multilevel selection, population genetics and cooperation ... · multilevel selection, ......

26 April 2016

Multilevel selection, pop gen and games in structured pops

Multilevel Selection, Population Genetics and Cooperation in Structured Populations

Jeremy Van Cleve University of Kentucky

UNIVERSITY OF KENTUCKY

NIMBioS Tutorial:Game Theoretical Modeling of

Evolution in Structured Populations

Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY

How do cooperative behaviors evolve?

Cooperation occurs when:

focal: cost to improve state of its partner

partner: benefit from improved state

Food improves state ⟶ higher fitness


Author's personal copy

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Author's personal copy

highest concentration. As more and more starve, theyconcentrate in great streams of dicty cells, flowing towarda center in a process called aggregation (Figure 3(a)).After a few hours, this center concentrates into a mound,which then elongates slightly and begins to crawl aroundtoward light and heat and away from ammonia (Figure 3(b)).

This translucent slug looks like a tiny worm, but differsfrom it in some important ways. As it crawls through asheath largely made up of cellulose, it drops cells at therear, and these cells can feed on any bacteria they discover,effectively recovering the solitary stage. The slug movesmore quickly and farther than any individual amoeba couldmove: an important advantage to the social stage. Thoughthe slug lacks a nervous system, there are differences amongthe constituent cells. Those at the front direct movementand ultimately become the stalk. There is a recently discov-ered class of cells called sentinel cells that sweep throughthe slug from front to back picking up toxins and bacteria,functioning simultaneously as liver, kidney, and innateimmune system, before they are shed at the rear of the slug.

Slugs move farther and for a longer time when theenvironment lacks electrolytes, when it is very moist,and when there is either directional light or no light.When they cease moving, the cells of the slug concen-trate into a tight form known as a Mexican hat. Then,

in a process called culmination, the cells that were atthe front of the slug begin to form cellulose walls andto rise up out of the mass as a very slender but rigidstalk (Figure 3(c)). These cells die. The remainingthree-quarters or so of the cells flow up this stalk,and at the top they form hardy spores. At this point,the spores, stalk, and basal disk comprise an erectstructure called a fruiting body (Figure 3(c)).

Thus, some of the cells sacrifice their lives so that theothers may rise up and sporulate a millimeter or so abovethe soil surface, or into a gap between soil particles.Others sacrifice themselves as sentinel cells picking uptoxins and bacteria as they made their way through theslug. Still others were shed from the rear of the slugduring their normal movement. If these do not encounterbacteria, or enough other shed cells to form a new, smallerfruiting body, then they also perish.

How Dictyostelids Are Obtained, Collected,and Cultured

Many studies can be performed using previously col-lected clones obtained from the stock center for theprice of postage. This stock center is accessed through

Figure 3 Multicellular stages of Dictyostelium discoideum. (a) Aggregation of formerly independent cells into a multicellular body.(b) Motile multicellular slug moving towards light. (c) Fruiting body consisting of a basal disc, a stalk, and a sorus, or spores. Thebasal disc and the stalk are formed of formerly living amoebae that have died to form this supporting structure. (d) Macrocysts, thesexual stage of D. discoideum. (Courtesy of Owen Gilbert).

Dictyostelium, the Social Amoeba 515

Encyclopedia of Animal Behavior (2010), vol. 1, pp. 513-519

Dictyostelium discoideum

Cost: dying as part of the stalk

Benefit: surviving as part of the spore


gene chromosomes

prokaryotes + mitochondria/

chloroplastseukaryotes

clonalreproduction

sexual reproduction

independent living cells

multicellular organisms

independent individuals

social groups (eusociality)

“Major transitions in evolution” or “transitions in individuality”(Maynard Smith and Szathmáry)


Cooperation occurs at two “scales”

“Within a group”

“Between/among groups”


Cooperation occurs at two “scales”

“Within a group”

“Between/among groups”

1. Plastic behaviors

2. Kin & Group selection


1. Plastic behaviors

Game theory(Nash equilibrium / ESS)

Tit for tat / punishment / reputation / etcReciprocity or responsiveness

2. Kin & Group selection

Multilevel selection combines responsiveness with kin/group processes through measures of population structure such as relatedness


Outline

The two scales of cooperation

Simple model of responsiveness within a population

Evolution in structured populations: the Price equation

Multilevel model with the Price equation

Evolution in structured populations: fixation probability & trait substitution

Social games in an island-model using fixation probability



Plastic behavior and repeated games

Plastic behavior requires repeated interactions(i.e., a repeated game)

Strategies in the repeated game determine how individuals respond to the actions of social partners


Cooperate (C) Defect (D)

Payoff to Cooperate (C)

Benefit – Cost – Cost

Payoff to Defect (D)

Benefit 0




Strategy Description

ALLD “always defect”

Always play D

GRIM“grim trigger”

Play C but switch to D once opponent defects

TFT“tit for tat”

Start with C and then repeat the opponent's last move

STFT “suspicious tit for tat”

Start with D and then repeat the opponent's last move

TF2T “tit for two tats”

Play C unless opponent played D in the last two moves

WSLS “win stay, lose shift”

Start with C and then play C if and only if the last payoff from the last round was R or T

Table 14.2 (Broom and Rychtář, 2013)



Payoff (fitness) is accumulated over the course of the interaction

Payoff in later games can be “discounted” due to probability the interaction is broken off

ω : probability of continuing interaction

Longer interactions allow individuals to obtain more information about their partner’s type


GRIM C D D D D D D D D

STFT D C D D D D D D D

ω ω ω ωω ω ω ω



Strategies vary both in

1. Propensity to lead to cooperation and defection

2. Responsiveness to the actions of their partner


TFT C C C C C C C C CTF2T C C C C C C C C C

TFT C D D D D D D D DALLD D D D D D D D D D

TF2T C C D D D D D D DALLD D D D D D D D D D


Responsiveness and the evolution of cooperation

Responsiveness = “direct reciprocity” or “reciprocal altruism”

Measure responsiveness ( ρ ) specifically to see its effect on the evolution of cooperation in a single population

Two types:

• “intrinsic cooperator” (C-type) cooperates with probability = 1 – ρ and reciprocates partner’s last action with probability = ρ

• “intrinsic defector” (D-type) defects with probability = 1 – ρ and reciprocates partner’s last action with probability = ρ


Two C-types always cooperate

Two D-types always defect

C-type vs D-type: [mij] = probability state i ⟶ jwhere i, j ∈ {(C, C), (C, D), (D, C), (D, D)}




(C, C) (C, D) (D, C) (D, D)(C, C)(C, D)(D, C)(D, D)

Equilibrium distribution of Markov chain is given by: v M = v

Payoffs: (focal, partner)

Assume there is no discounting: Fitness = w = payoffs from equilibrium actions in the game

Suppose that p is the frequency of the C-type




(C, C) (C, D) (D, C) (D, D)

B – C > 0 – C B 0

Condition for the increase in the C-type is wC > wD or

Bρ – C > 0 or B/C > 1/ρ

Similar to “Hamilton’s rule”

Same as ESS in “continuous iterated prisoner’s dilemma” (CIPD)(Taylor & Day, 2004; André and Day, 2007)





Outline









Measuring evolution in structured populations

Classical one/multi-locus theory measures Δ[gentoype frequency]

State space grows rapidly

Stability analysis of equilibria requires eigenvalues of large matrices

Stochastic models are hard to analyze


x1 A1B1

x2 A2B1

x3 A1B2

x4 A2B2

Pop1

x111 A1B1

x211 A2B1

x121 A1B2

x221 A2B2

Pop2

x112 A1B1

x212 A2B1

x122 A1B2

x222 A2B2



Alternative method looks more like “physics of many particles”

Initially, its more complex:

1. Track the frequency of each allele in each individual.

But then it simplifies:

2. Calculate the average allele frequency as a function of other statistical quantities (means, variances, etc).

3. This yields: Δ[mean allele frequency] = ΔE[p] = f (E[p],Var[p],etc)

4. If you only care about the mean, assume Var[p] doesn’t evolve

5. Otherwise, find ΔVar[p] and repeat.




This method allows the study of complex stochastic process

Generally called “moment closure” when used for dynamics

You may recognize it if you know quantitative genetics

“Breeder’s equation”: z = mean trait (e.g., allele frequency) s = strength of selection h2 = “heritability” (measures variance ins allele frequency)

Quantitative genetics often obtains moment closure through normality and constant genetic variances



Price equation

“Discovered” by George Price (partly also Alan Robertson)

Price equation is the general version of Δz

Often associated with analyses of group and multilevel selection

Useful for models with population structure

Both oversold and overly criticized



Price equation

Let zi be the value of some phenotype in individual i ~ e.g., allele frequency, body size, investment into cooperation

Δz = the change in the average value of zi over a single generation


piʹ ziʹ : frequency & phenotype of descendants of individual i in next generation

pi zi : frequency and phenotype of individual i incurrent generation


Price equation


Price equation


Price equation


Cov[wi, zi] : evolutionary change in z due to natural selection– statistical association between fitness and phenotype

E[wi Δzi] : evolutionary change in z due to imperfect transmission– e.g. mutation, non-random mating, segregation distortion


Price equation: one-locus selection w/ mutation



Price equation: one-locus selection w/ mutation


Mutation-selection balance


Conceptual applications of the Price equation

Using the Price equation, we can derive three of the most

fundamental expressions of evolutionary change due to natural

selection:

1. Fisher’s fundamental theorem of natural selection (FTNS)

2. Hamilton’s rule

3. Group or multilevel selection



Fisher’s fundamental theorem of natural selection

“The rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time.”


Ronald A. FisherThe Genetical Theory of Natural Selection (1930)


Fisher’s fundamental theorem of natural selection

“The rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time.”

No genetic variance implies no increase in fitness (equilibrium condition)

Variance > 0 implies that fitness is always increasing(stability condition: adaptive peaks)

Mathematization of the concept Darwinian natural selection(cf. Arrow & Debreu theorems of Welfare Economics and the Invisible Hand)



Price equation: FTNS

1. Regress phenotype on genes

• Properties of the linear regression




2. Plug zi into the Price equation

3. The FTNS measures Δw : ⟶ zi = wi = gi + δi




4. Fisher excluded changes in fitness due to “deterioration of the environment” = E[wiΔgi] = 0(no change breeding value due to mutation, etc)

5. FTNS:



Inclusive fitness and Hamilton’s rule

“a gene may receive positive selection even though disadvantageous to its bearers if it causes them to confer sufficiently large advantages on relatives.”


William D. HamiltonJournal of Theoretical Biology (1964)


Inclusive fitness and Hamilton’s rule

“a gene may receive positive selection even though disadvantageous to its bearers if it causes them to confer sufficiently large advantages on relatives.”Hamilton’s rule: – c + r b > 0 – cost + relatedness × benefit > 0

Effect of natural selection on a gene is a function all copies of a gene, not just that copy present in the focal individual.

Inclusive fitness effect = – c + r b

Not an “extension” or special kind of fitness; rather, method of accounting for social interactions (Akçay & Van Cleve, 2016, Phil. Trans. R. Soc. B)


1. Write fitness as a regression on focal phenotype and mean group phenotype (sensu Lande & Arnold 1983 and quant. evol. genet.)


Price equation: Hamilton’s rule


2. Plug expression for fitness into the Price equation






3. Set genetic relatedness =

4. Ignoring changes in breeding value due to mutation, etc⟶ E[wiΔgi] = 0

5. Hamilton’s rule:



Multilevel/Group selection

“...discussing the evolution of courage and self-sacrifice in man, [Darwin] left a difficulty apparent and unresolved. He saw that such traits would naturally be counterselected within a social group whereas in competition between groups the groups with the most of such qualities would be the ones best fitted to survive and increase.”

“A recent reformulation of natural selection can be adapted to show how two successive levels of the subdivision of a population contribute separately to the overall natural selection”


William D. HamiltonBiosocial Anthropology (1975)

George R. Price (author of “recent reformulation”)


Price equation: Multilevel/Group selection

Suppose there are n groups, each composed of N individuals

Let Wj be the mean fitness in group j and Zj the mean phenotype

Assume that E[wi Δzi] is zero (no mutation, etc)

Thus,


N N

NN

migration




: evolutionary change due to between group selection

: evolutionary change due to within group selection

equation is recursive: Covj[wji, zji] could be further partitioned


Conceptual applications of the Price equation

Evolutionary change has natural selection and transmission components

Natural selection can be partitioned in different ways due to population structure:

shared ancestry or group membership


Price equation (neglecting transmission term)

FTNS :

Inclusive fitness :

Group selection :


Outline









Price equation and responsiveness

Responsiveness model was for one population without structure

We can include population structure using the Price equation

Either the “inclusive fitness” or “group selection” version will work!

Start with inclusive fitness:

Write individual fitness wi so that it includes responsiveness



Inclusive fitness and responsiveness

Recall that in a single population (fitnesses are rescaled)

For individual i with genotype pi (pi = 1 if C-type, pi = 0 if D-type)

Compare with the fitness regression equation

Thus:


Plug cost and benefit:

Into Hamilton’s rule:

The invasion condition becomes

Notably

~ Symmetric in relatedness and responsiveness

~ Relatedness and responsiveness interact to create selection for cooperation when measured in terms of the payoffs of the game


Inclusive fitness and responsiveness



Group selection and responsiveness

Equally, we could start with the group selection expression

Suppose groups are of size N = 2. Between and within group components of selection are:

Adding in responsiveness:

Between group selection > 0:

Within group selection < 0:




Between group selection outweighs within group selection when

This simplifies to the same increase condition as for inclusive fitness

For interaction groups of size N




Between group selection outweighs within group selection when

No within group selection when r = 1 or ρ = 1

Perfect responsiveness or relatedness can lead to the emergence of groups as individuals and group-level adaptations

“Major transitions in evolution” or “transitions in individuality”


ρ = 1r = 1


Outline









Static versus dynamic models

Previous analysis with the Price equation was static

1. Assumed the full genotype distribution known

2. Calculated Δp over one generation

3. Maybe OK for equilibrium or increase conditions

4. But cannot calculate Δp in following generations without ΔVar[p] and potentially many other higher-order moments

5. Thus, no guarantee of convergence

Convergence requires a dynamic analysis and moment closure to make the analysis tractable



Weak selection and “separation of timescales”

Quantitative genetics often obtains moment closure throughassuming constant genetic variances

Implicit is in constant genetic variances is an assumption ofweak selection

Weak selection: coefficients that measure effect of genotype on fitness are “small”

These are called selection coefficients




Δp can be calculated as a Taylor or asymptotic series using a parameter that scales the selection coefficients, ω

First term in the expansion, Δp(0), corresponds to neutral evolution since selection coefficients are zero (ω = 0)

Under neutrality, the only forces changing gene frequencies aremutation, migration, recombination, and genetic drift

Δp(0) is usually easy to calculate (neutral models in pop. gen.)

E.g., Δp(0) = 0 (w/o mutation)




Typically,

Weak selection ( first-order in ω ) leads to very slow changes in higher-order moments of allele frequency (variance, LD, FST, etc)

“Separation of timescales” occurs where the mean changes slowly due to selection and higher order moments change quickly neutrally

Higher-order moments quickly reach “quasi-equilibrium” and can be assumed constant.

⟶ Moment closure and reduction of number of equations



Sequential fixation of mutations

Even with weak selection, coexisting mutations could lead to stable polymorphisms

This complicates analyses of convergence

If genetic drift is strong relative to mutation, then mutations will be fixed or lost before a new mutation arrives

“trait substitution sequence”or “sequential fixation”


AB

C

time

individu

als


Sequential fixation of mutations

“Adaptive dynamics” regime

Short-term evolution:fixed set of alleles

~ no phenotypic novelty

Long-term evolution:continuum of alleles

~ new novelty possible


AB

C

time

individu

als

total population size mutation rate

A a

A

a fixation (πa←A)

mutation (μ)fixation (πA←a)

AAA A

A AA

a aa

aaaa

NT

mutation (μ)


Short-term evolution & sequential fixation

Assume there are two possible alleles, A and a.

NT = total population sizeμ = mutation rate (A ⟶ a & a ⟶ A)

If NT μ log N ≪ 1, only need to track “monomorphic” populations: i.e., fixed for A or a.

As μ ⟶ 0, transitions between monomorphic states given by Λ.



Short-term evolution & sequential fixation

Stationary distribution ( λ ) of transition matrix ( Λ ) gives the fraction of time spent in each monomorphic population

Long-run frequency of A = E[p]:

A is more common than a when:

Determining which allele is more successful means comparing complementary fixation probabilities



Fixation probabilities under weak selection

Aim:

1. Calculate πA←a and πa←A under weak selection

2. Express in terms of population genetics quantitiese.g., coalescence times or coancestry probabilities

1. Write πA←a as a sum




2. Approximate πA←a with a first-order Taylor series

where after some work


Price equation!Neutral fixation probability, π°

=

Price equation



3. Write fitness wi(t) as a function of genotype and selection coefficients

Assume only pairwise interactions affect fitness


additive effects multiplicative effects

4. Combine fitness with fixation probability and “simplify”

Things to note:

1. “additive” interactions depend on genetic identity between gene pairs

2. “multiplicative” interactions depend on identity between gene triplets

3. no assumption so far about population structure or particular pairwise social game played between individuals

4. still time dependent






5. Express fixation probability in terms of coalescence times

After some rearranging…we get expected coalescence times

Analogous expression for πa←A allows us to evaluate



Fixation probability and Hamilton’s rule

“Canonical” version of Hamilton’s rule assumes

1. additive effects

2. two kinds of individuals: relatives and non-relatives

We’ll use a group -structured model like before (n group of size N)

Three classes of individual: self, group mates, non-group mates


N N

NN

migration

Plug wji into fixation probabilities to get

where


Fixation probability and Hamilton’s rule


Hamilton’s rule!


Outline









Social games in a population with island structure

To unpack c, b, and r from Hamilton’s rule, we need to fully specify the demography and the social interaction

Demography:

1. Adults interact socially, mate, and produce offspring. Fertility affected by the social interaction(e.g., neighbors provide resources)

2. Juveniles migrate at rate m to new groups or stay in home group (“hard selection”)

3. Juveniles compete to replace the N adults in each group (density dependent regulation)


N N

NN

m




Social interaction:

1. Pairwise interactions between adults 2. Individuals with allele A cooperate, those with allele a defect 3. Fertility is the average payoff from interaction within the group

B = benefit C = cost D = synergy Prisoner’s dilemma: 0 < D < C Stag hunt game: D > C > 0 Snow drift game: C > 0 & D < C


(C, C) (C, D) (D, C) (D, D)

B – C + D – C B 0




Social interaction:

1. Pairwise interactions between adults 2. Individuals with allele A cooperate, those with allele a defect 3. Fertility is the average payoff from interaction within the group

Fertility of individual i in group j =


(C, C) (C, D) (D, C) (D, D)

B – C + D – C B 0



Finally, calculate coalescence times(Notohara, 1990, JMB; Ladret & Lessard, 2007, TPB)

where M = n N m / (n – 1) = “effective number of migrants”

Putting it all together… (and dropping O(1/n), O(1/N), and O(m))




Observations:

1. No benefit term B !!!Classic result of Taylor (1990): benefits cancel due to competition within groups

2. Recover the “1/3 law” from Nowak et al. (2004, Nature) (M → ∞)

3. Recover “risk dominance” condition for unstructured populations from game theory more generally




This is a result is due to local competition within groups exactly canceling any effect of population structure

Holds for two specific demographic assumptions:(i) hard selection (ii) non-overlapping generations

Alternatives to hard selection:

Soft selection:density dependent regulation then migration (same number of migrants from each group)

Group competition:groups compete for resources then individuals compete within groups




Soft selection:

• worse for cooperation due to increased local competition

Group competition:

• way better for cooperation since there is no local competition!

• now population structure (small M) has a strong effect

• can rearrange in terms of r = FST = 1 / (1 + 2M)


More generally, we can write

where κ = “scaled relatedness”

Scaled relatedness takes into account local competition and other effects of demography on fitness

κ = (σ – 1) / (σ + 1) σ = “structure coefficient” of Tarnita et al. (2009)




Hard selection: κ = 0Soft selection: κ = –1/(N – 1)

Group competition: κ = 1/(1 + 2M) = FST



This is a general result for pairwise social interactions in an island model assuming “sequential fixation” (or “trait substitution”) holds

This shows that we can nicely summarize evolutionary success with:

1. Payoffs from the social game

2. Single index accounting for the effect of population structure (scaled relatedness, κ)


But what about the results from the “static” Price equation?

When using payoffs (B, C, and D), we know now that r → κ to account for effects of population structure, not just relatedness, so:

Moreover, complex population structures will introduce asymmetries due to migration and population size(e.g., irregular graphs, variation in population size)

N N

N

2 N

mm

m


Static versus dynamic models



Further reading

Van Cleve, Jeremy. 2015. Theoretical Population Biology 103:2--26. http://dx.doi.org/10.1016/j.tpb.2015.05.002

Tarnita, Corina E. and Taylor, Peter D.. 2014. American Naturalist 184:477--488. http://dx.doi.org/10.1086/677924

Van Cleve, Jeremy and Akçay, Erol. 2014. Evolution 68:2245--2258. doi:10.1111/evo.12438

Akçay, Erol and Van Cleve, Jeremy. 2012. American Naturalist 179:257-269. http://dx.doi.org/10.1086/663691

Lehmann, Laurent and Rousset, François. 2010. Philosophical Transactions B 365:2599-2617. http://dx.doi.org/10.1098/rstb.2010.0138

Tarnita, Corina E, Antal, Tibor, Ohtsuki, Hisashi, and Nowak, Martin A. 2009. PNAS 106:8601-4. http://dx.doi.org/10.1073/pnas.0903019106

Rousset, François. 2004. Genetic structure and selection in subdivided populations. Princeton University Press, Princeton, N.J..

Frank, Steven A.. 1998. Foundations of Social Evolution. Princeton University Press, Princeton, NJ.


http://dx.doi.org/10.1016/j.tpb.2015.05.002

http://dx.doi.org/10.1086/677924

http://dx.doi.org/10.1086/663691

http://dx.doi.org/10.1098/rstb.2010.0138

http://dx.doi.org/10.1073/pnas.0903019106

Santa Fe Institute


Group-level adaptations & transitions in individuality

How do group-level adaptations evolve?

Do transitions in individuality come before group-level adaptations?

cells multicells social groups societies

level of hierarchy (time?)

cells multicells social groups societies

level of hierarchy (time?)

betw

een

grou

p se

lect

ion

within group selection

betw

een

grou

p se

lect

ion

within group selection


Group-level adaptations & transitions in individuality

How do group-level adaptations evolve?

Do transitions in individuality come before group-level adaptations?

multilevel selection, population genetics and cooperation ... · multilevel selection, ......

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