Mutation, selection, and ancestry in the deterministic limit of the Moran model

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Moran model Ancestries References

Mutation, selection, and ancestry in thedeterministic limit of the Moran model

Sebastian Hummel

based on joint work (in progress) with Ellen Baake and Fernando Cordero andthanks to many discussions with Anton Wakolbinger

Bielefeld University

Genealogies of Interacting Particle Systems

08.08.2017

Mutation, selection, and ancestry in thedeterministic limit of the Moran model Sebastian Hummel Bielefeld University 08.08.2017 1 / 39

Mutation, selection, and ancestry in the deterministic limitof the Moran model

1 2-type Moran model and its deterministic limit2-type Moran modelDeterministic limitProperties of deterministic limit

2 Ancestries in the Moran model and in the deterministic limitAncestral selection graphKilled ancestral selection graphPruned lookdown ancestral selection graph

at rate 1

at rate s

at rate 𝑢𝑣0

at rate 𝑢𝑣1

Moran model with 2-typesHaploid population of fixed size NTypes: 0 (’fit’) and 1 (’unfit’)Individuals of type 1 reproduce at rate 1Individuals of type 0 reproduce at rate 1 + s, s ≥ 0Single offspring inherits parent’s type and replaces uniformlychosen individualParent-independent mutation at rate u > 0Resulting type: 0 with probability ν0; 1 with probability ν1

Graphical representation of 2-type Moran model

×0 tt

Type 0 (’fit’)Type 1 (’unfit’)

Mutation to type 0× Mutation to type 1

Selective arrowNeutral arrow

×0 tt

Deterministic limit of the Moran modelY

(N)t proportion of type 1 is Markov process on [0, 1]

y(t, y0) solution of IVP

dt(t) = −sy(t)(1− y(t))− uν0y(t) + uν1(1− y(t)) (t ≥ 0)

y(0) = y0 for y0 ∈ [0, 1]

If limN→∞ Y(N)

0 = y0, then ∀ε, T > 0,

limN→∞

supt≤T|Y (N)t − y(t, y0)| > ε

Convergence carries over to the stationary state (t→∞)Neither time nor parameters are rescaled

(N)t proportion of type 1 is Markov process on [0, 1]y(t, y0) solution of IVP

y(0) = y0 for y0 ∈ [0, 1]

If limN→∞ Y(N)

0 = y0, then ∀ε, T > 0,

limN→∞

y(0) = y0 for y0 ∈ [0, 1]

If limN→∞ Y(N)

0 = y0, then ∀ε, T > 0,

limN→∞

y(0) = y0 for y0 ∈ [0, 1]

If limN→∞ Y(N)

0 = y0, then ∀ε, T > 0,

limN→∞

Convergence carries over to the stationary state (t→∞)

Neither time nor parameters are rescaled

y(0) = y0 for y0 ∈ [0, 1]

If limN→∞ Y(N)

0 = y0, then ∀ε, T > 0,

limN→∞

Properties of the deterministic limit y(t; y0)If s = 0: unique equilibrium y = ν1 ⇒ stable

If s > 0, two equilibria

y = 12

√(1− u

)2+ 4u

∈ [0, 1]

y? = 12

√(1− u

)2+ 4u

If ν0 > 0,y ∈ [0, 1)→ stable; y? > 1→ unstableIf ν0 = 0,y = min

{us , 1

}→ stable; y? = max

{1, us

}→ unstable

Properties of the deterministic limit y(t; y0)If s = 0: unique equilibrium y = ν1 ⇒ stableIf s > 0, two equilibria

y = 12

√(1− u

)2+ 4u

∈ [0, 1]

y? = 12

√(1− u

)2+ 4u

{us , 1

{1, us

}→ unstable

y = 12

√(1− u

)2+ 4u

∈ [0, 1]

y? = 12

√(1− u

)2+ 4u

If ν0 > 0,y ∈ [0, 1)→ stable; y? > 1→ unstable

If ν0 = 0,y = min

{us , 1

{1, us

}→ unstable

y = 12

√(1− u

)2+ 4u

∈ [0, 1]

y? = 12

√(1− u

)2+ 4u

{us , 1

{1, us

}→ unstable

The equilibrium frequency

0 0.2 0.4 0.6 0.8 1 1.2 1.40

lim t→∞y

ν0 = 1100

ν0 = 0

Black line: stable. Grey line: unstable.

- error threshold -Mutation, selection, and ancestry in thedeterministic limit of the Moran model Sebastian Hummel Bielefeld University 08.08.2017 13 / 39

Properties of the deterministic limit y(t; y0)If y ∈ (0, 1) and y0 ∈ (0, 1), then

if y0 < y ⇒ y(t; y0) monotonically increases to y as t→∞if y0 > y ⇒ y(t; y0) monotonically decreases to y as t→∞

y0=0.5

y0=0.9

500 1000 1500 2000

Parameters: s = 1100 , u = 1

300 , ν0 = 11000 , ν1 = 1− 1

Ancestral selection graph (ASG)

Introduced by Krone and Neuhauser [1997]

Pecking order

D=Descendant C=Continuing I=Incoming

Descendant is of type 1 ⇔ all potential ancestors are of type 1

Pecking order

D=Descendant C=Continuing I=Incoming

Descendant is of type 1 ⇔ all potential ancestors are of type 1

ASG in the deterministic limit

No coalescence events, no collisionsBranching at rate s per existing lineMutation to type 0 at rate uν0 per existing lineMutation to type 1 at rate uν1 per existing line

Type of uniformly chosen individual at time tForward picture

y(t; y0)

Backward picture?

y(t; y0)r 0

The killed ASGType of uniformly chosen individual?Count potential ancestors of a single individualStop ASG if type is determined

qR(k, k + 1) = ks

qR(k, k − 1) = kuν1

qR(k,∆) = kuν0

→ (Rr)r≥0 counts potential ancestors until type is knownAbsorption states: 0 and ∆

qR(k, k + 1) = ks

qR(k, k − 1) = kuν1

qR(k,∆) = kuν0

qR(k, k + 1) = ks

qR(k, k − 1) = kuν1

qR(k,∆) = kuν0

qR(k, k + 1) = ks

qR(k, k − 1) = kuν1

qR(k,∆) = kuν0

qR(k, k + 1) = ks

qR(k, k − 1) = kuν1

qR(k,∆) = kuν0

qR(k, k + 1) = ks

qR(k, k − 1) = kuν1

qR(k,∆) = kuν0

qR(k, k + 1) = ks

qR(k, k − 1) = kuν1

qR(k,∆) = kuν0

qR(k, k + 1) = ks

qR(k, k − 1) = kuν1

qR(k,∆) = kuν0×

qR(k, k + 1) = ks

qR(k, k − 1) = kuν1

qR(k,∆) = kuν0×

TheoremFor t ≥ 0,

y(t, y0)n = E

0 | R0 = n]

∀n ∈ N0 ∪ {∆}, y0 ∈ [0, 1],

where y∆ := 0y(t, y0): proportion of type 1Rt: number of potential ancestors until descendant’s type is known

TheoremFor t ≥ 0,

y(t, y0)n = E

0 | R0 = n]

∀n ∈ N0 ∪ {∆}, y0 ∈ [0, 1],

TheoremFor t ≥ 0,

y(t, y0)n = E

0 | R0 = n]

∀n ∈ N0 ∪ {∆}, y0 ∈ [0, 1],

Absorption probabilities of Rr

wn := P (limr→∞Rr = 0 | R0 = n)w0 = 1 and w∆ = 0

First-step analysis ⇒ wk = su+swk+1 + uν1

u+swk−1

Independence of ancestries ⇒ wk = wk1

Hence,

(1 + u

s −√

(1− us )2 + 4us ν0

)if s > 0

ν1 s = 0

Duality ⇒ w1 = proportion of 1 at stationarity = y

wn := P (limr→∞Rr = 0 | R0 = n)w0 = 1 and w∆ = 0First-step analysis ⇒ wk = s

u+swk+1 + uν1u+swk−1

Hence,

(1 + u

s −√

(1− us )2 + 4us ν0

)if s > 0

ν1 s = 0

Hence,

(1 + u

s −√

(1− us )2 + 4us ν0

)if s > 0

ν1 s = 0

Hence,

(1 + u

s −√

(1− us )2 + 4us ν0

)if s > 0

ν1 s = 0

Hence,

(1 + u

s −√

(1− us )2 + 4us ν0

)if s > 0

ν1 s = 0

The equilibrium frequency and absorption probability

0 0.2 0.4 0.6 0.8 1 1.2 1.40

lim t→∞y

ν0 = 1100

ν0 = 0

Black line: stable. Grey line: unstable.

- error threshold -Mutation, selection, and ancestry in thedeterministic limit of the Moran model Sebastian Hummel Bielefeld University 08.08.2017 23 / 39

Representative ancestral typeDefinitionThe representative ancestral (RA) type at backward time r,denoted by Ir ∈ {0, 1}, is the type of the ancestor at backwardtime r of an individual uniformly chosen at time 0.

Quantities of interestg(y0, r) := Py0(Ir = 1)g∞(y0) := limr→∞ g(y0, r)→ conditional RA type distributiong∞(y)→ RA type distribution inequilibrium

Representative ancestral typeDefinitionThe representative ancestral (RA) type at backward time r,denoted by Ir ∈ {0, 1}, is the type of the ancestor at backwardtime r of an individual uniformly chosen at time 0.Quantities of interest

g(y0, r) := Py0(Ir = 1)g∞(y0) := limr→∞ g(y0, r)→ conditional RA type distributiong∞(y)→ RA type distribution inequilibrium

Pruned lookdown ASG (p-LD-ASG)In the diffusion case: Common ancestor type distributionFearnhead [2002] and Taylor [2007] ⇒ analytic argumentp-LD-ASG introduced by Lenz et al. [2015] ⇒ probabilisticargumentTranslation of p-LD-ASG to deterministic limit by Cordero[2017]

Idea of p-LD-ASG:Count potential ancestors of a single individualArrange potential ancestors in hierarchyMutations rule out some potential ancestors

Pruned lookdown ASG (p-LD-ASG)In the diffusion case: Common ancestor type distributionFearnhead [2002] and Taylor [2007] ⇒ analytic argumentp-LD-ASG introduced by Lenz et al. [2015] ⇒ probabilisticargumentTranslation of p-LD-ASG to deterministic limit by Cordero[2017]

Idea of p-LD-ASG:Count potential ancestors of a single individualArrange potential ancestors in hierarchyMutations rule out some potential ancestors

p-LD-ASG in deterministic limit

qL(k, k + 1) = ks

qL(k, k − 1) = (k

+ 1{k>1}uν0