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

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

1

0

0

1

1



1



1



1 1



0



0



at rate 𝑢𝑣0



0

at rate 𝑢𝑣0



at rate 𝑢𝑣1



at rate 𝑢𝑣1

1



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

×

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×

×0 tt

Type 0 (’fit’)Type 1 (’unfit’)

Mutation to type 0× Mutation to type 1

Selective arrowNeutral arrow




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×0 tt







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×0 tt







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×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

dy

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→∞

P(

supt≤T|Y (N)t − y(t, y0)| > ε

)= 0

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

dy


y(0) = y0 for y0 ∈ [0, 1]

If limN→∞ Y(N)

0 = y0, then ∀ε, T > 0,

limN→∞

P(


)= 0






dy


y(0) = y0 for y0 ∈ [0, 1]

If limN→∞ Y(N)

0 = y0, then ∀ε, T > 0,

limN→∞

P(


)= 0






dy


y(0) = y0 for y0 ∈ [0, 1]

If limN→∞ Y(N)

0 = y0, then ∀ε, T > 0,

limN→∞

P(


)= 0

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

Neither time nor parameters are rescaled





dy


y(0) = y0 for y0 ∈ [0, 1]

If limN→∞ Y(N)

0 = y0, then ∀ε, T > 0,

limN→∞

P(


)= 0




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

s−

√(1− u

s

)2+ 4u

sν0

∈ [0, 1]

y? = 12

1 + u

s+

√(1− u

s

)2+ 4u

sν0

≥ 1

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

s−

√(1− u

s

)2+ 4u

sν0

∈ [0, 1]

y? = 12

1 + u

s+

√(1− u

s

)2+ 4u

sν0

≥ 1


{us , 1


{1, us

}→ unstable




y = 12

1 + u

s−

√(1− u

s

)2+ 4u

sν0

∈ [0, 1]

y? = 12

1 + u

s+

√(1− u

s

)2+ 4u

sν0

≥ 1

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

If ν0 = 0,y = min

{us , 1


{1, us

}→ unstable




y = 12

1 + u

s−

√(1− u

s

)2+ 4u

sν0

∈ [0, 1]

y? = 12

1 + u

s+

√(1− u

s

)2+ 4u

sν0

≥ 1


{us , 1


{1, us

}→ unstable



The equilibrium frequency

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1

1.2

u/s

lim t→∞y

(t;y

0)

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=1

y0=0

y0=0.5

y0=0.9

y0= y

500 1000 1500 2000

0.2

0.4

0.6

0.8

1.0

Parameters: s = 1100 , u = 1

300 , ν0 = 11000 , ν1 = 1− 1

1000








Ancestral selection graph (ASG)

Introduced by Krone and Neuhauser [1997]

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t



Pecking order

D=Descendant C=Continuing I=Incoming

DC

I

1

1

1DC

I

0

1

0

DC

I

1

0

0DC

I

0

0

0

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



Pecking order

D=Descendant C=Continuing I=Incoming

DC

I

1

1

1DC

I

0

1

0

DC

I

1

0

0DC

I

0

0

0

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

0 t

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

[yRt

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

0 t

r 0

×

×



TheoremFor t ≥ 0,

y(t, y0)n = E

[yRt

0 | R0 = n]

∀n ∈ N0 ∪ {∆}, y0 ∈ [0, 1],


0 t

r 0

×

×



TheoremFor t ≥ 0,

y(t, y0)n = E

[yRt

0 | R0 = n]

∀n ∈ N0 ∪ {∆}, y0 ∈ [0, 1],


0 t

r 0

×

×



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,

w1 =

12

(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,

w1 =

12

(1 + u

s −√

(1− us )2 + 4us ν0

)if s > 0

ν1 s = 0








Hence,

w1 =

12

(1 + u

s −√

(1− us )2 + 4us ν0

)if s > 0

ν1 s = 0








Hence,

w1 =

12

(1 + u

s −√

(1− us )2 + 4us ν0

)if s > 0

ν1 s = 0








Hence,

w1 =

12

(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

0.2

0.4

0.6

0.8

1

1.2

u/s

lim t→∞y

(t;y

0)

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

×

×

r

y0

0



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

×

×

r

y0

0



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

)uν1

+ 1{k>1}uν0

qL(k, l) = uν0 where l ∈ {1, . . . , k − 2}

××

⇒ (Lr)r≥0 line-counting process, no absorbing states⇒ ancestor of type 1⇔ all potential ancestors in p-LD-ASG of type 1




qL(k, k + 1) = ks

qL(k, k − 1) = (k

− 1

)uν1

+ 1{k>1}uν0

qL(k, l) = uν0 where l ∈ {1, . . . , k − 2}

××





qL(k, k + 1) = ks

qL(k, k − 1) = (k

− 1

)uν1

+ 1{k>1}uν0

qL(k, l) = uν0 where l ∈ {1, . . . , k − 2}

××





qL(k, k + 1) = ks

qL(k, k − 1) = (k

− 1

)uν1

+ 1{k>1}uν0

qL(k, l) = uν0 where l ∈ {1, . . . , k − 2}

××





qL(k, k + 1) = ks

qL(k, k − 1) = (k

− 1

)uν1

+ 1{k>1}uν0

qL(k, l) = uν0 where l ∈ {1, . . . , k − 2}

××





qL(k, k + 1) = ks

qL(k, k − 1) = (k − 1)uν1

+ 1{k>1}uν0

qL(k, l) = uν0 where l ∈ {1, . . . , k − 2}

××





qL(k, k + 1) = ks

qL(k, k − 1) = (k − 1)uν1

+ 1{k>1}uν0

qL(k, l) = uν0 where l ∈ {1, . . . , k − 2}

××





qL(k, k + 1) = ks

qL(k, k − 1) = (k − 1)uν1

+ 1{k>1}uν0

qL(k, l) = uν0 where l ∈ {1, . . . , k − 2}

××





qL(k, k + 1) = ks

qL(k, k − 1) = (k − 1)uν1 + 1{k>1}uν0

qL(k, l) = uν0 where l ∈ {1, . . . , k − 2}

××





qL(k, k + 1) = ks

qL(k, k − 1) = (k − 1)uν1 + 1{k>1}uν0

qL(k, l) = uν0 where l ∈ {1, . . . , k − 2}

××

⇒ (Lr)r≥0 line-counting process, no absorbing states

⇒ ancestor of type 1⇔ all potential ancestors in p-LD-ASG of type 1




qL(k, k + 1) = ks

qL(k, k − 1) = (k − 1)uν1 + 1{k>1}uν0

qL(k, l) = uν0 where l ∈ {1, . . . , k − 2}

××




p-LD-ASG - exploiting the hierarchy

⇒ g(y0, r) = E[yLr0 | L0 = 1]

= 1− (1− y0)∑n≥0

P (Lr > n | L0 = 1)yn0

g∞(y0) := limr→∞ g(y0, r) ??

Proposition

1 If s = 0, Lr absorbs in 1 almost surely2 If u < s and ν0 = 0, Lr is transient, so Lr →∞ a.s. (r →∞)3 If u = s and ν0 = 0, Lr is null recurrent4 If u > s or ν0 > 0, Lr is positive recurrent and the stationary

distribution is geometric with parameter 1− p, where

p ={

suν1

y, if ν1 > 0,s

u+s , if ν1 = 0.




⇒ g(y0, r) = E[yLr0 | L0 = 1]

= 1− (1− y0)∑n≥0

P (Lr > n | L0 = 1)yn0

g∞(y0) := limr→∞ g(y0, r) ??

Proposition



p ={

suν1

y, if ν1 > 0,s

u+s , if ν1 = 0.




⇒ g(y0, r) = E[yLr0 | L0 = 1]

= 1− (1− y0)∑n≥0

P (Lr > n | L0 = 1)yn0

g∞(y0) := limr→∞ g(y0, r) ??

Proposition



p ={

suν1

y, if ν1 > 0,s

u+s , if ν1 = 0.Mutation, selection, and ancestry in thedeterministic limit of the Moran model Sebastian Hummel Bielefeld University 08.08.2017 27 / 39


Intuition behind the geometric lawLet an := P (L∞ > n)

n+ 1

n

×

n+ 1

n+ 2

n+ 1

Then,an = s

u+ san−1 + uν1

u+ san+1

⇒ lack of memory property




n+ 1

n

×

n+ 1

n+ 2

n+ 1

Then,an = s

u+ san−1 + uν1

u+ san+1





n+ 1

n

×

n+ 1

n+ 2

n+ 1

Then,an = s

u+ san−1 + uν1

u+ san+1




Recursion as FSA for absorption probabilitiesLet Dt be the continuous-time Markov chain on N ∪ {∆} withtransition rates

qD(d, j) =

(d− 1)s, if j = d− 1,(d− 1)uν1, if j = d+ 1,(d− 1)uν0, if j = ∆.

PropositionLt and Dt are Siegmund dual, i.e. for t ≥ 0,

P (m ≤ Lt | L0 = n) = P (Dt ≤ n | D0 = m), ∀n ∈ N, m ∈ N∪{∆}.



Recursion as FSA for absorption probabilitiesLet Dt be the continuous-time Markov chain on N ∪ {∆} withtransition rates

qD(d, j) =

(d− 1)s, if j = d− 1,(d− 1)uν1, if j = d+ 1,(d− 1)uν0, if j = ∆.

PropositionLt and Dt are Siegmund dual, i.e. for t ≥ 0,

P (m ≤ Lt | L0 = n) = P (Dt ≤ n | D0 = m), ∀n ∈ N, m ∈ N∪{∆}.



Consequences of Siegmund dualityCorollary

P (D∞ = 1 | D0 = n+ 1) = limr→∞

P1(Lr > n).

Corollary (null recurrent case)

If u = s and ν0 = 0,

limr→∞

P (Lr > n | L0 = 1) = 1.



Consequences of Siegmund dualityCorollary

P (D∞ = 1 | D0 = n+ 1) = limr→∞

P1(Lr > n).

Corollary (null recurrent case)

If u = s and ν0 = 0,

limr→∞

P (Lr > n | L0 = 1) = 1.



Conditional RA type distributionTheorem

1 If s = 0, g∞(y0) = y0 for all y0 ∈ [0, 1].

2 If u ≤ s and ν0 = 0, g∞(y0) ={

0, if y0 ∈ [0, 1),1, if y0 = 1.

3 If s > 0 and (u > s or ν0 > 0), g∞(y0) = 1−p1−py0

y0.



RA type distribution in equilibrium

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

u/s

g ∞(y

)

ν0 = 1100

ν0 = 0

•



Properties of RA type distribution

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1

1.2

u/s

lim t→∞y

(t;y

0)

ν0 = 1100

ν0 = 0

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1

u/s

g ∞(y

)

ν0 = 1100

ν0 = 0

•



RA type at backward time rBy means of non-absorbing (Lr)r≥0

××

Killed process? Absorption probability?

0 t

y(t; y0)



Piecewise-deterministic Markov processInspired by Taylor [2007]Yt piecewise-deterministic Markov process on [0, 1] withgenerator

AY f(y) = [−sy(1− y)− yuν0 + uν1(1− y)]∂f∂y

+ y

1− yuν0 [f(1)− f(y)] + 1− yy

uν1 [f(0)− f(y)]

with limy→1AY f(y) = lim

y→0AY f(y) = 0.

Absorbs in either 0 or 1



Piecewise-deterministic Markov process

200 400 600 800 1000 1200

0.2

0.4

0.6

0.8

1.0



TheoremThe piecewise-deterministic Markov processes Yt and theline-counting process of p-LD-ASG Lt are dual with respect toduality function yn, and hence for t ≥ 0,

E

[(Yt)n | Y0 = y0

]= E

[yLt

0 | L0 = n]

∀y0 ∈ [0, 1], n ∈ N.




E

[(Yt)n | Y0 = y0

]= E

[yLt

0 | L0 = n]

∀y0 ∈ [0, 1], n ∈ N.

××

200 400 600 800 1000 1200

0.2

0.4

0.6

0.8

1.0




E

[(Yt)n | Y0 = y0

]= E

[yLt

0 | L0 = n]

∀y0 ∈ [0, 1], n ∈ N.

Corollary

g∞(y0) = E

[yL∞

0 | L0 = 1]

= P (Yt absorbs in 1 | Y0 = y0)



Bibliography

F. Cordero. Common ancestor type distribution: A Moran modeland its deterministic limit. Stoch. Proc. Appl., 127:590 – 621,2017.

P. Fearnhead. The Common Ancestor at a nonneutral locus. J.Appl. Probab., 39:38–54, 2002.

S. M. Krone and C. Neuhauser. Ancestral processes with selection.Theoretical population biology, 51(3):210–237, 1997.

U. Lenz, S. Kluth, E. Baake, and A. Wakolbinger. Looking down inthe ancestral selection graph: A probabilistic approach to thecommon ancestor type distribution. Theor. Popul. Biol., 103:27– 37, 2015.

J. E. Taylor. The Common Ancestor Process for a Wright-FisherDiffusion. Electron. J. Probab., 12:808–847, 2007.



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

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