Applications of Game Theory in the Computational Biology Domain

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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 B Prisoner A

Cooperate Cooperate Defect

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

Share
Crocodile A

Fight

Share
Fight

V 2

/

V 2

0/V
V C 2 V C / 2

V/0

• 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   j 1  (ui , u j ) N j 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
 v  u  2  i n H i (u, N)   i   j 1  (ui , u j ) N j  d h exp  2     k (ui )  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'
Controls the mixing proportions of bound proteins Amount that site j is available for protein 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 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)
• The equilibrium condition is expressed in terms of just αj, so that overall occupancy is determined by which proteins are currently bound elsewhere
i

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


				
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