Chemical Kinetics, Stochastic Processes, and Irreversible Thermodynamics

139 Pages·2014·1.792 MB·English

by  Moisés Santillán (auth.)

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
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andcomprehensionmadethisbookpossible!
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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