Stochasticity and noise-induced transition of genetic toggle switch

Wei-Yin Chen

doi:10.1186/2195-5468-2-1

Abstract

The ability to predict and analyze the function of genetic circuits will enhance the design of autonomous, programmable, complex regulatory genetic structures. An abundance of modeling techniques has recently been developed to delineate simple genetic structures in terms of their constituents. Simple systems with characteristics of feedback inhibition, multi-stability, switching, and oscillatory expression have often been the focus. The present work is an attempt to improve existing deterministic models that fail to oblige to the crucial aspect of noise in genetic modeling. The objective of this work is to analyze, model, and simulate the protein populations in gene expression mechanisms by resorting to stochastic algorithms. The system involves two types of genes; the protein produced from the expression of one gene is capable of turning off the expression of the other gene. Rates of degradation of these proteins are assumed to be proportional to their concentrations. The master equation of this ‘genetic toggle switch’ is formulated using the probabilistic population balance around a particular state and by considering five mutually exclusive events. The efficacy of the present methodology is mainly attributable to the ability to derive the governing equations for the means, variances, and covariance of the random variables by the method of system-size expansion of the nonlinear master Equation. A less laborious approach based on Kurtz’s limit theorems for the derivation of the stochastic characteristics is also presented for comparison. Solving the resultant ordinary differential equations governing the means, variances, and covariance of the master equations simultaneously using the published data yield information concerning not only the means of the two populations of proteins but also the minimal uncertainties of the populations inherent in the expressions. It is demonstrated that systems with small populations are susceptible to large internal fluctuations (or uncertainties) in their population evolution. Large uncertainties are observed after the populations enter the proximity of the saddle node, which is likely to cause transition of system’s steady state from one to another. Independent Monte-Carlo simulation runs clearly demonstrates that the occurrence of such internal noise-induced transition.

Highlights

One of the earliest examples of a bistable genetic switch is represented in the rightward operator of bacteriophage lambda [1,2]
The master equations governing the numbers of the two types of protein are formulated from stochastic population balance
The stochastic pathways of the two proteins, i.e., their means and the fluctuations around these means, have been numerically simulated independently by the algorithm derived from the master equations, as well as by an event-driven Monte Carlo algorithm

Summary

Introduction

One of the earliest examples of a bistable genetic switch is represented in the rightward operator of bacteriophage lambda [1,2]. The essential elements of this type of genetic switch, are a pair of promoters that each produces a repressor protein capable of inhibiting the production of the opposing repressor. Overlayed on these essential elements are several layers of regulatory nuance. In the proximity of the bifurcation point, the final steady-state protein population possesses a bimodal distribution in their green fluorescent protein (GFP) fluorescence. It does not have a sharp jump from one fluorescence level to another, as the deterministic model predicts. The authors surmise that the stochastic nature of the dynamics blurs the bifurcation point

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Conclusion