Table Of ContentBranching Processes
in Biology
Marek Kimmel
David E. Axelrod
Springer
Interdisciplinary Applied Mathematics
Volume 19
Editors:
S.S.Antman J.E.Marsden
L.Sirovich S.Wiggins
MathematicalBiology
L.Glass,J.D.Murray
MechanicsandMaterials
R.V.Kohn
SystemsandControl
S.S.Sastry,P.S.Krishinaprasad
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Marek Kimmel David E. Axelrod
Branching Processes
in Biology
With54Illustrations
1 3
MarekKimmel DavidE.Axelrod
DepartmentofStatistics DepartmentofGenetics
RiceUniversity RutgersUniversity
Houston,TX77251 NelsonBiolabs
P.O.Box1892 904AllisonRoad
USA Piscataway,NJ08854
[email protected] USA
[email protected]
Editors
S.S.Antman J.E.Marsden
DepartmentofMathematics ControlandDynamicalSystems
and MailCode107-81
InstituteforPhysicalScienceandTechnology CaliforniaInstituteofTechnology
UniversityofMaryland Pasadena,CA91125
CollegePark,MD20742 USA
USA
L.Sirovich S.Wiggins
DivisionofAppliedMathematics ControlandDynamicalSystems
BrownUniversity MailCode107-81
Providence,RI02912 CaliforniaInstituteofTechnology
USA Pasadena,CA91125
USA
MathematicsSubjectClassification(2000):60J80,60J85,92DXX
LibraryofCongressCataloging-in-PublicationData
Kimmel,Marek,1953–
Branchingprocessesinbiology/MarekKimmel,DavidE.Axelrod.
p. cm.—(Interdisciplinaryappliedmathematics;19)
Includesbibliographicalreferences(p.).
ISBN0-387-95340-X(hc:alk.paper)
1.Biology—Mathematicsmodels. 2.Branchingprocesses. I.Title. II.Interdisciplinary
appliedmathematics;v.19.
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Preface
Thetheoryofbranchingprocessesisanareaofmathematicsthatdescribessitua-
tionsinwhichanentityexistsforatimeandthenmaybereplacedbyone,two,or
moreentitiesofasimilarordifferenttype.Itisawell-developedandactivearea
ofresearchwiththeoreticalinterestsandpracticalapplications.
The theory of branching processes has made important contributions to biol-
ogyandmedicinesinceFrancisGaltonconsideredtheextinctionofnamesamong
theBritishpeerageinthenineteenthcentury.Morerecently,branchingprocesses
have been successfully used to illuminate problems in the areas of molecular
biology, cell biology, developmental biology, immunology, evolution, ecology,
medicine,andothers.Fortheexperimentalistandclinician,branchingprocesses
havehelpedintheunderstandingofobservationsthatseemcounterintuitive,have
helped develop new experiments and clinical protocols, and have provided pre-
dictionswhichhavebeentestedinreal-lifesituations.Forthemathematician,the
challengeofunderstandingnewbiologicalandclinicalobservationshasmotivated
thedevelopmentofnewmathematicsinthefieldofbranchingprocesses.
The authors of this monograph are a mathematician and a cell biologist who
have collaborated on investigations in the field of branching processes for more
thanadecade.Inthismonograph,wehavecollectedexamplesofapplicationsof
branching processes from our own publications and from publications of many
otherinvestigators.Eachexampleisdiscussedinthecontextoftherelevantmath-
ematics. We have made an effort to collect and review much of the published
literature which has applied branching processes to problems in molecular and
cellularbiology,aswellasselectedexamplesfromthefieldsofhumanevolution
andmedicine.
Theintendedaudiencesforthismonographaremathematiciansandstatisticians
whohavehadanintroductiontostochasticprocessesbuthaveforgottenmuchof
theircollegebiology,andbiologistswhowishtocollaboratewithmathematicians
viii Preface
and statisticians. Both audiences will find many examples of successful appli-
cationsofbranchingprocessestobiologicalandmedicalproblems.Asanaidto
understandthespecificexamples,wehaveprovidedtwointroductorychapters,one
withbackgroundmaterialinmathematicsandtheotherwithbackgroundmaterial
inbiology,aswellastwoglossaries.Asadidacticaidwehaveincludedproblem
setsattheendofChapters3,4,and5.
The book is organized as follows: Chapter 1 provides a mathematical back-
ground and motivating examples of branching processes. Chapter 2 provides an
introduction to biological terms and concepts. The subsequent chapters are di-
videdintospecificareasofbranchingprocesses.Eachofthesechaptersdevelops
theappropriatemathematicsanddiscussesseveralapplicationsfromthepublished
literature.Chapter3discussestheGalton–Watsonprocess,theoldest,simplestand
bestknownbranchingprocess.Chapter4discussestheage-dependentprocess–
Markov case, the time-continuous branching process with exponential lifetime
distributions.Chapter5discussestheBellman–Harrisprocess,anage-dependent
process. Chapter 6 gives a more systematic treatment of multitype processes, in
which progeny may be of many types. Chapter 7 discusses branching processes
withinfinitelymanytypes,stressinginterestingpropertieswhicharedifferentfrom
thefinitemultitypesituation.Appendicesprovideinformationonprobabilitygen-
erating functions, construction of the probability space for the Bellman–Harris
process, as well as a brief introduction to the Jagers–Crump–Mode process (the
generalbranchingprocess).
Wehavemadeanefforttobroadlyreviewthepublishedliteratureonbranching
processesappliedtobiology.However,wehadtoselectspecificexamplesandwe
wishtoapologizetoourcolleagueswhoseworkhasnotbeencited.Wewelcome
commentsfromcolleaguesandstudentswhoareinterestedthefieldofbranching
processes.
AsearchofanyuniversitylibraryoranInternetbookstorewillrevealanumber
ofvolumesdevotedtobranchingprocesses.Amongthemostimportant,wecite
thefundamentalbooksbyHarris(1963)andbyAthreyaandNey(1972).Multi-
typebranchingprocesseswerefirstcoveredinthebookbyMode(1971).General
branchingprocesses,inasystematicway,wereexploredbyJagers(1975).Eachof
these classics, particularly Jagers (1975), includes some biological applications.
An important book concerning estimation of branching processes is by Guttorp
(1991).TheworkbyAsmussenandHerring(1983)involvesaverymathematical
approach.Inaddition,thereexistatleastadozenortwoofcollectionsofpapers
and more specialized volumes. For example, Yakovlev and Yanev (1989) deal
withcellproliferationmodels,mainlyusingbranchingprocesses.Recently,Pakes
(2000)preparedareportonbiologicalapplicationsofbranchingprocesses,which
is wider in scope (it has much on spatial branching and ecology, for example),
but less detailed, although an area of overlap with our book exists. We believe
thatthescopeofthepresentvolumeisuniqueinthatitillustratesaparadigm,in
whichtheoreticalresultsarestimulatedbybiologicalapplicationsandbiological
processesareilluminatedbymathematics.
Preface ix
Wegratefullyacknowledgesupportfromthefollowingsources:NationalInsti-
tutes of Health, National Science Foundation, Keck’s Center for Computational
Biology at Rice University, New Jersey Commission on Cancer Research, Can-
cerInstituteofNewJersey,PetersonFund,HydeandWatsonFoundation,Glazer
FamilyFund,RiceUniversity,SilesianTechnicalUniversity,theUniversityofPau,
andRutgersUniversity.MarekKimmelwasworkingonthefinaldraftofthisbook
while on a sabbatical leave at the Human Genetics Center at the University of
TexasinHouston.
We thank Dr. William Sofer and Dr. Navin Sinha and several anonymous re-
viewersforhelpfulsuggestionsonthemanuscript.Dr.AdamBobrowskiproofread
thebookforitsmathematicalcorrectness.Hiscriticalremarksimproveditsignif-
icantly. Remaining inaccuracies are our fault. Generations of graduate students
attheStatisticsDepartmentatRiceUniversityprovidedwelcomefeedback.No-
tably,DavidStivers,ShanePankratz,andChadShawdidtheirdoctoralresearch
in the area of branching processes. Professor Ovide Arino of the University of
Pau,France,hascontributed,overtheyears,totheunderstandingofconnections
between stochastic and deterministic population models. Professor Jim Thomp-
son encouraged teaching this material at Rice and provided much constructive
criticism. Professor Peter Jagers of Chalmers University in Gotheborg, Sweden,
hostedMarekKimmelonseveraloccasionsandprovidedmuchneededfeedback.
Our colleague Dr. Peter Olofsson was helpful with the more esoteric aspects of
thetheory.Ourfamiliesshowedwarmthandpatienceduringthegestationofthis
book.Jasiuhelpedwithpageproofs,andKasiaandDanieldidnothidewordsof
advice.
Wededicatethisbooktoourstudents,ourteachers,andourfamilies.
Houston,Texas,USA MarekKimmel
Piscataway,NewJersey,USA DavidE.Axelrod
