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

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

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

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

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

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

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 Collection of microscopic DNA probes attached to a solid surface such as glass, plastic, or silicon chip forming an array (matrix) To compare gene expression profile of a patient with that of a healthy one to identify which genes are involved in the disease. WHAT THEY DO? WHAT THEY ARE? 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 They allow to simultaneously examine the presence of many genes within a DNA sample

 This step is repeated for the wild type sample and for the patient sample RED: cDNA of cancerous cells GREEN: cDNA of wild type cells  Hybridization technique

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

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. 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. Non-Cooperative games: Individual behavior; agents cannot make agreement except for those which are established by the rules of the game. 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.  A n-person game in characteristic form with player set N={1,2,…,n} is a pair Γ= (N, v) (TU-games)  2 N subsets of N, i.e. coalitions N grand coalition; |S| cardinality of S  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

SOLUTION THE PIRATES GAMES

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

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

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

THE PIRATES GAMES a)x1 + x2 + x3=1000; b)xi ≥ 0, i = 1,2,3; c)x1 + x2 ≥ 1000; d)x1 + x3 ≥ 1000; e)x2 + x3 ≥ 1000.

THE SHAPLEY VALUE FOR MICROARRAY GAMES Shapley Value Core Microarray game are a class of cooperative games with transferable utility (TU-game) 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 Solution

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. Example 101g3 110g2 110g1 s3s2s1 These two sets are partnerships of genes in the corresponding microarray game 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.

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.

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 Partnership monotonicity DisjointEquivalentExhaustive

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. AXIOMATIC CHARACTERIZATION OF THE SHAPLEY VALUE FOR THE MICROARRAY GAMES Equal splitting Example

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

Level of expression of gene 5 in the sample 4 Array1Array2Array3 … 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. Decision rule: Microarray expression data from disease samples Microarray expression data from normal samples Cutoffs Discretized matrix

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

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

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): S. Moretti, F. Patrone and S. Bonassi. "The class of microarray games and the relevance index for genes.“, Top 15, (2007): L.S. Shapley, “A value for n-person games” Contributions to the Theory of Games II. Annals of Mathematics Studies, 28, (1953): 307–317.