Tom Wenseleers Dept. of Biology, K.U.Leuven

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Theoretical Modelling in Biology (G0G41A ) Pt I. Analytical Models IV. Optimisation and inclusive fitness models. Tom Wenseleers Dept. of Biology, K.U.Leuven. 28 October 2008. Aims. last week we showed how to do exact genetic models - PowerPoint PPT Presentation

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  • Theoretical Modelling in Biology (G0G41A ) Pt I. Analytical Models

    IV. Optimisation and inclusive fitness modelsTom WenseleersDept. of Biology, K.U.Leuven28 October 2008

  • Aimslast week we showed how to do exact genetic modelsaim of this lesson: show how under some limiting cases the results of such models can also be obtained using simpler optimisation methods (adaptive dynamics)discuss the relationship with evolutionary game theory (ESS)plus extend these optimisation methods to deal with interactions between relatives (inclusive fitness theory / kin selection)

  • General optimisation method: adaptive dynamics

  • Optimisation methodsin limiting case where selection is weak (mutations have small effect) the equilibria in genetic models can also be calculated using optimisation methods (adaptive dynamics)first step: write down invasion fitness w(y,Z) = fitness rare mutant (phenotype y) fitness of resident type(phenotype Z)if invasion fitness > 1 then fitness mutant > fitness resident and mutant can spreadevolutionary dynamics can be investigated using pairwise invasibility plots

  • Pairwise invasibility plots= contour plot of invasion fitnessResident trait ZMutant trait y invasion possible fitness rare mutant > fitness resident type

    invasion impossible fitness rare mutant > fitness resident type

    one trait substitution

    evolutionary singular strategy ("equilibrium")

  • Evolutionary singular strategySelection for a slight increase in phenotype is determined by the selection gradient

    A phenotype z* for which the selection differential is zero we call an evolutionary singular strategy. This represents a candidate equilibrium.

  • Reading PIPs: Evolutionary Stabilityis a singular strategy immune to invasions by neighbouring phenotypes? yes evolutionarily stable strategy (ESS) i.e. equilibrium is stable (local fitness maximum)Resident trait zMutant trait yyesResident trait zMutant trait ynoinvinvno invno inv

  • Reading PIPs: Invasion Potentialis the singular strategy capable of invading into all its neighbouring types?Resident trait ZMutant trait yyesResident trait ZMutant trait ynono invno invinvinvinvinvno invno inv

  • Reading PIPs: Convergence Stabilitywhen starting from neighbouring phenotypes, do successful invaders lie closer to the singular strategy? i.e. is the singular strategy attracting or attainable D(Z)>0 for ZB Resident trait ZMutant trait yyesResident trait ZMutant trait ynono invno invinvinvinvinvno invno inv

  • Reading PIPs: Mutual Invasibilitycan a pair of neighbouring phenotypes on either side of a singular one invade each other? w(y1,y2)>0 and w(y2,y1)>0, true when A>-BResident trait ZMutant trait yyesResident trait ZMutant trait ynono invno invinvinvinvinvno invno inv

  • Typical PIPsResident trait ZMutant trait yATTRACTORno invno invinvinvResident trait ZMutant trait yinvinvno invno invREPELLORstable equilibrium "CONTINUOUSLY STABLE STRATEGY"unstable equilibrium

  • Two interesting PIPsGARDEN OF EDENBRANCHING POINTevolutionarily stable, but not convergence stable (i.e. there is a steady state but not an attracting one)convergence stable,but not evolutionarily stable "evolutionary branching"Resident trait zMutant trait yinvinvno invno invResident trait zMutant trait yinvinv

  • Eightfold classification(Geritz et al. 1997)(1) evolutionary stable, (2) convergence stable, (3) invasion potential, (4) mutual invasibilityrepellor repellor"branching point" attractor attractor attractor "garden of eden" repellor

  • evol. branchingevol. attractorsevol. repellorsevolutionary stable, B < 0invasion potential, A > 0convergence stable A > BG. Edenmutually invasible A > -B

  • Application: game theory

  • Game theory"game theory": study of optimal strategic behaviour, developed by Maynard Smithextension of economic game theory, but with evolutionary logic and without assuming that individuals act rationallyfitness consequences summarized in payoff matrixhawk-dove game

  • Two types of equilibriaevolutionarily stable state: equilibrium mix between different strategies attained when fitness strategy A=fitness strategy Bevolutionarily stable strategy (ESS): strategy that is immune to invasion by any other phenotypecontinuously-stable ESS: individuals express a continuous phenotypemixed-strategy ESS: individuals express strategies with a certain probability (special case of a continuous phenotype)

  • Calculating ESSse.g. hawk-dove game earlier we calculated that evolutionarily stable state consist of an equilibrium prop. of V/C hawks what if individuals play mixed strategies? assume individual 1 plays hawk with prob. y1 and social interactant plays hawk with prob. y2, fitness of individual 1 is then w1(y1, y2)=w0+(1-y1).(1- y2).V/2+y1.(1- y2).V+y1. y2.(V-C)/2invasion fitness, i.e. fitness of individual playing hawk with prob. y in pop. where individuals play hawk with prob. Z is w(y,Z)=w1(y,Z)/w1(Z,Z)ESS occurs when

    true when z*=V/C, i.e. individuals play hawk with probability V/C This is the mixed-strategy ESS.

  • Extension for interactions between relatives: inclusive fitness theory

  • Problemin the previous slide the evolutionarily stable strategy that we found is the one that maximised personal reproductionbut is it ever possible that animals do not strictly maximise their personal reproduction?William Hamilton: yes, if interactions occur between relatives. In that case we need to take into account that relatives contain copies of one's own genes. Can select for altruism (helping another at a cost to oneself) = inclusive fitness theory or "kin selection"

  • Inclusive fitness theorycondition for gene spread is given by inclusive fitness effect = effect on own fitness + effect on someone else's fitness.relatedness

    relatedness = probability that a copy of a rare gene is also present in the recipient e.g. gene for altruism selected for whenB.r > C = Hamilton's rule

  • Calculating costs & benefits in Hamilton's rulee.g. hawk-dove game assume individual 1 plays hawk with prob. y1 and social interactant plays hawk with prob. y2, fitness of individual 1 is then w1(y1, y2)=w0+(1-y1).(1- y2).V/2+y1.(1- y2).V+y1. y2.(V-C)/2 and similarly fitness of individual 2 is given by w2(y1, y2)=w0+(1-y1).(1- y2).V/2+y2.(1- y1).V+y1. y2.(V-C)/2inclusive fitness effect of increasing one's probability of playing hawk ESS occurs when IF effect = 0 z*=(V/C)(1-r)/(1+r)

  • Calculating relatednessNeed a pedigree to calculate r that includes both the actor and recipient and that shows all possible direct routes of connection between the two Then follow the paths and multiply the relatedness coefficients within one path, sum across paths

  • r = 1/2 x 1/2 = 1/4

  • r = 1/2 x 1/2 + 1/2 x 1/2 = 1/2

  • r = 1/2 x 1/2 + 1 x 1/2 = 3/4(c) Full-sister in haplodiploid social insectsQueenHaploid father 1ABCAC, BCAC

  • Class-structured populationssometimes a trait affects different classes of individuals (e.g. age classes, sexes)not all classes of individuals make the same genetic contribution to future generationse.g. a young individual in the prime of its life will make a larger contribution than an individual that is about to die taken into account in concept of reproductive value. In Hamilton's rule we will use life-for-life relatedness = reproduce value x regression relatednesss

  • E.g. reproductive value of males and females in haplodiploidsMQxMQfrequency of allele in queens in next generation pf=(1/2).pf+(1/2).pm frequency of allele in males in next generation pm=pf

    if we introduce a gene in all males in the first generation then we initially have pm=1, pf=0; after 100 generations we get pm=pf=1/3 if we introduce a gene in all queens in the first generation then we initially have pm=0, pf=1; after 100 generations we get pm=pf=2/3 From this one can see that males contribute half as many genes to the future gene pool as queens. Hence their relative reproductive value is 1/2. Regression relatedness between a queen and a son e.g. is 1, but life-fore-life relatedness = 1 x 1/2 = 1/2

    Formally reproductive value is given by the dominant left eigenvector of the gene transmission matrix A (=dominant right eigenvector of transpose of A).

    (1) Is a singular phenotype immune to invasions by neighboring phenotypes? (2) When starting from neighboring phenotypes, do successful invaders lie closer to the singular one? (3) Is the singular phenotype capable of invading into all its neighboring types? (4) When considering a pair of neighboring phenotypes to both sides of a singular one, can they invade into each other?

    The eight possible generic local configurations of the pairwise invasibility plotand their relation to the second-order derivatives of sy(z). Inside the shaded regions within each separate plot sy(z) is >1, i.e the mutant can invade.