François Rousset - présentation MEE2013

Fitness for the impatient

Francois Rousset

May 2013

Francois Rousset Fitness for the impatient May 2013 1 / 33

The message

To understand the forces operating in social evolution, particularly inspatially structured populations:

Relatedness concepts under localized dispersalStability of kin recognition polymorphismsRelationship between inclusive fitness and evolutionary stabilityA fundamental impatience: reduce these problems to trivial building blocksDraw connections to other methods such as diffusion theory, multilocusmethods


The message

To understand the forces operating in social evolution, particularly inspatially structured populations:Relatedness concepts under localized dispersal

Stability of kin recognition polymorphismsRelationship between inclusive fitness and evolutionary stabilityA fundamental impatience: reduce these problems to trivial building blocksDraw connections to other methods such as diffusion theory, multilocusmethods


The message

To understand the forces operating in social evolution, particularly inspatially structured populations:Relatedness concepts under localized dispersalStability of kin recognition polymorphismsRelationship between inclusive fitness and evolutionary stability

A fundamental impatience: reduce these problems to trivial building blocksDraw connections to other methods such as diffusion theory, multilocusmethods


The message

To understand the forces operating in social evolution, particularly inspatially structured populations:Relatedness concepts under localized dispersalStability of kin recognition polymorphismsRelationship between inclusive fitness and evolutionary stabilityA fundamental impatience: reduce these problems to trivial building blocks

Draw connections to other methods such as diffusion theory, multilocusmethods


The message

To understand the forces operating in social evolution, particularly inspatially structured populations:Relatedness concepts under localized dispersalStability of kin recognition polymorphismsRelationship between inclusive fitness and evolutionary stabilityA fundamental impatience: reduce these problems to trivial building blocksDraw connections to other methods such as diffusion theory, multilocusmethods


Continuous evolutionary stability for the very impatient

Two alleles, a and A, inducing phenotypes za and zA



Two alleles, a and A, inducing phenotypes za and zA

Fitn

ess

ofA

alle

le

Phenotypes

za, zA1

HaL

z1 zm z2



Classification of different cases:

Continuously stablestrategy HCSSL

Branching point

Evolutionarily stable= noninvasible

Invasible

Unattainable

Convergence stable= attainable


Eshel 1983,1996; Christiansen, 1991; Abrams et al., 1993

Implicit assumptions

Resident phenotype z , mutant z + δ

∆p ∼ δstuff + δ2blob (1)

assumed to be essentially of the form

∆p ∼ δp(1− p)S(z) + δ2p(1− p)(1− 2p)blob(z) (2)

where S(z) and blob(z) are of constant sign wrt p.

Actually S(z) is independent from p in many models (strong claim!).Why?


Implicit assumptions

Resident phenotype z , mutant z + δ

∆p ∼ δstuff + δ2blob (1)

assumed to be essentially of the form

∆p ∼ δp(1− p)S(z) + δ2p(1− p)(1− 2p)blob(z) (2)

where S(z) and blob(z) are of constant sign wrt p.Actually S(z) is independent from p in many models (strong claim!).Why?


A study of frequency (p) dependence

The first-order termA conceptual deviceFitnessThe minimal algorithmTwo views of the Prisoner’s dilemmaMany views of inclusive fitnessThe more general logic: dominance, kin recognitionKin recognition

Glimpses of second-order resultsContinuous evolutionary stabilityKin recognition


A conceptual device

p′ =∑

parents i

AiXi

p′ =∑

gene copies g

AgXg

X indicator variable for an allele;A frequency of copies of parental genesConsider the function f giving (conditional) probabilities

E[p′| . . .] =∑i

ai (. . .)Xi

where ai is the probability that a descendant copy originates from parentalcopy i .


A conceptual device

p′ =∑

gene copies g

AgXg

X indicator variable for an allele;A frequency of copies of parental genesConsider the function f giving (conditional) probabilities

E[p′| . . .] =∑i

ai (. . .)Xi

where ai is the probability that a descendant copy originates from parentalcopy i .


Trivial example

Individuals share a resource in total amount R, their fecundity being equalto the amount of resource they consumeTheir share of resource is proportional to the value of some trait z :

sharei =zi∑i zi

= ai , fecundityi = Rzi∑i zi

Conflict between individual and group

fecundityi = (R − z)zi∑i zi

but ai is still zi∑i zi

.


Fitness

In terms of the number of adult offspring, or “fitness” W

p′ =

∑gene copies g XgWg∑

g Wg= mean(XgWg )

Define fitness functions so that

E[p′|p] =1

Ntot

∑g

Xgw(zg (p))

In a demographically stable population Ntota = w (e.g., w = ziz ).

A (minimal: W = 1) Price equation

p′ = X ′ =W X + Cov(wg ,Xg ),

E[∆p|z] = Cov(wg (z),Xg ).


Minimal algorithm

Express fitness in terms of indicator variablese.g., phenotype zi = z + XiδDifferentiate with respect to some measure of strength of selectionTo first order in δ,

wf(zf, zp) ∼ linear combination of (X ,X )

Collect products of indicator variables

p′ ∼ mean(wiH i ) =

linear combination of means of (H2,HH) =

p + δ∂w

∂zfmean(H2 − HH)

Take expectations E.g., a large population without (spatial) structure

p′ = p + δ∂w

∂zf(p − p2)


Two views of the Prisoner’s dilemma

(“dilemma”: T > R, P > S , R > P i.e. T > R > P > S)


Two views of the Prisoner’s dilemma

Phenotype (z): probability of cooperating in a prisoner’s dilemma

Fecundityf ∝ 1 + Rzfz◦ + Szf(1− z◦) + T (1− zf)z◦ + P(1− zf)(1− z◦)

No spatial structure:

wf =Fecundityf

Mean fecundity

phenotype zi = zres + Xiδ

∆p = Cov(wi ,Xi ) = δpq[S−P+(z+pδ)(R−S+P−T )]+O[δ3, (R, S ,T ,P)2]

View 1: R,T ,S ,P are given ecological constraints; evolution of z ⇒expansion in δ. ∆p ∼ δpq[S − P + z(R − S + P − T )]View 2: expansion in R,T , S ,P, not in δ.


Population structure

Color code: focal, neighbor, population

wf ∼ linear combination of (H,H,H)

so that

mean(H)t+1 ∼ mean(wiH i ) =

p + linear combination of means of (H2,HH,HH) =

p + δ

(∂w

∂zf(H2 − HH) +

∂w

∂zn

(HH − HH)

)Traditional population genetic argument:

E[H|H, p] = FSTH + (1− FST)p

for “relatedness”FST independent of p.


Genealogical interpretation: island model

past

. . . . . .

p p2

Lineages from distinct demes can be considered as draws of independentgenes copies, each A with frequency p.Probability that first event is coalescence: FST

Such coalescences are recent if migration“not too small”; p then consideredconstant if selection is “weak” and total population size is “large”.


Inclusive fitness under weak selection

Traditional argument about relatedness “r” (or FST) :

E[H|H, p] = rH + (1− r)p

for “relatedness” r independent of p. Hence

E[HH − HH|p] = p(r + (1− r)p)− p2 = rpq

hence (with E[HH − HH|p] = pq)

δ

(∂w

∂zf(H2 − HH) +

∂w

∂zn

(HH − HH)

)= pq δ

(∂w

∂zf+∂w

∂zn

r

)︸︷︷︸

inclusive fitness −c + rb

.

Selection gradient is independent of p.


Genealogical interpretation: “stepping stone”

past

. . . . . .

Lineages from distinct demes cannot be considered as draws ofindependent genes copies, each A with frequency p


Frequency-dependence in the stepping-stone model

(Circular stepping-stone model with 200 demes of 10 haploid individuals,dispersal rate 0.2, and a two allele model with mutation rate 10−5)


Weak selection under localized dispersal

Asymptotic results for large number of demes:(Infinite) island model

∆p ∼ δpqsIF = δpq(1− FST)φ

Localized dispersal

∆p ∼ δpqsIF(p) = δpq(1− FST(p))φ

FST(p) same as FST but for conditional probabilities

FST ≡E(XX )− E(XX )

E(XX )− E(XX )

FST(p) ≡ E(XX |p)− E(XX |p)

E(XX |p)− E(XX |p)

sIF: scaled inclusive fitness effect, selection gradientφ: p-independent localized selection gradient


Weak selection under localized dispersal

Asymptotic results for large number of demes:(Infinite) island model

∆p ∼ δpqsIF = δpq(1− FST)φ

Localized dispersal

∆p ∼ δpqsIF(p) = δpq(1− FST(p))φ

FST(p) same as FST but for conditional probabilities

FST ≡E(XX )− E(XX )

E(XX )− E(XX )FST(p) ≡ E(XX |p)− E(XX |p)

E(XX |p)− E(XX |p)

sIF: scaled inclusive fitness effect, selection gradientφ: p-independent localized selection gradient


What does that mean ?


The localized selection gradient φ and local FST’s

The finite population meaning and computation of φ

φ ≡ ∂π

∂δ= −

∑k 6=f

∂w

∂zk

Tk

T0

Tk

T0=

(limµ→0

E[p − XX k ]

E[p − XX 0]

)=

(limµ→0

1

1− FSTk

)

Localized interactions: short distances (k) only. Everythingunderstandable as the result of local interactions (global p doesn’tmatter!).

Expressed in terms of local population structure parameters relativelyeasy to estimate using genetic markers, and with genealogicalinterpretation(s)

Genealogical: FSTk(p) = FSTk at neutral genetic markers


Fitness costs and benefits

Effects B on neighbors’ fecundity and −C on focal’s fecundityrb − c = rB − C? In general No!

Local competition ⇒“inclusive fitness” = −C (Taylor, 1992).Actually ∆p ∼ −pq(1− FST)C in island model.

How to obtain rb − c ∝ rB − C?Version with spatially restricted dispersalIndividuals disperse as a group and compete as a group against othergroups for access to whole group breeding spots. The winners of suchgroup contests can then occupy whole demes⇒ ∆p ∼ (1− F )(RB − C )pq(Gardner & West 2006; Lehmann et al. 2006).





How to obtain rb − c ∝ rB − C?

Version with spatially restricted dispersalIndividuals disperse as a group and compete as a group against othergroups for access to whole group breeding spots. The winners of suchgroup contests can then occupy whole demes⇒ ∆p ∼ (1− F )(RB − C )pq(Gardner & West 2006; Lehmann et al. 2006).





How to obtain rb − c ∝ rB − C?Hamilton (1975): “groups break up completely and re-form in eachgeneration”“young animals take off to form a migrant pool”... [one type]assort[s] positively with its own type in settling from the migrant pool(...)to such a degree that the correlation of two separate randomly selectedmembers (...) is F”⇒ ∆p ∼ (RB − C )pq

Version with spatially restricted dispersalIndividuals disperse as a group and compete as a group against othergroups for access to whole group breeding spots. The winners of suchgroup contests can then occupy whole demes⇒ ∆p ∼ (1− F )(RB − C )pq(Gardner & West 2006; Lehmann et al. 2006).





How to obtain rb − c ∝ rB − C?Version with spatially restricted dispersalIndividuals disperse as a group and compete as a group against othergroups for access to whole group breeding spots. The winners of suchgroup contests can then occupy whole demes⇒ ∆p ∼ (1− F )(RB − C )pq(Gardner & West 2006; Lehmann et al. 2006).


More general logic: example of dominance

Phenotype zi = z + δ[2h(Xi1 + Xi2)/2 + (1− 2h)Xi1Xi2]To first order,

wf(zf, zp) ∼ linear combination of (X 1,X 2,X 1X2, ...)

so that

E[p′] = mean(wiX i ) =

linear combination of means of (X 21,X 1X 2,X 1X ,X 1X 1X 2, . . . ,X 1X1X2) =

Triplets generally lead to frequency-dependence, although there areintriguing exceptions:Helping among diploid full sibs and among haplodiploid sisters, partial sibmating (α)

∆p ∼ δpq(∂w

∂zf+∂w

∂zn

r

)f (h, α, /p)


Three gene lineages... island model

past

. . . . . .

p p2 p3

Lineages from distinct demes can be considered as draws of independentgenes copies, each A with frequency p.Probability that first event is coalescence: FST


More general logic: example of kin recognition

Demes of N individuals; two loci; indicator variables R,H for allelesConditional helping:

fecundityf = 1 +1

N − 1

∑neighbours k

[RfRk + (1− Rf)(1− Rk)](−CHf + BHk)

(two recognition alleles case)

wf =(1− d)ff

(1− d)fdeme + dfothers+ d

fffothers

(dispersal probability d ; regulation after dispersal)

mean(H)t+1 =mean(wiHi )

mean(wi )= mean(wiHi ) over individuals i .

(strict regulation)


Expectedly...

To first order in C and B,

wf ∼ linear combination of (H,H,RH,RH,RRH,RH,RRH,

H,RH,RRH,RH)

so that

mean(H)t+1 ∼ mean(wiH i ) =

linear combination of expectations of (H2,HH,RH2,RH2,RRH2,

RHH,RRHH,HH,RHH,RRHH,RHH)

RH RH R RH H


Meaning

An act of helping always involves the configurationHR R−C B


Meaning


RH RH

describes the probability that the receiver bears the helping allele;can be computed in terms of the probability of joint coalescence within thedeme


Meaning


R RH H

describes the probability that a third individual bears the helping allele.The fitness of this individual is reduced in proportion to (B-C), that is, inproportion to the increase in fitness of the pair of interacting individuals;can be computed in terms of the probability of joint coalescence within thedeme


Kin recognition: results

∆pH ∼− C (1− F ) (1− 2pqR) pqH

+

[−C F + B φ− (1−m)2(B − C )

[1

N(F + φ) +

(1− 1

N

)γ

]]∆pR ∼pHpqR(1− 2pR)

B − C

N[Z (N,m) < 0]

Polymorphism lost at the recognition locus.


Synthesis and developments

Many results follow mechanically from a trivial description ofallele-frequency changes in terms of fitness functions:

* Write a properly defined fitness function in terms of individualbehaviour

* Express behaviour in term of genotypes (indicator variables)

* Expand wiXi to appropriate order, and take expectations of productsof indicator variables (special case: “direct fitness” method, Taylor &Frank 1996)

Deterministic “multi”locus models, with a recombination step:

Diffusion methods (first order)

Diffusion with p-dependence (in first order, e.g. dominance)

Evolutionary stability







Deterministic “multi”locus models, with a recombination step:∆(mean(any thing X )) = ∆sel(mean(X )) + ∆recomb(mean(X )).Multilocus models often in terms of “centered” associations that are 0in expectation in neutral models, e.g. E[(R − pR)(H − pH)].








* Express behaviour in term of genotypes (indicator variables)* Expand wiXi to appropriate order, and take expectations of products

of indicator variables (special case: “direct fitness” method, Taylor &Frank 1996)

Deterministic “multi”locus models, with a recombination step:Diffusion methods (first order) e.g. approximation for fixationprobability

π ∼ 1− e−2Ntotφpini

1− e−2Ntotφ

for φ taken in an infinite-deme limit.

Diffusion with p-dependence (in first order, e.g. dominance)Evolutionary stability







Deterministic “multi”locus models, with a recombination step:





Continuous evolutionary stability in the island model

Second-order computation for any p feasible.Simpler computation for extreme p, and connexion to “number ofsuccessful emigrants” approach (Chesson 1984; Metz and Gyllenberg 2001)

B = ∂zf,zfw +2F∂zf,znw +K∂zn,zn

w +4F (N−1)(K∂zn

wp + F∂fwp

)wp∂zn

w

where wp: local offspringIn contrast to formula proposed by Day and Taylor (1998):(1) three-genes coefficient K ;(2) products of identity coefficients, product of derivatives: joint effect offirst-order change in number of offspring and first-order change in parentalpopulation structure.


Kin recognition: searching for stable polymorphisms

Second-order computation involves 30 associations, for up to 7 gene copiesin 5 individuals: R‖RH/R‖RH/RSearch for conditions for stable polymorphism: low migration and lowrecombinationConvergent orbits plus drift:


“Conclusions”

First order: Simple formalism provides connections between differentmodelling approaches (inclusive fitness, adaptive dynamics, diffusion,coalescence). Relatively simple pattern of frequency-dependence.

Second order: Essentially the same formalism previously used formultilocus models; Algebraically (and algorithmically) messy inspatially structured populations; still allows an analysis of the forcesacting on a trait.


More general coalescent interpretation of relatedness


Quasi equilibrium

Quasi linkage equilibrium

D(t + 1) = (1− r)D(t) + O(δ)⇒ D =O(s)

r

understood as

O(s)

r=

O(s)

1− (1− r)= (1− r)O(s) + (1− r)2O(s) + (1− r)3O(s) . . .

Quasi equilibrium

D(t + 1)− D◦ = λ(D(t)− D◦) + O(δ)⇒ D(t + 1)− D◦ =O(s)

1− λ

Actually

D(t + 1)−D◦ = A(D(t)−D◦) + O(δ)⇒ D−D◦ = (I− A)−1O(s)


François Rousset - présentation MEE2013

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