GENE FLOW ACCROSS A GEOGRAPHICAL BARRIER Raphal Forien Les - PowerPoint PPT Presentation

GENE FLOW ACCROSS A GEOGRAPHICAL BARRIER Raphaël Forien Les Probabilités de demain CMAP - École Polytechnique IHES - 11 mai 2017

GEOGRAPHICAL BARRIERS TO DISPERSAL ÉCOLE POLYTECHNIQUE – Gene flow accross a geographical barrier 2

GEOGRAPHICAL BARRIERS TO DISPERSAL ÉCOLE POLYTECHNIQUE – Gene flow accross a geographical barrier 2

GEOGRAPHICAL BARRIERS TO DISPERSAL ÉCOLE POLYTECHNIQUE – Gene flow accross a geographical barrier 2

GEOGRAPHICAL BARRIERS TO DISPERSAL ÉCOLE POLYTECHNIQUE – Gene flow accross a geographical barrier 2

STEPPING STONE MODEL OF DISPERSAL Figure: Stepping stone model with a barrier from Nagylaki 1976 At each generation, - the N individuals in each colony are replaced by new individuals - a proportion 1 − m of them are the offspring of (uniformly chosen) parents in the same colony, - a proportion m are the offspring of parents in neighbouring colonies ÉCOLE POLYTECHNIQUE – Gene flow accross a geographical barrier 3

STEPPING STONE MODEL OF DISPERSAL Figure: Stepping stone model with a barrier from Nagylaki 1976 At each generation, - the N individuals in ± ε are replaced by new individuals - a proportion 1 − 1+ c 2 m of them are the offspring of (uniformly chosen) parents in the same colony, - a proportion 1 2 cm are the offspring of parents in ∓ ε and a proportion 1 2 m come from colony ± 3 ε ÉCOLE POLYTECHNIQUE – Gene flow accross a geographical barrier 3

EVOLUTION OF ALLELE FREQUENCIES Individuals are of two types, 0 and 1. Parental type is inherited by the offspring. 1.00 frequency 0.75 0.50 Initial frequency 0.25 Frequency at time t=12 0.00 −10 0 10 Space Figure: Evolution of allele frequencies with a barrier ξ t : position of the ancestor of a (uniformly) sampled individual t generations in the past = random walk on Z with transition probabilities given by the migration matrix of the stepping stone model. p ( t, x ) = P x ( ξ t ∈ [0 , ∞ )) ÉCOLE POLYTECHNIQUE – Gene flow accross a geographical barrier 4

APPLICATIONS Goal : detect barriers to gene flow using genetic data by estimating the age of the most recent common ancestor for different pairs of individuals. ξ t : random walk not convenient, no explicit formulas for the law of ξ t . duality Stepping stone Random walk model on Z continuous approximation Partial di erential ?? equation ÉCOLE POLYTECHNIQUE – Gene flow accross a geographical barrier 5

MAIN RESULT For a sequence ( c n ) n ∈ N , let ( ξ n ( t )) t ≥ 0 be a random walk on Z with the 1 corresponding transition probabilities. Set X n ( t ) = √ n ξ n ( nt ) Theorem 1 Suppose √ nc n − n →∞ 2 γ ∈ [0 , + ∞ ] , then → sko X n n →∞ X. − → The process ( X ( t )) t ≥ 0 is (the projection on R of) a Markov process on ( −∞ , 0 − ] ∪ [0 + , + ∞ ) . When γ ∈ (0 , ∞ ) , we call X partially reflected Brownian motion. ÉCOLE POLYTECHNIQUE – Gene flow accross a geographical barrier 6

CONSTRUCTION OF PARTIALLY REFLECTED BM 1/2 Start from standard Brownian motion and keep only the excursions outside of [ − 1 2 γ , 1 2 γ ] . 25 0 position −25 −50 −75 0.0 2.5 5.0 7.5 10.0 time Figure: Speed and scale construction of partially reflected Brownian motion ÉCOLE POLYTECHNIQUE – Gene flow accross a geographical barrier 7

CONSTRUCTION OF PARTIALLY REFLECTED BM 1/2 Start from standard Brownian motion and keep only the excursions outside of [ − 1 2 γ , 1 2 γ ] . 25 1 γ 0 position −25 −50 −75 0.0 2.5 5.0 7.5 10.0 time Figure: Speed and scale construction of partially reflected Brownian motion ÉCOLE POLYTECHNIQUE – Gene flow accross a geographical barrier 7

CONSTRUCTION OF PARTIALLY REFLECTED BM 1/2 Start from standard Brownian motion and keep only the excursions outside of [ − 1 2 γ , 1 2 γ ] . 25 1 γ 0 position −25 −50 −75 0.0 2.5 5.0 7.5 10.0 time Figure: Speed and scale construction of partially reflected Brownian motion ÉCOLE POLYTECHNIQUE – Gene flow accross a geographical barrier 7

CONSTRUCTION OF PARTIALLY REFLECTED BM 1/2 Start from standard Brownian motion and keep only the excursions outside of [ − 1 2 γ , 1 2 γ ] . 25 1 γ 0 position −25 −50 −75 0.0 2.5 5.0 7.5 10.0 time Figure: Speed and scale construction of partially reflected Brownian motion ÉCOLE POLYTECHNIQUE – Gene flow accross a geographical barrier 7

CONSTRUCTION OF PARTIALLY REFLECTED BM 1/2 Start from standard Brownian motion and keep only the excursions outside of [ − 1 2 γ , 1 2 γ ] . 25 0 position −25 −50 −75 0.0 2.5 5.0 7.5 10.0 time Figure: Speed and scale construction of partially reflected Brownian motion ÉCOLE POLYTECHNIQUE – Gene flow accross a geographical barrier 7

CONSTRUCTION OF PARTIALLY REFLECTED BM 1/2 Start from standard Brownian motion and keep only the excursions outside of [ − 1 2 γ , 1 2 γ ] . 25 0 position −25 −50 −75 0.0 2.5 5.0 7.5 10.0 time Figure: Speed and scale construction of partially reflected Brownian motion ÉCOLE POLYTECHNIQUE – Gene flow accross a geographical barrier 7

CONSTRUCTION OF PARTIALLY REFLECTED BM 2/2 Start from reflected Brownian motion ( W t ) t ≥ 0 , and flip it when its local time at 0 reaches an exponential variable. 50 40 1 W(t) 30 20 10 0 40 20 2 X(t) 0 −20 −40 60 40 3 L(t) 20 0 0 1000 2000 3000 4000 5000 time ÉCOLE POLYTECHNIQUE – Gene flow accross a geographical barrier 8

SKETCH OF PROOF FOR THE CONVERGENCE RESULT ξ t random walk on Z with transition probabilities 1 X n ( t ) = √ n ξ ( nt ) √ n × number of visits of X n to {± 1 1 L n ( t ) : √ n } up to time t T n i : time of the i -th crossing of {± 1 √ n } by X n Proof of Theorem 1. 1. | X n | converges to reflected Brownian motion as n → ∞ , 2. { L n ( T n i +1 ) − L n ( T n i ) , i ≥ 0 } converges to an iid sequence of E (2 γ ) , 3. the two are asymptotically independent. ÉCOLE POLYTECHNIQUE – Gene flow accross a geographical barrier 9

TRANSITION DENSITIES We have an explicit formula for the transition densities of ( X t ) t ≥ 0 . 0.04 Transition density/probability Partially reflected Brownian motion 0.03 Random walk 0.02 0.01 0.00 −20 0 20 Space Figure: Comparison of transition probabilities for the random walk and transition densities for partially reflected Brownian motion ÉCOLE POLYTECHNIQUE – Gene flow accross a geographical barrier 10

SUMMING UP duality Stepping stone Random walk model on Z continuous approximation Partial di erential Partially re ected equation Brownian motion 25 1.00 frequency 0.75 0 position 0.50 − 25 0.25 Initial frequency Frequency at time t=12 − 50 0.00 − 10 0 10 − 75 Space 0.0 2.5 5.0 7.5 10.0 time ÉCOLE POLYTECHNIQUE – Gene flow accross a geographical barrier 11

SUMMING UP duality Stepping stone Random walk model on Z continuous approximation Partial di erential Partially re ected equation Brownian motion 25 1.00 frequency 0.75 0 position 0.50 − 25 0.25 Initial frequency Frequency at time t=12 − 50 0.00 − 10 0 10 − 75 Space 0.0 2.5 5.0 7.5 10.0 time � X t ∈ [0 + , + ∞ ) � p ( t, x ) = P x ÉCOLE POLYTECHNIQUE – Gene flow accross a geographical barrier 11

Thank you for your attention ! ÉCOLE POLYTECHNIQUE – Gene flow accross a geographical barrier 12