1. Ph.D. Student: Angelo Rosario Carotenuto XXVIII Cycle - Aa 2013/14 MSC Biomedical Engineering Università degli Studi di Napoli “Federico II” DICMAPI, piazzale Tecchio 80 DIST, via Claudio 21 Closing Seminar for the Ph.D. Course: “Game Theory and Analysis of Competitive Dynamics For Industrial Systems” Presented to: Prof. Lina Mallozzi

2. Angelo Rosario Carotenuto, Ph.D. Student Aa 2013/14 MSC Biomedical Engineering Università degli Studi di Napoli “Federico II” DICMAPI, piazzale Tecchio 80 / DIST, via Claudio 21  EVOLUTIONARY GAMES THEORY  EVOLUTIONARY STABLE STRATEGIES  REPLICATOR DYNAMICS  LOTKA – VOLTERRA DYNAMICS  APPLICATIONS TO BIOLOGY /OVERVIEW OF PAPERS 2

3. Angelo Rosario Carotenuto, Ph.D. Student Aa 2013/14 MSC Biomedical Engineering Università degli Studi di Napoli “Federico II” DICMAPI, piazzale Tecchio 80 / DIST, via Claudio 21 Difficult process to describe Game theory seen as a way of formally modeling natural selection Driven by Need Genetics “On the Origin of Species”, 1859 3

4. Angelo Rosario Carotenuto, Ph.D. Student Aa 2013/14 MSC Biomedical Engineering Università degli Studi di Napoli “Federico II” DICMAPI, piazzale Tecchio 80 / DIST, via Claudio 21 Evolutionary games is the youngest of mathematical tools used to describe evolution as a refinement of Nash equilibria games. EGT studies the behavior of large populations that repeatedly engage in strategic interactions, on which the survivor of a population depend. The most of interaction is based on the idea that the current state of a player depends on the presence of the other players. In classical games (single round or repetitive) players are rational as he considers the opponents strategies in making appropriate choices for maximizing utilities. In EGT, it is only required player to have a strategy (sometimes unwillingly) John Maynard Smith EGT players have a fitness function rather than utilities and do not know all details of the game. The organism with the best interaction strategy has the highest fitness, maximize their payoff and increase their ability to reproduce. 4

5. Angelo Rosario Carotenuto, Ph.D. Student Aa 2013/14 MSC Biomedical Engineering Università degli Studi di Napoli “Federico II” DICMAPI, piazzale Tecchio 80 / DIST, via Claudio 21 An EGT Player: 1) Define strategy (at birth) 2) Play strategy against opponents 3) Evaluate fitness based on value obtained through strategy 4) Convert fitness to replication, preserving the phenotype The genetic code of a player can’t change, but their offspring can exhibit mutations and therefore adopt a different strategy. Strategies that increase fitness are preferable in the Darwinian sense of evolution (invasion of a better strategist), and the new strategy will move to an Evolutionary Stable Strategy - ESS Payoff Matrix AB Au[A,A]u[A,B] Bu[B,A]u[B,B] 5

6. Angelo Rosario Carotenuto, Ph.D. Student Aa 2013/14 MSC Biomedical Engineering Università degli Studi di Napoli “Federico II” DICMAPI, piazzale Tecchio 80 / DIST, via Claudio 21 Payoff Matrix AB Au[A,A]u[A,B] Bu[B,A]u[B,B] Once at ESS, selection keeps us there A strategy is called ESS if it is not overtaken by rare (small) mutant strategies Assumptions of the game 1) Pairwise, symmetric game 2) Asexual inheritance 3) Infinite (large) population 4) Equal payoff, say (implying the same strategies space) Consider a 2 player symmetric game with ESS given by with payoff matrix Let p be a small percentage of population playing mutant strategy. Then is ESS if: with being the average fitness.Definition 6

7. Angelo Rosario Carotenuto, Ph.D. Student Aa 2013/14 MSC Biomedical Engineering Università degli Studi di Napoli “Federico II” DICMAPI, piazzale Tecchio 80 / DIST, via Claudio 21 Existence -Standard definition by J.M. Smith and G.R. Price ("The logic of animal conflict“, Nature, 1973 ) is ESS if for all OR -The strategy choice is a Nash Equilibrium -Alternative definition by Thomas B. ("On evolutionarily stable sets". J. Math. Biology, 1985) is ESS if for all AND Payoff Matrix AB Aab Bcd Prop1. A strategy is an ESS for all and in some neighborood of if 7

8. Angelo Rosario Carotenuto, Ph.D. Student Aa 2013/14 MSC Biomedical Engineering Università degli Studi di Napoli “Federico II” DICMAPI, piazzale Tecchio 80 / DIST, via Claudio 21 Payoff Matrix for Hawk/Dove Game HawkDove Hawk(V/2 - C/2, V/2 - C/2)(V,0) Dove(0,V)(V/2, V/2) V : value of the resource C: cost of the fight If a Hawk meets a Dove he gets the full resource V to himself (Uncontested Fight) If a Hawk meets a Hawk, each wins with probability ½ with a resulting outcome equal to (V – C)/2 (Perfectly Balanced Fight) If a Dove meets a Hawk he will back off and get nothing - 0 If a Dove meets a Dove both share the resource and get V/2 Competition can, in general, be modeled as a search for an ESS Example 8

9. Angelo Rosario Carotenuto, Ph.D. Student Aa 2013/14 MSC Biomedical Engineering Università degli Studi di Napoli “Federico II” DICMAPI, piazzale Tecchio 80 / DIST, via Claudio 21 Payoff Matrix for Hawk/Dove Game HawkDove Hawk(V/2 - C/2, V/2 - C/2)(V,0) Dove(0,V)(V/2, V/2) V : value of the resource C: cost of the fight Outcomes (1) Payoff Matrix for Hawk/Dove Game HawkDove Hawk(V/2 - C/2, V/2 - C/2)(V,0) Dove(0,V)(V/2, V/2) Payoff Matrix for Hawk/Dove Game HawkDove Hawk(V/2 - C/2, V/2 - C/2)(V,0) Dove(0,V)(V/2, V/2) Payoff Matrix for Hawk/Dove Game HawkDove Hawk(V/2 - C/2, V/2 - C/2)(V,0) Dove(0,V)(V/2, V/2) Payoff Matrix for Hawk/Dove Game HawkDove Hawk(V/2 - C/2, V/2 - C/2)(V,0) Dove(0,V)(V/2, V/2) Hawk strictly dominates Dove for both players, Thus, the players play (Hawk,Hawk) in equilibrium If (V/2 - C/2) >0 9

10. Angelo Rosario Carotenuto, Ph.D. Student Aa 2013/14 MSC Biomedical Engineering Università degli Studi di Napoli “Federico II” DICMAPI, piazzale Tecchio 80 / DIST, via Claudio 21 Payoff Matrix for Hawk/Dove Game HawkDove Hawk(V/2 - C/2, V/2 - C/2)(V,0) Dove(0,V)(V/2, V/2) V : value of the resource C: cost of the fight Outcomes (2) Payoff Matrix for Hawk/Dove Game HawkDove Hawk(V/2 - C/2, V/2 - C/2)(V,0) Dove(0,V)(V/2, V/2) Payoff Matrix for Hawk/Dove Game HawkDove Hawk(V/2 - C/2, V/2 - C/2)(V,0) Dove(0,V)(V/2, V/2) Payoff Matrix for Hawk/Dove Game HawkDove Hawk(V/2 - C/2, V/2 - C/2)(V,0) Dove(0,V)(V/2, V/2) Payoff Matrix for Hawk/Dove Game HawkDove Hawk(V/2 - C/2, V/2 - C/2)(V,0) Dove(0,V)(V/2, V/2) 2 PSNE: (Dove,Hawk) and (Hawk, Dove) The optimum mix of passive and aggressive behavior depends on the exact values of the costs and benefits. MS: it would be best to adopt a mixture of the strategies. But what mixture? Payoff Matrix for Hawk/Dove Game Hawk (p)Dove (1-p) Hawk(V/2 - C/2, V/2 - C/2)(V,0) Dove(0,V)(V/2, V/2) Payoff Matrix for Hawk/Dove Game Hawk (p)Dove (1-p) Hawk(V/2 - C/2, V/2 - C/2)(V,0) Dove(0,V)(V/2, V/2) Mixed player strategy; is an ESS being valid the. By taking If (V/2 - C/2) <0 10

11. Angelo Rosario Carotenuto, Ph.D. Student Aa 2013/14 MSC Biomedical Engineering Università degli Studi di Napoli “Federico II” DICMAPI, piazzale Tecchio 80 / DIST, via Claudio 21 Aims to study the actual evolutionary problem, the way how the fitter players will generate more replicas of themselves into the population and the less fit will be left out of the player population Replicator dynamics are a set of deterministic difference or differential equations Replicator Equation: 11 Non-linear RE is usually analyzed in terms of stability, by a set of evolutionarily stable states: x reaches an ES if its genetic composition is restored by selection after a disturbance, provided the disturbance is not too large (J.M. Smith, 1982)

12. Angelo Rosario Carotenuto, Ph.D. Student Aa 2013/14 MSC Biomedical Engineering Università degli Studi di Napoli “Federico II” DICMAPI, piazzale Tecchio 80 / DIST, via Claudio 21 Dynamics of n- dimensional population vector: The system will reach an equilibrium when. The solutions represents the rest points Convergent trajectories tend to stationary points in the state space. Equilibrium points are evolutionary equilibria if they result asymptotically stable, Stability of motions Stability of motions (A. Lyapunov, The Stability of Motion,1892) - - - Lyapunov Stable Lyapunov Asymptotically Stable (x* attractor) Definition & Stability 12

13. Angelo Rosario Carotenuto, Ph.D. Student Aa 2013/14 MSC Biomedical Engineering Università degli Studi di Napoli “Federico II” DICMAPI, piazzale Tecchio 80 / DIST, via Claudio 21 The Lotka-Volterra system describes evolution of two interacting populations with densities and : A.J. Lotka V. Volterra Main Readings. Lotka A.J., Elements of Physical Biology, 1924 Volterra V, Una teoria matematica sulla lotta per l’esistenza, 1962 Hofbauer J., On the occurrence of limit cycles in the Volterra- Lotka differential equation 1981 Equivalence of the RD and the L-V system through the transformation: Preys (e.g. Foxes) Predators (e.g. Rabbits) a>0 rate of intrinsic growth b<0 prey logistic term c<0 loss rate due to predation d<0 predators death rate e>0 predation rate f <0 logistic term Periodic solutions with no simple expressionDefinition 13

14. Angelo Rosario Carotenuto, Ph.D. Student Aa 2013/14 MSC Biomedical Engineering Università degli Studi di Napoli “Federico II” DICMAPI, piazzale Tecchio 80 / DIST, via Claudio 21 Dynamics Equilibrium Points by solving: Extintion Survivor of y Survivor of x 14

15. Angelo Rosario Carotenuto, Ph.D. Student Aa 2013/14 MSC Biomedical Engineering Università degli Studi di Napoli “Federico II” DICMAPI, piazzale Tecchio 80 / DIST, via Claudio 21 Stability What dynamics? Check the stability … 15 Predators Preys

16. Angelo Rosario Carotenuto, Ph.D. Student Aa 2013/14 MSC Biomedical Engineering Università degli Studi di Napoli “Federico II” DICMAPI, piazzale Tecchio 80 / DIST, via Claudio 21 In an organism, cells compete for various resources in their environment. Mutations occasionally occur in cell division due to various reasons Cancer is a disease where mutated (tumor) cells oust normal cells in a local population Assume cancer and healthy cells as populations in a evolutionary game, with fitness functions designed using LV functions Analysed Paper: - Gatenby and Vincent, Application of quantitative models from population biology and evolutionary game theory to tumor therapeutic strategies, Mol. Cancer Therapy, 2003; 2:919-927 …The problems: - Heterogeneity of cancer (i.e. different strategies) -Difficulty of controlling all system dynamics 16

17. Angelo Rosario Carotenuto, Ph.D. Student Aa 2013/14 MSC Biomedical Engineering Università degli Studi di Napoli “Federico II” DICMAPI, piazzale Tecchio 80 / DIST, via Claudio 21 Lotka-Volterra equations formed as follows: If the populations play a pair of strategies, the possible outcomes at the stable state are: – x, y = 0 Trivial, non-relevant result – x = k N, y = 0 All normal cells, tumor completely recessed – x = (k N - α NT k T )/(1 - α NT α TN ) y = (k T - α TN k N )/(1 - α NT α TN ) Normal and tumor cells living in equilibrium /benign tumor – x=0, y = k T All tumor cells, invasive cancer Recession Benign Invasive Ref #1 17

18. Angelo Rosario Carotenuto, Ph.D. Student Aa 2013/14 MSC Biomedical Engineering Università degli Studi di Napoli “Federico II” DICMAPI, piazzale Tecchio 80 / DIST, via Claudio 21 Key issue: cytotoxic terapies may eliminate large number of cells, but if invasive cancer remains the stable state solution to the state equation, even few surviving cells will re-invade the tissue Parameter changes can affect the equilibria reached. This suggests an easy cure for cancer, just by acting on parameters (e.g. opportunely coordinating anti-VEGF and MP therapies ) Starting from the same initial condition, the population trajectories result different by changing parameters Ref #1 y x 18

19. Angelo Rosario Carotenuto, Ph.D. Student Aa 2013/14 MSC Biomedical Engineering Università degli Studi di Napoli “Federico II” DICMAPI, piazzale Tecchio 80 / DIST, via Claudio 21 Ref #1 Parameters: – r i : maximum rate of proliferation for i th population – u i : strategy of i th population, defining its interaction with environmental factors – α(u i,u j ): competitive effect of u i versus u j – k(u i ): maximum size of i th population – The strategy for normal cells has σ i = 0 (static phenotype) In reality, the tumor species mutates quickly and changes strategy (phenotype) to adapt to the environment, making it independent from the previous system of equations. Basic idea: Assume n different populations of tumor cells can arise – Each population gets its own LV type fitness function 19

20. Angelo Rosario Carotenuto, Ph.D. Student Aa 2013/14 MSC Biomedical Engineering Università degli Studi di Napoli “Federico II” DICMAPI, piazzale Tecchio 80 / DIST, via Claudio 21 Normal cells don’t evolve and continue to die, being pressured by tumor cells The tumor cells appear to reach a steady state, by opportunely decreasing their self aggressiveness Paper claim. Can they be treated at this point with targeted cell-specific drug therapy by controlling parameters? 20 Strategies Densities

21. Angelo Rosario Carotenuto, Ph.D. Student Aa 2013/14 MSC Biomedical Engineering Università degli Studi di Napoli “Federico II” DICMAPI, piazzale Tecchio 80 / DIST, via Claudio 21 Extend fitness functions with a Gaussian, drug-specific term Parameters: – d h : dosage of drug h – σ h : variance in effectiveness of drug h – strategy (cell type) weakest against drug h Cell-specific treatment is effective at first, but evolving cells become resistant and invade 21

22. Angelo Rosario Carotenuto, Ph.D. Student Aa 2013/14 MSC Biomedical Engineering Università degli Studi di Napoli “Federico II” DICMAPI, piazzale Tecchio 80 / DIST, via Claudio 21 G...From the macro-scale to the µ-scale … 22

23. Thanks for your attention… 23