Stochastic Demography, Coalescents, and Effective Population Size

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1 Demography Stochastic Demography, Coalescents, and Effective Population Size Steve Krone University of Idaho Department of Mathematics & IBEST Demographic effects bottlenecks, expansion, fluctuating population size, population structure) affect polymorphism data Ex) Detecting selective sweeps confounded by demography and structure Effective population size: When is it meaningful? What is effect of demography? Appropriate scaling comes from what is observable in the coalescent i.e., what has an explicit effect on data). Wright Fisher model discrete time generations) constant population size panmictic no selection, no recombination ancestry: each individual chooses haploid) parent at random prob / each) from previous generation Effective population size Other population models reproduction, variable pop size, structure,...) sometimes behave in certain respects like a W-F model with an effective population size e. inbreeding effective size probability of identity by descent) variance effective size variance in reproductive success) eigenvalue effective size leading non-unit eigenvalue for allele frequency transition matrix) coalescent effective size if it exists) supersedes all of these. Exists when scaled ancestral process converges to linear time change of Kingman s coalescent; demographic fluctuations average out. The coalescent P indiv choose same parent) / Takes O) generations to find common ancestor per pair) Measure time in units of generations...[t] A τ) # ancestors τ generations in past A [t]) At)...Kingman coalescent All genetic information about a sample polymorphism data) is embedded in the coalescent. Fu and Li s F statistic F F π, η s,s) π n n )η s c S + c S where n sample size π ave. # pairwise differences influenced by deep branches) η s # singletons influenced by external branches) S # segregating sites

2 where a n n i i Tajima s D statistic π S a D Dπ, η s,s) n c S + c S Both statistics have mean, variance. Deviations from assumptions neutrality, constant pop size, panmixia,...) produce changes in F and D. Relative time scales Coalescence events have prob O/ ). Events that are faster have prob O/ α ), where α<. Effects appear in coalescent only in average sense. All demographic processes fast coalescent effective size exists.) Events with prob O/ ) areincorporatedinthe coalescent and affect pattern of variation in nonhomogeneous way. o coalescent effective size) Fluctuating population size backward) size process M ),M ),M 3),... Markov chain with state space {,,...} i x i How does this affect the coalescent? Depends on time it takes for large size changes i.e., O)) to occur. Fast size fluctuations linear time change Large pop size changes occur quickly e.g., every generation); size process stationary distribution γ,γ,...); W-F reproduction: P no coalescence in [t] generations) E [ [t] τ i ) ] M τ) γ i x i ) [t] exp{ t γi /x i } Limiting coalescent... linear time change of standard coalescent: A [t]) Act) where c γ i x i... pairwise coalescence rate pairwise coalescence prob γi x i e e c γ i ) i... harmonic mean of sizes This is the coalescent effective size : e /c Intermediate fluctuations stochastic time change What if macroscopic changes in pop. size i.e., O)) occur on coalescent time scale i.e., O) generations)? Pop. size τ generations in past Markov chain): M τ) X τ), where relative size proc. X [t]) M [t]) Xt)... cont-time Markov e.g., diffusion proc. or cont-time jump chain)

3 Reproduction Ex: Wright Fisher model Let c M τ ),M τ) ) denote prob. that two lineages coalesce when going from gen. τ to gen. τ in past). Assume c k, m) H k, m ), where H k, m ) Hx, y) as k/ x and m/ y. Time change becomes c M τ ),M τ) ) M τ) X τ) So c k, m) m H k, m ), where H x, y) y Hx, y) t HX s,x s )ds. Ex: Cannings model c M τ ),M τ) ) M τ )) M τ) i E [ ν τ) i ) ] ν τ) i... number of offspring produced by ith indiv in gen τ. With exchangeable reproduction, get k ) H, m k k ) ) md yd Hx, y) x Large size changes occur on same time scale as coalescence events; do not average out. Limiting coalescent is of form where the time change A [t]) AY t)), Y t) t HXs),Xs))ds is nonlinear and stochastic coalescence intensity). [WF case: Y t) t ds] Kaj and Krone 3; Donnelly and Kurtz Xs) 999; Griffiths and Tavaré 994.) o coalescent) effective size! Behavior different from any standard W-F model. Effects should show up in polymorphism data. Idea for W-F model Full convergence theorem P no coalescence in [t] generations {M )}) [t] τ [t] τ ) M τ) ) X τ) exp [t] τ ) t exp X τ) Xs) ds) X [t]),a [t])) Xt),AY t )) in D S {,...,n} [, ), whenever X ) X) in S. Transition semigroup T t fx, i) E x,i)[ fxt),ay t )) ] can be decomposed as T t fx, i) i j lj i C l i, j)e x,i)[ fxt),j) e l )Y t ]

4 Idea of Proof C l i, j) j+ s i ) s j r i,r l r ) l ) l ) )l j j l ) i) l j!l j)!i l), j l i. For i n, x Z and r, transition operator for X,A ): T r fx, i) E x,i)[ fx r),a r)) ], ) Show uniform convergence of semigroups: sup T[t] fx, i) T tfx, i), x,i Lfx, i) d dt T tfx, i) t ) i Lfx, i)+ Hx, x) fx, i ) fx, i) ) Local time interpretation of time change For any t fixed and j i n, as, P k,i) M [t]),a [t])) m, j) ) i C l i, j) lj P ) ) [t] ) l ˆP k, m)+o, L x t t... diffusion local time mdx)... speed measure ds X s E x Lx t mdx) Combinatorial term is same in discrete semigroup, so decomposition implies enough to show uniform convergence of discrete Feynman Kac semigroups to E x,i)[ fxt),j) e l )Y t ] Simulations for fluctuating size sizes, ; equal prob of size change q q q; mutation prob u.;, runs per data pt.; stationary starting size. Plot of Fu and Li s F 3, 5 3, 4 Rule of thumb: q, ) no averaging; too close to coalescent scale.

5 Dependence on initial size Why do polymorphism data always seem to suggest population expansion, and not population contraction? Top curve: initial size 3 Bottom curve: initial size 5 3, 5 ; q q 4. General Population Structure Many deviations from basic Wright Fisher model can be thought of as examples of population structure. geographic structure age classes diploidy ordborg and Donnelly 997) Population divided into groups that are connected by migration. Fast migration... effects only present in averaged sense. Slow migration... effects explicitly appear in coalescent. Differences in scaling... hierarchical structuring of population males and females a)ocollapse...structuredcoalescent. b) Full collapse to Kingman coalescent. Partial collapse to structured coalescent.

6 Rates depend on configuration of ancestral lineages While there are r ancestors r,...,n), the configuration process moves among the configurations in level r: S r {x,...,x L ):x + + x L r}. Ex. Ancestral urn for age structure. Starting with sample of size n, the state space for the configuration process is S S S n.forany configuration x,...,x L ) S, specify probabilities of jumping to other configurations due to migration and/or coalescence of ancestors. Proof of convergence Stationary distribution of backward migration process: γ γ,...,γ L ). Stationary distribution for level-r configuration process: π r x) r! x! x L! γx γx L L. Transition matrix for whole configuration process on S S S n : Π I + α B + ) C + o, where I is the identity matrix, B is a block diagonal matrix B B..... B B n,n B n,n... from backward migration jumps, Möhle s Lemma 998) C is a block matrix of the form C C C C C n,n C n,n... due to coalescence jumps. Case α : Π [t] A + C + o )) [t] P I + e tg, where P lim k A k and G PCP.

7 Structured Populations Population of total size, subdivided into L demes, connected by migration. Pop. size in deme k is k a k a + + a L ). Migration on same time scale as coalescence events i.e., migration prob. for lineage b ij β ij / ) limiting coalescent is structured. no averaging, no coalescent effective size) Fast migration i.e., b ij β ij / α, α<), and stationary distribution for locations γ,γ,...,γ L ) averaging occurs w/ coalescent time change c coalescent effective size is L k e c γ k k ) γ k a k. harmonic mean In case of fast migration, structured model can be thought of as panmictic W-F model with pop. size e. Simulations for population subdivision demes, equal size, equal migration rate β b 3 4 I. Kaj, Uppsala Univ. Mathematics) M. ordborg, USC Molecular and Computational Biology) M. Lascoux, Uppsala Univ. Cons. Biology & Genetics) P. Sjödin, Uppsala Univ. Cons. Biology & Genetics) SF DMS--798 IH P RR6448 P. Sjödin,I.Kaj,S.Krone,M.Lascoux,M.ordborg 5) On the meaning and existence of an effective population size. Genetics 69: 6-7. I. Kaj and S. Krone 3) The coalescent process in a population with stochastically varying size. J. Appl. Probab. 4: M. ordborg and S. Krone ) Separation of time scales and convergence to the coalescent in structured populations. In Modern Developments in Theoretical Population Genetics. M. Slatkin and M. Veuille eds.).