Genetic Variability in the Mendelian Diploid Model

Genetic Variability in the Mendelian Diploid Model

Loren Coquille (Institut Fourier, Grenoble)

joint work in progress with A. Bovier and R. Neukirch (University of Bonn)

Journ´ee de la chaire MMB, 13 avril 2016

Outline

1 Introduction

2 Clonal reproduction model Trait substitution sequence

3 Mendelian diploid model Genetic variability

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Outline

1 Introduction

2 Clonal reproduction model

3 Mendelian diploid model

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Clonal Reproduction versus Sexual Reproduction

Clonal Reproduction

individual reproduces out of itself

offspring = copy of the parent

offspring genom:

I 100% of genes of the parent

vs. Sexual Reproduction

individual needs a partner for reproduction

offspring genom:

I 50% genes of mother

I 50% genes of father

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Clonal Reproduction versus Sexual Reproduction

Clonal Reproduction

+ fast growing populations + no time loss for partner

selection

- genes stay constant over generations

- hard adaptation to changing environment

Sexual Reproduction

- slowly growing population - time cost for partner selection + genetic variability

+ possible adaptation to changing environment + correction of genetic defects + elimination of disadvantageous

mutations

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Outline

1 Introduction

2 Clonal reproduction model Trait substitution sequence

3 Mendelian diploid model

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Clonal reproduction model

Bolker, Pacala, Dieckmann, Law, Champagnat, M´el´eard,...

Θ'N: Trait space

n_i(t): Number of individuals of trait i∈Θ

Dynamics of the process

n(t) = (n0(t), n1(t), . . .)∈N^N : Each individual of traiti

reproduces clonally with rate bi(1−µ)

reproduces with mutation with ratebiµaccording to some mutation kernel M(i, j)

dies due to age or competition with rate di+P

jcijnj

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Clonal reproduction model

Bolker, Pacala, Dieckmann, Law, Champagnat, M´el´eard,...

Θ'N: Trait space

n_i(t): Number of individuals of trait i∈Θ

Dynamics of the process

n(t) = (n0(t), n1(t), . . .)∈N^N : The populationni

increases by 1 with ratebi(1−µ)ni

makesnj increase by 1 with ratebiµ·M(i, j)·ni

deceases by 1 with rate(d_i+P

jc_ijn_j)n_i

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Clonal reproduction model

Bolker, Pacala, Dieckmann, Law, Champagnat, M´el´eard,...

Θ'N: Trait space

ni(t): Number of individuals of trait i∈Θ

Dynamics of the rescaledprocess (with competitioncij/K)

n^K(t) = 1

K(n0(t), n1(t), . . .)∈(N/K)^N : The populationn^K_i

increases by 1/K with rateb_i(1−µ)·Kn^K_i

makesn^K_j increase by1/K with rateb_iµM(i, j)·Kn^K_i deceases by 1/K with rate(d_i+P

jc_ijn^K_j )·Kn^K_i

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Large populations limit

µ = 0 and K → ∞

Proposition

Let Θ ={1,2}, assume that the initial condition (n^K₁ (0), n^K₂ (0))

converges to a deterministic vector (x₀, y₀) for K → ∞. Then the process (n^K₁ (t), n^K₂ (t))converges in law, on bounded time intervals, to the

solution of d dt

x(t) y(t)

=X(x(t), y(t)) =

(b1−d1−c11x−c12y)x (b2−d2−c21x−c22y)y

. with initial condition (x(0), y(0)) = (x₀, y₀).

Monomorphic equilibria : n¯_i= ^bi_c^−di Invasion fitness : f_ij =b_i−d_i−cii_ij¯n_j.

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Large populations and rare mutations limit

µ = µ(K) (K log K)

Start with n^K_i (0) = ¯n_x1_i=x+_K¹1_i=y (time of the first mutation).

Ify is fitter than x, i.e. fyx>0 andfxy <0, then with proba→1:

O(logK) O(1) O(logK)

Supercritical BP LLN Subcritical BP 9 / 29

Trait Substitution Sequence

Proposition Assume

logK _Kµ¹ exp(cK)

∀(i, j)∈Θ², either fji <0 orfji >0 andfij <0.

Then the rescaled process

(n^K_t/Kµ)_t≥0 ⇒(¯nXt1Xt)_t≥0 where X_t is a Markov chain onΘ with transition kernel

P(i, j) =b_i[f_ji]₊

b_j M(i, j).

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Outline

1 Introduction

2 Clonal reproduction model

3 Mendelian diploid model Genetic variability

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Mendelian Diploid Model

Collet, M´el´eard, Metz, 2013

Let U be the allelic trait space, a countable set.

For example U ={a, A}.

An individualiis determined by two alleles out ofU :

I genotype: (uⁱ₁, uⁱ₂)∈ U²

I phenotype: φ((uⁱ₁, uⁱ₂)), withφ:U²→R⁺

Rescaled population:

let N_t be the total number of individuals at timet,

n^K_u1,u2(t) = 1 K

Nt

X

i=1

1_(uⁱ

1,uⁱ₂)(t)

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Reproduction

Reproduction rate of (uⁱ₁, uⁱ₂) with (u^j₁, u^j₂):

f_uⁱ

1uⁱ₂f_u^j

1u^j₂

Number of potential partners of(uⁱ₁, uⁱ₂)× their mean fertility

Reproduction without muta- tion: Probability1−µ_K

⇒ Mendelian rules: newborn get- ting genotype with coordinates that are sampled at random from each parent.

Ab aA bB

ab aB AB

Reproduction with mutation: Probability µ_K

⇒ changing one of the two allelic traits of the newborn from a to b according to the kernel M(a, b).

Aĉ aA cC

aĉ aĈ AĈ

âc Âc âC ÂC

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Reproduction

Reproduction rate of (uⁱ₁, uⁱ₂) with (u^j₁, u^j₂):

f_uⁱ

1uⁱ₂f_u^j

1u^j₂

Number of potential partners of(uⁱ₁, uⁱ₂)× their mean fertility

Reproduction without muta- tion: Probability1−µ_K

⇒ Mendelian rules: newborn get- ting genotype with coordinates that are sampled at random from each parent.

Ab aA bB

ab aB AB

Reproduction with mutation: Probability µ_K

⇒ changing one of the two allelic traits of the newborn from a to b according to the kernel M(a, b).

Aĉ aA cC

aĉ aĈ AĈ

âc Âc âC ÂC

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Reproduction

Reproduction rate of (uⁱ₁, uⁱ₂) with (u^j₁, u^j₂):

f_uⁱ

1uⁱ₂f_u^j

1u^j₂

Number of potential partners of(uⁱ₁, uⁱ₂)× their mean fertility

Reproduction without muta- tion: Probability1−µ_K

⇒ Mendelian rules: newborn get- ting genotype with coordinates that are sampled at random from each parent.

Ab aA bB

ab aB AB

Reproduction with mutation:

Probability µ_K

⇒ changing one of the two allelic traits of the newborn from a to b according to the kernel M(a, b).

Aĉ aA cC

aĉ aĈ AĈ

âc Âc âC ÂC

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Birth and Death Rate

Let U ={a, A}, andµ_K = 0.

The populations increase by 1 with rate

baa = ^{(faanaa+12faAnaA)}

2

f_aan_aa+f_aAn_aA+f_AAn_AA

b_aA = 2 ^(faanaa_f^{+12faAnaA)(fAAnAA+12faAnaA)}

aanaa+faAnaA+fAAnAA

bAA = ^{(fAAnAA+12faAnaA)}

2

f_aan_aa+f_aAn_aA+f_AAn_AA

The populations decrease by 1 with rate

daa = (Daa+Caa,aanaa+Caa,aAnaA+Caa,AAnAA)naa

d_aA = (D_aA+C_aA,aan_aa+C_aA,aAn_aA+C_aA,AAn_AA)n_aA dAA = (DAA+CAA,AAnAA+CAA,aAnaA+CAA,aanaa)nAA

aa

aa aa aa aA aA aa aA aA

1

1/4 1/2

1/2

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Birth and Death Rate

Let U ={a, A}, andµ_K = 0.

The populations increase by 1 with rate

baa = ^{(faanaa+12faAnaA)}

2

f_aan_aa+f_aAn_aA+f_AAn_AA

b_aA = 2 ^(faanaa_f^{+12faAnaA)(fAAnAA+12faAnaA)}

aanaa+faAnaA+fAAnAA

bAA = ^{(fAAnAA+12faAnaA)}

2

f_aan_aa+f_aAn_aA+f_AAn_AA

The populations decrease by 1 with rate

daa = (Daa+Caa,aanaa+Caa,aAnaA+Caa,AAnAA)naa

d_aA = (D_aA+C_aA,aan_aa+C_aA,aAn_aA+C_aA,AAn_AA)n_aA dAA = (DAA+CAA,AAnAA+CAA,aAnaA+CAA,aanaa)nAA

aa

aa aa aa aA aA aa aA aA

1

1/4 1/2

1/2

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Large populations limit

Proposition

Assume that the initial condition (n^K_aa(0), n^K_aA(0), n^K_AA(0)) converges to a deterministic vector (x0, y0, z0) for K→ ∞. Then the process

(n^K_aa(t), n^K_aA(t), n^K_AA(t))converges in law to the solution of

d dt

x(t) y(t) z(t)

!

=X(x(t), y(t), z(t)) =

b_aa(x, y, z)−d_aa(x, y, z) baA(x, y, z)−daA(x, y, z) bAA(x, y, z)−dAA(x, y, z)

! .

b_AA(x, y, z) = ^(fAAz+

1 2faAy)² faax+faAy+fAAz

d_AA(x, y, z) = (D_AA+C_AA,aax+C_AA,aAy+C_AA,AAz)z and similar expressions for the other terms

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The 3-System (aa, aA, AA)

Phenotypic viewpoint allele A dominant allele a recessive

The dominant allele A defines the phenotype⇒ φ(aA) =φ(AA)

cu1u2,v1v2 ≡c, ∀u₁u2, v1v2 ∈ {aa, aA, AA} f_AA=f_aA=f_aa≡f

D_AA=D_aA≡DbutD_aa =D+∆

⇒ type aA is as fit asAA and both are fitter than type aa

AA aA

aa

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The 3-System (aa, aA, AA)

Phenotypic viewpoint allele A dominant allele a recessive

The dominant allele A defines the phenotype⇒ φ(aA) =φ(AA)

cu1u2,v1v2 ≡c, ∀u₁u2, v1v2 ∈ {aa, aA, AA}

fAA=faA=faa≡f

D_AA=D_aA≡DbutD_aa =D+∆

⇒ type aA is as fit asAA and both are fitter than type aa

AA aA

aa

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The 3-System (aa, aA, AA)

Phenotypic viewpoint allele A dominant allele a recessive

The dominant allele A defines the phenotype⇒ φ(aA) =φ(AA)

cu1u2,v1v2 ≡c, ∀u₁u2, v1v2 ∈ {aa, aA, AA}

fAA=faA=faa≡f

D_AA=D_aA≡DbutD_aa =D+∆

⇒ type aA is as fit asAA and both are fitter than type aa

AA aA

aa

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Work of Bovier, Neukirch (2015)

Start with n^K_i (t) = ¯n_aa1_i=aa+_K¹1_i=aA then :

O(logK) O(1) ≥O(K^1/4+)

The system converges to(0,0,n¯_AA) ast→ ∞ butslowly!

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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 ?

We will study the deterministic sytem and start with initial condition n_i(0) = ¯n_AA1_i=AA+1_i=aA+²1_i=aa+³1_i=AB

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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 allelesa < A < B Differences in Fitness:

fertility: fa=f_A=f_B=f

natural death: Da=D+ ∆> DA=D > DB=D−∆

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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 allelesa < A < B

Differences in Fitness:

fertility: fa=f_A=f_B=f

natural death: Da=D+ ∆> DA=D > DB=D−∆

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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 allelesa < A < B Differences in Fitness:

fertility: fa=f_A=f_B=f

natural death: Da=D+ ∆> DA=D > DB=D−∆

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Birth Rates

No recombination between a and B

BB a

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Birth Rates

birth-rate of aa-individual:

baa=naa naa+¹₂naA

Pool(aa) +

1 2naB 1

2naA+¹₂naB

Pool(aB)

+

1

2n_aA n_aa+¹₂n_aA+ ¹₂n_aB Pool(aA)

Pools of potential partners:

aa aA aB

aa aA aa aA aA

AA AA aB AB AA aB AB

BB BB

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Competition

No competition between a and B

BB a

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Competition - first try

aa aA AA aB AB BB

aa c c c 0 0 0

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

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Competition - first try

0 100 200 300 400 500 600 700

1 2 3 4 5

aaaA AAaBABBB _{24 / 29}

Competition - first try

100 200 300 400 500 600 700

-35 -30 -25 -20 -15 -10 -5

aaaA AAaBABBB

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Competition - second try

competition felt by naA fromB-individuals

< competition felt by nAA fromB-individuals

the decay ofaA-population slows down aa-population can recover

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Competition - second try

aa aA AA aB AB BB

aa c c c 0 0 0

aA c c c c c c−η

AA c c c c c c

aB 0 c c c c c

AB 0 c c c c c

BB 0 c−η c c c c

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Competition - second try

0 100 200 300 400 500 600 700

1 2 3 4 5

aaaA AAaBABBB _{27 / 29}

Competition - second try

100 200 300 400 500 600 700

-35 -30 -25 -20 -15 -10 -5

aaaA AAaBABBB

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Recap - Dimorphism in two mutations

Mutation 1 Mutation 2

0 100 200 300 400 500 600 700

1 2 3 4 5

0 100 200 300 400 500 600 700

1 2 3 4 5

100 200 300 400 500 600 700

-35 -30 -25 -20 -15 -10 -5

100 200 300 400 500 600 700

-35 -30 -25 -20 -15 -10 -5

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Proof

1.Phase: Fixation of the mutant

AB grows to levelε0

aB, BB≤ε0

⇒ perturbation of the 3-system(aa, aA, AA) of at mostO(ε₀)

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Proof

2.Phase: Invasion of the mutant

aa, aA, aB≤ε₀

⇒ perturbation of the new 3-system (AA, AB, BB) of at most O(ε₀)

⇒ use results of Bovier, Neukirch (2015) n_aA+n_aB increases ifη >0

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Proof

3. Phase: Recovery ofaa

AAsmall enough aAbig enough

⇒ aastarts to reproduce out of itself as much as with the other partners.

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Proof

4. Phase: Coexistence

Delicate phase : aagrows out of itself and feels no competition with BB

⇒ convergence to coexistence-fixed-point n¯_aa,BB BUT meanwhile :

due to Mendelian recombination,aA, aB, AB have a ”bump” upwards, and due to competition with them BB has a ”bump” downwards.

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T H A N K

Y O U