Mathematical models in population genetics II

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1 Mathematical models in population genetics II Anand Bhaskar Evolutionary Biology and Theory of Computing Bootcamp January 1, 014

2 Quick recap Large discrete-time randomly mating Wright-Fisher population continuous-time coalescent process for the genealogy of a sample of size n drawn at present Continuous-time Markov chain with transitions from k ancestral lineages to k 1 at rate ( ) k Mutations can be superimposed on the genealogical tree to generate allelic types in the sample 1 / 34

3 Quick recap Large discrete-time randomly mating Wright-Fisher population continuous-time coalescent process for the genealogy of a sample of size n drawn at present Continuous-time Markov chain with transitions from k ancestral lineages to k 1 at rate ( ) k Mutations can be superimposed on the genealogical tree to generate allelic types in the sample But... What if the population size is changing or has a more complicated structure? How to extend these models to more than one genomic position/locus? 1 / 34

4 What might you do with all this theory? Generative model for sequence data that captures the most important biological mechanisms Infer biological parameters of the population: genome-wide recombination maps Infer demographic structure of the population Regions of the genome under selective pressure mutation rates, / 34

5 Demographic structure 3 / 34

6 Demography example 15Kya 45Kya 35-40Kya 50-60Kya 45Kya Kya Founder effect Source of founder effect Migration path (Henn et al., 01) Fig. 1. Ancient dispersal patterns of modern humans during the past 100,000 y. This map highlights demic events that began with a source population in southern Africa 60 to 100 kya and conclude with the settlement of South America approximately 1 to 14 kya. Wide arrows indicate major founder events during the demographic expansion into different continental regions. Colored arcs indicate the putative source for each of these founder events. Thin arrows indicate potential migration paths. Many additional migrations occurred during the Holocene (11). Expansion, there was a continuous decrease of genetic diversity with geographic distance from the place of origin in Africa (this takes account of the likely path of migration over land). The linear correlation between loss of genetic diversity and of several genomes indicated 1% to 7% differential archaic admixture among populations outside of Africa (3, 33). Importantly for the serial founder effect theory, a limited amount of archaic admixture does not destroy our power to and the successful expansion of humans 100,000 y later has been the subject of intense paleoanthropological debate. Genetic data can directly address the time and rate of population growth in the African ancestral population; however, 4 / 34

7 Importance of modelling demography Demographic processes influence genetic variation Population stratification can confound association studies Correct null model Forensics Historical interest 5 / 34

8 Wright-Fisher model with variable population size Randomly mating population of size N(t) in generation t t = 0 is present, t increasing in the past N(t + 1) N(t) At generation t, N(t) offspring generated, each picks a parent among N(t + 1) parents independently and uniformly 6 / 34

9 Coalescent with variable population size Take N(t) to at same rate for each t Choose rescaling parameter N for time such that η(t) = exists and is positive for all t 0 N( N t ) lim N, N N(t) Sample of size n randomly drawn at time 0 Each pair of lineages coalescences according to exponential distribution with time-variable rate 1 η(t) 7 / 34

10 Coalescent with variable population size Let T n,k be the waiting time while there are k ancestral lineages for a sample of size n drawn at time 0. η(t) t T 5, T 5,3 T 5,4 T 5, / 34

11 Coalescent with variable population size CDF of T n,n, P(T n,n t) = t 0 ( n ) η(τ) exp ( τ 0 ( n ) η(x) dx {T n,j } n j= are independent random variables when η(t) constant, but this is not true in general CDF of T n,k has a more complicated form for general k due to the varying population size But the CDF of T n,k conditional on n j=k+1 T n,j is simply, n t ( ( P(T n,k t T n,j = t +t k τ ( ) k ) = ) η(τ) exp ) η(x) dx dτ j=k+1 t ) t dτ 9 / 34

12 Wright-Fisher model with structure g subpopulations (aka demes), with Wright-Fisher random mating in each Lecture 1: November deme Figure 1.5: The two-dimensional stepping-stone model. Migration occurs only between adjacent islands. Population size N α in deme α Occasionally demes exchange individuals (migrations) is chosen to move to deme β in each generation. Considering this flux backwards in time, we see that the backwards migration parameters c αβ are related to f αβ by Per-generation probability an offspring in deme α has parent in c αβ = Nβfβα, deme β given by c N αβ α known as the backward migration fraction. For the symmetric island model, c αβ = f αβ = f βα. (One might wonder what happens if we insist upon independence for the migration of individuals in the population. An alternative formulation of the Wright-Fisher model with migration is as follows. Suppose N α individuals each migrate independently with probability f αβ to each deme β. After migration, deme α now has size N α + X α, where X α is the random number of net migrants. These N α + X α individuals then have N α offspring and die, to form the next generation. Notohara (1990) showed that this converges to the same structured coalescent process as the simpler deterministic formulation given above.) 10 / 34

13 Structured coalescent Take N α to at same rate for each α Let N = α N α be the rescaling parameter for time Sample n = (n α ) α drawn at time 0 Each pair of lineages in deme α coalesces at rate 1 N α/n Migration of lineages from deme α to β at rate m αβ, where m αβ = N c αβ 11 / 34

14 Structured coalescent Suppose we have two demes labelled α and β State of the Markov chain at time t given by n(t) = (n α (t), n β (t)) Transitions out of state n = (n α, n β ) (n α 1, n β ) at rate ( n α ) (n α, n β 1) at rate ( n β ) (n α, n β ) (n α 1, n β + 1) at rate nαm αβ (n α + 1, n β 1) at rate n βm βα 1 N α/n 1 N β /N 1 / 34

15 Structured coalescent 1- Example genealogy for demes and sample n = (5, 5) at time 0 13 / 34

16 Measure of population structure p w (θ) = probability that two individuals sampled from the same deme are IBD p b (θ) = probability that two individuals sampled from different demes are IBD p w = E[e θtw ], p b = E[e θt b ] where T w (resp. T b ) are the time to coalescence for two individuals sampled from the same (resp. different) deme Structure in the population summarized by F ST, defined as F ST = p w (θ) p(θ), 1 p(θ) where p(θ) is the probability of IBD when two individuals are sampled at random (across all demes) 14 / 34

17 Measure of population structure 0 F ST 1, F ST = 0 when no substructure, F ST = 1 when populations isolated For small θ, F ST 1 E[T w ] E[T ], where T is the time to coalescence for two individuals sampled at random (across all demes) For human population, some studies estimate F ST = / 34

18 Lecture 1: October Simple example symmetric island model Figure 1.: The symmetric island model. Subpopulation sizes are all equal, as are migration Suppose we have g demes of equal size rates between any two islands. Migration from any deme α to β is given by m αβ = m g 1 The system ( ) becomes Let θ/ be the mutation rate Conditioning one[t the w ]= most 1 recent m +1 + m genealogical m +1 E[T event, can write b], recurrences for p w (θ) and p b (θ) E[T b ]= 1 1 m + 1 g 1 E[T w]+ g g 1 E[T b], p w (θ) = 1 + θ + m + m 1 + θ + m p b(θ) which can be solved to obtain m/(g 1) p b (θ) = θ + m p m(g )/(g 1) w (θ) + p b (θ). Equations (1.5) deserves several remarks. E[T w ]=g, E[T b ]=g + g 1 m θ +. m (1.5) 16 / 34

19 General population structure 17 / 34

20 Recombination 18 / 34

21 Recombination Recombination is a major evolutionary mechanism responsible for generating genetic variation in sexual organisms Humans are diploid organisms We have two copies of every chromosome a maternal and a paternal copy. These are called homologous chromosomes In the synthesis phase of meiosis, each chromosome gets duplicated so that it is comprised of two identical sister chromatids joined at the centromere 19 / 34

22 Recombination During the subsequent prophase I stage of meiosis, homologous chromosomes come into contact and DNA is exchanged between chromatids on homologous chromosomes. 0 / 34

23 Recombination At the end of meiosis, 4 haploid daughter cells are produced. Some of these daughter cells have different haplotypes from either of the parental haplotypes. 1 / 34

24 Importance of recombination Creates new genetic variation by mixing alleles between different haplotypes Breaks down genealogical correlation between two positions on the same chromosome Implications for many computational problems in population genetics, including Phasing genotype data into haplotype data Imputing missing data Disease-association mapping Inferring local ancestry of admixed populations Detecting signatures of natural selection / 34

25 Wright-Fisher model with recombination Consider a population of N individuals at two loci For each offspring individual, with probability r, there is recombination between the loci and a parent is chosen for each locus independently and uniformly at random With probability 1 r, an individual chooses the same parent uniformly at random for both loci 3 / 34

26 Coalescent with recombination desiderata Suppose we have two loci A and B separated by recombination rate ρ/ We want a continuous-time process to model the genealogical structures at both loci Marginally, the genealogy at each locus must be given by the coalescent process we saw earlier If there was no recombination (ρ = 0), genealogies at both loci should be identical If there was free recombination (ρ = ), genealogies at both loci should be independent In general, the genealogies at both loci will be correlated due to non-trivial recombination 4 / 34

27 Ancestral recombination graph Sample of size 4 at two loci. Example joint genealogy at loci A and B 5 / 34

28 Ancestral recombination graph Sample of size 4 at two loci. Marginal genealogy at locus A 5 / 34

29 Ancestral recombination graph Sample of size 4 at two loci. Marginal genealogy at locus B 5 / 34

30 Coalescent with recombination r per-generation per-individual probability of recombination between A and B Population-scaled recombination rate ρ = Nr The configuration at time t in the ancestral process Markov chain will be specified by n(t) = (a(t), b(t), c(t), d(t)) a(t) (resp. b(t)) is the number of ancestors at time t that contribute genetic material to the original sample at locus A (resp. locus B) only c(t) is the number of ancestors at time t that contribute genetic material to the original sample at both locus A and locus B d(t) is the number of ancestors at time t that do not contribute any genetic material to the original sample 6 / 34

31 P(Cn Coalescent with recombination A (t) =α,cn B (t) =β) = When ρ =, the two loci are completely independent and P(C A n (t) =α) if α = β, 0 otherwise. Suppose the current sample configuration is (a, b, c, d) There are 10 kinds of coalescence events: (17.5) P(C A n (t) =α,c B n (t) =β) =P(C A n (t) =α)p(c B n (t) =β). (17.6) We can obtain the transition rates for U t by enumerating the various kinds of coalescent and recombination events. There are ten kinds of coalescent events. (a, b, c, d) (a, b, c 1,d) Rate: c (a, b, c, d) (a 1, b, c, d) Rate: a (a, b, c, d) (a, b 1, c, d) Rate: b (a, b, c, d) (a, b, c, d 1) Rate: d (a, b, c, d) (a, b, c, d 1) Rate: bd (a, b, c, d) (a 1,b 1,c+1,d) Rate: ab (a, b, c, d) (a 1, b, c, d) Rate: ac (a, b, c, d) (a, b 1, c, d) Rate: bc (a, b, c, d) (a, b, c, d 1) Rate: ad (a, b, c, d) (a, b, c, d 1) Rate: cd There are four kinds of recombinations events. (a, b, c, d) (a +1,b+1,c 1,d) (a, b, c, d) (a, b, c, d + 1) 7 / 34

32 Rate: b Coalescent with recombination (a, b, c, d) (a, b, c, d 1) Rate: d (a, b, c, d) (a, b, c, d 1) Rate: bd There are 4 kinds of recombination events: There are four kinds of recombinations events. Rate: bc (a, b, c, d) (a, b, c, d 1) Rate: ad (a, b, c, d) (a, b, c, d 1) Rate: cd (a, b, c, d) (a +1,b+1,c 1,d) Rate: cρ (a, b, c, d) (a, b, c, d + 1) Rate: bρ (a, b, c, d) (a, b, c, d + 1) Rate: aρ (a, b, c, d) (a, b, c, d + 1) Rate: cρ 8 / 34

33 Coalescent with recombination Summarizing the transitions out of state (a, b, c, d) (a, b, c 1, d) at rate ( ) c (a 1, b 1, c + 1, d) at rate ab (a 1, b, c, d) at rate ( a ) + ac (a, b, c, d) (a, b 1, c, d) at rate ( b ) + bc (a, b, c, d 1) at rate (a + b + c)d + ( ) d (a + 1, b + 1, c 1, d) at rate cρ (a, b, c, d + 1) at rate (a+b+d)ρ Absorbing states {(0, 0, 1, d) d 0} Can impose mutations on the genealogy at each locus as usual 9 / 34

34 Coalescent with recombination reduced representation The fourth component of the state representation (the d component) need not be tracked since these ancestors do not contribute any genetic material to the original sample, and hence have no influence Let the reduced state be (a, b, c). state are (a, b, c 1) (a, b, c) The transitions out of this at rate ( ) c (a 1, b 1, c + 1) at rate ab (a 1, b, c) at rate ( a ) + ac at rate ( b ) + bc (a, b 1, c) (a + 1, b + 1, c 1) at rate cρ Absorbing state is (0, 0, 1) 30 / 34

35 Ancestral recombination graph Example joint genealogy at loci A and B (0, 0, 1) (0, 1, 1) (0, 1, ) (1,, 1) (0, 1, ) (1, 1, ) (0, 0, 3) (0, 0, 4) 31 / 34

36 Coalescent with recombination The coalescent process at just locus A is embedded in the coalescent with recombination for loci A and B For state (a, b, c), there are a+c ancestors that contribute genetic material at locus A Transitions out of states in S m = {(a, b, c) a + c = m} are { (a, b, c 1) at rate ( ) c (a, b, c) (a 1, b, c) at rate ( a ) + ac Total rate of transitions out of S m is ( ( a ) + c ( ) + ac = m ), which agrees with the transition rates of the coalescent process at locus A 3 / 34

37 Computational challenges due to recombination The two-locus model can be extended to multiple loci with different recombination rates between them For K loci, the state space of the Markov chain would have K 1 components for the number of ancestors that contribute genetic material to the sample at different subsets of the K loci State space size O((n + K ) (K 1) ) Even simulating data under these models can be expensive for large K and large ρ Computing probability of an observed sample under this model is prohibitive in practice even for K =, two alleles per locus and a sample size of a few hundred haplotypes 33 / 34

38 What about natural selection? Easy to incorporate an allelic advantage in the discrete Wright- Fisher model Can construct a continuous time coalescent model Genealogical structure ancestral selection graph However, easier to develop a forwards-in-time continuum model (coming up) 34 / 34

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