4.2 The Moran model with selection and recombination
In order to gain a better understanding of Eq. (4.9) and to prepare for the genealogical arguments to follow in Section 4.5, we briefly recall the Moran model with selection and recombination. This is a stochastic model that describes selection and recombination in a finite population, from which (4.9) is recovered via a dynamical law of large numbers. We will use the representation as aninteracting particle system (IPS). The Moran IPS works with N individuals, labelled 16α6N, each equipped with a (random) type Ξt(α)∈X(of (2.1)) at timet, which behaves as follows.
Every individual β reproduces asexually at a fixed rate according to its fitness. That is, unfit individuals reproduce at rate 1 whereas fit individuals reproduce at rate 1 +s, where s > 0 is again the selection intensity. Upon reproduction, the single offspring inherits the parent’s type and replaces a uniformly chosen individualαin the population (possibly its own parent). We will realise the different reproduction rates of the two types by distinguishing between neutral reproduction events, which happen at rate 1 to all individuals regardless of their type, and selective reproduction events, which are additionally performed by fit individuals at rates. This distinction is a crucial ingredient in the ancestral selection graph [KN97].
At rate ̺i, i ∈ S∗, individual β reproduces sexually, choosing a partner γ uniformly at random, possibly β itself. (Biologically, this means that we include the possibility of selfing.) The offspring is of type ΞCi(β),ΞDi(γ) and replaces another uniformly chosen individual α, possibly one of its own parents.
Formally, we can thus define the Moran IPS as a continuous-time Markov chain with states ξ= ξ(α)16α6N ∈XN and the following transitions when Ξt= Ξt(α)16α6N =ξ: where, for 16ε6N, the new state vectors explicitly read
ξneutα←β(ε) =
Remark 4.3. The reader may wonder at this point why we include both sexual and asexual reproduction in our model. However, the ‘asexual’ reproduction events are actually sexual ones in which no recombination has occurred; that is,C=∅andD=S, so the offspring is a full copy of the first parent, and the second parent is irrelevant. Selective reproduction never occurs together with recombination due to the independence built into the SRE. ♦
For our purpose, it is particularly profitable to take advantage of thegraphical representation of the Moran IPS, see Figure 4.2. Here, every individual is represented by a horizontal line, lines are labelled 16α6N from bottom to top, and reproduction events are represented by arrows between the lines with the parent at the tail, the offspring at the tip, and the offspring replacing the individual at the target line (arrows pointing to their own tails have no effect and are omitted). In line with (4.13) and for reasons to become clear when taking the ancestral perspective in Section 5, we distinguish two types of arrows: neutral arrows (with normal arrowheads), which appear between every ordered pair of lines at rate 1/N regardless of the types of the lines; and selective arrows (with star-shaped arrowheads), which are laid down at rate s/N between every ordered pair of lines, again regardless of the types. Similarly, a recombination event in which the individual at lineαis replaced by the joint offspring of linesβ and γ is encoded as a square (on theα-th line) in which the recombination siteiis inscribed and which is accompanied by two arms connecting to the parents and labelled C or D, indicating which of the parents contributes thei-head andi-tail, respectively. These graphical elements appear at rate̺i/N2 for every ordered triple of lines and everyi∈S∗. If both arms connect to the same parent, the recombination event turns into a neutral reproduction event.
Remark 4.4. In view of this graphical construction, another perspective on the transition rates in the Moran IPS is natural. We can say that, with rates̺i, each individual is replaced by the joint offspring of two uniformly chosen parents with the crossover point at site i.
Likewise, at rate 1, each individual is replaced by the offspring of a single uniformly chosen parent individual; and with rates|{16α6N : Ξi∗,t(α) = 0}|, it is replaced by the offspring of a parent individual chosen uniformly from the subset of fit individuals. This point of view will be particularly useful when looking back in time in Section 4.5. ♦
The fact that we use different kinds of arrows for the two types of reproduction events (rather than simply letting fit individuals shoot reproduction arrows at a faster rate) reflects the distinction between neutral and selective reproduction. The advantage of this strategy is that it allows for an untyped construction of the Moran IPS; that is, we first lay down the graphical elements between the lines regardless of the types and only then assign an initial type configuration. This type configuration is finally propagated forward in time under the rule that only individuals of beneficial type use the selective arrows to place their offspring, while neutral arrows and the arms of recombination events are used by all individuals, regardless of type.
4.2 The Moran model with selection and recombination 39
Consider now the processZ(N):= (Zt(N))t>0, whereZt(N) is the empirical measure Zt(N):= 1
N XN α=1
δΞt(α);
Proposition 3.1 in [Cor17b] in combination with Theorem 2.1 from [Ess16] (see also [BEP16]) shows that, asN → ∞without rescaling of parameters or time, the processes Z(N) converge almost surely locally uniformly to the solutionω = (ωt)t>0 of the deterministic SRE (4.9) for every finite time horizon, wheneverZ0(N)converges to ω0. This is because the Moran models, indexed with population size, form a density-dependent family, for which a dynamical law of large numbers applies; see [EK86, Thm. 3.2, Ch. 11].
For completeness, we will quote this theorem and see how it applies to our situation. In the terminology of [EK86], adensity-dependent family is a sequence (X(N))N>1 of Markov chains in continuous time, each defined on its own state space E(N) which is assumed to be of the form
E(N) =E∩ 1 NZd,
whereE is some subset of Rd. It is further assumed that there exists a collection (βℓ)ℓ∈Zd of non-negative functions on E which describe the transition ratesq(N)(e1, e2) of these Markov chains via
q(N)(e1, e2) =N βN(e2−e1)(e1).
For such a family of Markov chains, the following theorem holds.
Theorem 4.2 ([EK86, Thm. 3.2, Ch. 11]). Let (X(N))N>1 be a density-dependent family as above. Suppose that, for each compact K ⊆E,
X
ℓ∈Zd
|ℓ| sup
e1∈K
βℓ(e1)<∞ (4.14)
and that G := Pℓ∈Zdℓβℓ is Lipschitz continuous. If X0(N) → x0 ∈ E as N → ∞, then for every t>0
N→∞lim sup
06s6t
|Xs(N)−Xs|= 0 almost surely, where X is the unique solution of
X˙t=G(Xt) (4.15)
with initial condition X(0) =x0.
Remark 4.5. Clearly, Eq. (4.14) ensures that G is a well-defined continuous function onE.
Implicitly, we assume that a global solution to (4.15) exists; its uniqueness is guaranteed by the assumed Lipschitz continuity ofGvia the Picard–Lindelöf Theorem. ♦ For us, the role ofE will be played by the simplexP(X) of probability measures, which can be thought of as a subset of R2n. Clearly, as Z(N) records the relative frequencies of types
1
Figure 4.2. Graphical representation of the Moran IPS. Time runs from left to right. Arrows corresponding to neutral reproduction events are depicted with normal arrowheads, selective arrows with star-shaped arrowheads; recombination events are symbolised by squares con-taining the recombination point, and arms connecting to the parents that contribute the head (C) and tail (D) segments. The selected site is marked in light brown.
in a population of total size N, this process takes values in P(X)∩ N1Z2n. Its transition rates can be described as follows (see also Remark 4.4). A transition from state ζ ∈ P(X) to ζ +N1(δy −δx) occurs if an individual of type x is replaced by an individual of type y.
This happens either due to neutral reproduction (4.10) with rate N ζ(y)ζ(x), due to selective reproduction (4.11) with rate s(1−yi∗)N ζ(y)ζ(x) and due to recombination (4.12) at sitei with rate ̺iRi(ζ)(y)N ζ(x). Thus, the total transition rate is given by
q(N)(ζ, ζ+ 1
Since individuals are replaced one by one, all possible transitions are of this form and q(N)(ζ, ζ′) = 0 for all ζ′ which are not of the form ζ +δy −δx for some x, y ∈ X. Due to the finiteness of X, the summability condition (4.14) is trivially fulfilled and we have
X
x,y∈X
βδy−δx(ζ)(δy −δx) = (ζ−ζ) +s F ζ−f(ζ)ζ+ X
i∈S∗
̺i(Riζ−ζ) = Ψsel(ζ) + Ψrec(ζ).
That the right-hand side is Lipschitz continuous follows from the Lipschitz continuity of the recombinators (compare [BBS16, Prop. 1]) and thus, of Ψrec. The Lipschitz continuity of Ψsel is immediate. Thus, Theorem 4.2 yields
N→∞lim sup
06s6t
|Zs(N)−ωs|= 0,
under the assumption that Z0(N)→ω0 asN → ∞, whereω= (ωt)t>0 solves Eq. (4.9).