Biodiversity arising from evolutionary games between mutants

Transcription

1 Biodiversity and Environment: Viability and Dynamic Games Perspectives November 7 th, 03 Biodiversity arising from evolutionary games between mutants Arne Traulsen Research Group for Evolutionary Theory MPI for Evolutionary Biology August-Thienemann-Str Plön traulsen@evolbio.mpg.de

2 Biodiversity and Environment: Viability and Dynamic Games Perspectives November 7th, 03 Biodiversity arising from evolutionary games between mutants Arne Traulsen Research Group for Evolutionary Theory MPI for Evolutionary Biology August-Thienemann-Str Plön

3 Outline 3 Motivation Mutant games Fixation probabilities Average diversity Does fitness increase or decrease? Another approach with truly novel mutations

4 Games appear on all levels For this reason, evolutionary game theory is ideally suited for modeling frequency-dependent selection. In general, game theoretic models are more manageable and tractable than their quantitative genetic counterparts. Brown & Vincent, 987 Emerging fields as diverse as metabolic control networks within cells and evolutionary psychology, for example, should benefit from game theory. The main themes of evolutionary psychology include cooperation and communication among individuals and are therefore intrinsically game theoretic. Nowak & Sigmund, An alternative to treating organisms as evolving in a fixed environment or fitness landscape is coevolution. In coevolution, organisms interact and their interactions drive each to evolve. Goldenfeld & Woese, 0 References

5 Usually, we study finite strategy sets P P 0% P = 0.0 t = 99.0 D C C C 0% = 0.08 t = 50.5 D D 00% D C Replicator dynamics Low mutation rates High mutation rates 5 Hauert et al., Science (007); Traulsen et al., PNAS (009); Hilbe & Traulsen, Sci. Rep. (0); Garcia & Traulsen, PLoS One (0)

6 Evolutionary branching into known types a 0 b 3 8 Time Time 0 4 Mating probability Ecological character A Ecological character Evolutionary branching Figure Convergence to disruptive selection a, Evolutionary branching in the individual-based asexual model: at the branching point x 0 ¼ 0, the population splits into two morphs. Three insets show x-ε x* fitness x+ε functions x-ε x* x+ε (continuous curves) generated by the ecological interactions at different points in time (indicated by horizontal dotted lines). Selection changes from directional to disruptive when investment evolution reaches x 0. The resource distribution K(x) has its maximum at x 0 and is x x+ε x* x x+ε shown for comparison (dashed curve). b, As in a, but with multilocus genetics for Attractor the ecological character and random mating. Shading represents phenotype distributions (5 diploid and diallelic B loci result in possible phenotypes). Despite ESS no invasion invasion investment Difference in ecological or marker character Mating character disruptive selection at the branching point (see insets), branching does not occur. (close to 0) correspond to random mating. x x+ε x* x x+ε Fig. 3. Examples of simultaneous occurrences of a branchingp point (dashed line) and a repellor (dashdotted line) in the continuous snowdrift game. (A) BðxÞ0 b ffiffi x, C(x) 0 ln(cx þ ) (b 0 and c 0 0.6); Repellor 6 if the population Dieckmann starts to the& right Doebeli, of the repellor Nature at(999); xˆ, 3:9, cooperative Doebeli et investments al., Science continue (004) to increase until the upper limit of the trait interval is reached (inset). However, if started below xˆ, Cooperation dominates - 0 Figure Mating probabilities as determined by mating character and differe in ecological or marker character between mates. The mating character m scaled to vary between (all alleles) and + (all þ alleles). Mating probabil vary with differences in either ecological or marker character, depending on scenario. If the mating character in the focal individual is close to +, it has a h probability of mating with similar individuals. If its mating character is close to is more likely to mate with dissimilar individuals. Intermediate mating charac - 0 R EPORTS

7 Some referee s comments But, given that the authors argue that mutations mimick exploration in a cultural context, then I find the model too restrictive, as here exploration is limited; it means re-injection of more of the same (at most) 4-types of players. For large mutation rates, this becomes kin to random choice within a very limited set of options. 3rd referee, 008 Reducing EGT to analyzing "predefined" strategies is like saying: Every possible species was created at the beginning and only those who survived the competition are left over and explain the species we see now... 3rd referee, 03 Most existing approaches to formulating evolutionary dynamics mathematically [...] share the limitation that the space in which evolution takes place is fixed. A bona fide game theory approach to evolution would allow the game rules themselves to change as a function of the state of the players and their intrinsic dynamics. Goldenfeld & Woese, 0 7

8 Innovative mutations Finite alleles models Mutations can only lead to types that are predefined Back-mutations are possible Infinite alleles models Every mutation leads to something new No back-mutations µ C!D µ D!C µ D!C µ P!C µ C!D µ D!C µ P!C µ P!D µ C!P µ D!P µ P!C µ P!D A 8

9 Game theoretical models of many alleles? Usual assumptions in population genetics Mutations have a fixed fitness advantage They arise with a certain mutation rate Mutations either compete with each other or reach fixation sequentially Evolutionary game dynamics The fitness of mutations depends on type and on the abundance of other types Fixed fitness is a particular special case 9 The dynamics is comparable to population genetics

10 Outline Motivation Mutant games Fixation probabilities Average diversity Does fitness increase or decrease? Another approach with truly novel mutations 0 References

11 Mutant games Constant selection Frequency dependent selection Probability density fitness values F W W a M a M a a Fitness is replaced by a linear function of the relative abundance

12 Only finite populations allow extinction In infinite populations, all types are always present In finite populations, any new mutation can take over the population, go extinct or coexist Evolutionary game dynamics as birth death process Discrete time Recovers the replicator dynamics in the appropriate limit Nowak et al., Nature (004); Traulsen et al., PRL (005); PRL (0)

13 Dynamics in mutant games Mutation event S S S 3 S S S 3 Mutation S S S 3 S 4 S X S X S 3 X S 4 X X X X Extinction event S S S 3 S 4 S S S 3 S 4 Extinction S S S 3 S 4 S S 3 S 4 Consider an individual based, stochastic model with extinction and mutation 3

14 Outline Motivation Mutant games Fixation probabilities Average diversity Does fitness increase or decrease? Another approach with truly novel mutations 4

15 Fixation probabilities: Weak selection Fixation probability f ¼ ARTICLE IN PRESS X g, A. Traulsen / Journal of Theoretical Biology 63 (00) 6 68 Expectation for random games X þ P N k ¼ exp½þb P k i ¼ ðp B p A ÞŠ ARTICLE IN PRESS Instead, we have to ensure bndp5. ZZZ Eðf Þ¼ þ P N k ¼ exp½þb P Eq. (). In this case, we have k Instead, we have to ensure bndp5.! For a first order expansion to be meaningful, b5 is not enough. In principle, we could assume any function for p B p A. The g, A. Traulsen / Journal of Theoretical Biology 63 (00) 6 68 most important case, however, are games, which lead to a. 0) Traulsen 6 68 / Journal of Theoretical Biology 63 (00) 6 68 linear dependence of the payoff difference Dp ¼ p B p A ¼ uiþv, cf. For a first order expansion to be meaningful, b5 is not enough. xpansion to be meaningful, b5 is not enough. C ¼In XN X principle, k ðuiþvþ¼ we could XN uk assume any function for p B p A. The most important Eðf Þ case, however, N b EðC þ ðuþvþk ¼ u NðN Þ to ensure bndp5. þv NðN Þ : are games, 6 which lead k ¼ i ¼ k ¼ e could assume any function for p to a B p A. The ase, however, are linear games, dependence which of lead the payoff to a Þ difference EðC Þ N b Dp ¼ p B p A ¼ uiþv, cf. Eq. (). In this case, we have e of the payoff difference For the Dpsecond ¼ p B porder A ¼ uiþv, term incf. Eq. (0), we obtain! N se, we have C ¼ XN X k ðuiþvþ¼ XN! uk þ ðuþvþk ¼ u NðN! Þ þv NðN Þ : C ¼ XN X k 6 k ¼ i ¼ ðuiþvþ k ¼ ¼ XN uk þ ðuþvþk! k ¼ i ¼ k ¼ Þ¼ XN uk C ð3þ þ ðuþvþk ¼ u NðN Þ þv NðN Þ For¼ the u XN second k 4 þorder u term in Eq. (0), we obtain 8 4 þ uv : 6 N X k 3 þ u 8 þ uv þ v N k ¼ X k k ¼! k ¼ ð3þ! k ¼ C XN X k ðuiþvþ ¼ XN uk þ ðuþvþk ¼ NðN Þ u 3N3 þ3n N þuv 3N þn þv N der term in Eq. (0), we obtain : om payoff values under 0 4 k ¼ i ¼ k ¼! results,! lines are the 5 ¼ u XN k 4 þ u 8 4 þ uv N X k 3 þ u 8 þ uv X þ v N ð4þ ian iþvþ distributed ¼ with XN uk the þ ðuþvþk k i ¼ ðp B p A ÞŠ þb EðC Þ ð3þ N 3 pðaþpðbþpðcþ da db dc: Dp ¼ p B p A ¼ uiþv, Huang & Traulsen, JTB (00) ð

16 Fixation probabilities in mutant games Fixation probability φ Including coexistence game Excluding coexistence game Strong selection limit Weak selection approximation Fraction of beneficial mutants Probability density fitness values F Selection intensity β Moment expansion of the payoff distribution W W a M a M a a 6 Huang & Traulsen, JTB (00)

17 Outline Motivation Mutant games Fixation probabilities Average diversity Does fitness increase or decrease? Another approach with truly novel mutations 7

18 Computational model for many strategies S S S 3 S S S 3 Mutation S S S 3 S 4 S X S X S 3 X S 4 X X X X S S S 3 S 4 S S S 3 S 4 Extinction S S S 3 S 4 S S 3 S 4 Simulate this process for a finite population of size N and some fixed intensity of selection with Gaussian distributed payoff entries! 8

19 How are mutant games different? Constant selection Frequency dependent selection 000 a w=0.000 b w=0.000 Number of individuals c d w=0 w=0 Weak selection 500 Strong selection Time (generations) 9 Huang et al., Nature Communications (0)

20 Measuring diversity Weak selection and any mutation rate: Ewen s sampling formula tion under neutral select Pm ( )= N m / S ( ) m q m ), N can q be calculated by Stirling numbers of the fi S N Probability to observe m coexisting types ( q)= ( q i) Π N i = 0 +, d 3,39. For a haploid Mo Strong selection and weak mutation: Measure the transitions between different levels of diversity d w =0 Strong selection ,000

21 A Markov chain of diversity The model allows an arbitrary number of types, but only a handful coexist! Huang et al., Nature Communications (0)

22 How diverse is the population on average? Constant selection Frequency dependent selection.0 a type µ =5 0 5 b type µ =5 0 5 Density of numbers of types types 3 types c type types µ =5 0 4 types 3 types 4 types d µ =5 0 4 type types High mutation rate Low mutation rate types 4 types 3 types 4 types >= 5 types Selection Intensity Huang et al., Nature Communications (0)

23 Stationary mutant games W W a M a M a a If we assume that the game is always sampled from the same distribution, the system tends to increase payoff within that distribution If we sample from a distribution around the wild type s payoff, payoff increases indefinitely Workaround-solution: Normalize payoffs to highest payoff entry in the game 3

24 Outline Motivation Mutant games Fixation probabilities Average diversity Does fitness increase or decrease? Another approach with truly novel mutations 4

25 How does the average payoff change? Use replicator dynamics and assume extinction at a threshold /N = R d Probability density dxf(x) fitness values F 5 Huang et al., BMC Evolutionary Biology (0)

26 x mutant games under strong selection A coordination game does not change anything Dominant mutations maintain diversity but can increase fitness or decrease it (Prisoner s dilemma games) Coexistence games always increase diversity, but they can increases or decreases fitness (Snowdrift games) d w =0 Strong selection ,000 6 Gokhale & Traulsen, PNAS (00); Han et al., TPB (0); Wu et al., PLoS CB (forthcoming)

27 Static properties of random games How many equilibria are there in random multiplayer games? What is the probability that the rank abundance changes with the intensity of selection under low mutation rates? A exponential mapping A B linear mapping change. changes. 3 changes. 4 changes. B change. changes. 3 changes. 4 changes. 5 changes. 5 changes. 6 changes. 6 changes. 7 changes. 7 changes. 8 changes. 8 changes. 7 C D Gokhale & Traulsen, PNAS (00); Han et al., TPB (0); Wu et al., PLoS CB (forthcoming)

28 Dynamic generation of random games Change in average payoff Changes Changes in the in average the average fitness fitness Gaussian distribution , c is greater 8 than the wild type fitness d is = R dxf(x) = F (d) Uniform distribution Gaussian distribution Gaussian distribution Uniform distribution Uniform distribution density function f(x). As the probability that one of the new payo d 4 4 Maintaining diversity Increasing diversity Decreasing diversity Huang et al., BMC Evolutionary Biology (0)

29 Random generation of dominance games B A Case I: Mutant takes over and increases fitness e expressed as p(w() >d a > c, b > d) = p(a > d a > c, b > d) }{{ p(a > d, a > c, b > d) = }}{{} p(a > c, b > d) p(a > d, a > c) =. (4) p(a > c) p(w() >d a > c, b > d) = θ θ ew payoff ent 9 Case II: Mutant takes over and decreases fitness alculation above, we have p(w() <d a > c, b > d) = = p(a < d, a > c, b > d) p(a > c, b > d) p(a < d, a > c) p(a > c) = ( θ). () Huang et al., BMC Evolutionary Biology (0)

30 Random generation of coexistence W(x ) = ax + b( x ) = bc ad b d a + c. B A Case I: Mutant invades and increases fitness p(w(x )>d a < c, b > d) = p ((b d)(c d) > 0 a < c, b d > 0) = p (c > d a < c) p (c > a, c > d) = = θ θ p (c > a) Case II: Mutant invades and decreases fitness oexistence game, we find p(w(x )<d a < c, b > d) = ( θ). Thus, using a calculation similar to Eq. ( 30 Huang et al., BMC Evolutionary Biology (0)

31 Fitness can increase or decrease Probability of changing average fitness Constant selection Average fitness increasing Average fitness decreasing Frequency dependent selection Average fitness increasing Average fitness decreasing s = R d dxf(x) Huang et al., BMC Evolutionary Biology (0)

32 Another modeling approach Add random number to the column and the line of the new mutant strategy This game has the property of equal gains from switching (a+d=b+c) and thus does not allow coexistences In this case, only increase or decrease of fitness can be addressed based on the payoff distributions 3 Nowak & Sigmund, Acta Appl. Math (990); WIld & Traulsen, JTB (007); Huang et al., BMC Evolutionary Biology (0)

33 Diploid populations In the simplest case, this just means to consider partnership games A B A w w AA AB B w w AB BB..0 a type µ = 0 6 This approach favors coexistences (heterozygote advantage) and thus leads to more diversity Density of numbers of types b types 3 types type types 3 types 4 types µ = 0 4 High mutation rate Low mutation rate types Selection Intensity

34 Outline Motivation Mutant games Fixation probabilities Average diversity Does fitness increase or decrease? Another approach with truly novel mutations 34

35 Novel strategies in repeated games Modeling repeated games with finite state automata 35 Garcia, Van Veelen et al., PNAS (0)

36 Novel strategies in repeated games Population structure and repetition Region 5: =0.35, = % % A l l C G r i m % % T F T % Region : =0., =0. 5 % A l l D % % % 36 % Van Veelen, Garcia et al., PNAS (0)

37 Open questions How can we understand the dynamics of such systems better? Analytical approaches? What happens when we couple evolutionary dynamics with population dynamics? How would such a dynamics be reflected in genetic data? Have we been looking for the right things? 37

38 Summary Evolutionary game theory typically considers a predefined set of mutations Truly novel mutations can lead to much more complex dynamics in evolutionary games than the usual case of constant selection S S S 3 S S S 3 S S S 3 S 4 S S S 3 S 4 Mutation Extinction S S S S 3 S 4 S X S X S 3 X S 4 X X X X S S 3 S 4 S S 3 S 4 Are such mutations out there?? 38

39 Acknowledgement Maria Abou Chakra Benedikt Bauer Kathrin Büttner Julian Garcia Chaitanya Gokhale Kristin Hagel Christian Hilbe Laura Hindersin Cindy Li Yuiry Pichugin Weini Huang Torsten Röhl Yixian Song Benjamin Werner Bin Wu Bernhard Haubold (Plön) Christoph Hauert (Vancouver) Jorge Pacheco (Braga) Martin Nowak (Harvard) Karl Sigmund (Vienna) 39