Population Genetics and Evolution III

Transcription

1 Population Genetics and Evolution III The Mechanisms of Evolution: Drift São Paulo / January 2019 SMRI (Italy) luca@peliti.org 1

2 Drift 2

3 The Population Genetics Triad Sewall Wright Ronald A. Fisher Motoo Kimura 3

4 Finite population The Wright-Fisher model Population size N, number n k of individuals of type k, k = 1,..., r, with fitness w k Nonoverlapping generations Given the composition vector x = (x i ), x i = n i /N, the numbers n k in the next generation are distributed according to where Prob(n 1,..., n N! r) = n 1! n r! ξn 1 1 ξn r r ξ k = x kw k j x jw j Thus n k is approximately distributed as a Gaussian with mean Nξ k and variance Nξ k (1 ξ k ) 4

5 Finite population The Wright-Fisher model 4

6 Finite population The Wright-Fisher model: one realization (neutral) xk t 4

7 Finite population The Wright-Fisher model: several realizations (neutral) fixation 0.6 x(t) extinction t 4

8 Finite population The Wright-Fisher model: one realization (selective: N = , w k {1.0, 1.1}, x k (0) = 0.1) xk t 4

9 Finite population The Wright-Fisher model: several realizations (selective: N = 500, s = 0.01, x(0) = 0.1) x(t) t Fixation in 5 cases out of 10 4

10 Drift it is often convenient to consider a natural population not so much as an aggregate of living individuals as an aggregate of gene ratios. Such a change of viewpoint is similar to that familiar in the theory of gases R. A. Fisher,

11 Drift We will start our discussion from the simplest situation where the gene frequency fluctuates from generation to generation because of the random sampling of gametes in a finite population. Since Wright s work, the term drift has become quite popular among biologists. However, in the mathematical theory of Brownian motion, the term drift originally connotes directional movement of the particle; therefore in our context the adjective random should be attached to it. M. Kimura, 1964 (abridged) 5

12 Drift Finite population implies different outcomes for different experiments in the same conditions (lack of self-averaging) Necessity to describe an ensemble of populations Use of the theory of Markov processes Simplification by means of diffusion equations 5

13 Random drift in the neutral case Population of N haploid individuals, 2 neutral alleles: A, a Frequency of the A allele: x = n A /N Wright-Fisher model: At each time step, each individual i of the new generation picks up a parent at random and copies it 6

14 Random drift in the neutral case The Wright-Fisher model t t+1 6

15 Random drift in the neutral case Probability that n A (t + 1) = n, given n A (t) = Nx(t): ( ) N p n (t + 1) = (x(t)) n (1 x(t)) N n n 1 Assume N 1, N x 1 1 N, then ) (x x(t))2 Prob (x(t + 1)=x) exp ( 2Nx(t)(1 x(t)) x(t) = x(t + 1) x(t): x(t) = 0 ( x(t)) 2 = x(t)(1 x(t)) N 6

16 The diffusion equation Fokker-Planck equation: p(x, t) = t x ( x x p(x, t)) ( x 2 x 2 p(x, t)) x In our case p t = 1 2 (x(1 x) p(x, t)) 2N x2 7

17 The solution in the neutral case Set p(x, t x 0, 0) = n c n(x 0 )χ n (x) e λnt/(2n) Eigenvalue equation: x(1 x)χ n(x) + (1 2x)χ n(x) + λ n χ n (x) = 0 Boundary conditions: x = 0, 1 are singular points; we require χ n (0, 1) finite n Initial condition: p(x, 0 x 0, 0) = n c n (x 0 )χ n (x) = δ(x x 0 ) Solution in terms of hypergeometric functions: χ n (x) = F (1 n, n + 2, 2, x) λ n = n(n + 1) 8

18 The solution in the neutral case p x,t x t = 0.05N 8

19 The solution in the neutral case p x,t x t = 0.1N 8

20 The solution in the neutral case p x,t x t = 0.2N 8

21 The solution in the neutral case p x,t x t = 0.5N 8

22 The solution in the neutral case p x,t x t = N 8

23 The solution in the neutral case p x,t x t = 1.5N 8

24 Initial condition x(0) = 0.1 p x,t x t = 0.05N 9

25 Initial condition x(0) = 0.1 p x,t x t = 0.1N 9

26 Initial condition x(0) = 0.1 p x,t x t = 0.2N 9

27 Initial condition x(0) = 0.1 p x,t x t = 0.5N 9

28 Initial condition x(0) = 0.1 p x,t x t = N 9

29 Initial condition x(0) = 0.1 p x,t x t = 1.5N 9

30 Results p(x, t) decays exponentially: p(x, t) 6x(0)(1 x(0))e t/n for t N Probability that A and a coexist at generation t: Ω(t) = 1 dx p(x, t) decays with the same rate (p(x, t) is flat) 0 However, p(x, t) becomes flat later when x(0) 1 2 What is the probability of fixation of allele A as a function of x(0)? 10

31 The backward equation p(x, t x 0, t 0 ): Conditional probability that x(t) = x given that x(t 0 ) = x 0 Consider the effect of a single-generation sampling near t 0 : x(t 0 + 1) = x 0 + x 0 Equation for p(x, t x 0, t 0 ): p p = x 0 t x0 + 1 x 2 0 x x 2 p 0 x 2 0 In our case p = x 0(1 x 0 ) 2 p t 0 2N x

32 The fixation probability P (t, x 0, t 0 ) = p(1, t x 0, t 0 ): probability of being fixed by time t Ultimate fixation probability: p fix (x 0 ) = lim t P (t, x 0, t 0 ) From the backward equation we obtain d 2 p fix dx 2 0 = 0 x [0, 1] Boundary conditions: p fix (x 0 =0) = 0 and p fix (x 0 =1) Solution: p fix (x 0 ) = x 0 12

33 Wright-Fisher model with selection Population of N haploid individuals, two alleles A and a Fitnesses: w A, w a Probability that an individual with allele A is chosen as a parent: ξ A = n Aw A N j=1 w j = n A w A n A w A + n a w a = Probability that n A (t + 1) = n: ( ) N p n (t + 1) = ξa n (1 ξ A ) N n n Average and variance: x w A xw A + (1 x)w a x A (t + 1) = ξ A (x A (t + 1) x A (t + 1) ) 2 = ξ A (1 ξ A ) /N 13

34 Selection and drift If the first human infant with a gene for levitation were struck by lightning in its pram, this would not prove the new genotype to have low fitness, but only that the particular child was unlucky. John Maynard Smith 14

35 Selection and drift Set w A = 1 + s, w a = 1, s 1 Then ξ A = xw A /(xw A + w a (1 x)) = (1 + s)x/(1 + sx) Then x x = x(t + 1) x = sx(1 x)/(1 + sx) sx(1 x) and x 2 (x(1 x)/n) Diffusion equation for p(x, t): p t = s 1 2 (x(1 x)p) + (x(1 x)p) x 2N x2 Solution in terms of spheroidal functions Asymptotically p(x, t) χ(x) e λt/n 14

36 Solution with selection The long-living eigenfunction: Ns = 0.1 Ns = Ns = χ(x) x 1 The leading eigenfunction χ(x) for several values of s 15

37 Solution with selection The decay rate: λ s Leading eigenvalue λ as a function of Ns; decay rate: λ/n 15

38 The fixation probability with selection The backward equation: Stationary solution: p = sx 0 (1 x 0 ) p + x 0(1 x 0 ) t 0 x 0 2N p fix x 0 = C 1 e 2Nsx0 p fix (x 0 ) = C 0 C 1 e 2Nsx0 In particular, for s 0, p fix x 0 = 1 e 2Nsx0 1 e 2Ns 2 p x

39 The fixation probability with selection p fix (x) Ns s = 0 Ns = 1 Ns = 3 Ns =

40 Fixation probability of a single mutant For a single mutant x 0 = 1 N Thus Limits: p fix = 1 e 2s 1 e 2Ns s > 0, Ns 1: p fix 1 e 2s (for s 1, p fix 2s) s < 0, Ns 1, p fix 0 Ns 1, p fix 1 N 17

41 Fixation probability of a single mutant p fix (x) N = 10 N = s 1 17

42 Frequency needed to obtain fixation How large must be x to be almost sure that a beneficial mutant fixes? Solve p fix (x ) = 1 γ For Ns 1 we have p fix (x) 1 e 2Nsx, thus x = log γ 2Ns or n = log γ 2s The fate of the mutant is determined in its initial phase, where it undergoes a branching process the size of N is irrelevant! 18

43 Substitution rate For a new mutant, x 0 = 1 N For a neutral mutant, s = 0, thus p fix = x 0 = 1 N If u is the mutation probability per genome and generation, the expected number of mutants per generations is un Of these, only a fraction 1 N substitution reaches fixation, i.e., produces a Therefore the rate ν of neutral substitutions in a population with mutation rate u is equal to u: substitution rate = mutation rate independently of the population size N 19

44 The Moran model Overlapping generations individual-based model: Initial population Select for reproduction Select for death Replace 20

45 The Moran model Selection: p kill (A) = 1 s, p kill (a) = 1 t = 1 N ; n A { 1, 0, +1} Probabilities: P 1 = n a N }{{} Prob repr(a) (1 s) n A }{{ N } Prob kill (A) = (1 s)x(1 x) n a P +1 = n A = x(1 x) N N P 0 = 1 (P +1 + P 1 ) 20

46 The Moran model Thus, for t = 1 N, s 1: n A = P +1 P 1 = sx(1 x) ( n A ) 2 = P +1 + P 1 = (2 s)x(1 x) 2 x(1 x) The diffusion equation for the Moran model: p t = x (sx(1 x)p) (x(1 x)p) }{{} N x2 = 1/2N for WF The devil (or God?) is in the details 20

47 Adaptation and drift Mustonen and Lässig, Finite population of size N, r alleles, Moran model. Effects of mutation and selection: dx j dt = k Φ Γ jk ; x k Φ = f x + α µ α log x α Γ jk (x) = { xj x k, if j k x j (1 x j ), if j = k Γ positive definite 21

48 Adaptation and drift Mustonen and Lässig, Random drift: x x + ξ ξ j x = 0; ξ j ξ k = 2 Γ jk(x) N Fokker-Planck equation for the pdf P (x): P = [ Φ (Γ jk P ) + 1 ] (Γ jk P ) t x j x k N x k jk = ( Γ jk Φ ) P + 1 P x j x k N x k jk 21

49 Adaptation and drift Mustonen and Lässig, Φ = Φ 1 N log det Γ; det Stationary solution: Γ = α x α P eq (x) e N Φ = (det Γ) 1 e NΦ = P 0 e N f x P 0 (x) α x 1+Nµα Thus, for a static fitness function f, [N f x ] eq av = dx P eq (x) log P eq (x) P 0 (x) = D KL (P eq P 0 ) }{{} Kullback-Leibler divergence D KL (p q) = k p k log p k q k (1) 21

50 camp-response protein binding loci in E. Coli Mustonen and Lässig, 2005 Factor binding sites are short DNA sequences which bind activating factors Small mutation rates: µn 1 Population becomes monomorphic (x = (x α ) δ αβ ) p β = Prob ( x = δ αβ ) e Nf β It is reasonable to assume that their fitness depends on their binding energy E One can expect a linear model for E(σ), σ = (σ 1,..., σ l ), σ i {A, T, G, C} E(σ) = l ϵ i (σ i ) i=1 p 0 (σ): background nucleotide frequency with ϵ i (σ) = ϵ 0 log q i(σ) p 0 (σ) 22

51 camp-response protein binding loci in E. Coli Mustonen and Lässig, 2005 Log histogram P (E) of binding energy E for CRP-binding loci in E. Coli. Compared with P (E) = (1 λ)p 0 (E) + λp 0 (E)e 2NF (E). The inferred form of 2N F (E) is also plotted. (W-F model) 22

52 Thank you! 22

53 References i G. H. Hardy, Mendelian proportions in a mixed population, Science (1908) W. Weinberg, Über den Nachweis der Vererbung beim Menschen, Jahreshefte des Vereins für vaterländische Naturkunde in Württemberg (1908) B. de Finetti, Considerazioni matematiche sull ereditarietà mendeliana, Metron (1926) John H. Gillespie, Population Genetics: A Concise Guide (2nd ed.) (Baltimore: Johns Hopkins Press, 2004) M. Eigen, Self-organization of matter and the evolution of biological macromolecules, Naturwissenschaften (1973) 23

54 References ii C. O. Wilke, Quasispecies theory in the context of population genetics, BMC Evolutionary Biology 5 44 (2005) J. J. Bull and C. O. Wilke, Theory of Lethal Mutagenesis for Viruses, Journal of Virology (2006) J. J. Bull and C. O. Wilke, Lethal Mutagenesis of Bacteria, Genetics (2008) S. Crotty, C. E. Cameron and R. Andino, RNA virus error catastrophe: Direct molecular test by using ribavirin Proc. Natl. Acad. Sci. USA (2001) M. Kimura, The neutral theory of molecular evolution (Cambridge: Cambridge U. P., 1983) R. A. Fisher, The Genetical Theory of Natural Selection (Oxford: Clarendon Press, 1930) 24

55 References iii R. A. Fisher, Population genetics, Proc. Roy. Soc. London, Ser. B (1953) S. Wright, Evolution in Mendelian populations, Genetics (1931) M. Kimura, Diffusion models in population genetics, J. Appl. Prob (1964) J. Maynard Smith, Evolutionary Genetics (Oxford: Oxford U. P., 1989) P. A. P. Moran, Random processes in genetics, Mathematical Proceedings of the Cambridge Philosophical Society (1958) T. Ohta, Population size and rate of evolution, J. Mol. Evol (1972) 25

56 References iv V. Mustonen and M. Lässig, From fitness landscapes to seascapes: Non-equilibrium dynamics of selection and adaptation, Trends Genet (2009) V. Mustonen and M. Lässig, Fitness flux and ubiquity of adaptive evolution, Proc. Natl. Acad. Sci. USA (2010) 26