GAME THEORY F Ascione R Liuzzi R DApolito

GAME THEORY F. Ascione, R. Liuzzi, R. D’Apolito, A. Carciati, C. Taddei APPLIED TO GENE EXPRESSION ANALYSIS

OUTLINES Ø Biology introduction: from DNA to protein Ø Microarray technology and introduction to game theory Ø Cooperative games Ø Axiomatic characterization of the Shapley value for the microarray games Ø Cooperative games applied to microarray analysis

base pairs Ø A gene is a segment of a chromosome made up of DNA Ø Genes contain instructions for making proteins

FROM GENE TO PROTEIN Cells are governed by a chain of command DNA RNA protein Transcription: the synthesis of RNA under the direction of DNA; produces messenger RNA (m. RNA) Translation: the actual synthesis of a polypeptide, which occurs under the direction of m. RNA Each gene specifies a protein via transcription and translation

THE GENETIC CODE The genetic code consists of 64 triplets of nucleotides. These triplets are called codons. Gene 2 DNA molecule Gene 1 Gene 3 DNA strand (template) 3 A C C A A A C C G U G G U U U G G C U A G T 5 TRANSCRIPTION m. RNA 5 C A Codon TRANSLATION Protein Trp Phe Gly Ser Amino acid The genetic code is redundant but not ambiguous; no codon specifies more than one amino acid 3 Each codon specifies one of the 20 aminoacids used in the synthesis of proteins

PROTEIN STRUCTURE AND FUNCTION Ø Proteins are fundamental components of all living cells, performing a variety of biological tasks Ø Each protein has a particular 3 D structure that determines its biological function DNA gets all the glory, but proteins do all the work!

DNA MUTATIONS ORIGINAL DNA SEQUENCE BASE SUBSTITUTION BASE INSERTION BASE DELETION

SICKLE CELL ANEMIA The change in amino acid sequence causes hemoglobin molecules to crystallize when oxygen levels in the blood are low. As a result, red blood cells sickle and get stuck in small vessels.

MICROARRAY WHAT THEY ARE? Collection of microscopic DNA probes attached to a solid surface such as glass, plastic, or silicon chip forming an array (matrix) WHAT THEY DO? They allow to simultaneously examine the presence of many genes within a DNA sample HOW DO THEY WORK? Hybridization technique: it consists in fixing all segments of DNA (probes) on a support and label the nucleic acid that we want to identify (target). OBJECTIVE To compare gene expression profile of a patient with that of a healthy one to identify which genes are involved in the disease.

Ø This step is repeated for the wild type sample and for the patient sample RED: c. DNA of cancerous cells GREEN: c. DNA of wild type cells Ø Hybridization technique

Ø This step is repeated for the wild type sample and for the patient sample RED: the mutated gene is present GREEN: the wild type gene is present YELLOW: both genes are present at different levels

FROM MICROARRAY TO GAMES Game theory is a branch of mathematics that deals with interactive decision making, namely with situations where two or more individuals (Players) make decision that affect each other. Game theory is widely applied also in biomedical field. Non-Cooperative games: Individual behavior; agents cannot make agreement except for those which are established by the rules of the game. Cooperative games: There is negotiations among rational agents who can make binding agreements about how to play the game. The emphasis is on the COALITIONS of the players. 1) With whom to cooperate? 2) How to share profits/costs?

CHARACTERISTIC FORM GAME A cooperative game with transferable utility or TU-game, is a pair (N, v), where N denotes the finite set of players and v : 2 N → R the characteristic function, with v(ø) = 0. Ø A n-person game in characteristic form with player set N={1, 2, …, n} is a pair Γ= (N, v) (TU-games) Ø 2 N subsets of N, i. e. coalitions N grand coalition; |S| cardinality of S Ø v: 2 N → R s. t. v(ø) = 0 characteristic function describes how much collective payoff a set of players can gain by forming a coalition

THE PIRATES GAMES SOLUTION

AXIOMS SOLUTION There exist several solution concepts based on different notions of fairness.

THE SHAPLEY VALUE for each i ∈ N, where s=|S| and n=|N| are the cardinality of coalitions S and N, respectively. PROPERTIES

THE PIRATES GAMES If you want to go fast, go alone. If you want to go far, go together. - African proverbe -

THE CORE An imputation is a vector x ϵ RN satisfying 1. Efficency 2. Individual rationality An imputation that satisfies the coalition rationality assumption is in the CORE PROPERTIES

THE PIRATES GAMES a) b) c) d) e) x 1 + x 2 + x 3=1000; xi ≥ 0, i = 1, 2, 3; x 1 + x 2 ≥ 1000; x 1 + x 3 ≥ 1000; x 2 + x 3 ≥ 1000.

THE SHAPLEY VALUE FOR MICROARRAY GAMES Microarray game are a class of cooperative games with transferable utility (TU-game) Solution Core Shapley Value The Shapley value of a microarray game has been proposed as an index suitable to evaluate the role covered by each gene in realizing the association between the expression property and the biological condition of the original cell. But we are sure to use the Shapley value for microarray games? Axiomatic characterization

PARTNERSHIP OF GENES In order to characterize the Shapley value by means of properties with genetic interpretation, the definition of partnership of genes takes a basic role. In other words a group of genes S such that does not exist a proper ( ) subset of S which contributes in changing the worth of genes outside S. Example These two sets are partnerships of genes in the corresponding microarray game s 1 g 1 0 g 2 0 g 3 1 s 2 1 1 0 s 3 1 1 1

AXIOMATIC CHARACTERIZATION OF THE SHAPLEY VALUE FOR THE MICROARRAY GAMES Partnership rationality The PR property determines a lower bound of the power of a partnership, i. e. , the total relevance of a partnership of genes in determining the onset of the tumor in the individuals should not be lower than the average number of cases of tumor enforced by the partnership itself.

AXIOMATIC CHARACTERIZATION OF THE SHAPLEY VALUE FOR THE MICROARRAY GAMES Partnership feasibility The PF properties determines an upper bound of the power of a partnership, i. e. , the total relevance of a partnership of genes in determining the tumor onset in the individuals should not be greater than the average number of cases of tumor enforced by the grand coalition.

AXIOMATIC CHARACTERIZATION OF THE SHAPLEY VALUE FOR THE MICROARRAY GAMES Partnership monotonicity Disjoint Equivalent Exhaustive The genes in the smaller partnership should receive not less power index than genes in the larger partnership.

AXIOMATIC CHARACTERIZATION OF THE SHAPLEY VALUE FOR THE MICROARRAY GAMES Equal splitting Each sample should receive the same level of reliability. So the power of a gene on two samples should be equal to the sum of the power on each sample divided by two. Example

AXIOMATIC CHARACTERIZATION OF THE SHAPLEY VALUE FOR THE MICROARRAY GAMES Null gene A gene which does not contribute to change the value (of activations of the tumor) of any coalition of genes, should receive zero power. The aim of these properties is to state how a relevance index should behave in very simple situations of genes interaction. Theorem The Shapley value is the unique solution which satisfies NG, ES, PM, PR, PF on the class of Microarray Games. How can we calculate the Shapley value of thousands of genes?

THE SHAPLEY VALUE FOR MICROARRAY GAMES The general formula of the Shapley value for the game (N, v) is:

MICROARRAY GAMES … Array 1 Array 2 Array 3 Level of expression of gene 5 in the sample 4

MICROARRAY GAMES GOAL: Quantifying the relative relevance of genes on the basis of the information provided by microarray experiments, taking into account the level of interaction among the genes. Microarray expression data from disease samples Discretized matrix Microarray expression data from normal samples Decision rule: Cutoffs

MICROARRAY GAMES The most important gene is gene 2 followed by gene 1 and 3 with the same score But…

MICROARRAY GAMES. . . a typical experiment consists of a table of numbers with more than 22000 rows (genes) and 60 of arrays (samples). … 2 22000 coalitions… Specific software for microarray analysis

MICROARRAY GAMES Top ten genes with highest Shapley value on the microarray game Some of these genes were previously observed in association with other cancer type

REFERENCES 1. S. Moretti, "Statistical analysis of the Shapley value for microarray games. " Computers & Operations Research 37. 8 (2010): 1413 -1418. 2. S. Moretti, F. Patrone and S. Bonassi. "The class of microarray games and the relevance index for genes. “, Top 15, (2007): 256 -280. 3. L. S. Shapley, “A value for n-person games” Contributions to the Theory of Games II. Annals of Mathematics Studies, 28, (1953): 307– 317.