Genome evolution

1. Genome evolution Lecture 2: population genetics I: models and drift

2. Reading • Course slides on the web • Hartl and Clark, Principle of population genetics • Rick Durret – Probability models for DNA sequence evolution • Gillespie – Pop. Genetics, A concise guide • Hein et al. - Gene Genealogies, Variation and Evolution: A Primer in Coalescent Theory • Wakeley: Coalescent Theory, an introduction • Graur and Li: Intro to molecular evolution • Classics: Kimura, Dobzhanski..

3. Studying Populations Models A set of individuals, genomes Ancestry relations or hierarchies Experiments Fields studies, diversity/genotyping Experimental evolution mtDNA human migration patterns Åland Islands, Glanville fritillary population

4. Human population Growth:

5. The Data: the hapmap project 1 million SNPs (single nucleotide polymorphisms) 4 populations: 30 trios (parents/child) from Nigeria (Yoruba - YRI) 30 trios (parents/child) from Utah (CEU) 45 Han chinease(Beijing - CHB) 44 Japanease (Tokyo - JPT) Haplotyping – each SNP/individual. No just determining heterozygosity/homozygosity – haplotyping completely resolve the genotypes (phasing)

6. The Data: the hapmap project Because of linkage, the partial SNP Map largely determine all other SNPs!! The idea is that a group of “tag SNPs” Can be used for representing all genetic Variation in the human population. This is extremely important in association studies that look for the genetic cause of disease.

7. Correlation on SNPs between populations

8. Modeling…

9. The Hardy-Weinberg Model • Diploid organisms • Two copies of each allele/gene/base • Homozygous / Heterozygous • Sexual Reproduction • Mating haplotypes • Large population, No migration • Fixed size, closed system • Non-overlapping generations • Synchronous process • Not as bad as it may look like • Random mating • New generation is being selected from the existing haplotypes with replacement • No mutations, no selection (will add these later)

10. The Hardy-Weinberg Model • Non-overlapping generations • Synchronous process • Not as bad as it may look like • Random mating • New generation is being selected from the existing haplotypes with replacement • No mutations, no selection (will add these later) Hardy-Weinberg equilibrium: Random mating AA aa AA aa Aa aA Aa aA Non overlapping generations With the model assumption, equilibrium is reached within one generation

11. Frequency estimates We will be dealing with estimation of allele frequencies. To remind you, when sampling n times from a population with allele of frequency p, we get an estimate that is distributed as a binomial variable. This can be further approximated using a normal distribution: When estimating the frequency out of the number of successes we therefore have an error that looks like:

12. Testing Hardy-Weinberg(HW) using chi-square statistics HW is over simplifying everything, but can be used as a baseline to test if interesting evolution is going on for some allele Classical example is the blood group genotypes M/N (Sanger 1975) (this genotype determines the expression of a polysaccharide on red blood cell surfaces – so they were quantifiable before the genomic era..): Observed Expected (HW) Chi-square significance can be computed from the chi-square distribution with df degrees of freedom. Here: df = #classes - #parameters – 1 = 3(MN/NN/MM) – 1 (p) – 1 = 1

13. Modeling population: the Wright-Fisher model Haploid model Generation t 1 2 3 4 2N ….. Generation t+1 ….. 1 2 3 4 2N Diploid model Nf Nm ….. ….. Nf Nm ….. …..

14. Wright-Fisher model for genetic drift We follow the frequency of an allele in the population, until fixation (f=2N) or loss (f=0) We can model the frequency as a Markov process on a variable X (the number of A alleles) with transition probabilities: Sampling jalleles from a population 2N population with i alleles. In larger population the frequency would change more slowly (the variance of the binomial variable is pq/2N – so sampling wouldn’t change that much) 0 1 Loss 2N 2N-1 Fixation

15. Coalescent and fixation

16. Drift and fixation probability Since 0 and 2N are absorbing states, given sufficient time, the wright-Fisher process will converge to either 0 or 2N. Define: Theorem (fixation in drift): In the Wright-Fisher model, the probability of fixation in the A’s allele state, given a population of 2N alleles out of which i are A, is: Proof: The mean of the binomial sample in the n’th step is np: Which means that the expected number of A’s is constant in time. Intuitively: More formally:

17. Drift Number of populations Figure 7.4 Number of bw75alleles Generation Experiments with drifting fly populations: 107 Drosophila melanogaster populations. Each consisted originally of 16 brown eys (bw) heterozygotes. At each generation, 8 males and 8 females were selected at random from the progenies of the previous generation. The bars shows the distribution of allele frequencies in the 107 populations

18. The geometric distribution: reminder Rolling a dice, and recording the time until first appearance of k (waiting time) Lack of memory: Moments: “Intersection”:

19. The exponential distribution: reminder The limit of the geometric distribution when the time step is going to 0: M=2 M=4 Probability: Density: Moments: t Dt P=at Memory less! “Intersection”:

20. Coalescence Coalescent at time -1? Coalescent at time -T? No coalescence for k samples? Distribution of time from k to k-1:

21. The continuous time coalescent When sampling K new individuals, the chances of peaking up the same parent twice is roughly: When looking at k individuals, we can trace their coalescent backwards and ask when did they had k-1,k-2, or one common ancestor. Theorem: The amount of time during which there are k lineages, tk has approximately an exponential distribution with mean 2N * (2/(k(k-1))) Proof: the probability of not merging k lineages in n generations is: Past We scale the time by N, so it is like an exponential : The expected value is This is correct for any k, so going backward from present time, we can estimate the time to coalescent at each step Present 1 2 3 4 5

22. The coalescent The expected time to the common ancestor ofnindividuals: When looking atnindividuals, we can trace their coalescent backwards and ask when did they had n-1,n-2, or one common ancestor. Theorem: The probability that the most recent common ancestor of a sample of size n is the same as that of the population converges to (n-1)/(n+1) as the population size increase. Past 4N is the magic number Present 1 2 3 4 5

23. Diffusion approximation and Kimura’s solution Fisher, and then Kimura approximated the drift process using a diffusion equation (heat equation): The density of population with frequency x..x+dx at time t The flux of probability at time t and frequency x The change in the density equals the differences between the fluxes J(x,t) and J(x+dx,t), taking dx to the limit we have: The if M(x) is the mean change in allele frequency when the frequency is x, and V(x) is the variance of that change, then the probability flux equals: Heat diffusion Fokker-Planck Kolmogorov Forward eq.

24. Diffusion approximation and Kimura’s solution Fisher, and then Kimura approximated the drift process using a diffusion equation (heat equation). We start with working on the time step dy and frequency step dx The probability that the population have allele frequency x time t the probability that the frequency increased from x by dx, due to mutation/selection The probability of dx increase or decrease due to drift We limit changes from t to t+dt and x+-dx. The population can be on x at t+dt if: It was at x and stayed there: It was at x-dx and moved to x: It was at x+dx and moved to x:

25. Diffusion approximation and Kimura’s solution Fisher, and then Kimura approximated the drift process using a diffusion equation (heat equation). We start with working on the time step dy and frequency step dx The probability that the population have allele frequency x time t the probability that the frequency increased from x by dx, due to mutation/selection The probability of dx increase or decrease due to drift For drift the variance is binomial: And we assume no selection: Still not easy to solve analytically…

26. Changes in allele-frequencies, Fisher-Wright model After about 4N generations, just 10% of the cases are not fixed and the distribution becomes flat.

27. Absorption time and Time to fixation According to Kimura’s solution, the mean time for allele fixation, assuming initial probability p and assuming it was not lost is: The mean time for allele loss is (the fixation time of the complement event):

28. Effective population size 4N generations looks light a huge number (in a population of billions!) But in fact, the wright-Fisher model (like the hardy-weinberg model) is based on many non-realistic assumption, including random mating – any two individuals can mate The effective population size is defined as the size of an idealized population for which the predicted dynamics of changes in allele frequency are similar to the observed ones For each measurable statistics of population dynamics, a different effective population size can be computed For example, the expected variance in allele frequency is expressed as: But we can use the same formula to define the effective population size given the variance:

29. Effective population size: changing populations If the population is changing over time, the dynamics will be affect by the harmonic mean of the sizes: So the effective population size is dominated by the size of the smallest bottleneck Bottlenecks can occur during migration, environmental stress, isolation Such effects greatly decrease heterozygosity (founder effect – for example Tay-Sachs in “ashkenazim”) Bottlenecks can accelerate fixation of neutral or even deleterious mutations as we shall see later. Human effective population size in the recent 2My is estimated around 10,000 (due to bottlenecks). (so when was our T1?)

30. Effective population size: unequal sex ratio, and sex chromosomes If there are more females than males, or there are fewer males participating in reproduction then the effective population size will be smaller: Any combination of alleles from a male and a female So if there are 10 times more females in the population, the effective population size is 4*x*10x/(11x)=4x, much less than the size of the population (11x). Another example is the X chromosome, which is contained in only one copy for males.

31. Population genetics Drift: The process by which allele frequencies are changing through generations Mutation: The process by which new alleles are being introduced Recombination: the process by which multi-allelic genomes are mixed Selection: the effect of fitness on the dynamics of allele drift Epistasis: the drift effects of fitness dependencies among different alleles “Organismal” effects: Ecology, Geography, Behavior