Brownian Dynamics at Boundaries and Interfaces: In Physics, Chemistry, and Biology

340 Pages·2013·4.352 MB·English

by  Zeev Schuss (auth.)

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