Lecture Notes on Mathematical Modelling in the Life Sciences Moisés Santillán Chemical Kinetics, Stochastic Processes, and Irreversible Thermodynamics Lecture Notes on Mathematical Modelling in the Life Sciences SeriesEditors AngelaStevens MichaelC.Mackey Forfurthervolumes: http://www.springer.com/series/10049 Moisés Santillán Chemical Kinetics, Stochastic Processes, and Irreversible Thermodynamics 123 MoisésSantillán CentrodeInvestigaciónydeEstudios AvanzadosdelIPN(Cinvestav) UnidadMonterrey VíadelConocimiento201,ParquePIIT ApodacaNL,México ISSN2193-4789 ISSN2193-4797(electronic) ISBN978-3-319-06688-2 ISBN978-3-319-06689-9(eBook) DOI10.1007/978-3-319-06689-9 SpringerChamHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2014941453 MathematicsSubjectClassification(2010):92C40,92C45,80A30,82C31,60H35,34F05 ©SpringerInternationalPublishingSwitzerland2014 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 and executed on a computer system, for exclusive use by the purchaser of the work. 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.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. While the advice and information in this book are believed to be true and accurate at the date of publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with respecttothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) ToMatilda,ourmostrecentfamilymember, andthesourceofallinspiration! ToMariana,mywife,whoselove,support, andcomprehensionmadethisbookpossible! Tomyparents,whohavepavedmypath throughlifewiththeirboundlessloveand wisdom! ToPedroSantillán, LourdesA.Vega-Acosta, FernandoAngulo-Brown,andMichaelC. Mackey,mymentors! Preface Thereisalongtraditionregardingmathematicalmodelingofbiologicalphenomena, which dates back to the very beginning of modern science. However, there has been a revival in the last decade and a half, in which much effort has been made to elucidate the dynamic behavior of intracellular processes. This has been due in part to the development of impressive novel experimental techniques that permit precisemeasurementsatthesinglecell,andevenatthesinglemoleculelevels.But alsothankstotheavailabilityoflow-costandhighlyefficientcomputationalpower, which permits to numerically explore the dynamics of quite complex systems. One particular problem that has generated big interest is the stochasticity of gene expression. Over the last years it has become clear that the dynamics of most biological phenomena can be studied via the techniques of either nonlinear dynamics or stochastic processes. In either case, the biological system is usually visualized as asetofinterdependentchemicalreactionsandthemodelequationsarederivedout ofthispicture.Deterministic,nonlineardynamicmodelsrelyonchemicalkinetics, while stochastic models are developed from the chemical master equation. Recent publicationshavedemonstratedthatdeterministicmodelsarenothingbutanaverage description of the behavior of unicellular stochastic models. In that sense, the mostdetailedmodelingapproachisthatofstochasticprocesses.However,boththe deterministicandthestochasticapproachesarecomplementary.Thevastamountof availabletechniquestoanalyticallyexplorethebehaviorofdeterministic,nonlinear dynamicalmodelsisalmostcompletelyinexistentfortheirstochasticcounterparts. On the other hand, the only way to investigate biochemical noise is via stochastic processes. It is my impression that the excitement of developing a new science (that has been termed systems biology) has made people unaware of a large amount of related discoveries performed in the late nineteenth and early twentieth centuries by people like Planck, Nernst, de Groot, Prigogine, etc., which we have classified as physical chemistry and irreversible thermodynamics. Moreover, although not as extensive a dismissal, there is no general acknowledgment on the part of systems biologists of the common mathematical grounds shared by biochemical vii viii Preface andelectrophysiologicalmodels;thelatteronesareusuallyclassifiedintherealmof biophysics.Oneoftheobjectivesofthepresentbookistoshowthattheapproaches of deterministic nonlinear dynamics, stochastic chemical master equation, and irreversiblethermodynamicalchemistryareallcomplementary,andthattheirproper combination allows a deeper understanding of the dynamic behavior of a large varietyofbiologicalsystems.Inparticular,wetackleinthisbookgeneexpression andiontransportacrossmembranes. There are many great books on nonlinear dynamics, stochastic processes, irre- versible thermodynamics, physical chemistry, and biochemistry. Nonetheless, to the best of my knowledge, there is none that brings all of these theories together, in an introductory but formal and comprehensive manner, for people interested in modelingbiologicalphenomena.Thepresentbookisaimedatfilling,atleastinpart, thisgap.InordertoachievethisgoalIdecidedtofollowahands-onconstructivist approach. The theory is developed stepwise, starting from the simplest concepts, and building upon them to derive, one step at a time, a more general framework. Butinsteadoffirstdevelopingthetheoryandlaterstudyingitspossibleapplications to biological systems, the examples are introduced right away. On the one hand, eachtheoreticaldevelopmentismotivatedbyspecificbiologicalexamples.Butalso, everynewmathematicalderivationisimmediatelyappliedtooneormorebiological systems. Thetargetaudienceofthisbookaremainlylastyearundergraduateandgraduate students with a solid mathematical background (physicists, mathematicians, and engineers),aswellaswithbasicnotionsofbiochemistryandcellularbiology,who are interested in learning the previously described techniques to model biological phenomena.Thebookcanalsobeusefultostudentswithabiologicalbackground, who are interested in mathematical modeling, and have a working knowledge of calculus,differentialequations,andbasicnotionsofprobabilitytheory. Thebookisorganizedasfollows.Chapters1and2areintroductoryandinthem somebasicnotionsofchemicalkineticsandthermodynamicsare,respectively,pre- sented.ThereadersalreadyfamiliarwiththismaterialcanjumpdirectlytoChap.3. However,IencourageeveryonetoatleasttakealookatChaps.1and2,becausethe materialthereintroducediswidelyemployedintherestofthebook.InChap.3,the so-called telegraph stochastic process is analyzed from all the three perspectives discussed above, and the results are employed to discuss some aspects of ion channelgating,promoterrepressionandactivation,andproteinphosphorylationand dephosphorylation.InChap.4thefollowingstochasticprocessesareintroducedand analyzedinconnectionwiththeproductionanddegradationofbiologicalmolecules: Poissonprocess,exponentialdecay,andbirth–deathprocess.InChap.5,birth–death processes are generalized to account for enzyme kinetics. The concept of quasi- stationary approximation for stochastic processes is also introduced in Chap.5. Chapter 6 is advocated to studying a generalization of the telegraph process in the context of the chemical interaction between one receptor and several ligands. Chapter7furthergeneralizedtheresultsinChap.6toaccountforcooperativity.In Chap.8, all the results and developments introduced in the previous chapters are Preface ix appliedtothestudyofgeneexpression.Finally,inChap.9,thedevelopedtheoryis appliedtostudyingiontransportacrossmembranes. Likealllong-termprocesses,thewritingofthepresentbookinvolvednumerous peopleandinstitutions.Inparticular,Iamindebtwithmyworkingplace,theCentro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional, for grantingmeasabbaticalleaveinwhichIfoundthenecessarytimetowrite.Ialso thank the Department of Physiology of McGill University and the International Centre for Theoretical Physics for their hospitality and for providing me with the mostadequateenvironmenttocarryoutthisproject.Finally,ofallthepeoplewho wereinvolved,onewayoranother,intheprocessofwritingthepresentbook,Iwish to emphasize Michael C. Mackey, Emanuel Salazar-Cabazos, Román U. Zapuién- Campos, and Luis U. Aguilera-de-Lira. I am deeply grateful with all the persons who contributed to the present book, but the support and advice of the above- mentionedpeopleweresoimportantthat,instrictsense,madethisbookpossible.In spiteofseveralreviewsitispossiblethatsomemistakesarefoundalongthebook. I,andnooneelse,amtheonlyresponsibleforallofthem. Apodaca,Mexico MoisésSantillán