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 that model the functions of biological micro devices such as protein ionic channels of biological membranes, cardiac myocytes, neuronal synapses, and many more. Stochastic differential equations 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. This book takes the readers on a journey that starts with the rigorous definition of mathematical Brownian motion, and ends with the explicit solution of a series of complex problems that have immediate applications. It is aimed at applied mathematicians, physicists, theoretical chemists, and physiologists who are interested in modeling, analysis, and simulation of micro devices of microbiology. The book contains exercises and worked out examples throughout.
Author(s): Zeev Schuss
Series: Applied Mathematical Sciences
Publisher: Springer Science & Business Media
Year: 2013
Language: English
Pages: 340
Preface
List of Figures
List of Symbols
List of Acronyms
Contents
Chapter
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 TerminatingTrajectories
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
Chapter
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
2.4 Absorbing Boundaries
2.4.1 Unidirectional Flux and the Survival Probability
2.5 Reflecting and Partially Reflecting Boundaries
2.5.1 Reflection and Partial Reflection in One Dimension
2.6 Partially Reflected Diffusion in Rd
2.6.1 Partial Reflection in a Half-Space: Constant DiffusionMatrix
2.6.2 State-Dependent Diffusion and Partial ObliqueReflection
2.6.3 Curved Boundary
2.7 Boundary Conditions for the Backward Equation
2.8 Discussion and Annotations
Chapter
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 SmoluchowskiEquation
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
Chapter
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 Spentat 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)
Chapter
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 UnbindingReactions
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
Chapter
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 abs(D), MFPT "426830A (D)"526930B , anEigenvalue 1(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
Chapter
7 Narrow Escape in R2
7.1 Introduction
7.1.1 The NET Problem in Neuroscience
7.1.2 NET, Eigenvalues, and Time-Scale Separation
7.2 A Neumann–Dirichlet Boundary Value Problem
7.2.1 The Neumann Function and an Integral Equation
7.3 The NET Problem in Two Dimensions
7.4 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
7.5 A Composite Domain with a Bottleneck
7.5.1 The NET from Domains with Bottlenecks in R2and R3
7.6 The Principal Eigenvalue and Bottlenecks
7.6.1 Connecting Head and Neck
7.6.2 The Principal Eigenvalue in Dumbbell-ShapedDomains
7.7 A Brownian Needle in Dire Straits
7.7.1 The Diffusion Law of a Brownian Needle in a PlanarStrip
7.7.2 The Turnaround Time LR
7.8 Applications of the NET
7.9 Annotations
7.9.1 Annotation to the NET Problem
Chapter
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
Bibliography
Index
