Selection intensity and the time to fixation in evolutionary systems [Elektronische Ressource] / vorgelegt von Philipp Martin Friedhelm Altrock

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Selection intensity and the time to fixationin evolutionary systemsKumulative Dissertationzur Erlangung des Doktorgradesder Mathematisch-Naturwissenschaftlichen Fakultätder Christian-Albrechts-Universität zu KielVorgelegt vonPhilipp Martin Friedhelm AltrockGeboren am 18.04.1982in Frankfurt am MainKiel, 2011Referent : Dr. Arne Traulsen - Max-Planck-Institutfür Evolutionsbiologie, PlönKoreferenten : Prof. Dr. Gerd Pfister - Christian-Albrechts-Universitätzu KielProf. Dr. Joachim Krug - Universität zu KölnTag der Mündlichen Prüfung: 9. März 2011Zum Druck genehmigt: 9. März 2011gez. Prof. Dr. Lutz Kipp, DekanContentsKurzfassung 1Abstract 3Thesis overview 51 Introduction 61.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2 Deterministic evolutionary game dynamics . . . . . . . . . . . . . . . . . . . 111.3 Stochasticry game dynamics . . . . . . . . . . . . . . . . . . . . . 261.4 Population genetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 Fixation events in well mixed populations of finite size 462.1 Fixation times in evolutionary games under weak selection . . . . . . . . . . 462.2 Universality of weak selection . . . . . . . . . . . . . . . . . . . . . . . . . 672.3 Stochastic slowdown in evolutionary processes . . . . . . . . . . . . . . . . 812.4 Deterministic evolutionary game dynamics in finite populations . . . . . . .
Publié le : samedi 1 janvier 2011
Lecture(s) : 87
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Source : D-NB.INFO/1011452790/34
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Selection intensity and the time to fixation
in evolutionary systems
Kumulative Dissertation
zur Erlangung des Doktorgrades
der Mathematisch-Naturwissenschaftlichen Fakultät
der Christian-Albrechts-Universität zu Kiel
Vorgelegt von
Philipp Martin Friedhelm Altrock
Geboren am 18.04.1982
in Frankfurt am Main
Kiel, 2011Referent : Dr. Arne Traulsen - Max-Planck-Institut
für Evolutionsbiologie, Plön
Koreferenten : Prof. Dr. Gerd Pfister - Christian-Albrechts-Universität
zu Kiel
Prof. Dr. Joachim Krug - Universität zu Köln
Tag der Mündlichen Prüfung: 9. März 2011
Zum Druck genehmigt: 9. März 2011
gez. Prof. Dr. Lutz Kipp, DekanContents
Kurzfassung 1
Abstract 3
Thesis overview 5
1 Introduction 6
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Deterministic evolutionary game dynamics . . . . . . . . . . . . . . . . . . . 11
1.3 Stochasticry game dynamics . . . . . . . . . . . . . . . . . . . . . 26
1.4 Population genetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2 Fixation events in well mixed populations of finite size 46
2.1 Fixation times in evolutionary games under weak selection . . . . . . . . . . 46
2.2 Universality of weak selection . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.3 Stochastic slowdown in evolutionary processes . . . . . . . . . . . . . . . . 81
2.4 Deterministic evolutionary game dynamics in finite populations . . . . . . . 90
3 Stability in structured populations with heterozygote disadvantage 102
3.1 Using underdominance to bi-stably transform local populations . . . . . . . 104
3.2 Heterozygote disadvantage in subdivided populations of finite size . . . . . . 120
4 Conclusions 130
Bibliography 133
Danksagung 146
Curriculum Vitae 148
Eidesstattliche Erklärung 150
iKurzfassung
Prozesse Darwinscher Evolution sind dynamisch, nichtlinear, und unterliegen Fluktu-
ationen. Systeme Darwinscher Evolution können mit wohl etablierten Methoden der
statistischen Physik analysiert werden. Die für evolutionäre Veränderungen wesentlichen
Mechanismen sind Reproduktion, Mutation und Selektion. Individuen reproduzieren sich
und vererben Gene und Merkmale, so dass die Population evolviert. Mutationen treten
spontan auf, z.B. durch Fehler in der Reproduktion, wodurch verschiedene neue Typen
von Genen oder Merkmalen entstehen können. Selektion wirkt auf verschiedene Typen.
Diese Arbeit konzentriert sich auf Selektion in Systemen, welche den Prinzipien Darwin-
scher Evolution, sowie Fluktuationen unterliegen. Die Wechselwirkungen verschiedener
Typen untereinander können die jeweiligen reproduktiven Raten beeinflussen. Eine
wichtige Disziplin, welche solche Wechselwirkungen betrachtet, ist die Spieltheorie. In
der evolutionären Spieltheorie identifiziert man verschiedene Typen mit verschiedenen
Strategien. Der (spieltheoretische) Erfolg einer Strategie beeinflusst deren reproduktiven
Erfolg. Eine wichtige Eigenschaft evolutionärer Spiele ist, dass der evolutionäre Erfolg
einer Strategie im Allgemeinen mit der Zusammensetzung der Population variiert.
Der Begriff Fixierung bezeichnet das Ereignis der Übernahme einer Population durch
eine Mutation. Hauptsächlich werden in dieser Arbeit die Fixierungszeiten einer mutierten
Strategie betrachtet. Sie sind ein Maß für die Zeit, die eine Population benötigt, um
von einem Zustand mit nur wenigen zu einem Zustand mit ausschließlich Mutanten zu
gelangen.
Selektion kontrolliert die Erfolgsdifferenz zwischen Typen. Dies ermöglicht die Def-
inition verschiedener Regime der Selektion. Ohne Selektion ist Evolution neutral und
Fluktuationen dominieren. Ein wichtiger Grenzfall ist schwache Selektion, welche eine
gerichtete Veränderung zu diesen zufälligen evolutionären Veränderungen hinzufügt. In
dieser Arbeit spielt die Analyse der schwachen Selektion eine bedeutende Rolle in der
Klassifikation verschiedener evolutionärer Prozesse. Sie erlaubt eine Vereinfachung der
nichtlinearen Systeme und damit eine analytische Beschreibung. Es werden approxima-
tive Formulierungen der Fixierungszeiten unter schwacher Selektion präsentiert und die
1Kurzfassung
Universalität dieses Grenzfalls betrachtet. Auf Zwischenskalen kann man beobachten,
dass die Fixierungszeit einer vorteilhaften Mutation mit der Selektion ansteigt, obwohl
die entsprechende Fixierungswahrscheinlichkeit ebenso größer wird.
Davon ausgehend kann man zur Betrachtung starker Selektion übergehen, so dass
Selektion die Dynamik auch in kleinen Systemen dominiert. Hierbei lassen sich Segre-
gationseffekte beobachten: Das Schicksal der Population ist deterministisch durch die
Anfangsbedingung bestimmt.
Ein weiterer wichtiger Mechanismus der Evolution ist der Genfluss, welcher z.B. durch
Migration zwischen Population der selben Art erzeugt wird. In diesem Zusammenhang
kann Migration der Selektion entgegenwirken. In Systemen bistabiler evolutionärer
Dynamik kann solch ein Migrations-Selektionsgleichgewicht zu lang stabiler Koexistenz
führen. Die vorliegenden Arbeit gibt hier eine quantitative Analyse der dynamischen und
statistischen Eigenschaften. Zu diesem Zweck werden die Austerbe- oder Fixierungszeiten
des nichtlinear gekoppelten Populationssystems analysiert.
2Abstract
Processes of Darwinian evolution are dynamic, nonlinear, and underly fluctuations. A
way to analyze systems of Darwinian evolution is by using methods well established in
statistical physics. The main mechanisms that are responsible for evolutionary changes
are reproduction, mutation, and selection. Individuals reproduce and inherit genes and
traits, such that a population evolves. Mutations occur spontaneously, e.g., by errors in
reproduction, whereby different new types of genes or traits can emerge. Selection acts
on different types.
This thesis focuses on selection in systems that underlie the principles of Darwinian
evolution, as well as fluctuations. Once there are different types, their interactions with
each other can influence their reproductive rates. One important framework to look
at such interactions is game theory. In evolutionary game theory, different types are
identified with different strategies, and the payoff of a strategy affects the reproductive
success. An important property of evolutionary games is that, in general, the evolutionary
success of a strategy varies with the composition of the population.
The event of a mutation taking over a population is called fixation. The quantities
mainly considered in this thesis are the fixation times of a mutant strategy. They are
a measure for the time a population spends reaching the state of only mutants, when
starting from a few.
The role of selection is to control the payoff differences between types, which gives
rise to several regimes of selection. In the absence of selection evolution is neutral and
fluctuations dominate. An important limit case is weak selection, which introduces a
small bias to the random evolutionary changes. In this thesis, weak selection analysis
plays an important part in the classification of different evolutionary processes. This
allows to simplify the nonlinear dynamical system and thus an analytical description.
Here, approximative formulations of the fixation times under weak selection are presented,
and the universality of the weak selection regime is addressed. On intermediate scales,
one can observe that the average fixation time of an advantageous mutation increases
with selection, although the probability of fixation also increases.
3Abstract
One can then move on to strong selection, such that selection dominates the dynamics
even in small systems. Here, one can observe segregation effects, where the initial
condition determines the fate of the finite population in a deterministic way.
Another important evolutionary mechanism is gene flow, e.g., caused by migration
between populations of the same species. In this context, migration can counterbalance
selection. In systems with bi-stable evolutionary dynamics, the migration-selection
equilibrium can lead to coexistence that is stable for a long time. This thesis gives a
quantitative analysis of the dynamical and statistical properties of such a system. To
this end, the extinction (fixation) times are analyzed also in the nonlinearly coupled
population system.
4Thesis overview
This thesis includes the following manuscripts.
• Philipp M. Altrock and Arne Traulsen,
Fixation times in evolutionary games under weak selection,
New Journal of Physics 11, 013012 (2009).
• Bin Wu, Philipp M. Altrock, Long Wang and Arne Traulsen,
Universality of weak selection,
Physical Review E 82, 046106 (2010).
(Virtual Journal of Biological Physics Research 20(8), 2010).
• Philipp M. Altrock, Chaitanya S. Gokhale and Arne Traulsen,
Stochastic slowdown in evolutionary processes,
Physical Review E 82, 011925 (2010).
(Virtual Journal of Biological Physics Research 20(3), 2010).
• Philipp M. Altrock and Arne Traulsen,
Deterministic evolutionary game dynamics in finite populations,
Physical Review E 80, 011909 (2009).
(Virtual Journal of Biological Physics Research 18(2), 2009).
• Philipp M. Altrock, Arne Traulsen, R. Guy Reeves and Floyd A. Reed,
Using underdominance to bi-stably transform local populations,
Journal of Theoretical Biology 267, 62 (2010).
• Philipp M. Altrock, Arne Traulsen and Floyd A. Reed,
A stochastic model for heterozygote disadvantage in subdivided populations,
to be submitted, January 2011.
5CHAPTER1
Introduction
The step from describing inert matter
to describing biological life seems enormous,
but maybe it isn’t.
Per Bak (How Nature Works)
1.1 Motivation
Fluctuations play an important role in physical and biological systems. In particular,
far from equilibrium fluctuations are one of the most fundamental properties commonly
observed in many processes of the living world. The rigorous quantitative description
of fluctuations is performed in the field of stochastic processes. Its origins lie in the
description of Brownian motion, first systematically observed by R. Brown [Brown,
1866]. The mathematical analysis, at least from a physical perspective, started about
half a century later, conducted by A. Einstein, M. von Smoluchowski, and P. Langevin
[Einstein, 1905; Langevin, 1908; von Smoluchowski, 1906]. What first came along as a
rather heuristic chain of arguments guided by genius and intuition led to fundamental
physical equations. These equations describe a stochastic process, e.g., position or
velocity of a Brownian particle, by microscopic and macroscopic laws accounting for
thermal fluctuations. The formalism was later put on a more rigorous basis [Feller,
1968; Kolmogorov, 1956]. Since then, stochastic processes have become unavoidable in
successfully describing fundamental processes in physics, chemistry and biological physics,
where thermal or quantum fluctuations have to be accounted for, see, e.g., [Gammaitoni
etal.,1998;Gardiner,2008;Krug,1997;Reimann,2002;Schnakenberg,1976;vanKampen,
1997]. Typically, the interactions taken into account here are (quantum-)mechanic or
electromagnetic etc., i.e. they are based on the fundamental interactions in physics.
61 Introduction
It is, however, notable that in other fields like quantitative social science or biology, a
mathematical description can often be formulated as if such physical interactions occur
and as if fluctuations are of thermal origin (from a physical point of view, in most cases
they are not). Fluctuations can be of thermal origin, but are often generally referred
to as demographic fluctuations that simply originate from discreteness and finite size.
In the quantitative description of collective motion, social (hierarchical) interactions
govern the onset (e.g., movement is induced by imitation), patterns, and phase transitions
[Helbing and Molnár, 1995; Romanczuk et al., 2010; Sumpter, 2010; Vicsek et al., 1995].
In Darwinian evolution, the outcome of mutation and selection depends on interactions
on many levels, e.g., between molecules, individuals, and/or populations of different
species. In both non-physical examples stochasticity can act in a way very similar to
physical systems: the formation of patterns in motion can stabilize in a certain regime
of fluctuations, or populations can benefit from fluctuations maintaining ’evolutionary
freedom’ or allowing ’evolutionary revolutions’ [Lenormand et al., 2009].
Complexity is another important concept in the natural and social sciences. Systems in
whichmacroscopicpropertiescannotbepredictedbythepropertiesofitsmicroscopicparts
alone are characterized as complex. Such complex behavior typically arises in nonlinear
dynamical systems, or when problems across a wide range of scales are considered. This is
commonly the case in physics, chemistry, and engineering [Haken, 2006], traffic modeling
[Helbing, 2001], or modeling stock markets [Farmer et al., 2005], but also omnipresent in
adaptive systems, e.g., in the human brain, evolving populations, and ecological networks
[Levin, 2002; Schuster, 2002]. When microscopically defined entities such as charged
particles, molecules, neurons, or animals, interact in a complex way to form macroscopic
patterns in space and/or time, mathematical methods from theoretical physics are well
established to make predictions. Moreover, in Darwinian evolution, the fate of biological
traits in terms of reproductive success can depend on the interactions with others and
with the environment in a complex way.
In a broad sense, this thesis deals with nonlinear dynamics and fluctuations in mathe-
matical models of Darwinian evolution. Although the ’theory of evolution’ is now rather
a fact than a theory, there are theoretical approaches, which intend to make quantitative
predictions that can be tested experimentally. In the beginning of the 20th century a
(first) mathematical description of Darwin’s exciting approach to biology was on its
way. In the century before, C. Darwin and others formulated the fundamental principles
of evolution to explain life’s diversity by evolutionary mechanisms, which is based on
G. Mendel’s observations and laws of inheritance [Bateson, 1909]. For the great luck of
others, this work was neither very mathematical, nor could Darwin and Mendel know
about the molecular mechanisms of inheritance.
7

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