Applications of Game Theory in the Computational Biology Domain

1. Applications of Game Theory in the Computational Biology Domain Richard Pelikan April 13, 2008 CS 3110

2. Overview • The evolution of populations • Understanding mechanisms for disease and regulatory processes • Models of cancer development • Competition for limited resources, e.g. protein site binding • Many biological processes can be tied to game theory

3. Evolution • Difficult process to describe • Game theory seen as a way of formally modeling natural selection

4. Evolutionary Game Theory • Evolution revolves around a fitness function • Frequency based, success is measured primitively by number present. • Strategies exist because of this function • Difficult to define the entire game with just the strategy.

5. Prisoner’s Dilemma • Players have strategies for obtaining the payoffs • But we are so lucky to know this information! Prisoner B Prisoner A

6. Crocodile’s Dilemma • V: The value of a resource • C: The cost to fight for a resource, C > V >0 • Negative payoff results in death • But who defines V and C? These variables are unclear for real-life competitions. Crocodile B Crocodile A

7. Population’s Dilemma • Population members play against each other • Natural selection favors the better strategists at the game • Key: strategies are really genetically encoded and do not change

8. Strategy and Genetics • Idea: An organism’s strategy is encoded at birth by its genetic code • The fitness of a phenotype is determined by its frequency in the population • The genetic code of a player can’t change, but their offspring can have mutated genes (and therefore a different strategy).

9. Population’s Dilemma • Consider 2 scenarios from crocodile’s dilemma: • A population of purely aggressive crocodiles • A population of purely docile crocodiles • In both scenarios, a mutation results in an “invasion” of better strategists.

10. Evolutionarily Stable Strategy (EES) • An EES is a strategy used by a population of players • Once established, it is not overtaken by rare (or “mutant”) strategies • These are similar but not equivalent to Nash equilibria

11. Formal Definition of EES • Let S be an evolutionary strategy and T be any alternative strategy. S is an EES if either of these conditions hold: • Payoff(S,S) > Payoff(T,S) or • Payoff(S,S) = Payoff(T,S) and Payoff(S,T) > Payoff(T,T) • T is a neutral strategy against S, but S always maintains an advantage over T.

12. Difference between EES and Nash • In a Nash equilibrium, • Players know the structure of the game and the potential strategies of opponents. • In an EES, • Strategies are genetically encoded, cannot change, and the structure of the game is unclear. Opponent strategies are not exhaustively defined.

13. Current applications of ESS to evolutionary theory • Competition can, in general, be modeled as a search for an EES • Hard to explain all of evolution at once • Step down from the population to the organism (cellular) level.

14. Mechanisms of Disease • 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

15. Applied Game Theory for Cancer Therapeutics • Claim: To effectively treat cancer, all system dynamics responsible for the invasion must be controlled • The problems: • Heterogeneity of cancer (i.e. different strategies) • Unfeasability of controlling all system dynamics

16. Modeling competition between tumor and normal cells • Assume tumor and normal cells are players in a game • Create equations which define a competition between normal and a certain type of tumor cells • These equations incorporate system dynamics variables which can favor either normal or tumor cells

17. Lotka-Volterra Equations • Used to model population competition • Parameters: • x: number of prey (normal cells) • y: number of predators (tumor cells) • : parameters representing interaction btwn species, open to design by user of model • Equations represent population growth rates over time

18. In the tumor vs. normal setting • Lotka-Volterra equations formed as follows: • If the populations play a pair of strategies, the possible outcomes at the stable state (where dx/dt = dy/dt = 0) are: • x, y = 0 • Trivial, non-relevant result • x = kN, y = 0 • All normal cells, tumor completely recessed • x = (kN - βkT)/(1 - βδ), y = (kT - δkN)/(1 - βδ) • Normal and tumor cells living in equilibrium (benign tumor) • x=0, y = kT • All tumor cells, invasive cancer

19. Finding Equilibria Recession Benign Invasive

20. Defining the multi-strategy case • Until now, the tumor population had a constant strategy (mutation requires a different set of parameters) • The new question is, where can the equilibria be when the strategy space is exhausted? • In practice, a population of tumor cells is already present; can the progress be reversed?

21. Heterogeneity of Cancer • Parameter changes can affect the equilibria reached. This suggests an easy cure for cancer, just by changing parameters. • In reality, the tumor population mutates quickly and changes strategy, making it independent from the previous system of equations

22. Heterogeneity of Cancer • Basic idea: Assume n different populations of tumor cells can arise • Each population gets its own fitness function (i.e. own set of Lotka-Volterra functions) • Parameters: • αi: maximum rate of proliferation for ith population • ui: strategy of ith population • β(ui,uj): competitive effect of ui versus uj • k(ui): maximum size of ith population

23. Tumor Evolution • A strategy evolves according to: • σi= chance for mutation in ith population • v = auxillary variable over strategy space • The strategy for normal cells has σi= 0

24. Tumor Evolution vs. Normal • Normal cells don’t evolve (bottom) and continue to die, being pressured by tumor cells (top) • The tumor cells appear to reach a steady state. Can they be treated at this point with a cell-specific drug?

25. Augmenting system with specific drug targets • Extend fitness functions with a Gaussian, drug-specific term • Parameters: • dh: dosage of drug h • σh: variance in effectiveness of drug h • : strategy weakest against drug h

26. Cell-specific treatment is effective at first, but evolving cells become resistant and invade

27. In Summary • Population fitness functions can be designed using the Lotka-Volterra functions • Drug-specific therapies alone won’t work • Trajectories of tumor evolution need to be changed by systemic, outside factors • Angiogenesis inhibitors, TNF, etc.

28. Game Theory in Molecular Biology • Binding game • Inputs: • Protein classes (players) • Sites (other set of players) which compete and coordinate for proteins • Players decide which sites to send proteins to, based on • How occupied sites are • Availability of proteins • Chemical equilibrium (sites have affinities for particular proteins up to a certain constant) • Output: allocation of proteins to sites

29. Formal definition of binding game • fj = concentration of protein i • pij= amount of protein i allocated to site j • sij = amount of time for site j to bind protein i • Eij = affinity of protein i to site j • Utility of protein assignment is defined as:

30. Formal definition of binding game • fj = concentration of protein i • pij= amount of protein i allocated to site j • sij = amount of time for site j to bind protein i • Eij = affinity of protein i to site j • Utility of protein assignment to set of sites s: Amount that site j is available for protein i Controls the mixing proportions of bound proteins

31. Formal definition of binding game • fj = concentration of protein i • pij= amount of protein i allocated to site j • sij = amount of time for site j to bind protein i • Eij = affinity of protein i to site j • Kij = chemical equilibrium constant between protein i and site j • Utility of site player j binding to a set of proteins p Amount of available protein to site j Amount of “free time” that site j has

32. Finding the equilibrium • It turns out, finding the equilibrium between protein and site player’s utilities reduces to finding site occupancies αj • The equilibrium condition is expressed in terms of justαj, so that overall occupancy is determined by which proteins are currently bound elsewhere

33. Algorithm • Start with all sites empty (αj =0; j = 1…n) • Repeat until convergence: • pick one site • maximize its occupancy time in the context of available proteins and sites • algorithm is monotone and guaranteed to find equilibrium

34. Simulation model for • iuiu RNA gene CI2 gene CRo

35. Validation of simulated model • Increasing concentration at different receptors leads to different equilibrium • validated using studied concentrations in literature (shaded region)

36. Summary • Many potential applications of game theory to biological domain • Most methods include intuitive and simplistic reasoning about how biological entities compete • Despite simplicity, the models often explain initial beliefs about behavior