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Applications of Game Theory in the Computational Biology Domain Richard Pelikan April 13, 2008 CS 3110 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 Evolution • Difficult process to describe • Game theory seen as a way of formally modeling natural selection 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. Prisoner’s Dilemma • Players have strategies for obtaining the payoffs Prisoner A Prisoner B Cooperate Defect Cooperate Defect 3/3 5/0 0/5 1/1 • But we are so lucky to know this information! Crocodile’s Dilemma • V: The value of a resource • C: The cost to fight for a resource, C > V >0 Crocodile B Crocodile A Share Share Fight V 2 / V 2 V/0 Fight 0/V V C 2 V C / 2 • Negative payoff results in death – But who defines V and C? These variables are unclear for reallife competitions. 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 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). 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. 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 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. 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. 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. 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 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 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 Lotka-Volterra Equations • Used to model population competition dx x(a y ) dt dy y ( x) dt • 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 In the tumor vs. normal setting • Lotka-Volterra equations formed as follows: x y dx x 1 dt kN • y x dy y 1 dt kT 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 Finding Equilibria Recession Benign Invasive 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? 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 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) Ni Ni H i (u, N) i n H i (u, N) i (ui , u j ) N j j 1 k (ui ) • 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 Tumor Evolution • A strategy evolves according to: H (u, N ) ui i |v ui v • σi= chance for mutation in ith population • v = auxillary variable over strategy space • The strategy for normal cells has σi= 0 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 cellspecific drug? Augmenting system with specific drug targets • Extend fitness functions with a Gaussian, drugspecific term 2 v u i n H i (u, N) i (ui , u j ) N j d h exp j 1 k (ui ) 2 h • Parameters: – dh: dosage of drug h – σh: variance in effectiveness of drug h – u : strategy weakest against drug h • Cell-specific treatment is effective at first, but evolving cells become resistant and invade 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. 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 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: ui ( pi , s) pij Eij (1 sij ) H ( pi ) j i' 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: ui ( pi , s) pij Eij (1 sij ) H ( pi ) j i' Amount that site j is available for protein i Controls the mixing proportions of bound proteins 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 u ( s j , p) sij K ij ( pij f i sij )(1 sij ) i i' s i Amount of available protein to site j Amount of “free time” that site j has Finding the equilibrium • It turns out, finding the equilibrium between protein and site player’s utilities reduces to finding site occupancies αj j sij (a) i • The equilibrium condition is expressed in terms of just αj, so that overall occupancy is determined by which proteins are currently bound elsewhere 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 Simulation model for • iuiu RNA gene CI2 gene CRo Validation of simulated model • Increasing concentration at different receptors leads to different equilibrium • validated using studied concentrations in literature (shaded region) 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