Non-equilibrium theory of the allele frequency spectrum ... - ICMS

Non-equilibrium theory of the allele frequency spectrumSteven N. EvansDepartments of Statistics and Mathematicsand Graduate Group in Computational and Genomic BiologyUniversity of California at Berkeleyevans@stat.berkeley.eduhttp://www.stat.berkeley.edu/users/evans(with Yelena Shvets and Montgomery Slatkin)Research supported in part by NSF and NIH grants

Contemporary traces of past human population growth?It is suggested in Reich and Lander (2001) and Williamson et al.(2005) that the effective size of the modern human populationwas stable at around 10 4 until 1.5 × 10 5 years ago.The current effective size is 1.6 × 10 9 .If the population grew according to a given schedule over thisperiod (e.g. exponential growth at a constant rate), how wouldthis manifest itself in the genetic composition of contemporaryhumans?

Equilibria under stable conditionsIf mutations are not reversible at a polymorphic locus, then oneallele will eventually fix there (i.e. no other alleles will be presentin the population).However, the distribution of allele frequencies across polymorphicloci (the frequency spectrum) reaches a non-trivial equilibrium ifboth population size and selection intensities are constant.The theory for the equilibrium frequency spectrum under irreversiblemutation was developed in Fisher (1930), Wright(1938), Kimura (1964). Kimura (1969), Sawyer and Hartl(1992), Bustamante et al. (2001), Williamson et al. (2004),...

What if population size or selection intensities change?Nei et al. (1975) showed that rapid growth resulted in more lowfrequency alleles than expected under neutrality.Tajima (1989) confirmed that conclusion and examined the effectof past population growth on other aspects of the frequencyspectrum.Griffiths and Tavaré (1998) developed the coalescent theory forthe frequency spectrum of neutral alleles in a population that hasexperienced arbitrary changes in population size (related work byNielsen (2000), Wooding and Rogers (2002) and Polanski andKimmel (2003)).

What if population size or selection intensities change? –continuedGriffiths (2003): the frequency spectrum in a neutral populationof variable size could be derived from the spectrum for a populationof constant size when a transformation of the time scalereduces a suitable backwards equation to one for a populationof constant size.We present a forward equation approach that computes the frequencyspectrum for arbitrary population growth and changes inselection intensity.

Discrete time finite population modelsMonoecious, randomly-mating, diploid population containingN(t) individuals in generation t ∈ {0, 1, 2, . . .}.Independent loci.At each locus there are only two alleles A, the derived allele, anda the ancestral allele. Mutation only occurs from ancestral toderived, at loci that haven’t seen mutations before (infinite sitesassumption).Large number of loci – effectively infinite pool.

Discrete time finite population models – continuedPut f j (t) := expected number of loci at which A is found on jchromosomes, 1 ≤ j ≤ 2N(t).Put p ij (t) := conditional probability that a locus with i copies ofA in generation t will have j copies in generation t + 1.The change in f j (t) because of genetic drift and mutation atrate µ is described by the set of “forward” difference equationsf j (t + 1) =2N(t)∑i=1f i (t)p ij (t) + 2N(t)µδ 1,j , 1 ≤ j ≤ 2N(t + 1).

An analogous forward equationRecallf j (t + 1) =2N(t)∑i=1f i (t)p ij (t) + 2N(t)µδ 1,j , 1 ≤ j ≤ 2N(t + 1).If π j (t) := probability of a given locus having j copies of A ingeneration t, thenπ j (t + 1) =2N(t)∑i=1π i (t)p ij (t).

The diffusion limit for a single locusAssume for now that p ij (t) = p ij – time-homogeneous case.Assume for now # chromosomes constant at 2Nρ.Suppose that if we shrink space by a factor of 2Nρ and speedtime up by a factor of 2N, then the chain converges to a diffusionprocess on [0, 1] (call it the 0-diffusion) with generatorG = a(x)dx d + 1 d22b(x) dx 2.This is the scaling regime that is appropriate for models such asWright-Fisher with or without selection.

The single locus diffusion forward equationWrite π(y, t) for the probability density of the 0-diffusion at frequencyy ∈ (0, 1) and time t > 0.The Kolomogorov forward equation∂∂π(y, t) = −∂t ∂y [a(y)π(y, t)] + 1 2∂y2[b(y)π(y, t)].holds with suitable boundary conditions.∂ 2

The diffusion frequency spectrum without mutationSuppose at time 0 that there are countably many loci at whichderived alleles are present, with respective frequencies x 1 , x 2 , . . ..Assume for now there is no further mutation from the ancestralstate.After passage to the diffusion limit, the frequency spectrum attime t is just the intensity measure of the point process thatcomes from starting independent copies of the 0-diffusion processat each of the x i and letting them run to time t.

The forward equation without mutationThe frequency spectrum is obtained by taking a sum of pointmasses at the x i and moving it forwards with the 0-diffusionsemigroup.Set f o (y, t) := frequency spectrum, then∂∂t f o (y, t) = − ∂∂y [a(y)f o (y, t)] + 1 2with suitable boundary conditions.∂ 2∂y 2[b(y)f o (y, t)].

Introducing mutation from the ancestral typeIn the Markov chain model, new mutants arise at rate 2Nρµ = θ 2 ρper generation.The initial number of mutants at a locus is 1.Hence mutants appear at rate 2N θ 2ρ per unit of rescaled time,with the initial proportion of mutants at a locus being 12Nρ .

Introducing mutation from the ancestral type – continuedPass to the diffusion limit for the allele frequencies, but for nowstill work with a finite N for the description of the appearanceof new mutants.The evolving point process has new points added at location 12Nρat rate θ 2 2Nρ.The points then evolve as independent copies of the diffusion.

Introducing mutation from the ancestral type – continuedSet P t (x, dy) := 0-diffusion semigroup.Contribution to the frequency spectrum at y ∈ (0, 1) from mutationsthat appear after time 0 =2N θ ∫ ( )t 12 ρ P t−s0 2Nρ , dy ds.

Entrance boundary theoryChoose a scale function s for the 0-diffusion such that s(0) = 0and s ′ (0) = 1.The Doob h-transformP ↑ u(x, dy) := 1s(x) P u(x, dy)s(y), 0 < x, y ≤ 1,is the semigroup of a diffusion that never hits 0 (the ↑-diffusion)The ↑-semigroup can be extended to allow starting at 0 by settingP ↑ u(0, dy) = limx↓0P ↑ u(x, dy).The extended process can start at 0 but it never returns to 0.

Entrance boundary theory – continuedPutlimN→∞ 2NρP u(12Nρ , dy )= P ↑ u(0, dy)s(y)=: λ u (dy),then ∫ λ s (dx)P t (x, dy) = λ s+t (dy) and (λ u ) u>0 has densities thatsatisfy the forward equation.Hencelim 2N θ ∫ ( )t 1N→∞ 2 ρ P t−s0 2Nρ , dyds = θ 2∫ t0 λ t−s(dy) dsalso has densities φ t (x) that satisfy the forward equation.What are the boundary conditions?

Entrance boundary theory – continuedThe ↓-diffusion is the 0-diffusion conditioned to hit 0 before 1.It has the Doob h-transform semigroup(Pt ↓ (x, dy) = 1 − s(x)) −1 (P t (x, dy) 1 − s(y)).s(1)s(1)From Williams (1974), the ↑-diffusion started at 0 and killed atthe last time it visits y > 0 is the time-reversal of the ↓-diffusionstarted at y and killed when it first hits 0.Write (Q ↓ t ) t≥0 := semigroup of killed ↓-diffusion.

Finding the boundary conditionSince s(y) ≈ y for y close to 0,lim yφ t (y) = lim s(y)φ t (y)y↓0 y↓0= limy↓0s(y)= θ 2 limy↓0= θ 2 limy↓0= θ 2 limy↓0= θ 2 limy↓0∫ t0∫ t0∫ ∞0∫ ∞0∫ t0θ λ t−s (dy)ds2 dyPt−s ↑ (0, dy)dyP ↑ s (0, dy)dyP ↑ s (0, dy)dydsdsdsQ ↓ s(y, dy)ds.dy

Finding the boundary condition – continuedIf (B t ) t≥0 is a standard Brownian motion and T := inf{t ≥ 0 :B t = 0}, then∫ ∞limy↓0 0P y {B s ∈ dy, T > s}dy∫ ∞ 1ds = lim √ − √ 1 e −(2y)2 /2s dsy↓0 0 2πs 2πs= 2y.By Itô–McKean theory,∫θ ∞2 limy↓0 0Q ↓ s(y, dy)dyds = θ limy↓0yb(y) .

Conclusion for the time-homogeneous casePut f(x, t) := for the frequency spectrum of the model with mutationfrom ancestral type, with everything time-homogeneous.Note that f(x, t) = f o (x, t) + φ t (x).Hence∂∂f(x, t) = −∂t ∂x [a(x)f(x, t)] + 1 2∂x2[b(x)f(x, t)].with lim x↓0 xf(x, t) = θ limx x↓0 b(x) and lim x↑1 f(x, t) finite.∂ 2

Conclusion for the time-inhomogeneous caseSuppose that everything is allowed to depend on time.Write f(x, t) for the frequency spectrum.Allowing a, b, θ and ρ to be piecewise constant, using the aboveanalysis, and then taking limits gives∂∂f(y, t) = −∂t ∂y [a(y, t)f(y, t)] + 1 2∂y2[b(y, t)f(y, t)]with lim y↓0 yf(y, t) = θ(t) limx x↓0 b(x,t) and lim y↑1 f(y, t) finite.∂ 2

Structured populationsThe same approach works for systems of populations interactingvia migration.A single PDE with boundary conditions is replaced by a familyof coupled PDE with boundary conditions.

ExampleTake θ(t) = θ, a(x, t) = Sx(1 − x), and b(x, t) = x(1 − x)/ρ(t) ≡Wright-Fisher diffusion with constant mutation, additive selection,and varying population size (2Nρ(t) in generation 2Nt).Set g(x, t) := x(1 − x)f(x, t).The forward equation is∂∂tg(x, t) = −Sx(1 − x)∂∂xwith lim x↓0 g(x, t) = θρ(t).[g(x, t)] +x(1 − x)2ρ(t)∂ 2[g(x, t)]∂x2

Example – continuedPut µ n (t) := ∫ 10 x n g(x, t) dx = ∫ 10 x n x(1 − x)f(x, t) dx.Integrating by parts gives the coupled system of ODEsandµ ′ 0 (t) = θ 2 − 1ρ(t) µ 0(t) + S (µ 0 (t) − 2µ 1 (t))µ ′ n(t) = 12ρ(t)[(n + 1)nµn−1 (t) − (n + 2)(n + 1)µ n (t) ]+ S ( (n + 1)µ n (t) − (n + 2)µ n+1 (t) ) , n ≥ 1.

Frequency spectrum in a finite sampleIn a sample of n chromosomes the expected number of loci atwhich the derived allele is found on i chromosomes isf i (t) = ( ni= ( ni) ∫ 10 xi (1 − x) n−i f(x, t) dx) n−i−1 ∑j=0(−1) j( n − i − 1j)µj+i−1 (t).

Recent human population growthAssume an effective size N 0 = 10 4 until 1.5 × 10 5 ya (t = 0).Assume a generation time of 25 years.Measuring time in units of 2N 0 , the present is at t = 0.3.The current effective population size is 1.6×10 9 = 10 5 ×e 40×0.3 .Assume exponential growth, so ρ(t) = e 40t , and θ(t) = 1.Assume that the spectrum at t = 0 is the equilibriumf(x, 0) = e2S ( 1 − e −2S(1−x))(e 2S − 1 ) x(1 − x)

10090f(x,0)f(x,0.3))8070f(x)60504030201000 0.1 0.2 0.3 0.4 0.5x0.6 0.7 0.8 0.9 1Plots of f(x, t) for S = 0

10090f(x,0)f(x,0.3))8070f(x)60504030201000 0.1 0.2 0.3 0.4 0.5x0.6 0.7 0.8 0.9 1Plots of f(x, t) for S = +2

10090f(x,0)f(x,0.3))807060f(x)504030201000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1xPlots of f(x, t) for S = −2

43.5f i(n=20,t=0)f i(n=20,t=0.3))32.5f i21.510.500 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1i/n = number of derived alleles / sample sizePlots of f i (t) for S = 0 and n = 20

54.5f i(n=20,t=0)f i(n=20,t=0.3))43.53f i2.521.510.500 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1i/n = number of derived alleles / sample sizePlots of f i (t) for S = +2 and n = 20

3f i(n=20,t=0)f i(n=20,t=0.3)2.52f i1.510.500 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1i/n = number of derived alleles/ sample sizePlots of f i (t) for S = −2 and n = 20

76f i(n=40,t=0)f i(n=40,t=0.3)) from truncated system of ODEsf i(n=40, t=0.3) from simulation algorithm54f i32100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1i/n = number of derived alleles / sample sizePlots of f i (t) for S = 0 and n = 40

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