Surfing genes. On the fate of neutral mutations in a spreading population

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1 Surfing genes On the fate of neutral mutations in a spreading population Oskar Hallatschek David Nelson Harvard University ohallats@physics.harvard.edu

2 Genetic impact of range expansions Population expansions have a profound effect on the genealogy of a species, because the gene pool for the new habitat is provided only by a small number of founder individuals. The genetic footprints of these pioneers are often recognizable today and provide information about the migrational history of the species. E.g.: In agreement with the out-of- Africa hypothesis, it is found that the neutral* genetic diversity of humans decreases roughly linearly with the distance from Africa. See, e.g., Ramachandran S, et al. (005) PNAS. (*Neutral mutations have no effect on the phenotype; act like inheritable

3 Abstract To assist the inference of migrations from spatially resolved genetic data, we study the dynamics of neutral gene frequencies in a population undergoing a continual range expansion in one dimension. During such a colonization period, lineages can fix at the wave front by means of a surfing mechanism*. We quantify this phenomenon in terms of (i) the spatial distribution of lineages that reach fixation and, closely related, (ii) the continual loss of genetic diversity at the wave front, characterizing the approach to fixation. Our simulations show that an effective population size can be assigned to the wave that controls the (observable) gradient in genetic diversity left behind the colonization process. This effective population size is markedly higher in pushed waves than in pulled waves, and increases only sub-linearly with deme size. We develop an analytical approach, based on the physics of reaction-diffusion systems, that yields simple predictions for any deterministic population dynamics. O. H. and David Nelson (007) *Edmonds, C. A., Lillie, A.S., & Cavalli-Sforza LL. (004) PNAS. S. Klopfstein, M. Currat, and L. Excoffier, Mol. Biol. Evol. 3, (006)

4 Surfing neutral mutations t 0 A neutral mutant (label red) arises in the wave front at lattice site i=9. Due to random number fluctuations, the genetic diversity is strongly reduced. By chance, red mutants dominate. t t f Fixation in the co-moving frame of descendents of red. ( successful surfing event) Numbers in these sketches represent inheritable labels that are used in our simulations to trace back the spatial origin of individuals in the wave front. For instance, descendents of red are associated with position 9 in the co-moving frame. The dashed blue frame indicates the comoving simulation box.

5 Statistical mechanics of gene surfing x Where do successful surfers originate? How fast does the genetic diversity decrease at the wave front?

6 Ancestral distribution (Simulation) P i Fig. 1. Distributions of the spatial origin of mutations that reach fixation (bell-curves) together with averaged wave profiles n i (sigmoidal curves) for two discrete lattice models of propagating fronts. In the continuum limit, (a) describes a stochastic Fisher wave with linear growth rate of s 0.1, diffusion constant of 1 and carrying capacity N. (b) Identical to (a) except that the effective growth rate is set to zero for ni N c.

7 Theory vs. simulation Deterministic Theory: P n expv i i (See Eq. (1), below) i Fig.. The data from Fig. b (thin lines) superimposed with predictions (thick lines) for the ancestral distribution function P i based on the theoretical expression (red box) that depends on the wave profile n i and the velocity v. The apparent systematic deviations in the pulled case ( N c 0 ) are caused by fluctuations at the tip of the wave.

8 Decrease in genetic diversity Fig. 3. H t Pr t c t is the probability that one has to go back at least a time t c (coalescence time) to find the most recent common ancestor of two individuals randomly chosen from the simulation box. Both models show that an exponential decay is reached asymptotically. By analogy with well-mixed populations, in which H (t) decays exponentially with rate / N (Moran model), it is convenient to express the decay rate as / Ne, i.e., in terms of an effective populations size N e.

9 Effective population size Theory 1 N e Pi / i n (See Eq. (), below) i Fig. 5. The measured effective population size as a function of carrying capacity N on a log-log scale for pulled waves ( N c 0, asterisks) and pushed waves ( N c 10, crosses). The dashed and dotted lines have slope.30 and.4, respectively, i.e., significantly smaller than 1. Triangles represent effective populations sizes as inferred from the strong migration approximation (red box). The inset shows the behavior of N e for varying cutoffvalue and fixed carrying capacity N=1000, again, on a log-log scale. N c N e

10 (Theory) Ancestral distribution Backward in time: Drift & diffusion, no reaction time t x,t G, x, t, space G, x, t : probability that a x,t particle at had an ancestor at, Fokker-Planck description: G, x, t D G VG D, Diffusion constant V U P v U, Drift coefficient Dlog lim G, x, t exp c, Deterministic approx.: c, c P : q-bio/ U / D c expv D s / Ancestral distribution function s (1)

11 (Theory) Coalescence time t 0 distribution of two lineages Pr Pr Coalescence rate of two lineages. t Prt G Probability that the lineages have not coalesced earlier. c c s c Probability of meeting at the same place. (Strong migration approximation*) s d / Ne 1 t c e with N P / c O. H. and David Nelson (007) e Rate of coalescence given the lineages are at the same place. d Effective population size *Nagylaki, T. (1980) J. Math. Biol. 9, s t c x ()

12 (Appendix) No surfing on Fisher waves In the case of the deterministic Fisher wave (Fisher 1937), the wave profile obeys in the foot of the wave c exp vx / D const., for x. As a consequence, Eq. (1) is not normalizable and 0. P I.e., gene surfing is not possible on a deterministic Fisher wave, in contrast to our simulations of a stochastic Fisher wave, cf. Fig.1a. For an approximate description of the stochastic case, a heuristic cutoff* (function) in the growth rates may be introduced to ensure that the survival probability u of a mutant does not exceed the theoretical limit (O. Hallatschek, unpublished). P / c *Brunet, E. and Derrida, B. (1997) Phys. Rev. E 56,

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