Fractional-order Gene Regulatory Network Research Articles

Bifurcation control of a novel fractional-order gene regulatory network with incommensurate order and time delay

Gene regulatory networks play an extremely important role in human life activities. This paper presents a novel fractional-order gene regulatory network model (NFGRNM). The model employs incommensurate fractional orders and introduces time delay arising from the binding of miRNAs to RNA. Time delay often triggers instability such as oscillation and bifurcation, so in this paper we introduce a generalized hybrid control to control the unstable dynamical behavior induced by time delay. First we derive the NFGRNM using fractional order theory, after which the equilibrium point of the network model is calculated. Next, a systematic dynamics analysis of the NFGRNM is performed to derive sufficient conditions to generate the Hopf bifurcation. Most importantly, we introduce the extended hybrid control for the periodic oscillations generated by the time delay in the NFGRNM. Finally, numerical simulation results show that the order of the fractional order can have a great impact on the dynamical behavior of the network model and that the hybrid control has a significant role in delaying (or advancing) the Hopf bifurcation. Since infectious diseases such as SARS-Cov-2, influenza A, and HIV are still prevalent today, the study of gene regulatory networks not only promotes the study of the dynamics of complex networks but also has profound implications for the study of virus transmission in the host and various life activities in humans.

New results for dynamical analysis of fractional-order gene regulatory networks with time delay and uncertain parameters

In this paper, a novel class of networks named fractional-order gene regulatory networks with time delay and uncertain parameters (FGRNTDUP) are formulated, and some new results are obtained for the dynamics of FGRNTDUP. Firstly, a general fractional-order differential equality is built to supply a new viewpoint about the study of finite-time stability, stabilization, and synchronization for many fractional systems. Secondly, based on contraction mapping principle, the existence and uniqueness of the equilibrium point are strictly proved for the FGRNTDUP. Meanwhile, by virtue of fractional Lyapunov method and inequality techniques, some sufficient conditions are derived to guarantee the global Mittag–Leffler stability of FGRNTDUP. Moreover, based on a newly developed inequality, some novel criteria are yielded to realize the finite-time synchronization of FGRNTDUP by designing a suitable adaptive controller. Finally, two numerical examples are provided to verify the effectiveness of the obtained results.

Stability and adaptive control-based synchronization of delayed uncertain fractional-order gene regulatory networks

This paper is devoted to investigating stability and synchronization issues for delayed uncertain fractional-order gene regulatory networks (DUFOGRNs). First, the existence and uniqueness of the equilibrium point of the DUFOGRNs are explored via constructing contraction map, whereafter, a stability criterion is studied by means of the fractional-order Lyapunov–Razumikhin approach and property of Mittag-Leffler function. Second, a novel fractional-order Lemma is established by using the theory of fractional calculus showing that the derivative of Lyapunov functions in the fractional-order systems (FOSs) satisfy certain form of convergence, which can be able to discuss the problems about finite-time stability and synchronization of FOSs. Third, adaptive control strategies are designed based on the established Lemma, which are different control protocols anent the synchronization of gene regulatory networks proposed in previous work. Furthermore, sufficient synchronization criteria are derived by probing fractional calculus theory and inequality analysis techniques for guaranteeing the complete synchronization (CS) and finite-time synchronization (F-TS) of DUFOGRNs. Finally, three numerical examples are provided to verify the availability of the proposed method.

Bifurcation Analysis and Fractional PD Control of Gene Regulatory Networks with sRNA

This paper investigates the problem of bifurcation analysis and bifurcation control of a fractional-order gene regulatory network with sRNA. Firstly, the process of stability change of system equilibrium under the influence of the sum of time delay is discussed, the critical condition of Hopf bifurcation is explored, and the effect of fractional order on the system stability domain. Secondly, aiming at the system’s instability caused by a large time delay, we design a controller to improve the system’s stability and derive the parameter conditions that satisfy the system’s stability. It is found that changing the parameter values of the controller within a certain range can control the system’s nonlinear behaviours and effectively expand the stability range. Then, a numerical example is given to illustrate the results of this paper.

Novel Nonlinear Control for a Class of Non-Integer Order Time Lag Gene System

This study presents the bifurcation analysis and stabilization via the impulsive control of a fractional-order gene regulatory network with time delay. First, the author chooses the transcription rate k as the bifurcation parameter and obtains the Hopf bifurcation condition by analyzing its characteristic equation. The research shows that Hopf bifurcation occurs when the transcription rate k exceeds a critical value. This bifurcation behavior may destabilize the system. Subsequently, the author designs an impulsive controller to stabilize the system. Finally, simulation examples are used to verify our theory.

Robust global synchronization of delayed incommensurate fractional-order gene regulatory networks

This paper investigates the issue of robust synchronization for delayed incommensurate fractional-order gene regularity networks (IFOGRNs). The common stability analyzing methods for commensurate fractional-order models cannot be used for incommensurate cases. Therefore, the synchronization of the GRNs with incommensurate order models is more challenging than the commensurate ones. This paper employs a novel fractional Lyapunov-based scheme that can be used for incommensurate cases without conversion to a commensurate model. Moreover, this method can analyze the IFOGRNs without linearization, which guarantees global stability. Firstly, some sufficient conditions are proposed for robust controller design when the parameter uncertainties are similar between two such IFOGRNs. It is shown that asymptotic stability is not guaranteed when the parameter uncertainties are not the same. In this paper, a novel adaptive based controller is introduced, which leads to asymptotic stability even for dissimilar parameter uncertainties. Finally, to evaluate the efficiency of the proposed method, some numerical examples are investigated.

Fractional-Order Impulsive Delayed Reaction-Diffusion Gene Regulatory Networks: Almost Periodic Solutions

The paper is oriented on the existence of almost periodic solutions of factional-order impulsive delayed reaction-diffusion gene regulatory networks. Caputo type fractional-order derivatives and impulsive disturbances at not fixed instants of time are considered. New almost periodic and perfect Mittag–Leffler stability criteria are proposed. Lyapunov’s like impulsive functions, the properties of the fractional derivatives and comparison principle are the main tools in the investigation. Illustrative examples are also presented to demonstrate the proposed criteria. Our results contribute to the development of qualitative the theory of fractional-order gene regulatory networks.

Design of [formula omitted]/passive state feedback control for delayed fractional order gene regulatory networks via new improved integral inequalities

This work deals with H∞/passivity analysis and control design for fractional-order gene regulatory networks (FOGRNs) with delays. The results are derived by structuring a new class of Lyapunov Krasovskii functional (LKF). To deal with these LKF and time delay terms, two new fractional-order integral inequalities (FOIIs) are established. Firstly, a fractional-order free weighting matrix based integral inequality (FOFMBII) is derived. This FOFMBII includes fractional-order Wirtinger’s integral inequality (FOWII) as a particular case. Secondly, another improved FOII is constructed to facilitate the estimation of the fractional derivative of Lyapunov functional. Then, using the derived FOIIs and constructed LKF, a new set of sufficient conditions is established in terms of linear matrix inequalities (LMIs) for ensuring the stability of FOGRNs with a prescribed mixed H∞/passivity or H∞ or passivity level. The theoretical results are validated on a genetic network model. Three control gain matrices for the considered model are calculated based on H∞, passivity, and mixed H∞/passivity performances. The efficiencies of these three controllers are compared. Finally, the simulation results are given to exhibit the validity of the proposed results.

Synchronization of fractional-order gene regulatory networks mediated by miRNA with time delays and unknown parameters

To investigate the regulatory function of miRNA on gene expression at the post-transcriptional level, the synchronization of fractional-order gene regulatory networks (FOGRNs) mediated by miRNA with time delays and unknown parameters is explored. First, a more realistic gene regulatory network model mediated by miRNA with leakage, time-varying delays and unknown parameters is established. Second, the feedback controllers and parameter adaptive laws are designed to induce synchronization and identify unknown parameters. Third, some synchronization conditions are presented and the unknown parameters are accurately identified by using the designed feedback controllers, the parameter adaptive laws and the constructed Lyapunov function. It is found that the smaller the value of the fractional order, the longer it takes to synchronize, and it takes more time to complete the identification of the unknown parameters. Finally, MATLAB simulations are given to verify the theoretical results.

Stability and bifurcations in fractional-order gene regulatory networks

A fractional-order mathematical model of a large-scale GRN is presented, focusing on the positivity of solutions and on the stability analysis of equilibria based on Matignon’s theorem. With special emphasis on cyclic GRNs, necessary and sufficient conditions are obtained in terms of inequalities involving the system’s parameters and the fractional order, which guarantee the stability of equilibrium points. Based on the stability analysis, the critical values of the fractional order for which Hopf-type bifurcations may occur are detected. In this context, it is shown that oscillatory behavior due to a Hopf-type bifurcation (considering the fractional order as bifurcation parameter) is expected only in the case of an odd number of repressive genes. Numerical examples are given in order to demonstrate the usefulness of the proposed results and to exemplify the effect of the fractional order on the dynamics of cyclic GRNs, additionally revealing that period-doubling bifurcations of limit cycles may also occur in sufficiently large cyclic GRNs.

Impulsive effects on fractional order time delayed gene regulatory networks: Asymptotic stability analysis

This paper address the fractional-order gene regulatory networks (FOGRNs) with both time delays and impulsive effects. Inspired by the integer-order gene regulatory network models, a general class of fractional-order gene regulatory network model is further investigated via fractional-order Lyapunov-Razumikhin stability theory. Firstly, impulsive effects, feedback regulation, and translation time delays are taken well into account and effective analysis techniques are used to reflect the system’s practically dynamic behavior. Secondly, the existence and uniqueness of the equilibrium point of the fractional-order gene regulatory networks are derived based on the Banach contraction mapping principle. Besides, we derived analytically the condition under which the solution of this network is bounded through generalized Gronwall inequality. Thirdly, some novel delay-free sufficient conditions are derived to ensure the global asymptotic stability of the considered model with the help of the fractional-order Lyapunov-Razumikhin method and properties of Mittag-Leffler functions. Finally, an example with numerical simulation is provided to illustrate the validity and effectiveness of the proposed results.

Global dissipativity and adaptive synchronization for fractional‐order time‐delayed genetic regulatory networks

AbstractThis manuscript examines global dissipativity and synchronization criteria for time‐delayed fractional‐order gene regulatory networks (FGRNs). The Lyapunov stability theory is used to examine a class of FGRNs with delay arguments that are inspired by the integer‐order delayed gene regulatory network model. Some sufficient criteria are established using the fractional‐order comparison principle and stability theorem for fractional‐order time‐delayed systems to ensure the global dissipativity solution of FGRNs with both feedback regulation and translation time delays. Furthermore, for FGRNs, the global asymptotic synchronization condition is accomplished by constructing an adaptive delayed feedback control method for the master–slave system. Finally, two numerical examples are provided to demonstrate the applicability and superiority of the conclusions obtained.

Delay-independent stability criteria for fractional order time delayed gene regulatory networks in terms of Mittag-Leffler function

This paper investigates the finite-time delay-free stability criteria for fractional-order gene regulatory networks (FOGRNs) with feedback regulation time delays. The existence of unique equilibrium points of considered systems is established based on the Banach fixed point theorem and Cauchy–Schwarz inequality. Further, when the order γ satisfies 0<γ<1 and 1<γ<2, some novel delay-free sufficient conditions are derived to ensure the finite-time stability for addressing fractional order systems by using the ideas of generalized Gronwall–Bellman inequality, equivalent norm techniques, and Laplace transform. In the end, the article comes up with two numerical cases to manifest the applicability and superiority of the obtained theoretical results.

Mixed [formula omitted]/passivity based stability analysis of fractional-order gene regulatory networks with variable delays

In this paper, we derive delay and fractional-order dependent sufficient conditions for fractional-order gene regulatory networks (FOGRNs) with time-varying delays to be stable with H∞/passivity performance. Distinct from the existing works on delayed FOGRNs, the Lyapunov-Krasovskii functional (LKF) is suitably structured so that the delay functions can be non-differentiable or even have jump type discontinuities. Further, the positive-definiteness condition is relaxed for the matrices involved in LKF. A new set of linear matrix inequality (LMI) conditions that ensure the stability of FOGRNs with certain H∞/passivity performance level is derived using various results on fractional derivatives/integrals and convex property of LMIs. Finally, the results obtained are verified with existing FOGRNs via numerical simulations.

Impulsive Fractional Differential Inclusions and Almost Periodic Waves

In the present paper, the concept of almost periodic waves is introduced to discontinuous impulsive fractional inclusions involving Caputo fractional derivative. New results on the existence and uniqueness are established by using the theory of operator semigroups, Hausdorff measure of noncompactness, fixed point theorems and fractional calculus techniques. Applications to a class of fractional-order impulsive gene regulatory network (GRN) models are proposed to illustrate the results.

Global uniform asymptotical stability for fractional-order gene regulatory networks with time-varying delays and structured uncertainties

In this paper, we investigate a class of fractional-order gene regulatory networks with time-varying delays and structured uncertainties (UDFGRNs). First, we deduce the existence and uniqueness of the equilibrium for the UDFGRNs by using the contraction mapping principle. Next, we derive a novel global uniform asymptotic stability criterion of the UDFGRNs by using a Lyapunov function and the Razumikhin technique, and the conditions relating to the criterion depend on the fractional order of the UDFGRNs. Finally, we provide two numerical simulation examples to demonstrate the correctness and usefulness of the novel stability conditions. One of the most interesting findings is that the structured uncertainties indeed have an impact on the stability of the system.

Design of &lt;i&gt;H&lt;/i&gt; &lt;sub&gt;∞&lt;/sub&gt;/Passive State Feedback Control for Delayed Fractional Order Gene Regulatory Networks Via New Improved Integral Inequalities

Design of &lt;i&gt;H&lt;/i&gt; &lt;sub&gt;∞&lt;/sub&gt;/Passive State Feedback Control for Delayed Fractional Order Gene Regulatory Networks Via New Improved Integral Inequalities

Design of impulsive controllers and impulsive control strategy for the Mittag-Leffler stability behavior of fractional gene regulatory networks

In this paper, we consider fractional-order gene regulatory networks. The impulsive effects on their Mittag-Leffler stability properties are investigated and new criteria are proposed. The main results are proved by using the fractional Lyapunov method. Examples are given to demonstrate the efficiency of the proposed impulsive control strategy. Our results are the first step in the development of the theory of impulsive control fractional-order gene regulatory networks.

Stability Analysis and Bifurcation Control of a Delayed Incommensurate Fractional-Order Gene Regulatory Network

In this paper, stability analysis and bifurcation control for a novel incommensurate fractional-order delayed gene regulatory network are investigated. Firstly, the associated characteristic equation is analyzed by taking time delay as a bifurcation parameter, and the conditions of creation for Hopf bifurcation are established. It is demonstrated that the time delay can profoundly affect the dynamics of the proposed system and each order has a significant influence on the creation of bifurcation simultaneously. Then we study the stability and bifurcation behavior of the fractional-order networks by adding a nonlinear feedback controller. Finally, numerical simulations of two examples validate the obtained results.

Finite-time synchronization of fractional-order gene regulatory networks with time delay

As multi-gene networks transmit signals and products by synchronous cooperation, investigating the synchronization of gene regulatory networks may help us to explore the biological rhythm and internal mechanisms at molecular and cellular levels. We aim to induce a type of fractional-order gene regulatory networks to synchronize at finite-time point by designing feedback controls. Firstly, a unique equilibrium point of the network is proved by applying the principle of contraction mapping. Secondly, some sufficient conditions for finite-time synchronization of fractional-order gene regulatory networks with time delay are explored based on two kinds of different control techniques and fractional Lyapunov function approach, and the corresponding setting time is estimated. Finally, some numerical examples are given to demonstrate the effectiveness of the theoretical results.