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.


Introduction
In recent years, tremendous progress has been made in modern biology because of the development of gene sequencing, especially the third generation gene sequencing technology. Based on gene sequencing and other experimental methods, it has been found that most biological functions are controlled not only by several molecules or genes, but also by complex interactions between many components. A large number of DNAs, RNAs, proteins, and small molecules in an organism together with the mechanisms which regulate the expression of genes form gene regulatory networks (GRNs) [1]. Now GRNs comprise one of the important research directions in the field of system biology. Researchers have established various GRN models from their respective perspectives, such as Boolean networks [2], Petri networks [3], Bayesian networks [4], differential equation models [5][6][7], etc.
A lot of research has focused on integer-order differential equation model of GRNs [8][9][10]. Fractional-order calculus is a generalization of integer-order derivative and integral theory to arbitrary real order. Because fractional derivative operators are nonlocal, possess memory and hereditary properties, they have more advantages than integer-order systems in modeling GRNs. So with the development of fractional differential equation theory, the research on GRNs has gradually shifted from integer-order model to fractional-order model [1,5,11,12].
As GRNs are modeled from time-series data of real-world gene expression, it has been well recognized that the modeling errors and parameter fluctuations are unavoidable primarily due to certain limitations to the current experimental techniques in GRNs. Moreover, it has been pointed out that the system parameters identified from experimental data may form an unknown, but bounded time-varying function, which incorporates structured or parameter uncertainties (also called variations or fluctuations) [13]. It is obtained that structured uncertainties in GRNs may cause poor performance or even instability of real genetic networks [13][14][15][16][17]. Therefore, it may be essential to take the structured uncertainties into account when investigating the dynamical behaviors of DFGRNs.
According to bioinformatics theory, some time is needed to complete the process of transcription and translation of gene information in GRNs. For example, in eukaryotic cells, it takes time for RNA and protein synthesis and to transport RNA and protein from nucleus to cytoplasm at different locations. Therefore, time delay has become a key factor affecting gene expression. Many literature sources have reported the effects of time delay in gene regulatory networks [8,10,[18][19][20]. These results show that the stability of a gene regulatory network system may even be affected due to the existence of time delay. In recent years, fractional-order gene regulatory networks with constant delays have attracted more and more researchers' interest [6,10,19,21]. Moreover, some researchers have paid attention to the fractional-order dynamic systems with time-varying delays [7,[22][23][24][25][26]. Zhang et al. [7] discussed the stability for a fractional-order gene regulatory network with time-lag by using Jensen and Wirtinger inequalities, etc., and obtained the model stability results. Recently, Zhang et al. put forward a novel stability condition on fractional-order composite systems with time delay based on the vector Lyapunov function [27], and investigated the asymptotic stability of nonlinear fractional-order systems with multiple time delays via two new control methods [28], which provide us a new idea to find stability conditions and control of fractional-order dynamic systems with time-varying delays in the future research. In [22], Razumikhin-type stability for fractional-order differential equation with time-varying delays is investigated. The authors obtained global uniform asymptotic stability results for the considered systems. However, the stability conditions didn't include the fractional order of the systems. Wu et al. [21] investigated global asymptotic stability for fractional-order GRNs (FGRNs) with constant time delay by using the Lyapunov method and comparison theorem. Wu et al. [26] studied the finite-time stability for DFGRNs with structured uncertainties and controllers by using a generalized Gronwall inequality and norm technique. The results showed that the structured uncertainties can shorten the "estimated time" of finite-time stability. We naturally ask whether the structured uncertainties affect the global asymptotic stability of DFGRNs. What are the global uniform asymptotic stability conditions related to fractional order for FGRNs with time-varying delays and structured uncertainties?
This paper answers the above questions. We investigate a class of FGRNs with timevarying delays and structured uncertainties by virtue of a Lyapunov function and Razumikhin technique. Compared with some recent results [21,22,26], the chief contributions of our study are as follows: (1) we obtain sufficient conditions related to fractional order of the global uniform asymptotic stability for FGRNs with time-varying delays and structured uncertainties, and illustrate the advantages of our stability conditions in numerical examples; (2) It is shown that bigger structured uncertainties can make DFGRNs unstable in numerical examples. The rest of this paper is organized as follows: Sect. 2 mainly introduces the model studied in this paper and some necessary concepts and lemmas. In Sect. 3, the existence and uniqueness of the equilibrium point are obtained, and the sufficient conditions on global uniform asymptotical stability for the DFGRNs are given. In Sect. 4, two examples are provided to show the effectiveness of the obtained results. Finally, some conclusions are drawn in Sect. 5.

Preliminaries
Definition 1 ([29]) The fractional integral of order q for a function f (t) is defined as where t ≥ a, a ∈ R, q > 0. The Gamma function (q) is defined by the integral (q) = ∞ 0 t q-1 e -t dt.

Definition 2 ([29])
The Riemann-Liouville fractional derivative of order q for a function f is defined as where t ≥ a and n is a positive integer such that n -1 < q < n.

Definition 3 ([29])
Caputo's fractional derivative of order q for a function f is defined by where t ≥ a and n is a positive integer such that n -1 < q < n.
For convenience, we choose the notation In [26], we have investigated the finite-time stability of the following FGRNs with timevarying delays and structured uncertainties: + (e i + e i (t))m i (tτ 2 (t)), i = 1, 2, . . . , n, where q ∈ (0, 1), m i (t), p i (t) ∈ R + are the concentrations of mRNA and protein of the ith node, respectively; the parameters a i > 0 and c i > 0 are the decay rates of mRNA and protein, respectively; d i > 0 and e i ≥ 0 are the translation rates; a i (t), c i (t), d i (t), e i (t), w ij (t), k ij (t) are the norm-bounded unknown functions; the transcriptional delay τ 1 (t) and translational delay τ 2 (t) are bounded continuous functions on R with 0 ≤ τ i (t) ≤ τ * i (i = 1, 2), here τ * i are positive constants. Both f j (p j (t)) and g j (p j (t -τ 1 (t))) represent the feedback regulation of the protein on the transcription. Generally, both functions are nonlinear but monotonic. As monotonic increasing or decreasing regulatory functions, f j and g j usually have the Michaelis-Menten or Hill forms [30]; B i = j∈I i b ij + j∈Ī ib ij , b ij andb ij are bounded constants which are respectively the dimensionless transcriptional rate of transcription factor j to i at time t and tτ 1 (t), and I i ,Ī i , respectively, are the sets of all j's where the transcription factor j is a repressor of gene i at time t and tτ 1 (t); the matrices W = (w ij ) ∈ R n×n , K = (k ij ) ∈ R n×n are the coupling matrices of the gene network, which are defined as follows: if transcription factor j is an activator of gene i, In order to investigate whether structured uncertainties can affect the global asymptotical stability of FGRNs and get the global stability conditions related to the fractional order, we consider the global uniform asymptotical stability of DFGRN (1).
The initial conditions for system (1) are as follows: where are the given initial functions with

Main results
In order to prove our theorems, we need the following lemmas.
is a continuously differentiable function, then the following inequality holds almost everywhere:
In order to prove our theorems, we still need the following lemmas. Consider the following system:
Proof See the Appendix.
In addition, we introduce the following assumptions: for all x, y ∈ R with x = y.
The inequality q 2 -q+1 ) qμ 2 ≤ 0 can imply μ 1 < μ 2 in [22], but not the converse. If we choose μ 1 and μ 3 , there exists μ 2 satisfying moreover, we find that μ 2 satisfying the above inequality is unstable in the numerical example, which shows that the condition of Lemma 4 is better than that of Theorem 3.2 in [22].
Remark 2 The condition of Lemma 4 depends on the fractional-order q.

The existence and uniqueness of the equilibrium point of DFGRNs
By using the contraction mapping theorem, we have the following conclusion.
Remark 4 Corollary 1 is still new.
Subsequently, we use the improved predictor-corrector method [32] to calculate the numerical solutions of DFGRN (17). The trajectories of variables m i (t) and p i (t) (i = 1, 2, 3) with q = 0.95, q = 0.6 and different initial values are shown in Figs. 1 and 2, respectively. The convergence behaviors are obvious.
Subsequently, we also calculate the numerical solutions of DFGRN (18). The trajectories of variables m i (t) and p i (t) (i = 1, 2, 3) with q = 0.95, q = 0.6 and different initial values are shown in Figs. 3 and 4, respectively. The convergence behaviors are obvious.
From Figs. 1-4, we find that the effect of small structured uncertainties on stability of DFGRNs is not obvious. Meanwhile, we also find that the fractional-order q can affect the time of stability of DFGRNs. To illustrate the bigger structured uncertainties' effect on the stability of DFGRN (17), we set A(t) = diag a 1 (t), a 2 (t), a 3 (t) = diag 0.33 sin(t), 0.10 cos(t), 0.17 sin(t) ,

Conclusion
In this paper, we have proposed a class of fractional-order gene regulatory network models with time-varying delays and structured uncertainties, and we have obtained the following results related to DFGRNs: (1) By using the contraction mapping theorem, we obtained that DFGRNs have a unique equilibrium point; (2) Based on Lyapunov function and Razumikhin technique, we proved that DFGRNs are globally uniformly asymptotically stable.
Furthermore, numerical simulations showed the stability condition q 2 -q+1 μ q (1-q) +K (1-τ * μ ) qλ ≤ 0, which depends on the fractional-order q, is better than that of Theorem 3.2 in [22]; (3) We found that the influence of smaller structured uncertainties on the stability is not obvious, but bigger structured uncertainties can change the stability of DFGRNs in a numerical example.