The effects of a weak selection pressure in a spatially structured population

Transcription

1 The effects of a weak selection pressure in a spatially structured population A.M. Etheridge, A. Véber and F. Yu CNRS - École Polytechnique

2 Motivations Aim : Model the evolution of the genetic composition of a geographically structured population. Space is continuous (and in 2d, most of the time). In particular : What happens when blue flowers are more sexy to bees?

3 The spatial Λ-Fleming-Viot process with selection Geographical space : R d. Type/allele space : {0, 1}. We assume that type 0 is more sexy (favoured by natural selection). Population at time t : Measure M t on R d {0, 1} whose first marginal is Lebesgue measure (uniform density of individuals). Define w t(x) := M t({x} {1}) = frequency of type 1 s at x at time t. Evolution : Fix an impact u (0, 1], a finite measure µ(dr) on (0, ) and some s > 0. Neutral events : Π N = Poisson point process on R + R d (0, + ) with intensity measure dt dx µ(dr) ; Selective events : Π S = Poisson point process on R + R d (0, + ) with intensity measure s dt dx µ(dr).

4 The spatial Λ-Fleming-Viot process with selection Geographical space : R d. Type/allele space : {0, 1}. We assume that type 0 is more sexy (favoured by natural selection). Population at time t : Measure M t on R d {0, 1} whose first marginal is Lebesgue measure (uniform density of individuals). Define w t(x) := M t({x} {1}) = frequency of type 1 s at x at time t. Evolution : Fix an impact u (0, 1], a finite measure µ(dr) on (0, ) and some s > 0. Neutral events : Π N = Poisson point process on R + R d (0, + ) with intensity measure dt dx µ(dr) ; Selective events : Π S = Poisson point process on R + R d (0, + ) with intensity measure s dt dx µ(dr).

5 Evolution rules Neutral event : If (t, x, r) Π N, at time t a reproduction event occurs within B(x, r). A parent is chosen uniformly at random from B(x, r) [type k] ; For every y B(x, r), w t(y) = (1 u)w t (y) + u1 {k=1}. Selective event : If (t, x, r) Π S, at time t a reproduction event occurs within B(x, r). Two potential parents are chosen uniformly at random from B(x, r) [types k and k ] ; For every y B(x, r), w t(y) = (1 u)w t (y) + u1 {k=k =1}.

6 Evolution with u = 1, s = 0 and r fixed (sim. by H. Saadi) Initial configuration :

7 Evolution with u = 1, s = 0 and r fixed (sim. by H. Saadi) After events :

8 Evolution with u = 1, s = 0 and r fixed (sim. by H. Saadi) After events :

9 Evolution with u = 1, s = 0 and r fixed (sim. by H. Saadi) After events :

10 Evolution with u = 1, s = 0 and r fixed (sim. by H. Saadi) After events :

11 Questions Does the favourable type always invade the population? When it does, shape of the invaded area after a large time? Links with the F-KPP equation? Existence of a front and a speed? New behaviour when large-scale extinction-recolonisation events occur? Fisher-KPP equation with noise : dw t = { } 1 wt swt(1 wt) dt wt(1 wt) Ẇ, N where N is an effective population size and W a space-time white noise. This type of noise can be included only in 1d. Admits travelling wave solutions with constant speed.

12 Precise framework Case 1 : Fixed radius and impact All events have the same radius R > 0 and the same impact u (0, 1]. Most natural first case... Asymptotic behaviour equivalent to that of the (biased) voter model. Case 2 : Radii with an α-stable distribution We fix α (1, 2) and set µ(dr) = 1 {r>1} dr. r d+1+α All events have the same impact u (0, 1]. Allows very large but very rare events. Rescaled ancestral lineages follow α-stable processes.

13 Rescalings Assume the local density of individuals is large (u 1), the selective advantage is weak (s 1). We look at the long-term behaviour of the population. Rk : Of course, many other scenarios could be considered!! Hence, set : In fact, we rather consider u n = u n γ, sn = s n δ, w n t (x) = w nt(n β x). w n t (x) = nβd V 1 B(x,n β ) w n t (y) dy.

14 Asymptotics - Case 1 u n = un 1/3, s n = sn 2/3, w n t (x) = w nt(n 1/3 x). Theorem 1 (Fixed radius) Suppose the initial condition ( w n 0 ) n 1 converges to w 0. Then as n, w n converges weakly towards a process w characterized by : For every f C c(r d ) and every t 0, t { } wt uγr, f = w 0, f w s, f usv R ws (1 ws ), f ds + M t, where Γ R > 0 depends only on d, and (M t) t 0 is 0 when d 2, and is a martingale with quadratic variation when d = 1. u 2 V 2 R t 0 w s (1 w s ), f 2 ds

15 In words... We have identified a range of parameters for which the frequency of the less favoured type can be precisely understood. The limit is a weak solution to the F-KPP equation (with noise in 1d). The influence of u and s is clearly identified. When d 2, it is impossible to keep a noise in the limit. However, for n 1, w n can be seen as a solution to F-KPP with small noise. And this process is not too difficult to handle.

16 Asymptotics - Case 2 β = 1 2α 1, γ = α 1 2α 1, δ = α 2α 1. Theorem 2 (Stable radii) Suppose the initial condition ( w n 0 ) n 1 converges to w 0. Then as n, w n converges weakly towards a process w characterized by : For every f C c(r d ) and every t 0, t { wt, f = w 0, f + u ws, ( ) α/2 f usv } 1 0 α w s (1 ws ), f ds+m t, where ( ) α/2 is the generator of a symmetric α-stable process, and (M t) t 0 is 0 when d 2, and is a martingale with quadratic variation when d = 1. 4u 2 α 1 t 0 w s (1 w s ), f 2 ds

17 Consequences Again, we have identified an interesting range of parameters. The limit is a weak solution to an F-KPP-style equation, where the underlying motion allows very large (but rare) jumps Long-range correlations between the local frequencies due to the fact that individuals can send their offspring at very large distances. The effects of selection and noise are still local (surprising, at first sight). Known behaviour of the limit : the front speed is no longer constant, but grows exponentially with time.

18 Genealogies We can associate a process of potential ancestors to this model. Suppose we sample k individuals from our population at time T. Then, if (T t, x, r) is an event in the past containing some of their ancestors, Neutral events : Each ancestor in B(x, r) was in the fraction of the pop. replaced with proba u, indep. of each other. All those lying in this fraction merge into a common ancestor, whose location is uniformly distributed over B(x, r). Selective events : Each ancestor in B(x, r) was in the fraction of the pop. replaced with proba u, indep. of each other. All those lying in this fraction are replaced by two ancestors indep. and uniformly distributed over B(x, r). If N t is the number of ancestors and ξt i is the location of the i-th ancestor at time T t, then (ξ t) t 0 = ({ξt 1,..., ξ N t t }) t 0 is a system of branching and coalescing jump processes.

19 Genealogies We can associate a process of potential ancestors to this model. Suppose we sample k individuals from our population at time T. Then, if (T t, x, r) is an event in the past containing some of their ancestors, Neutral events : Each ancestor in B(x, r) was in the fraction of the pop. replaced with proba u, indep. of each other. All those lying in this fraction merge into a common ancestor, whose location is uniformly distributed over B(x, r). Selective events : Each ancestor in B(x, r) was in the fraction of the pop. replaced with proba u, indep. of each other. All those lying in this fraction are replaced by two ancestors indep. and uniformly distributed over B(x, r). If N t is the number of ancestors and ξt i is the location of the i-th ancestor at time T t, then (ξ t) t 0 = ({ξt 1,..., ξ N t t }) t 0 is a system of branching and coalescing jump processes.

20 Duality relation For every k N, ϕ C c((r d ) k ) and t 0, we have [ ( k ) ] E w0 ψ(x 1,..., x k ) w t(x j ) dx 1 dx k (R d ) k That is, for a.e. x 1,..., x k, j=1 = ψ(x 1,..., x k ) E {x1,...,x k } (R d ) k [ Nt ] w 0 (ξ j t ) dx 1 dx k. j=1 [ k ] [ Nt E w0 w t(x j ) = E {x1,...,x k } w 0 (ξ j t ]. ) j=1 j=1 Usually, the proof of the convergence of (w n t ) t 0 is based on that of the rescaled process (n β ξ 1 nt,..., n β ξ N nt nt ) t 0. But not in this case!

21 Duality relation For every k N, ϕ C c((r d ) k ) and t 0, we have [ ( k ) ] E w0 ψ(x 1,..., x k ) w t(x j ) dx 1 dx k (R d ) k That is, for a.e. x 1,..., x k, j=1 = ψ(x 1,..., x k ) E {x1,...,x k } (R d ) k [ Nt ] w 0 (ξ j t ) dx 1 dx k. j=1 [ k ] [ Nt E w0 w t(x j ) = E {x1,...,x k } w 0 (ξ j t ]. ) j=1 j=1 Usually, the proof of the convergence of (w n t ) t 0 is based on that of the rescaled process (n β ξ 1 nt,..., n β ξ N nt nt ) t 0. But not in this case!

22 Convergence of the ancestries Under the same scalings as before, we have : Theorem 3 (Fixed radius) The sequence of processes (ξ n ) n 1 converges in distribution to a system of independent Brownian motions with speed uγ R, branching at rate usv R into two particles. Furthermore, when d = 1 each pair of particles coalesces at rate u 2 V 2 R times their local time together (no coalescence when d 2). Theorem 4 (Stable radii) The processes (ξ n ) n 1 converges in distribution to a system of independent symmetric α-stable motions, branching at rate usv 1 /α into two particles. Furthermore, when d = 1 each pair of particles coalesces at rate 4u 2 /(α 1) times their local time together (no coalescence when d 2).

23 To go further Open door to the study of several loci (= locations on the same chromosome) and of genetic hitchhiking in particular. Other regimes of parameters? Shape of the front when selection is strong? Relation to models of range expansion?

24 Range expansion Extreme case of selection : only type 0 s reproduce. Expanding population of Pseudomonas aeruginosa (courtesy of Kevin Foster), and a simulation of the modified SLFV, by J. Kelleher.

25 Thanks for your attention!