Recovery of a recessive allele in a Mendelian diploid model

Transcription

1 Recovery of a recessive allele in a Mendelian diploid model Loren Coquille joint work with A. Bovier and R. Neukirch (University of Bonn) Young Women in Probability and Analysis 2016, Bonn

2 Outline 1 Introduction 2 Clonal reproduction model 3 Mendelian diploid model 2 / 27

3 Outline 1 Introduction 2 Clonal reproduction model 3 Mendelian diploid model 3 / 27

4 Clonal Reproduction versus Sexual Reproduction Clonal Reproduction vs. Sexual Reproduction individual reproduces out of itself offspring = copy of the parent offspring genom: 100% of genes of the parent individual needs a partner for reproduction offspring genom: 50% genes of mother 50% genes of father 4 / 27

5 Outline 1 Introduction 2 Clonal reproduction model 3 Mendelian diploid model 5 / 27

6 Clonal reproduction model Bolker, Pacala, Dieckmann, Law, Champagnat, Méléard,... Θ N : Trait space n i (t) : Number of individuals of trait i Θ Dynamics of the process n(t) = (n 0 (t), n 1 (t),...) N N : Each individual of trait i reproduces clonally with rate b i (1 µ) reproduces with mutation with rate b i µ according to kernel M(i, j) dies due to age or competition with rate d i + j c ijn j 6 / 27

7 Clonal reproduction model Bolker, Pacala, Dieckmann, Law, Champagnat, Méléard,... Θ N : Trait space n i (t) : Number of individuals of trait i Θ Dynamics of the process n(t) = (n 0 (t), n 1 (t),...) N N : The population n i increases by 1 with rate b i (1 µ)n i makes n j increase by 1 with rate b i µ M(i, j) n i deceases by 1 with rate (d i + j c ijn j )n i 6 / 27

8 Clonal reproduction model Bolker, Pacala, Dieckmann, Law, Champagnat, Méléard,... Θ N : Trait space n i (t) : Number of individuals of trait i Θ Dynamics of the rescaled process (with competition c ij /K) n K (t) = 1 K (n 0(t), n 1 (t),...) (N/K) N : The population n K i increases by 1/K with rate b i (1 µ) Kn K i makes n K j increase by 1/K with rate b i µm(i, j) Kn K i deceases by 1/K with rate (d i + j c ijn K j ) KnK i 6 / 27

9 Large population and NO mutation K Fournier, Méléard, 2004 Proposition Let Θ = {1, 2}, assume that the initial condition (n K 1 (0), nk 2 (0)) converges to a deterministic vector (x 0, y 0 ) for K. Then the process (n K 1 (t), nk 2 (t)) converges in law, on bounded time intervals, to the solution of ( ) ( ) d x(t) (b1 d = X(x(t), y(t)) = 1 c 11 x c 12 y)x. dt y(t) (b 2 d 2 c 21 x c 22 y)y with initial condition (x(0), y(0)) = (x 0, y 0 ). Monomorphic equilibria : n i = b i d i c ii Invasion fitness : f ij = b i d i c ij n j. 7 / 27

10 Large population and ONE mutation Champagnat, 2006 Start with n K i (0) = n x1 i=x + 1 K 1 i=y (time of the first mutation). If y is fitter than x, i.e. f yx > 0 and f xy < 0, then with proba 1: O(log K) O(1) O(log K) Supercritical BP LLN Subcritical BP 8 / 27

11 Large population and RARE mutations Champagnat, 2006 Proposition (Trait Substitution Sequence) Assume log K 1 Kµ (i, j) Θ 2, either (f ji < 0) or (f ji > 0 and f ij < 0). Then the rescaled process (n K t/kµ ) t 0 ( n X t 1 Xt ) t 0 where the TSS X t is a Markov chain on Θ with transition kernel [f ji ] + P (i, j) = b i M(i, j). b j Jumps towards higher fitness 9 / 27

12 Outline 1 Introduction 2 Clonal reproduction model 3 Mendelian diploid model 10 / 27

13 Mendelian Diploid Model Collet, Méléard, Metz, 2013 Let U be the allelic trait space, a countable set. For example U = {a, A}. An individual i is determined by two alleles out of U : genotype: (u i 1, u i 2) U 2 phenotype: φ((u i 1, u i 2)), with φ : U 2 R + Rescaled population: let N t be the total number of individuals at time t, n K u 1,u 2 (t) = 1 K N t i=1 1 (u i 1,u i 2 )(t) 11 / 27

14 Reproduction Reproduction rate of (u i 1, ui 2 ) with (uj 1, uj 2 ): f u i 1 u i f 2 u j 1 uj 2 Number of potential partners of (u i 1, ui 2 ) their mean fertility which means, for one individual (u i 1, ui 2 ) : Exp(f u i 1 u i )-Clock rings, 2 choose a partner at random, reproduce (proba according to fertility of the partner) 12 / 27

15 Birth and Death Rate For U = {a, A} The populations increase by 1 with rate 1 aa aa b aa = (faanaa+ 1 2 f aan aa ) 2 f aan aa+f aa n aa +f AA n AA b aa = 2 (faanaa+ 1 2 f aan aa )(f AA n AA f aan aa ) f aan aa+f aa n aa +f AA n AA aa 1/2 1/2 aa aa aa aa b AA = (f AAn AA f aan aa ) 2 f aan aa+f aa n aa +f AA n AA 1/4 aa aa 13 / 27

16 Birth and Death Rate For U = {a, A} The populations increase by 1 with rate 1 aa aa b aa = (faanaa+ 1 2 f aan aa ) 2 f aan aa+f aa n aa +f AA n AA b aa = 2 (faanaa+ 1 2 f aan aa )(f AA n AA f aan aa ) f aan aa+f aa n aa +f AA n AA aa 1/2 1/2 aa aa aa aa b AA = (f AAn AA f aan aa ) 2 f aan aa+f aa n aa +f AA n AA 1/4 aa aa The populations decrease by 1 with rate d aa d aa d AA = (D aa + C aa,aa n aa + C aa,aa n aa + C aa,aa n AA )n aa = (D aa + C aa,aa n aa + C aa,aa n aa + C aa,aa n AA )n aa = (D AA + C AA,AA n AA + C AA,aA n aa + C AA,aa n aa )n AA 13 / 27

17 Large populations limit Collet, Méléard, Metz, 2013 Proposition Assume that the initial condition (n K aa(0), n K aa (0), nk AA (0)) converges to a deterministic vector (x 0, y 0, z 0 ) for K. Then the process (n K aa(t), n K aa (t), nk AA (t)) converges in law to the solution of ( ) ( ) d x(t) baa (x, y, z) d aa (x, y, z) y(t) = X(x(t), y(t), z(t)) = b aa (x, y, z) d aa (x, y, z). dt z(t) b AA (x, y, z) d AA (x, y, z) b AA (x, y, z) = (f AAz+ 1 2 f aay) 2 f aax+f aa y+f AA z d AA (x, y, z) = (D AA + C AA,aa x + C AA,aA y + C AA,AA z)z and similar expressions for the other terms 14 / 27

18 ONE mutation : 3-System (aa, aa, AA) Phenotypic viewpoint allele A dominant allele a recessive The dominant allele A defines the phenotype φ(aa) = φ(aa) 15 / 27

19 ONE mutation : 3-System (aa, aa, AA) Phenotypic viewpoint allele A dominant allele a recessive The dominant allele A defines the phenotype φ(aa) = φ(aa) c u1 u 2,v 1 v 2 c, u 1 u 2, v 1 v 2 {aa, aa, AA} f AA = f aa = f aa f D AA = D aa D but D aa = D + 15 / 27

20 ONE mutation : 3-System (aa, aa, AA) Phenotypic viewpoint allele A dominant allele a recessive The dominant allele A defines the phenotype φ(aa) = φ(aa) c u1 u 2,v 1 v 2 c, u 1 u 2, v 1 v 2 {aa, aa, AA} f AA = f aa = f aa f D AA = D aa D but D aa = D + type aa is as fit as AA and both are fitter than type aa AA aa aa 15 / 27

21 Work of Bovier, Neukirch (2015) Start with n K i (t) = n aa1 i=aa + 1 K 1 i=aa then : O(log K) O(1) O(K 1/4+ɛ ) The system converges to (0, 0, n AA ) as t but slowly! 16 / 27

22 Genetic Variability? Polymorphism? Suppose a new dominant mutant allele B appears before aa dies out. Suppose that phenotypes a and B cannot reproduce. Can the aa-population recover and coexist with the mutant population? 17 / 27

23 Model with a second mutant Mutation to allele B U = {a, A, B} 18 / 27

24 Model with a second mutant Mutation to allele B U = {a, A, B} aa aa AA ab AB BB 6 possible genotypes: aa, aa, AA, ab, AB, BB Dominance of alleles a < A < B 18 / 27

25 Model with a second mutant Mutation to allele B U = {a, A, B} aa aa AA ab AB BB 6 possible genotypes: aa, aa, AA, ab, AB, BB Dominance of alleles a < A < B Differences in Fitness: fertility: f a = f A = f B = f natural death: D a = D + > D A = D > D B = D 18 / 27

26 Birth Rates No recombination between a and B a B 19 / 27

27 Birth Rates birth-rate of aa-individual: b aa = n ( aa naa n aa) + Pool(aa) + Pools of potential partners: 1 2 n ( aa naa n aa n ab Pool(aA) 1 2 n ( 1 ab 2 n aa n ab) Pool(aB) aa aa ab ) aa aa aa AB aa AB aa ab ab AA AA AA BB BB 20 / 27

28 Competition No competition between a and B a B 21 / 27

29 Competition aa aa AA ab AB BB aa c c c aa c c c c c c AA c c c c c c ab 0 c c c c c AB 0 c c c c c BB 0 c c c c c 22 / 27

30 Result Theorem (Bovier, Neukirch, C. 2016/10/07 (yesterday!)) The deterministic system started with initial condition n i (0) = n AA 1 i=aa + ɛ1 i=aa + ɛ 2 1 i=aa + ɛ 3 1 i=ab and under the following assumptions on the parameters 1 sufficiently small, 2 f sufficiently large, converges polynomially fast to the fixed point ( n a, 0, 0, 0, 0, n B ). 23 / 27

31 Simulation aa aa AA ab AB BB 24 / 27

32 Large population : deterministic system aa aa AA ab AB BB 25 / 27

33 Large population : deterministic system aa aa AA ab AB BB 25 / 27

34 Recap - Dimorphism in two mutations Mutation 1 Mutation / 27

35 Proof Phase 1 : Exponential growth of AB at rate inside the 3-system (aa, aa, AA) Phase 2 : Effective new 3-system (AA, AB, BB) until AA aa Phase 3 : Exponential growth of aa at rate f D Phase 4 : Convergence to ( n a, 0, 0, 0, 0, n B ) 27 / 27

36 T H A N K Y O U