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
Outline
1
Introduction
2
Clonal reproduction model
3
Mendelian diploid model
2 / 27
Outline
1
Introduction
2
Clonal reproduction model
3
Mendelian diploid model
3 / 27
Clonal Reproduction versus Sexual Reproduction
Clonal Reproduction
vs.
Sexual Reproduction
individual reproduces out of
itself
offspring = copy of the
parent
offspring genom:
I
100% of genes of the
parent
individual needs a partner for
reproduction
offspring genom:
I
I
50% genes of mother
50% genes of father
4 / 27
Outline
1
Introduction
2
Clonal reproduction model
3
Mendelian diploid model
5 / 27
Clonal reproduction model
Bolker, Pacala, Dieckmann, Law, Champagnat, Méléard,...
Θ ' N : Trait space
ni (t) : Number of individuals of trait i ∈ Θ
Dynamics of the process
n(t) = (n0 (t), n1 (t), . . .) ∈ NN
:
Each individual of trait i
reproduces clonally with rate bi (1 − µ)
reproduces with mutation with rate bi µ according to kernel M (i, j)
P
dies due to age or competition with rate di + j cij nj
6 / 27
Clonal reproduction model
Bolker, Pacala, Dieckmann, Law, Champagnat, Méléard,...
Θ ' N : Trait space
ni (t) : Number of individuals of trait i ∈ Θ
Dynamics of the process
n(t) = (n0 (t), n1 (t), . . .) ∈ NN
:
The population ni
increases by 1 with rate bi (1 − µ)ni
makes nj increase by 1 with rate bi µ · M (i, j) · ni
P
deceases by 1 with rate (di + j cij nj )ni
6 / 27
Clonal reproduction model
Bolker, Pacala, Dieckmann, Law, Champagnat, Méléard,...
Θ ' N : Trait space
ni (t) : Number of individuals of trait i ∈ Θ
Dynamics of the rescaled process (with competition cij /K)
nK (t) =
1
(n0 (t), n1 (t), . . .) ∈ (N/K)N
K
:
The population nK
i
increases by 1/K with rate bi (1 − µ) · KnK
i
K
makes nK
j increase by 1/K with rate bi µM (i, j) · Kni
P
K
deceases by 1/K with rate (di + j cij nK
j ) · Kni
6 / 27
Large population and NO mutation
K→∞
Fournier, Méléard, 2004
Proposition
K
Let Θ = {1, 2}, assume that the initial condition (nK
1 (0), n2 (0))
converges to a deterministic vector (x0 , y0 ) for K → ∞. Then the process
K
(nK
1 (t), n2 (t)) converges in law, on bounded time intervals, to the
solution of
d x(t)
(b1 − d1 − c11 x − c12 y)x
= X(x(t), y(t)) =
.
(b2 − d2 − c21 x − c22 y)y
dt y(t)
with initial condition (x(0), y(0)) = (x0 , y0 ).
i
Monomorphic equilibria : n̄i = bic−d
ii
Invasion fitness : fij = bi − di − cij n̄j .
7 / 27
Large population and ONE mutation
Champagnat, 2006
1
Start with nK
i (0) = n̄x 1i=x + K 1i=y (time of the first mutation).
If y is fitter than x, i.e. fyx > 0 and fxy < 0, then with proba → 1:
O(log K)
Supercritical BP
O(1)
LLN
O(log K)
Subcritical BP
8 / 27
Large population and RARE mutations
Champagnat, 2006
Proposition (Trait Substitution Sequence)
Assume
log K ∀(i, j) ∈
1
Kµ
Θ2 ,
either (fji < 0) or (fji > 0 and fij < 0).
Then the rescaled process
(nK
t/Kµ )
t≥0
⇒ (n̄Xt 1Xt )t≥0
where the TSS Xt is a Markov chain on Θ with transition kernel
P (i, j) = bi
[fji ]+
M (i, j).
bj
Jumps towards higher fitness
9 / 27
Outline
1
Introduction
2
Clonal reproduction model
3
Mendelian diploid model
10 / 27
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 :
I
I
genotype:
(ui1 , ui2 ) ∈ U 2
phenotype: φ((ui1 , ui2 )), with φ : U 2 → R+
Rescaled population:
let Nt be the total number of individuals at time t,
nK
u1 ,u2 (t)
Nt
1 X
=
1(ui ,ui ) (t)
1 2
K
i=1
11 / 27
Reproduction
Reproduction rate of (ui1 , ui2 ) with (uj1 , uj2 ):
fui ui fuj uj
1 2
1 2
Number of potential partners of (ui1 , ui2 ) × their mean fertility
which means, for one individual (ui1 , ui2 ) :
Exp(fui ui )-Clock rings,
1 2
choose a partner at random,
reproduce (proba according to fertility of the partner)
12 / 27
Birth and Death Rate
For U = {a, A}
The populations increase by 1 with rate
baa =
(faa naa + 12 faA naA )2
faa naa +faA naA +fAA nAA
baA = 2
bAA =
(faa naa + 12 faA naA )(fAA nAA + 12 faA naA )
faa naa +faA naA +fAA nAA
(fAA nAA + 21 faA naA )2
faa naa +faA naA +fAA nAA
aa
aa
aa
aA
aA
aa
aA
aA
1
1/2
aa
1/2
1/
4
13 / 27
Birth and Death Rate
For U = {a, A}
The populations increase by 1 with rate
baa =
(faa naa + 12 faA naA )2
faa naa +faA naA +fAA nAA
baA = 2
bAA =
(faa naa + 12 faA naA )(fAA nAA + 12 faA naA )
faa naa +faA naA +fAA nAA
aa
aa
aa
aA
aA
aa
aA
aA
1
1/2
aa
(fAA nAA + 21 faA naA )2
1/2
1/
4
faa naa +faA naA +fAA nAA
The populations decrease by 1 with rate
daa = (Daa + Caa,aa naa + Caa,aA naA + Caa,AA nAA )naa
daA = (DaA + CaA,aa naa + CaA,aA naA + CaA,AA nAA )naA
dAA = (DAA + CAA,AA nAA + CAA,aA naA + CAA,aa naa )nAA
13 / 27
Large populations limit
Collet, Méléard, Metz, 2013
Proposition
K
K
Assume that the initial condition (nK
aa (0), naA (0), nAA (0)) converges to a
deterministic vector (x0 , y0 , z0 ) for K → ∞. Then the process
K
K
(nK
aa (t), naA (t), nAA (t)) converges in law to the solution of
!
d x(t)
y(t) = X(x(t), y(t), z(t)) =
dt z(t)
bAA (x, y, z) =
baa (x, y, z) − daa (x, y, z)
baA (x, y, z) − daA (x, y, z)
bAA (x, y, z) − dAA (x, y, z)
!
.
(fAA z+ 12 faA y)2
faa x+faA y+fAA z
dAA (x, y, z) = (DAA + CAA,aa x + CAA,aA y + CAA,AA z)z
and similar expressions for the other terms
14 / 27
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
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)
cu1 u2 ,v1 v2 ≡ c, ∀u1 u2 , v1 v2 ∈ {aa, aA, AA}
fAA = faA = faa ≡ f
DAA = DaA ≡ D but Daa = D + ∆
15 / 27
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)
cu1 u2 ,v1 v2 ≡ c, ∀u1 u2 , v1 v2 ∈ {aa, aA, AA}
fAA = faA = faa ≡ f
DAA = DaA ≡ D but Daa = D + ∆
⇒ type aA is as fit as AA and both
are fitter than type aa
AA
aA
aa
15 / 27
Work of Bovier, Neukirch (2015)
Start with nK
i (t) = n̄aa 1i=aa +
O(log K)
O(1)
1
K 1i=aA
then :
≥ O(K 1/4+ )
The system converges to (0, 0, n̄AA ) as t → ∞ but slowly !
16 / 27
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
Model with a second mutant
Mutation to allele B → U = {a, A, B}
18 / 27
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
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: fa = fA = fB = f
natural death: Da = D + ∆ > DA = D > DB = D − ∆
18 / 27
Birth Rates
No recombination between a and B
a
B
19 / 27
Birth Rates
birth-rate of aa-individual:
baa
1
1
1
naa naa + 21 naA
2 naB 2 naA + 2 naB
=
+
Pool(aa)
Pool(aB)
1
1
1
2 naA naa + 2 naA + 2 naB
+
Pool(aA)
Pools of potential partners:
aa
aa
aA
AA
aA
aA
AB
aa aB
AA BB
aB
aA AB
aB
AA BB
20 / 27
Competition
No competition between a and B
a
B
21 / 27
Competition
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
22 / 27
Result
Theorem (Bovier, Neukirch, C. 2016/10/07
(yesterday!))
The deterministic system started with initial condition
ni (0) = n̄AA 1i=AA + 1i=aA + 2 1i=aa + 3 1i=AB
and under the following assumptions on the parameters
1
∆ sufficiently small,
2
f sufficiently large,
converges polynomially fast to the fixed point (n̄a , 0, 0, 0, 0, n̄B ).
23 / 27
Simulation
aa aA AA aB AB BB
24 / 27
Large population : deterministic system
5
4
3
2
1
0
200
400
600
800
1000
aa aA AA aB AB BB
1200
1400
25 / 27
Large population : deterministic system
200
400
600
800
1000
1200
1400
-5
-10
-15
-20
aa aA AA aB AB BB
25 / 27
Recap - Dimorphism in two mutations
Mutation 1
Mutation 2
5
5
4
4
3
3
2
2
1
1
0
100
100
200
200
300
300
400
400
500
500
600
700
600
700
0
200
200
400
400
600
600
800
800
1000
1000
1200
1400
1200
1400
-5
-10
-5
-15
-10
-20
-25
-15
-30
-35
-20
26 / 27
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 (n̄a , 0, 0, 0, 0, n̄B )
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T
H
A
N
Y
O
U
K