Effect of molecular noise on the dynamics of tryptophan operon system in Escherichia coli

Abstract

Noise in gene expression, or the variation in gene expression in an isogenic population under a homogeneous environment, has been of much interest in recent years. Differences in gene expression of two isogenic cells could be attributed to the variation in factors determining gene expression in these cells, such as transcription factors, the concentration of operators, RNA polymerase, the cell cycle, etc., which is termed extrinsic noise. However, variation could still persist even when all extrinsic noise is eliminated, due to the limited number of molecules for typical molecular species involved. The latter is termed intrinsic noise. However, the implications of stochastic gene expression are still not clear. There is very little knowledge about the consequences of stochasticity on particular systems. Here, we seek to better understand what differences may result from stochastic and deterministic kinetic approaches to modelling genetic regulatory systems by considering a model system of tryptophan (Trp) operon system in Escherichia coli. This genetic regulatory network is responsible for the production of tryptophan amino acid inside the cells. The molecular basis of the system is presented in the introduction part of the paper. The development and analysis of two stochastic models for the tryptophan operon system are discussed in section 2 and 3. In the first model we introduce molecular noise by setting up stochastic differential equations using the Langevin approach in which molecular fluctuation in the form of white noise is explicitly considered. The second stochastic model is based on the Gillespie method. Due to the lack of data on kinetic rates for elementary reaction steps of molecular processes, the implementation of the Gillespie method is carried out without decomposing the deterministic mechanism into detailed reaction steps. Simulation results from two versions of the stochastic regimes are compared to their deterministic counterpart. We found that intrinsic fluctuations resulted from molecular noise can destroy stable oscillatory behaviour. In this case, a new value for the bifurcation point is established, which is far from the corresponding deterministic bifurcation point. Moreover, we demonstrate that intrinsic noise can enable the system to obtain qualitatively different dynamics compared to when noise is absent. Specifically, stable sustained oscillations are obtained only when molecular noise is incorporated. Quantification of noise strength for key molecular species indicates that the transcription process exhibits high fluctuation levels which subsequently suggests that in order to reduce noise at the tryptophan output level, one may consider speeding up mRNA transcripts degradation.