Stochastic models for regulatory networks of the genetic toggle switch

Abstract

Bistability arises within a wide range of biological systems from the lambda phage switch in bacteria to cellular signal transduction pathways in mammalian cells. Changes in regulatory mechanisms may result in genetic switching in a bistable system. Recently, more and more experimental evidence in the form of bimodal population distributions indicates that noise plays a very important role in the switching of bistable systems. Although deterministic models have been used for studying the existence of bistability properties under various system conditions, these models cannot realize cell-to-cell fluctuations in genetic switching. However, there is a lag in the development of stochastic models for studying the impact of noise in bistable systems because of the lack of detailed knowledge of biochemical reactions, kinetic rates, and molecular numbers. In this work, we develop a previously undescribed general technique for developing quantitative stochastic models for large-scale genetic regulatory networks by introducing Poisson random variables into deterministic models described by ordinary differential equations. Two stochastic models have been proposed for the genetic toggle switch interfaced with either the SOS signaling pathway or a quorum-sensing signaling pathway, and we have successfully realized experimental results showing bimodal population distributions. Because the introduced stochastic models are based on widely used ordinary differential equation models, the success of this work suggests that this approach is a very promising one for studying noise in large-scale genetic regulatory networks.

Fig. 1.

The genetic toggle switch interfaced with the SOS signaling pathway.

Fig. 2.

Simulations of the genetic toggle switch interfaced with the SOS signaling pathway. (A) A deterministic simulation of unsuccessful switching (s = 1.7). (B) A stochastic simulation of unsuccessful switching based on s = 1.7. (C) A stochastic simulation of successful switching also based on s = 1.7. (D) Percentages of switched cells in stochastic simulations based on different degradation parameter s and percentages obtained by a Hill function p(s) = 1.2364 × (s − s₀)⁴/(0.25⁴ + (s − s₀)⁴) × 100%, where 1.2364 is used to match the simulated percentage when s = 1.85.

Fig. 3.

Comparison of simulation results with experimental observations for the genetic toggle switch interfaced with the SOS signaling pathway. Numbers of cells with different LacR molecular numbers are based on 1,000 simulations, and experimental observations in fluorescence signal are derived from figure 3B in ref. using the top and right labels. (A) s = 1.3; no cell has high LacR expression level, and no MMC was applied in experiments. (B) s = 1.7; 35.9% of cells have high LacR expression levels, and 1 ng/ml MMC was applied. (C) s = 1.75; 67.1% of cells have high LacR expression level, and 10 ng/ml MMC was applied. (D) s = 2.0; all cells have high LacR expression levels, and 500 ng/ml MMC was applied.

Fig. 4.

The genetic toggle switch interfaced by a quorum-sensing signaling pathway.

Fig. 5.

Simulations of the genetic toggle switch interfaced by the quorum-sensing signaling pathway. (A) A deterministic simulation with 3,080 cells. (B) A stochastic simulation of successful switching with 3,080 cells. (C) A stochastic simulation of unsuccessful switching with 3,080 cells. (D) Percentages of switched cells obtained by simulations or from a Hill function p(c) = 1.0232 × c⁵/(3,560⁵ + c⁵) × 100%, where c is the cell number and 1.0232 is used to ensure p(7,560) = 100%.

Fig. 6.

Comparison of simulation results with experimental observations for the genetic toggle switch interfaced by the quorum-sensing signaling pathway. Numbers of cells with different molecular numbers of LacR are based on simulations with different cell population densities in the culture, and experimental observations in fluorescence signal are derived from figure 6B in ref. using the top and right labels. (A) n = 840; only nine cells have high LacR expression levels, and A₆₀₀ = 0.06. (B) n = 1,400; 68 cells have high LacR expression levels, and A₆₀₀ = 0.10. (C) n = 3,080; 1,042 cells have high LacR expression levels, and A₆₀₀ = 0.22. (D) n = 7,560; all cells have high LacR expression levels, and A₆₀₀ = 0.54.