Evolution in a spatial continuum

Transcription

1 Evolution in a spatial continuum Drift, draft and structure Alison Etheridge University of Oxford Joint work with Nick Barton (Edinburgh) and Tom Kurtz (Wisconsin) New York, Sept p.1

2 Kingman s Coalescent Kingman (1982) Neutral (haploid) population of constant size N Wright-Fisher model: new generation determined by multinomial sampling with equal weights New York, Sept p.2

3 Kingman s Coalescent Kingman (1982) Neutral (haploid) population of constant size N Wright-Fisher model: new generation determined by multinomial sampling with equal weights Equivalently offspring choose their parent at random New York, Sept p.2

4 Kingman s Coalescent Kingman (1982) Neutral (haploid) population of constant size N Wright-Fisher model: new generation determined by multinomial sampling with equal weights Equivalently offspring choose their parent at random Time in units of population size and let N New York, Sept p.2

5 Kingman s Coalescent Kingman (1982) Neutral (haploid) population of constant size N Wright-Fisher model: new generation determined by multinomial sampling with equal weights MRCA Equivalently offspring choose their parent at random Time in units of population size and let N Coalescence rate ( ) k 2 Coalesent time Wright Fisher time New York, Sept p.2

6 Forwards in time Two types a and A. p(t) = proportion of type a. New York, Sept p.3

7 Forwards in time Two types a and A. p(t) = proportion of type a. Forwards in time, E[ p] = 0 (neutrality) E[( p) 2 ] = δtp(1 p) E[( p) 3 ] = O(δt) 2 dp t = p t (1 p t )dw t New York, Sept p.3

8 Forwards in time Two types a and A. p(t) = proportion of type a. Forwards in time, E[ p] = 0 (neutrality) E[( p) 2 ] = δtp(1 p) E[( p) 3 ] = O(δt) 2 Backwards in time MRCA Coalesent time Wright Fisher time dp t = p t (1 p t )dw t New York, Sept p.3

9 Forwards in time Two types a and A. p(t) = proportion of type a. Forwards in time, E[ p] = 0 (neutrality) E[( p) 2 ] = δtp(1 p) E[( p) 3 ] = O(δt) 2 Backwards in time MRCA Coalesent time Wright Fisher time dp t = p t (1 p t )dw t 1 dp τ = N p τ(1 p τ )dw τ, Coalescence rate 1 N ( ) k 2 New York, Sept p.3

10 Basic observation Genetic diversity is orders of magnitude lower than expected from census numbers and genetic drift. Something else is going on... New York, Sept p.4

11 Genetic hitchhiking When a selectively advantageous allele arises, it is either lost or it sweeps to fixation What about a neutral allele on the same chromosome? New York, Sept p.5

12 Genetic hitchhiking When a selectively advantageous allele arises, it is either lost or it sweeps to fixation What about a neutral allele on the same chromosome? A D A C B D MEIOSIS B C Selected locus Neutral Locus Germ cell Gametes New York, Sept p.5

13 ... genetic draft Gillespie (2000) The frequency of the neutral allele linked to the new mutation will be boosted 1 selected frequency neutral u 1/N time New York, Sept p.6

14 ... genetic draft Gillespie (2000) The frequency of the neutral allele linked to the new mutation will be boosted Probability p, p u + p(1 u) Probability 1 p, p p(1 u) 1/N 1 frequency selected neutral u time New York, Sept p.6

15 ... genetic draft Gillespie (2000) The frequency of the neutral allele linked to the new mutation will be boosted Probability p, p u + p(1 u) Probability 1 p, p p(1 u) 1/N 1 frequency selected neutral u time E[ p] = 0, E[( p) 2 ] = p(1 p)e[u 2 ] New York, Sept p.6

16 ... genetic draft Gillespie (2000) The frequency of the neutral allele linked to the new mutation will be boosted Probability p, p u + p(1 u) Probability 1 p, p p(1 u) 1/N 1 frequency selected neutral u time E[ p] = 0, E[( p) 2 ] = p(1 p)e[u 2 ] E[( p) 3 ] = O(1) = multiple coalescences New York, Sept p.6

17 Λ-coalescents Pitman (1999), Sagitov (1999) If there are currently p ancestral lineages, each transition involving j of them merging happens at rate β p,j = 1 Λ a finite measure on [0, 1] Kingman s coalescent, Λ = δ 0 0 u j 2 (1 u) p j Λ(du) New York, Sept p.7

18 Forwards in time Bertoin & Le Gall (2003) The Λ-coalescent describes the genealogy of a sample from a population evolving according to a Λ-Fleming-Viot process. Poisson point process intensity dt u 2 Λ(du) individual sampled at random from population proportion u of population replaced by offspring of chosen individual New York, Sept p.8

19 Spatial structure Kimura s stepping stone model dp i = j 1 m ij (p j p i )dt+ p i (1 p i )dw i N e System of interacting Wright-Fisher diffusions New York, Sept p.9

20 Spatial structure Kimura s stepping stone model dp i = j 1 m ij (p j p i )dt+ p i (1 p i )dw i N e System of interacting Wright-Fisher diffusions Genealogy described by system of coalescing random walks New York, Sept p.9

21 Evolution in a spatial continuum? For many biological populations it is more natural to consider a spatial continuum Can we replace the stepping stone model by a stochastic pde? New York, Sept p.10

22 Evolution in a spatial continuum? For many biological populations it is more natural to consider a spatial continuum Can we replace the stepping stone model by a stochastic pde? dp = 1 2 pdt + p(1 p)dw W a space-time white noise New York, Sept p.10

23 Evolution in a spatial continuum? For many biological populations it is more natural to consider a spatial continuum Can we replace the stepping stone model by a stochastic pde? dp = 1 2 pdt + p(1 p)dw W a space-time white noise In two dimensions the equation has no solution Diffusive rescaling leads to the heat equation But anyway local populations are finite New York, Sept p.10

24 Another basic observation Real populations experience large scale fluctuations in which the movement and reproductive success of many individuals are correlated New York, Sept p.11

25 An individual based model Start with Poisson point process intensity λdx New York, Sept p.12

26 An individual based model Start with Poisson point process intensity λdx At rate µ(dr) dx dt throw down ball centre x, radius r. New York, Sept p.12

27 An individual based model Start with Poisson point process intensity λdx At rate µ(dr) dx dt throw down ball centre x, radius r. Each individual in region dies with probability u New York, Sept p.12

28 An individual based model Start with Poisson point process intensity λdx At rate µ(dr) dx dt throw down ball centre x, radius r. Each individual in region dies with probability u New individuals born according to a Poisson λudx New York, Sept p.12

29 An individual based model Start with Poisson point process intensity λdx At rate µ(dr) dx dt throw down ball centre x, radius r. Each individual in region dies with probability u New individuals born according to a Poisson λudx New York, Sept p.12

30 An individual based model Start with Poisson point process intensity λdx At rate µ(dr) dx dt throw down ball centre x, radius r. Each individual in region dies with probability u x r New individuals born according to a Poisson λudx New York, Sept p.12

31 An individual based model Start with Poisson point process intensity λdx At rate µ(dr) dx dt throw down ball centre x, radius r. Each individual in region dies with probability u x r New individuals born according to a Poisson λudx New York, Sept p.12

32 An individual based model Start with Poisson point process intensity λdx At rate µ(dr) dx dt throw down ball centre x, radius r. Each individual in region dies with probability u x r New individuals born according to a Poisson λudx New York, Sept p.12

33 Two limits Let the local population density λ New York, Sept p.13

34 Two limits Let the local population density λ Rescale space and time to investigate large scale effects New York, Sept p.13

35 The spatial Λ-Fleming-Viot process State {ρ(t, x, ) M 1 (K), x R 2, t 0}. New York, Sept p.14

36 The spatial Λ-Fleming-Viot process State {ρ(t, x, ) M 1 (K), x R 2, t 0}. π Poisson point process rate µ(dr) dx dt. New York, Sept p.14

37 The spatial Λ-Fleming-Viot process State {ρ(t, x, ) M 1 (K), x R 2, t 0}. π Poisson point process rate µ(dr) dx dt. For each r > 0, ν r (du) M 1 ([0, 1]). New York, Sept p.14

38 The spatial Λ-Fleming-Viot process State {ρ(t, x, ) M 1 (K), x R 2, t 0}. π Poisson point process rate µ(dr) dx dt. For each r > 0, ν r (du) M 1 ([0, 1]). Dynamics: for each (t, x, r) π, New York, Sept p.14

39 The spatial Λ-Fleming-Viot process State {ρ(t, x, ) M 1 (K), x R 2, t 0}. π Poisson point process rate µ(dr) dx dt. For each r > 0, ν r (du) M 1 ([0, 1]). Dynamics: for each (t, x, r) π, u ν r (du) New York, Sept p.14

40 The spatial Λ-Fleming-Viot process State {ρ(t, x, ) M 1 (K), x R 2, t 0}. π Poisson point process rate µ(dr) dx dt. For each r > 0, ν r (du) M 1 ([0, 1]). Dynamics: for each (t, x, r) π, u ν r (du) z U(B r (x)) New York, Sept p.14

41 The spatial Λ-Fleming-Viot process State {ρ(t, x, ) M 1 (K), x R 2, t 0}. π Poisson point process rate µ(dr) dx dt. For each r > 0, ν r (du) M 1 ([0, 1]). Dynamics: for each (t, x, r) π, u ν r (du) z U(B r (x)) k ρ(t, z, ). New York, Sept p.14

42 The spatial Λ-Fleming-Viot process State {ρ(t, x, ) M 1 (K), x R 2, t 0}. π Poisson point process rate µ(dr) dx dt. For each r > 0, ν r (du) M 1 ([0, 1]). Dynamics: for each (t, x, r) π, u ν r (du) z U(B r (x)) k ρ(t, z, ). For all y B r (x), ρ(t, y, ) = (1 u)ρ(t, y, ) + uδ k. New York, Sept p.14

43 Conditions (1) ρ(t, x, ) experiences jump of size u A [0, 1] at rate πr 2 ν r (u)µ(dr). (0, ] A New York, Sept p.15

44 Conditions (1) ρ(t, x, ) experiences jump of size u A [0, 1] at rate πr 2 ν r (u)µ(dr). (0, ] A Λ(du) = (0, ) u 2 r 2 ν r (du)µ(dr) M F ([0, 1]) New York, Sept p.15

45 Conditions (1) ρ(t, x, ) experiences jump of size u A [0, 1] at rate πr 2 ν r (u)µ(dr). (0, ] A Λ(du) = (0, ) u 2 r 2 ν r (du)µ(dr) M F ([0, 1]) A single ancestral lineage evolves in series of jumps with intensity dt ( x /2, ) [0,1] L r (x) πr 2 on R + R 2 where L r (x) = B r (0) B r (x). u ν r (du)µ(dr)dx New York, Sept p.15

46 Conditions (2) ( (1 x 2 ) R 2 ( x /2, ) [0,1] L r (x) πr 2 u ν r (du)µ(dr) ) dx <. New York, Sept p.16

47 Conditions (2) ( (1 x 2 ) R 2 ( x /2, ) [0,1] L r (x) πr 2 u ν r (du)µ(dr) ) dx <. Two lineages currently at separation y R 2 coalesce at instantaneous rate ( ) L r (y) u 2 ν r (du) µ(dr). ( y /2, ) [0,1] New York, Sept p.16

48 Generalisations Replace R 2 by an arbitrary Polish space New York, Sept p.17

49 Generalisations Replace R 2 by an arbitrary Polish space Choose a Poisson number of parents at each reproduction event New York, Sept p.17

50 Generalisations Replace R 2 by an arbitrary Polish space Choose a Poisson number of parents at each reproduction event Choose spatial position of parents non-uniformly. New York, Sept p.17

51 Generalisations Replace R 2 by an arbitrary Polish space Choose a Poisson number of parents at each reproduction event Choose spatial position of parents non-uniformly. Impose spatial motion of individuals not linked directly to the reproduction events. New York, Sept p.17

52 Generalisations Replace R 2 by an arbitrary Polish space Choose a Poisson number of parents at each reproduction event Choose spatial position of parents non-uniformly. Impose spatial motion of individuals not linked directly to the reproduction events. Instead of replacing a portion u of individuals from a ball centred on x, replace individuals sampled according to some distribution (e.g. Gaussian) centred on x. New York, Sept p.17

53 Generalisations Replace R 2 by an arbitrary Polish space Choose a Poisson number of parents at each reproduction event Choose spatial position of parents non-uniformly. Impose spatial motion of individuals not linked directly to the reproduction events. Instead of replacing a portion u of individuals from a ball centred on x, replace individuals sampled according to some distribution (e.g. Gaussian) centred on x.... and many more. New York, Sept p.17