Uppsala University Department of Information Technology

Licentiate thesis 2008-003

Numerical Simulation of Well Stirred Biochemical Reaction Networks Governed by the Master Equation

Andreas Hellander

15 October 2008


Numerical simulation of stochastic biochemical reaction networks has received much attention in the growing field of computational systems biology. Systems are frequently modeled as a continuous-time discrete space Markov chain, and the governing equation for the probability density of the system is the (chemical) master equation. The direct numerical solution of this equation suffers from an exponential growth in computational time and memory with the number of reacting species in the model. As a consequence, Monte Carlo simulation methods play an important role in the study of stochastic chemical networks. The stochastic simulation algorithm (SSA) due to Gillespie has been available for more than three decades, but due to the multi-scale property of the chemical systems and the slow convergence of Monte Carlo methods, much work is currently being done in order to devise more efficient approximate schemes.

In this thesis we review recent work for the solution of the chemical master equation by direct methods, by exact Monte Carlo methods and by approximate and hybrid methods. We also describe two conceptually different numerical methods to reduce the computational time when studying models using the SSA. A hybrid method is proposed, which is based on the separation of species into two subsets based on the variance of the copy numbers. This method yields a significant speed-up when the system permits such a splitting of the state space. A different approach is taken in an algorithm that makes use of low-discrepancy sequences and the method of uniformization to reduce variance in the computed density function.

Note: Included papers available at http://dx.doi.org/10.1016/j.jcp.2007.07.020, http://dx.doi.org/10.1007/s10543-008-0174-z, and http://dx.doi.org/10.1063/1.2897976

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Uppsala Universitet