of the Mean Time Spent
at a Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 The Exit Density and Probability Flux Density . . . . . . . . . .
4.4 Conditioning . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Conditioning on Trajectories that Reach A Before B . .
4.5 Application of the FPT to Diffusion Theory . . . . . . . . . . .
4.5.1 Stationary Absorption Flux in One Dimension . . . . .
4.5.2 The Probability Law of the First Arrival Time . . . . . .
4.5.3 The First Arrival Time for Steady-State Diffusion in R3
4.5.4 The Next Arrival Times . . . . . . . . . . . . . . . . .
4.5.5 The Exponential Decay of G(r, t) . . . . . . . . . . . .

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132
133


Contents

xix

5 Brownian Models of Chemical Reactions in Microdomains
5.1 A Stochastic Model of a Non-Arrhenius Reaction . . . . . . .
5.2 Calcium Dynamics in Dendritic Spines . . . . . . . . . . . . .
5.2.1 Dendritic Spines and Their Function . . . . . . . . . .
5.2.2 Modeling Dendritic Spine Dynamics . . . . . . . . . .
5.2.3 Biological Simplifications of the Model . . . . . . . .
5.2.4 A Simplified Physical Model of the Spine . . . . . . .
5.2.5 A Schematic Model of Spine Twitching . . . . . . . .
5.2.6 Final Model Simplifications . . . . . . . . . . . . . .
5.2.7 The Mathematical Model . . . . . . . . . . . . . . . .
5.2.8 Mathematical Simplifications . . . . . . . . . . . . .
5.2.9 The Langevin Equations . . . . . . . . . . . . . . . .
5.2.10 Reaction–Diffusion Model of Binding and Unbinding
5.2.11 Specification of the Hydrodynamic Flow . . . . . . .
5.2.12 Chemical Kinetics of Binding and Unbinding
Reactions . . . . . . . . . . . . . . . . . . . . . . . .
5.2.13 Simulation of Calcium Kinetics in Dendritic Spines . .
5.2.14 A Langevin (Brownian) Dynamics Simulation . . . .
5.2.15 An Estimate of a Decay Rate . . . . . . . . . . . . . .
5.2.16 Summary and Discussion . . . . . . . . . . . . . . . .
5.3 Annotations . . . . . . . . . . . . . . . . . . . . . . . . . . .
6 Interfacing at the Stochastic Separatrix
6.1 Transition State Theory of Thermal Activation . . .
6.1.1 The Diffusion Model of Activation . . . . .
6.1.2 The FPE and TST . . . . . . . . . . . . .
6.2 Reaction Rate and the Principal Eigenvalue . . . .
6.3 MFPT . . . . . . . . . . . . . . . . . . . . . . . .
6.3.1 The Rate κ abs (D), MFPT τ (D) , an
Eigenvalue λ1 (D) . . . . . . . . . . . . .

6.3.2 MFPT for Domains of Types I and II in Rd
6.4 Recrossing, Stochastic Separatrix, Eigenfunctions .
6.4.1 The Eigenvalue Problem . . . . . . . . . .
6.4.2 Can Recrossings Be Neglected? . . . . . .
6.5 Accounting for Recrossings and the MFPT . . . . .
6.5.1 The Transmission Coefficient kTR . . . . .
6.6 Summary and Discussion . . . . . . . . . . . . . .
6.6.1 Annotations . . . . . . . . . . . . . . . . .

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186
188
192
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194

7 Narrow Escape in R2
199
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
7.1.1 The NET Problem in Neuroscience . . . . . . . . . . . . . 199
7.1.2 NET, Eigenvalues, and Time-Scale Separation . . . . . . . . 203


xx

Contents
7.2
7.3
7.4

7.5

7.6

7.7

7.8
7.9

A Neumann–Dirichlet Boundary Value Problem . . . . . . .
7.2.1 The Neumann Function and an Integral Equation . .
The NET Problem in Two Dimensions . . . . . . . . . . . .
Brownian Motion in Dire Straits . . . . . . . . . . . . . . .
7.4.1 The MFPT to a Bottleneck . . . . . . . . . . . . . .
7.4.2 Exit from Several Bottlenecks . . . . . . . . . . . .
7.4.3 Diffusion and NET on a Surface of Revolution . . .
A Composite Domain with a Bottleneck . . . . . . . . . . .
7.5.1 The NET from Domains with Bottlenecks in R2
and R3 . . . . . . . . . . . . . . . . . . . . . . . .
The Principal Eigenvalue and Bottlenecks . . . . . . . . . .
7.6.1 Connecting Head and Neck . . . . . . . . . . . . .
7.6.2 The Principal Eigenvalue in Dumbbell-Shaped
Domains . . . . . . . . . . . . . . . . . . . . . . .
A Brownian Needle in Dire Straits . . . . . . . . . . . . . .
7.7.1 The Diffusion Law of a Brownian Needle in a Planar
Strip . . . . . . . . . . . . . . . . . . . . . . . . . .
7.7.2 The Turnaround Time τL→R . . . . . . . . . . . . .
Applications of the NET . . . . . . . . . . . . . . . . . . .
Annotations . . . . . . . . . . . . . . . . . . . . . . . . . .
7.9.1 Annotation to the NET Problem . . . . . . . . . . .

8 Narrow Escape in R3
8.1 The Neumann Function in Regular Domains in R3 . . . .
8.1.1 Elliptic Absorbing Window . . . . . . . . . . . .
8.1.2 Second-Order Asymptotics for a Circular Window
8.1.3 Leakage in a Conductor of Brownian Particles . .
8.2 Activation Through a Narrow Opening . . . . . . . . . . .
8.2.1 The Neumann Function . . . . . . . . . . . . . . .
8.2.2 Narrow Escape . . . . . . . . . . . . . . . . . . .
8.2.3 Deep Well: A Markov Chain Model . . . . . . . .
8.3 The NET in a Solid Funnel-Shaped Domain . . . . . . . .
8.4 Selected Applications in Molecular Biophysics . . . . . .
8.4.1 Leakage from a Cylinder . . . . . . . . . . . . . .
8.4.2 Applications of the NET . . . . . . . . . . . . . .
8.5 Annotations . . . . . . . . . . . . . . . . . . . . . . . . .

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281

Bibliography

285

Index

307


Chapter 1

Mathematical Brownian
Motion
1.1 Definition of Mathematical Brownian Motion
The basic concepts in the axiomatic definition of the one-dimensional Brownian
motion as a mathematical object are a space of events Ω, whose elementary events
are real-valued continuous functions ω = ω(·) on the positive axis R+ . The construction of the set of events F in Ω and of Wiener’s probability measure Pr{A} for
A ∈ F is given in Schuss (2010b). A continuous stochastic process is a function
w(t, ω) : R+ × Ω → R such that for all ω ∈ Ω, the function w(t, ω) is a continuous
function of t and for all x ∈ R and t ∈ R+ , the set {ω ∈ Ω : w(t, ω) ≤ x} is an
event in F . Mathematical Brownian motion (MBM), often referred to as the Wiener
process, is defined as follows.
Definition 1.1.1 (The MBM). A real-valued stochastic process w(t, ω) defined on
R+ ×Ω is an MBM if (1) w(0, ω) = 0 with probability 1, (2) w(t, ω) is a continuous
function of t for almost all ω ∈ Ω, and (3) For every t, s ≥ 0, the increment
Δw(s, ω) = w(t + s, ω) − w(t, ω) is independent of w(τ, ω) for all τ ≤ t, and is a
zero mean Gaussian random variable with variance
2

E |Δw(s, ω)| = s.

(1.1)

The first part of property (3) (independence of increments of the MBM) means that
the conditional probability Pr{ω ∈ Ω : Δw(s, ω) ≤ x | ω ∈ Ω : w(τ, ω) ≤ y} is
independent of the condition, that is,
Pr{ω ∈ Ω : Δw(s, ω) ≤ x | ω ∈ Ω : w(τ, ω) ≤ y}
= Pr{ω ∈ Ω : Δw(s, ω) ≤ x}.

Z. Schuss, Brownian Dynamics at Boundaries and Interfaces: In Physics, Chemistry,
and Biology, Applied Mathematical Sciences 186, DOI 10.1007/978-1-4614-7687-0__1,
© Author 2013

1


2

Chapter 1. Mathematical Brownian Motion

The second part of property (3) means that the probability distribution function
(PDF) of an MBM is
Fw (x, t) = Pr{ω ∈ Ω : w(t, ω) ≤ x | w(0, ω) = 0}
x

1
=√
2πt

e−y

2

/2t

dy

(1.2)

∂
1 −x2 /2t
Fw (x, t) = √
e
.
∂x
2πt

(1.3)

−∞

and the probability density function (pdf) is
fw (x, t) =

It is well known (and easily verified) that fw (x, t) is the solution of the initial value
problem for the diffusion equation
∂fw (x, t)
1 ∂ 2 fw (x, t)
=
,

∂t
2
∂x2

lim fw (x, t) = δ(x).
t↓0

(1.4)

It can be shown that a stochastic process satisfying these axioms actually exists
(Schuss 2010b). Some of the properties of MBM follow from the axioms in a
straightforward manner. For example, note that (1) and (2) are not contradictory,
despite the fact that not all continuous functions vanish at time t = 0. Property
(1) asserts that all trajectories of the Brownian motion that do not start at the origin
are assigned probability 0. In view of the above, the Brownian paths are those continuous functions that take the value 0 at time 0. That is, the Brownian paths are
conditioned on starting at time t = 0 at the point x0 = w(0, ω) = 0. To emphasize
this point, we modify the notation of the Wiener probability measure (1.2) to Pr0 {·}
Wiener (1923).
If (1.2) is replaced by
Fw (x, t) = Pr{ω ∈ Ω : w(t, ω) ≤ x | w(0, ω) = x0 }
x

1
=√
2πt

e−(y−x0 )

2

/2t

dy,

(1.5)

−∞

the initial condition is replaced with w(0, ω) = x0 with probability 1 and then
Prx0 w(0, ω) = x0 = 1 under the modified Wiener probability measure, now
denoted by Prx0 {·} (Schuss 2010b).
Thus conditioning reassigns probabilities to the Brownian paths. The set of
trajectories {ω ∈ Ω : w(0, ω) = x0 }, which was assigned the probability 0 under
the measure Pr0 {·}, is now assigned the probability 1 under the measure Prx0 {·}.
Similarly, replacing the condition t0 = 0 with t0 = s and conditioning on w(s, ω) =
x0 in (1.5) shifts the Wiener probability measure, now denoted by Prx0 ,s , so that
Prx0 ,s {ω ∈ Ω : w(t, ω) ∈ [a, b]} = Pr0 {ω ∈ Ω : w(t − s, ω) ∈ [a − x0 , b − x0 ]}.


1.1. Definition of Mathematical Brownian Motion

3

This means that for all positive t, the increments of the MBM Δw(s, ω) = w(t +
s, ω)− w(t, ω), as functions of s, are MBMs, so that the probabilities of any Brownian event of Δw(s, ω) are independent of t, that is, the increments of the MBM are
stationary. Accordingly, the moments of the MBM are
∞

Ew(t, ω) =
−∞

x −x2 /2t
√
dx = 0
e
2πt

1
Ew (t, ω) = √
2πt

∞

2

x2 e−x

2

/2t

dx = t.

(1.6)

−∞

We recall that the autocorrelation function of a stochastic process x(t, ω) is defined
as the expectation Rx (t, s) = Ex(t, ω)x(s, ω).
Exercise 1.1 (Property 4). Using the notation t ∧ s = min{t, s}, prove that the

autocorrelation function of the MBM w(t, ω) is
Ew(t, ω)w(s, ω) = t ∧ s.

(1.7)
✷

1.1.1 Mathematical Brownian Motion in Rd
The set of events F in the product space
d

Ω=

Ωj ,
j=1

where Ωj are probability spaces for one-dimensional MBMs, is endowed with the
product probability measure. The elementary events in Ω are all Rd -valued continuous functions of t ∈ R+ . That is, ω(t) ∈ Ω means that
⎞
⎛
ω1 (t)
⎜ ω2 (t) ⎟
⎟
⎜
ω(t) = ⎜ . ⎟ ,
⎝ .. ⎠
ωd (t)
where ωj (t) ∈ Ωj , j = 1, . . . d. If w1 (t, ω1 ), w2 , (t, ω2 ), . . . , wd (t, ωd ) are independent Brownian motions, the vector process
⎞
⎛
w1 (t, ω1 )

⎜ w2 (t, ω2 ) ⎟
⎟
⎜
w(t, ω) = ⎜
⎟
..
⎠
⎝
.
wd (t, ωd )
is defined as the d-dimensional Brownian motion w(t, ω).


4

Chapter 1. Mathematical Brownian Motion

Consider the so called “cylinder” event for times 0 ≤ t1 < t2 < · · · < tk and
open sets I j in Rd , j = 1, 2, . . . , k,
C t1 , . . . , tk ; I 1 , . . . , I k = ω(t) ∈ Ω : ω(tj ) ∈ I j , j = 1, . . . , k .

(1.8)

The open sets I j can be, for example, open boxes or balls in Rd (see Schuss 2010b,
Sect. 2.2).
Definition 1.1.2 (The Wiener probability measure for a d-dimensional MBM).
The d-dimensional Wiener probability measure of a cylinder is defined as
Pr C t1 , . . . , tk ; I 1 , . . . , I k
k

···

=
I1

Ik

j=1

|xj − xj−1 |2
dxj
exp −
d/2
2(tj − tj−1 )
[2π(tj − tj−1 )]

.

(1.9)

The PDF of the d-dimensional MBM is
Fw (x, t) = Pr{ω ∈ Ω : w(t, ω) ≤ x | w(0, ω) = 0}
1
=
(2πt)n/2

x1

xd

···
−∞

e−|y|

2

/2t

dy1 · · · dyd ,

(1.10)

−∞

and the pdf is
fw (x, t) =

2
∂ d Fw (x, t)
e−|x| /2t
=
.
∂x1 ∂x2 · · · ∂xd
(2πt)d/2

(1.11)

Equation (1.4) implies that fw (x, t) satisfies the d-dimensional diffusion equation
and the initial condition

1
∂fw (x, t)
= Δfw (x, t),
∂t
2

lim fw (x, t) = δ(x).
t↓0

(1.12)

It can be seen from (1.9) that every rotation of the d-dimensional Brownian motion
is a d-dimensional Brownian motion. Higher-dimensional stochastic processes are
defined as vector-valued processes.
Definition 1.1.3 (Vector-valued processes). A vector-valued function x(t, ω) :
R+ ×Ω→ Rd is called a stochastic process in (Ω, F ) with continuous trajectories if (i) x(t, ω) is a continuous function of t for every ω ∈ Ω, and (ii) for every
t ≥ 0 and x ∈ Rd , the sets ω ∈ Ω : x(t, ω) ≤ x are Brownian events, that is,
ω ∈ Ω : x(t, ω) ≤ x ∈ F .
The PDF of x(t, ω) is defined as
Fx (y, t) = Pr{ω ∈ Ω : x(t, ω) ≤ y},

(1.13)