Analysis of Systems of Chemical Equations with Decision Diagrams A Decision DiagramThe Goddess Durga

inputsoutput Large domain, small range? Analysis of Systems of Chemical Equations N m possibilities Yes/No possibilities Chemical Equations Yes/No ? For m species, each with max. quantity N : Initial State

Biochemical Reactions Lingua Franca of computational biology. 1 molecule of type A combines with 2 molecules of type B to produce 2 molecules of type C. Reaction Reaction is annotated with a rate constant and physical constraints (localization, gradients, etc.)

Biochemical Reactions Lingua Franca of computational biology. Elementary molecules (e.g., hydrogen, phosphorous,...) Complex molecules (e.g., proteins, enzymes, RNA...) Species: Reaction: Elementary step (e.g., ) Conglomeration of steps (e.g., transcription of gene product) Reaction

Coupled Set Reactions R1R1 R2R2 R3R3 Goal: given initial conditions, analyze (predict) the evolution of such a system. Lingua Franca of computational biology. Biochemical Reactions

Can a certain state, S 1, be transformed into another state, S 2 ? If so, in how many different ways? S 2 System of Chemical Equations Discrete Quantities of Molecular Species Types of Questions: Can S 1 be transformed into S 2 without passing through a third state S 3 ? Can S 1 be reached from at least one state in a set of states T? From all the states in a set of states U?

Decision Diagrams States ABC S1S1 S2S2 S3S3 e.g., set of possible initial states

state beforestate after R 1 occurs R 2 occurs R 3 occurs or State Evolution System of Chemical Equations

Decision Diagrams CBA22  R1:R1: reaction 1 occurs

Decision Diagrams reaction 2 occurs R2:R2: ACB2 

Decision Diagrams reaction 3 occurs R3:R3: BCA2 

S1S1 or Reachable States After The Next Reaction Decision Diagrams S2S2 S3S3

Evolution of Reachable States Decision Diagrams S S1S1 or S2S2 S3S3 T T1T1 T2T2 T3T3 U... Track evolution of a large number of states “in parallel”.

Yes/No Questions Decision Diagram Can ask (and answer) arbitrarily complicated yes/no questions pertaining to reachability: Yes if C 1 or not(C 2 ) C 1 : state S is reachable after 100 reactions C 2 : state T is reachable from state U or from state V but not from both C 3 : state X is never reachable Yes if not C 1 and (C 2 or C 3 )