Impulsive control strategies of mRNA and protein dynamics on fractional-order genetic regulatory networks with actuator saturation and its oscillations in repressilator model

Abstract

In genetic regulatory networks (GRNs), the control strategies of messenger RNA (mRNA) and protein play a key role in regulatory mechanisms of gene expression, especially in translation and transcription. However, the influence of impulsive control strategies on oscillatory gene expression is not well understood. In this article, by considering the impulsive control strategies of mRNA and protein, a novel fractional-order genetic regulatory networks with actuator saturation is proposed. By applying polytopic representation technique, the actuator saturation term is first considered into the design of impulsive controller, and less conservative linear matrix inequalities (LMIs) criteria that guarantee finite-time Mittag-Leffler stabilization problem for fractional-order genetic regulatory networks are given. The derived sufficient conditions can easily be verified by designing impulsive control gains and solving simple LMIs. Finally, to investigate the effectiveness and applicability of the control strategies, an interesting simulation example as a synthetic oscillatory network of transcriptional regulators in Escherichia coli is illustrated.

Introduction

Genetic regulatory networks (GRNs) are biochemical networks that regulate gene expression and perform complex biological functions (via direct or indirect interactions between deoxyribonucleic acid (DNA), ribonucleic acid (RNA), proteins, and small molecules) as shown in Fig. 1. GRNs are a significant topic in bioscience and biomedical engineering, as they can help many biologists, engineers, and scientists understand a variety of complex challenges in living cells [1], [2], [3]. Because many traits and diseases are linked to dysfunctional transcriptional regulators or mutations in regulatory sequences, understanding gene expression regulation has an immediate impact on biology and medicine. Acquiring precise information about the states of GRNs is particularly useful in biological and biomedical sciences for applications such as gene identification and medical diagnosis/treatment [4], [5]. One of the key challenges in this area is to (i) understand the cells behavior and control their operations; and (ii) discover how cellular systems fail in disease. Mathematical modeling and simulation tools aid in understanding how complex GRNs, which are made up of numerous genes and their tangled interactions, control the functioning of living systems. Understanding the dynamics and predicting the behavior of GRNs is critical in cell and molecular biology, namely different GRNs models have been developed. Hence, the research on GRNs includes the following aspects: gene circuit control design [6], modeling [7], and stability analysis [8]. Stability analysis is one of the most noticeable aspects of many dynamic systems, including GRNs. Various researchers have dedicated their efforts to the stability mechanism and biological rhythms, and both theoretical analysis and biotic experiments have contributed a huge quantity of beneficial research results. In [9], a simple gene circuit consisting of the regulator and transcriptional repressor modules in Escherichia coli was built, and the stability gain produced by negative feedback was demonstrated. It has been widely researched that time delay is an unavoidable factor in modeling, designing, and controlling GRNs because they naturally occur as a result of transcription, transcript splicing, processing, and protein synthesis [10], [11], [12], [13], [14], [15], [16]. Therefore, the biological scopes of GRNs, it is of great significance to study the dynamic behavior of messenger RNA (mRNA) and protein in regulatory mechanisms with time-varying delays.

The involvement of memory and hereditary properties in dealing with fractional-order derivatives provides a more realistic way to biological models [17]. Because of the memory effect, non-integer models incorporate all previous information from the past, allowing them to more accurately predict and translate molecular models [18]. Fractional-order genetic regulatory networks (FGRNs) have two advantages over integer-order GRNs: more degrees of freedom and infinite memory [19], [20], [21]. Furthermore, experiments on yeast cell cycle gene expression data show that the proposed mathematical model is better suited to modeling genetic regulatory mechanisms. Although fractional-order differential equations have been used to GRNs model due to their lower data fitting error on test data than integer-order models, few articles have been published on FGRNs. Therefore, FGRNs have formulated numerous molecular models of fractional derivative to study the transmission dynamics during the past few years [22], [23]. All the results above have shown that FGRNs are of great importance in enlightening the mechanism of multistability and biological rhythms.

In recent decades, the impulsive control approach has been intensively researched and applied to the analysis of nonlinear system dynamics [24], [25], [26]. In some practical applications, impulsive control is really valuable, such as biological models [27], multi-agent systems [28], neural networks [29], and so on. Because it has some excellent characteristics, impulsive control has recently received a lot of attention [30]. It is reasonable and powerful to introduce the ideas and methods from system and control theory to underly the complicated biological functions of living organisms in their entirety [31], [32], [33], [34]. Environmental cues, differentiation cues, and disease all cause regulatory circuits controlling gene expression to constantly rewire [35]. GRN states are frequently impulsively changed in response to transient environmental stimuli. Indeed, the gene regulatory mechanism is always exposed to intrinsic noise caused by the random births and deaths of individual molecules, as well as extrinsic noise caused by environmental variations. Because environmental noises can affect the stability of equilibrium states, it is critical to investigate impulsive generalizations of the GRNs in which the states of the models are abruptly changed [36]. Few authors have developed impulsive control strategies to investigate the stability issue of GRNs models to date [37], [38]. Although more and more experts recognize the significance of actuator saturation, the findings of saturated impulsive control are extremely rare. This is because it is very challenging to deal with saturation nonlinearity and the estimate of domain of attraction. The authors [39] investigated the impact of saturation on network performance. Actuator saturation can degrade dynamic performance and even destabilize the system under study. An impulse saturation can have a significant impact on system dynamics [40], [41], [42]. However, despite its practical significance, the finite-time Mittag-Leffler stabilization (FTMLS) problem for FGRNs via impulsive control with actuator saturation has not been investigated yet.

Inspired by above, this article addresses the finite-time Mittag-Leffler stabilization problem of fractional-order genetic regulatory networks via impulsive control with actuator saturation. The main contributions are:

• A novel the controller that involves saturated impulsive control has been designed to achieve FTMLS problem of FGRNs for the first time in this article.

• The sufficient criteria that ensure the FTMLS of the proposed FGRNs are determined using the novel Lyapunov functional, and the proposed conditions are represented in terms of solvable LMIs.

• Furthermore, we take advantage of how polytopic representation approaches handle saturation nonlinearity.

• Finally, to illustrate and demonstrate the efficiency of our obtained results, we present some new simulation results that reveal the time responses of the state variables with and without the inclusion of impulsive actuator saturation into the repressilator model.

To better illustrate the biological scopes of GRNs and major contributions of theoretical as well as practical significance of this article, we provide Table 1 for comparison with other research works on GRNs, where fractional-order, impulsive control, impulsive actuator saturation, linear matrix inequalities (LMIs), finite-time Mittag-Leffler stabilization (FTMLS), and repressilator model. Moreover, $\surd$ means this item is included in that paper, $\times$ means it is not.

Notation: The notes of the symbols appearing in the article is as follows: $\mathcal{C}$ the complex numbers; $\mathcal{R}$ the real numbers; ${\mathcal{R}}_{+}$ the real numbers; ${\mathcal{Z}}_{+}$ the positive integers. ${\mathcal{C}}^q$ and ${\mathcal{R}}^q$ denotes the set of all $q$-dimensional complex-valued vectors and real-valued vectors, respectively. ${\mathcal{R}}^{m\times m}$ denotes the set of all $m\times m$ real matrices. $\mathrm{diag}\left\{\cdot \cdot \cdot \right\}$ is a block diagonal matrix. ${\mathcal{I}}_n$ stands for the $\mathrm{n}\times \mathrm{n}$ identity matrix. ${}_{t_0}^{ℂ}{D}_t^{\gamma }$ a denotes the Caputo fractional derivative with order $\gamma$. ${\mathbf{E}}_{\gamma }\left(\cdot \right)$ denotes the Mittag-Leffler function of $\left(\cdot \right)$. For a real matrix $\Omega$, ${\Omega }^T$ stands for its transpose, and ${\lambda }_{max}\left(\Omega \right)$. ${\lambda }_{min}\left(\Omega \right)$ are maximum and minimum eigenvalues of $\Omega$ respectively. The saturation function $\mathbf{sat}(ħ\left(t\right))={(\mathbf{sat}\left({ħ}_1\left(t\right)\right),\mathbf{sat}\left({ħ}_2\left(t\right)\right),\dots ,\mathbf{sat}\left({ħ}_q\left(t\right)\right))}^T$ with $\mathbf{sat}\left({ħ}_1\left(t\right)\right)=sign\left(h_1\left(t\right)\right)min\{{ħ}_{0\tau },\left|{ħ}_{\tau }\left(t\right)\right|\}$ $(\tau \in \mathcal{M})$, where ${ħ}_{0\tau }\in {\mathcal{R}}_{+}$ is the ${\tau }^{\mathrm{th}}$ element of the vector ${ħ}_0\in {\mathcal{R}}_{+}^q$ and is the know saturation level. $\mathbf{co}\left\{\nu \right\}$ represents the convex hull defined by the vertices $\nu$. Let $\Re =\{{\Re }_{ȷ}:ȷ\in \Lambda \}$ be the set of $ȷ\times ȷ$ diagonal matrices whose diagonal element take value $1$ or $0$. The notation $\ast$ is used to denote the symmetric term in a matrix