Diffusion Models in Population Genetics Laura Kubatko kubatko.2@osu.edu MBI Workshop on Spatially-varying stochastic differential equations, with application to the biological sciences July 10, 2015 Laura Kubatko Diffusion Models in Population Genetics July 10, 2015 1 / 24
Population Genetics Population genetics: Study of genetic variation within a population Assume that a gene has two alleles, call them A and a Population is composed of N individuals who have two copies of each gene so possible genotypes are: The population evolves over time AA Aa aa We are interested in the composition of the population at generation t Need a model for how a generation is derived from the previous generation Laura Kubatko Diffusion Models in Population Genetics July 10, 2015 2 / 24
Wright-Fisher Model Assumptions: Population of 2N gene copies Discrete, non-overlapping generations of equal size Parents of next generation of 2N genes are picked randomly with replacement from preceding generation (genetic differences have no fitness consequences) Probability of a specific parent for a gene in the next generation is 1 2N Laura Kubatko Diffusion Models in Population Genetics July 10, 2015 3 / 24
Wright-Fisher Model Source: Popvizard, a python program to simulate evolution under the WF and other models, written by Peter Beerli http://people.sc.fsu.edu/ pbeerli/popvizard.tar.gz Laura Kubatko Diffusion Models in Population Genetics July 10, 2015 4 / 24
The Wright-Fisher Model View Wright-Fisher model as a discrete-time Markov process Let Y t = number of alleles of type A in population at generation t, 0 Y t 2N for t = 0,1,... Define p ij = P(Y t+1 = j Y t = i). Then, {( 2N ) j ( i 2N p ij = )j ( 2N i 2N )2N j, j = 0,1,...,2N 0, otherwise States 0 and 2N are absorbing states we can never leave these states Laura Kubatko Diffusion Models in Population Genetics July 10, 2015 5 / 24
The Wright-Fisher Model Note that: E(Yt+1 Y t = i) = 2N( i 2N ) = i Var(Yt+1 Y t = i) = 2N( i 2N )(1 i 2N ) So the expected number of A alleles remains the same, but the actual number may vary between 0 and 2N Classical approach: Look at the limit as the population size N Kingman s Coalescent Process Widely used in population genetics and phylogenetics Difficult to extend to handle features of the evolutionary process, such as selection Laura Kubatko Diffusion Models in Population Genetics July 10, 2015 6 / 24
Wright-Fisher Model as a Diffusion Process Define a diffusion process {X t } t 0 as a continuous-time Markov process with approximately Guassian increments over small time intervals and for which the following three conditions hold for small δt and X t = x: E(Xt+δt X t X t = x) = µ(t,x)δt +o(δt) E((Xt+δt X t) 2 X t = x) = σ 2 (t,x)δt +o(δt) E((Xt+δt X t) k X t = x) = 0 for k > 2 From Radu s slides, we had: dx t = S(X t )dt +σ(x t )dw t, where S(X t ) is the drift coefficient and σ(x t ) is the diffusion coefficient. For standard Brownian Motion, µ(t,x) = 0 and σ 2 (t,x) = 1. Laura Kubatko Diffusion Models in Population Genetics July 10, 2015 7 / 24
Wright-Fisher Model as a Diffusion Process Let Y t be the number of A alleles in the population at generation t Let X t = proportion of A alleles in population at generation t; X t = Yt 2N Let X t represent the continuous-time process (eventually measure time in units of 2N generations, as before) Define Y t = Y t+1 Y t and X t = X t+1 X t Then E(Y t+1 X t = x) = 2Nx E( Y t X t = x) = 0 E[( Y t ) 2 X t = x)] = 2Nx(1 x) E( X t X t = x) = 0 = µ(t,x) = µ(x) E(( X t ) 2 X t = x) = x(1 x) 2N = σ 2 (t,x) = σ 2 (x) Laura Kubatko Diffusion Models in Population Genetics July 10, 2015 8 / 24
Wright-Fisher Model as a Diffusion Process Now re-define Y t = Y t+ t Y t and X t = X t+ t X t, where t = 1 2N and let N, so that E(( X t) 2 X t ) = X t (1 X t ) t The corresponding SDE is dx t = X t (1 X t )dw t, X t [0,1] where W t is standard Brownian Motion (See Pardoux, 2009, for a rigorous proof) Laura Kubatko Diffusion Models in Population Genetics July 10, 2015 9 / 24
The Wright-Fisher Model with Selection Model for selection: Suppose that allele A is superior to allele a so that p x = 2Nx(1+s) 2Nx(1+s)+(2N 2Nx) As before, let N and define s = β/(2n). E( Xt X t) (βx t(1 X t)) t E(( Xt) 2 X t) X t(1 X t) t The corresponding SDE is dx t = βx t (1 X t )dt + X t (1 X t )dw t, X t [0,1] Laura Kubatko Diffusion Models in Population Genetics July 10, 2015 10 / 24
The Wright-Fisher Diffusion with Selection: Intuition Use the Euler Method (see Radu s lectures) to simulate from the WF Diffusion model X(t i+1 ) = X(t i )+βx(t i )(1 X(t i ))(t i+1 t i )+ t i+1 t i X(ti )(1 X(t i ))Z where Z N(0,1) Python code to simulate this: T = 0.05 Define 0 = t0 < t 1 < < t N 1 < t N = T, equally spaced Vary β Laura Kubatko Diffusion Models in Population Genetics July 10, 2015 11 / 24
The Wright-Fisher Diffusion with Selection: Intuition β = 0, varying N Laura Kubatko Diffusion Models in Population Genetics July 10, 2015 12 / 24
The Wright-Fisher Diffusion with Selection: Intuition N = 1000, vary β Laura Kubatko Diffusion Models in Population Genetics July 10, 2015 13 / 24
Application: Inferring Selection From Genome-scale Data Diffusion models are currently becoming more widely used in analyzing genome-scale data. Example: Williamson, S. H. et al. 2005. Simultaneous inference of selection and population growth from patterns of variation in the human genome. PNAS: 120(22): 7882-7887. Data: NIEHS Environmental Genome Project web site (http://egp.gs.washington. edu) Sequenced 301 genes associated with variation in response to environmental exposure 90 individuals: 24 African Americans, 24 Asian Americans, 24 European Americans, 12 Mexican Americans, and 6 Native Americans Goal: Detect selection in different types of mutations; distinguish selection from other demographic factors, such as population size change Laura Kubatko Diffusion Models in Population Genetics July 10, 2015 14 / 24
Application: Inferring Selection From Genome-scale Data Data are recorded as SNPs bases in the DNA sequence at which there is variation across individuals Example data: this would be Taxon Sequence (A) Human GCCGATGCCGATGCCGAA (B) Chimp GCCGTTGCCGTTGCCGTT (C ) Gorilla GCGGAAGCGGAAGCGGAA Taxon Sequence (A) Human CATCATCAA (B) Chimp CTTCTTCTT (C ) Gorilla GAAGAAGAA Laura Kubatko Diffusion Models in Population Genetics July 10, 2015 15 / 24
Application: Inferring Selection From Genome-scale Data Example SNP data is Taxon Sequence (A) Human CATCATCAA (B) Chimp CTTCTTCTT (C ) Gorilla GAAGAAGAA Record this as the site frequency spectrum (SFS), denoted by the vector u, where entry u i = number of SNP sites with i copies of the derived allele For the example, we have (assuming that the ancestral state is that found in Gorilla), u = (4,5) If we let Human be ancestral, we d have u = (9,0) Laura Kubatko Diffusion Models in Population Genetics July 10, 2015 16 / 24
Application: Inferring Selection From Genome-scale Data Idea of analysis: Write the likelihood function and obtain MLEs of the parameters of interest Likelihood function for a sample of K SNPs: L(β) = K Pr(i k,n k β) where Pr(i k,n k ) is the probability of that SNP k is at frequency i k nk k=1 Pr(i k,n k ) comes from the diffusion model how? Williamson et al. (2005): Use numerical methods to approximate the diffusion Today: use a naive sampling method based on the Euler approximation Ongoing work (with Radu Herbei and Jeff Gory): use exact sampling from the WF diffusion to implement a Bayesian version of the model Laura Kubatko Diffusion Models in Population Genetics July 10, 2015 17 / 24
Application: Inferring Selection From Genome-scale Data Naive method: 1 Use the Euler method to simulate a path from the WF diffusion with selection parameter β, and record the final allele frequency, q. 2 For the q from step 1, simulate the data for a SNP by drawing Y Bin(2n,q). n is the number of people in the sample. 3 Repeat steps 1-2 a large number of times, say M (the larger, the better), to generate a set of observed Y values, Y 1,Y 2,,Y M. 4 Form the estimates ˆP i (β) = 1 M M m=1i(ym = i) The approximate likelihood is then K ˆL(β) = ˆP ik (β) k=1 Laura Kubatko Diffusion Models in Population Genetics July 10, 2015 18 / 24
Application: Inferring Selection From Genome-scale Data Does it work? Simulate data for 15 people and 100 SNPs with various values of β and M β = 0.2 Laura Kubatko Diffusion Models in Population Genetics July 10, 2015 19 / 24
Application: Inferring Selection From Genome-scale Data Does it work? Simulate data for 15 people and 100 SNPs with various values of β and M β = 2.0 Laura Kubatko Diffusion Models in Population Genetics July 10, 2015 20 / 24
Application: Inferring Selection From Genome-scale Data Does it work? Simulate data for 15 people and 100 SNPs with various values of β and M β = 10.0 Laura Kubatko Diffusion Models in Population Genetics July 10, 2015 21 / 24
Application: Inferring Selection From Genome-scale Data Does it work? Take the maximum value of the approximate likelihood as the MLE Repeat the simulation multiple times and look at properties of the MLEs True β Number of reps Mean MLE MSE 2.0 30 2.10 2.13 10.0 15 10.25 3.58 Laura Kubatko Diffusion Models in Population Genetics July 10, 2015 22 / 24
Conclusions Diffusion models are being increasingly used for data analysis in population genetics. Methods used for estimation are mostly based on numerical approximations, rather than on statistical techniques. Promising area of application as availability of whole-genome sequence data is increasing rapidly. Laura Kubatko Diffusion Models in Population Genetics July 10, 2015 23 / 24
References Wakeley, J. (2009) Coalescent Theory: An Introduction. Robert and Company. Williamson, S. et al. (2005) Simultaneous inference of selection and population growth from patterns of variation in the human genome. PNAS 102(22): 7882-7887. Gutenkunst, R. et al. (2009) Inferring the Joint Demographic History of Multiple Populations from Multidimensional SNP Frequency Data. PLoS Genetics 5(10): e1000695. Pardoux, E. (2009). Probabilistic models of population genetics. http://www.cmi.univ-mrs.fr/ pardoux/enseignement/cours genpop.pdf dadi: Diffusion Approximation for Demographic Inference. https://code.google.com/p/dadi/ Thank you! Laura Kubatko Diffusion Models in Population Genetics July 10, 2015 24 / 24