Biochemical Reactions: how types of molecules combine. Playing by the Rules + + 2a2a b c

Biochemical Reactions cell speciescount Discrete chemical kinetics; spatial homogeneity.

Biochemical Reactions slow medium fast Relative rates or (reaction propensities): Discrete chemical kinetics; spatial homogeneity.

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

Assumes that molecular quantities are continuous values that vary deterministically over time. Convential Approach: numerical calculations based on coupled ordinary differential equations. d[A]/dt = 2k_2 [B][C] -- k_1 [A][B]^2 -- k_3 [A][C] d[B]/dt = 2k_3 [A][C] -- k_1 [A][B]^2 -- k_2 [B][C] d[C]/dt = 2k_1 [A][B]^2 -- k_2 [B][C]^-- k_3 [A][C]

R1R1 R2R2 R3R3 See D. Gillespie, “Stochastic Chemical Kinetics”, The probability that a given reaction is the next to fire is proportional to: Its rate. The number of ways that the reactants can combine. Discrete Stochastic Kinetics

Choose the next reaction according to: RiRi let For each reaction Stochastic Kinetics

Track precise (integer) quantities of molecular species. “States” ABC S1S1 S2S2 S3S3 A reaction transforms one state into another: e.g., Gillespie’s Framework Reactions R1R1 R2R2 R3R3

S 1 = [5, 5, 5] 0 Choose the next reaction according to: Stochastic Simulation RiRi where R1R1 R2R2 R3R3

Stochastic Simulation RiRi R1R1 R2R2 R3R3 Choose the time of the next reaction according to: S 1 = [5, 5, 5] 0

Stochastic Simulation R1R1 R2R2 R3R3 See D. Gillespie, “Exact Stochastic Simulation of Coupled Chemical Reactions”, J. Phys. Chem S 1 = [5, 5, 5] 0

Stochastic Simulation S 1 = [5, 5, 5] 0 S 2 = [4, 7, 4] Choose R 3 and t = 3 seconds. R1R1 R2R2 R3R3 S 3 = [2, 6, 7] 4 Choose R 1 and t = 1 seconds. S 4 = [1, 8, 6] 6 Choose R 3 and t = 2 seconds. 3 Choose R 2 and t = 1 seconds.

Stochastic Simulation S 1 = [5, 5, 5] 0 S 2 = [4, 7, 4] Choose R 3 and t = 3 seconds. S 3 = [2, 6, 7] 4 Choose R 1 and t = 1 seconds. S 4 = [1, 8, 6] 6 Choose R 3 and t = 2 seconds. Choose R 2 and t = 1 seconds. 37