Variants of Stochastic Simulation Algorithm Henry Ato Ogoe Department of Computer Science Åbo Akademi University

The Stochastic Framework Assume N molecular species {s 1,...,S N } Assume N molecular species {s 1,...,S N } State Vector State Vector X (t) = (X 1 (t),…,X N (t)) where X i (t) is the number of molecules of species S i at time t. M reaction channels R 1,…,R N Assume system is well-stirred and in thermal equilibrium

Dynamics of reaction channel R i is characterized by its Dynamics of reaction channel R i is characterized by its propensity function a j, and propensity function a j, and state change vector v j =(v 1j,…,v Nj ), where v ij gives change in population of S i induced by R j, such that state change vector v j =(v 1j,…,v Nj ), where v ij gives change in population of S i induced by R j, such that a j (x)dt is the probability that, given X(t) = x, one reaction will occur in the next infinitesimal time interval [t,t + dt] a j (x)dt is the probability that, given X(t) = x, one reaction will occur in the next infinitesimal time interval [t,t + dt] R(x) is a jump Markov process R(x) is a jump Markov process The Stochastic Framework

The time evolution of the probabilities of each state is defined by the Chemical Master Equation (CME); The time evolution of the probabilities of each state is defined by the Chemical Master Equation (CME); where P(x,t|x 0,t 0 ) is the probability that X(t)=x given X(t 0 ) = x 0 where P(x,t|x 0,t 0 ) is the probability that X(t)=x given X(t 0 ) = x 0 CME is impractical to solve especially for large systems CME is impractical to solve especially for large systems Alternative approaches??? Alternative approaches???

Alternative Approaches to the CME Exact Simulations Exact Simulations Inexact Simulations/Approximations Inexact Simulations/Approximations

Exact Stochastic Simulation Starting from the initial states, X(t 0 ) the SSA simulated the trajectory by repeatedly updating the states after estimating Starting from the initial states, X(t 0 ) the SSA simulated the trajectory by repeatedly updating the states after estimating 1. τ, the time the next reaction will fire, and 2. μ, the index of the firing reaction Both τ and μ can be estimated probabilistically from the probability density function P(μ,τ) that the next reaction is μ and it occurs at τ. Both τ and μ can be estimated probabilistically from the probability density function P(μ,τ) that the next reaction is μ and it occurs at τ.

Exact Stochastic simulation Let Let It can be shown that It can be shown that Integrating P(μ,τ) over all τ from 0 to ∞ Integrating P(μ,τ) over all τ from 0 to ∞ P(μ = j) = a j /a 0 P(μ = j) = a j /a 0 Summing P(μ,τ) over all μ Summing P(μ,τ) over all μ The two distributions above leads to Gillespie‘s SSA and other mathematically equivalent

variants with different computational efficiency

First Reaction Method (FRM)- Gillespie, 1977 Generate a putative time τ k for each reaction channel R k according to Generate a putative time τ k for each reaction channel R k according to where k = 1,…,M; r 1,…,r M are M statistically independent random samplings of U(0,1) where k = 1,…,M; r 1,…,r M are M statistically independent random samplings of U(0,1) τ = min{τ 1,…,τ M } τ = min{τ 1,…,τ M } μ = index of min{τ 1,…,τ M } μ = index of min{τ 1,…,τ M } Update X X + V μ Update X X + V μ

Flaws ???? Uses M random numbers per time step Uses M random numbers per time step Uses O (M) to update the a k ’s Uses O (M) to update the a k ’s Uses O (M) to identify smallest τ μ Uses O (M) to identify smallest τ μ

Direct Method (DM)-Gillespie, 1977 Draw two independent samples r 1 and r 2 from U(0,1) Draw two independent samples r 1 and r 2 from U(0,1) The index of the firing reaction is the smallest integer satisfying The index of the firing reaction is the smallest integer satisfying

Flaws???? Unnecessary recalculation of all propensities Unnecessary recalculation of all propensities Slow, search depth (the no. of steps taken to identify ) ≈ O (M) Slow, search depth (the no. of steps taken to identify ) ≈ O (M)

Next Reaction Method (NRM) – Gibson & Bruck (2000)