Stochastic Modelling of Reaction-Diffusion Processes

321 Pages·2020·14.276 MB·English

by  Radek Erban, S. Jonathan Chapman

StochasticModellingofReaction–DiffusionProcesses
Thispracticalintroductiontostochasticreaction–diffusionmodellingisbasedon
coursestaughtattheUniversityofOxford.Theauthorsdiscusstheessenceof
mathematicalmethodsthatappear(underdifferentnames)inanumberof
interdisciplinaryscientificfieldsbridgingmathematicsandcomputationswithbiology
andchemistry.
Thebookcanbeusedbothforself-studyandasasupportingtextforadvanced
undergraduateorbeginninggraduate-levelcoursesinappliedmathematics.New
mathematicalapproachesareexplainedusingsimpleexamplesofbiologicalmodels,
whichrangeinsizefromsimulationsofsmallbiomoleculestogroupsofanimals.The
bookstartswithstochasticmodellingofchemicalreactions,introducingstochastic
simulationalgorithmsandmathematicalmethodsforanalysisofstochasticmodels.
Differentstochasticspatio-temporalmodelsarethenstudied,includingmodelsof
diffusionandstochasticreaction–diffusionmodelling.Themethodscoveredinclude
moleculardynamics,Browniandynamics,velocity-jumpprocessesand
compartment-based(lattice-based)models.
radek erbanisProfessorofMathematicsattheUniversityofOxford,aFellowof
MertonCollege,Oxford,andaRoyalSocietyUniversityResearchFellow.
s. jonathan chapmanisProfessorofMathematicsanditsApplicationsatthe
UniversityofOxford,andaFellowofMansfieldCollege,Oxford.
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Stochastic Modelling of
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Reaction–Di usion Processes
RADEK ERBAN
UniversityofOxford
S. JONATHAN CHAPMAN
UniversityofOxford
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Informationonthistitle:www.cambridge.org/9781108498128
DOI:10.1017/9781108628389
(cid:2)c RadekErbanandS.JonathanChapman2020
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Firstpublished2020
PrintedinSingaporebyMarkonoPrintMediaPteLtd
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary.
LibraryofCongressCataloging-in-PublicationData
Names:Erban,Radek,author.|Chapman,Jon,author.
Title:Stochasticmodellingofreaction–diffusionprocesses/RadekErban(Universityof
Oxford),S.JonathanChapman(UniversityofOxford).
Description:Cambridge;NewYork,NY:CambridgeUniversityPress,[2020]|Series:
Cambridgetextsinappliedmathematics|Includesbibliographicalreferencesandindex.
Identifiers:LCCN2019019436|ISBN9781108498128(alk.paper)
Subjects:LCSH:Stochasticprocesses–Textbooks.|Reaction–diffusion
equations–Textbooks.
Classification:LCCQA274.E732020|DDC515/.3534–dc23
LCrecordavailableathttps://lccn.loc.gov/2019019436
ISBN978-1-108-49812-8Hardback
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Contents
Preface pageix
1 StochasticSimulationofChemicalReactions 1
1.1 StochasticSimulationofDegradation 1
1.2 StochasticSimulationofProductionandDegradation 8
1.3 Higher-OrderChemicalReactions 14
1.4 StochasticSimulationofDimerization 16
1.5 GillespieAlgorithm 25
Exercises 29
2 DeterministicversusStochasticModelling 33
2.1 SystemswithMultipleFavourableStates 34
2.2 Self-InducedStochasticResonance 37
2.3 StochasticFocusing 42
2.4 DesigningStochasticChemicalSystems 49
Exercises 55
3 StochasticDifferentialEquations 59
3.1 AComputationalDefinitionofSDE 60
3.2 ExamplesofSDEs 62
3.3 Fokker–PlanckEquation 66
3.4 BoundaryConditionsontheFokker–PlanckEquation 72
3.5 KolmogorovBackwardEquation 75
3.6 SDEswithMultipleFavourableStates 76
3.7 ChemicalFokker–PlanckEquation 80
3.8 AnalysisofProblemfromSection2.1 85
3.9 AnalysisofProblemfromSection2.2 88
Exercises 92
4 Diffusion 95
4.1 DiffusionModelledbySDEs 96
4.2 Compartment-BasedApproachtoDiffusion 100
v
vi Contents
4.3 DiffusionandVelocity-JumpProcesses 107
4.4 DiffusiontoAdsorbingSurfaces 117
4.5 ReactiveBoundaryConditions 125
4.6 Einstein–SmoluchowskiRelation 130
Exercises 133
5 EfficientStochasticModellingofChemicalReactions 137
5.1 ASimpleMultiscaleProblem 139
5.2 MultiscaleSSAwithPartialEquilibriumAssumption 142
5.3 MultiscaleModelling 148
5.4 First-ReactionSSA 151
5.5 ExactEfficientSSAs 152
Exercises 157
6 StochasticReaction–DiffusionModels 160
6.1 ACompartment-BasedReaction–DiffusionAlgorithm 161
6.2 A Reaction–Diffusion SSA Based on the
SDEModelofDiffusion 164
6.3 Compartment-BasedSSAforHigher-Order
Reactions 166
6.4 AChoiceofCompartmentSizeh 169
6.5 Molecular-Based Approaches for Second-Order
Reactions 174
6.6 ReactionRadiusandReactionProbability 177
6.7 ModellingReversibleReactions 183
6.8 BiologicalPatternFormation 186
Exercises 190
7 SSAsforReaction–Diffusion–AdvectionProcesses 192
7.1 SSAsforDiffusion–AdvectionProcesses 193
7.2 Reaction–Diffusion–AdvectionSSAs 196
7.3 BacterialChemotaxis 199
7.4 CollectiveBehaviourofLocusts 206
7.5 IonsandIonChannels 211
7.6 Metropolis–HastingsAlgorithm 216
Exercises 222
8 MicroscopicModelsofBrownianMotion 226
8.1 One-ParticleSolventModel 227
8.2 GeneralizedLangevinEquation 233
8.3 SolventasHarmonicOscillators 242
8.4 SolventasPointsCollidingwiththeDiffusingParticle 246
8.5 ForcesBetweenAtomsandMolecules 252
Contents vii
8.6 MolecularDynamics 257
Exercises 265
9 MultiscaleandMulti-ResolutionMethods 268
9.1 CouplingSDE-BasedandCompartment-BasedModels 270
9.2 CouplingMolecularDynamicswithLangevin
Dynamics 278
9.3 Multi-ResolutionMolecularandBrownianDynamics 285
Exercises 289
Appendix 293
AppendixA DeterministicModellingofChemicalReactions 293
AppendixB DiscreteProbabilityDistributions 295
AppendixC ContinuousProbabilityDistributions 296
References 297
Index 305

Preface
In this textbook, we provide an introduction to stochastic modelling of
reaction–diffusion processes presented in a way that is aimed at advanced
undergraduateorbeginninggraduatestudents.Weassumethatthereaderhas
abasicunderstandingofdifferentialequations,butwedonotassumeanyprior
knowledgeofadvancedprobabilitytheoryorstochasticanalysis.Weintroduce
and explain some common stochastic simulation methods using illustrative
examples. At the same time we present some basic theoretical tools that are
usedtoanalysethemethods.Newtheoryisintroducedwheneveritprovidesa
better insight into a particular example. In our experience, such an example-
based approach is more accessible to students than a theory-first approach.
To make this textbook self-contained, we also summarize some basic facts
aboutdeterministicmodelling(basedondifferentialequations)andprobability
distributionsintheAppendix.
The material in this textbook has been developed (and used in teaching
students in Oxford) over a period of 12 years. It started as a short 30-
page-long set of lecture notes on stochastic modelling of reaction–diffusion
processeswrittenbyErban,ChapmanandMaini(Erbanetal.,2007),wherewe
explainedsimplealgorithmsonsimple(linear)models.Manyimportanttopics
hadtobeomittedduetolackofspace.Overthenext12years,wesignificantly
extendedthatmaterialtothepresentform.Theresultingbookcoversstochas-
tic models of chemical reactions, stochastic differential equations (SDEs),
andstochasticmodelsofdiffusion,reaction–diffusionandreaction–diffusion–
advection processes. We explain how stochastic models can be derived from
moredetailedmoleculardynamicsapproachesandprovideanintroductionto
methods designed for modelling processes on multiple spatial and temporal
scales.Weintroducemethodsforanalysisofstochasticmodels(forexample,
the chemical master equation and the Fokker–Planck equation) and efficient
ix