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Pitt CS 3150 - Applications of Game Theory in the Computational Biology

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Applications of Game Theory in the Computational Biology DomainOverviewEvolutionEvolutionary Game TheoryPrisoner’s DilemmaCrocodile’s DilemmaPopulation’s DilemmaStrategy and GeneticsSlide 9Evolutionarily Stable Strategy (EES)Formal Definition of EESDifference between EES and NashCurrent applications of ESS to evolutionary theoryMechanisms of DiseaseApplied Game Theory for Cancer TherapeuticsModeling competition between tumor and normal cellsLotka-Volterra EquationsIn the tumor vs. normal settingFinding EquilibriaDefining the multi-strategy caseHeterogeneity of CancerSlide 22Tumor EvolutionTumor Evolution vs. NormalAugmenting system with specific drug targetsSlide 26In SummaryGame Theory in Molecular BiologyFormal definition of binding gameSlide 30Slide 31Finding the equilibriumAlgorithmSimulation model forValidation of simulated modelSummaryApplications of Game Theory in the Computational Biology DomainRichard PelikanApril 13, 2008CS 3110Overview•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 theoryEvolution•Difficult process to describe•Game theory seen as a way of formally modeling natural selectionEvolutionary 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•But we are so lucky to know this information!Cooperate DefectCooperate3/3 0/5Defect5/0 1/1Prisoner APrisoner BCrocodile’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.Share FightShare/ 0 / VFightV / 0 /Crocodile ACrocodile B2V2V2CV 2CV 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 changeStrategy 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 equilibriaFormal 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 populationApplied 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 dynamicsModeling 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 cellsLotka-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)( yaxdtdx)( xydtdy,,,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Nkyxxdtdx1Tkxyydtdy1Finding EquilibriaRecession Benign InvasiveDefining 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 equationsHeterogeneity 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

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