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 ssa@math.umd.edu P.J Holmes pholmes@math.princeton.edu K.R Sreenivasan katepalli.sreenivasan@nyu.edu Advisors L Greengard J Keener R.V Kohn B Matkowsky R Pego C Peskin A Singer A Stevens A Stuart For further volumes: http://www.springer.com/series/34 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 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer Permissions for use may be obtained through RightsLink at the Copyright Clearance Center Violations are liable to prosecution under the respective Copyright Law The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made The publisher makes no warranty, express or implied, with respect to the material contained herein Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) 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 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 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 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 200 200 201 201 ix 308 autonomous SDE, 29, 115 average rate, 118 Büttiker, M., 196, 288 Bénichou, O., 248, 282, 286 backward binding processes, 159 backward binding rate, 137, 158 backward binding reaction, 151 backward integral, 18 backward Kolmogorov equation, 32, 82, 83, 112 backward Kolmogorov operator, 13, 24–26, 28, 30, 84, 116, 121, 130, 171, 188, 225, 237 Bajaj, C.L., 86, 304 Baker, N.A., 86, 304, 305 Bally, V., 88, 285 Barenbrug, Th.M.A.O.M., 87, 301 Bartsch, T., 195, 285, 286 Barzykin, A.V., 248, 287 Bats, C., 248, 305 Batsilas, L., 86, 286 Baylog, L.R., vii, 286 Baylor, D.A., 136, 302 Beccaria, M., 87, 286 Belch, A.C., 107, 108, 286 Belopolskaya, Y.I., 248, 286 Ben-Jacob, E., 193, 194, 197, 286, 299 Bender, C.M., vii, 286 Benson, S.H., 194, 286 Berezhkovskii, A.M., 86, 194, 248, 282–284, 286, 287, 289, 293, 301 Bergman, D.J., 194, 286 Berkowitz, M., 107, 108, 286, 287 Berne, B.J., 151, 194, 287, 301 Berry, R.S., 107, 108, 195, 287, 296 Biess, A., 247, 287 bind to receptors, 145 Binder, K., 107, 294 binding calcium, 148–150 binding fluctuations, 136 binding kinetics, 137 binding molecules, 46, 136, 149 binding molecules to receptors, 136 Index binding of neurotransmitter to receptor, 199, 201 binding probability, 136 binding proteins, 151 binding rate, 158 binding reaction, 135, 136, 154 binding sites, 136, 139, 141, 151, 158 binding sites on substrate, 137 binding sites on the boundary, 142 binding to substrate, 154, 159 biorthogonal eigenfunction of the FPO, 264 bistable dynamics, 173, 177–179, 182, 186, 194 bistable Langevin dynamics, 182 bistable potential, 168, 171, 187, 288, 290 BKE (backward Kolmogorov equation), xv Blomberg, F., 145, 287 Bobrovsky, B.Z., 194, 196, 287 Boda, D., 247, 287 Boltyanskii, V.G., 283, 301 Bond, S.D., 86, 305 Bonhoeffer, T., 144–147, 287 Bordewijk, P., 126, 133, 287 Borgdorff, A.J., 245, 247, 248, 287, 288 Borkovec, M., 90, 158, 163, 234, 248, 283, 293 Bossy, M., 88, 287 bottleneck, 168, 192, 218, 219, 223, 224, 227, 230–232, 248, 276 bound proteins, 140 boundary concentration, vi, 94–96, 98–100, 104, 106, 109 boundary conditions for the BKE, 82, 84, 85 boundary flux, 66, 67, 98 boundary layer, 47–49, 54–57, 60–62, 66–68, 70–72, 74, 87, 88, 101, 104–106, 172, 180, 181, 183, 184, 189, 190, 193, 194, 205, 210, 217, 224, 229, 234, 240, 241, 248, 262, 274, 276, 281–283, 290 Index boundary reflection of a rod, vi, 85 boundary value problem, 26–29, 116, 119–122, 124, 143, 180, 205, 213, 221, 223–225, 228, 229, 236, 237, 250, 263, 264, 266, 272, 281–283 Bourne, J.N., 247, 287 Bray, D., vii, 247, 277, 285 Brinkhaus, H., 147, 291 Brinkman, H.C., 196, 287 Brooks, C.L., III, 107, 108, 287 Brown, E., 158, 162, 163, 247, 298 Brown, T.H., 144, 306 Brownian bridge, 125 Brownian dynamics, v, vi, 45, 86, 100, 101, 162, 280, 289, 292, 294, 297, 300, 301 Brownian dynamics simulations, v, vi, 85, 100, 202, 204, 242, 243, 299 Brownian motion, v, xii, 2–5, 7, 9, 12, 14, 22, 23, 31, 59, 90, 91, 94, 124, 125, 152, 159, 165, 166, 179, 199, 207, 212–215, 217–219, 224–226, 228, 230, 232, 233, 244, 284, 285, 287, 297, 304 Brownian motion in dire straits, 218, 294 Brownian needle in a planar strip, 235, 238 Brownian needle in dire straits, 234 Brownian paths, 2, 7, 289 Brownian scaling, 21 Brush, S.G., 107, 287 Bryan, G.H., 88, 288 Burbanks, A., 195, 306 Burger, M., 247, 288 Caillol, J.M., 107, 288 calcineurin binding, 151 calcineurin proteins, 149, 151, 152 calcium, 136, 144, 145, 148 calcium binding, 136, 152, 159 calcium channels, 146 309 calcium concentration, 147 calcium dynamics, v, 136, 144, 145, 147 calcium ions, 145–148 calcium pump, 149, 298 calcium unbinding, 136 calcium-saturated proteins, 149 calmodulin binding, 151 calmodulin calcium complex (CaMCa4 ), 148 calmodulin protein kinase (CaMK), 136, 148 calmodulin proteins (CaM), 136, 146, 148, 149, 151, 152, 158 Carslaw, H.S., 88, 288 Cartling, B., 172, 196, 288 Casella, G., vii, 302 center of curvature, 239 Chandrasekhar, S., 17, 100, 108, 133, 158, 196, 288 channel gating, 202, 247 channel gating noise, 135 channel simulation, 104, 106 Chapman, J., 47, 87, 291 chemical kinetics of binding, 157 Chen, D.P., 247, 288, 290 Cheviakov, A., 248, 258, 276, 288 Choquet, D., 245–248, 287, 288, 302, 305 Christiansen, J.A., 195, 196, 288 Chung, S.H., 107, 282, 289 Ciccotti, G., 107, 108, 289 CKE (Chapman–Kolmogorov equation), xv, 8, 9, 31, 32, 49, 55 Clifford, P., 87, 289 Coalson, R.D., 247, 293 Cognet, L., 248, 305 Cohen, B.J., 134, 289 Cohen, R.S., 145, 287 Cole, J.D., vii, 295 Collins, F.C., 85–88, 289 Collins, W.D., 281, 283, 289 colored noise, 15, 31 concentration fluctuations, 95, 96 conditional expectation, xi, 10, 24, 25 310 conditional moments of the displacement process, 16 conditioned diffusion, 121, 124 conditioned MBM, 124 conditioning, 2, 120, 121, 123, 270 conditioning on a future interval, 124 conformal mapping, 211–214, 218–220, 239, 240, 273 connecting discrete simulation to continuum, 35, 42, 45, 99, 100, 106 conormal reflection, 86, 235 conservation law, 42 convergence of Euler’s scheme, 36 convergence of Itô sums in probability, convergence of refinements in L2 , 7, convergence of simulation, vii convergence of simulation on a fixed Brownian trajectory, 37 convergence of simulation on infinite time intervals, 36 Coombs, D., 248, 289 Corry, B., 107, 282, 289 Costantini, C., 88, 289 Courant, R., 282, 289 Cowan, W.M., vii, 289 Crick, F., 146, 148, 163, 289 Curci, G., 87, 286 curvature, 276 curvature parameter, 218, 223 Curvin, M.S., 300 cyclic guanosine monophosphate (cGMP), 136, 301 cylinder event, 4, 124 Dagdug, L., 289 Dalecky, Y.L., 248, 286 dark noise, 136 Dauge, M.R., 282, 289 Day, M.V., 188, 196, 290 Dembo, A., 283, 290 dendrite, 144–147, 150, 162, 163, 199, 202, 281, 284, 297 dendritic shaft, 147, 150, 159, 201, 277 Index dendritic spine, 135, 144, 145, 147–152, 158, 159, 162, 199, 230, 281, 284, 287, 289, 291, 294, 296, 299, 300, 302, 303, 305 Denk, W., 144, 163, 247, 305, 306 density fluctuations, 107, 108 density of time spent at a point, 116, 267 diffusion approximation, 47 diffusion approximation to a Markovian jump process, 47, 54, 245, 296 diffusion approximation to the FPE, 92 diffusion between two baths, 95–97, 99, 100, 104–108, 125–127, 129, 168–170, 195, 196, 303 diffusion coefficient, 16 diffusion equation, 2, 4, 31, 100, 154, 160, 162, 163, 244, 305 diffusion matrix, 13, 22, 23, 76, 87, 123, 165 diffusion model of activation, 169 diffusion of a Brownian needle, 235, 236 diffusion of a stiff rod, vi, 85, 234–237, 272 diffusion of shaped objects, vi, 35, 85 diffusion on a membrane with obstacles, 242, 244–246, 248 diffusion process, vii, 22, 23, 45, 65, 85–87, 90, 121, 124, 163, 188, 195, 196, 283, 291, 302, 304 diffusion with obstacles, 150, 152, 200, 202, 234, 242, 245, 246 Dirichlet boundary condition, 49, 53, 86, 176, 231 Doering, C.R., 197, 290, 293 Doucet, A., vii, 290 dumbbell-shaped domain, 166, 168, 231, 232, 234, 248, 284 Dunaevsky, A., 147, 148, 290 Dygas (Kłosek), M.M., 194, 290 Dzougoutov, A., 88, 290 Index Edidin, M., 248, 290 effective diffusion coefficient, 99, 139, 171, 200, 234, 242, 244–246, 248, 294 efflux, 54 Ehlers, M.D., 247, 300 eigenfunction expansion, 262 eigenfunction expansion of the pdf, 174, 176, 177 eigenfunction of the reduced problem, 262 eigenfunctions of the FPO, 114, 171, 177, 179, 184, 302 eigenvalue and decay rate, 113 eigenvalue of the FPO, 174 eigenvalue of the reduced problem, 259 eigenvalue problem, 182, 186 eigenvalues, 233, 264 eigenvalues of a Markov chain, 231 eigenvalues of the FPO, 114, 171, 177 Einstein’s fluctuation–dissipation principle, 15 Einstein, A., v, 14, 15, 17, 89, 91, 100, 153, 290 Eisenberg, R.S., viii, 92, 96, 104–107, 247, 248, 282, 285, 287, 288, 290, 303–306 Eisinger, J., 248, 291 elliptic boundary value problem, 27, 28, 118 elliptic operator, 113, 283, 301 Engl, H.W., 247, 288 epdf Equilibrium probability density function, xv equilibrium, v, 170 equilibrium density fluctuations, 108 Erban, R., 47, 87, 291 escape from a bounded domain, 140, 142, 173 escape of MBM through a narrow window, 262, 263, 287, 293, 294 escape probability, 23 escape process, 188 escape rate, 175, 194, 196 311 escape time, vi, 123, 244, 245 Euler scheme, vii, 18, 35, 36, 39, 41, 46, 54, 60, 87, 100, 285, 292 Euler scheme and the Feynman–Kac formula, 38 Euler scheme as a Markovian jump process, 53, 105 Euler scheme for killed diffusions, 292 Euler scheme for Langevin dynamics, 105 Euler scheme for reflected diffusion, 287 Euler scheme for reflected SDEs, 298 Euler scheme in Rd , 39 Euler scheme reflection rule, 67 Euler scheme with a larger time step, 88 Euler scheme with absorbing boundary, 46, 84 Euler scheme with boundary behavior, 87 Euler scheme with conormal reflection, 77 Euler scheme with normal reflection, 81 Euler scheme with oblique reflection, 60 Euler scheme with partial oblique reflection, 68 Euler scheme with partial reflection, 60, 66, 77 Euler scheme with reflection, 75, 77 Euler scheme’s pdf, 37 Euler simulation with absorption and reflection, 67 Euler’s scheme convergence, 39 Euler’s scheme convergence at partially reflecting boundaries, 54 Euler’s scheme convergence for absorbing boundary, 49 Evans, M.G., 196, 291 exit density, 119 exit distribution, 27, 111, 303 exponentially distributed FPT, 132, 231, 233, 245 exponentially distributed interarrival times, 126, 129, 132, 233 312 Eyring, H., 167, 169, 187, 194–196, 291, 292 Fabrikant, V.I., 247, 281–283, 291 Feller, W., 34, 45, 86, 190, 291 Feynman, R.P., 23 Feynman–Kac representation formula, 23, 25, 38 Fichera, G., 121, 291 Fife, P., 27, 291 Fifkova, E., 147, 148, 151, 299 first arrival time, 126, 127, 129, 130, 132 first eigenfunction of the FPO, 114 first exit time, 20 Fischer, M., 147, 148, 291 Fleming, G.R., 192, 196, 291 Flores, J., 248, 291 fluctuations in number of bound particles, 142 fluctuations in number of bound sites, 143 fluctuations in number of open channels, 136 flux, 96, 107, 109, 113, 117 flux boundary condition, 96 flux density, 42, 92 flux density vector, 50, 52, 97, 98 flux in 1-D, 51, 93 flux in the FPE, 35 flux of eigenfunction of the FPO, 259 flux over population formula, 117 Fokker, A.D., xv, 30, 31, 37, 40, 42, 51, 52, 55, 57, 66, 84, 86, 87, 92, 97, 112–115, 117, 119, 130, 165, 169–173, 175–177, 194, 196, 236, 263, 288, 290, 299, 302 Fokker–Planck equation (FRE), 30, 31, 37, 40, 42, 51, 52, 55, 57, 66, 86, 87, 92, 112, 113, 117, 119, 130, 170, 172, 173, 175–177, 194, 196, 236, 263, 288, 290, 299, 302 Fokker–Planck equation for MBM, 31 Index Fokker–Planck equation for OU, 31 Fokker–Planck operator, 30, 84, 113, 114, 165, 171, 172, 176, 177, 299, 302 Ford, G.W., 195, 196, 291 forward binding rate, 137, 139, 142, 158 forward binding reaction, 157 FPT (first passage time), xv Frauenfelder, H., 192, 196, 291 free binding sites, 137, 138, 142, 154 free Brownian motion, 15–18, 91, 100, 105, 133, 204, 242 Freidlin, M., 283, 291 Freidlin, M.A., 156, 196, 197, 292 Freitas, N.D., vii, 290 Friedman, A., vii, 31, 84, 292 Friedman, R.A., 107, 292 Fujiwara, T., 248, 297 funnel, 200–202, 218, 225, 227, 230, 231, 233, 248, 272, 275, 276, 284, 294 gambler’s ruin paradox, 33 Gamkrelidze, R.V., 283, 301 Gandolfi, A., 247, 292 Garabedian, P.R., 208, 249, 282, 292 Gardiner, C.W., 43, 178, 292 gating molecules, 136 Gaussian white noise, 14, 170, 263 generalized Langevin equation (GLE), 91, 169, 170, 195, 306 Gerardi, A., 247, 292 Geyer, T., 108, 292 Ghoniem, A.F., 86, 292 Gihman, I.I., vii, 20, 27, 292 Gillespie, D., 247, 287 Giraudo, M.T., 88, 292 Glasstone, S., 167, 169, 187, 194, 195, 292 glutamate neurotransmitter, 147, 150 Glynn, P., 88, 285 Gobet, E., 81, 88, 287, 292 Goodrich, F.C., 86, 292 Gorba, C., 108, 292 Gordon, N., vii, 290 Index Grabert, H., 193, 196, 301 Graf, P., 247, 293 Graham, R., 178, 293 Grant, S.G.N., 277, 305 Green’s formula, 229 Green, N.J.B., 87, 289, 293 Grigoriev, I.V., 248, 282–284, 293 Hänggi, P., 90, 158, 163, 167, 195, 234, 248, 283, 293, 305 Hänngi, P., 193, 196, 301 Hagan, P.S., 197, 290, 293 Haken, H., 194, 293 Hamiltonian, 169 harmonic oscillator, 169 Harris, E.P., 196, 288 Harris, K.M., 247, 287, 293 Haugh, J.M., 86, 299 Haynes, L.W., 163, 293 Helmholtz integral equation, 205, 249, 253, 254, 256, 282 Helmholtz lemma, 253, 256, 257 Helmholtz, H.L.F von, 205, 249, 253, 256, 282, 283, 305 Helms, V., 108, 292 Henderson, D., 247, 287 Henshaw, W.D., 248, 281, 306 Hernandez, R., 195, 285, 286 Herrmann, A., 247, 294 Hida, T., 7, 293 higher-order asymptotics of the MFPT, 256, 258 Hilbert, D., 282, 289 Hille, B., vii, 106, 202, 214, 247, 293 Hille, E., 293 hippocampal dendritic spine, 287 hippocampal neuron, 297, 305 hippocampal spine, 298 Hoffman, B.M., 194, 296 Holcman, D., 85, 149, 151, 158, 159, 162, 163, 211, 213, 217, 234, 243, 247, 248, 281, 282, 284, 287, 293, 294, 297, 303, 304 Holst, M.J., 86, 305 Honerkamp, J., vii, 87, 294 313 Hoogenraad, C.C., 294 Horbach, J., 107, 294 Hotulainen, P., 294 Hoyle, M., 282, 289 Hoze, N., 234, 243, 248, 284, 294 Huang, Q., 247, 294 hydrodynamic drag, 151, 152 hydrodynamic effect on binding, 151 hydrodynamic flow, 152, 155, 157, 160, 162 hydrodynamic movement, 150 Im, W., 107, 108, 247, 282, 294 Imry, Y., 194, 286 increments of the MBM, 1, 3, 5, 9, 10, 17, 22, 36 independence of increments, infinite unidirectional flux, 94 initial condition, 2, 4, 5, 16, 19–22, 24, 25, 31, 33, 37, 38, 54, 55, 115, 116, 127, 130, 133, 134, 140, 143, 155, 157, 170, 171, 174–177, 216, 241, 263, 306 initial value problem, 2, 19, 37, 40, 46, 49, 86 instantaneous unidirectional flux , 51 interarrival times, 126, 132 interface between simulation and continuum, v, 42, 100, 101, 104, 106 ion exchanger, 137, 199, 201, 202 ion pump, 87, 137, 146, 150, 154, 159, 163, 201, 202, 277, 278 ionic flux, vi Itô differential, 12 Itô equation, vii, 18, 20, 21, 36, 53, 82, 124, 225, 236 Itô integral, 9, 11, 12, 18, 19, 22 Itô’s formula, 12–14, 20, 26, 30, 36, 67, 75, 78, 111 Itô, K., vii, 7, 9–15, 18–26, 29, 30, 35, 36, 46, 53, 67, 75, 76, 78, 82, 86, 111, 121, 124, 165, 195, 225, 235, 236, 295 314 Jackson, J.D., 249, 281–283, 295 Jacobson, K., 248, 302 Jaeger, J.C., 88, 288 Jansons, K.M., 88, 295 Jessell, T.M., 144, 150, 161, 247, 295 Jimbo, S., 284, 295 John, F., 248, 295 joint pdf, 17 Kłosek, M.M., 92, 184, 194, 197, 290, 296, 300 Kłosek-Dygas, M.M., 194, 296 Kac, M., 23, 195, 196, 291 Kaech, S., 147, 148, 291 Kandel, E.R., 144, 150, 161, 247, 295 Karatzas, I., 19, 23, 295 Karlin, S., 45, 101, 121, 125, 188, 295 Karplus, M., 107, 108, 287 Kasai, R.S., 248, 297 Katz, A., 194, 295 Kauer, J.A., 149, 282, 284, 298 Kay, A.R., 163, 293 Keck, J.C., 196, 295 Keller, J.B., 86, 248, 262, 281, 295, 306 Kellog, O.D., 253, 282, 295 Kelman, R.B., 283, 295 Kevorkian, J., vii, 295 killing rate, 25, 37 killing time, 25, 37 Kimball, G.E., 85–88, 289 King, G., 107, 108, 295 Kloeden, P.E., vii, 88, 295 Knapp, E.W., 247, 294 Knessl, C., 47, 296 Knutti, D., 147, 148, 291 Kob, W., 107, 294 Koch, C., vii, 144, 163, 296, 306 Kolmogorov’s representation formula, 23–25, 27, 28, 111 Kolmogorov, A.N., xv, 13, 23–28, 30, 32, 38, 39, 61, 82–84, 88, 111, 112, 116, 171, 174, 225, 283, 296 Kolokolnikov, T., 248, 281, 296 Komatsuzaki, T., 195, 296, 298 Index Kondo, J., 248, 297 Korenbrot, J.I., 163, 301 Korkotian, E., 145, 147–149, 151, 158, 159, 162, 163, 247, 282, 284, 287, 294, 297, 305 Kosugi, S., 284, 295 Kozlov, V.A., 282, 297 Kramers rate, 186, 193, 196, 197 Kramers’ formula, 158, 267, 269 Kramers’ method, 194–196 Kramers, H.A., 158, 163, 168, 173, 175, 186, 187, 193–197, 267, 269, 286, 293, 296, 297 Kreevoy, M.M., 192, 296 Kubo, R., 195, 196, 297 Kuo, S.C., 248, 290 Kupka, I., 247, 294 Kurnikova, M.G., 247, 293 Kushmaro, A., 247, 298 Kusumi, A., 248, 297 Kuyucak, S., 107, 282, 289 Lépingle, D., 88, 298 Ladd, A.J.C., 86, 305 Laidler, K.J., 167, 169, 187, 194, 195, 292 Lamm, G., 86–88, 297 Landau, L.D., 155, 297 Landauer, R., 196, 288, 297 Landauer,R., 196 Langer, J.S., 194, 196, 297 Langevin equation, 14, 15, 51, 52, 89–91, 94, 97, 104, 125, 152, 153, 170, 280 Langevin’s overdamped equation, 89, 90, 93, 125, 193 Langevin, P., v, 297 Laplace–Beltrami operator, 205, 208, 213, 214, 248 large fluctuations in small reactions, 135, 147 last passage time, 123 leakage, 258, 277 leakage flux, 201, 258 Lear, J., 247, 288 Index Leis, A., 247, 298 level crossing, 7, 89 Levermore, C.D., 197, 290 Levermore, C.D., 197, 293 Levesque, D., 107, 288 Lewis, J., vii, 247, 277, 285 Lewis, R., 134, 289 Li, C.B., 195, 298 Lieber, A., 247, 298 lifetime of a trajectory, 204 Lifshitz, E.M., 155, 297 ligand binding, 87, 135 ligand-gated ion channels, 145, 199, 201 ligands binding to channels, 163 Linse, P., 107, 300 Liptser, R.S., 19, 298 Lisman, J., 136, 148, 163, 298 local mapping of the boundary, 76 Lounis, B., 248, 305 Ludwig, D., 181, 196, 197, 298 Lurie, A.I., 253, 282, 283, 298 Lythe, G.D., 88, 295 315 matching condition, 49, 57, 59, 63, 65, 182, 241, 274 matching in the Wiener–Hopf equation, 72 mathematical Brownian motion (MBM), 1–9, 17, 18, 21, 27, 29, 33, 39, 89 Matkowsky, B.J., 47, 163, 184, 193, 194, 196, 197, 269, 271, 283, 286, 290, 296, 299, 300, 303 Matus, A., 147, 148, 291 maximum principle, 39, 49 Maxwell distribution of velocities, 15, 16 Mazur, P., 195, 196, 291 Mazya, V.G., 282, 297 McCammon, J.A., 86, 107, 108, 287, 300, 304, 305 McKean, H.P., Jr, vii, 7, 195, 248, 295, 299 McLaughlin, D.W., 86, 281, 295 mean exit time, 244 mean exit time of an OU, 27 mean exit time of MBM, 27 mean first passage time, 26, 27, 51, Möbius transformation, 219, 220, 273 113–116, 118, 158, 172, 176, MacKinnon, R., 202, 247, 298 177, 188, 190, 191, 196, 199, MacMillan, H.R., 86, 305 213–218, 221, 223, 225, 226, Majewska, A., 147, 148, 158, 162, 163, 228–232, 236, 237, 242, 243, 247, 290, 298 245, 249, 253, 256, 272, 275, Makhnovskii, Y.A., 86, 248, 282–284, 281–284 286, 293 mean first passage time, absorption rate, Malenka, R.C., 149, 282, 284, 298 and principal eigenvalue, 175 Mandel, I., 251, 252, 281, 303 mean interarrival time, 280 Mandl, P., 45, 298 mean time spent at a point, vi, 115, 117, Mangel, M., 181, 197, 298 266 Mannella, R., 87, 299 mean time spent in a domain, 166, 173, Maravall, M., 159, 302 179, 271, 286 Marchetti, F., 247, 292 mean time spent in a well, 269 Marchewka, A., 85, 86, 299 Medalia, O., 247, 298 Markov chain model, 269, 271 Meissner, G., 247, 288 Markov process, 8, 9, 22, 23, 90, 138, Melnikov, V.I., 196, 299 232, 233, 269, 291, 298, 302 Markovian jump process, 53, 56, 61, 66, membrane potential fluctuations, 136 memory kernel, 169 83, 85–87, 233, 245 Menozzi, S., 88, 292 Mason, C., 147, 148, 290 316 Meshkov, S.V., 196, 299 metastable state, 167, 168, 173, 293, 297 Mezei, M., 107, 292 MFPT to a bottleneck, 218, 219 MFPT to an elliptic absorbing window, 253, 258, 267 MFPT to the boundary of a square, 243 Miller, W.H., 196, 299 Milshtein, G.N., 88, 299 Milstein, G.N., vii, 88 Minsky, A., 247, 298 Mishchenko, E.F., 283, 296, 301 mixed boundary value problem, 84, 87, 134, 159, 203, 204, 207, 215, 216, 224, 228, 231, 236, 244, 248, 258, 259, 262 model of spine twitching, 150, 151, 159, 162 modeling dendritic spine dynamics, 148–152, 163 Moix, J.M., 195, 286 Monine, M.I., 86, 286, 299 Moon, K.-S., 88, 290 Morales, M., 147, 148, 151, 299 Murakoshi, H., 248, 297 Murase, K., 248, 297 myosin molecules, 147 myosin protein, 146 N-methyl-D-aspartate (NMDA), 136, 145, 148, 150, 199, 201, 277, 279 Nadler, B., 96, 104, 105, 107, 108, 125, 133, 158, 300, 303, 304 Naeh, T., 107, 108, 125, 158, 184, 197, 300 Nakada, C., 248, 297 narrow escape problem, vii, 199–201, 207, 247, 248, 258, 267, 281–284, 286, 288, 294, 303, 304 narrow escape time (NET), 199, 203–208, 210–213, 215–219, 227, 228, 230–232, 242, 243, Index 247–249, 257, 258, 262, 267, 270, 276, 279, 282–284, 286 narrow neck, 146, 150, 155–157, 159–162, 165, 166, 168, 179, 199–203, 219, 220, 223, 224, 226–228, 230–234, 240, 243, 248, 272, 275, 276, 281, 284 Nemenman, I., 251, 252, 281, 303 Nernst, W.H., 94–96, 98, 126, 132, 154, 160, 247, 288, 289, 293 Nernst–Planck equation, 154 net flux, 35, 43, 44, 93, 94, 100, 106 NET from a composite domain, 227, 228, 231, 232, 248, 276, 284 NET from a domain with a long neck, 228, 232, 248, 284 NET from domains with corners, 211, 213, 248, 269, 282, 284, 289, 297 NET from domains with cusps, 202, 212, 213, 218, 226, 248, 269, 276, 284 NET on a surface of revolution, 224, 228, 272 NET on the sphere, 213, 215 net probability flux, 42 NET solid funnel-shaped domain, 233, 272 Neumann function, 205, 207, 208, 211, 229, 249, 250, 253, 264, 265, 267, 268, 276, 282 Neumann problem, 156, 232, 233 Neumann–Dirichlet boundary conditions, 87, 203, 207, 231, 244, 248, 258, 263, 272, 281, 282 neuron, 136, 146, 199, 242, 245 neuronal activity, 146 neuronal cleft, 145, 201, 247, 279 neuronal dendrite, 144, 245 neuronal dendritic spine, 136 neuronal membrane, 201, 246 neuronal spine, 144, 147 neuronal spine neck, 277 Index neuronal synapse, v, 136, 145, 162, 199, 200, 277, 279 neurotransmitter (NT), 147, 201, 277 neurotransmitter receptors, 199 Newpher, T.M., 247, 300 next arrival time, 132 Nicoll, R.A., 149, 282, 284, 298 Nimchinsky, E.A., 147, 300 Nitzan, A., 194, 195, 247, 293, 296, 300 no-flux boundary condition, 53–55, 66, 86 Noble, B., 48, 300 noise amplitude, 136 noise generation, 163 noise in microdomains, 163 noise intensity, 165 noise matrix, 13, 165 noise-induced escape, 36, 194 noise-induced transitions, 165 noiseless dynamics, 165 non-Arrhenius reaction, 137 nondifferentiable, 7, 18 Nonner, W., 247, 287 normal boundary flux, 76 normal flux, 66, 97 normal flux density, 51, 98 normal reflection, 69, 82, 83 normalized eigenfunction of the FPO, 174, 264 Northrup, S.H., 300 numerical simulations, v, vii, 18, 35, 36, 77–81, 87, 88, 101, 286, 288, 295, 296, 299 Nymand, T.M., 107, 300 O’Malley, R.E., Jr, vii, 300 oblique reflection, 60, 67, 68, 77–80, 82, 83, 88 occupied binding sites, 154 Opitz, R., 247, 294 Oppenheim, I., 173, 300 Ornstein–Uhlenbeck process (OU), xv, 15, 27, 31 Orszag, S.A., vii, 286 317 Osipov, A., 304 outer expansion, 48, 62, 70, 183, 184 Pacchiarotti, B., 88, 289 Paley, R.N., 7, 300 Parnas, H., 163, 305 partial differential equation (PDE), vi, vii, 5, 23, 30, 45, 84, 127, 162, 281 partial reflection, vi, 53, 85 partial reflection in a half-space, 60, 67, 74 partially absorbing boundary, 35, 52, 53 partially reflected trajectories, 53, 54, 59, 86, 87, 298 partially reflecting boundary, vi pdf (probability density function), xv PDF (probability distribution function), xv pdf of the displacement process, 16 PDF of the FPT, 29, 111, 112, 115, 116, 125 PDF of the FPT and maximum, 33 PDF of the FPT and survival probability, 32, 33, 51 Pecora, R., 151, 287 Perkel, D.J., 149, 282, 284, 298 Perrin, J., 16, 301 Peters, E.A.J.F., 87, 301 Petersen, W.P., 248, 291 Picones, A., 163, 301 Pinsky, M., 290 Pinsky, R.G., 283, 301 Pitman, J., 88, 285 Planck, M.K.E.L., xv, 30, 31, 37, 42, 51, 52, 55, 57, 66, 84, 86, 87, 92, 94–97, 112–115, 117, 119, 126, 130, 132, 154, 160, 165, 169–173, 175–177, 194, 196, 236, 247, 263, 288–290, 293, 299, 302 Platen, E., vii, 88 Poisson equation, 95, 152, 156, 203, 214, 248, 282 Poisson process, 132, 300 318 Poisson, S.D., 247, 288, 289, 293 Poisson–Nernst–Planck equations, 203, 247, 288, 289, 293 Pollak, E., 193–196, 301 Pontryagin, L.S., 23, 26, 27, 204, 213, 219, 225, 283, 296, 301 Pontryagin–Andronov–Vitt (PAV), xv, 23, 26, 27, 204, 213, 219, 225 Popov, I.Yu., 249, 256, 258, 266, 281, 301 positive flux, 50 postsynaptic density (PSD), 147–149, 200, 201, 277, 279, 280 postsynaptic neuron, 199, 201 potential barrier, 158, 165, 170, 181, 186, 193–195, 226, 231, 267, 268, 270, 303 potential barrier height, 182, 187 potential well, 123, 124, 151, 158, 159, 167, 170–172, 182, 194, 196, 231, 234, 262, 269–271, 302 presynaptic neuron, 201 principal curvatures, 250, 251, 257, 260 principal eigenfunction of the FPO, 178, 183, 185, 194 principal eigenvalue and absorption rate, 176 principal eigenvalue and NET, 203 principal eigenvalue and reaction rate, 172 principal eigenvalue and the MFPT, 175 principal eigenvalue in a domain with a bottleneck, 231 principal eigenvalue in a dumbbell-shaped domain, 232, 248, 284, 285 principal eigenvalue of the FPO, 114, 159, 165, 172, 185, 194, 196 principal eigenvalue of the mixed problem in a domain with a narrow neck, 231 principal eigenvalue of the Neumann problem a domain with a bottleneck, 231 Index principal eigenvalue of the Neumann problem in a domain with a bottleneck, 232 principal eigenvalue of the Neumann–Dirichlet problem, 204, 231, 244, 258 principal eigenvalue of the Neumann-Dirichlet problem, 281 principal eigenvalue, MFPT, and rate, 176 probability current, 43 probability density function, vi, 2, 4, 5, 8, 9, 15, 17, 23, 28, 30, 32, 33, 37–41, 45–47, 49, 50, 90, 92, 101, 104, 105, 112, 114, 115, 119, 121, 122, 125–127, 132, 133, 140, 170–175, 177, 178, 193–196, 236, 263 probability distribution function, 51 probability flux density, vi, 42, 119, 173, 176 protein, 45, 135, 137, 139–141, 147, 148 protein channels, v, vi, 42, 95, 100, 104, 106, 142, 150 proteins, 149 Protter, P., vii, 302 pyramidal neurons, 144 quasi-equilibrium density, 186, 196 quasi-equilibrium flux, 185, 186 quasi-equilibrium pdf, 172 quasi-steady state, 114 quasi-steady-state rate, 113 quasistationary density, 175, 184, 194 quasistationary pdf, 193 radiation boundary condition, 52, 86 radius of curvature, 218, 230, 233, 239 Raff, M., vii, 247, 277, 285 Rall, W., 145, 303 Ramón y Cajal, S., 144, 302 random binding, 136 random walk, 86–88, 233, 244, 292 Ratner, M., 194, 296 Index Ravaioli, U., 282, 305 Rayleigh, J.W.S., vii, 199, 247, 253, 282, 283, 302 reactant population, 171, 185, 186 reaction boundary condition, 52, 86 reaction–diffusion equation, 154, 159 recrossing, 89, 94, 166, 168, 172, 179, 186, 188, 192–196, 286, 301, 303 reflected trajectories, 53, 60, 61, 69, 75–77, 81, 142, 204, 287 reflecting boundary, vi, vii, 52–54, 66, 75, 76, 82, 83, 85–88, 107, 108, 129, 133, 137, 140–142, 155, 159, 160, 166, 199, 200, 204, 205, 207, 212, 215, 217–219, 221, 227, 228, 231, 232, 241, 242, 244, 247, 253, 258, 262, 263, 274, 277, 283, 284 reflection at a curved boundary, 67, 75–77, 87, 237, 238 reflection in a half-space, 45 reflection probability, 53, 54 Regehr, W.G., 145, 306 regular expansion of eigenvalues, 259 reinjection of trajectories, 118 renewal equation, 228, 271 renewal theory, 101 Renner, M., 246, 302 Rice, S., 107, 108, 287 Rieke, F., 136, 302 Risken, H., vii, 196, 302 Ritchie, K., 248, 297 Robert, C., 302 Robert, E., vii Roberts, K., vii, 247, 277, 285 Robin boundary condition, 52–54, 65, 66, 74, 80, 85–88 Robin boundary conditions for the BKE, 85 Rogers, L.C.G., 7, 302 Roman, F.L., 107, 302 Ross, J., 107, 108, 158, 162, 163, 247, 287, 298 319 Rossmann, J., 282, 297 Roux, B., 107, 108, 247, 282, 294 Roy, R., 255, 285 Ryter, D., 197, 302 Südhof, T.C., vii, 289, 303 Sabatini, B., vii, 303 Sabatini, B.L., 147, 159, 300, 302 Sacerdote, L., 88, 292 saddle point, 166–170, 184, 186, 187, 192, 194–196 Sako, Y., 248, 297 Saraniti, M., 282, 285, 306 Sartoretto, F., 88, 289 saturated protein, 151, 155–157, 160–162 Saxton, M.J., 248, 302 scaled white noise, 90 Schulten, K., 86–88, 297 Schumaker, M., 88, 302 Schuss, Z., vii, viii, 1, 2, 4, 6, 7, 9, 11–13, 15, 17, 20, 23–25, 27–29, 31, 32, 34, 36, 37, 42, 43, 47, 52, 85–87, 91, 92, 96, 101, 104, 105, 107, 108, 125, 149, 151, 158, 159, 162, 163, 169–171, 174, 175, 178, 180, 183, 184, 188, 191–197, 204, 211, 213, 217, 228, 231, 233, 234, 243–245, 247, 248, 269, 271, 281–284, 286, 287, 290, 292, 294–296, 299–301, 303, 304 Schwartz, J.H., 144, 150, 161, 247, 295 second arrival time, 132 Seefeld, S., 107, 108, 247, 294 Segal, M., 145, 147, 148, 163, 247, 282, 284, 294, 297, 305 Segev, I., vii, 144, 145, 163, 303 separation of time scales, 179, 203 separatrix, 166–168, 170, 182, 187, 193, 195 Shamm, Y.Y., 107, 303 shape fluctuations, 147 Sheetz, M.P., 248, 290, 303, 304 320 Shen, T., 86, 304 Sheng, M., vii, 303 Shepherd, G.M., 145, 303 Sherman, F.S., 86, 292 Shiryayev, A.N., 19, 298 Shoujiguchi, A., 195, 298 Shreve, S.E., 19, 23, 295 Shuler, K.E., 173, 300 Shvartsman, S.Y., 86, 282–284, 286, 289 Siekevitz, P., 145, 287 Silbergleit, A., 251, 252, 281, 303 simulation of calcium kinetics, 158, 159, 162, 163 simulation scheme, vi, vii, 86, 107 Singer, A., 42, 52, 85–87, 96, 104, 105, 108, 211, 213, 247, 248, 282, 284, 300, 303, 304 Skorokhod, A.V., vii, 20, 27, 36, 46, 88, 292, 304 small noise, vii, 124 Smith, P.D., 282, 305 Smoluchowski approximation, 93 Smoluchowski density fluctuation theory, 108 Smoluchowski equation, 91, 93, 154, 262, 280, 304 Smoluchowski limit, 90–92, 98, 153 Smoluchowski limit of free Brownian motion, 91 Smoluchowski solution, 88 Smoluchowski’s formula, 17, 158 Smoluchowski’s probability aftereffect, 133 Smoluchowski, M., 17, 100, 289, 304 Smoluchowski–Fokker–Planck equation, 92 Smoluchowski–Nernst-Planck equation, 98 Sneddon, I.N., 281–283, 304 sojourn time, 166, 179 solutions of parabolic equations that exist for all times, 27 Song, Y., 86, 304 source of trajectories, 95, 96, 98, 101, Index 117, 118, 175, 186, 195, 196, 201, 258, 277–279 source strength, 95, 96, 98, 101, 106, 118 spine head, 145, 147, 149, 150, 155, 157–159, 161, 199, 277, 281 spine neck, 146, 147, 149, 150, 154, 157, 159, 277, 298 Spitzer, F., 29, 304 Spivak, A., 197, 303 Stafford, D., 284, 306 state-dependent diffusion, 67, 237 state-dependent noise, 18 stationary absorption flux, 125 stationary arrival process, 300 stationary bath concentration, 126 stationary density, 129, 258, 262 stationary flux, 186 stationary flux method, 175, 194, 196 stationary FPE, 97, 130 stationary Gaussian process, 91, 170 stationary increments, stationary Maxwellian pdf, 90 stationary pdf, 97, 104 stationary process, 170 stationary relative rate of change, 176 stationary solution of the FPE, 98 stationary substrate, 137 steady flux, 195, 196 steady state, 90, 100, 107, 114, 140, 141, 168, 195, 286, 295, 298, 304 steady state diffusion, 129 steady-state bath, 129 steady-state concentration, 127 steady-state density, 117, 125, 126, 129, 172 steady-state density with a source, 117 steady-state flux, 132 steady-state flux density, 119 steady-state rate, 115, 117 Stevens C.F., 289 Stevens, C.F., vii Stevens, J.K., 247, 293 stochastic differential, 11 Index stochastic differential equation, v, vii, viii, 14, 18, 19, 22, 23, 27, 28, 30, 32, 35, 37, 46, 67, 75, 123, 125, 165, 285, 295, 298, 299 stochastic differential equation (SDE), xv stochastic dynamics, vii, 18, 23–25, 31, 39, 59, 86, 121, 178 stochastic dynamics with killing, 25, 37 stochastic process, viii, 1–4, 7, 9–11, 19, 108, 120, 156, 292, 295, 300, 303 stochastic separatrix, 165, 179, 180, 186, 193, 194, 197, 232, 234, 296, 303 stochastic separatrix for discontinuous drift, 181, 182 Stokes’s formula, 15, 156 Stratonovich integral, 11, 12, 18 Stratonovich, R.L., 11, 12, 18 Straube, R., 248, 258, 276, 288, 289 Stroock, D.W., 248, 304 survival probability, vi, 32, 33, 50, 51, 77, 80, 81, 87, 112, 115, 127, 158 Suzuki, K., 248, 297, 304 Svoboda, K., 147, 159, 247, 300, 302, 305 Swanson, J.A., 196, 297 synaptic cleft, 199, 201, 277, 279 synaptic vesicles, 201, 277 Szepessy, A., 88, 290 Szymczak, P., 86, 305 Tai, K., 86, 305 Talay, D., 88, 285, 287 Talay,D., 88 Talkner, P., 90, 158, 163, 195, 234, 248, 283, 293, 301, 305 Tang, J , 282, 305 Tank, D.W., 247, 305 Tardin, C., 248, 305 Tashiro, A., 147, 148, 163, 290, 298 Taylor, H.M., 45, 101, 121, 125, 188, 295 321 Tel, T., 178, 293 telegraph process, 233 Tempone, R., 88, 290 Tenenbaum, A., 107, 108, 289 terminal boundary value problem, 29, 30, 112 terminal value problem, 24–26, 32, 112, 116 terminated trajectories, 23, 25, 32, 45, 49, 53–55, 68, 86, 95, 96, 100, 102, 104, 109, 135, 159, 175, 179, 269 termination probability, 54, 60, 87 termination probability and the radiation constant, 88 thermal activation, 158, 163, 167, 193, 195, 196, 286, 288 Tier, C., 47, 193, 196, 197, 271, 296, 299 Tildesley, D.J., vii, 107, 285 Titcombe, M., 248, 281, 296 Toda, M., 195, 298 Toresson, H., 277, 305 total population, 50, 112, 113, 117, 126, 176, 185 trajectories between fixed concentrations, 94, 95, 100, 105, 300 trajectory, xi, xii transient calcium, 147 transition PDF, 8, 233 transition state theory (TST), xv, 166–168, 170–172, 187, 195 transmission factor, 193, 196 Tretyakov, M.V., vii, 88 Triller, A., 245–248, 293, 302, 305 Tripathy, A., 247, 288 Truhlar, D.G., 192, 296 Tucker, S.C., 194, 301 turnaround time, 234, 237, 247 unbinding calcium, 151 unbinding kinetics, 151 unbinding probability, 138 unbinding reaction, 135 322 unbound binding sites, 137 unconditional displacement variance, 16 uncorrelated white noise, 18 unidirectional and net flux, 42–44 unidirectional current, 93 unidirectional flux, 35, 42–45, 86, 93, 101, 104 unidirectional flux and the survival probability, 50 unidirectional Langevin flux, 51 uniform convergence of refinements, uniform convergence with probability one, uniform expansion, 183, 299 Uzer, T., 195, 285, 286 Valisko, M., 247, 287 Valleau, J.P., 107, 305 Van De Velde, E., 248, 281, 306 van der Straaten, T.A., 282, 305 Varadhan, S.R.S., 248, 304 Velasco, S., 107, 302 vesicles, 200 Vicere, A., 87, 286 Vinogradov, S.S., 282, 305 Vinogradova, E.D., 282, 305 Viterbi, A.J., 194, 305 Vitt, A.A., 23, 26, 27, 204, 213, 219, 225 Voituriez, R., 248, 282, 286 Volfovsky, N., 163, 305 voltage fluctuations, 95 von Schwerin, E., 88, 290 Waalkens, H., 195, 306 Wagner, U., 147, 291 Walker, M., 282, 289 Ward, M., 248, 289 Ward, M.J., 248, 258, 262, 276, 284, 288, 296, 306 Warshel, A., 107, 108, 295, 303 Watson, J.D., vii, 247, 277, 285 Weber, H., 281, 306 Weiss, G.H., 173, 282–284, 289, 300 Index Weiss, J.J., 107, 288 Wentzell, A.D., 156, 196, 197, 292 white noise, 9, 14, 18, 170, 263 White, J.A., 107, 302 white-noise approximation, 18 white-noise limit, 18 Whittington, S.G., 107, 305 Wiener path integral, vi, 18, 19, 46, 47 Wiener probability measure, 2, 4, 7, 35 Wiener probability measure of a cylinder event, 4, 124 Wiener process, Wiener, N., vi, 1, 2, 4, 7, 35, 300, 306 Wiener–Hopf equation, 48, 56, 64, 71–74 Wiener–Hopf method, 58, 65, 300 Wiggins, S., 195, 306 Wigner, E., 196, 306 Williams, D., 7, 302 WKB (Wentzel–Kramers–Brillouin), 178, 183 Wolynes, P.G., 192, 196, 291 Wong, E., 11 Wong–Zakai correction, 11 Xu, L., 247, 288 Yamamoto, M., 248, 297 Yau, K.W., 163, 293 Yuste, R., vii, 144–148, 158, 162, 163, 247, 287, 290, 298, 306 Zador, A., 144, 163, 296, 306 Zakai, M., 11 Zauderer, E., vii, 306 Zeitouni, O., 283, 290 Zhang, Y., 86, 304 Zitserman, V.Yu., 86, 194, 248, 282–284, 286, 287, 293 Zucca, C., 88, 292 Zucker, R.S., 145, 306 Zwanzig, R., 86, 306 Zygmund, A., 7, 300 ... 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 Chapter Mathematical Brownian. .. 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:... Matkowsky R Pego C Peskin A Singer A Stevens A Stuart For further volumes: http://www.springer.com/series/34 Zeev Schuss Brownian Dynamics at Boundaries and Interfaces In Physics, Chemistry, and