Dynamical Modeling and Qualitative Analysis of a Delayed Model for CD8 T Cells in Response to Viral Antigens

Although the immune effector CD8 T cells play a crucial role in clearance of viruses, the mechanisms underlying the dynamics of how CD8 T cells respond to viral infection remain largely unexplored. Here, we develop a delayed model that incorporates CD8 T cells and infected cells to investigate the functional role of CD8 T cells in persistent virus infection. Bifurcation analysis reveals that the model has four steady states that can finely divide the progressions of viral infection into four states, and endows the model with bistability that has ability to achieve the switch from one state to another. Furthermore, analytical and numerical methods find that the time delay resulting from incubation period of virus can induce a stable low-infection steady state to be oscillatory, coexisting with a stable high-infection steady state in phase space. In particular, a novel mechanism to achieve the switch between two stable steady states, time-delay-based switch, is proposed, where the initial conditions and other parameters of the model remain unchanged. Moreover, our model predicts that, for a certain range of initial antigen load: 1) under a longer incubation period, the lower the initial antigen load, the easier the virus infection will evolve into severe state; while the higher the initial antigen load, the easier it is for the virus infection to be effectively controlled and 2) only when the incubation period is small, the lower the initial antigen load, the easier it is to effectively control the infection progression. Our results are consistent with multiple experimental observations, which may facilitate the understanding of the dynamical and physiological mechanisms of CD8 T cells in response to viral infections.

State estimation for delayed genetic regulatory networks with reaction diffusion terms and Markovian jump

Robust state estimation for delayed genetic regulatory networks with reaction–diffusion terms and uncertainties terms under Dirichlet boundary conditions is addressed in this article. The main purpose of the problem investigation is to design a novel state observer for estimate the true concentrations of mRNA and protein by available measurement outputs. Based on Lyapunov–Krasovskii functions and linear matrix inequalities (LMI), sufficient conditions are given to ensure the robust stability of the estimation error networks. Two examples are presented to illustrate the effectiveness of the proposed approach.

Global Asymptotic Stability of Competitive Neural Networks with Reaction-Diffusion Terms and Mixed Delays

In this article, a new competitive neural network (CNN) with reaction-diffusion terms and mixed delays is proposed. Because this network system contains reaction-diffusion terms, it belongs to a partial differential system, which is different from the existing classic CNNs. First, taking into account the spatial diffusion effect, we introduce spatial diffusion for CNNs. Furthermore, since the time delay has an essential influence on the properties of the system, we introduce mixed delays including time-varying discrete delays and distributed delays for CNNs. By constructing suitable Lyapunov–Krasovskii functionals and virtue of the theories of delayed partial differential equations, we study the global asymptotic stability for the considered system. The effectiveness and correctness of the proposed CNN model with reaction-diffusion terms and mixed delays are verified by an example. Finally, some discussion and conclusions for recent developments of CNNs are given.

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.

Mathematical Modeling of ceRNA-Based Interactions

Competing endogenous RNA (ceRNA) molecules have emerged as key players in regulating gene expression, increasing the complexity of the range of possible dynamics within a cell. The actions of competing RNA typically are sponging behaviors, in a manner that fine-tunes gene expression, but there are particular network structures that may show destabilization due to ceRNA interactions. In this chapter, we discuss how these interactions can be modeled and probed from a mathematical, first-principles perspective.

Bifurcation and oscillatory dynamics of delayed CDK1-APC feedback loop

Extensive experimental evidence has been demonstrated that the dynamics of CDK1-APC feedback loop play crucial roles in regulating cell cycle processes, but the dynamical mechanisms underlying the regulation of this loop are still not completely understood. Here, the authors systematically investigated the stability and bifurcation criteria for a delayed CDK1-APC feedback loop. They showed that the maximum reaction rate of CDK1 inactivation by APC can drive sustained oscillations of CDK1 activity ([inline-formula removed]) and APC activity ([inline-formula removed]), and the amplitude of these oscillations is increasing with the increase of the reaction rate over a wide range; a certain range of the self-activation rate for CDK1 is also significant for generating these oscillations, for too high or too low rates the oscillations cannot be generated. Moreover, they derived the sufficient conditions to determine the stability and Hopf bifurcations, and found that the sum of time delays required for activating CDK1 and APC can induce [inline-formula removed] and [inline-formula removed] to be oscillatory, even when the [inline-formula removed] and [inline-formula removed] settle in a definite stable steady state. Furthermore, they presented an explicit algorithm for the properties of periodic oscillations. Finally, numerical simulations have been presented to justify the validity of theoretical analysis.

Oscillatory Dynamics of p53-Mdm2 Circuit in Response to DNA Damage Caused by Ionizing Radiation

Although the dynamical behavior of the p53-Mdm2 loop has been extensively studied, the understanding of the mechanism underlying the regulation of this pathway still remains limited. Herein, we developed an integrated model with five basic components and three ubiquitous time delays for the p53-Mdm2 interaction in response to DNA damage following ionizing radiation (IR). We showed that a sufficient amount of activated ATM level can initiate the p53 oscillations with nearly the same amplitude over a wide range of the ATM level; a proper range of p53 level is also required for generating the oscillations, for too high or too low levels it would fail to generate the oscillations; and increased Mdm2 level leads to decreased amplitude of the p53 oscillation and reduced expression of the p53 activity. Moreover, we found that the negative feedback loop formed between p53 and nuclear Mdm2 plays a dominant role in determining the p53 dynamics, whereas when interaction strength of the negative feedback loop becomes weaker, the positive feedback loop formed between p53 and cytoplasmatic Mdm2 can induce different types of dynamics. Furthermore, we demonstrated that the total time delay required for protein production and nuclear translocation of Mdm2 can induce p53 oscillations even when the p53 level is at a certain stable high steady state or at a certain stable low steady state. In addition, the two important features of the oscillatory dynamics-amplitude and period-can be controlled by such time delay. These results are in agreement with multiple experimental observations and may enrich our understanding of the dynamics of the p53 network.

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.

Oscillatory Dynamics of p53 Genetic Network Induced by Feedback Loops and Time Delays

DNA damage caused by γ -irradiation initiates oscillatory expression of the p53 genetic network. Although many studies revealed the effects of the p53-Mdm2 circuit on p53 dynamics, a few studies explored the contribution of upstream kinases to p53 oscillation. In this paper, an integrated mathematical model of the p53 network in response to γ -irradiation is studied, which consists of five basic components, two ubiquitous time delays, and two negative feedback loops. It is found that recurrent p53 pulses are externally initiated by ataxia telangiectasia mutated (ATM), and the negative feedback loop formed between ATM and p53, via Wip1, plays a dominant role in generating p53 oscillation. In addition, p53 oscillation requires not only an appropriate Mdm2 negative strength but also a threshold level of Wip1 negative strength. Furthermore, the time delays required for transcription and translation of Mdm2 and Wip1 proteins are essential for p53 oscillation. In particular, the critical value of time delay for inducing oscillation and the properties of delay-driven Hopf bifurcation are theoretically analyzed. As expected, the results are clearly in consistence with biological experiments and observations.

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.