Genetic Inference
vorgelegt von
Dipl.-Inform. Matthias Steinr¨ucken aus Wuppertal
Von der Fakult¨at II – Mathematik und Naturwissenschaften der Technischen Universit¨at Berlin
zur Erlangung des akademischen Grades Doktor der Naturwissenschaften
– Dr. rer. nat. – genehmigte Dissertation
Promotionsausschuss
Vorsitzender: Prof. Dr. G¨unter M. Ziegler Gutachter: Prof. Dr. Jochen Blath
Prof. Dr. Ellen Baake Dr. Matthias Birkner
Tag der wissenschaftlichen Aussprache: 28. September 2009
Berlin 2009 D 83
Matthias Steinr¨ucken
Mathematische Populationsgenetik besch¨aftigt sich mit der Konstruktion von Modellen um biologisch interessante Ph¨anomene zu beschreiben und zu analysieren. Der klassische Fleming-Viot Prozess beschreibt die Evolution einer Population in vorw¨artsgerichteter Zeit. Er ist dual zu Kingman’s Koaleszenten, welcher die Genealogie einer Stichprobe von Individuen, also die R¨uckw¨artsdynamik beschreibt.
Pitman [P99] und Sagitov [S99] haben eine allgemeinere Klasse von Koaleszenten eingef¨uhrt, die im Gegensatz zu Kingman’s Koaleszent das Verschmelzen von mehr als zwei Ahnenlinien auf einmal erlaubt. Diese Λ-Koaleszenten sind dual zu den Λ-Fleming-Viot Prozessen. Donnelly und Kurtz [DK96, DK99] f¨uhrten die sogenannte Lookdown-Konstruktion ein um zu zeigen, das diese Dualit¨aten auch in einem pfadweisen Sinne gelten.
In der vorliegenden Arbeit erweitern wir diese Lookdown-Konstruktion um damit den Fleming-Viot Prozess zu konstruieren und seine pfadweise Dualit¨at zum Ξ-Koaleszenten zu etablieren. Der Ξ-Koaleszent zeichnet sich dadurch aus, dass er das simultane Verschmelzen von mehren Ahnenlinen erlaubt.
Der zweite Teil der Arbeit besch¨aftgt sich mit der statistischen Inferenz von evolu-tion¨aren Parametern unter dem Λ-Koaleszenten und dem sogenannten infinitely-many-sites Mutationsmodel. Wir pr¨asentieren Rekursionformeln mit denen sich die Wahr-scheinlichkeit eine gegebene Stichprobe von der Population zu ziehen berechnen l¨asst (eingef¨uhrt in [BB08]). Mit Hilfe dieser Rekursion l¨asst sich ein Monte-Carlo-Simulations-verfahren beschreiben, das zur Sch¨atzung der Wahrscheinlichkeit f¨ur große Stichproben benutzt werden kann. Aufbauend auf der Arbeit [SD00] und [HUW08] interpretieren wir diese Methode als ein
”importance sampling“-Schema und f¨uhren weitere Schemata ein, um die Genauigkeit der Sch¨atzungen zu erh¨ohen.
Wir wenden diese Verfahren an, um evolution¨are Parameter f¨ur reale Datens¨atze mit-ochondrialer DNA vom atlantischen Kabeljau zu bestimmen. Wir argumentieren, dass die Λ-Koaleszenten besser geeignet sind die Genealogien der Datens¨atze zu beschreiben als der klassische Kingman-Koaleszent.
Ich danke meinem Betreuer Jochen Blath, der mir die M¨oglichkeit geben hat an diesem interessanten Thema zu arbeiten und mir bei Problemen stets hilfsbereit zur Seite stand. Des weiteren danke ich meinem Zweitbetreuer Matthias Birkner, dass er mich beim anfertigen dieser Arbeit unterst¨utzte und mit wertvollen Ratschl¨agen zu ihrem Gelingen beigetragen hat.
Dank geht auch an Ellen Baake, dass sie mein Interesse an der mathematischen Biologie geweckt hat und sich bereit erkl¨art hat ein weiteres Gutachten zu verfassen.
Dem internationalen Graduiertenkolleg Stochastische Modelle Komplexer Prozesse danke ich f¨ur die finanzielle Unterst¨utzung in den letzten drei Jahren.
Besonderer Dank gilt meinen Freunden und Kollegen in der Arbeitsgruppe Stochastik und im Gradiertenkolleg SMCP, die es mir erm¨oglichten in einer freundlichen Atmo-sph¨are zu arbeiten und bei Gelegenheit auch mal eine Pause vom Arbeiten zu machen. Ferner danke ich meiner Familie und meinen Freunden, die mich auf meinem Weg begleitet und unterst¨utzt haben, und auf diese Weise ihren Beitrag zu dieser Arbeit leisteten.
I would like to express my deep and sincere gratitude to Mimi Tsuruga who provided me with food, help on English grammar, and understanding during the preparation of this work.
1. Introduction 1
1.1. Motivation & Overview . . . 1
1.2. Neutral population models . . . 3
1.2.1. Cannings-models . . . 3
1.2.2. Moran Model with Large Family Sizes . . . 4
1.2.3. Fleming-Viot Process . . . 4 1.3. Exchangeable Coalescents . . . 7 1.3.1. Kingman’s Coalescent . . . 7 1.3.2. Λ-coalescents . . . 9 1.3.3. Ξ-coalescents . . . 10 1.4. Mutation models . . . 13
1.5. Duality and Lookdown Construction . . . 15
1.5.1. Classical and Modified Lookdown Construction for Fleming-Viot Process . . . 17
1.5.2. Classical and Modified Lookdown Construction for Λ-Fleming-Viot Processes . . . 18
2. A Lookdown Construction for the Ξ-Fleming-Viot Process 21 2.1. Path Properties and Generator . . . 22
2.2. Exchangeable E∞-valued Particle Systems . . . . 25
2.2.1. The Canonical (Ξ, B)-Moran Model . . . 25
2.2.2. The Ordered Model and Exchangeability . . . 26
2.2.3. The Limiting Population . . . 32
2.3. Pathwise Convergence: Proof of Theorem 2.1 . . . 35
2.4. The Hille-Yosida Approach . . . 41
2.4.1. Two Representations of the Ξ0-Fleming-Viot Generator . . . 41
2.4.2. Construction of the Markov Semigroup and Proof of Proposition 2.3 44 2.5. Dualities . . . 48
2.5.1. Distributional Duality versus Pathwise Duality . . . 48
2.5.2. The Function-valued Dual of the (Ξ, B)-Fleming-Viot Process . . 49
2.5.3. The Dual of the Block Counting Process . . . 50
2.6. Examples . . . 53
2.6.1. An Example Involving Recurrent Bottlenecks . . . 53
2.6.2. The Poisson-Dirichlet Case . . . 55
3. Computing Likelihoods 57 3.1. Finitely-Many-Alleles . . . 58
3.2. Infinitely-Many-Alleles . . . 60
3.2.1. M¨ohle’s Recursion . . . 60
3.3. Infinitely-Many-Sites . . . 65
3.3.1. Combinatorics of Samples . . . 66
3.3.2. Recursion for Sampling Probabilities . . . 76
3.3.3. Likelihood Based on the Frequency-spectrum . . . 86
3.3.4. Likelihood Based on Complete Tree . . . 87
4. Importance Sampling in the Infinitely-Many-Sites-Model 89 4.1. Optimal Proposal Distribution . . . 91
4.2. Importance Sampling Schemes . . . 96
4.2.1. Griffiths & Tavar´e’s Scheme . . . 96
4.2.2. Stephens & Donnelly’s Scheme . . . 99
4.2.3. Hobolth, Uyenoyama & Wiuf’s Scheme . . . 101
4.3. Performance Comparison . . . 113
4.3.1. Concrete Examples . . . 113
4.3.2. Empirical Runtime Distributions - Random Samples . . . 115
4.3.3. Empirical Runtime Distributions - Random Scenarios . . . 119
4.3.4. Discussion . . . 122
5. Inference of Evolutionary Parameters from Datasets 125 5.1. Computational Tricks . . . 125
5.1.1. Driving Value . . . 125
5.1.2. Likelihood Pre-calculation . . . 126
5.2. Coalescent Models . . . 128
5.2.1. Eldon & Wakeley’s ψ-Coalescent . . . 128
5.2.2. Schweinsberg’s Beta-coalescent . . . 129
5.3. Maximum-Likelihood Estimation . . . 130
5.3.1. Approximating Maximum of Likelihood Surface . . . 130
5.4. Time to the Most Recent Common Ancestor . . . 132
5.5. Empirical Distributions of Maximum Likelihood Estimators on Simulated Data . . . 133
5.6. Real data . . . 139
5.6.1. Nuh-Chah-Nulth Data . . . 140
5.6.2. Boom, Boulding, Beckenbach’s Oyster Data . . . 141
5.6.3. ´Arnason’s Atlantic Cod Data . . . 144
5.6.4. Possible biological causes for shallow genealogies . . . 149
5.6.5. TMRCA for Atlantic Cod data . . . 152
5.7. Concluding Remarks . . . 153
A. Appendix 155 A.1. MetaGeneTree . . . 155
A.2. TreeCount . . . 159
1.1. Motivation & Overview
A fundamental goal in mathematical population genetics is the construction of pop-ulation models to describe and analyse certain phenomena which are of interest for biological applications. Usually these models are constructed such that they describe the evolution of a population under consideration forward in time. A classical and widely used model of this type is the Fisher-Wright diffusion, which can be used to approximate the evolution of the fraction of individuals carrying a particular allele in large populations. On the other hand, it is often helpful to look from the present backwards into the past to trace the ancestry of a sample of n individuals, genes or particles. In many situations, the Kingman coalescent, introduced in [K82a, K82b], turns out to be an appropriate tool to approximate the ancestry of a sample taken from a large population. The Fisher-Wright diffusion is dual to the block counting process of the Kingman coalescent (c.f. [D86, M01]). More generally, the Fleming-Viot pro-cess [FV79], a measure-valued extension of the Fisher-Wright diffusion, is dual to the Kingman coalescent. This duality will be presented in Section 1.5. It can also be es-tablished using the so-called lookdown construction introduced by Donnelly and Kurtz in [DK96]. This construction and the corresponding duality results have been extended to the Λ-Fleming-Viot process which we introduce in Section 1.3.2 and which is the measure-valued dual of a coalescent process allowing for multiple collision of ancestral lineages [DK99, BLG03, BLG05, BLG06].
There exists a broader class of coalescent processes in which more than one multiple collisions can occur with positive probability simultaneously at the same time [MS01, S00a, S03b]. These processes can be characterized by a measure Ξ on an infinite simplex and are thus called Ξ-coalescents and they are introduced in Section 1.3.3. It is natural to further extend the above constructions and results to this full class of coalescent processes and, in particular, to provide constructions of the dual processes, called Ξ-Fleming-Viot processes. Although such extensions have been briefly indicated in [DK99] and [BLG03], these extensions have yet to be carried out in detail.
In Chapter 2 we explicitly construct the Ξ-Fleming-Viot process and provide a rep-resentation via empirical measures of an exchangeable particle system in the spirit of Donnelly and Kurtz [DK96, DK99]. Furthermore, we establish corresponding conver-gence results and pathwise duality to the Ξ-coalescent. We also provide an alternative, more classical, functional-analytic construction of the Ξ-Fleming-Viot process based on the Hille-Yosida theorem including neutral mutations and present representations for the generator of the Ξ-Fleming-Viot process. The results give insights into the pathwise structure of the Ξ-Fleming-Viot process and its dual, the Ξ-coalescent. In Section 2.6
examples and situations are presented in which certain Ξ-Fleming-Viot processes and their dual Ξ-coalescents occur naturally.
The pathwise duality between the Fleming-Viot processes and the coalescent pro-cesses introduced in Section 1.5 and Chapter 2 can be employed to investigate the stationary distribution of the Λ-Fleming-Viot process under different mutation mod-els and to derive recursions for the sampling probabilities. In the case of Kingman’s coalescent see for example [H84, KM72, E72] for the infinitely-many-alleles model and [EG87] for the infinitely-many-sites model. These approaches were generalised for the Λ-coalescent and the finitely-many-alleles mutation model in [BB09]. In [M06] the gen-eralisation for the infinitely-many-alleles model is given and the authors give recursive formulae for the infinitely-many-sites model in [BB08]. We present the recursion for all three mutation models in the Λ-coalescent case in Chapter 3 with a particular emphasis on the infinitely-many-sites model and its combinatorial properties in Section 3.3. The sampling recursions can be extended to the Ξ-coalescent, but the details look rather tedious and an application to biological data seems to be impractical. Furthermore it seems unlikely that significant statistical evidence in favour of the Ξ-coalescents could be presented.
The recursions can only be computed exactly for small samples, thus in [GT94] Grif-fiths and Tavar´e introduced a Monte Carlo method to estimate the sampling probabil-ities in the Kingman case. This method was generalised to the Λ-coalescent case by Birkner and Blath in [BB08]. We review their method in Section 4.2.1 from a different perspective.
For Kingman’s coalescent Stephens and Donnelly showed in [SD00] that this “canon-ical” approach can be improved when the problem is approached in the importance sampling framework. We present part of their results in Chapter 4 and describe the importance sampling framework in the Λ-coalescent case. The authors in [HUW08] present an approach that, in the case of the infinitely-many-sites mutation model, fur-ther improves the results of Stephens and Donnelly. In Section 4.2.3 we extend the ideas of [HUW08] to the Λ-coalescent and present several importance sampling schemes based on their ideas. Furthermore, we compare the performances of the different schemes in Section 4.3.
In Chapter 5 we apply the methods for the likelihood computation introduced in Chapter 3 and 4 to infer evolutionary parameters via maximum likelihood. We motivate two one-parameter families of coalescent processes introduced in [EW06] and [S03b] suitable for parametric inference. In Section 5.5 we apply our methods to simulated data to gain insight on the empirical distributions of the maximum likelihood estimators. Real datasets from Humans (c.f. [WFDJP91]), Pacific oysters (c.f. [BBB94]) and Atlantic cod (c.f. [A04]) are analysed in Section 5.6. We argue that there is evidence that the marine (especially cod) datasets are better described by a multiple merger coalescent rather than the standard model Kingman’s coalescent.
The results from Chapter 4 are provided in the manuscript [BBS09b] and part of the results from Chapter 5 in the manuscript [BBS09a].
1.2. Neutral population models
In this section we introduce two general classes of neutral population models. Popula-tion models in the first class evolve in discrete time with non-overlapping generaPopula-tions. The second class consists of models in which time is continuous, that is generations can overlap. The modelled population is assumed to be of fixed size N ∈ N and each indi-vidual is of a certain type, where the space E of possible types is assumed to be compact and Polish. The population evolves neutrally so the type of an individual has no impact on the reproductive success. Individuals can give birth to offspring and exhibit muta-tion events. The continuous-time models can be transferred to the discrete-time models and vice versa. Therefore, in the sequel we use the class of models that is convenient for the task at hand. Furthermore, we investigate the limit of the empirical measure of the continuous time population model when the population size tends to infinity. We give proper scalings so that this limit is non-degenerate.
1.2.1. Cannings-models
In [C74] and [C75] Cannings introduced a class of discrete-time population models with non-overlapping generations. The evolution of the population vector (YN(k))
k∈N0 =
(Y1N(k), . . . , YNN(k))k∈N0 taking values in EN is characterised by a sequence of random
vectors
ν(k)= (ν1(k), . . . , νN(k)),
of natural numbers each of which is exchangeable for every k ∈ N. For k′ 6= k the vectors ν(k′) and ν(k) are independent and PNi=1νi(k) = N holds for every k. The random variable νi(k) gives the number of offspring in generation k that descend from individual i in generation k − 1 and inherit its type. The indices of the offspring are chosen uniformly at random. These conditions guarantee that the population is of constant size N and in every generation the reproduction is independent of the other generations. Furthermore, no individual is favoured since the offspring numbers are exchangeable and thus the population vector is exchangeable. The Fisher-Wright model (see [F22] and [W31]) falls into the class of Cannings-models. In this case the distribution of the offspring vector ν is given by the multinomial distribution, so
P(ν1(k), . . . , νN(k)) = (n1, . . . , nN) =
n n1· · · nN
holds. The classical discrete-time Moran model introduced in [M58] is also a Cannings-model where the offspring distribution is the uniform distribution on (2, 1, . . . , 1, 0) and all its permutations. An example of a Cannings-model is depicted in Figure 1.1. Further investigations concerning the Fisher-Wright model and its properties are given for example in [TZ04, Section 2].
1 2 3 4 5 k′
1 2 3 4 5 k′+ 1
1 2 3 4 5 k′+ 2
Figure 1.1.: An excerpt of two generations from a realisation of a Cannings-model with population size N = 5. In this example ν(k′+1)
= (0, 1, 0, 3, 1) and ν(k′+2)
= (1, 0, 2, 1, 1).
1.2.2. Moran Model with Large Family Sizes
The Moran models with large family sizes or Moran-type models (in the sequel occa-sionally just Moran models) are a general class of continuous-time population mod-els. The population is again of a fixed size N and the evolution of the vector of types (YN(t))
t≥0 = (Y1N(t), . . . , YNN(t))t≥0 is given as follows: Assume for each vec-tor k = (k1, k2, . . .) of integers satisfying k1 ≥ k2 ≥ · · · ≥ 0 and P∞i=1ki ≤ N, a non-negative real quantity rN(k) ≥ 0 is given. With rate rN(k), m groups of sizes k1, . . . , km are chosen uniformly from the present population. Inside each of these m groups a parental particle is chosen uniformly and this parental particle superimposes its type on all other individuals of the group. Note that the types do not change in-between resampling events. The classical Moran model in continuous time corresponds to r(2, 0, . . . , 0) = N2 and all other rates are zero. An example for the evolution of a Moran-type model is given in Figure 1.2.
The continuous-time Moran-type models resemble a Markovian embedding of the Cannings-models with exponentially distributed waiting times between the resampling events. In other words, the skeleton chains of the continuous-time models are time-discrete Cannings-models. This correspondence holds in particular for the time-contin-uous and the time-discrete version of the classical Moran model from [M58].
We now turn to the limiting population model in continuous time as the population size tends to infinity.
1.2.3. Fleming-Viot Process
The Fleming-Viot process describes the evolution of an infinitely large population. It was introduced and studied in the literature (see for example [FV79] and [EK93]) in the continuous-time setting. Thus we follow those lines and consider continuous-time population models in this section.
First, we investigate the case where the type space consists of only two elements, so assume E = {a, A} without loss of generality. It is often more convenient to approxi-mate the dynamics of large populations by the limiting dynamics of finite populations where the population size tends to infinity. Let WN(k) denote the relative frequency of individuals of type A in generation k, so WN(k)∈ [0, 1]. If the time is multiplied by
t t1
t2 t3
Figure 1.2.: Example of the evolution of a Moran-type model of size N = 6 encountering a series of resampling events. In the event at t1 there is one family with k1 = 2. The left hand individual of the family attains the type of the rightmost individual. At t2 two families with k1 = 3 and k2 = 2 are involved in the resampling event. Only one family of size 3 is involved in the third event.
the the population size N it can be shown that the convergence WN([N t])t≥0⇒ W (t)t≥0
holds in DE[0,∞) as N tends to infinity. Here [Nt] denotes the integer part of Nt. The limiting process (W (t))t≥0 is a diffusion process (see [KT81, Chapter 15] for an introduction on diffusion theory). It can be shown that the infinitesimal mean is given by µ(w) = 0 and the infinitesimal variance is given by σ2(w) = w(1− w). Thus the generator of the limiting diffusion is given by
Lf(w) = 1
2w(1− w)f ′′(w)
for f (w)∈ C2([0, 1]). This process is known as the Fisher-Wright diffusion. We refer to [D08, Chapter 7 & 8] for a more rigorous treatment of this diffusion approximation. For a finite number of types this approach can be generalised but this generalisation is not straightforward if the type space is infinite.
Before we turn to the treatment of more general type spaces we need to introduce the concept of exchangeable random sequences.
Definition 1.1. A sequence of random variables (Y1, Y2, . . .) taking values in E is said
to be exchangeable if for all n∈ N and all permutations σn∈ Sn the sequences (Y1, . . . , Yn, Yn+1, . . .)
and
have the same distribution.
The following theorem can be shown for exchangeable sequences.
Theorem 1.2 (De Finetti). Let (Y1, Y2, . . .) be an exchangeable sequence of random
variables taking values in E. The empirical measures n1Pni=1δYi converge almost surely
as n tends to infinity to a random measure Z taking values in M1(E).
We refer to [K07, Chapter 12] for a more thorough treatment of the notion of ex-changeability and in particular [K07, Theorem 12.26] combined with [K07, Remark 12.27] for a proof of Theorem 1.2. We now turn to the generalisation of the diffusion approach from the previous paragraph.
In [FV79] Fleming and Viot introduced more general measure-valued diffusions in the population genetic setting to cope with a larger class of type spaces. Due to the sym-metric reproduction mechanisms of the abovementioned classes of population models exchangeability is retained in both classes, that is if the the initial distribution of the population vector YN(0) is exchangeable, then YN(t) is exchangeable for all t. Thus the relevant information on the type frequencies is contained in the empirical measure
ZN(t) := 1 N N X i=0 δYN i (t).
If we now assume the underlying population model to be the classical continuous time Moran model with r(2, 0, . . . , 0) = N2and all other rates zero and let N tend to infinity, then we obtain a limiting process
Zδ0(t) := lim N →∞ 1 N N X i=0 δYN i (t),
which is a measure-valued diffusion taking values in M1(E). The limit Zδ0(t) exists for every t due to Theorem 1.2. In Section 2.3 we prove path properties of Zδ0(t) in a
more general setup. For µ∈ M1(E) and n∈ N consider test functions of the form
Gf(µ) :=hf, µ⊗ni = Z
En
f (x)dµ⊗n(x), (1.1)
where f ∈ B(En) is bounded and measurable. The generator of the Fleming-Viot process acts on functions of the form (1.1) as follows:
Lδ0G
f(µ) = X 1≤i<j≤n
hf ◦ ηj,i− f, µ⊗ni, (1.2)
where ηj,i(x1, . . . , xi, . . . , xj, . . . , xn) = (x1, . . . , xi, . . . , xi, . . . , xn) duplicates the entry xi and places it at position j. The Fleming-Viot process has been extended to incorpo-rate other evolutionary forces such as recombination and selection. We refer to [EK93] for an overview. Furthermore, it is possible to define Fleming-Viot processes for more general reproduction mechanisms, but let us first introduce the associated genealogies.
1.3. Exchangeable Coalescents
Coalescent processes describe the ancestral relationships in large populations. They have been introduced and extensively studied for time-discrete populations models, thus we also use this framework in this section. The presented results can be modified to hold in the time-continuous setting in a similar form. We introduce Kingman’s coalescent which evolves due to binary merging of ancestral lines. This idea is then generalised to the Λ-coalescents where multiple lines can merge into one at a time. Coalescents with simultaneous multiple mergers are called Ξ-coalescents and form the most general class exchangeable coalescents and.
1.3.1. Kingman’s Coalescent
Assume that the Fisher-Wright model is in equilibrium and take a sample of size n < N from the present generation. Define the ancestral partition process (ΠN
n(k))k∈N0 as
fol-lows: two integers i and j are in the same block of the ancestral partition ΠNn(k) if individual i and j drawn from population alive at time 0 have a common ancestor k generations ago. The genealogy of the sample is then described by the ancestral parti-tion process (ΠNn(k))k∈N0 starting in a the partition where all numbers are singletons.
It evolves by merging blocks of the partition until it eventually reaches the absorbing state{1, . . . , n} of all individuals having the same common ancestor.
Now fix the sample size an let the population size N tend to infinity. Kingman showed in [K82b] that the finite dimensional distributions of the ancestral partition
ΠNn([N t])t≥0,
in the Fisher-Wright model converge to the Kingman’s n-coalescent as N tends to infinity. Note that the time is multiplied by N .
The n-Kingman-coalescent (Πδ0
n(t))t≥0is a continuous-time Markov chain taking val-ues in Pn, the partitions of the numbers {1, . . . , n}. For ξ, η ∈ Pn write ξ ≺ η if η can be obtained from ξ by combining two blocks. The transition rates qξ,η are ei-ther 1 if ξ ≺ η or 0 otherwise. In the initial state all numbers are singletons, thus Πδ0
n(0) =
{1}, . . . , {n} . Since all transitions involve binary merging of blocks, the Markov chain will reach its absorbing state {1, . . . , n} in finite time. We call the absorbing state the most recent common ancestor.
It is possible to give necessary and sufficient conditions for the convergence of the ancestral partition to hold weakly in DPn[0,∞) for general Cannings-models. Kingman
already proved necessary moment conditions, but we state M¨ohle’s Lemma (c.f. [MS03, 5.4]) since it gives necessary and sufficient conditions. Denote by
cN := E
ν1(ν1− 1) N − 1
(1.3) the probability that two uniformly chosen individuals from the present have a common ancestor in the previous generation. It turns out that c−1N is the right timescale to obtain a non-trivial limit of the ancestral partition process. For k ≤ ν denote by
1 2 3 4 5 {1}, {2}, {3}, {4}, {5} {1}, {2}, {3}, {4, 5} {1, 2}, {3}, {4, 5} {1, 2}, {3, 4, 5} {1, 2, 3, 4, 5}
Figure 1.3.: A realisation of Kingman’s 5-coalescent. The ancestral partitions are shown at the transition times. Note that times runs upwards and at each transition time two blocks merge.
(ν)k= ν(ν− 1) · · · (ν − k + 1) the k-th falling factorial of ν. M¨ohles Lemma states that if and only if lim N →∞ E (ν1)3 N E (ν1)2 → 0 (1.4) the convergence ΠNn([c−1N t])t≥0 ⇒ Πδ0 n(t) t≥0
holds weakly as N tends to infinity. Informally, condition (1.4) ensures that the variance of the offspring distribution remains small when compared to the total population size. The n-Kingman-coalescent is a robust model which occurs as the limiting genealogy in many population models, most notably the Fisher-Wright model and the classical discrete-time Moran model. The Kingman coalescent is well studied and has found numerous applications, see for example [D08] or [TZ04]. A realisation of Kingman’s n-coalescent is depicted in Figure 1.3.
The presented rates for Kingman’s n-coalescents define the process for all n∈ N as they are time-continuous Markov chains on a finite state space. These can be used to construct an P∞-valued coalescent (Πδ0(t))t≥0 containing all n-coalescents. Here P∞ denotes the partitions of the natural numbers. The proof is provided in [K82a] and [K82b], and we give a short sketch here. Let ρn,m : Pn → Pm for m < n and ρn : P∞ → Pn denote the natural restrictions of partitions to a smaller number of elements. Using Theorem 2 from [R59] the rates of the coalescent induced by this restriction can be calculated and used to show that
ρn,m Πδn0(t) t≥0 d = Πδ0 m(t) t≥0
holds. An application of Kolmogorov’s extension theorem guarantees the existence of theP∞-valued coalescent process Πδ0 with the property that ρn(Πδ0) is an n-coalescent
for every n ∈ N. The notion exchangeable coalescent stems from the fact that the distribution of the process does not change under renumbering of the ancestral lines.
1.3.2. Λ-coalescents
The only possible transitions in Kingman’s coalescent are due to binary merging of two blocks. More general coalescents, where multiple blocks merge into one at a time, were introduced independently in [P99] and [S99].
Pitman approached the problem in [P99] by defining the Λ-coalescent ΠΛas a Markov process on P∞ evolving due to multiple merging of blocks. Again P∞ denotes the partitions of N. In analogy to Kingman’s coalescent Pitman imposes the consistency condition that every restriction to a finite number of blocks again has to be a time-homogenous Markov chain sharing the same rates. As above, [R59, Theorem 2] can be used to show that the rates of the projected coalescents have to fulfill the relation
λb,k= λb+1,k+ λb+1,k+1, (1.5)
where λb,k is the rate of the event that if the current state has b blocks, then k of them merge into one block.
This relation can be derived as follows: Assume the process ρb+1(ΠΛ) is in a state where {b + 1} is a singleton block. Possible transitions include that the block {b + 1} is involved in a k + 1 merger with certain blocks or that the same blocks perform a k-merger not including{b + 1}. However, since the number b + 1 does not exist in the process ρb+1,b ρb+1(ΠΛ), both transitions map to the same k-merger. Thus the sum of the former rates have to be equal to the latter rates, resulting in equation (1.5).
Pitman used condition (1.5) to establish a one-to-one correspondence between a con-sistent array of rates and a finite measure Λ on the Borel σ-algebra of the interval [0, 1]. The rates can be calculated from this measure via the formula
λb,k = Z 1
0
xk−2(1− x)b−kΛ(dx). (1.6)
The necessity of condition (1.5) can be checked by inserting the integral form (1.6) of the rates, whereas the sufficiency can be proved using the Hausdorff moment prob-lem. The details of the proofs can be found in [P99, Section 3]. The existence of the P∞-valued Markov process (ΠΛ(t))t≥0 can be proved as before using Kolmogorov’s ex-tension theorem. This class of coalescent processes is called Λ-coalescents because each finite measure Λ uniquely determines a coalescent process. The Dirac measure in zero δ0 corresponds to Kingman’s coalescent (which can be checked using equation (1.6)). Denote by λb := Pbk=2 kbλb,k the total rate that any merging event happens if the coalescent is in a state with b blocks.
In [S99] Sagitov pursued a different approach to introduce multiple merger coales-cents. Like Kingman, he investigated the limit of the ancestral partition process for a sample of size n in certain Cannings-models. He considered more general conditions on the moments of the offspring distribution than the ones M¨ohle [M00] proved necessary
and sufficient for the convergence to Kingman’s coalescent given in equation (1.4). In [S03b, Proposition 3] Schweinsberg proved that if and only if
(i) cN → 0, as N → ∞,
(ii) for all y∈ (0, 1) with Λ({y}) = 0, we have N cNP{ν1 > N y} −→ Z (y,1] 1 x2Λ(dx), as N → ∞ and (1.7) (iii) for i6= j, E νi(νi− 1) νj(νj− 1) N2c N −→ 0 , as N → ∞. (1.8)
the ancestral partition process
ΠNn([c−1N t])t≥0
properly rescaled converges to the n-Λ-coalescent. Though the proof was essentially given in [S99, Theorem 3.1], Schweinsberg stated this more concise form of the theorem. Intuitively speaking condition (1.7) allows that a substantial fraction of the individuals in the current generation descended from one individual in the previous generation and this fraction does not vanish in the limit. The distribution of this fraction is governed by the measure x12Λ(dx). These huge reproduction events correspond to the multiple
merging of ancestral lines in the genealogy. Though one individual can have a large number of offspring, condition (1.8) guarantees that the probability of two individuals replacing a substantial fraction of the next generation at the same time remains small and thus no simultaneous (multiple) merger occurs in the genealogy. It can be shown that this convergence holds in the Skorohod sense (c.f. [MS01, Theorem 2.1]). A realisation of a Λ-coalescent is depicted in Figure 1.4.
1.3.3. Ξ-coalescents
More general exchangeable coalescents that allow for the simultaneous merging of mul-tiple ancestral lines were introduced by M¨ohle and Sagitov in [MS01] and Schweinsberg in [S00a]. Again one possible way to introduce these processes that was pursued by the latter is to consider a continuous-time Markov process with state space P∞ and imposing the consistency condition that the projection processes to the finite partitions Pn are time-homogenous Markov chains sharing the same rates. Let λb;k1,...,kr;s be the
rate of the event that if the current state has b blocks, then km merge into a single block for every m ∈ {1, . . . , k} and s blocks remain unchanged. Thus the number of blocks before the coalescent event is given by b = s +Prm=1kmand the number of blocks after the event is r + s. Similar to the result from equation (1.5), Schweinsberg shows in [S00a, Lemma 18] that the rates have to satisfy
λb;k1,...,kr;s = r X m=1
1 2 3 4 5 6 {1}, {2}, {3}, {4}, {5}, {6} {1}, {2}, {3}, {4, 5}, {6} {1, 2}, {3}, {4, 5}, {6} {1, 2}, {3, 4, 5, 6} {1, 2, 3, 4, 5, 6}
Figure 1.4.: A realisation of a 6-Λ-coalescent. The respective ancestral partitions are shown at the transition times. Note that in the third coalescent event three ancestral lines merge at once.
due to the consistency condition since in the pre-image of the event the additional lineage {n + 1} can either participate in any of the mergers, be in an unseen merger with one of the single lines or be all by itself. This relation is then used in Theorem 2 of [S00a] to prove that there is a one-to-one correspondence between finite measures Ξ = Ξ0+ aδ0 on the infinite simplex
∆ :=n(ζ1, ζ2, . . .) : ζ1 ≥ ζ2 ≥ · · · ≥ 0, ∞ X i=1 ζi ≤ 1 o
and exchangeable coalescents with simultaneous multiple mergers, where the rates can be calculated by λb;k1,...,kr;s=a1 {r=1,k1=2} + Z ∆ s X l=0 s l (1− |ζ|)s−l X i16=···6=ir+l ζk1 i1 · · · ζ kr irζir+1· · · ζir+l Ξ0(dζ) (ζ, ζ) , (1.10)
with (u, v) :=P∞i=1uivi and the measure Ξ is splitted into a Kingman part aδ0 and a non-Kingman part with Ξ0({0}) = 0. Again the necessity of the consistency condition can directly be verified by inserting equation (1.10) into equation (1.9) and the suffi-ciency is shown in [S00a] by a generalised version of the Hausdorff moment problem. Because of this one-to-one correspondence, the exchangeable coalescents with simulta-neous multiple mergers are called Ξ-coalescents. Again, the same limiting procedure as before can be used to show the existence of the Ξ-coalescent ΠΞ. M¨ohle and Sagitov followed another way to introduce the n-Ξ-coalescents in [MS01]. As in [S99] they in-vestigated the limit of the ancestral partition process in certain Cannings-models under even more relaxed moment conditions than (1.8) allowing for a stronger dependence between large families. By taking the limit of the transition probabilities of the ances-tral partition process, they showed in Lemma 3.3 of [MS01] that the transition rates of
1 2 3 4 5 6 7 8 {1}, {2}, {3}, {4}, {5}, {6}, {7}, {8} {1}, {2}, {3, 4, 5}, {6}, {7}, {8} {1}, {2}, {3, 4, 5}, {6}, {7, 8} {1, 2}, {3, 4, 5, 6, 7, 8} {1, 2}, {3, 4, 5, 6, 7, 8}
Figure 1.5.: A realisation of a 8-Ξ-coalescent. The ancestral partitions are shown at the transition times. In the first coalescent event three ancestral lines merge at once. In the third coalescent event two lines merge into one and at the same time three other lines merge into one.
the limiting genealogy fulfill a consistency condition similar to (1.9). However, unlike Schweinsberg in [S00a], they characterise the rates of the limiting process in terms of a sequence of symmetric measures (Fr)r∈N, where Fr is concentrated on the finite simplex
∆r:=
(ζ1, . . . , ζr)∈ [0, 1]r : ζ1+ . . . ζr≤ 1
.
In [S03a, Theorem 2.1] Sagitov reformulated the results of [MS01] and stated a con-vergence theorem for the ancestral partition process in terms of the finite measure Ξ from (1.10). He denoted the factor to scale the time by TN, which matches with the timescale from (1.3) since TN ∼ c−1N as N tends to infinity. Let σ2N = V(ν1) and ΦN(dζ) be the joint distribution of the vector (ν(1)
N , . . . , ν(N )
N ) of the ranked relative frequencies for sizes of N sibling groups constituting a generation. If σN2 ∈ o(N) then the time-scaled genealogical process ΠNn(Tnt)
t≥0 converges weakly to the n-Ξ-coalescent if and only if the weak convergence condition
TNΦN(dζ)⇒ Ξ(dζ) (ζ, ζ)
holds on ∆∩ {ζ1 > ǫ} for all ǫ > 0 as N tends to infinity. This can be explained intuitively since the limit of the ranked relative frequencies yields the asymptotic family sizes ordered decreasingly and in this more general case two or more simultaneous big families are allowed at a time, thus leading to the simultaneous multiple mergers in the genealogy. Figure 1.5 shows a realisation of a Ξ-coalescent. In the sequel we will occasionally refer to n-coalescent processes just as coalescent processes, if it is clear from the context.
After describing the possible genealogies embedded in neutral population models we now introduce neutral mutations into the evolution of the population.
1.4. Mutation models
In this section we briefly introduce three models of neutral mutation. We focus on the finitely-many-alleles model, the infinitely-many-alleles model, and most significantly the infinitely-many-sites model, all of which will be introduced and analysed in Chapter 3 in more detail.
The models differ in the actual space of genetic types the individuals attain as well as in the transition mechanism in this type space. However, the basic mechanism deter-mining the time points at which mutation events happen is the same for all mutation models in the time-discrete Cannings-models and in the continuous-time Moran-type models.
Let N ∈ N again denote the size of the population. In discrete time, in addition to the usual resampling step, each individual has a chance of getting a mutant gene instead of inheriting the parental gene. The mutant gene may depend on the parental gene. The probability of inheritting the parental gene is 1− uN and the probability to get a mutant gene is uN for some real number uN ∈ [0, 1]. Since the mutations are assumed to be neutral in the sense that they do not affect the reproductive success of individuals, they do not alter the genealogy. Thus the limiting genealogy is still given by an exchangeable coalescent when the population size tends to infinity. However, if the mutation probability is not scaled correctly in the limiting procedure, then the sample of size n may either show no variability at all, or the types will be completely uncorrelated. Assume that limN →∞c−1N uN = r holds. The probability that no mutation occurred along a genealogical line within the last [c−1N t] generations is given by
P{no mut.} = (1 − uN)[c −1 N t] = 1− r c−1N [c−1N t]
and the last expression tends to e−rt as N tends to infinity since c−1N → ∞ holds. Note that e−rt is the cumulative distribution function of an exponentially distributed random variable and this idea can be made rigorous to show that the mutations occur independently along the branches of the coalescent-tree with rate r.
In the continuous time Moran-type models every individual mutates at rate r. We will see in Section 1.5 that the genealogy of a sample can be obtained by reversing the Moran model dynamics. Thus the mutations in the coalescent tree are again given by a Poisson process along the branches with rate r.
Figure 1.6 shows a multiple merger coalescent tree with mutations along the branches. The following paragraphs introduce the three different mutation models and the tran-sition mechanisms in the type space are given by the corresponding generators which can be readily transformed into the discrete time setting.
Finitely-Many-Alleles. In the finitely-many-alleles mutation model, individuals carry types from a finite type space |E| < ∞. Whenever a mutation event happens to an individual of type x ∈ E, the probability that the mutation leads to type y ∈ E is given by px,y. The transition probabilities are subsumed in the stochastic matrix
1 2 3 4 5 6 x x x x x x x
Figure 1.6.: The Λ-coalescent-tree realisation from Figure 1.4 with mutations along the branches marked by an x.
P = (px,y)x,y∈E. Thus the mutation process for one individual in continuous time is given by the generator
BFMAf (x) = r X y∈E
px,y f (y)− f(x), (1.11)
where f is a bounded and measurable function on E. Some examples are E ={a, A}, the DNA nucleotides EDNA = {A, C, G, T}, or the amino acid alphabet EAA with |EAA| = 20 (or 21). Substitution-matrices for these type spaces exist in the literature. The finitely-many-alleles mutation model can also be used to model sequences of length k with E = EDNAk or E = EAAk . However, these type spaces grow very fast as k increases.
Infinitely-Many-Alleles. In the infinitely-many-alleles model the type space assumed to be infinite. Whenever a mutation event happens, the new type of the affected individual is chosen such that it is a type that has never occurred before in the evolution of the population. The new type is chosen independently of the parental type. This mechanism can be modelled using the unit interval as the type space E = [0, 1] and whenever a mutation occurs the new type is chosen uniformly from that interval. Thus no type x∈ [0, 1] occurs twice almost surely. The action of the generator on functions f ∈ B([0, 1]) is given by
BIMAf (x) = r Z
[0,1]
f (y)− f(x)dy. (1.12)
Infinitely-Many-Sites. The infinitely-many-sites model (c.f. [K69, W75]) was intro-duced to model sequences with an infinite number of sites (see [EG87, Section 3] for a more rigorous treatment of this idea). Whenever a mutation event occurs we assume that this mutation happens at a site that has never mutated before. During reproduc-tion the infomareproduc-tion about the sites at which an individual bears mutareproduc-tions gets passed
on to the children and thus a type in the present generation is a list of sites that have mutated in the ancestral history of this individual. The different sites are referred to by a real number from the unit interval [0, 1] and the type space is modelled as list of such values E = [0, 1]∞. Note that this type space is compact in the product topology. A mutation event adds a new mutated site to the list of mutated sites the parental individual bears. The site is chosen uniformly from the unit interval and thus there has never been a mutation at this site almost surely. The action of the generator is given by BIMSf (x1, x2, . . .) = r Z [0,1] f (y, x1, x2, . . .)− f(x1, x2, . . .)dy, (1.13) for f ∈ B(E).
In Chapter 3 we investigate the properties of populations and sampled individuals under the different mutation models in more detail and present recursions that can be used to calculate the probability to observe a certain sample under the stationary distribution of the population model. Note that all three mutation models introduced so far fit into the framework of a compact polish type space E and the transition in the mutation space is given by a pure jump process with a generator of the form
Bf (x) = r Z
E
f (y)− f(x)q(x, dy), (1.14)
where r is the global mutation rate, f a bounded and measurable function on E (f ∈ B(E)) and q(x, dy) a Feller transition function on E × B(E) that has to be chosen according to the mutation model. Note that this does not include diffusions like Brownian motion which was the mutation mechanism used for example in [FV79]. Some of the results presented in this work, in particular those about Fleming-Viot processes given in Chapter 2, can be generalised to this case but since in Chapter 3 we will focus on the three abovementioned mutation mechanisms that fit into the framework of the (1.14), we restrict our investigations to this case.
In the next section we go back to investigating neutral population models without mutation. This allows us to introduce the concept of duality and show that the Fleming-Viot process is dual to Kingman’s coalescent. The same holds true for more general population models that feature a Λ-coalescent as their limiting genealogy. This duality will play an important role in the subsequent Chapters and we will also re-introduce mutation into the picture.
1.5. Duality and Lookdown Construction
It can be shown that the coalescent processes introduced in Section 1.3 are dual to certain Fleming-Viot processes. For a more complete introduction on duality of Markov processes see for example [E00], [EK86] or [L85].
First compare the evolution of the classical continuous-time Moran model and the dynamics of Kingman’s coalescent and note that the dynamics of the former coincides with the dynamics of the latter under a reversal of the time. In the Moran model each
time present
Figure 1.7.: A visualisation of the duality between the classical Moran model and King-man’s coalescent. The Moran model evolves to the right. The thick lines give the genealogy of the individuals sampled at the present resembling Kingman’s coalescent.
pair of individuals encounters a birth event with rate 1 whereas each pair of lines merges with a rate of 1 in Kingman’s coalescent. This observation can be made more precise.
Recall that the action of the generator of the Fleming-Viot process (1.2) on functions of the form Gf(µ) is given by
X 1≤i<j≤n
hf ◦ ηj,i− f, µ⊗ni. (1.15)
For π = {P1, . . . , Pm} ∈ Pn and x ∈ En define x[π] = π(x) to be the vector with entries (x[π])i := xmin{Pj} if i ∈ Pj for j = 1, . . . , m. For i < j note that if π
i,j is the partition of{1, . . . , n} into singleton blocks except for the indices i and j belonging to the same block, then f◦ ηj,i(x) = f (x[πi,j]) holds for f : En→ R. Substituting this into expression (1.15) allows to re-interpret hf ◦ π, µ⊗ni as a function acting on partitions and the sum as the sum over all possible transitions in Kingman’s coalescent. Thus expression (1.15) also gives the action of the generator of Kingman’s coalescent. The martingale problem representation for both generators and Theorem 1.23 from [E00] can be used to show that the functional analytic duality
EZ(0)Df◦ Π(0), Z(t)⊗nE= EΠ(0)Df◦ Π(t), Z(0)⊗nE (1.16)
holds between Kingman’s coalescent Π = Πδ0 and the Fleming-Viot process Z = Zδ0.
This duality is only a statement about the one-dimensional distributions of the pro-cesses, but we will see in the following section that duality also holds in a pathwise sense.
1.5.1. Classical and Modified Lookdown Construction for Fleming-Viot Process
In [DK96] Donnelly and Kurtz extended the observation (1.16) and, by introducing the lookdown construction, showed that it is possible to construct theP∞-valued Kingman-coalescent and an E∞-valued particle system on the same probability space. The Fleming-Viot process is then obtained as the empirical measure of this particle system. The construction is based on a family of independent Poisson processes (Ni,j)1≤i<j on R≥0, each with rate 1.
Looking forward in time, the lookdown process (X1(t), X2(t), . . .) describes the evo-lution of an infinitely large population in E∞. It evolves as follows: whenever a point of increase of the Poisson process Ni,j is encountered the type of the individual at level i remains unchanged whereas the individual at level j looks down and copies the type from the individual at level i. Donnelly and Kurtz showed that due to this mechanism the evolution of the population below a certain level N does not depend on the evo-lution of the individuals above this level and thus the E∞-valued particle model can be constructed inductively. They showed that this model retains initial exchangeability and thus Theorem 1.2 can again be applied to obtain the Fleming-Viot process defined in Section 1.2.3 as the limit of the empirical measure.
The authors also introduced a modified version of this lookdown construction. In the classical version during a birth event the type of the particle on the lower level is copied and overwrites the type of the higher level particle, thus the type residing on level j vanishes from the population. However, in the modified construction we do not delete the type, but rather insert it one level above. Since there are types residing on the levels above the level involved in the birth event, they get shifted upwards to make room for the new type. Thus no type vanishes from the population. This modified construction has advantages if one considers population models where the population size is allowed to vary over time. Again the Fleming-Viot process can be obtained as the limit of the empirical measure.
By exchangeability, taking a sample of size n from the Fleming-Viot process repre-senting the infinitely large population corresponds to examining the types of the first n levels in the lookdown construction. The two different versions of the lookdown construction are depicted in Figure 1.8.
We now embed a genealogy into the lookdown process. Suppose the population evolved up to a given time T . Define NT
n(t) to be the level at time T− t of the ancestor of the individual occupying level n at time T . Now define the process (Πδ0
n(t))0≤t≤T taking values in the partitions Pn as follows: two integers i, j ∈ {1, . . . , n} are in the same block of the partition Πδ0
n(t) if and only if NiT(t) = NjT(t), that is individual i and j have a common ancestor at time T − t. By the construction of the Poisson processes each pair of ancestral lines merges at rate 1 and thus Πδ0
n(t) is Kingman’s coalescent run for time T . In a similar fashion the P∞-valued Kingman coalescent Πδ0 can be constructed from these Poisson processes. Since we have constructed Kingman’s coalescent and the dual E∞-valued Moran-type model from the same Poisson processes and identified the paths in the forward model with their time reversed counterpart, we
time
(a) The type residing at the upper level in-volved in the resampling event vanishes from the population in the classical lookdown con-struction. The new type is copied from the family member residing on the lower level.
time
(b) The modified lookdown construction. Upon a resampling event types get shifted up-wards and the topmost type vanishes in the modified lookdown construction. Again the family member on the lower level superim-poses its type on the upper level involved in the family.
Figure 1.8.: Example of the classical and modified lookdown construction for the Fleming-Viot process. Both pictures base on the same realisations of the driving Poisson processes.
say that the duality between the processes holds in a pathwise sense. Furthermore we can now make sense of sampling from and embedding a genealogy into the Fleming-Viot process. The lookdown construction has only been defined for t ≥ 0, but this construction can be extended to the whole real line.
1.5.2. Classical and Modified Lookdown Construction for Λ-Fleming-Viot Processes
The functional analytic and pathwise duality also holds in the more general setting of multiple merger coalescents which are dual to the Λ-Fleming-Viot processes. The dualities were established in [DK99] where the lookdown construction was extended to the more general framework. We give a brief overview of the more general lookdown construction here and refer to [DK99], [BBC+05, Section 2] or [BB09, Section 1.2.4] for a more detailed introduction. The term Λ-Fleming-Viot process was coined in [BLG03] by Bertoin and Le Gall who constructed these processes as the duals of certain flows of bridges that closely resemble the Λ-coalescents. Here we follow the lines of Donnelly and Kurtz.
Kingman’s coalescent is the Λ-coalescent with Λ equal to δ0. Like for the measure Ξ in equation (1.10) we can decompose an arbitrary finite measure Λ = aδ0+ Λ0 on the unit interval into a Kingman part δ0 scaled by a and a part Λ0 having no mass at zero that governs the multiple mergers.
The dynamic of the E∞-valued lookdown process with occasional large families is given by the dynamic of the lookdown process with just binary families introduced in
Section 1.5.1 time-scaled by the factor a and independent of this a dynamic for the large families that is governed by Λ0. This dynamic strongly resembles the Poisson process construction of the Λ-coalescent introduced in [P99]. The dynamic is based on a Poisson point process on
R≥0× [0, 1] × [0, 1]N with intensity measure
dt⊗ 1
x2Λ0(dx)⊗ 1[0,1](t)dt
⊗N ,
where dt denotes the Lebesgue measure. It works as follows: if t, x, (u1, u2, . . .) is a point of the Poisson point process, then at time t a resampling event occurs. The ui are interpreted as uniform coin tosses for each level i. The probability for the coins to come up heads is given by the random fraction x. So if ui ≤ x, then the coin for level i came up heads whereas it came up tails otherwise. All levels where the coins came up heads participate in the resampling event at time t and the other levels do not participate. In analogy to the lookdown mechanism introduced in the previous section the individual on the lowest level retains its type. All individuals on higher levels dismiss their previous type, look down, and copy the type residing on the lowest participating level. It might be the case that the measure x12Λ0(dx) is infinite and thus the points of the Poisson
process are dense on the time axis. However, it can be shown by an application of the Master formula (see for example Proposition 1.10 in [RY05, Chapter XII.§1]) that in a finite time interval for each level n there is only a finite number of timepoints where two or more levels below n are involved in a birth event.
The same argument as in Section 1.5.1 can be used to show that the lookdown process X1(t), X2(t), . . . can be constructed inductively since the evolution of the n-th level only depends on the dynamics of the levels below n.
Again this construction can be modified so that the pre-birth types on the levels of the children do not vanish, but get shifted upwards as well as the other types above to make room for the new individuals. Figure 1.9 shows the two versions of the lookdown construction based on the same realisation of the driving Poisson point process. Fur-thermore, it can again be shown that this construction retains initial exchangeability and thus we can define the Λ-Fleming-Viot process
ZΛ(t) := lim n→∞ 1 n n X i=1 δXi(t)
properly since this limit exists due to Theorem 1.2. It can be shown that the generator is given by the following expression.
Definition 1.3 (Λ-Fleming-Viot process). Let f ∈ B(Ep), µ ∈ M
1(E) and Gf(µ) = hf, µ⊗pi as before. For these test functions the generator of the Λ-Fleming-Viot process
without mutation is given by (see [BBC+05, Equation (1.11)])
LΛGf(µ) = X J⊂{1,...,p},|J|≥2 λp;|J|;p−|J| Z f (xJ)− f(x)µ⊗p(dx), (1.17)
time
(a) The types residing at the upper levels in-volved in the resampling event vanish from the population in the classical lookdown con-struction. The levels obtain their new types from the lowest level involved in the resam-pling event.
time
(b) At a resampling event types get shifted upwards and the topmost types vanish in the modified lookdown construction. The vacant levels obtain the new type from the lowest level involved in the resampling event.
Figure 1.9.: Classical and modified lookdown constructions for the Λ-Fleming-Viot pro-cess based on the same realisation of the driving Poisson point propro-cesses.
where (xJ)i = ( xmin(J) if i∈ J, xi otherwise. (1.18)
Remark 1.4. Note that (1.17) includes the generator of the classical Fleming-Viot
pro-cess (1.2) if the summation is restricted to sets J satisfying|J| = 2.
Introducing the process NnT(t) in this setting to keep track of the ancestors and again defining that i ∼ΠΛ
n(t) j if and only if N
T
i (t) = NjT(t), the process (ΠΛn(t))0≤t≤T is a Λ-coalescent run for time T since by Pitman’s Poisson process construction the merging rates equal those of the n-Λ-coalescent. Thus the same pathwise duality holds for the Λ-coalescent and the Λ lookdown process respectively the Λ-Fleming-Viot process. As before the Poisson process construction can be used to construct the P∞-valued Λ-coalescent ΠΛ.
In the next chapter we extend the functional analytic and pathwise duality to the Ξ-coalescent framework with simultaneous large families. Furthermore, we introduce the Ξ-Fleming-Viot process in different ways and show some properties of this process.
Ξ
-Fleming-Viot Process
In this chapter we present the modified lookdown construction for a measure-valued process that we will call the Ξ-Fleming-Viot process with mutation, or the (Ξ, B)-Fleming-Viot process, where B is an operator describing the mutation process. We will establish its duality to the Ξ-coalescent with mutation. The modified lookdown construction will also enable us to establish some path properties of the (Ξ, B)-Fleming-Viot process. This chapter largely coincides with [BBM+09].
As for the Λ-coalescent, the Ξ-lookdown construction closely resembles a Poissonian construction of the Ξ-coalescent that we will employ later.
Poisson Process Construction of theΞ-coalescent
Schweinsberg shows in [S00a, Section 3] that the Ξ-coalescent can be constructed from a family of Poisson processes{NK
i,j}i,j∈N,i<j and a Poisson point process MΞ0 on R+× ∆×[0, 1]N
. The processes NKij have rate a = Ξ({0}) each and govern the binary mergers of the coalescent. The process MΞ0 has intensity measure
dt⊗Ξ0(dζ) (ζ, ζ) ⊗ (1
[0,1](t)dt)⊗N. (2.1)
These processes can be used to construct the Ξ-coalescent as follows: Assume that before the time tm the process ΠΞ is in a state{B1, B2, . . .}. If tm is a point of increase of one of the processes NK
i,j (and there are at least i∨ j blocks), then the corresponding blocks Bi and Bj are merged into a single block. The blocks are renumbered after the event.
The non-Kingman collisions are governed by the points
(tm, ζm, um) = tm, (ζm,1, ζm,2, . . .), (um,1, um,2, . . .)
of the Poisson process MΞ0. The random vector ζ
m denotes the respective asymptotic family sizes in the multiple merger event at time tm and the um are “uniform coins”, determining the blocks participating in the respective merger groups; see (2.8) or [S00a, Section 3] for details.
A Modified Lookdown Construction of the (Ξ, B)-Fleming-Viot Process
Recall that in the continuous-time Moran model a population is described by a vector YN(t) = (YN
1 (t), . . . , YNN(t)) with values in EN, where YiN(t) is the type of individual
i at time t. In addition to reproduction, each particle undergoes mutation according to the bounded linear mutation operator
Bf (x) = r Z
E
(f (y)− f(x))q(x, dy), (2.2)
where f is a bounded measurable function on E, q(x, dy) is a Feller transition function on E× B(E), and r ≥ 0 is the global mutation rate.
As for the Λ-lookdown construction, the resampling of the population is governed by the Poisson processes NK
ij and the Poisson point process MΞ0, which were introduced as a driving processes for the Ξ-coalescent. In particular, the resampling events allow for the simultaneous occurrence of one or more large families. The resampling procedure is described in detail in Section 2.2. An important fact is that this resampling is made such that it again retains exchangeability of the population vector.
In Section 2.2, we introduce the lookdown particle system XN = (X1N, . . . , XNN) again with values in EN. Each particle mutates according to the same generator (2.2) as before. For the resampling event, we will use the same driving Poisson processes, but we will use the modified lookdown construction of Donnelly and Kurtz introduced in Section 1.5.2, suitably adapted to our scenario. This (Ξ, B)-lookdown process will be introduced in Section 2.2.2. Again, it is crucial that the resampling events retain exchangeability of the population vector and that the process {XN(t)} has the same empirical measure PNi=1δXN
i (t) as the process{Y
N(t)}.
The construction of the resampling events allows us to pass to the limit as N tends to infinity and obtain an E∞-valued particle system X = (X1, X2, . . .). Since this particle system is also exchangeable, the almost sure limit of the empirical measure as N tends to infinity can be accessed by the De Finetti Theorem (c.f. Theorem 1.2). This is not possible for the YN.
2.1. Path Properties and Generator
Let D(B) denote the domain of the mutation generator B and let f1, f2, . . . ∈ D(B) be functions that separate points of M1(E) in the sense that
R
fkdµ = R
fkdν for all k ∈ N implies that µ = ν. Such sequences exist, see for example Section 1 of [DK96] (Lemma 1.1 in particular). Define the metric d onM1(E) via
d(µ, ν) := X k 1 2k Z fkdµ− Z fkdν , µ, ν ∈ M1(E)
and equip the topology of locally uniform convergence on DM1(E)([0,∞)) with the metric
dp(µ, ν) := Z ∞
0
e−td(µ(t), ν(t)) dt. (2.3)
Theorem 2.1. The M1(E)-valued process (ZΞ(t))t≥0, defined in terms of the ordered particle system X = (X1, X2, . . . ) by ZΞ(t) := lim n→∞Z n(t) = lim n→∞ 1 n n X i=1 δXi(t), t≥ 0,
is called the Ξ-Fleming-Viot process with mutation operator B or simply the (Ξ,
B)-Fleming-Viot process. Moreover, the empirical processes (Zn(t))t≥0 converge almost
surely on the path space DM1(E)([0,∞)) to the c`adl`ag process (ZΞ(t))t≥0.
Since the empirical measures of XN and YN are identical, the following corollary holds true.
Corollary 2.2. Define, for each n,
˜ Zn(t) := 1 n n X i=1 δYn i (t), t≥ 0,
the empirical process of the n-th unordered particle system, and assume that ˜Zn(0) ⇒ ZΞ(0) weakly as n→ ∞. Then, ( ˜Zn(t))t≥0 converges weakly on DM1(E)([0,∞)) to the
(Ξ, B)-Fleming-Viot process (ZΞ(t))t≥0.
The Markov process (ZΞ(t))t≥0 is characterized by its generator as follows.
Proposition 2.3. The (Ξ, B)-Fleming-Viot process (ZΞ(t))t≥0is a strong Markov
pro-cess. Its generator, denoted by L, acts on test functions of the form Gf(µ) = hf, µ⊗ni
defined in equation (1.1) via
LGf(µ) := Laδ0Gf(µ) + LΞ0Gf(µ) + LBGf(µ), (2.4) where Laδ0G f(µ) := a X 1≤i<j≤n Z En f ηj,i(x) − f(x)µ⊗n(dx), (2.5) LΞ0G f(µ) := Z ∆ Z EN Gf (1− |ζ|)µ +P∞i=1ζiδxi − Gf(µ) µ⊗N(dx)Ξ0(dζ) (ζ, ζ) ,(2.6) LBGf(µ) := r n X i=1 Z En Bi(f (x))µ⊗n(dx), (2.7)
and Bif is the mutation operator B, defined in (1.14), acting on the i-th coordinate
of f and ηj,i is as in equation (1.2).
Remark 2.4. (i) In Section 1.5.2 we presented the construction and pathwise duality for
the Λ-Fleming-Viot established by Donnelly & Kurtz in [DK99]. In some sense, their paper works under the general assumption “allow simultaneous and/or multiple births and deaths, but we assume that all the births that happen simultaneously come from
the same parent” (p. 166), even though in [DK99, Section 2.5] they very briefly mention a possible extension to scenarios with simultaneous multiple births to multiple parents. In the present Chapter we convert these ideas into theorems.
(ii) Note that in a similar direction, Bertoin & Le Gall remark briefly [BLG03, p. 277] how their construction of the Λ-Fleming-Viot process via flows of bridges can be ex-tended to the simultaneous multiple merger context (but leave details to the reader). We are not following this approach, as it is hard to combine with a general type space and general mutation process.
(iii) The Ξ-Fleming-Viot process has recently been independently constructed by Taylor and V´eber (personal communication, 2008) via Bertoin and Le Gall’s flow of bridges [BLG03] and Kurtz and Rodriguez’ Poisson representation of measure-valued branch-ing processes [KR08]. In this context we refer to Taylor and V´eber [TV09] for a larger study of structured populations, in which Ξ-coalescents appear under certain limiting scenarios.
(iv) Note that the modified lookdown construction of the Λ-Fleming-Viot process con-tains all information available about the genealogy of the process and therefore also provides a pathwise embedding of the Λ-coalescent measure tree considered by Greven, Pfaffelhuber and Winter [GPW09]. We conjecture that a similar statement holds for the Ξ-coalescent.
The rest of this Chapter is organised as follows: In Section 2.2 we use the Poisson point process MΞ0 to introduce the finite unordered (Ξ, B)-Moran model YN and the finite
ordered (Ξ, B)-lookdown model XN. It is shown that the ordered model is constructed in such a way that a well defined limit is obtained as N tends to infinity. We will also show that the reordering preserves the exchangeability property, which will be crucial for the proof in Section 2.3. In this section, we will introduce the empirical measures of the process YN and XN, show that they are identical and converge to a limiting process having nice path properties, which is the statement of Theorem 2.1.
Section 2.4.2 will be concerned with the generator of the Ξ0-Fleming-Viot process. We will give two alternative representations and show that it generates a strongly continuous Feller semigroup. Furthermore, we will show that the process constructed in Section 2.3 solves the martingale problem for this generator.
One representation of the generator will then be used in Section 2.5 to establish a functional duality between the Ξ-coalescent and the Ξ-Fleming-Viot process on the genealogical level. Due to the Poissonian construction, this duality can also be extended to a “pathwise” duality. We will also give a function-valued dual, which incorporates mutation.
In Section 2.6, we look at two examples: The first example is concerned with a population model with recurrent bottlenecks. Here, a particular Ξ-coalescent, which is a subordination of Kingman’s coalescent, arises as a natural limit of the genealogical process. The second example discusses the Poisson-Dirichlet-coalescent and obtains explicit expressions for some quantities of interest.
2.2. Exchangeable E
∞-valued Particle Systems
We now introduce the particle systems YN and XN and show that their empirical measures coincide.
2.2.1. The Canonical (Ξ, B)-Moran Model
We can use the Poisson processes NKij and MΞ0 governing the Ξ-coalescent to describe
a corresponding forward population model with occasional large families in a canonical way, simply reversing the construction of the coalescent by interpreting the merging events as resampling events. For the moment we just consider MΞ0, but the Kingman
part can be incorporated later. Consider the points
(tm, ζm, um) = tm, (ζm,1, ζm,2, . . .), (um,1, um,2, . . .)
of MΞ0 defined by (2.1). The t
m denote the times of reproduction events. Define
g(ζ, u) := (
min{j | ζ1+· · · + ζj ≥ u} if u ≤Pi∈Nζi,
∞ otherwise. (2.8)
At time tm, the N particles are grouped according to the values g(ζm, um,l), l = 1, . . . , N as follows: For each k ∈ N, all particles l ∈ {1, . . . , N} with g(ζm, um,l) = k form a family. Among each non-trivial family a parental particle is chosen uniformly and it superimposes its type on the other family members. Note that although the jump times (tm) may be dense in R+, the condition
Z ∆ X i ζi2Ξ0(dζ) (ζ, ζ) = Ξ(∆) < ∞
guarantees that in a finite population in each finite time interval only finitely many non-trivial reproduction events occur. Each particle follows an independent mutation process between reproductive events, according to the generator (1.14).
The reproduction mechanism yields the dynamics of a Moran-type population model. Since in addition the particles evolve due to mutation we refer to this model as the N -particle (Ξ, B)-Moran model. For a given time t ≥ 0 it is described by a random vector
YN(t) := Y1N(t), . . . , YNN(t), taking values in EN.
Remark 2.5. Note that this model is completely symmetric, thus, for each t, the
