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

Raphaël Forien CMAP - École Polytechnique Les Probabilités de demain IHES - 11 mai 2017

GENE FLOW ACCROSS A GEOGRAPHICAL BARRIER

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

2cm are the offspring of parents in ∓ε and a

proportion 1

2m come from colony ±3ε

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

• ffspring.

0.00 0.25 0.50 0.75 1.00 −10 10

Space frequency

Initial frequency Frequency at time t=12

Figure: Evolution of allele frequencies with a barrier

ξt : 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) = Px (ξt ∈ [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. ξt : random walk not convenient, no explicit formulas for the law of ξt.

Stepping stone model Random walk

• n Z

Partial di erential equation ??

duality continuous approximation

ÉCOLE POLYTECHNIQUE – Gene flow accross a geographical barrier 5

MAIN RESULT

For a sequence (cn)n∈N, let (ξn(t))t≥0 be a random walk on Z with the corresponding transition probabilities. Set Xn(t) =

1 √nξn(nt)

Theorem 1

Suppose √ncn − →

n→∞ 2γ ∈ [0, +∞], then

Xn

sko

− →

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

• utside of [− 1

2γ , 1 2γ ].

−75 −50 −25 25 0.0 2.5 5.0 7.5 10.0

time position

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

• utside of [− 1

2γ , 1 2γ ]. 1 γ

−75 −50 −25 25 0.0 2.5 5.0 7.5 10.0

time position

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

• utside of [− 1

2γ , 1 2γ ]. 1 γ

−75 −50 −25 25 0.0 2.5 5.0 7.5 10.0

time position

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

• utside of [− 1

2γ , 1 2γ ]. 1 γ

−75 −50 −25 25 0.0 2.5 5.0 7.5 10.0

time position

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

• utside of [− 1

2γ , 1 2γ ].

−75 −50 −25 25 0.0 2.5 5.0 7.5 10.0

time position

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

• utside of [− 1

2γ , 1 2γ ].

−75 −50 −25 25 0.0 2.5 5.0 7.5 10.0

time position

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 (Wt)t≥0, and flip it when its local time at 0 reaches an exponential variable.

1 W(t) 2 X(t) 3 L(t) 1000 2000 3000 4000 5000 10 20 30 40 50 −40 −20 20 40 20 40 60

time

ÉCOLE POLYTECHNIQUE – Gene flow accross a geographical barrier 8

SKETCH OF PROOF FOR THE CONVERGENCE RESULT

ξt random walk on Z with transition probabilities Xn(t) =

1 √nξ(nt)

Ln(t) :

1 √n× number of visits of Xn to {± 1 √n} up to time t

T n

i : time of the i-th crossing of {± 1 √n} by Xn

Proof of Theorem 1.

• 1. |Xn| converges to reflected Brownian motion as n → ∞,
• 2. {Ln(T n

i+1) − Ln(T n i ), i ≥ 0} converges to an iid sequence of E(2γ),

• 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 (Xt)t≥0.

0.00 0.01 0.02 0.03 0.04 −20 20

Space Transition density/probability

Partially reflected Brownian motion Random walk

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

0.00 0.25 0.50 0.75 1.00 −10 10

Space frequency

Initial frequency Frequency at time t=12 −75 −50 −25 25 0.0 2.5 5.0 7.5 10.0 time position

Stepping stone model Random walk

• n Z

Partial di erential equation Partially re ected Brownian motion

duality continuous approximation

ÉCOLE POLYTECHNIQUE – Gene flow accross a geographical barrier 11

SUMMING UP

0.00 0.25 0.50 0.75 1.00 −10 10

Space frequency

Initial frequency Frequency at time t=12 −75 −50 −25 25 0.0 2.5 5.0 7.5 10.0 time position

Stepping stone model Random walk

• n Z

Partial di erential equation Partially re ected Brownian motion

duality continuous approximation

p(t, x) = Px

• Xt ∈ [0+, +∞)
• ÉCOLE POLYTECHNIQUE – Gene flow accross a geographical barrier

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Thank you for your attention !

ÉCOLE POLYTECHNIQUE – Gene flow accross a geographical barrier 12