Table Of Content

Foreword Computation, Modeling, and Biology in the Twenty-First Century Overthepastquartercentury,extraordinaryadvanceshavebeenroutineinthebiological sciences and the computer sciences. For both, the popular press applies the term revolu- tionaryadvancesandtheacademicworldspeaksofnewparadigms.Ourunderstandingof livingsystemsandthedevelopmentofcomputationalscience(alongwithitssibling,infor- mationscience)haveadvancedtothepointwhereeachisideallysuitedfortheother,and thegrandestadvancesinbothªeldswillbemadeattheinterface.Whilethenatureofthe progressseemslikelytoexceedevenourdreams,pressarticlesforthenextquartercentury will surely describe truly spectacular advances that arise from integrating computational and information technology with biological science research. Livingsystemsareverycomplexandtheircharacterizationwillremainachallengefor theforeseeablefuture.Totrytoimaginewhatwemaydiscoverintheªrstfewdecadesof the twenty-ªrst century, consider where the biological sciences stood a few decades ago. Shockingtograduatestudentsatthetime,thirty-someyearsago,awell-known,productive biologist, considered insightful, proclaimed that biology was now so well understood at themolecularlevelthatnogreatnewdiscoverieslayoverthehorizonandthusbrightsci- entistsshouldlookelsewhereforchallenges.Thencamethedeluge:thetoolsofmolecular biology uncovered unexpected feature after feature of model organisms and began to lay openthevastcomplexityofmetazoans.Today,theincrediblediversenatureoflivingsys- tems, often called biocomplexity, is clear to all of us. Thecompletionofthecompletegenomesequenceforhumanandmodelorganismshas given us the most powerful biological data set ever assembled, but even this information onlytellsushowªniteouractualknowledgeis.Therichnessofexperimentalobservations andtheirscalerequireadata-drivenapproach.Biologicalscientistswillneedtoembrace computational tools to uncover the meaning implicit in the DNA sequence. Large challengesforcomputationalsciencelieaheadinextendingknowledgerepresentationand delivering data-driven computing, in contrast to numeric computing, which characterizes many physical science applications. Throughoutthissamequartercenturyofremarkableprogress,biologistshavedistrusted modeling,theory,andanalyticalandmathematicalanalysis.Researchershavefocusedon simplesystems,thesimplestpossibletoaddresstheirownareaofinterest,andwhenever possible, asked questions believed to yield only binary answers, not quantitative ones. CompletionandanalysisoftheDNAsequenceoftheªrstmodelorganism,H.inºuenza, markedtheendofourinnocence;similarly,entryintothecontemporaryhighthroughput era, initiated by the human genome project, unambiguously demonstrated the limits of qualitative biology and introduced the vision of quantitative biology at a system level. Whilewearejustbeginningtoappreciatehowwecanapproachquantitativeanalysisfor biology,outlininghowresearchcanberefocusedandsimultaneouslytraininganewgener- ationofscientistsisessential.ThecoursedevelopedattheCaliforniaInstituteofTechnol- viii Foreword ogy and represented in this book is a ªrst step, much like the famous original moon landing,asmallstepforeachaspectbutalargestepforbiology.Thedecisiontodeliverthe outcome of that course to a broader audience is very important and the effort should be praised.Timingissaidtobeeverything,andthetimingofintroducingatexttoaccelerate progressatthefrontierofcomputingandbiologyisperfect.Aboutadecadeago,theNa- tional Science Foundation established the ªrst government program directly and exclu- sively aimed at funding computational biology research and, simultaneously, a program aimedatfundingbiologicalknowledgerepresentationandanalysisorwhattodayiscalled bioinformatics. At that time, experimentalists and even bureaucrats considered computa- tionalbiologytobeanoxymoron,oratleastthatlivingsystemsmightbeagreatmodelfor computerscientiststoemulate,butcomputerswouldneverbesigniªcantresearchtoolsin the biological sciences. The ever-pragmatic, hypothesis-driven, productive, experimental biologist,seekingtobesuccessful,tohaveanimpact,andtoberespected,avoidedmathe- matical or analytical modeling and any experimental approach requiring quantitative insight.Afamousbiologist,ongettingtenure,wastold(inessence)byhischairman:well, youhavedonegreatexperimentalwork,thereferenceletterssayso,butnowthatyouhave been promoted, surely you will stay away from that theory and computation you were (also)doing.Iviewthisbookastheanswertothatchallenge.Today,everybiologistusesa computertosearchforafavoritegeneanditsattributes.Today,graduatestudentswhoex- ploit the access to the daily updates to the world’s research ªndings leap ahead of those ploddingintheirlittlecornerofalab.Bioinformaticsisthe“hottestcommodity”forstart- ingdepartments,obtainingjobs,ormakingheadlines,andincreasingly,acentralcompo- nentofacademicstrategicplansforgrowth.Informationtechnologyprovidesuswiththe ability to create computational tools to manage and probe the vast data based on experimental discoveries, from sequence to transcription regulation, from protein structure to function,andfrommetabolitetoorganphysiology.Today,justasthemethodsofmolecular biologytransformedalloflifesciencesresearch,themethodsofcomputationalandinfor- mationsciencewillestablishnextgenerationbiology.Tomorrow,Ibelieve,everybiologist willuseacomputertodeªnetheirresearchstrategyandspeciªcaims,managetheirexper- iments,collecttheirresults,interprettheirdata,incorporatetheªndingsofothers,dissemi- natetheirobservations,andextendtheirexperimentalobservations—throughexploratory discovery and modeling—in directions completely unanticipated. Findingpatterns,insight,andexperimentalfocusfromthegloballiterature,ordiscovery science,providesanewapproachforbiologicalresearch.Wenowhavetheabilitytocreate modelsoftheprocessesoflivingcells,modelsforhowmacromoleculesinteractandeven howcellsactuallywork.Foranexamplefromthistext,modelingdifferentialgeneexpres- sion as a function of time and space—speciªcally, the role of cis-regulatory control ele- mentsduringseaurchindevelopment—isanexemplaradvanceforwhatwillbecomenext- Foreword ix generation biology. We have all been schooled in the regulation of the lac operon in the classicbacteriumofchoiceandnowmustbeawedbyaªrstglimpseintotheintricaciesof eukaryotic gene regulation, incomplete as the model must be at this stage. Thelimitationsofexperimentalknowledgeaboutbiologyandthelimitedpowerofcom- puters led earlier generations of modelers to hit a brick wall seemingly inªnite in height andthickness.Hereinyouwillreadnumerousexamplesofcracksinthewall,whichwill tumbleintheyearsahead,intheeraofcompletelysequencedgenomesandhighthrough- putbiologicalscienceexperiments.Despitetheclichénatureoftheexpression,whichwe haveallheardrepeatedlyfordecades,biologistsarenowindeedworkingandlivinginex- citingtimes,andcomputationalbiologyalreadyallowsustolive/work“inthefuture.”The extensive,in-depthcontributionsinthisbook,alltrulyexceptionalinbreadth,originality andinsight,speakloudly,shoutout,thatbiologyisbecomingaquantitativescience,that biologycanbeapproachedsystematically,thatentirebiologicalsystemssuchasmetabo- lism, signal transduction, and gene regulation, their interacting networks, or even higher levelsoforganization,canbestudiedandfullycharactierizedthroughthecombinationof theory,simulationandexperiment.Thenewlyestablishedfundingofcomputationalbiol- ogybyprivateresearchfoundationsandbytheNationalInstitutesofHealthisessentialto provide the fuel through sustained, adequate funding. As the former government ofªcial who established those ªrst programs for bringing the tools of computer and information sciencetobearonbiologicalsciencesresearch,Iammyselfhonoredtohavebeenevena small part of this revolution and excited about what lies ahead. Justreadthisbook.Now.Foronce,readtheentirebook,whichgivesdirectchallenges for experimentalists and clear directions for current graduate students. John C. Wooley Associate Vice Chancellor, Research University of California, San Diego Introduction: Understanding Living Systems In1982,atasymposiumcommemoratingthetenthanniversaryofthedeathofthebiolo- gistandmodelerAharonKatzir-Katchalsky,GeorgeOsterrecalledKatchalskyjokingthat “Biologistscanbedividedintotwoclasses:experimentalistswhoobservethingsthatcan- notbeexplained,andtheoreticianswhoexplainthingsthatcannotbeobserved.”1Thejoke is still funny because it sums up in one statement the traditionally awkward alliance be- tweentheoryandexperimentinmostªeldsofbiologyaswellasscienceingeneral.The statementthatexperimentalists“observethingsthatcannotbeexplained”mirrorstheview of many traditional theorists that biological experiments are for the most part simply de- scriptive(“stampcollecting”asRutherfordmayhaveputit)andthatmuchoftheresulting biological data are either of questionable functional relevance or can be safely ignored. Conversely, the statement that theorists “explain things that cannot be observed” can be takentoreºecttheviewofmanyexperimentaliststhattheoristsaretoofarremovedfrom biological reality and therefore their theories are not of much immediate usefulness. In fact,ofcourse,whenpresentingtheirdata,mostexperimentalistsdoprovideaninterpre- tation and explanation for the results, and many theorists aim to answer questions of biological relevance. Thus, in principle, theorists want to be biologically relevant, and experimentalistswanttounderstandthefunctionalsigniªcanceoftheirdata.Itistheprem- iseofthisbookthatconnectingthetworequiresanewapproachtobothexperimentsand theory. We hope that this book will serve to introduce practicing theorists and biologists and especially graduate students and postdocs to a more coordinated computational approach to understanding biological systems. Why Is Modeling Necessary? Beforeconsideringtheessentialfeaturesofthisnewapproach,itmaybeworthwhiletoin- dicatewhyitasneeded.Putasdirectlyaspossible,thesheercomplexityofmolecularand cellular interactions currently being studied will increasingly require modeling tools that canbeusedtoproperlydesignandinterpretbiologicalexperiments.Theadventofmolecu- lar manipulation techniques such as gene knockouts has made it very clear to most experimentaliststhatthesystemstheyarestudyingarefarmorecomplexanddynamicthan had often previously been assumed. Concurrently, the recent development of large-scale assaytechnologiessuchascDNAarraysisalreadyprovidingexperimentalistswithanal- most overwhelming quantity of data that defy description or characterization by conven- tionalmeans.Asourunderstandingofthecomplexityofthesystemswestudygrows,we willincreasinglyhavetorelyonsomethingotherthanintuitiontodeterminethenextbest experiment. When the question of the role of modeling in biological investigations is raised, it is quite common for experimentalists to state that not enough is yet known to construct a xiv Introduction model of their system. Of course, properly formulated, a model is often most useful as a way to decide what data are now necessary to advance understanding. As already stated, the complexity of biological systems makes it increasingly difªcult to identify the next bestexperimentwithoutsuchatool.However,inreality,experimentalistsalreadyhaveand usemodels.Indeed,itisstandardpracticeforexperimentaliststobeginandendresearch seminars with a “block and arrow” diagram. These diagrams constitute the experimentalists’understanding of how the components of their system interact. Accordingly, theyareakindofmodel.However,“models”inthisformhavenoexplicitformorstructure butthediagramitself,withnoassociatedformalspeciªcation.Withoutrealparametersand mathematically deªned relationships between the components, they cannot be falsiªed; theycannotmakequantiªablytestablepredictions,orserveasamechanismforconveying detailed information to other researchers on how the experimentalist is really thinking about the system under study. Fromthepointofviewofatheorist,theenormousandgrowingamountofdetailedin- formationavailableaboutbiologicalsystemsdemandsanintegrativemodelingapproach. Atomicdescriptionsofmolecularstructureandfunction,aswellasmolecularexplanations ofcellularbehavior,arenowincreasinglypossible.Accesstomoreandmoresophisticated high-resolutioninvivoandinvitroimagingtechnologiesnowprovidesthepossibilityof testing theoretical ideas at a fundamentally new level. At the atomic scale, technologies suchaselectronmicroscopy,X-raycrystallography,andnuclearmagneticresonancespec- troscopyallowdirectmeasurementofthewayinwhichatomicinteractionsdeterminethe three-dimensionalgeometry,thechemicalandphysicalcharacteristics,andthefunctionof biomolecules. At the molecular level, confocal microscopy, calcium imaging, and ºuorescenttaggingofproteinshavemadeitpossibletotrackthemovementandreactions of molecules within single living cells. Combined with the new technologies for large- scalegeneexpressionassays,thesetechnologiesarenowyieldingvastamountsofquanti- tative data on such cellular processes as developmental events, the cell cycle, and meta- bolic and signal transduction pathways. We now have the technology to track the expressionpatternofthousandsofgenesduringthelifetimeofacell,andtotracetheinter- actionsofmanyoftheproductsofthesegenes.Weareenteringanerawhenourabilityto collect data may no longer be the primary obstacle to understanding biology. Whileourcapacityfortestingtheoreticalideasisexpandingdramatically,takingadvan- tage of that capacity requires models that are directly linked to measurable biological structuresandquantities.Atthesametime,thecomplexityofbiologicalprocessesmakes itlessandlesslikelythatsimpliªed,abstractmodelswillhavethecomplexitynecessaryto capturebiologicalreality.Tobeusefultoexperimentalists,modelsmustbedirectlylinked tobiologicaldata.Forthisreason,modelersinterestedinbiologywillhavetodevelopnew mathematicaltoolsthatcandealwiththecomplexityofbiologicalsystems,ratherthantry to force those systems into a form accessible to the simpler tools now available. Introduction xv What Type of Modeling Is Appropriate? It is our belief that to make progress in understanding complex biological systems, experimentalistsandtheoreticiansneedeachother.However,tomakethisconnection,we needtorethinktheroleofmodelsinmodernbiologyandwhattypesofmodelsaremost appropriate.Traditionally,biologicalmodelshaveoftenbeenconstructedtodemostratea particulartheorist’sideasofhowaparticularsystemorsubsystemactuallyworks.Inthis type of modeling, the functional idea exists before the model itself is constructed. While modelsofthissortcanmakeexperimentallytestablepredictions,itisusuallyeasyenough to adjust the model’s parameters to protect the original functional idea. When presented withsurprisingnewresults,oftenthetheoristcanadaptthemodeltoªtthenewdatawith- out having to reevaluate the basic premises of the model. Inourview,recentadvancesinbiologyrequireamoredirectconnectionbetweenmod- eling and experiment. Instead of being a means to demonstrate a particular preconceived functionalidea,modelingshouldbeseenasawaytoorganizeandformalizeexistingdata onexperimentallyderivedrelationshipsbetweenthecomponentsofaparticularbiological system.2Insteadofgeneratingpredictionsthatcanbeconªrmedbytheirmodel,modelers shouldbemoreconcernedwithensuringthattheassumptionsandstructureofaparticular modelcanbefalsiªedwithexperimentaldata.Asanaidinunravelingbiologicalcomplex- ity, what a model cannot do is often more important than what it can do. It follows that models will have to generate experimentally measurable results and include experimentally veriªable parameters. Models of this kind should not be seen or presented as repre- sentations of the truth, but instead as a statement of our current knowledge of the phenomenonbeingstudied.Usedinthisway,modelsbecomeameansofdiscoveringfunc- tionally relevant relationships rather than devices to prove the plausibility of a particular preexisting idea. Manyoftheseobjectivescanbemorereadilyobtainedifthereisaclosestructuralcon- nectionbetweenbiologicaldataandthemodelitself.Bystructuralwemeanthatthemodel itselfreºectsthestructureofthesystembeingstudied.Suchanarrangementmakesitmore likely that the modeler will learn something new (taken from the biology in some sense) fromconstructingthemodel.Infact,allmodelersshouldbepreparedtoanswertheques- tion: What do you know now that you did not know before? If the answer is “that I was correct,” it is best to look elsewhere. Experimentally, models of this type almost always provide extremely important infor- mationonwhattypesofexperimentaldataarecurrentlylacking.Thus,ironically,whilewe have said that the classical criticism of biological modeling is that we do not yet know enough to construct the model, model construction often makes it very clear that one knowsevenlessthanonethoughtaboutaparticularsystem.Whenitisspeciªcandexperi- mentally addressable, This knowledge is extremely valuable in planning research. It xvi Introduction follows that models should be evaluated by their ability to raise new experimental questions, not by their ability to account for what is already known or suspected. In conse- quence, understanding biological systems becomes an integrated, iterative process, with models feeding experimental design while experimental data in turn feed the model. Thistypeofinteractionbetweenmodelsandexperimentwillinevitablyalterthenature ofbothenterprisesandthescientiststhatusethem.Ifmodelsaredesignedtodirectlyinte- grateexperimentaldata,thenexperimentalinvestigationswillalsohavetoprovidedatafor testingmodels.Thisinturnwillrequirethedevelopmentofnewexperimentaltechniques. Themorecloselymodelsarelinkedtoexperimentaldata,thelessobviousthedistinction betweenatheoristandanexperimentalist.Mostoftheauthorsinthisbookareavantgarde inthattheyaretheoristswhocollaboratecloselywithexperimentalists.However,itistime to train a new generation of biologists who are equally comfortable with models and experiments. What Is the Purpose of This Book? Thisbookarosefromagraduatecourseofthesametitlethatweorganizedandranatthe CaliforniaInstituteofTechnologyinthewintertermof1998.Theobjectiveofthecourse wastoprovidebiologygraduatestudentsandpostdoctoralscholarswithanintroductionto modelingtechniquesappliedtoarangeofdifferentmolecularandcellularbiologicalques- tions.Theorganizationofthecourseanditspedagogywereinformedbyeffortsoverthe past15yearstointroduceasimilarapproachtomodelingneurobiology.Asinourefforts to support the growth and extension of computational neuroscience,2 this book reºects a convictionthatmodelingtechniquesmustinevitablybecomearequiredfeatureofgraduate traininginbiology.Wehopethatthisbookwillserveasanintroductiontothosemolecular and cellular biologists who are ready to start modeling their systems. How Should this Book Be Used? The book is aimed at the graduate and advanced-level student and is intended to provide instructionintheapplicationofmodelingtechniquesinmolecularandcellbiology.Since thebookwasoriginallyusedtosupportteachingatCaltech,wehopeitwillproveuseful for others organizing similar courses. While we hope the book will be a useful instruc- tional tool, we have also attempted to make it useful as a stand-alone introduction to the ªeld for interested researchers. We expect the readers of this book to include cell biologists,developmentalbiologists,geneticists,structuralbiologists,andmathematicalbiolo- gists, as well as the computational neuroscience community. Introduction xvii Whileeachchapterstartswithanoverviewofthebiologicalsysteminquestion,noef- fortismadetoprovideacompleteenoughbiologicaldescriptiontoreallyunderstandthe detailedobjectivesofthemodelsdescribed.Instead,attentionisprincipallyfocusedonthe modeling approach itself. We regard our principal audience as biologists interested in modeling.However,wewouldhopethatnonbiologistsinterestedinmakingusefulbiologi- calmodelswouldalsoªndsupporthere.Nonbiologists,however,arestronglyencouraged totakeadvantageofthereferencesattheendofeachchaptertoªllintheirunderstanding of the biological systems being studied. Thespanofdisciplinescontributingtothisemergingªeldisvast.Noreaderwillbefully conversant with all of them. Nor can this volume cover all aspects of them in sufªcient depthandbreadth.Wethereforepresentthisbook,notasacomprehensivetreatise,butasa samplerandastartingpointforthosewithspeciªcinterests.Forthisreason,wehavetaken caretoincludeextensivereferencestoimportantworkineachsubjectarea.However,we anticipateexplosivegrowthinthisªeld,whichwillalmostcertainlyrequirecontinuedvig- ilant searching of the latest peer-reviewed literature for new applications of modeling techniques. Organization of the Book The book is organized in two parts: models of gene activity, i.e., genetic networks, and models of interactions among gene products, i.e., biochemical networks. Each part in- cludestreatmentsofmodelingatseveraldifferentscales,fromthesmallesttothelargest. Ourhopeisthatinthiswaythereaderwillbecomeawareoftheadvantagesanddisadvan- tages of modeling at ªner scales, before considering more simpliªed models aiming at larger scales. It is our view that ultimately these levels of scale must be brought together and articulated into one large framework for understanding biology. For this reason, we have included a chapter by Bard Ermentrout that addresses the issue of scaling between models at different levels of description. The inclusion of this chapter also serves to em- phasizetheimportanceweplaceonthefurtherdevelopmentofmathematicaltoolstosup- port biological modeling. Part I. Modeling Genetic Networks Theªrstpartofthebookdealswithmodelsofgeneregulation,thatis,protein–DNAand, indirectly,DNA–DNAinteractions.Themodelingapproachespresentedinthesechapters rangefrommodelsofindividualgenes,throughmodelsofinteractionsamongafewgenes, to techniques intended to help unravel the interactions among hundreds or thousands of genes. It is well understood that regulated gene expression underlies cellular form and xviii Introduction functioninbothdevelopmentandadultlife.Changesingeneexpressionpatternsgenerate the phenotypic variation necessary for evolutionary selection. An understanding of gene regulationisthereforefundamentaltoallaspectsofbiology.Yet,atpresent,knowledgeof generegulation,itsevolution,anditscontrolofcellfunctionispatchyatbest.Thefirstpart ofthisbookreviewsseveraltoolsandmethodsthatarecurrentlyavailableforunraveling genetic regulatory networks at various levels of resolution and abstraction. Chapter 1. Modeling the Activity of Single Genes How is the activity of individual genesregulated?Thistutorialchapterbeginsbyrelatingtheregulationofgeneactivityto thebasicprinciplesofthermodynamicsandphysicalchemistry.Theauthorsthengoonto discusstheneedfordifferenttypesofmodelstosuitdifferentmodelsystems,theavailabil- ity of experimental data, and the computational resources needed. Chapter 2. A Probabilistic Model of a Prokaryotic Gene and Its Regulation This chapterprovidesadetailedinvestigationofanexamplemodelsysteminwhichthenumber ofregulatorymoleculesdetermininggeneactivationissmallandgeneregulationisthere- fore stochastic. In addition to presenting a detailed model, it discusses the need for stochastic modeling, the advantages it provides in this case, and various examples of stochastic models applied to other systems. Chapter3.ALogicalModelofcis-RegulatoryControlinaEukaryoticSystem The differential regulation of eukaryotic genes permits cell differentiation in space and time andistypicallymorecomplexthangeneregulationinprokaryotes.Buildingonthetheo- reticalframeworkpresentedinchapter1,thischapterpresentsadetailedcharacterization oftranscriptionalregulationofaeukaryoticgeneinwhichlargenumbersofregulatoryfac- tors interact in modular ways to control the transcription rate. Chapter4.TrainableGeneRegulationNetworkswithApplicationtoDrosophilaPat- tern Formation Chapters 1 to 3 address the regulation of individual genes. However, gene regulation by its nature also involves interactions of whole systems of interacting genes. While we may someday be able to explore these interactions based on detailed models of individual genes, this chapter presents a framework for building phenomeno- logicalmodelsthatself-organizetoreplicatelarge-scalegeneactivitypatterns.Suchmod- els can also be valuable aids in forming hypotheses and planning experiments. Chapter5.GeneticNetworkInferenceinComputationalModelsandApplicationsto Large-Scale Gene Expression Data As already discussed, several emerging technologies now permit the simultaneous determination of the expression levels of thousands of genes in cell tissues. How can we unravel and reconstruct regulatory interactions among genesinsuchlarge-scaleexpressionassays?Thischapterreviewsanumberofpotentially Introduction xix usefulmeasuresforuncoveringcorrelatedandpotentiallycausallyrelatedevents.Itpres- ents several studies that used these methods. Part II. Modeling Biochemical Networks The second part of this book considers interactions among the proteins produced by ge- neticregulation.Itstartswithmodelingtechniquesthathelpusunderstandtheinteractions betweensinglemolecules,andthengoesontoconsidermodelsaimedatunderstandingre- actions and diffusion by large numbers of molecules. The later chapters emphasize the richness and complexity of the behavior that can arise in such networks and are a strong testament to the need for modeling in molecular biology. Chapter 6. Atomic-Level Simulation and Modeling of Biomacromolecules Ulti- mately,wewillwanttobuildourunderstandingofbiologicalsystemsfromtheªrstprinci- ples of the physics of atomic interactions. At present, however, the computational resources required to explain molecular function in terms of fundamental quantum me- chanicslimitoureffortstosystemsofatmostafewhundredinteractingatoms.Thischap- terpresentsaseriesofapproximationstoquantummechanics,anddemonstratestheiruse inmodelinginteractionsofhundredsofthousandsofatoms.Examplesofpredictingmo- lecular structure and activity for drug design are presented. The chapter also provides a justiªcationforthesimpler,phenomenologicalmodelsofmolecularinteractionusedinthe restofthisbookbypointingoutthatatomicinteractionmodelswouldbeseveralordersof magnitude too computationally intensive. Chapter 7. Diffusion Diffusionistheprocessbywhichrandommovementofindividual moleculesorionscausesanoverallmovementtowardregionsoflowerconcentration.Dif- fusionprocesseshaveaprofoundeffectonbiologicalsystems;forexample,theystrongly affect the local rate of chemical reactions. Historically, diffusion has frequently been ne- glectedinmolecularsimulations.However,asthischapterillustrates,diffusionoftenhasa crucial impact on modeling results. The chapter describes a number of approximation methodsformodelingdiffusion,withparticularreferencetocomputationalload.Example simulations are presented for diffusion and buffering in neural dendritic spines. Chapter 8. Kinetic Models of Excitable Membranes and Synaptic Interactions Whilechapter7looksatthefreemovementofmoleculesinasolvent,thischapteriscon- cernedwithselectivemoleculartransportacrosscellmembranes.Itconcentratesonmod- els of inorganic ion transport channels. The stochastic mechanisms underlying voltage- dependent, ligand-gated, and second-messenger-gated channels are discussed. In each case,arangeofrepresentations,frombiophysicallydetailedtohighlysimpliªedtwo-state (open/closed)models,arepresented.Althoughsuchchannelsarestructur-allyverydiffer- entfromproteintransportchannels,themodelingframeworkpresentedisapplicableinall