Stability and bifurcation analysis of new coupled repressilators in genetic regulatory networks with delays

Delay-dependent Lurie-Postnikov type Lyapunov-Krasovskii functionals for stability analysis of discrete-time delayed neural networks

This paper addresses the influence of time-varying delay and nonlinear activation functions with sector restrictions on the stability of discrete-time neural networks. Compared to previous works that mainly focuses on the influence of delay information, this paper devotes to activation nonlinear functions information to help compensate the analysis technique based on Lyapunov-Krasovskii functional (LKF). A class of delay-dependent Lurie-Postnikov type integral terms involving sector constraints of nonlinear activation function is proposed to complement the LKF construction. The less conservative criteria for the stability analysis of discrete-time delayed networks is given by using improved LKF. Numerical examples show that conservatism can be reduced by the delay-dependent integral terms involving nonlinear activation functions.

State Observer Design of Coupled Genetic Regulatory Networks With Reaction-Diffusion Terms via Time-Space Sampled-Data Communications

In this paper, state observation of coupled reaction-diffusion genetic regulatory networks (GRNs) with time-varying delays is investigated under Dirichlet boundary conditions. First, the above GRNs are constructed to model gene regulatory properties, where the feedback regulation function of the GRNs is assumed to exhibit the Hill form and a novel method to deal with it is introduced. Then a time-space sampled-data state observer is designed for the mentioned networks and new criteria are established by utilizing the Lyapunov stability theory and the inequality techniques of Halanay et al. Finally, the validity of the theoretical results is proved by numerical examples.

Stochastic quasi-synchronization of heterogeneous delayed impulsive dynamical networks via single impulsive control

This paper investigates the quasi-synchronization problem of the stochastic heterogeneous complex dynamical networks with impulsive couplings and multiple time-varying delays. It is shown that this kind of dynamical networks can achieve exponential quasi-synchronization by exerting impulsive control added on only one chosen pinning node. By employing the Lyapunov stability theory, some sufficient criteria on quasi-synchronization for this dynamical network are established, revealing the relationship between the quasi-synchronization performance and the stochastic perturbations as well as the frequency and strength of impulsive coupling. Finally, some numerical examples are used to illustrate the effectiveness of the main results.

Multi-stability in cellular differentiation enabled by a network of three mutually repressing master regulators

Identifying the design principles of complex regulatory networks driving cellular decision-making remains essential to decode embryonic development as well as enhance cellular reprogramming. A well-studied network motif involved in cellular decision-making is a toggle switch-a set of two opposing transcription factors A and B, each of which is a master regulator of a specific cell fate and can inhibit the activity of the other. A toggle switch can lead to two possible states-(high A, low B) and (low A, high B)-and drives the 'either-or' choice between these two cell fates for a common progenitor cell. However, the principles of coupled toggle switches remain unclear. Here, we investigate the dynamics of three master regulators A, B and C inhibiting each other, thus forming three-coupled toggle switches to form a toggle triad. Our simulations show that this toggle triad can lead to co-existence of cells into three differentiated 'single positive' phenotypes-(high A, low B, low C), (low A, high B, low C) and (low A, low B, high C). Moreover, the hybrid or 'double positive' phenotypes-(high A, high B, low C), (low A, high B, high C) and (high A, low B, high C)-can coexist together with 'single positive' phenotypes. Including self-activation loops on A, B and C can increase the frequency of 'double positive' states. Finally, we apply our results to understand cellular decision-making in terms of differentiation of naive CD4⁺ T cells into Th1, Th2 and Th17 states, where hybrid Th1/Th2 and hybrid Th1/Th17 cells have been reported in addition to the Th1, Th2 and Th17 ones. Our results offer novel insights into the design principles of a multi-stable network topology and provide a framework for synthetic biology to design tristable systems.

New Method to Global Exponential Stability Analysis for Switched Genetic Regulatory Networks With Mixed Delays

In this paper, the sufficient conditions for the global exponential stability of the switched genetic regulatory networks with mixed time delays are obtained. The proposed method does not need the construction of Lyapunov-Krasovskii functional, but is directly proceeded by the definition of global exponential stability. The derived sufficient conditions can easily be verified by checking the eigenvalues of a constant matrix or solving several simple linear matrix inequalities. Finally, two numerical examples are presented to illustrate that the obtained global exponential stability criteria are available.

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.

Multisynchronization of Coupled Heterogeneous Genetic Oscillator Networks via Partial Impulsive Control

This paper focuses on the collective dynamics of multisynchronization among heterogeneous genetic oscillators under a partial impulsive control strategy. The coupled nonidentical genetic oscillators are modeled by differential equations with uncertainties. The definition of multisynchronization is proposed to describe some more general synchronization behaviors in the real. Considering that each genetic oscillator consists of a large number of biochemical molecules, we design a more manageable impulsive strategy for dynamic networks to achieve multisynchronization. Not all the molecules but only a small fraction of them in each genetic oscillator are controlled at each impulsive instant. Theoretical analysis of multisynchronization is carried out by the control theory approach, and a sufficient condition of partial impulsive controller for multisynchronization with given error bounds is established. At last, numerical simulations are exploited to demonstrate the effectiveness of our results.

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

Many biological systems have the conspicuous property to present more than one stable state and diverse rhythmic behaviors. A closed relationship between these complex dynamic behaviors and cyclic genetic structures has been witnessed by pioneering works. In this paper, a typical structure of inhibitory coupled cyclic genetic networks is introduced to further enlighten this mechanism of stability and biological rhythms of living cells. The coupled networks consist of two identical cyclic genetic subnetworks, which inhibit each other directly. Each subnetwork can be regarded as a genetic unit at the cellular level. Multiple time delays, including both internal and coupling delays, are considered. The existence of positive equilibriums for this kind of coupled systems is proved, and the stability for each equilibrium is analyzed without or with delays. It is shown that the coupled networks with positive cyclic genetic units have an ability to show multistability, while the coupled networks with negative units may present a series of Hopf bifurcations with the variation of time delays. Several numerical simulations are made to prove our results.