Selection and Population Genetics

Transcription

1 Selection and Population Genetics Evolution by natural selection can occur when three conditions are satisfied: Variation within populations - individuals have different traits (phenotypes). height and weight are normally distributed in human populations coat color varies from light to dark in deer mice (P. maniculatus) Selection - traits influence fecundity and survivorship (fitness). larger body size may be beneficial in cold environments differences between coat color and ground color may affect deer mouse vulnerability to avian predators Heritability - offspring are similar to their parents. human height is affected by hundreds of genetic variants coat color differences are affected by variation in the expression of the Agouti gene Jay Taylor (ASU) Selection 23 Feb / 33

2 Example: Beak size in the Medium Ground Finch (Geospiza fortis) Restricted to the Galapagos Islands. Forages mainly on seeds. Large seeds are handled more efficiently by birds with larger bills. Large seeds predominate following drought years (e.g., 1977). Jay Taylor (ASU) Selection 23 Feb / 33

3 Population genetics focuses on understanding evolution at the molecular level: how does natural selection affect the dynamics of gene frequencies? There are two fundamental issues. 1 To what extent do population genetical processes influence adaptation? How strongly does demography (population size and structure) affect adaptation? Does adaptation rely mainly on standing variation or on new mutations? Have genetic systems evolved to facilitate adaptation? 2 How much impact does selection have on genetic variation? What proportion of a genome is directly under selection? What proportion is affected by selection at linked sites? How robust are methods that use genetic variation to infer demography to deviations from neutrality? Jay Taylor (ASU) Selection 23 Feb / 33

4 Selection in an Infinite Haploid Population We begin by formulating a simple model of selection in a haploid population containing two alleles, A 1 and A 2, where 1 Individuals with genotype A i survive to adulthood with probability v i and then give birth to r i offspring. 2 Genotypes are transmitted without mutation. 3 Genetic drift can be ignored. 4 We will let n i denote the number of A i alleles in the current generation and p denote the frequency of A 1. Notice that differences in v i will give rise to viability selection, while differences in r i will result in fecundity selection. Their product will be denoted w i = v i r i and can be considered the fitness of allele A i. Jay Taylor (ASU) Selection 23 Feb / 33

5 Because we are ignoring genetic drift, the number of A i individuals alive at the beginning of the next generation is n i = n i v i r i = n i w i, This shows that the frequency of A 1 in the next generation is p = n 1 n 1 + n 2 = n 1w 1 n 1w 1 + n 2w 2 = (n 1/n)w 1 (n 1/n)w 1 + (n 2/n)w 2 = = p w1 w, pw 1 pw 1 + (1 p)w 2 where w = pw 1 + (1 p)w 2 is the mean fitness of the population. Jay Taylor (ASU) Selection 23 Feb / 33

6 This is one of the few models with selection that can be solved explicitly. Letting p(t) denote the frequency of A 1 in generation t and setting p(0) = p, one can show that Selection in an infinite haploid model p(t) = { p p ( w2 w 1 ) t } 1 In particular, this shows that the fitter of the two alleles will spread towards fixation, e.g., if w 1 > w 2, then p(t) will approach 1 as time progresses. On the other hand, this model makes several predictions that are false. The beneficial allele is never actually fixed in the population. Any beneficial allele, no matter how rare initially, will spread. To remedy these shortcomings, we need to formulate a model that incorporates both selection and genetic drift. Jay Taylor (ASU) Selection 23 Feb / 33

7 Selection in Finite Populations We can incorporate selection into the haploid Wright-Fisher model by relaxing the assumption that parents are chosen uniformly at random. 1 The alleles A 1 and A 2 have relative fitnesses 1 + s and 1, respectively, where the selection coefficient s = s(p) may depend on the frequency of A 1. 2 The parents of the N individuals alive in generation t + 1 are chosen at random and with replacement, but each A 1-type individual is (1 + s)-times more likely to be chosen than an A 2-type individual. 3 Mutation occurs at birth, but after selection, at rates v (A 1 A 2) and u (A 2 A 1). Jay Taylor (ASU) Selection 23 Feb / 33

8 In this model, the expected frequency of A 1 is changed by both mutation and selection: E[ p t] = u(1 p) vp + (1 u v) s(p) p(1 p) 1 + s(p) p Neutrality N =1000, µ = Directional Selection s = Balancing Selection s =0.01*(0.5 p) p 0.5 p 0.5 p Generation Generation Generation Jay Taylor (ASU) Selection 23 Feb / 33

9 Fixation Probabilities of Selected Alleles If we neglect mutation, then eventually one of the two alleles will be fixed in the population. In this case, we would like to know how the fixation probability of an allele depends on its fitness. Fixation probability of a selected allele in a haploid population If s(p) = s is constant, the probability that A 1 is fixed given that its initial frequency is p is approximately P(A 1 is fixed) = 1 e 2Nsp 1 e 2Ns. Remark: This result is obtained with the help of the diffusion approximation and is exact in that limit. For models other than the Wright-Fisher model, we need to replace N by N e. Jay Taylor (ASU) Selection 23 Feb / 33

10 The most important case is when a single copy of a new allele is introduced into a population, either by mutation or immigration. In this case, the initial frequency is p = 1/N and we can use the preceding result to show that Fixation probability of a new allele P fix = 1 e 2s 1 e 2Ns 2s if 1/N s 1 1/N if 1/N s 1/N 2 s e 2N s if 1 s 1/N In particular, Novel beneficial mutations are likely to be lost from a population. Selection is dominated by genetic drift when s < 1. N Deleterious mutations can be fixed, but only if N s is not too large. Jay Taylor (ASU) Selection 23 Feb / 33

11 Selection and genetic drift As a general rule, genetic drift reduces the efficiency of natural selection, especially in small populations. Not only are beneficial mutations likely to be lost, but deleterious mutations can become fixed in small populations. Fixation Probabilities of New Mutants s = 0.01 s = prob E-05 1E-06 s = s = 0 1E-07 1E-08 s = E-09 1E N Jay Taylor (ASU) Selection 23 Feb / 33

12 Substitution Rates The substitution rate is the rate at which new mutations are fixed in a population. Substitution rates depend on population size, mutation and selection. Divergence between populations or species occurs when different mutations are fixed in these populations. Thus, substitution rates can sometimes be estimated from divergence. Even when selection cannot be observed directly, it can sometimes be inferred from the effect that it has on divergence. Substitution rates are difficult to calculate exactly. However, we can find a good approximation if we assume that the mutation rate is low enough that each new mutation is likely to be lost or fixed before another mutation enters the population. Jay Taylor (ASU) Selection 23 Feb / 33

13 Under this assumption, we can approximate the substitution rate (per generation) by the expression ( ) 1 ρ Nµ u, N where Nµ is the expected number of new mutations per generation, while u(1/n) is the probability that any one of these is fixed in the population. This is accurate when Nµ 1. Neutral substitutions: For neutral mutations, we have ρ = Nµ 1 N = µ, which shows that the neutral substitution rate is equal to the mutation rate and is independent of population size. Jay Taylor (ASU) Selection 23 Feb / 33

14 Beneficial substitutions: If s 1/N, then ρ 2Nµs, and so the beneficial substitution rate is greater than the mutation rate and increases with population size. Deleterious substitutions: If s 1/N, then ρ 2Nµ s e 2N s, and so the deleterious substitution rate is less than the mutation rate and decreases with population size. Moral: The substitution rate at a locus under selection is usually different from the mutation rate and does depend on population size. Jay Taylor (ASU) Selection 23 Feb / 33

15 Protein-coding sequences appear to be under stronger selective constraints in larger populations. Jay Taylor (ASU) Selection 23 Feb / 33

16 Selection-Mutation-Drift Balance When selection and mutation are both incorporated into the model, neither allele will be permanently fixed and so we instead investigate the stationary distribution of the allele frequencies. Using diffusion theory, we can show that Stationary distribution with mutation and selection Provided that N is sufficiently large, the stationary distribution can be approximated by the following density π(p) = 1 { p } C p2nv 1 (1 p) 2Nu 1 exp 2N s(q)dq, 0 p 1. 0 Depending on the sign of s(q), the exponential term will either increase or decrease very rapidly as N increases: this too reflects the competing influences of drift and selection. Jay Taylor (ASU) Selection 23 Feb / 33

17 Stationary distribution with mutation and purifying selection π(p) = 1 C p2nv 1 (1 p) 2Nu 1 e 2Nsp, 0 p 1. Purifying selection has two consequences: It shifts the stationary distribution in the direction of the favored allele. It tends to reduce the amount of variation present at the selected locus Ns = Ns = p p Jay Taylor (ASU) Selection 23 Feb / 33

18 Purifying selection on human protein-coding genes Variation is greatly reduced in exons relative to introns in the human genome. Variation is slightly reduced in the 5 and 3 UTR s. Human-chimp divergence is proportional to human polymorphism. This suggests that most deleterious variants in coding regions are strongly selected against. Source: Durbin et al. (2010): 1000 Genomes Project Jay Taylor (ASU) Selection 23 Feb / 33

19 Selection in Finite Populations: Some Implications Adaptations that are under weak selection are less likely to evolve in populations with small N e. This could explain why codon bias is often more pronounced in bacteria and single-celled eukaryotes than in vertebrates (Bulmer 1991). Muller s ratchet: The repeated fixation of deleterious alleles on non-recombining chromosomes can lead to their eventual degeneration (Muller 1964). Mutational meltdown: The fixation of deleterious alleles in very small populations may lead to futher population size declines that eventually result in extinction. Lynch & Conery (2003) propose that many features of eukaryotic genomes (e.g., expansion of repetitive DNA, introns) have evolved through the accumulation of very weakly deleterious mutations in small populations. Jay Taylor (ASU) Selection 23 Feb / 33

20 Purifying selection can lead to overestimates of recent divergence times Estimates of divergence times are often affected both by fixed differences and polymorphism. Because deleterious mutations are more likely to contribute to polymorphism than to substitutions, this can lead to overestimates of divergence times. This bias will be greatest for recently diverged taxa Ho et al. Rate of Change (changes/site/ma) (a) (b) (c) Ho et al. (2005) split, 3 lack of chimpa chimpa of mol 2002; For the from N calibrat tal sequ bers AY Ra implem 2002; D assump through rate in bution This m (2002) to be s vious r Kishino Sander user-sp multipl sets), a these c size an estimat highest 1,000,0 Jay Taylor (ASU) Selection 23 Feb / 33

21 Selection on Diploid Loci Selection at diploid loci is complicated by two additional factors. To predict the effects of selection at a diploid locus, we need to know the fitness of each genotype. For example, with two alelles, A 1 and A 2, we need genotype relative fitness A 1A 1 w 11 = 1 + s 11 A 1A 2 w 12 = 1 + s 12 A 2A 2 w 22 = 1 + s 22 In sexually reproducing taxa, the genotype frequencies are also affected by meiosis and mating. In particular, because of segregation, there is no guarantee that the fittest allele will spread to fixation even in an infinite population. Jay Taylor (ASU) Selection 23 Feb / 33

22 Marginal Fitness In many cases, the effect of selection at a diploid locus can be predicted by determing the marginal fitnesses of the alleles, i.e., the average fitness of an allele weighted by the frequency with which it occurs in each diploid genotype before selection acts: w 1 = P(A 1 A 1)w 11 + P(A 2 A 1)w 12 w 2 = P(A 1 A 2)w 12 + P(A 2 A 2)w 22 Here P(A j A i ) is the conditional probability that an A i allele is contained in an A i A j genotype: P(A j A i ) = freq(a ij) freq(a i ) Jay Taylor (ASU) Selection 23 Feb / 33

23 Under random mating, we have P(A j A i ) = p j and so the marginal fitnesses are: w 1 = pw 11 + (1 p)w 12 w 2 = pw 12 + (1 p)w 22 In this case, because the marginal fitnesses are functions of p, we have p = p w1 w2 p = p(1 p) w where w = pw 1 + (1 p)w 2 is the population mean fitness and w 1 w 2 = p(w 11 w 12) + (1 p)(w 12 w 22) = w 12 w 22 + p(w 11 2w 12 + w 22). Notice that this last expression depends on p except when w 11 2w 12 + w 22 = 0. In other words, selection is frequency-dependent unless the fitness of the heterozygote is equal to the average fitness of the two homozygous genotypes (additive selection). Jay Taylor (ASU) Selection 23 Feb / 33

24 Selection at a diploid locus segregating two alleles can have very different consequences depending on the relative fitnesses of the genotypes. fitnesses consequence w 11 > w 12 > w 22 directional selection in favor of A 1 w 11 < w 12 < w 22 directional selection in favor of A 2 w 12 > max{w 11, w 22} overdominance: selection maintains both alleles w 12 < min{w 11, w 22} underdominance: selection can favor either homozygote Jay Taylor (ASU) Selection 23 Feb / 33

25 Dominance and Directional Selection Suppose that A 1 is deleterious compared to A 2 and that the selection coefficients have the form where s > 0 and h [0, 1]. s 11 = s s 12 = hs s 22 = 0, The constant h is called the dominance coefficient because it quantifies the contribution of the A 1 allele to the fitness of the heterozygote. A 1 is said to be dominant if h (1/2, 1] recessive if h [0, 1/2) additive if h = 1/2. Jay Taylor (ASU) Selection 23 Feb / 33

26 In this case, the stationary distribution of the diffusion process has density π(p) = 1 C p4ne v 1 (1 p) 4Ne u 1 e 2Ne sp+(1 2h)2Ne sp(1 p). Because the exponent is a decreasing function of h, recessive deleterious alleles tend to be more common than dominant deleterious alleles Equilibrium Frequency of Deleterious Alleles p Ns = h (dominance coefficient) Jay Taylor (ASU) Selection 23 Feb / 33

27 Selection and Genealogies The simplicity of the neutral coalescent stems from the fact that the genealogy of a random sample depends only on demographic events involving lineages ancestral to the sample. With selection, these statements are no longer true since the reproductive contribution of the ancestral lineages depends also on lineages that are not ancestral to the sample. In this case, we must keep track of additional information that allows us to account for the effects of selection on the genealogy. This can be done in two ways: with the ancestral selection graph (Krone & Neuhauser 1997) or with a genetically structured coalescent (Hudson et al. 1988). Jay Taylor (ASU) Selection 23 Feb / 33

28 The Ancestral Selection Graph The main insight underlying the ancestral selection graph (ASG) is that selection leads to additional deaths that can be modeled backwards in time by branching events. If there are two alleles, A 1 and A 2, the graph can be simulated as follows: Coalescent events occur at rate ( n 2) when there are n lineages. Branching events occur at rate nσ, where σ = Ns > 0 is the selection coefficient of A 2. One is the incoming branch and the other is the continuing branch. This process is simulated until there is only one lineage (the ultimate ancestor, UA). The type of the UA is determined and then mutation is simulated along the graph. The genealogy is extracted from the graph by starting at the UA and working forward. When branch events are encountered, the incoming branch replaces the continuing branch whenever its type is the fitter allele. Jay Taylor (ASU) Selection 23 Feb / 33

29 Two simulations of the ASG for a sample of 3 chromosomes (Krone & Neuhauser 1997, Fig. 5). Ancestral Processes with Selection 221 FIG. 5. The ancestral selection graph without mutation events when the ultimate ancestor is (a) of type 1 and (b) of type 2. (Thick lines represent the true genealogy.) the case of two genotypes, there are exactly two (not necessarily distinct) trees embedded in the ancestral selection graph which describe the genealogy of the constructed sample. Before we proceed with analyzing T MRCA, we would like to mention how the above procedure needs to be call the particle a virtual particle. The rules are now as follows: If a real particle reaches a branching point, it splits into a real particle and into a virtual particle. If a virtual particle reaches a branching point, it splits into two virtual particles. If two particles reach a coalescing point, the resulting particle is real if and only if at least modified in the non-stationary case. In this case, one of the two particles is real, otherwise the resulting instead of running the mutation selection process along particle is virtual. We are now ready to state our next thejay branches, Taylor (ASU) simply put mutation events along the Selection result. 23 Feb / 33 Remark: When σ is large, the ASG contains many branching events and is expensive to simulate.

30 Coalescents Structured by Genetic Background Key ideas: Think of the population as being subdivided into different genetic backgrounds defined by the allele carried at the selected locus. Chromosomes that carry the same allele are neutrally equivalent. Consequently, coalescence within genetic backgrounds can be described by a modification of Kingman s coalescent. To be concrete, suppose that we are interested in the genealogy at a neutral marker locus which is linked to a locus segregating two alleles, A 1 and A 2 under selection. Our goal is to describe the ancestral process, G t = (n 1(t), n 2(t), p(t)), where n i (t) is the number of ancestral lineages in the A i background and p(t) is the ancestral frequency of A 1 at time t in the past. Jay Taylor (ASU) Selection 23 Feb / 33

31 Changes to the ancestral process can occur through the following events: past% Common%ancestor% coal% Two A 1 lineages can coalesce. 2 % 2 % Two A 2 lineages can coalesce. 0.3 Each lineage can migrate between backgrounds, through: mutation at the selected locus; recombination between the selected and marker loci. 0.2 coal% mut% 2 % coal% The allele frequencies at the selected locus change as we go backwards in time. 0.1 rec% 1 % 1 % 1 % 2 % present% p Jay Taylor (ASU) Selection 23 Feb / 33

32 When time is measured in units of 2N e generations, these events will occur at the following rates: Transition two A 1 lineages coalesce two A 2 lineages coalesce a lineage mutates from A 1 to A 2 a lineage mutates from A 2 to A 1 a lineage recombines from A 1 to A 2 a lineage recombines from A 2 to A 1 Rate ( n1 ) 1 2 p ( n2 ) 1 2 q n 1µ 1(q/p) n 2µ 2(p/q) n 1rq n 2rp Furthermore, the ancestral allele frequencies at the selected locus will be changed by genetic drift, mutation and selection. Jay Taylor (ASU) Selection 23 Feb / 33

33 The structured coalescent can be extended to handle selective sweeps, frequency-dependent selection and other complicated scenarios. Dealing with selection at multiple loci, however, is still challenging. Unless the deleterious mutation rate is very large, the effect of purifying selection at a single locus on the genealogy at a linked locus is fairly modest. Purifying selection does tend to shift deleterious mutations towards the top of the tree, i.e., they tend to be more recent than neutral mutations. nd A. M. Etheridge Selection in Fluctua Figure 15. The effect of purifying selection on mean coalescence time, plotted against recombination rate, R; U 0.5, p 0.5. The vertical axis shows decreases from the neutral value, 1 E[ ], on a logarithmic scale. The top thick curve shows the deterministic limit for S 8, in which allele fre- Figure 16. The effect of purifying selection on mean coalescence time, plotted against selection, S, for U 0.25, 0.5, 1; p 0.5. There is complete linkage (R 0). Jay Taylor (ASU) Selection 23 Feb / 33