Multilevel Selection, Population Genetics and Cooperation in Structured Populations
NIMBioS Tutorial: Game Theoretical Modeling of Evolution in Structured Populations
Jeremy Van Cleve University of Kentucky
26 April 2016
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: beneft from improved state
Food improves state ⟶ higher ftness
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY rt noatgtfr nw saMxcnht Then, hat. Mexican a as concen- slug light. known the no form of tight or cells a light into the trate directional moving, moist, cease either very they is is When there it when when and electrolytes, lacks environment slug. the of innate rear the at and shed are kidney, they before liver, system, bacteria,immune and as toxins up simultaneously picking back throughfunctioning to sweep front that from cells slug sentinel the called cells of class discov- recently ered movement a is direct There stalk. front the become the ultimately and at Those cells. constituent among the differences are there Though system, nervous stage. a lacks social slug the the to advantage important an couldmove: moves amoeba individual any slug than farther The and quickly stage. more the solitary discover, they at the bacteria any cells recovering a on effectively drops feed through can it crawls cells cellulose, these it and of rear, As up ways. made important largely sheath some in it around from crawl to begins ( and ammonia from away and heat mound, slightly and light toward a elongates into ( concentrates then center aggregation this which called hours, few process a a After they in toward starve, flowing center cells, more dicty a of and streams great more in concentrate As concentration. highest 3 Figure aa icadtesakaefre ffrel iigaobeta aede ofr hsspotn tutr.()Mcoyt,the Macrocysts, (d) structure. The supporting spores. this or form sorus, to a died and have stalk, that a amoebae disc, living basal formerly a of of of formed stage consisting are sexual body stalk Fruiting the (c) and light. disc towards basal moving slug multicellular Motile (b) lg oefrhradfralne iewe the when time longer a for and farther move Slugs differs but worm, tiny a like looks slug translucent This
utclua tgsof stages Multicellular .discoideum. D. Cuts fOe Gilbert). Owen of (Courtesy itotlu discoideum Dictyostelium Encyclopedia of Animal Behavior
Dictyostelium discoideum A iue3(a) Figure u iue3(b) Figure t h o r ' a grgto ffrel needn el noamliellrbody. multicellular a into cells independent formerly of Aggregation (a) . Author's personal copy s
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Dictyostelium, the Social Amoeba 515 r
s ors poto h asa eysedrbtrigid but slender very a as and mass walls the cellulose ( of form at out to were stalk begin up that slug rise cells the to the of culmination, front called the process a in rc fpsae hssokcne sacse through the accessed for is center center stock stock This col- the postage. previously from of using price obtained performed clones be lected can studies Many Cultured and Collected, Obtained, Are Dictyostelids How perish. slug also smaller new, they the a then form body, of to fruiting cells rear shed encounter other not the enough do or these from bacteria, If the movement. shed normal through their were way during particles. their others made soil up Still they picking between slug. as cells gap bacteria sentinel and a as toxins into themselves sacrifice or above Others so surface, or millimeter soil erect a sporulate the an and up rise comprise may others point, disk ( this stalk, body basal At this fruiting spores. a and up hardy called stalk, flow structure form they cells spores, top the the the of at so and or three-quarters o (2010), vol. 1, pp. 513-519 513-519 (2010), vol. 1, pp. hs oeo h el arfc hi ie ota the that so lives their sacrifice cells the of some Thus, n a iue3(c) Figure l
c Dictyostelium o p y
remaining The die. cells These ).
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Cost: dying as part of the stalk
Beneft: surviving as part of the spore 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. The basal disc and the stalk are formed of formerly living amoebae that have died to form this supporting structure. (d) Macrocysts, the sexual stage of D. discoideum. (Courtesy of Owen Gilbert). NIVERSITY OF ENTUCKY Multilevel selection, pop gen and games in structured pops U K highest concentration. As more and more starve, they in a process called culmination, the cells that were at concentrate in great streams of dicty cells, flowing toward the front of the slug begin to form cellulose walls and a center in a process called aggregation (Figure 3(a)). to rise up out of the mass as a very slender but rigid After a few hours, this center concentrates into a mound, stalk (Figure 3(c)). These cells die. The remaining which then elongates slightly and begins to crawl around three-quarters or so of the cells flow up this stalk, toward light and heat and away from ammonia (Figure 3(b)). and at the top they form hardy spores. At this point, This translucent slug looks like a tiny worm, but differs the spores, stalk, and basal disk comprise an erect from it in some important ways. As it crawls through a structure called a fruiting body (Figure 3(c)). sheath largely made up of cellulose, it drops cells at the Thus, some of the cells sacrifice their lives so that the rear, and these cells can feed on any bacteria they discover, others may rise up and sporulate a millimeter or so above effectively recovering the solitary stage. The slug moves the soil surface, or into a gap between soil particles. more quickly and farther than any individual amoeba could Others sacrifice themselves as sentinel cells picking up move: an important advantage to the social stage. Though toxins and bacteria as they made their way through the the slug lacks a nervous system, there are differences among slug. Still others were shed from the rear of the slug the constituent cells. Those at the front direct movement during their normal movement. If these do not encounter and ultimately become the stalk. There is a recently discov- bacteria, or enough other shed cells to form a new, smaller ered class of cells called sentinel cells that sweep through fruiting body, then they also perish. the slug from front to back picking up toxins and bacteria, functioning simultaneously as liver, kidney, and innate immune system, before they are shed at the rear of the slug. How Dictyostelids Are Obtained, Collected, Slugs move farther and for a longer time when the and Cultured environment lacks electrolytes, when it is very moist, and when there is either directional light or no light. Many studies can be performed using previously col- When they cease moving, the cells of the slug concen- lected clones obtained from the stock center for the trate into a tight form known as a Mexican hat. Then, price of postage. This stock center is accessed through
Encyclopedia of Animal Behavior (2010), vol. 1, pp. 513-519
gene chromosomes
prokaryotes + mitochondria/ eukaryotes chloroplasts clonal sexual reproduction reproduction
independent multicellular living cells organisms
independent social groups individuals (eusociality)
“Major transitions in evolution” or “transitions in individuality” (Maynard Smith and Szathmáry)
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Cooperation occurs at two “scales”
“Within a group”
“Between/among groups”
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Cooperation occurs at two “scales”
1. Plastic behaviors “Within a group”
“Between/among groups”
2. Kin & Group selection
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY 1. Plastic behaviors
Game theory (Nash equilibrium / ESS)
Tit for tat / punishment / reputation / etc Reciprocity or responsiveness
2. Kin & Group selection
Multilevel selection combines responsiveness with kin/group processes through measures of population structure such as relatedness
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY 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: fxation probability & trait substitution
Social games in an island-model using fxation probability
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY 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: fxation probability & trait substitution
Social games in an island-model using fxation probability
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Plastic behavior and repeated games
Plastic behavior requires repeated interactions (i.e., a repeated game)
Cooperate (C) Defect (D) Payoff to Strategies in the repeated game Beneft – Cost – Cost determine how individuals respond Cooperate (C) to the actions of social partners Payoff to Beneft 0 Defect (D)
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Plastic behavior and repeated games
Strategy Description ALLD Always play D “always defect” GRIM Play C but switch to D once opponent defects “grim trigger” TFT Start with C and then repeat the opponent's last move “tit for tat” STFT Start with D and then repeat the opponent's last move “suspicious tit for tat” TF2T Play C unless opponent played D in the last two moves “tit for two tats” WSLS Start with C and then play C if and only if the last payoff “win stay, lose shift” from the last round was R or T Table 14.2 (Broom and Rychtář, 2013)
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Plastic behavior and repeated games
Payoff (ftness) 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 ω ω ω ω ω ω ω ω
GRIM C D D D D D D D D STFT D C D D D D D D D
Longer interactions allow individuals to obtain more information about their partner’s type
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Plastic behavior and repeated games
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 C TF2T C C C C C C C C C TFT C D D D D D D D D ALLD D D D D D D D D D TF2T C C D D D D D D D ALLD D D D D D D D D D
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Responsiveness and the evolution of cooperation
Responsiveness = “direct reciprocity” or “reciprocal altruism”
Measure responsiveness ( ρ ) specifcally 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
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Responsiveness and the evolution of cooperation
Two C-types always cooperate
Two D-types always defect
C-type vs D-type: [mij] = probability state i ⟶ j where i, j ∈ {(C, C), (C, D), (D, C), (D, D)}
(C, C) (C, D) (D, C) (D, D) (C, C) (C, D) . ӝ ӝ (D, C) (D, D)
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Responsiveness and the evolution of cooperation
Equilibrium distribution of Markov chain is given by: v M = v W Ӟ ࠀ џ ࠀ ࠀ Ӟ Ӟ Ӟ Ӟ ෝ џ ࠀ џ ࠀ џ ࠀ џ ࠀ ෞ Payoffs: (focal, partner) (C, C) (C, D) (D, C) (D, D) B – C > 0 – C B 0 Assume there is no discounting: Fitness = w = payoffs from equilibrium actions in the game
Suppose that p is the frequency of the C-type
# $ $ # X Q # $ Q ӝ $ ඛ ග ӝ Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Responsiveness and the evolution of cooperation
# $ $ # Q # $ X Q # $ Q ӝ X% $ ඛ ග ӝ Q # $ # $
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)
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY 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: fxation probability & trait substitution
Social games in an island-model using fxation probability
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Measuring evolution in structured populations
Classical one/multi-locus theory measures Δ[gentoype frequency]
x111 A1B1
x1 A1B1 x211 A2B1 Pop1 x2 A B x121 A1B2 2 1 x221 A2B2 x3 A1B2 x112 A1B1 x4 A2B2 x212 A2B1 Pop2 x122 A1B2 State space grows rapidly x222 A2B2
Stability analysis of equilibria requires eigenvalues of large matrices
Stochastic models are hard to analyze
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Measuring evolution in structured populations
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 simplifes: 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, fnd ΔVar[p] and repeat.
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Measuring evolution in structured populations
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
[ TI ӝ “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
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY 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
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY 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
[ Q [ QJ[J ࿊J ࿊J ၹJ ၹJ
piʹ ziʹ : frequency & phenotype of descendants of individual i in next generation
pi zi : frequency and phenotype of individual i in current generation
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Price equation
[ Q [ QJ[J ࿊J ࿊J ၹJ ၹJ QJXJ Q XJ ĕUOFTT PG JOEJWJEVBM J ࿊J X [ [J [J ࿊J
X [ QJXJ [J [J QJX [J ၹJ අ ආ ၹJ
QJ[J XJ X QJXJ [J ၹJ අ ආ ၹJ
$PW XJ [J & XJ [J ඌ උ ඌ උ Price equation
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Price equation
X [ $PW XJ [J & XJ [J ඌ උ ඌ උ
Cov[wi, zi] : evolutionary change in z due to natural selection – statistical association between ftness and phenotype
E[wi Δzi] : evolutionary change in z due to imperfect transmission – e.g. mutation, non-random mating, segregation distortion
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Price equation: one-locus selection w/ mutation
[J QJ #FSOPVMMJ Q Q QJ QJ ࿊J XJ ྡྷTQJ QJ ސ QJ ސ ސ $PW XJ QJ & XJ QJ ඌ උ ඌ උ
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Price equation: one-locus selection w/ mutation
[J QJ #FSOPVMMJ Q Q QJ QJ ࿊J XJ ྡྷTQJ QJ ސ QJ ސ ސ $PW XJ QJ TQ Q & XJ QJ Q T ආ අ ސ උ ඌ ඌ උ
X Q $PW XJ QJ & XJ QJ ඌ උ Q ඌ T TQ උ Q ސඅ ආ *G Q T Q Mutation-selection balance T ྭސෳ ෴ T XIFSF ސ ࠔ ࠔ Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Conceptual applications of the Price equation
X [ $PW XJ [J & XJ [J ඌ උ ඌ උ
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
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Fisher’s fundamental theorem of natural selection
“The rate of increase in ftness of any organism at any time is equal to its genetic variance in ftness at that time.”
Ronald A. Fisher The Genetical Theory of Natural Selection (1930)
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Fisher’s fundamental theorem of natural selection
“The rate of increase in ftness of any organism at any time is equal to its genetic variance in ftness at that time.”
No genetic variance implies no increase in ftness (equilibrium condition)
Variance > 0 implies that ftness 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)
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Price equation: FTNS
1. Regress phenotype on genes
[J CKYJK J HJ J ၹK ܤ ܤ
CKYJK HJ iCSFFEJOH WBMVFw ၹK
• Properties of the linear regression
$PW HJ J උ ܤඌ & SFTJEVBM & J උ ඌ උܤ ඌ
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Price equation: FTNS
2. Plug zi into the Price equation
[ H J J J ܤ
X [ X H X X H
PW XJ HJ & XJ HJ$ ܤ ඌ උ ඌ උ
3. The FTNS measures Δw : ⟶ zi = wi = gi + δi
X X X H $PW HJ J HJ & XJ HJ ඌ උ ඌ ܤ උ 7BS HJ & XJ HJ උ ඌ උ ඌ
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Price equation: FTNS
4. Fisher excluded changes in ftness due to “deterioration of the environment”
= E[wiΔgi] = 0
(no change breeding value due to mutation, etc)
5. FTNS: X X 7BS HJ HFOFUJD WBSJBODF උ ඌ
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Inclusive ftness 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. Hamilton Journal of Theoretical Biology (1964)
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Inclusive ftness 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 × beneft > 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 ftness effect = – c + r b
Not an “extension” or special kind of ftness; rather, method of accounting for social interactions (Akçay & Van Cleve, 2016, Phil. Trans. R. Soc. B)
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Price equation: Hamilton’s rule
1. Write ftness as a regression on focal phenotype and mean group phenotype (sensu Lande & Arnold 1983 and quant. evol. genet.)
XJ [J D Z C J Z HSPVQ NFBO QIFOPUZQF ܀ ܰ H D HC SFTJEVBMT J SFHSFTTJPO PG ĕUOFTT D DPTU ܀ PO GPDBM QIFOPUZQF SFHSFTTJPO PG ĕUOFTT C PO NFBO HSPVQ CFOFĕU QIFOPUZQF
H NFBO HSPVQ CSFFEJOH WBMVF
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Price equation: Hamilton’s rule
2. Plug expression for ftness into the Price equation
XJ [J D Z C J ܀ ܰ HJ D HC SFTJEVBMT ܀
X [ X H $PW XJ HJ & XJ HJ ඌ උ ඌ උ $PW J HJ D HC SFTJEVBMT HJ & XJ HJ ඌ උ ඌ ܀උ D 7BS HJ C $PW HJ H & XJ HJ ඌ උ ඌ උ ඌ උ
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Price equation: Hamilton’s rule
$PW H H 3. Set genetic relatedness = S J 7BSඋ H ඌ J උ ඌ X H D S C & XJ HJ ࿙ උ ඌ 4. Ignoring changes in breeding value due to mutation, etc
⟶ E[wiΔgi] = 0
5. Hamilton’s rule: X [ D S C ல
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Multilevel/Group selection
“...discussing the evolution of courage and self-sacrifce 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 ftted 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”
George R. Price William D. Hamilton (author of “recent Biosocial Anthropology (1975) reformulation”)
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Price equation: Multilevel/Group selection
Suppose there are n groups, each composed of migration N individuals N N
Let Wj be the mean ftness in group j and Zj the mean phenotype
Assume that E[wi Δzi] is zero (no mutation, etc) N N Thus, X [ $PW XJ [J ඌ උ
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Price equation: Multilevel/Group selection
X [ $PW XJ [J XKJ[KJ XKJ [KJ /O /O /O ၹK J ၹK J ၹK J ඌ උ ˷ ˷ ˷ X [ X [ /O KJ KJ O / KJ / KJ ၹK J ၹK ၹJ ၹJ ˷ XKJ [KJ XKJ [KJ O / / /O /O ၹK ၹJ ၹJ ၹK J ၹK J ˷ ˷
$PW 8K ;K & $PWK XKJ [KJ භ ම භ භ මම
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Price equation: Multilevel/Group selection
X [ $PW XJ [J $PW 8K ;K & $PWK XKJ [KJ මම ම භ භ භ ඌ උ
$PW 8K ;K : evolutionary change due to between group selection භ ම
& $PWK XKJ [KJ : evolutionary change due to within group selection භ භ මම equation is recursive: Covj[wji, zji] could be further partitioned
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Conceptual applications of the Price equation
Price equation (neglecting transmission term)
FTNS : X X 7BS HJ උ ඌ Inclusive ftness : X [ D S C ࿙ Group selection : X [ $PW 8K ;K & $PWK XKJ [KJ මම ම භ භ භ
Evolutionary change has natural selection and transmission components
Natural selection can be partitioned in different ways due to population structure:
shared ancestry or group membership
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Outline
The two scales of cooperation
Simple model of responsiveness within a population
Evolution in structured populations: the Price equation
Multilevel model with the Price equation
Evolution in structured populations: fxation probability & trait substitution
Social games in an island-model using fxation probability
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Price equation and responsiveness
Responsiveness model was for one population without structure
We can include population structure using the Price equation
Either the “inclusive ftness” or “group selection” version will work!
Start with inclusive ftness: X [ D S C ࿙ Write individual ftness wi so that it includes responsiveness
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Inclusive ftness and responsiveness
Recall that in a single population (ftnesses are rescaled)
X$ Q # $ # $ X% Q # $ For individual i with genotype pi (pi = 1 if C-type, pi = 0 if D-type)
XJ QJ # $ Q # $ අ ආ Compare with the ftness regression equation
XJ HJ D HC SFTJEVBMT ܀ Thus: D # $C# $
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Inclusive ftness and responsiveness
Plug cost and beneft: D # $C# $ Into Hamilton’s rule: X [ D S C $ # ࿙# $ S # S අ$ ආS The invasion condition becomes අ ආ අ ආ
# S $ S 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
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Group selection and responsiveness
Equally, we could start with the group selection expression
X [ $PW 8K ;K & $PWK XKJ [KJ මම ම භ භ භ Suppose groups are of size N = 2. Between and within group components of selection are:
$PW 8K ;K S C D & $PWK XKJ [KJ S C D භ ම ࿙ භ භ මම ࿙ Adding in responsiveness: D # $C# $ Between group selection > 0: $PW 8K ;K S # $ භ ම ࿙
Within group selection < 0: & $PWK XKJ [KJ S # $ භ භ මම ࿙
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Group selection and responsiveness
Between group selection outweighs within group selection when
S # $ S # $ This simplifes to the same increase condition as for inclusive ftness
# S $ S For interaction groups of size N
# S / $ / S S / අ ආ
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Group selection and responsiveness
Between group selection outweighs within group selection when
S # $ S # $ No within group selection when r = 1 ρ = 1 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”
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY 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: fxation probability & trait substitution
Social games in an island-model using fxation probability
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY 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
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Weak selection and “separation of timescales”
Quantitative genetics often obtains moment closure through assuming constant genetic variances
Implicit is in constant genetic variances is an assumption of weak selection
Weak selection: coefficients that measure effect of genotype on ftness are “small”
These are called selection coefficients
XJ T QJ XJ HJ D HC ܀
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Weak selection and “separation of timescales”
Δp can be calculated as a Taylor or asymptotic series using a parameter that scales the selection coefficients, ω
FH Q Q Q Q 0 ࿊ ࿊࿊ ӝ ӗ ި ඛި ග ި 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 are mutation, 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)
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Weak selection and “separation of timescales”
Typically, JG Q Q Q ࿊ ި ྭ UIFO 7BS Q 7BS Q 7BS Q ࿊ ި උ ඌ උ ඌ ྭ උ ඌ Weak selection ( frst-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
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Sequential fxation of mutations
Even with weak selection, coexisting mutations could lead to stable polymorphisms
This complicates analyses of convergence
time If genetic drift is strong relative A to mutation, then mutations B will be fxed or lost before a C new mutation arrives individuals
“trait substitution sequence” or “sequential fxation”
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Sequential fxation of mutations
“Adaptive dynamics” regime /5 MPH /5 ސ ཫ total population size mutation rate Short-term evolution: fxed set of alleles ~ no phenotypic novelty time A B Long-term evolution: C
continuum of alleles individuals ~ new novelty possible
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Short-term evolution & sequential fxation
mutation ( ) fxation ( ) Assume there are two possible μ a πa A alleles, A and a. ← A A a a a N A A a a N = total population size T A A a T A A a a μ = mutation rate (A ⟶ a & a ⟶ A) A fxation (πA a) mutation (μ) ← If NT μ log N ≪ 1, only need to track “monomorphic” populations: i.e., fxed for A or a.
ɢ ɢ As μ ⟶ 0, transitions between ͌ B " B " ɢ ɢ monomorphic states given by Λ. ญ " B " Bฎ ɢB " QSPCBCJMJUZ PG POF B öYJOH JO B QPQVMBUJPO PG " ɢ" B QSPCBCJMJUZ PG POF " öYJOH JO B QPQVMBUJPO PG B Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Short-term evolution & sequential fxation
Stationary distribution ( λ ) of transition matrix ( Λ ) gives the fraction of time spent in each monomorphic population
ɢ ɢ ѥ͌ ͌ ѥ " B B " ɢ ɢ ɢ ɢ ஔ ෳ " B B " " B B " ෴ Long-run frequency of A = E[p]: ɢ & Q " B ɢ ɢ උ ඌ " B B " A is more common than a when:
& Q ɢ" B ɢB " උ ඌ ல Determining which allele is more successful means comparing complementary fxation probabilities
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Fixation probabilities under weak selection
Aim:
1. Calculate πA←a and πa←A under weak selection 2. Express in terms of population genetics quantities e.g., coalescence times or coancestry probabilities
1. Write πA←a as a sum
ɢ" B & Q Q උ ࿘ ] ඌ
Q ࿘ & ͅQ U Q ၹU ඡ ຨ ජ
Q ࿘ 1S Q U Q & ͅQ U Q U ၹU QၹU උ ] ඌ ඡ ຨ ජ Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Fixation probabilities under weak selection
ɢ" B Q ࿘ 1S Q U Q & ͅQ U Q U ၹU QၹU උ ] ඌ ඡ ຨ ජ Neutral fxation probability,ඪ π° วศศษศศสPrice equation!
2. Approximate πA←a with a frst-order Taylor series
Eɢ ɢ ɢ " B ѱ 0 ѱ " B Eѱ ඛ ග where after some work
Eɢ" B E ࿘ & & ͅQ U Q U Eѱ U Eѱ
ၹ ญ භ ඡ = ຨ ජමฎ Price equation E ࿘ & $PW X U Q U Eѱ J J ၹU ญ භ උ ඌමฎ Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Fixation probabilities under weak selection
Eɢ" B E ࿘ & $PW X U Q U Eѱ Eѱ J J ၹU ญ භ උ ඌමฎ 3. Write ftness wi(t) as a function of genotype and selection coefficients
Assume only pairwise interactions affect ftness
/5 /5 XJ ѱ TJ L QL TJ L L QL QL 0 ѱ ၹL ˷ Lၹ L ˷ ඛ ග วศษศส วศศศษศศศส
additive effects multiplicative effects
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Fixation probabilities under weak selection
4. Combine ftness with fxation probability and “simplify”
ѱ /5 /5 /5 ࿘ ɢ" B ɢ TJ L & QJQL TJ LL & QJQLQL 0 ѱ /5 ၹU ၹJ ၹL ˷ භ ම Lၹ L ˷ භ ම ඛ ග 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
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Fixation probabilities under weak selection
5. Express fxation probability in terms of coalescence times
& Q Q 1S BMMFMF PSJHJOBMMZ XBT " 1S DPBMFTDFODF CFGPSF UJNF U J L භ ම උ ඌ < > 1S 5 U / JL 5 භ ම After some rearranging…we get expected coalescence times
ѱ /5 /5 /5 ɢ T & 5 & 5 T 0 ѱ " B J L JL JLL J LL /5 / 5 ၹJ ၹL ˷ භ ම Lၹ L භ ම ˷ ඛ ග
Analogous expression for πa←A allows us to evaluate
ɢ" B ɢB " Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY 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 migration N N 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 /5 XKJ ѱ TJ L QL 0 ѱ ၹL ˷ ඛ ග ѱ DQKJ CQK J C D Q K 0 ѱ න = අ ආ = ඛ ග Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Fixation probability and Hamilton’s rule
XKJ ѱ DQKJ CQK J C D Q K 0 ѱ න = අ ආ = ඛ ග
Plug wji into fxation probabilities to get
ɢ" B ɢB " D S C லHamilton’sวศศษศศส rule! where
& 5 & 5 S ' ඡ & ජ 5 ඡ ˷ ජ 45 ඡ ජ & 5 DPBMFTDFODF UJNF CFUXFFO BMMFMFT JO EJòFSFOU HSPVQT
& 5ඡ ජ DPBMFTDFODF UJNF CFUXFFO BMMFMFT JO UIF TBNF HSPVQ ඡ ˷ ජ
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY 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: fxation probability & trait substitution
Social games in an island-model using fxation probability
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY 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: m 1. Adults interact socially, mate, and produce offspring. Fertility affected by the social interaction N N (e.g., neighbors provide resources)
2. Juveniles migrate at rate m to new groups or stay in home group (“hard selection”) N N 3. Juveniles compete to replace the N adults in each group (density dependent regulation)
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY 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
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 (C, C) (C, D) (D, C) (D, D)
B – C + D – C B 0 B = beneft C = cost D = synergy Prisoner’s dilemma: 0 < D < C Stag hunt game: D > C > 0 Snow drift game: C > 0 & D < C
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY 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
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 (C, C) (C, D) (D, C) (D, D)
B – C + D – C B 0
Fertility of individual i in group j = #QK J $QKJ %QKJQK J = =
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Social games in a population with island structure
Finally, calculate coalescence times (Notohara, 1990, JMB; Ladret & Lessard, 2007, TPB)
& 5 / & 5 / 5 5 / . . / ඡ ˷ ජ ඡ ජ ෳ ෴ 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))
ɢ ѱ $ % " B / . 5 ෳ ෳ ෴෴ ɢ ѱ $ % B " / . 5 ෳ ෳ ෴෴
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Social games in a population with island structure
ɢ ѱ $ % " B / . 5 ෳ ෳ ෴෴ ɢ ѱ $ % B " / . 5 ෳ ෳ ෴෴ Observations: 1. No beneft term B !!! Classic result of Taylor (1990): benefts cancel due to competition within groups 2. Recover the “1/3 law” from Nowak et al. (2004, Nature) (M → ∞) ɢ % $ " B / 5 ல 3. Recover “risk dominance” condition for unstructured populations from game theory more generally
ɢ" B ɢB " % $ Multilevel selection, pop gen and games in structured pops ல UNIVERSITY OF KENTUCKY Social games in a population with island structure
ɢ" B ɢB " % $ ல This is a result is due to local competition within groups exactly canceling any effect of population structure
Holds for two specifc 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
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Social games in a population with island structure
Soft selection: # $ % ɢ ɢ % $ " B B " / ல • worse for cooperation due to increased local competition
Group competition: # $ % ɢ ɢ % $ " B B " . ல • 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) S ɢ ɢ $ S# % " B B " ல
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Social games in a population with island structure
More generally, we can write Ѥ ɢ ɢ $ Ѥ# % " B B " ல where κ = “scaled relatedness”
Scaled relatedness takes into account local competition and other effects of demography on ftness
κ = (σ – 1) / (σ + 1) σ = “structure coefficient” of Tarnita et al. (2009)
Hard selection: κ = 0 Soft selection: κ = –1/(N – 1)
Group competition: κ = 1/(1 + 2M) = FST
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Social games in a population with island structure
Ѥ ɢ ɢ $ Ѥ# % " B B " ல This is a general result for pairwise social interactions in an island model assuming “sequential fxation” (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)
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Static versus dynamic models
But what about the results from the “static” Price equation?
# S $ S When using payoffs (B, C, and D), we know now that r → κ to account for effects of population structure, not just relatedness, so:
# ѪѤ N N $ Ѫ Ѥ m m Moreover, complex population structures will introduce N asymmetries due to migration and population size (e.g., irregular graphs, variation in population size) m 2 N
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Further reading
Van Cleve, Jeremy. 2015. Teoretical 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.
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY 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?)
Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY Multilevel selection, pop gen and games in structured pops pop gen and games in structured Multilevel selection, adaptations? come before group-level transitionsDo inindividuality adaptations evolve? How dogroup-level Group-level adaptations &
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