Multistability and Bifurcation Analysis of Inhibitory Coupled Cyclic Genetic Regulatory Networks With Delays

Stability and Bifurcation Analysis of a Diffusive miR-9/Hes1 Network With Time Delay

In this paper, a model of miR-9/Hes1 interaction network involving one time delay and diffusion effect under the Neumann boundary conditions is studied. First of all, the stability of the positive equilibrium and the existence of local Hopf bifurcation and Turing-Hopf bifurcation are investigated by analyzing the associated characteristic equation. Second, a algorithm for determining the direction, stability and period of the corresponding bifurcating periodic solutions is presented. The obtained results suggest that the quiescent progenitors (high steady-state Hes1) can be easily excited into oscillation by time delay whereas the differentiated state (low steady-state Hes1) is basically unaffected, and the integrated effect of delay and diffusion can induce the occurrence of spatially inhomogeneous patterns. More importantly, spatially homogeneous/inhomogeneous periodic solutions can exist simultaneously when the diffusion coefficients of Hes1 mRNA and Hes1 protein are appropriately small, conversely, there is only spatially homogeneous periodic solutions. Intriguingly, both temporal patterns and spatial-temporal patterns show that time delay can prompt Hes1 protein to shift from the high concentration state to the low concentration one ("ON" → "OFF"), where Hes1 protein shows low level whereas miR-9 shows high level. Finally, some numerical examples are presented to verify and visualize theoretical results.

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.

Bifurcation and Oscillatory Dynamics of Delayed Cyclic Gene Networks Including Small RNAs

It has been demonstrated in a large number of experimental results that small RNAs (sRNAs) play a vital role in gene regulation processes. Thus, the gene regulation process is dominated by sRNAs in addition to messenger RNAs and proteins. However, the regulation mechanism of sRNAs is not well understood and there are few models considering the effect of sRNAs. So it is of realistic biological background to include sRNAs when modeling gene networks. In this paper, sRNAs are incorporated into the process of gene expression and a new differential equation model is put forward to describe cyclic genetic regulatory networks with sRNAs and multiple delays. We mainly investigate the stability and bifurcation criteria for two cases: 1) positive cyclic genetic regulatory networks and 2) negative cyclic genetic regulatory networks. For a positive cyclic genetic regulatory network, it is revealed that there may exist more than one equilibrium and the multistability can appear. Sufficient conditions are established for the delay-independent stability and fold bifurcations. It is found that the dynamics of positive cyclic gene networks has no bearing on time delays, but depends on the biochemical parameters, the Hill coefficient and the equilibrium itself. For a negative cyclic genetic regulatory network, it is proved that there exists a unique equilibrium. Delay-dependent conditions for the stability are derived, and the existence of Hopf bifurcations is examined. Different from the delay-independent stability of positive gain networks, the stability of equilibrium is determined not only by the biochemical parameters, the Hill coefficient and the equilibrium itself, but also by the total delay. At last, three illustrative examples are provided to validate the major results.

State Estimation for Discrete Time-Delayed Genetic Regulatory Networks With Stochastic Noises Under the Round-Robin Protocols

This paper investigates the problem of state estimation for discrete time-delayed genetic regulatory networks with stochastic process noises and bounded exogenous disturbances under the Round-Robin (RR) protocols. The network measurement outputs obtained by two groups of sensors are transmitted to two remote sub-estimators via two independent communication channels, respectively. To lighten the communication loads of the networks and reduce the occurrence rate of data collisions, two RR protocols are utilized to orchestrate the transmission orders of sensor nodes in two groups, respectively. The error dynamics of the state estimation is governed by a switched system with periodic switching parameters. By constructing a transmission-order-dependent Lyapunov-like functional and utilizing the up-to-date discrete Wirtinger-based inequality together with the reciprocally convex approach, sufficient conditions are established to guarantee the exponentially ultimate boundedness of the estimation error dynamics in mean square with a prescribed upper bound on the decay rate. An asymptotic upper bound of the outputs of the estimation errors in mean square is derived and the estimator parameters are then obtained by minimizing such an upper bound subject to linear matrix inequality constraints. The repressilator model is utilized to illustrate the effectiveness of the designed estimator.