Applications of Game Theory in the Computational Biology Domain Richard Pelikan April 16, 2008 CS 3110

Overview The evolution of populations Understanding mechanisms for disease and regulatory processes –Models of cancer development –Protein and drug interactions, resource competition 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 –Fitness function is often unknown –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 Earlier in the course, we knew just about everything about the game But we are so lucky to know this information! CooperateDefect Cooperate 3/30/5 Defect 5/01/1 Prisoner A Prisoner B

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. ShareFight Share /0 / V Fight V / 0/ Crocodile A Crocodile B

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

Evolutionary algorithm 1) Obtain strategy (at birth) 2) Play strategy against environmental 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 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 (ESS) An ESS 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 ESS Let S be an evolutionary strategy and T be any alternative strategy. S is an ESS 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 ESS and Nash In a Nash equilibrium, –Players know the structure of the game and the potential strategies of opponents. In an ESS, –Strategies are not exhaustively defined –Payoffs are uncertain –Strategies can’t change –Everyone adopts the same strategy

Current applications of ESS to evolutionary theory Competition can, in general, be modeled as a search for an ESS ES strategies used to explain altruism, animal conflict, market competition, etc. Modeling evolution entirely through EES is hard. –On the smaller scale of cell populations, it’s easier to see the practical applications.

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 Paper: –Gatenby and Vincent, Application of quantitative models from population biology and evolutionary game theory to tumor therapeutic strategies, Mol. Cancer Therapy, 2003; 2: Claim: To effectively treat cancer, all system dynamics responsible for the tumor 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 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

Lotka-Volterra Equations Used to model population competition Basically, Rate of growth = # in population * (environmental help to population – rate of destruction by opponent) 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: 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 = k N, y = 0 All normal cells, tumor completely recessed –x = (k N - βk T )/(1 - βδ), y = (k T - δk N )/(1 - βδ) Normal and tumor cells living in equilibrium (benign tumor) –x=0, y = k T 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, tumor cells from many different populations are 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) Parameters: –α i : maximum rate of proliferation for i th population –u i : strategy of i th population –β(u i,u j ): competitive effect of u i versus u j –k(u i ): maximum size of i th population

Tumor Evolution A strategy evolves according to: σ i = chance for mutation in i th population v = auxillary variable over strategy space The strategy for normal cells has σ i = 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 cell- specific drug?

Augmenting system with specific drug targets 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

A Bleak Outcome 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 Paper claims: –Targeted drug therapies alone won’t work –Trajectories of tumor evolution need to be changed by systemic, outside factors –Angiogenesis inhibitors, TNF, etc.

Following this, Lots of interest regarding drug interactions and how they affect cells Usually dependent on how much of, or for how long, a drug molecule is in contact (binds) with a cell structure Computational approaches can be used to conduct drug simulations in silico –Paper: Perez-Breva et. al, Game theoretic algorithms for protein-DNA binding, NIPS 2006

Game Theory in Molecular Biology Binding game –Inputs: Protein classes (players) Sites (other set of players) which compete and coordinate for proteins –Players decide how much protein is allocated to each site, 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 f j = concentration of protein i p ij = amount of protein i allocated to site j s ij = amount of protein I bound to site j E ij = affinity of protein i to site j Utility of protein assignment is defined as:

Formal definition of binding game f j = concentration of protein i p ij = amount of protein i allocated to site j s ij = amount protein i bound to site j E ij = affinity of protein i to site j Utility of protein assignment to set of sites s: Amount of time that site j is available for protein i Controls the mixing proportions of bound proteins

Formal definition of binding game f j = concentration of protein i p ij = amount of protein i allocated to site j s ij = amount of protein i bound to site j E ij = affinity of protein i to site j K ij = chemical equilibrium constant between protein i and site j Utility of site player j binding to a set of proteins p Amount of protein i bound to site j Proportion of protein i that’s just floating around

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

Algorithm Start with all sites empty (α j =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 λ-phage virus gene CRogene CI 2 Switch Sites Virus genes are embedded in a cell’s DNA

Simulation model for λ-phage virus gene CRogene CI 2 RNA Switch Sites During normal function, cell requires RNA to transcribe genes to proteins

Simulation model for λ-phage virus gene CRogene CI 2 RNA Switch Sites RNA unknowingly transcribes viral genes, producing virus proteins

Simulation model for λ-phage virus gene CRogene CI 2 RNA Virus proteins are produced by first gene

Simulation model for λ-phage virus gene CRogene CI 2 RNA Virus proteins bind to available sites

Simulation model for λ-phage virus gene CRogene CI 2 Virus proteins prevent transcription of later genes, keeping virus dormant.

Simulation model for λ-phage virus gene CRogene CI 2 Virus proteins bind and block transcription

Simulation model for λ-phage virus gene CRogene CI 2 Stress changes the affinities of binding sites

Simulation model for λ-phage virus gene CRogene CI 2 RNA is free to bind to later genes

Simulation model for λ-phage virus gene CRogene CI 2 “clearing” virus proteins are produced

Simulation model for λ-phage virus gene CRogene CI 2 Clearing proteins release viral proteins from the switches

Simulation model for λ-phage virus gene CRogene CI 2 Replicated virus At this stage, cell breaks open and releases the replicated virus

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