布朗运动与边界接口：生物物理中的扩散动力学

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"《Brownian Dynamics at Boundaries and Interfaces》由Zeev Schuss撰写，是Applied Mathematical Sciences系列的一部分，探讨了在物理、化学和生物学中边界与界面上的布朗动力学。本书深入研究了随机微分方程（SDEs）在模拟生物微观设备功能中的应用，如生物膜上的蛋白质离子通道、心肌细胞、神经元突触等。" 在物理、化学和生物学的交叉领域，布朗动力学是一种关键的数学模型，用于描述微小颗粒在气体、液体或固体环境中的扩散运动。这种现象对于理解分子和细胞生物物理学至关重要，因为离子和分子的扩散是所有生命活动的基础驱动力。随着对这一领域的兴趣日益增长，布朗运动模拟已经成为研究基于随机微分方程的数值实现的常用工具。随机微分方程（SDEs）在书中被用来模拟生物系统中的复杂行为。例如，这些方程可以用来描述蛋白质离子通道如何允许特定离子在生物膜上通过，从而维持细胞的电生理特性。此外，它们还用于理解心肌细胞的电信号传递，这是心脏功能的基础。在神经科学中，布朗动力学帮助我们解析神经元间的突触传递，这是大脑信息处理的关键过程。本书不仅关注基本的理论概念，还涵盖了在实际计算中实施布朗动力学模拟的技术细节。作者Zeev Schuss是一位来自以色列特拉维夫大学数学科学学院的专家，他提供了丰富的数学背景和生物学应用的见解。书中可能涵盖了如何处理边界条件，这对于理解粒子在受限空间内的行为至关重要，比如在细胞内部或者在膜结构附近。数学分类包括60-Hxx（概率论基础）、60H30（随机微分方程）、62P10（统计应用）、65Cxx（数值随机过程）、82C3（相变和临界现象）、92C05（一般生物数学）、92C37（细胞生物学）、92C40（生理学的数学问题）以及涉及偏微分方程的多个分类，强调了布朗动力学与多学科的紧密联系。通过深入的分析和实例，这本书不仅适合概率论、统计学、计算数学和生物物理的学者，也适合那些希望将这些理论应用于解决实际生物问题的研究人员和工程师。它提供了一个全面的平台，将理论研究与实际应用相结合，推动了对生命科学中微观运动机制的理解。

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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

xviii Contents

2.4 Absorbing Boundaries ........................ 46

2.4.1 Unidirectional Flux and the Survival Probability ...... 50

2.5 Reﬂecting and Partially Reﬂecting Boundaries ............ 52

2.5.1 Reﬂection and Partial Reﬂection in One Dimension ..... 53

2.6 Partially Reﬂected Diffusion in R

.................. 59

2.6.1 Partial Reﬂection in a Half-Space: Constant Diffusion

Matrix ............................. 60

2.6.2 State-Dependent Diffusion and Partial Oblique

Reﬂection ........................... 67

2.6.3 Curved Boundary ....................... 75

2.7 Boundary Conditions for the Backward Equation .......... 82

2.8 Discussion and Annotations ..................... 85

3 Brownian Simulation of Langevin’s 89

3.1 Diffusion Limit of Physical Brownian Motion ............ 90

3.1.1 The Overdamped Langevin Equation ............ 90

3.1.2 Diffusion Approximation to the Fokker–Planck

Equation ........................... 92

3.1.3 The Unidirectional Current in the Smoluchowski

Equation ........................... 93

3.2 Trajectories Between Fixed Concentrations ............. 94

3.2.1 Trajectories, Fluxes, and Boundary Concentrations ..... 96

3.3 Connecting a Simulation to the Continuum .............. 99

3.3.1 The Interface Between Simulation and the Continuum ...100

3.3.2 Brownian Dynamics Simulations ..............100

3.3.3 Application to Channel Simulation ..............104

3.4 Annotation ..............................107

4 The First Passage Time to a Boundary 111

4.1 The FPT and Escape from a Domain .................111

4.2 The PDF of the FPT and the Density of the Mean Time Spent

at a Point ...............................115

4.3 The Exit Density and Probability Flux Density ............119

4.4 Conditioning .............................120

4.4.1 Conditioning on Trajectories that Reach A Before B ....121

4.5 Application of the FPT to Diffusion Theory .............125

4.5.1 Stationary Absorption Flux in One Dimension .......125

4.5.2 The Probability Law of the First Arrival Time ........126

4.5.3 The First Arrival Time for Steady-State Diffusion in R

. . 129

4.5.4 The Next Arrival Times ...................132

4.5.5 The Exponential Decay of G(r,t) ..............133

Contents xix

5 Brownian Models of Chemical Reactions in Microdomains 135

5.1 A Stochastic Model of a Non-Arrhenius Reaction ..........137

5.2 Calcium Dynamics in Dendritic Spines ................144

5.2.1 Dendritic Spines and Their Function .............144

5.2.2 Modeling Dendritic Spine Dynamics .............148

5.2.3 Biological Simpliﬁcations of the Model ...........149

5.2.4 A Simpliﬁed Physical Model of the Spine ..........150

5.2.5 A Schematic Model of Spine Twitching ...........150

5.2.6 Final Model Simpliﬁcations .................151

5.2.7 The Mathematical Model ...................152

5.2.8 Mathematical Simpliﬁcations ................152

5.2.9 The Langevin Equations ...................152

5.2.10 Reaction–Diffusion Model of Binding and Unbinding ...154

5.2.11 Speciﬁcation of the Hydrodynamic Flow ..........155

5.2.12 Chemical Kinetics of Binding and Unbinding

Reactions ...........................157

5.2.13 Simulation of Calcium Kinetics in Dendritic Spines .....158

5.2.14 A Langevin (Brownian) Dynamics Simulation .......159

5.2.15 An Estimate of a Decay Rate .................159

5.2.16 Summary and Discussion ...................162

5.3 Annotations ..............................163

6 Interfacing at the Stochastic Separatrix 165

6.1 Transition State Theory of Thermal Activation ............168

6.1.1 The Diffusion Model of Activation ..............169

6.1.2 The FPE and TST ......................170

6.2 Reaction Rate and the Principal Eigenvalue .............172

6.3 MFPT .................................175

6.3.1 The Rate κ

abs

(D), MFPT τ(D),an

Eigenvalue λ

(D) ......................176

6.3.2 MFPT for Domains of Types I and II in R

.........177

6.4 Recrossing, Stochastic Separatrix, Eigenfunctions ..........179

6.4.1 The Eigenvalue Problem ...................182

6.4.2 Can Recrossings Be Neglected? ...............186

6.5 Accounting for Recrossings and the MFPT ..............188

6.5.1 The Transmission Coefﬁcient k

..............192

6.6 Summary and Discussion .......................193

6.6.1 Annotations ..........................194

7 Narrow Escape in R

199

7.1 Introduction ..............................199

7.1.1 The NET Problem in Neuroscience .............199

7.1.2 NET, Eigenvalues, and Time-Scale Separation ........203

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布朗运动与边界接口：生物物理中的扩散动力学

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