Oscillatory expression in Escherichia coli mediated by microRNAs with transcriptional and translational time delays

Oscillation dynamic mechanism driven by time delays in the competent gene regulatory circuit of B. subtilis

Bacillus subtilis with competent states absorbs DNA and may improve the growth of bacteria by integrating new genetic material. Therefore, it is important to clarify how the genes interact in the circuit so that cells enter into a competent state or return to a vegetative state. The gene regulatory circuit consists of two positive feedback loops and one negative feedback loop. In this paper, a mathematical model is developed by considering transcription time delays to further study dynamic behavior of the B. subtilis competent gene regulatory network. Combined with theoretical calculation and numerical simulation, it is verified that the time delay in indirect transcription inhibition indeed has the effect of inducing the periodic oscillation of the B. subtilis competent system. In addition, some important chemical reaction rates can also regulate system dynamic behavior. However, under the control of time delay, the effects of the important chemical reaction rates have changed significantly. In particular, the time delay can advance critical value of the important chemical reaction rates where vibration occurs and can also weaken or even eliminate the effect of the important chemical reaction rates. These results will help us to analyze the competent state of B. subtilis.

Stability and Oscillation Analysis of a Gene Regulatory Network With Multiple Time Delays and Diffusion Rate

In genetic regulatory networks (GRNs), the diffusion rate of mRNA and protein play a key role in regulatory mechanisms of gene expression, especially in translation and transcription. However, the influence of diffusion rate on oscillatory gene expression is not well understood. In this paper, by considering the diffusion rate of mRNA and protein, a novel GRN is proposed. Then, two basic problems of such network, i.e. stability and oscillation, are solved in detail. Moreover, the properties of oscillation are also investigated. it is found that the total biochemistry reaction time can affect the stability of the positive equilibrium and give rise to the oscillation. The diffusion rate of mRNA and proteins have a major impact on the oscillation properties. Finally, two examples not only verify the theoretical results, but also show that a slight diffusion rate increasing may lead to huge change in oscillatory gene expressions.

Parametric Sensitivity Analysis of Oscillatory Delay Systems with an Application to Gene Regulation

A parametric sensitivity analysis for periodic solutions of delay-differential equations is developed. Because phase shifts cause the sensitivity coefficients of a periodic orbit to diverge, we focus on sensitivities of the extrema, from which amplitude sensitivities are computed, and of the period. Delay-differential equations are often used to model gene expression networks. In these models, the parametric sensitivities of a particular genotype define the local geometry of the evolutionary landscape. Thus, sensitivities can be used to investigate directions of gradual evolutionary change. An oscillatory protein synthesis model whose properties are modulated by RNA interference is used as an example. This model consists of a set of coupled delay-differential equations involving three delays. Sensitivity analyses are carried out at several operating points. Comments on the evolutionary implications of the results are offered.