1. UNIVERSITA’ DEGLI STUDI NAPOLI FEDERICO II DOTTORATO IN INGEGNERIA DEI MATERIALI E DELLE STRUTTURE Brunella Corrado Filomena Gioiella Bernadette Lombardi EVOLUTIONARY GAME THEORY OF CARCINOGENESIS Ref: Robert A., Gatenby and thomas L. Vincent, Cancer Research, 2003

2. Summary Overview on the Evolutionary Game Theory The mathematical model Case study Conclusion

3.  Principles  Continuous time repetitive game  The players are randomly chosen in a large population  The choice of strategies is based on the interaction between players  The payoffs depend on the strategies of the players Evolutionary game theory  Symmetric two-player normal formal game Г = The payoffs are described by the matrix A є R nxn A ij : payoff obtained when a player chooses strategy i and his opponent chooses the strategy j. Symmetric Nash Equilibrium A ii ≥ A ji (n = 2)u : payoff Each of two players chooses a strategy from the finite set X = {x 1,…x n } for all j є X

4.  Mixed strategies m = (p, 1-p) on X = {x 1,x 2 } p є [0,1]  Evolutionarity Stable Strategy (ESS) u(x, [(1-є)x + єx’]) > u(x’, [(1- є)x + єx’]) є : mutant x > x’ є < є’ invasion barrier Population in which all members play this strategy is resistant to invasion by a small group of mutants who play an alternative mixed strategy.

5. Dynamic systems  Evolutionary equilibrium (EE)  Replicator dynamic If strategy x is performing better than the average, the agents who play it will see their proportion increase in the total population Mathematical aspects Equilibrium point z, first order, autonomous, ordinary differential equation z is asymptotically stable if: 1.Lyapnov stable: given any neighbourhood I 1 of z there exist I2 neighbourhood of z s.t. all trajectories with x(0) є I 1 satisfy x(t) є I 2 for all t ≥ 0 attractive: exist a neighbourhood I of z s.t. all trjectories starting in I satisfy x(t) z as t ∞ Dynamic systems f(z) = 0 s(x): agents u(x) = s(x)u(x,x)+s(x’)u(x,x’) payoff

6. Evolutionary model of carcinogenesis Here we present a mathematically model of carcinogenesis that frames the Fearon-Vogelstein model of colorectal carcinogenesis. The model demonstrates carcinogenesis is an emergent phenomenon requiring a sequence of evolutionary steps as cellular proliferation follows successful adaptation to varying environmental constraints. This approach describes a system that proceeds linearly: 1.parameters that control cell proliferation and clonal expansion? 2.the interactive dynamics that produce a malignant phenotype? 3.the environmental effects? 1.parameters that control cell proliferation and clonal expansion? 2.the interactive dynamics that produce a malignant phenotype? 3.the environmental effects? What are…

7. 1.control cellular and clonal proliferation: Tissue communication Cell-cell interaction Cell-matrix interaction Soluble growth factors Uptake and availability of substrate Michaelis-Menten kitenics

8. What are… 2.the interactive dynamics that produce a malignant phenotype: Genetic mutations from single to multiple Simultaneous genic action by Oncogenes Tumor suppressor genes A gene that causes the transformation of normal cells into cancerous tumor cells, especially a viral gene that transfor ms a host cell into a tumor cell Is a gene that protects a cell from one step on the path to cancer

9. What are… 3.the environmental effects: Increase of substrate delivery Complete destruction of steady-state and cellular balance Normal tissue dynamics Differents normal population have differents strategies so, they are non- competitive and can coexist …

10. Tumoral tissue dynamics ….. an evolving system 1° phase : one mutation the mutant are derived by the same G-function, they do not change the shape of the adaptive landscape and they will simply coexist at low numbers. 2° phase : more mutations change in tissue carrying capacity, the mutant’s fitness can not be determined from the normal cell’s G-function and a second G-function is introduced for the mutant. Cellular proliferation is constrained by substrate availability. 3° phase : new strategies to maximize substrate availability the evolutionary constrains on the tumor cells are removed by assuming an increased mutation rate due to microsatellite instability, chromosomal instability or mutagenic environmental.

11. Reference Model: Colorectal carginoma Colorectal cancer (also known as colon cancer, rectal cancer, bowel cancer or colorectal adenocarcinoma) is a cancer of the colon or rectum (parts of the large intestine). It is due to the abnormal growth of cells that have the ability to invade or spread to other parts of the body.

12. Dynamics in multicellular organisms Here we use a form of general equations where are assumed two control mechanisms:  Cell-cell and environmental interactions  Availability of substrate (glucose) sufficient for formation of new cells The hypothesis are:  Each volume of tissue contains p different population with N p individuals  Neoplastic cells retain proliferative capacity  Michealis –Menten substrate kinetic The normal cell population dynamics is given by: (1) The multiple tumor population (produced by mutations) is given by: (2)

13. Fitness generating function (G-function)

14. But, does this solution is an efficient way to model evolution within a population? The individuals within a population may be evolutionary identical The fitness generating function (G-function) Fitness generating function (G-function)

15. Condition promoting carcinogenesis u* is determined by solving equations (7) and (8)

16. For a given u* there are two equilibrium solutions possible, N and E. Condition promoting carcinogenesis Equilibrium N = 0

17. For a given u* there are two equilibrium solutions possible, N and E. Condition promoting carcinogenesis Equilibrium E = 0

18. Conclusions  The solution we search for have to satisfy a ESS maximum principle.  This principle require that the equilibrium solutions be ecologically stable (return to equilibrium for a fixed u) and evolutionarily stable (return to equilibrium under changes in u).  Equilibrium solutions that do not satisfy the ESS maximum principle are, in general, subject to invasion.  The mutant populations now need to evolve new strategies to maximize fitness within an adaptive landscape dominated by substrate limitation. Available strategies include increasing substrate delivery through angiogenesis, increased efficiency of substrate uptake, or both

19. Thanks for your attention