Critical population and error threshold on the sharp peak landscape for a Moran model

100 Pages·2014·1.328 MB·English

by  Raphael Cerf

MEMOIRS
of the
American Mathematical Society
Volume 233 • Number 1096 (second of 6 numbers) • January 2015
Critical Population and Error
Threshold on the Sharp Peak
Landscape for a Moran Model
Raphae¨l Cerf
ISSN 0065-9266 (print) ISSN 1947-6221 (online)
American Mathematical Society
MEMOIRS
of the
American Mathematical Society
Volume 233 • Number 1096 (second of 6 numbers) • January 2015
Critical Population and Error
Threshold on the Sharp Peak
Landscape for a Moran Model
Raphae¨l Cerf
ISSN 0065-9266 (print) ISSN 1947-6221 (online)
American Mathematical Society
Providence, Rhode Island
Library of Congress Cataloging-in-Publication Data
Cerf,Rapha¨el.
Critical population and error threshold on the sharp peak landscape for a Moran model /
Rapha¨elCerf.
pages cm. – (Memoirs of the AmericanMathematicalSociety, ISSN 0065-9266; volume 233,
number1096)
Includesbibliographicalreferencesandindex.
ISBN978-1-4704-0967-8(alk. paper)
1. Moleculardynamics–Mathematicalmodels. 2. Macromolecules–Mathematicalmodels. 3.
Chromosomereplication–Mathematicalmodels. I.Title.
QP517.M65C47 2015
547(cid:2).7–dc23 2014033062
DOI:http://dx.doi.org/10.1090/memo/1096
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Contents
Chapter 1. Introduction 1
Chapter 2. The Model 9
Chapter 3. Main Results 13
Chapter 4. Coupling 19
Chapter 5. Normalized Model 21
Chapter 6. Lumping 25
6.1. Distance process 25
6.2. Occupancy process 28
6.3. Invariant probability measures 30
Chapter 7. Monotonicity 33
7.1. Coupling of the lumped processes 33
7.2. Monotonicity of the model 35
Chapter 8. Stochastic Bounds 39
8.1. Lower and upper processes 39
8.2. Dynamics of the bounding processes 40
8.3. A renewal argument 41
8.4. Bounds on ν 43
Chapter 9. Birth and Death Processes 47
9.1. General formulas 47
9.2. The case of (Ztθ)t≥0 47
9.3. Persistence time 49
9.4. Invariant probability measure 51
Chapter 10. The Neutral Phase 55
10.1. Ancestral lines 55
10.2. Mutation dynamics 58
10.3. Falling to equilibrium from the left 61
10.4. Falling to equilibrium from the right 66
10.5. Discovery time 70
Chapter 11. Synthesis 79
Appendix A. Appendix on Markov Chains 81
Bibliography 85
Index 87
iii

Abstract
The goal of this work is to propose a finite population counterpart to Eigen’s
model,whichincorporatesstochasticeffects. WeconsideraMoranmodeldescribing
theevolutionofapopulationofsizemofchromosomesoflength(cid:3)overanalphabet
of cardinality κ. The mutation probability per locus is q. We deal only with the
sharp peak landscape: the replication rate is σ > 1 for the master sequence and
1 for the other sequences. We study the equilibrium distribution of the process in
the regime where
(cid:3)→+∞, m→+∞, q →0,
m
(cid:3)q →a∈]0,+∞[, →α∈[0,+∞].
(cid:3)
We obtain an equation αφ(a) = lnκ in the parameter space (a,α) separating the
regime where the equilibrium population is totally random from the regime where
a quasispecies is formed. We observe the existence of a critical population size
necessary for a quasispecies to emerge and we recover the finite population coun-
terpart of the error threshold. Moreover, in the limit of very small mutations, we
obtain a lower bound on the population size allowing the emergence of a quasis-
pecies: if α < lnκ/lnσ then the equilibrium population is totally random, and a
quasispecies can be formed only when α ≥ lnκ/lnσ. Finally, in the limit of very
large populations, we recover an error catastrophe reminiscent of Eigen’s model:
if σexp(−a) ≤ 1 then the equilibrium population is totally random, and a quasis-
pecies can be formed only when σexp(−a) > 1. These results are supported by
computer simulations.
ReceivedbytheeditorMay19,2012,and,inrevisedform,November26,2012.
ArticleelectronicallypublishedonMay19,2014.
DOI:http://dx.doi.org/10.1090/memo/1096
2010 MathematicsSubjectClassification. Primary60F10,Secondary92D25.
Key wordsand phrases. Moran,quasispecies,errorthreshold.
The author thanks an anonymous Referee for his careful reading and his remarks, which
helpedtoimprovethepresentation.
Affiliationattimeofpublication: Universit´eParisSud,Math´ematique,Bˆatiment425,91405
OrsayCedex–France;email: [email protected].
(cid:3)c2014 American Mathematical Society
v

CHAPTER 1
Introduction
In his famous paper [13], Eigen introduced a model for the evolution of a pop-
ulationofmacromolecules. Inthismodel,themacromoleculesreplicatethemselves,
yet the replication mechanism is subject to errors caused by mutations. These two
basic mechanisms are described by a family of chemical reactions. The replication
rateofamacromoleculeisgovernedbyitsfitness. AfundamentaldiscoveryofEigen
is the existence of an error threshold on the sharp peak landscape. If the mutation
rate exceeds a critical value, called the error threshold, then, at equilibrium, the
populationiscompletelyrandom. Ifthemutationrateisbelowtheerrorthreshold,
then, at equilibrium, the population contains a positive fraction of the master se-
quence (the most fit macromolecule) and a cloud of mutants which are quite close
to the master sequence. This specific distribution of individuals is called a quasis-
pecies. ThisnotionhasbeenfurtherinvestigatedbyEigen,McCaskillandSchuster
[15] and it had a profound impact on the understanding of molecular evolution
[11]. Ithasbeenarguedthat, atthepopulationlevel, evolutionaryprocesses select
quasispecies rather than single individuals. Even more importantly, this theory
is supported by experimental studies [12]. Specifically, it seems that some RNA
viruses evolve witharather high mutation rate, which is adjusted tobe close toan
error threshold. It has been suggested that this is the case for the HIV virus [38].
Some promising antiviral strategies consist of using mutagenic drugs that induce
anerrorcatastrophe [2,8]. Asimilar error catastrophecould alsoplay arolein the
development of some cancers [36].
Eigen’s model was initially designed to understand a population of macro-
molecules governed by a family of chemical reactions. In this setting, the number
of molecules is huge, and there is a finite number of types of molecules. From
the start, this model is formulated for an infinite population and the evolution is
deterministic (mathematically, it is a family of differential equations governing the
timeevolutionofthedensitiesofeachtypeofmacromolecule). Theerrorthreshold
appears when the number of types goes to ∞. This creates a major obstacle if
one wishes to extend the notions of quasispecies and error threshold to genetics.
Biological populations are finite, and even if they are large so that they might be
considered infinite in some approximate scheme, it is not coherent to consider situ-
ations where the size of the population is much larger than the number of possible
genotypes. Moreover, it haslong beenrecognizedthatrandom effectsplay amajor
role in the genetic evolution of populations [24], yet they are ruled out from the
startinadeterministicinfinitepopulationmodel. Therefore,itiscrucialtodevelop
a finite population counterpart to Eigen’s model, which incorporates stochastic ef-
fects. ThisproblemisalreadydiscussedbyEigen,McCaskillandSchuster[15]and
morerecentlybyWilke[41]. Numerousworkshaveattackedthisissue: Demetrius,
Schuster and Sigmund [9], McCaskill [27], Gillespie [19], Weinberger [40]. Nowak
1
2 1. INTRODUCTION
and Schuster [32] constructed a birth and death model to approximate Eigen’s
model. This birth and death model plays a key role in our analysis, as we shall
see later. Alves and Fontanari [1] study how the error threshold depends on the
population in a simplified model. More recently, Musso [29] and Dixit, Srivastava,
Vishnoi [10] considered finite population models which approximate Eigen’s model
when the population size goes to ∞. These models are variants of the classical
Wright–Fisher model of population genetics. Although this is an interesting ap-
proach, it is already a delicate matter to prove the convergence of these models
towards Eigen’s model. We adopt here a different strategy. Instead of trying to
prove that some finite population model converges in some sense to Eigen’s model,
wetrytoprovedirectlyinthefinitemodelanerrorthresholdphenomenon. Tothis
end, we look for the simplest possible model, and we end up with a Moran model
[28]. Themodelwechoosehereisnotparticularlyoriginal, thecontributionofthis
work is rather to show a way to analyze this kind of finite population model.
Let us describe informally the model (see chapter 2 for the formal definition).
We consider a population of size m of chromosomes of length (cid:3) over the alphabet
{A,T,G,C}. We work only with the sharp peak landscape: there is one specific
sequence, called the master sequence, whose fitness is σ > 1, and all the other
sequences have fitness equal to 1. The replication rate of a chromosome is propor-
tional to its fitness. When a chromosome replicates, it produces a copy of himself,
which is subject to mutations. Mutations occur randomly and independently at
each locus with probability q. The offspring of a chromosome replaces a chromo-
some chosen at random in the population.
The mutations drive the population towards a totally random state, while the
replication favors the master sequence. These two antagonistic effects interact in a
complicated way in the dynamics and it is extremely difficult to analyze precisely
the time evolution of such a model. Let us focus on the equilibrium distribution
of the process. A fundamental problem is to determine the law of the number
of copies of the master sequence present in the population at equilibrium. If we
keep the parameters m,(cid:3),q fixed, there is little hope to get useful results. In order
to simplify the picture, we consider an adequate asymptotic regime. In Eigen’s
model, the population size is infinite from the start. The error threshold appears
when (cid:3) goes to ∞ and q goes to 0 in a regime where (cid:3)q = a is kept constant. We
wish to understand the influence of the population size m, thus we use a different
approach and we consider the following regime. We send simultaneously m,(cid:3) to ∞
and q to 0 and we try to understand the respective influence of each parameter on
the equilibrium law of the master sequence. By the ergodic theorem, the average
number of copies of the master sequence at equilibrium is equal tothe limit, as the
timegoesto∞, ofthetimeaverageofthenumber ofcopiesofthemastersequence
present through thewhole evolutionof the process. Inthe finite population model,
the number of copies of the master sequence fluctuates with time. Our analysis of
thesefluctuationsreliesonthefollowingheuristics. Supposethattheprocessstarts
with a population of size m containing exactly one master sequence. The master
sequence is likely to invade the whole population and become dominant. Then the
master sequence will be present in the population for a very long time without
interruption. We call this time the persistence time of the master sequence.
The destruction of all the master sequences of the population is quite unlikely,
nevertheless it will happen and the process will eventually land in the neutral
1. INTRODUCTION 3
region consisting of the populations devoid of master sequences. The process will
wander randomly throughout this region for a very long time. We call this time
thediscoverytimeofthemastersequence. Becausethecardinalityofthepossible
genotypesisenormous, themastersequenceisdifficulttodiscover, neverthelessthe
mutationswilleventuallysucceedandtheprocesswillstartagainwithapopulation
containing exactly one master sequence. If, on average, the discovery time is much
larger thanthe persistence time, then theequilibrium statewill be totallyrandom,
while a quasispecies will be formed if the persistence time is much larger than the
discovery time. Let us illustrate this idea in a very simple model.
1− θ θ 1 1
2 2 1 1 2 2
··· 2 2 ···
(cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0)
0 1 2 i−1 i i+1 (cid:3)−1 (cid:3)
Figure 1. Random walk example
We consider the random walk on {0,...,(cid:3)} with the transition probabilities de-
pending on a parameter θ given by:
θ θ 1
p(0,1)= , p(0,0)=1− , p((cid:3),(cid:3)−1)=p((cid:3),(cid:3))= ,
2 2 2
1
p(i,i−1)=p(i,i+1)= , 1≤i≤(cid:3)−1.
2
The integer (cid:3) is large and the parameter θ is small. Hence the walker spends its
time either wandering in {1,...,(cid:3)} or being trapped in 0. The state 0 plays the
roleofthequasispecieswhiletheset{1,...,(cid:3)}playstheroleoftheneutralregion.
With this analogy in mind, the persistence time is the expected time of exit from
0, it is equal to 2/θ. The discovery time is the expected time needed to discover 0
starting for instance from 1, it is equal to 2(cid:3). The equilibrium law of the walker is
the probability measure μ given by
1 θ
μ(0) = , μ(1) = ··· = μ((cid:3)) = .
1+θ(cid:3) 1+θ(cid:3)
We send (cid:3) to ∞ and θ to 0 simultaneously. If θ(cid:3) goes to ∞, the entropy factors
win and μ becomes totally random. If θ(cid:3) goes to 0, the selection drift wins and μ
converges to the Dirac mass at 0.
In order to implement the previous heuristics, we have to estimate the per-
sistence time and the discovery time of the master sequence in the Moran model.
For the persistence time, we rely on a classical computation from mathematical
genetics. Supposewestartwithapopulationcontainingm−1copiesofthemaster
sequence and another non master sequence. The non master sequence is very un-
likelytoinvade the whole population, yet ithas asmall probability todoso, called
the fixation probability. If we neglect the mutations, standard computations yield
that, in a population of size m, if the master sequence has a selective advantage of
σ,thefixationprobabilityofthenonmastersequenceisroughlyoforder1/σm (see
for instance [31, Section6.3]). Now the persistence time can be viewed as the time
needed for non master sequences to invade the population. This time is approxi-
mately equal to the inverse of the fixation probability of the non master sequence,