An Efficient Algorithm for Generating Trajectories of Stochastic Gene Regulation Reactions

Abstract

Systems of weakly coupled chemical equations occur in gene regulation and other biological systems. For small numbers of molecules (as in a small cell), the usual differential equations approach to chemical kinetics must be replaced with a stochastic approach. To deal with this kind of system, one generates trajectories through stochastic phase space. By generating a large enough number of trajectories, one can understand the statistics of the behavior of the complex, non-linear system. The algorithms for dealing with sparsely connected stochastic processes are not as advanced as those for sparse deterministic processes. In particular. the existing algorithm of choice for generating trajectories, which is not optimized in any way for sparseness, is O(rE), where r is the number of reactions and E is the number of reaction events in the trajectory. \Ye present two algorithms of O(r + Elogr), one of which is a simple extension of the existing algorithm, and the other of which is more subtle. The latter is more easily extended to include stochastic processes of different types. We apply our faster algorithm to a model of bacteriophage lambda and are able to run the same calculations on a cluster of desktop workstations that previously required a supercomputer. This allows us to run more complicated calculations than could be done previously. As an example of this, we analyse the sensitivity of the lambda model to the values of several of its parameters. We find that the model is relatively insensitive to changes in the translation rate, protein dimerization rates and protein degradation rates; is somewhat sensitive to the transcription rate. and is extremely sensitive to the average number of proteins per mRNA transcript.

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