The fate of beneficial mutations in a range expansion

Transcription

1 The fate of beneficial mutations in a range expansion R. Lehe (Paris) O. Hallatschek (Göttingen) L. Peliti (Naples) February 19, 2015 / Rutgers

2 Outline Introduction The model Results Analysis

3 Range expansion reduces genetic diversity Out of Africa: Cavalli-Sforza et al., 1994

4 Range expansion reduces genetic diversity Out of Africa: Prugnolle et al., 2005

5 Gene surfing Hallatschek and Nelson, 2007

6 Gene surfing Hallatschek and Nelson, 2007

7 Gene surfing Hallatschek and Nelson, 2007

8 Gene surfing Excoffier and Ray, 2008

9 Experiments on a Petri dish Haploid CFP and RFP-marked S. Cerevisiae Hallatschek and Nelson, 2007

10 The question How does range expansion interfere with selection? If a mutant arises with an offset x wrt an advancing Fisher wave, what is the probability u(x) that it eventually fixes? How does u(x) depend on the growth rate r m of the mutant and on the local carrying capacity of the environment?

11 The stepping-stone model Kimura and Weiss, 1964 W K deme v At each step: Replication in a deme: p rep K W 0 x Death: p death (1 r w )p rep Diffusion

12 The ancestor location in the neutral case Hallatschek and Nelson, 2007 Two competing effects: The fixation probability u(x) increases as x increases The population density n(x) decreases as x increases

13 Introducing the mutants W, M K 0 x A single mutant is added at location x wrt the center of the wave W : number of wildtypes, M number of mutants Replication in a deme: p rep K (W + M) Death: p death (1 r m )p rep (no advantage in a full deme)

14 Possible outcomes N.B.: The window is placed so that the total number of individuals is kept roughly constant Fixation: The window is filled by mutants Extinction: All the mutants die off in the window Failure: At least one mutant crosses the left boundary (but they disappear from the window)

15 A stochastic reaction-diffusion system w = W /K m = M/K r w,m 1 w t m t = r w w(1 m w) + 2 w x 2 w(1 m w) + 2 η w (x, t) 2 mw K K ηw,m (x, t) = r m m(1 m w) + 2 m x 2 m(1 m w) + 2 η m (x, t) + 2 mw K K ηw,m (x, t) Can be made adimensional (depending only on K e = K r w and α = r m /r w ) by setting X = r w x T = r w t

16 The Fisher wave WK x v = 2 Dr (Fisher, 1937)

17 The fixation probability u(x) L' u 0.06 L h x N.B.: Data for r w = 0.1, r m = 0.11

18 Fixation and failure ux vx u(x): probability of fixation v(x): probability of failure

19 Limit behavior of u(x) r m 0.1 r m 0.2 r m 0.5 u x For x, u(x) 0: neutral dynamics against an infinite population For x +, u(x) r m : fixation provided the mutant survives sampling fluctuations

20 Dependence on K u K 100 K K

21 Dependence of L on K and α Α 0.3 Α 0.5 Α 1 Α

22 The basic scenario Mutants are established when their population is large enough for its advantage to be felt In an empty deme, this requires m > m c (K e, α) In a full deme, the mutant s offspring undergoes neutral drift against an infinite wild-type population: fixation probability vanishes u(x) 0 for x < 0 For x 0 the mutant population can get established, but is rapidly surrounded by the wild-types: it will disappear on a much longer time scale: thus u(x) 0, but v(x) > 0 For x > L, the mutant population can get established before the wildtype population reaches it: thus u(x) > 0 for x +

23 The large-x limit For x +, it only matters if the mutant population avoids stochastic death at the beginning, and the wave of advancing wildtypes is irrelevant In a well-mixed population with growth rate r m, this probability equals r m (Moran model) This also holds true in our model with spatial structure and local logistic growth

24 The argument The only relevant dynamical variable is m tot = i m i Diffusion events do not change m tot Death events are (1 r m ) less likely than birth events Thus the survival probability P mtot satisfies P mtot = 1 r m 2 r m P mtot r m P mtot+1 with boundary conditions Thus and P 0 = 0 P = 1 P mtot = 1 (1 r m ) mtot P 1 = r m

25 The length L Can we estimate the behavior of L? Speed of the advancing wildtype wave: v 2 r w (diffusion constant D = 1) Time available for a mutant to grow: t 0 = x 0 /v Assume free exponential growth on average for the mutant population: N m e rmt Thus N m = exp(r m t)/r m Success requires N m (t 0 ) m c K / r m Thus survival may take place if x 0 > L v ln(m c K rw r m ) ln(k r m ) f (α, K e ) r m r m Actually this is just an upper bound (one can check that lim k L < )

26 A differential equation for the survival probability u(x)? The survival probability u(x) apparently satisfies 2 u x 2 v u w x + r m(1 w )u u 2 = 0 with the boundary conditions lim u(x) = 0 x lim x + u(x) = r m where w(x) is the average profile of the wildtype wave

27 A heuristic derivation Assumptions: The fate of the mutant population is settled when m is still very small (does not hold for nearly neutral, neutral, or disadvantageous mutations) We consider a local average growth rate r m (1 w(x) ) Define p(x, t ξ, τ) as the probability to find a mutant at x at time t, given that one was introduced at ξ at time τ Assume t > τ, but very close: then w(x, t) w(x, τ) = w init (x) Then u(x) is the probability that the mutant in x fixes If all events are independent, u(x) satisfies u(ξ) = + dx u(x) p(x, t ξ, τ)

28 Differentiating wrt t we obtain + Setting r m mw r m m w init : Thus p t dx u(x) p(x, t ξ, τ) = 0 t = r m (1 w init ) p + v p x + 2 p x 2 r m (1 w init ) u v u x + 2 u x 2 = 0 valid for u 1 The u 2 term imposes lim x + u(x) = r m

29 Comparison with the data r m 0.1 r m 0.2 r m 0.5 u x The lines correspond to the solution of the differential equation The differential equation correctly evaluates L

30 The substitution rate Substitution rate R vs. the beneficial mutation rate U b : R = U b dx u(x)w (x) = U b G For K 1, u(x)w (x) u(x) W (x) Since W (x) = Kw(x), we expect G K GK s s s

31 Results for G G 20 K K 316 K

32 Effects of finite density By exploiting the analogy with travelling waves (s = r m /r w 1) G 2πK 1 π/ s ln 2 K e+π 2 e ( ). π 2 / ln 2 K e + s Asymptotics is reached very slowly!

33 Summary Range expansion suffocates beneficial mutations, unless they arise enough far ahead of the expanding waves The substitution rate is strongly decreased for weakly beneficial mutations However, even a small amount of drift substantially increases the substitution rate A similar phenomenon favors individuals with larger D (but equal fitness) (Pigolotti and Benzi, 2014) The moral: Rather than the fittest or most deserving Evolution favors those who are in the right place at the right moment!!