Brownian Dynamics at Boundaries and Interfaces: In Physics, Chemistry, and Biology PDF
Preview Brownian Dynamics at Boundaries and Interfaces: In Physics, Chemistry, and Biology
Applied Mathematical Sciences Zeev Schuss Brownian Dynamics at Boundaries and Interfaces In Physics, Chemistry, and Biology Applied Mathematical Sciences Volume 186 FoundingEditors FritzJohn,JosephLaselleandLawrenceSirovich Editors S.S.Antman [email protected] P.J.Holmes [email protected] K.R.Sreenivasan [email protected] Advisors L.Greengard J.Keener R.V.Kohn B.Matkowsky R.Pego C.Peskin A.Singer A.Stevens A.Stuart Forfurthervolumes: http://www.springer.com/series/34 Zeev Schuss Brownian Dynamics at Boundaries and Interfaces In Physics, Chemistry, and Biology 123 ZeevSchuss SchoolofMathematicalSciences TelAvivUniversity TelAviv,Israel ISSN0066-5452 ISBN978-1-4614-7686-3 ISBN978-1-4614-7687-0(eBook) DOI10.1007/978-1-4614-7687-0 SpringerNewYorkHeidelbergDordrechtLondon LibraryofCongressControlNumber:2013944682 Mathematics Subject Classification (2010): 60-Hxx, 60H30, 62P10, 65Cxx, 82C3, 92C05, 92C37, 92C40,35-XX,35-B25,35Q92 ©Author2013 Thisworkissubjecttocopyright. AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered andexecuted onacomputer system, forexclusive usebythepurchaser ofthework. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer. PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter. Violations areliabletoprosecutionundertherespectiveCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc. inthispublica- tiondoesnotimply,evenintheabsenceofaspecificstatement, thatsuchnamesareexemptfromthe relevantprotectivelawsandregulationsandthereforefreeforgeneraluse. Whiletheadviceandinformationinthisbookarebelievedtobetrueandaccurateatthedateofpub- lication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityforany errorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,withrespect tothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface Browniandynamicsserveas mathematicalmodelsforthe diffusivemotionof mi- croscopicparticlesofvariousshapesingaseous,liquid,orsolidenvironments.The renewedinterestinBrowniandynamicsisdueprimarilytotheirkeyroleinmolec- ular and cellular biophysics: diffusion of ions and molecules is the driver of all life. Brownian dynamics simulations are the numerical realizations of stochastic differentialequations (SDEs) that model the functions of biological microdevices suchasproteinionicchannelsofbiologicalmembranes,cardiacmyocytes,neuronal synapses, and many more.SDEs are ubiquitousmodelsin computationalphysics, chemistry,biophysics,computerscience,communicationstheory,mathematicalfi- nance theory, and many other disciplines. Brownian dynamics simulations of the random motion of particles, be it molecules or stock prices, give rise to mathe- matical problems that neither the kinetic theory of Maxwell and Boltzmann nor Einstein’sandLangevin’stheoriesofBrownianmotioncouldpredict. Kinetic theory, which assigns probabilities to configurations of ensembles of particles in phase space, assumes that the ensembles are in thermodynamic equi- librium,whichmeansthatnonetcurrentisflowingthroughthesystem. Thusitis not applicable to the description of nonequilibrium situations such as conduction of ions through protein channels, nervous signaling, calcium dynamics in cardiac myocytes,theprocessofviralinfection,andcountlessothersituationsinmolecular biophysics. Themotionofindividualparticlesintheensembleisnotdescribedinsufficient detail to permit computer simulations of the atomic or molecular individual mo- tionsinawaythatreproducesallmacroscopicphenomena. TheEinsteinstatistical characterization of the motion of a heavy particle undergoing collisions with the muchsmallerparticlesofthesurroundingmediumlaysthefoundationforcomputer simulationsof the Brownianmotion. However,pushingEinstein’sdescriptionbe- yonditsrangeofvalidityleadstoartifactsthatbafflethesimulators:particlesmove without velocity, so there is no telling when they enter or leave a given domain. Theoretically, they crossand recrossinterfacesan infinite numberof times in any finite time interval. Thus the simulation of Brownian particles in a small domain surroundedbyacontinuumbecomesproblematic.TheLangevindescription,which includesvelocity,partiallyremediestheproblem.Thereis,however,apricetopay: thedimension,andthereforethecomputationalcomplexity,isdoubled. 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 com- plicated. Both the discrete reflection and partial reflection laws of the simulated trajectories are not very intuitive in their peculiar dependenceon the geometryof the boundaryand 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 prioriclear what should be the reflection law of the rod when one of its ends hits theimpermeableboundaryoftheconfiningdomain. Thisissuehasbeenathornin the side ofsimulatorsfora longtime, whichmay be explainedby the unexpected mathematicalcomplexityoftheproblem.ItisresolvedinSects.2.5and2.6. Thebehaviorofrandomtrajectoriesnearboundariesofthesimulationimposesa varietyofboundaryconditionsontheprobabilitydensityoftherandomtrajectories anditsfunctionals. The quiteintricateconnectionbetweentheboundarybehavior ofrandomtrajectoriesandtheboundaryconditionsforthepartialdifferentialequa- tionsistreatedherewithspecialcare. Theanalysisofthemathematicalissuesthat arise in Brownian dynamics simulations relies on Wiener’s discrete path integral representation of the transition probability density of the random trajectories that arecreatedbythediscretesimulation. Asthesimulationisrefined,theWienerin- tegralrepresentationleadstoinitialandboundaryvalueproblemsforpartialdiffer- entialequationsofellipticandparabolictypesthatdescribeimportantprobabilistic 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’sfunctionand its functionalsplay a centralrole in expressingthesequantitiesanalyticallyandindeterminingtheirinterrelationships. Theanalysisprovidesthemeansfordeterminingtherelationshipbetweenthetime stepinasimulationandtheboundaryconcentrations. Key mathematical problemsin runningBrownian or Langevinsimulations in- cludethefollowingquestions: Whatisthe“correct”boundarybehavioroftheran- dom trajectories? What is the effectof their boundarybehavioron statistics, e.g., onthepdf?Whatboundarybehaviorshouldbechosentoproduceagivenboundary behaviorofthe pdf? Howcanthehigher-dimensionalLangevindynamicsbeade- quatelyapproximatedbycoarserBrowniandynamics? Howshouldonechoosethe timestepinasimulation?Anothercurseofcomputersimulationsofrandommotion is the ubiquitousphenomenonof rare events. It is particularlyacute 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 throughproteinchannelsofbiologicalmembranes. Findingasmalltargetisanim- portantproblemin Browniandynamicssimulations. Can the computationaleffort bereducedbyprovidinganalyticalinformationabouttheprocess?Whilenumerical analysisgiveserrorestimatesforgivensimulationschemesonfinitetimeintervals, simulationsareoftenrequiredtoproduceestimatesofunlimitedrandomquantities such as first passage times or their moments. Thus we need to know how much computationaleffortis neededforan estimate of therandomescapetime froman attractororaconfiningdomain. Preface vii In this book, we address these and additionalmathematical problemsof com- puter 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,butratherwiththemorefundamentalquestionsmentionedabove.The analysispresentedinthisbooknotonlyisapplicabletotheEulerscheme,butcan also be applied to many other simulation schemes. While the singular perturba- tion 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 eventsdueto the geometryof the confiningdomainrequiresnew mathemati- calmethods. The“narrowescapeproblem”indiffusiontheory,whichgoesbackto LordRayleigh,istocalculatethemeanfirstpassagetimeofadiffusionprocesstoa smallabsorbingtargetonanotherwisereflectingboundaryofaboundeddomain.It includesalsotheproblemofdiffusingfromonecompartmenttoanotherthrougha narrowpassage,asituationthatisoftenencounteredinmolecularandcellularbio- physicsand frustrates numericalsimulations. The new mathematicalmethods for resolvingthisproblemarepresentedhereingreatanalyticaldetail. Theexpositioninthisbookiskeptatanintermediatelevelofmathematicalrigor. Experience shows that mathematical rigor and applications can hardly coexist in thesame course; excessiverigorleavesnoroomforin-depthdevelopmentof ana- lytical methods and tends to turn off students interested in scientific applications. Therefore,thebookcontainsonlytheminimalmathematicalrigorrequiredforun- derstanding the mathematical concepts and for enabling the students to use their own judgment of what is correct and what requires further theoretical study. All topicsrequireabasicknowledgeofSDEsandofasymptoticmethodsinthetheory ofpartialdifferentialequations,aspresented,forexample,inSchuss(2010b). The introductoryreview of stochastic processes in Chap.1 should not be mistaken for an expositorytext on the subject. Its role it to establish terminologyand to serve as a refresher on SDEs. The role of the exercises is give the reader an opportu- nitytoexaminehis/hermasteryofthesubject. Othertextsonstochasticdynamics include,amongothertitles, (Arnold1998;Friedman2007;GihmanandSkorohod 1972;McKean1969;Øksendal1998;Protter1992). Textsonnumericalanalysis ofstochasticdifferentialequationsinclude(AllenandTildesley1991;Kloedenand Platen 1992;Milstein 1995;Risken 1996;Robertand Casella 1999;Doucet et al. 2001; Kloeden 2002; Milstein and Tretyakov 2004; Honerkamp 1994). A solid traininginpartialdifferentialequationsofmathematicalphysicsandintheasymp- toticmethodsofappliedmathematicscanbederivedfromthestudyofclassicaltexts suchas(Zauderer1989;O’Malley1974;KevorkianandCole1985)or(Benderand Orszag1978). Manyoftheapplicationsandexamplesinthisbookconcernmolec- ular and cellular biophysics, especially in the context of neurophysiology. Basic facts on these subjects should not be acquiredfrom mathematiciansor physicists, butratherfromprofessionalelementarytextsonthesubjects,suchas(Albertsetal. 1994;Hille2001;Koch1999;KochandSegev2001;Shengetal.2012;Cowanet al.2003;Yuste 2010;Baylog2009). Wikipediashouldbeconsultedforclarifying biochemicalandphysiologicalterminology. viii Preface Thisbookisaimedatappliedmathematicians,physicists,theoreticalchemists, and physiologistswho are interested in modeling, analysis, and simulation of mi- crodevicesof microbiology. A specialtopicscoursefromthisbookrequiresgood preparation in the theory of SDEs, such as can be found in Schuss (2010b). Alternatively,someofthetopicsdiscussedinthisbookcanbeinterspersedbetween the topicsof a moregeneralcourseas applicationsand illustrationsof thegeneral theory. The book contains exercises and worked-out examples. Hands-on training in stochasticprocesses,asmylongteachingexperienceshows,consistsinsolvingthe exercises,withoutwhichunderstandingisonlyillusory. 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. TelAviv,Israel ZeevSchuss List of Figures 2.1 Reflectedtrajectories . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.2 Obliqueandnormalreflections . . . . . . . . . . . . . . . . . . . . 69 2.3 Marginaldensityofx(T)withobliquereflection . . . . . . . . . . 78 2.4 Marginaldensityofy(T)withobliquereflection . . . . . . . . . . 79 2.5 NumericalsolutiontheFPEwithobliquereflection . . . . . . . . . 79 2.6 AnothernumericalsolutionoftheFPEwithobliquereflection . . . 80 2.7 ThereflectionlawofX inΩ . . . . . . . . . . . . . . . . . . . . . 81 t 2.8 Marginaldensityofx(T)withnormalandobliquereflections. . . . 82 2.9 Marginaldensityofy(T)withnormalandobliquereflections . . . . 83 3.1 Typicalbathsseparatedbymembranewithchannel . . . . . . . . . 96 3.2 Simulationin(0,1)withnormalinitialdistribution . . . . . . . . . 102 3.3 Simulationin(0,1)withinitialresidualofthenormal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 3.4 Concentrationprofileswithtime-step-independentinjectionrate . . 103 3.5 Concentrationprofilewithtime-step-dependentinjectionrate . . . . 103 3.6 Concentrationvs.displacementofaLangevindynamics simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.1 ThedomainDanditscomplementinthesphereD . . . . . . . . 130 R 5.1 Varianceoffluctuationsinthefractionofboundsites . . . . . . . . 143 5.2 Schematicdrawingofasynapsebetweentwoneurons . . . . . . . . 145 5.3 Modelofadendriticspine . . . . . . . . . . . . . . . . . . . . . . 146 6.1 Double-wellpotentialsurface . . . . . . . . . . . . . . . . . . . . . 166 6.2 Contoursandtrajectories . . . . . . . . . . . . . . . . . . . . . . . 167 6.3 Apotentialwellwithasinglemetastablestate . . . . . . . . . . . . 167 6.4 Dumbbell-shapeddomain . . . . . . . . . . . . . . . . . . . . . . . 168 7.1 EscapingBrowniantrajectory. . . . . . . . . . . . . . . . . . . . . 200 7.2 Compositedomains . . . . . . . . . . . . . . . . . . . . . . . . . . 200 7.3 Receptormovementontheneuronalmembrane . . . . . . . . . . . 201 7.4 Anidealizedmodelofthesynapticcleft . . . . . . . . . . . . . . . 201 ix
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