Population Genetics: a tutorial

Transcription

1 : a tutorial Institute for Science and Technology Austria ThRaSh 2014

2 provides the basic mathematical foundation of evolutionary theory allows a better understanding of experiments allows the development of new concepts, and their empirical usefulness. allows us to ask what if questions. PG s Basic Problem: How do the processes of selection, inheritance, mutation, among others, affect the evolution of biological populations.

3 The problem of evolutionary change T 1 : Genotype-phenotype map T 2 : Selection, mating, migration... T 3 : Inference of the genotype distribution from the phenotype distribution T 4 : Genetic rules (Mendel) of inheritance

4 Fitness Fitness Fitness is the total contribution of an individual to population growth. Absolute Fitness is the expected number of surviving offspring produced by a parent genotype. Relative Fitness is just a scaling of absolute fitness by some number.

5 The mathematical models of differential and difference equations selection, mutation, migration Stochastic Models Markov Chains, stochastic differential equations, partial differential equations All evolutionary forces

6 The replicator equation Consider two types with distinct growth rates: na t+1 = W a na t n t+1 A = W A na t p t+1 = = na t+1 na t+1 + n t+1 A W a p t W a p t + W A (1 p t ) = W a W pt This is why fitness can be scaled by an arbitrary scalar.

7 The replicator equation p t = p t+1 p t = W a W A W pt (1 p t ) = W W pt (1 p t ) The replicator equation appears in many contexts.

8 The replicator equation is for distinct types. In general, with recombination, we would have to track all genotype frequencies. For a 2 gene genotype: x 00 = 1 w [x 00(w 00 w) rd] x 01 = 1 w [x 01(w 01 w) + rd] x 10 = 1 w [x 10(w 10 w) + rd] x 11 = 1 w [x 11(w 11 w) rd] where D = x 00 x 11 x 01 x 10 is a measure of the linkage disequilibrium, and r is the rate of recombination between the two loci.

9 Linkage disequilibrium If there is no selection: D t+1 = (1 r)d t recombination quickly eliminates linkage disequilibrium with selection: D = (w 01 + w 10 )D + x 00 x 11 E 2wD rd where E = w 00 + w 11 w 10 w 01 is a measure of epistasis.

10 Equilibrium At equilibrium: xˆ 00 x 11 ˆE ˆD = 2ŵ w 01 w 10 + r ŵ = w ij ± rḓ x ij Fitness is only maximized at an internal equilibrium if D = 0. However, if epistasis exists, this condition can be violated.

11 Linkage equilibrium If recombination rate is fast compared to selection, the system is always very close to Linkage Equilibrium (also known as product distribution, or Wright manifold) (Nakylaki 1993). x ij = p i p j where p i = k x ik. and again we can write the dynamics in terms of allele frequencies.

12 Wright s equation (Wright, 1937) In general, under linkage equilibrium: Several insights: dp i dt = p i(1 p i ) 2 ln W p i allele frequencies will evolve in the direction of increasing mean fitness all stable equilibria are local maxima of w. the change in allele frequencies is basicaly the slope of the fitness landscape, times the genetic variance (in this case two variants: p(1 p)).

13 Applications: Mutation-Selection Balance The problem of variation: how is variation maintained in natural populations? Question At what frequency are deleterious alleles kept in a population, under recurrent mutation? p t+1 = 1 a µ A 1 s 1 s (1 s)p + (1 p) pt + µ(1 p)

14 Solving for the equilibrium we find: ˆp = 1 µ ˆp = s(1 + µ) µ s Checking stability we find that ˆp = µ s s is stable for µ < 1 s. At this equilibrium, mean fitness is w 1 µ < w max = 1. We say that there is a mutational load.

15 Applications: Modifiers Typically, we consider cases where parameters such as: mutation rate recombination rate migration rate... are held constant. However, we can imagine that alleles at certain loci can change these parameters

16 The pronounced tendency of the mutant gene to be recessive, to the gene of wild type from which it arises, calls for explanation Fisher (1930) The Genetical Theory of Natural Selection Modifier theory was first introduced to understand dominance. In modifier theory, one checks which modifiers can invade the current genotype. In general this process does not maximize fitness. Even if some parameter value is optimal, it is not guaranteed that it will evolve.

17 Two types of modifiers: Selected modifiers Neutral modifiers Selected modifiers evolve in response to their own fitness effect and the associations they generate with other (fitness) loci. Neutral modifiers evolve strickly in response to associations they generate with other (fitness) loci.

18 Example: evolution of mutation rate A locus that modifies the mutation rate in our mutation-selection balance example: 1 a µ A 1 s Will only invade if it reduces the mutation rate (Reduction Principle) Reduction Principle Given a choice, evolution will favor modifiers that make transmision more reliable.

19 Stochastic Models

20 The neutral Wright-Fisher Model Typical assumptions in PG N (haploid) individuals non-overlapping generations Sampling with replacement (large number of gametes) We can describe a population by the number of copies of a given allele in the population, and its dynamics by a Markov process: n t+1 = P n t where the elements of the transition matrix P ij are given by: P ij = ( ) ( ) N j i ( 1 j ) N i i N N

21 We are typically interested in the frequency of alleles: p = i N. ( ) N P(i) = p i (1 p) N i i From this we see that which means that E[i] = Np V [i] = Np(1 p) E[p] = p V [p] = p(1 p) N

22 The Forward Equation Any continuous stochastic system can in principle be described by the so-called Chapman-Kolmogorov equation: ψ(t + t, x) = φ(x ɛ, ɛ)ψ(t, x ɛ)dɛ where ψ(t, x) represents the probability of finding the stochastic system at state x at time t (more formally, it is the probability density function of the stochastic system at time t), and φ(x, ɛ) represents the transition probability from state x to state x + ɛ.

23 We can always expand the integrand (φ.ψ) of this equation as a Taylor series around x (The diffusion approximation basically allows us to disregard higher order terms in this expansion): ψ(t + t, x) = ψ(t, x)φ(x, ɛ) ɛ ψ(t, x)φ(x, ɛ) x 2 + ɛ2 ψ(t, x)φ(x, ɛ) 2 x 2 + dɛ

24 We can now integrate these terms one by one. Notice that in the integrad, ψ does not depend on ɛ 1 2 ψ(t, x)φ(x, ɛ)dɛ = ψ(t, x) ɛ 2 2 φ(x, ɛ)dɛ = ψ(t, x) ɛ ψ(t, x)φ(x, ɛ)dɛ = ψ(t, x) ɛφ(x, ɛ)dɛ x x = ψ(t, x) φ x x 2 ψ(t, x)φ(x, ɛ)dɛ = 1 2 ψ(t, x) ɛ 2 φ(x, ɛ)dɛ 2 x 2 = 1 2 [ 2 x 2 ψ(t, x) Var(φ) + φ 2] 2 1 ψ(t, x)var(φ) 2 x 2

25 Putting all these terms together we get the forward diffusion equation (also called Fokker-Planck or Kolmogorov forward equation in physics): The Diffusion Equation ψ(t, x) t 2 = x [ψ(t, x)m(x)] + 1 [ψ(t, x)v (x)] 2 x 2 where M(x) and V (x) are the mean and variance of the transition probability φ at state x. This equation describes the forward evolution of the pdf, given some initial conditions.

26 steady state solution The typical use of this equation is to derive the steady state distribution for the process in question. this can be done by settting ψ t = 0 x [ψ(t, x)m(x)] [ψ(t, x)v (x)] 2 x 2 = 0 Solving this equation leads to: Steady State Solution ˆψ = C M(x V (x) exp 2 ) V (x ) dx This formally gives us the probability of finding the system in any state, in the long term.

27 The diffusion equation is a way to integrate all the processes in population genetics: selection mutation migration genetic drift (finite sampling effects) It allows explicit solutions only for simple cases, but it allows us to gain insight about which (compound) parameters are relevant.

28 The diffusion equation allows us to probe many questions: How small does a population needs to be for sampling effects to dominate over selection? How much migration between populations is necessary to make them evolve in concert? How much genetic variation is maintained by the interaction of all the processes of evolution? What is the probability that a mutation will go to fixation?

29 Example One gene with selection and mutation 1 a A 1 s M(p) = s p(1 p) + (1 p)µ νp w V (p) = p(1 p) N

30 Example ˆψ = e 2Nsp p Nν 1 (1 p) Nµ 1 = w 2N ψ 0 The steady state distribution is the neutral distribution (induced by mutation and drift) biased by selection

31 Ψ p Ψ p Ψ p N Ψ p N s = 0.01 µ = ν = 0.01

32 general equilibrium distribution In general, Wright derived an expression for arbitrary number of loci, under linkage equilibrium: Wright 1937 ˆψ = w 2N i p 2Nν 1 i (1 p i ) 2Nµ 1

33 the backwards equation A related equation is obtained by considering the time evolution of the pdf of the system conditional on being in a certain state in the past. In order to derive this equation, we consider the following equation: ψ(t + t, x x 0 ) = φ(x 0, ɛ)ψ(t, x x 0 + ɛ)dɛ The meaning of this equation is that the probability of finding the system at a particular state x in the next time-step, conditional on the fact that it was a some other state x 0 at time 0 is the probability that the system was at some intermediate state x 0 + ɛ, conditional on the fact that it was at x 0 before, times the probability that the system jumped from x 0 to x 0 + ɛ, summed over all possible intermediate states.

34 In order to derive the backwards equation we expand only ψ(t, x x 0 + ɛ) in Taylor series ψ(t, x x 0 + ɛ) = ψ(t, x x 0 ) + ɛ ψ + ɛ2 2 ψ x 0 2 x0 2 + Since now only φ(x, ɛ) depends on ɛ, only φ needs to be integrated, which leads to the equation: ψ(t + t, x x 0 + ɛ) = ψ(t, x x 0 ) + ψ x ψ 2 φ(x 0, ɛ)dɛ } {{ } 1 φ(x 0, ɛ)ɛdɛ }{{} M(x 0 ) φ(x 0, ɛ)ɛ 2 dɛ }{{} x0 2

35 This leads to the PDE: ψ(t, x x 0 ) t = M(x 0 ) ψ x V (x 0) 2 ψ x 2 0 The main use of this equation is to get an expression for the probability that the system will reach some later state, as a function of the initial state. Typically, one is interested in the probability that a boundary state is reached. In order to get such expression, we again calculate the steady state of this equation and, after application of the appropriate boundary conditions, one gets the probability of absortion.

36 In general, the probability of fixation of an allele that first arises at a single copy is given by: p fix (s) = 1 exp ( 2s) 1 exp ( 2Ns) if the allele is neutral(s = 0), p fix = 1 N (its initial frequency). If N is very large and s is small, then p fix = 2s (the probability of surviving stochastic loss).

37 probability of fixation p fix p fix Applying these techniques to EAs should be a matter of being able to calculate the expected advance (as in drift analysis) and the variance.

38 A framework for EA analysis on functions of unitation? Assume that a genotype is defined by a bitstring X = (X 1,..., X N ) X i {0, 1} X p = N = δ X Assume that only one bit is flipped at any time and that mutations are accepted with probability: α(w old, w new )

39 One now needs to calculate the expectation and variance of the increase in p. E( p) = (1 p)α( w)δ pα( α)δ Var( p) = (1 p)α 2 ( w)δ 2 + pα 2 ( w)δ 2 E[ p] 2 1 p is the probability that the mutation flips a 0 α( w) is the probability that this change is accepted δ is the increase if the mutation is accepted

40 if α( w) = 1 α( w) then these expressions simplfy: E( p) = δ (α( w) p) Var( p) = δ 2 p(1 p)

41 Example Let s try to apply this to a problem w = (p a) 2 w

42 In order to make progress we need to choose a functional form for α( w) 1 α( w) : α( w) = 1 + e β w/δ E p p Var p p

43 The critical point is being able to calculate M V We can try to linearize α( w): 0 w < 2δ β 1 α( w) = 2 + β 2δ 2δ 4δ w β < w < β 1 w > 2δ β

44 we can solve the integral M V in a piecewise way: 1 ( δ log(1 ) p) + C 1 1 2δ βa log p + 2δ 2 1 δ log(p) + C 3 ( ) 1 2δ + β (a 1) log(1 p) + C 2δ 2 2 p < p min p min < p < p ma p > p max where p min = a δ β and p max = a + δ β. The constants could be found by requiring continuity and that this function is normalized. This function could now be analized to extract the probability that this EA is found at some partcular state in the long term The backwards equation could be used to find the probability that a particular optimum is found. Because these yield analytical expressions, one could find the dependencies of these quantities with the several parameters

45 Conclusion PG offers a variety of ways of looking at evolution but always restricted to a specific reproduction/sampling scheme modifier methods seem appropriate to analyse self adapting algorithms The diffusion approximation approach seems promising to look at the evolution of EAs (at least in functions of unitation) Quantitative genetics is yet another way to look at evolution of EAs/GAs (Paixao and Barton, GECCO 2013)

46 THANK YOU