Bistability, Probability Transition Rate and First-Passage Time in an Autoactivating Positive-Feedback Loop

Xiu-Deng Zheng,Yi Tao,Xiao-Qian Yang,Giuseppe Chirico

doi:10.1371/journal.pone.0017104

Abstract

A hallmark of positive-feedback regulation is bistability, which gives rise to distinct cellular states with high and low expression levels, and that stochasticity in gene expression can cause random transitions between two states, yielding bimodal population distribution (Kaern et al., 2005, Nat Rev Genet 6: 451-464). In this paper, the probability transition rate and first-passage time in an autoactivating positive-feedback loop with bistability are investigated, where the gene expression is assumed to be disturbed by both additive and multiplicative external noises, the bimodality in the stochastic gene expression is due to the bistability, and the bistability determines that the potential of the Fokker-Planck equation has two potential wells. Our main goal is to illustrate how the probability transition rate and first-passage time are affected by the maximum transcriptional rate, the intensities of additive and multiplicative noises, and the correlation of additive and multiplicative noises. Our main results show that (i) the increase of the maximum transcription rate will be useful for maintaining a high gene expression level; (ii) the probability transition rate from one potential well to the other one will increase with the increase of the intensity of additive noise; (iii) the increase of multiplicative noise strength will increase the amount of probability in the left potential well; and (iv) positive (or negative) cross-correlation between additive and multiplicative noises will increase the amount of probability in the left (or right) potential well.

Highlights

Bistability arises within a wide range of biological systems from the bacteriophage l to cellular signal transduction pathways in mammalian cells [1,2]
Basic model It is well known that the simplest circuit motif able to exhibit multiple stable states is the autoactivating positive-feedback loop [5,9,10,11], in which a single gene encodes a protein, and the activator monomers bind into dimers that subsequently bind to the upstream regulatory site of the gene, activating production of the activator monomers
We have that R{ will increase with the increase of kmax but Rz will decrease with the increase of kmax. This means that the increase of kmax will promote the probability transfer from the left well to the right well, or the increase of kmax will be useful for maintaining a high gene expression level

Summary

Introduction

Bistability arises within a wide range of biological systems from the bacteriophage l to cellular signal transduction pathways in mammalian cells [1,2]. The relationship between the depth of potential well and kmax is plotted, i.e. the system state should be more attracted by the left well with the increase of the maximum transcription rate.

Results

Conclusion