A practical guide to stochastic simulations of reaction-diffusion processes

A practical introduction to stochastic modelling of reaction-diffusion processes is presented. No prior knowledge of stochastic simulations is assumed. The methods are explained using illustrative examples. The article starts with the classical Gillespie algorithm for the stochastic modelling of chemical reactions. Then stochastic algorithms for modelling molecular diffusion are given. Finally, basic stochastic reaction-diffusion methods are presented. The connections between stochastic simulations and deterministic models are explained and basic mathematical tools (e.g. chemical master equation) are presented. The article concludes with an overview of more advanced methods and problems.

[1]  S. Chandrasekhar.  Stochastic problems in Physics and Astronomy , 1943 .

[2]  N. Rashevsky,et al.  Mathematical biology , 1961, Connecticut medicine.

[3]  F. Schlögl.  Chemical reaction models for non-equilibrium phase transitions , 1972 .

[4]  D. Gillespie.  Exact Stochastic Simulation of Coupled Chemical Reactions , 1977 .

[5]  J. Schnakenberg,et al.  Simple chemical reaction systems with limit cycle behaviour. , 1979, Journal of theoretical biology.

[6]  Joshua Kiddy K. Asamoah,et al.  Fractal–fractional age-structure study of omicron SARS-CoV-2 variant transmission dynamics , 2022, Partial Differential Equations in Applied Mathematics.

[7]  H. Berg.  Random Walks in Biology , 2018 .

[8]  J. Elgin.  The Fokker-Planck Equation: Methods of Solution and Applications , 1984 .

[9]  B. Øksendal.  Stochastic Differential Equations , 1985 .

[10]  W. J. Freeman,et al.  Alan Turing: The Chemical Basis of Morphogenesis , 1986 .

[11]  H. Othmer,et al.  Models of dispersal in biological systems , 1988, Journal of mathematical biology.

[12]  A. M. Turing,et al.  The chemical basis of morphogenesis , 1952, Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences.

[13]  D. Gillespie.  Markov Processes: An Introduction for Physical Scientists , 1991 .

[14]  L. Arnold.  Stochastic Differential Equations: Theory and Applications , 1992 .

[15]  C. Lumsden,et al.  Stochastic Simulation of Coupled Reaction-Diffusion Processes , 1996 .

[16]  Lewis Wolpert,et al.  Principles of Development , 1997 .

[17]  S. Shreve,et al.  Stochastic differential equations , 1955, Mathematical Proceedings of the Cambridge Philosophical Society.

[18]  Pll Siinksen,et al.  Control , 1999 .

[19]  J. Collins,et al.  Construction of a genetic toggle switch in Escherichia coli , 2000, Nature.

[20]  M. Ehrenberg,et al.  Stochastic focusing: fluctuation-enhanced sensitivity of intracellular regulation. , 2000, Proceedings of the National Academy of Sciences of the United States of America.

[21]  D. Gillespie.  The chemical Langevin equation , 2000 .

[22]  Michael A. Gibson,et al.  Efficient Exact Stochastic Simulation of Chemical Systems with Many Species and Many Channels , 2000 .

[23]  Desmond J. Higham,et al.  An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations , 2001, SIAM Rev..

[24]  T. Kepler,et al.  Stochasticity in transcriptional regulation: origins, consequences, and mathematical representations. , 2001, Biophysical journal.

[25]  Jeff Hasty,et al.  Engineered gene circuits , 2002, Nature.

[26]  C. Rao,et al.  Control, exploitation and tolerance of intracellular noise , 2002, Nature.

[27]  C. W. Gear,et al.  Equation-Free, Coarse-Grained Multiscale Computation: Enabling Mocroscopic Simulators to Perform System-Level Analysis , 2003 .

[28]  Radek Erban,et al.  From Individual to Collective Behavior in Bacterial Chemotaxis , 2004, SIAM J. Appl. Math..

[29]  D. Bray,et al.  Stochastic simulation of chemical reactions with spatial resolution and single molecule detail , 2004, Physical biology.

[30]  H. Meinhardt,et al.  A theory of biological pattern formation , 1972, Kybernetik.

[31]  Radek Erban,et al.  From Signal Transduction to Spatial Pattern Formation in E. coli: A Paradigm for Multiscale Modeling in Biology , 2005 .

[32]  E Weinan,et al.  Self-induced stochastic resonance in excitable systems , 2005 .

[33]  H. Othmer,et al.  Facilitated Transport of a Dpp/Scw Heterodimer by Sog/Tsg Leads to Robust Patterning of the Drosophila Blastoderm Embryo , 2005, Cell.

[34]  Stanislav Y Shvartsman,et al.  Computational analysis of EGFR inhibition by Argos. , 2005, Developmental biology.

[35]  Johan Hattne,et al.  Stochastic reaction-diffusion simulation with MesoRD , 2005, Bioinform..

[36]  H. Othmer,et al.  A stochastic analysis of first-order reaction networks , 2005, Bulletin of mathematical biology.

[37]  Osamu Shimmi,et al.  Facilitated Transport of a Dpp/Scw Heterodimer by Sog/Tsg Leads to Robust Patterning of the Drosophila Blastoderm Embryo , 2005, Cell.

[38]  Samuel A. Isaacson,et al.  Incorporating Diffusion in Complex Geometries into Stochastic Chemical Kinetics Simulations , 2006, SIAM J. Sci. Comput..

[39]  Ioannis G Kevrekidis,et al.  Spatially distributed stochastic systems: Equation-free and equation-assisted preconditioned computations. , 2006, The Journal of chemical physics.

[40]  Eric Vanden-Eijnden,et al.  Non-meanfield deterministic limits in chemical reaction kinetics. , 2005, The Journal of chemical physics.

[41]  Ioannis G Kevrekidis,et al.  Gene regulatory networks: a coarse-grained, equation-free approach to multiscale computation. , 2005, The Journal of chemical physics.

[42]  J. Timmer,et al.  Supporting Online Material Material and Methods , 2022 .

[43]  Adam P Arkin,et al.  Deviant effects in molecular reaction pathways , 2006, Nature Biotechnology.

[44]  R. Erban,et al.  Reactive boundary conditions for stochastic simulations of reaction–diffusion processes , 2007, Physical biology.

[45]  Radek Erban,et al.  Time scale of random sequential adsorption. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[46]  H. Othmer,et al.  Taxis equations for amoeboid cells , 2006, Journal of mathematical biology.

[47]  Jeffrey W. Smith,et al.  Stochastic Gene Expression in a Single Cell , 2022 .