Information Geometry and Population Genetics: The Mathematical Structure of the Wright-Fisher Model

323 Pages·2017·4.477 MB·English

by  Julian Hofrichter, Jürgen Jost, Tat Dat Tran (auth.)

Understanding Complex Systems
Julian Hofrichter
Jürgen Jost
Tat Dat Tran
Information
Geometry and
Population
Genetics
The Mathematical Structure of the
Wright-Fisher Model
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Julian Hofrichter (cid:129) JuRrgen Jost (cid:129) Tat Dat Tran
Information Geometry
and Population Genetics
The Mathematical Structure
of the Wright-Fisher Model
123
JulianHofrichter JuRrgenJost
MathematikindenNaturwissenschaften MathematikindenNaturwissenschaften
Max-Planck-Institut MaxPlanckInstitut
Leipzig,Germany Leipzig,Germany
TatDatTran
MathematikindenNaturwissenschaften
MaxPlanckInstitut
Leipzig,Germany
ISSN1860-0832 ISSN1860-0840 (electronic)
UnderstandingComplexSystems
ISBN978-3-319-52044-5 ISBN978-3-319-52045-2 (eBook)
DOI10.1007/978-3-319-52045-2
LibraryofCongressControlNumber:2017932889
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Preface
Populationgeneticsisconcernedwiththedistributionofalleles,thatis,variantsat
ageneticlocus,inapopulationandthedynamicsofsuchadistributionacrossgen-
erations under the influences of genetic drift, mutations, selection, recombination
andotherfactors[57].TheWright–Fishermodelisthebasicmodelofmathematical
populationgenetics.ItwasintroducedandstudiedbyRonaldFisher,SewallWright,
Motoo Kimura and many other people. The basic idea is very simple. The alleles
in the next generation are drawn from those of the current generation by random
samplingwith replacement.When thisprocessis iterated acrossgenerations,then
byrandomdrift,asymptotically,onlyasingleallelewillsurviveinthepopulation.
Once this allele is fixed in the population,the dynamicsbecomes stationary. This
effectcan be counteredby mutationsthat might restore some of those alleles that
had disappeared.Or it can be enhancedby selection that mightgive one allele an
advantageoverthe others,thatis, ahigherchanceofbeingdrawnin thesampling
process. When the alleles are distributed over several loci, then in a sexually
recombiningpopulation,theremayalsoexistsystematicdependenciesbetweenthe
allele distributions at different loci. It turns out that rescaling the model, that is,
lettingthepopulationsizegotoinfinityandthetimestepsgoto0,leadstopartial
differentialequations,called the Kolmogorovforward(orFokker–Planck)andthe
Kolmogorovbackwardequation.These equationsare well suited for investigating
theasymptoticdynamicsoftheprocess.Thisiswhatmanypeoplehaveinvestigated
beforeusandwhatwealsostudyinthisbook.
So, what can we contribute to the subject? Well, in spite of its simplicity,
the model leads to a very rich and beautiful mathematical structure. We uncover
this structure in a systematic manner and apply it to the model. While many
mathematicaltools,fromstochasticanalysis,combinatorics,andpartialdifferential
equations,havebeenappliedtotheWright–Fishermodel,webringinageometric
perspective. More precisely, information geometry, the geometric approach to
parametric statistics pioneered by Amari and Chentsov (see, for instance, [4, 20]
and for a treatmentthat also addresses the mathematical problemsfor continuous
sample spaces [9]), studies the geometry of probability distributions. And as a
remarkable coincidence, here we meet Ronald Fisher again. The basic concept
v
vi Preface
of information geometry is the Fisher metric. That metric, formally introduced
by the statistician Rao [102], arose in the context of parametric statistics rather
than in population genetics, and in fact, it seems that Fisher himself did not see
thistightconnection.Anotherfundamentalconceptofinformationgeometryisthe
Amari–Chentsovconnection[3,10].Asweshallargueinthisbook,thisgeometric
perspective yields a very natural and insightful approach to the Wright–Fisher
model,andwithitshelpwecaneasilyandsystematicallycomputemanyquantities
of interest, like the expected times when alleles disappear from the population.
Also, information geometry is naturally linked to statistical mechanics, and this
will allow us to utilize powerfulcomputationaltools from the latter field, like the
freeenergyfunctional.Moreover,thegeometricperspectiveisaglobalone,andit
allows us to connect the dynamics before and after allele loss events in a manner
thatismoresystematicthanwhathashithertobeencarriedoutintheliterature.The
decisiveglobalquantitiesare the momentsof the process,and with their help and
with sophisticated hierarchical schemes, we can construct global solutions of the
Kolmogorovforwardandbackwardequations.
Letusthussummarizesomeofourcontributions,inadditiontoprovidingaself-
containedandcomprehensiveanalysisoftheWright–Fishermodel.
(cid:129) We providea new setof computationaltoolsforthe basic quantitiesofinterest
of the Wright–Fisher model, like fixation or coexistence probabilities of the
differentalleles.Thesewillbespelledoutindetailforvariouscasesofincreasing
generality,startingfromthe2-allele,1-locuscasewithoutadditionaleffectslike
mutationorselectiontocasesinvolvingmorealleles,severallociand/ormutation
andselection.
(cid:129) We develop a systematic geometric perspective which allows us to understand
results like the Ohta–Kimura formula or, more generally, the properties and
consequencesofrecombination,inconceptualterms.
(cid:129) Free energyconstructionswill yield new insightinto the asymptotic properties
oftheprocess.
(cid:129) Our hierarchical solutions will preserve overall probabilities and model the
phenomenonof allele loss duringthe processin more geometricand analytical
detailthanpreviouslyavailable.
Clearly, the Wright–Fisher model is a gross simplification and idealization
of a much more complicated biological process. So, why do we consider it
then? There are, in fact, several reasons. Firstly, in spite of this idealization, it
allows us to develop some qualitative understanding of one of the fundamental
biologicalprocesses.Secondly,mathematicalpopulationgeneticsis a surprisingly
powerfultool both for classical genetics and modern molecular genetics. Thirdly,
asmathematicians,wearealsointerestedintheunderlyingmathematicalstructure
for its own sake. In particular, we like to explore the connectionsto several other
mathematicaldisciplines.
Asalreadymentioned,ourbookcontainsaself-containedmathematicalanalysis
oftheWright–Fishermodel.Itintroducesmathematicalconceptsthatareofinterest
and relevance beyond this model. Our book therefore addresses mathematicians
Preface vii
and statistical physicists who want to see how concepts from geometry, partial
differential equations (Kolmogorov or Fokker–Planck equations) and statistical
mechanics(entropy,freeenergy)canbedevelopedandappliedto oneofthemost
importantmathematicalmodelsinbiology;bioinformaticianswhowanttoacquire
a theoreticalbackgroundin populationgenetics;and biologistswho are notafraid
of abstract mathematical models and want to understand the formal structure of
populationgenetics.
Our book consists essentially of three parts. The first two chapters introduce
the basic Wright–Fisher model (random genetic drift) and its generalizations
(mutation,selection,recombination).The nextfew chaptersintroduceandexplore
thegeometrybehindthemodel.Wefirstintroducethebasicconceptsofinformation
geometry and then look at the Kolmogorov equations and their moments. The
geometricstructurewillprovideuswithasystematicperspectiveonrecombination.
And we can utilize moment-generating and free energy functionals as powerful
computational tools. We also explore the large deviation theory of the Wright–
Fisher model. Finally, in the last part, we develop hierarchical schemes for the
constructionofglobalsolutionsinChaps.8and9andpresentvariousapplicationsin
Chap.10.Mostofthoseapplicationsareknownfromtheliterature,butourunifying
perspectiveletsusobtaintheminamoretransparentandsystematicmanner.
From a differentperspective, the first four chapters contain general material, a
description of the Wright–Fisher model, an introductionto informationgeometry,
and the derivation of the Kolmogorov equations. The remaining five chapters
containourinvestigationofthemathematicalaspectsoftheWright–Fishermodel,
the geometry of recombination, the free energy functional of the model and its
properties, and hierarchical solutions of the Kolmogorov forward and backward
equations.
This book contains the results of the theses of the first [60] and the third
author [113] written at the Max Planck Institute for Mathematics in the Sciences
in Leipzig under the direction of the second author, as well as some subsequent
work.Followingtheestablishedcustominthemathematicalliterature,theauthors
are listed in the alphabetical order of their names. In the beginning,there will be
someoverlapwiththesecondauthor’stextbookMathematicalMethodsinBiology
and Neurobiology [73]. Several of the findings presented in this book have been
publishedin[61–64,114–118].
The research leading to these results has received funding from the European
Research Council under the European Union’s Seventh Framework Programme
(FP7/2007–2013)/ERCgrantagreementno.267087.Thefirstandthethirdauthors
havealsobeensupportedbytheIMPRS“MathematicsintheSciences”.
We would like to thank Nihat Ay for a number of inspiring and insightful
discussions.
Leipzig,Germany JulianHofrichter
Leipzig,Germany JürgenJost
Leipzig,Germany TatDatTran
Contents
1 Introduction................................................................. 1
1.1 TheBasicSetting..................................................... 1
1.2 Mutation,SelectionandRecombination............................ 3
1.3 LiteratureontheWright–FisherModel............................. 8
1.4 Synopsis .............................................................. 12
2 TheWright–FisherModel................................................. 17
2.1 TheWright–FisherModel ........................................... 17
2.2 TheMultinomialDistribution ....................................... 19
2.3 TheBasicWright–FisherModel .................................... 20
2.4 TheMoranModel.................................................... 23
2.5 ExtensionsoftheBasicModel ...................................... 24
2.6 TheCaseofTwoAlleles............................................. 27
2.7 ThePoissonDistribution............................................. 28
2.8 ProbabilitiesinPopulationGenetics ................................ 29
2.8.1 TheFixationTime.......................................... 29
2.8.2 TheFixationProbabilities.................................. 30
2.8.3 ProbabilityofHaving.kC1/Alleles(Coexistence)..... 30
2.8.4 Heterozygosity.............................................. 30
2.8.5 LossofHeterozygosity..................................... 31
2.8.6 RateofLossofOneAlleleinaPopulation
Having.kC1/Alleles..................................... 31
2.8.7 AbsorptionTimeofHaving.kC1/Alleles .............. 31
2.8.8 ProbabilityDistributionattheAbsorption
TimeofHaving.kC1/Alleles............................ 31
2.8.9 ProbabilityofaParticularSequenceofExtinction ....... 31
2.9 TheKolmogorovEquations.......................................... 32
2.10 LookingForwardandBackwardinTime........................... 33
2.11 NotationandPreliminaries........................................... 35
2.11.1 NotationforRandomVariables............................ 35
2.11.2 MomentsandtheMomentGeneratingFunctions......... 36
ix
x Contents
2.11.3 NotationforSimplicesandFunctionSpaces.............. 38
2.11.4 Notation for Cubes and Corresponding
FunctionSpaces ............................................ 41
3 GeometricStructuresandInformationGeometry...................... 45
3.1 TheBasicSetting..................................................... 45
3.2 TangentVectorsandRiemannianMetrics........................... 46
3.3 Differentials,Gradients,andtheLaplace–BeltramiOperator...... 50
3.4 Connections .......................................................... 51
3.5 TheFisherMetric .................................................... 56
3.6 ExponentialFamilies................................................. 58
3.7 TheMultinomialDistribution ....................................... 64
3.8 TheFisherMetricastheStandardMetricontheSphere........... 66
3.9 TheGeometryoftheProbabilitySimplex .......................... 68
3.10 TheAffineLaplacian................................................. 70
3.11 TheAffineandtheBeltramiLaplacianontheSphere.............. 73
3.12 TheWright–FisherModelandBrownianMotion
ontheSphere......................................................... 74
4 ContinuousApproximations .............................................. 77
4.1 TheDiffusionLimit.................................................. 77
4.1.1 Convergenceof Discrete to Continuous
SemigroupsintheLimitN !1.......................... 77
4.2 TheDiffusionLimitoftheWright–FisherModel.................. 88
4.3 MomentEvolution ................................................... 91
4.4 MomentDuality...................................................... 99
5 Recombination.............................................................. 103
5.1 RecombinationandLinkage......................................... 103
5.2 RandomUnionofGametes.......................................... 105
5.3 RandomUnionofZygotes........................................... 107
5.4 DiffusionApproximation............................................ 109
5.5 Compositionality..................................................... 110
5.6 TheGeometryofRecombination.................................... 111
5.7 TheGeometryofLinkageEquilibriumStates...................... 114
5.7.1 LinkageEquilibriainTwo-LociMulti-AllelicModels... 115
5.7.2 Linkage Equilibria in Three-Loci
Multi-AllelicModels....................................... 117
5.7.3 TheGeneralCase........................................... 120
6 MomentGeneratingandFreeEnergyFunctionals..................... 123
6.1 MomentGeneratingFunctions ...................................... 123
6.1.1 TwoAlleles ................................................. 124
6.1.2 TwoAlleleswithMutation................................. 128
6.1.3 TwoAlleleswithSelection................................. 130
6.1.4 nC1Alleles................................................ 132