Diploid biparental Moran model with large offspring numbers and recombination Bjarki Eldon New mathematical challenges from molecular biology and genetics BIRS workshop Sept 6, 2009
Mendel s Laws The First Law : Law of segregation - gametes are haploid The Second Law : Law of independent assortment - alleles of different genes assort independently into gametes
Usual haploid Wright-Fisher model parents offspring If population size is N then timescale is also N
Diploid Wright-Fisher population of size N - two ancestral lines: 2N N 2N 2N N 2N 0 0 Replacing N with 2N follows from Mendel s laws
Earlier work by Kämmerle (989) on bisexual Moran model ; but has only one type: Two individuals are born. They choose their parents at random. A pair is chosen randomly, removed and replaced by the two newborn individuals who form a new pair. Each event is independent of the other.
Möhle (994) distinguishes between males and females in a model of nonoverlapping generations. The population is composed of N females and N males at all times. Each generation the offspring choose their parents at random. Möhle and Sagitov (2003) classify diploid ancestral processes in a two-sex population with non-overlapping generations
A diploid biparental model accounts for gametes coming from two distinct parents parent offspring parent
A diploid biparental Moran model - two ancestral lines 0 N N Π N = N(N ) 2 N(N ) N(N ) 0 0 A separation of timescales problem arises in the ancestral process
Π N = I + N 0 + N 2 2 ( ) + O N 3
A similar problem was considered by Nordborg and Krone (2002) in the context of fast migration (0 α<): Π N = I + N α M + N C + o ( ) N in which M contains the rates for migration (irreducible and aperiodic)
Ancestral process for a finite sample size n k: number of individuals with two ancestral lines m: number of individuals with one ancestral line P((k, m) (k, m + 2)) = k ( ) N + O N 2 ( m ( ) 2) P((0, m) (0, m )) = N + O 2 N 3 ( m ( ) 2) P((0, m) (, m 2)) = N + O 2 N 3
A model of large offspring numbers Under usual Moran model: one offspring, U = Large number of offspring: U = ψn, 0<ψ< A simple mixture distribution for U, γ>0: P(U = u) = /N γ if u = /N γ if u = ψn If 0 <γ<2: λ k,n = ( ) n k ψ k ( ψ) n k If γ = 2: λ k,n = ( ( n 2) + n ) k ψ k ( ψ) n k If γ>2: usual Kingman coalescent
Hedgecock (994) and Beckenbach (994) suggested sweepstakes-style recruitment when considering data on Pacific oysters Williams (975) argued for Sisyphean genotypes - sweepstakes style reproduction due to selection on genotypes rather than chance matching of reproduction with favorable environmental conditions
Ancestral process for two ancestral lines under modified sampling: N N 0 ψ ψ 0 Π N = ( N γ ) N(N ) 2 N(N ) N(N ) + N γ ψ N 2ψ N ψ N 0 0 0 0
In case 0 <γ<-usemöhle (998) result on separation of timescales: with Π(t) = lim Π [Nγ t] N N = Pe tg 0 0 0 ψ ψ P = 0 0 0 0, G = 0 ψ ψ
In case γ = : Π N = I + N ψ + ψ 0 + N 2 0 + ψ 2 2ψ + ψ ( ) + O N 3
In case <γ<2 0 ψ ψ 0 Π N = I + N + N 2 2 + ( ) + O N 2+γ + N γ N +γ 0 ψ 2ψ ψ
Ancestral process for sample size n under modified sampling (U = ψn, 0 <ψ<) Probability of sampling l 2 + l offspring, andl 2 of them carry two ancestral lines, while l contain one ancestral line, when k individuals carry two ancestral lines, and m one P((k, m) (k l 2, m l )) = ( k + m l 2 + l ) ( k )( m ) ( ) ψ l 2+l ( ψ) k+m l 2 l l 2 l ( k+m ) + O N l 2 +l
A largeoffspring number event Ancestral lines are labelled a, b, c, and d ab c d bd ac a a } {{ } offspring parent parent
Including recombination in a sample of size two of two loci Sample states are ( ) ( ) ab ab, ab ( ) ab, a ( b ) ( ) ab, ( )( ) ( ) a a a, b b a ( ) ab ( a b ( b b ) ( ) ab, ) ( ) a, b ( ) a, ( ) a, ( ) b ( b and all ancestral states (not shown) ) ( ) a, a ( ) b, ( b ) ( ) a, ( ) a, ( ) b, ( ) b
Ancestral process is similar as in the single locus case First, a dispersion mode - sending all into single gamete case by rate matrix A on timescale N A = ) ( ab ab ( ab )( ab ) ( ab )( b ) a ( ab )( a )( b ) ( a )( a ) b b ( a )( a )( b ) b ( a )( a )( b )( b ) ( ab ab ) ( ab )( ab ) ( ab )( b a ) ( ab )( a )( b ) ( a )( a b b ) ( a )( a )( b b ) ( a )( a )( b )( b 0 0 0 0 0 0 0 2 2 0 0 0 0 )
The process can be written as Π N = I + N A + ( ) N (S + W) + O 2 N 3 where W is the same rate matrix as one obtained from Wright-Fisher and S holds transitions for revisiting the A process Recombination scaled as c = ρ/n, ρ [0, )
Allowing large offspring numbers: U = ψn Π N = I + Ψ + cψ + O ( ) N in which Ψ represents the sampling from ψn offspring and c is the per timestep recombination probability Scaling of c?
Conclusions A diploid biparental Moran model results in a separation of timescales problem Large offspring numbers result in a coalescent process with simultaneous multiple mergers Scaling recombination with N - as opposed to N 2 - results in convergence to same process as for Wright-Fisher, apart from a dispersion phase Ancestral process with recombination and large offspring numbers results in dispersion and coalescence on same timescale O() Relevance for biology?
Acknowledgments Many thanks to my old boss, John Wakeley, and my new boss, Alison Etheridge, for helpful discussions