Intensive Course on Population Genetics Ville Mustonen
Population genetics: The study of the distribution of inherited variation among a group of organisms of the same species [Oxford Dictionary of Biology] http://www.exploratorium.edu/exhibits/mutant_flies/normal.gif http://www.emacswiki.org/emacs/emacsicons
Cross- and intra-species comparisons at the molecular level m ingroup samples outgroup AACTGTCCACGTTCTTCCGATGCAGCCTGA AACTGTCCACGTTCTTCCGATGCAGCCTGA AACTGTCCACGTTCTTCCGATGCAGCCTGA AACTGTCCACGTTCTTCCGATGCAGCCTGA AACTGTCCACGTTCTTCCGATGCAGCCTGA AACTGTCCAGGTTCTTCCGATGCAGCCTGA AACTGTCCAGGTTCTTCCGATGCAGCCTGA AACTGTCCAGGTTCTTCCGATGCAGCCTGA AACTGTCCAGGTTCTTCCGACGCAGCCTGA frequency counts 000000000500000000008000000000 polymorphic locus substitution event
Let s study fundamental evolutionary forces: 1. genetic drift (noise in reproduction) 2. selection (differential reproductive success of individuals) 3. mutation (process providing variation) 4....
Genetic drift: noise in reproduction time in generations individuals
0.4 0.3 0.2 0.1 40 0.0 5 30 10 20 15 10 20
Wright - Fisher process p b = N b /N = x p a = N a /N =1 x P (m, N, t + 1) = ( ) N x m m t (1 x t ) N m m = Nx t m 2 m 2 = Nx t (1 x t ) x t+1 = x t x 2 t+1 x t+1 2 = x t(1 x t ) N
Wright - Fisher process x t 1.0 0.8 0.6 τ d N N=10 0.4 0.2 0.0 0 2 4 6 8 10 1.0 N=100 t 0.8 x t 0.6 0.4 0.2 0.0 0 2 4 6 8 10 t
Selection: differential reproductive success of individuals N a = F a N a 7 6 5 N b = F b N b 4 3 2 2 4 6 8 10 t ẋ = d dt N b (t) N a (t)+n b (t) N b N b = N a + N b N a + N b = (F b F a )x(1 x) N b + N a N a + N b
Selection ẋ = F ba x(1 x) x(t) = x 0 exp(f ba t) 1+x 0 (exp(f ba t) 1) 1.0 0.8 x t 0.6 0.4 τ s 1/F ba 0.2 0.0 0 2 4 6 8 10 12 14 t
Mutation: source of variation time in generations individuals
Mutation: source of variation N a = µn b µn a N b = µn a µn b µ µ ẋ = d dt N b (t) N a (t)+n b (t) N b = N a + N b = µ(1 2x) N b N a + N b N b + N a N a + N b
Mutation ẋ = µ(1 2x) x(t) = 1 2 (1 + exp( 2µt)(2x 0 1)) 1.0 τ 0.8 m 1/µ x 0.6 t 0.4 0.2 0.0 0 2 4 6 8 10 12 14 t
Drift, Selection and Mutation 1.0 τ d N N=10 0.8 x t 0.6 0.4 0.2 0.0 0 2 4 6 8 10 t
Drift, Selection and Mutation 1.0 0.8 x t 0.6 0.4 0.2 τs 1/Fba 0.0 0 2 4 6 8 10 12 14 t
Drift, Selection and Mutation 1.0 0.8 x t 0.6 τ m 1/µ 0.4 0.2 0.0 0 2 4 6 8 10 12 14 t
Drift, Selection and Mutation Langevin equation: ẋ(t) =F ba x(t)[1 x(t)] + µ[1 2x(t)] + χ x (t) χ x (t) =0 χ x (t)χ x (t ) =(x(1 x)/n ) δ(t t )
One locus two alleles model Wright-Fisher process with mutation and selection Ratio of time scales: N 1/µ = µn 1 lighter shading indicates the fitter state population fraction of allele b µt
Diffusion equation p(x, t) probability density of populations g(x, ɛ, dt) probability of change x x + ɛ, dt p(x, t + dt) = p(x ɛ, t)g(x ɛ, ɛ, dt)dɛ = = p(x, t) p(x, t)g(x, ɛ, dt) ɛ x pg + ɛ2 2 g(x, ɛ, dt)dɛ x p 2 pg +... dɛ x2 ɛgdɛ + 1 2 2 x 2 p ɛ 2 gdɛ [SH Rice, Evolutionary Theory (2004)]
= p(x, t) g(x, ɛ, dt)dɛ x p ɛgdɛ + 1 2 2 x 2 p ɛ 2 gdɛ = p(x, t) p(x, t)m(x) x dt + 1 2 2 p(x, t)v (x) x 2 dt g(x, ɛ, dt)dɛ = 1 ɛg(x, ɛ, dt)dɛ = ɛ rate of directional change of allele frequency at point x = M(x)dt ɛ 2 g(x, ɛ, dt)dɛ = ɛ 2 + var(ɛ) = V (x)dt variance of allele frequency change due to nondirectional effects
p(x, t + dt) p(x, t) dt = p(x, t)m(x) x + 1 2 2 p(x, t)v (x) x 2 p(x, t) t = p(x, t)m(x) x + 1 2 2 p(x, t)v (x) x 2 Now we just need to plug in our earlier results to get Kimura s diffusion equation for one locus two alleles case: p(x, t) t = 1 2 2 x(1 x) x 2 p(x, t) N x [F bax(1 x)+µ(1 2x)]p(x, t)
One locus two alleles model: looking at the averages 1 Equilibrium distributions G t (x x0) 0.1 0.01 0.001 0.0 0.2 0.4 0.6 0.8 1.0 x
One locus two alleles model: looking at the averages 1 after infinite time 0.1 Q(x) 0.01 0.001 0.0 0.2 0.4 0.6 0.8 1.0 x
Substitution rate from Kimura s diffusion equation u/µ 4 u(2nf ba )= µ2nf ba 1 exp( 2NF ba ) 3 2 1 4 2 0 2 4 2NF ba
Substitution dynamics N 1/µ = µn 1 p a = u ba p b u ab p a p b = u ab p a u ba p b
Substitution dynamics p a = p b = u ba u ba + u ab u ab u ba + u ab p a = p b exp( 2NF ba ) N deleterious = N beneficial exp(2nf ba )
Conclusions Fundamental evolutionary forces: 1. genetic drift, diffusive force (!), effect strongest in small populations, changes allele frequencies at time-scale 2N 2. selection, directed force, increases the fitter phenotypes, timescale 1/F 3. mutation, directed force, restarts the process from monomorphic states, time-scale 1/µ More complex scenarios, recombination, linkage, diploid genomes, demographics, frequency/time-dependent selection... Further reading, Gillespie: Population Genetics, Hartl and Clark: Principles of Population Genetics