Non-fragile state estimation for reaction-diffusion genetic regulatory networks with mode-dependent timevarying delays

1. INTRODUCTION

Recently, genetic regulatory networks (GRNs) have gained significant interest in the fields of biology, mathematics, and medicine ^[1–3]. This is because mathematical models of GRNs can describe the complex dynamic behaviors and mutual regulation of mRNAs and proteins within genes during transcription and translation. The models describing GRNs are mainly classified into two types: Boolean models ^[4,5] and differential equation models ^[6,7]. In the Boolean model, the state of the gene is viewed as either on or off. The differential equation model can describe the regulatory relationship between mRNAs and proteins in detail and accurately. Therefore, differential equation models are more suitable for modeling GRNs than Boolean network models.

The diffusion phenomenon is widespread; e.g., molecules propagate in the air, and mRNAs and proteins diffuse orderly within genes. However, in ^[8,9], the authors default that mRNA and protein concentrations are uniformly distributed in space, which does not consider the existence of diffusion phenomena, making the constructed model inaccurate. Therefore, this paper introduces the reaction-diffusion term in the GRN model. During the modeling process, in addition to taking the diffusion phenomenon into account, it is crucial not to ignore the significance of time delay. This is because delay phenomena are present in the process of gene expression and demand attention. Based on the above analysis, we incorporate time delays in reaction-diffusion GRNs (RDGRNs) ^[10–13]. In ^[13], Zhang et al. constructed RDGRNs with constant delays. However, since constant delays cannot accurately describe the gene regulation process, many scholars investigate GRNs with time-varying delays ^[14–16]. Furthermore, compared with common time-varying delays ^[17], mode-dependent time-varying delays can reduce the conservativeness of the stability criterion. Thus, Tian et al. in ^[18] focused on the stability analysis of Markovian jump neural networks with mode-dependent time-varying delays. To the best of our knowledge, there has been no relevant research on RDGRNs with mode-dependent time-varying delays, which is one of the motivations for this paper.

For RDGRNs with time-varying delays, the mode of the system may switch when there are mutations in the gene sequence or changes in the external environment, and the probabilities of switching are described by the Markov chain ^[19–21]. Thus, the investigations of RDGRNs with Markovian jump parameters and time-varying delays have received much attention ^[22,23]. For example, in ^[23], Zou et al. studied the robust stochastic stability of delayed RDGRNs with Markovian jump parameters. However, it is known that the switching probability of the system is difficult to obtain. Yet, in ^[24] and ^[25], the authors have assumed that the switching probabilities are completely known, which is not realistic. Therefore, it is worthwhile for Markovian jump RDGRNs with partially unknown transfer probabilities ^[26–28] to be further scrutinized.

So far, for some applications of RDGRNs, it is a prerequisite to know the exact concentrations of mRNA and protein within genes. Nevertheless, due to the specificity of RDGRNs and the cost issue, the mRNA and protein concentrations are often only partially available, so it is extremely investigative to utilize the partially available state information to estimate the system state. In research on state estimation of RDGRNs ^[29–31], there are defaults that the estimator parameters can be executed exactly. However, in general, the parameters may be perturbed to some extent due to factors such as component aging and internal noise, leading to system vulnerability. Therefore, some scholars have begun to focus on the non-fragile state estimation and synchronization of GRNs, and several results have been achieved ^[32–34]. For example, Li et al. in ^[32] investigated the issue of non-fragile state estimation for delayed GRNs. Nevertheless, the design of non-fragile state estimators for RDGRNs has not been investigated, which is another research motivation for this paper.

In light of the above discussion, this work studies the non-fragile state estimation for Markovian jump RDGRNs with partially unknown transfer probabilities and mode-dependent time-varying delays. The main innovations of this paper are as follows: (1) The reaction-diffusion term, mode-dependent time-varying delays, and Markovian jump parameters are considered in the model, which makes this model investigated in this paper more generalized compared to the existing studies on GRNs ^[8]; (2) The partially unknown transfer probabilities ^[26] are used in this work, which is more realistic compared to Markovian switching, where the transfer probabilities ^[22] are fully known; and (3) The estimation performance of conventional state estimators ^[29] is affected when facing cases such as component aging and internal noise, whereas the non-fragile state estimator designed in this paper for RDGRNs makes the estimation performance non-fragile to gain variations.

Notations: $$ \Re^{n\times n} $$ denote the space of $$ n\times n $$ real matrices, col$$ (\cdot) $$ denotes the column vector, diag$$ (\cdot) $$ indicates the diagonal matrix, and $$ \text{sym} \{\mathbb{M}\}=\mathbb{M}+\mathbb{M}^T $$; $$ (g_{hq})_{n\times n} $$ represents the $$ n\times n $$ matrix consisting of the elements $$ g_{hq} $$; $$ \Omega $$ denotes the set $$ \{\nu_1\leq y\leq\nu_2\} $$. In addition, this paper uses the following abbreviations: $$ \tilde{m}_h\triangleq \tilde{m}_h(x,y) $$, $$ \tilde{p}_h\triangleq \tilde{p}_h(x,y) $$, $$ \bar{m}\triangleq \bar{m}(x,y) $$, $$ \bar{p}\triangleq \bar{p}(x,y) $$, $$ L_m\triangleq L_m(x,y) $$, $$ L_p\triangleq L_p(x,y) $$, $$ \hat{m}\triangleq \hat{m}(x,y) $$, and $$ \hat{p}\triangleq \hat{p}(x,y) $$.

2. NETWORK MODEL AND PRELIMINARIES

Consider the Markovian jump RDGRNs with mode-dependent time-varying delays as follows:

where the border and initial conditions are given as follows:

in which $$ h=1,2,\ldots,n $$; $$ \partial\Omega $$ indicates the boundary of $$ \Omega $$, $$ \varphi(x,y) $$ and $$ \theta(x,y) $$ denote the initial conditions of $$ \tilde{m}_h $$ and $$ \tilde{p}_h $$, respectively. $$ \tilde{m}_h $$ and $$ \tilde{p}_h $$ are the mRNA and protein concentrations; $$ x $$ and $$ y\in\Omega $$ denote time and space position, respectively; $$ \xi_q(\tilde{p}_q) $$ denotes the monotonic feedback regulation function, which is described in the form $$ \xi_q(\tilde{p}_q)= \frac{(\frac{\tilde{p}_q}{\varrho_q})^{H_q}}{1+(\frac{\tilde{p}_q}{\varrho_q})^{H_q}} $$, where $$ H_q $$ is the Hill coefficient representing the degree of cooperativity, $$ \varrho_q $$ is a positive constant, and $$ q= 1,\ldots,n $$; the meaning and form of $$ \xi_q(\tilde{p}_q) $$ are explained in detail by ^[35-37], so they are omitted here; $$ \delta_h=\sum_{q\in\mu_h}g^*_{hq,\varepsilon(x)} $$, and $$ \mu_h $$ is the set containing all the repressors of gene $$ h $$; $$ \alpha_{\varepsilon(x)}(x) $$ and $$ \beta_{\varepsilon(x)}(x) $$ are time-varying delays that satisfy the following conditions

with known scalars $$ \alpha_{\varepsilon(x)} $$, $$ \beta_{\varepsilon(x)} $$, $$ \psi_{1,\varepsilon(x)} $$, and $$ \psi_{2,\varepsilon(x)} $$. $$ e_h $$ and $$ \bar{e}_h $$ represent the diffusion rate; $$ b_{h,\varepsilon(x)} $$ and $$ \bar{b}_{h,\varepsilon(x)} $$ denote the mRNA and protein degradation rate, respectively; $$ \bar{g}_{h,\varepsilon(x)} $$ indicates the translation rate; $$ G_{\varepsilon(x)}= (g_{hq,\varepsilon(x)})_{n\times n} $$ with

denotes the coupling matrix, and $$ g^{*}_{hq,\varepsilon(x)} $$ is the dimensionless transcriptional rate.

Define $$ \varepsilon(x)\in\mathbb{U}\triangleq\{1,2,\ldots,U\}(t\geq0) $$ to be a Markov chain, the corresponding transfer probability matrix $$ \prod\triangleq (\pi_{ab})_{U\times U} $$ is given as

where $$ \Delta t>0 $$, $$ \lim\limits_{\Delta t\rightarrow 0}\frac{0(\Delta t)}{\Delta t}=0 $$, $$ \pi_{a b} $$ is the transfer probability from mode $$ a $$ to $$ b $$, and $$ \pi_{aa}=-\sum\limits_{a=1,a\neq b}^U \pi_{a b} $$. Moreover, the transfer matrix with partially unknown probabilities can be expressed as

where $$ "?" $$ represents the unknown transfer probabilities. Define $$ \Re_r^a\triangleq\{b:\pi_{a b}\ \text{is}\ \text{known}\} $$, $$ \Re_{ur}^a\triangleq\{b:\pi_{a b}\ \text{is}\ \text{unknown}\} $$, and $$ \Re\triangleq\Re_r^a+\Re_{ur}^a $$.

Let $$ \tilde{m}^*(y)=\text{col}\left(\tilde{m}^*_1(y),\tilde{m}^*_2(y),\ldots,\tilde{m}^*_n(y)\right) $$ and $$ \tilde{p}^*(y)=\text{col}\left(\tilde{p}^*_1(y),\tilde{p}^*_2(y),\ldots,\tilde{p}^*_n(y)\right) $$ be the unique equilibrium point of the system (1), that is

Define $$ \bar{m}_h=\tilde{m}_h-\tilde{m}^*_h(y) $$ and $$ \bar{p}_h=\tilde{p}_h-\tilde{p}^*_h(y) $$, the system (1) is converted into

where

Additionally, the measured outputs of (5) are

where $$ N\in\Re^{n\times n} $$ and $$ \bar{N}\in\Re^{n\times n} $$ are known matrices; $$ L_{m} $$ and $$ L_{p} $$ represent the measured outputs of mRNA and protein, respectively.

To estimate the states of (5) by using the measured outputs in (6), one can construct the estimator as follows:

where $$ \hat{m} $$ and $$ \hat{p} $$ represent the estimations of $$ \bar{m} $$ and $$ \bar{p} $$, $$ K_{1,\varepsilon(x)} $$ and $$ K_{2,\varepsilon(x)} $$ are the estimator gain matrices, and $$ \Delta K_{1,\varepsilon(x)} $$ and $$ \Delta K_{2,\varepsilon(x)} $$ denote the gain perturbation, which takes the following form:

where $$ A_{1,\varepsilon(x)} $$, $$ C_{1,\varepsilon(x)} $$, $$ A_{2,\varepsilon(x)} $$, and $$ C_{2,\varepsilon(x)} $$ stand for the given matrices; $$ \Lambda_{1,\varepsilon(x)}(x) $$ and $$ \Lambda_{2,\varepsilon(x)}(x) $$ are unknown matrices satisfying $$ \Lambda_{1,\varepsilon(x)}^{\text{T}}(x)\Lambda_{1,\varepsilon(x)}(x)\leq I $$ and $$ \Lambda_{2,\varepsilon(x)}^{\text{T}}(x)\Lambda_{2,\varepsilon(x)}(x)\leq I $$, respectively.

Set $$ \varepsilon(x)=a $$, we denote $$ B_a\triangleq B_{\varepsilon(x)} $$, $$ G_a\triangleq G_{\varepsilon(x)} $$, $$ \alpha_a\triangleq \alpha_{\varepsilon(x)} $$, $$ \bar{B}_a\triangleq \bar{B}_{\varepsilon(x)} $$, $$ \bar{G}_a\triangleq \bar{G}_{\varepsilon(x)} $$, $$ \beta_a\triangleq \beta_{\varepsilon(x)} $$, $$ K_{da}\triangleq K_{d,\varepsilon(x)} $$, $$ \Delta K_{da}\triangleq \Delta K_{d,\varepsilon(x)} $$, $$ A_{da}\triangleq A_{d,\varepsilon(x)} $$, $$ \Lambda_{da}\triangleq \Lambda_{d,\varepsilon(x)} $$, $$ C_{da}\triangleq C_{d,\varepsilon(x)} $$, $$ (d=1,2) $$. Define $$ \phi_m=\bar{m}-\hat{m} $$ and $$ \phi_p=\bar{p}-\hat{p} $$ to be the error state vectors; based on (5)–(7), the following error system is derived,

where $$ f(\phi_p(x-\alpha_a(x),y))=f(\bar{p}(x-\alpha_a(x),y)) $$$$-f(\hat{p}(x-\alpha_a(x),y)) $$, $$ \phi_m(x-\beta_a(x),y)=$$$$ \bar{m}(x-\beta_a(x),y)-\hat{m}(x-\beta_a(x),y) $$, $$ \Delta K_{1a}=A_{1a}\Lambda_{1a}(x)C_{1a} $$, and $$ \Delta K_{2a}=A_{2a}\Lambda_{2a}(x)C_{2a} $$.

Because $$ f_h(\cdot) $$ is a monotonically increasing function, which satisfies the following inequality

where $$ \vartheta_h $$ is a known positive real number.

The following lemmas are important tools for obtaining the principal results of this article.

Lemma 1.^[38] Assume $$ \hbar\geq0\in \Re^{n\times n} $$ and the function $$ \ell\in\mathcal{H}_l^n([r_1,r_2]) $$ satisfying $$ \ell(r_1)=0 $$ or $$ \ell(r_2)=0 $$, then

Further, if $$ \ell(r_1)=\ell(r_2)=0 $$, one has

Lemma 2.^[39] With real matrices $$ H $$ and $$ J $$ of appropriate dimensions, the inequality satisfies

where $$ \gamma $$ is a positive scalar.

Lemma 3.^[40] The existence of matrices $$ \Psi=\Psi^\text{T} $$, $$ H $$ and $$ J $$ of appropriate dimensions, then

for all satisfying $$ \Lambda^{\text{T}}(x)\Lambda(x)\leq I $$, the sufficient condition for $$ \Lambda(x) $$ to hold is the existence of a positive constant $$ \gamma $$ such that the following equation holds:

3. MAIN RESULTS

For this section, establishing new criteria by choosing appropriate Lyapunov-Krasovskii functionals (LKFs), which assures that system (9) is asymptotically stable.

Theorem 1. Given constants $$ \psi_{1a} $$, $$ \psi_{2a} $$, $$ \varpi $$, matrix $$ \mathbb{N} $$, and estimator gain matrices $$ K_{1a} $$ and $$ K_{2a} $$, the system (9) is asymptotically stable if there exist positive matrices $$ P_{l a} $$, $$ F_l $$, $$ Q_l $$, $$ (l=1, 2) $$ with suitable dimensions, symmetric matrices $$ W_a\in\Re^{n\times n} $$ and $$ \bar{W}_a\in\Re^{n\times n} $$, and the diagonal matrix $$ R_1>0\in\Re^{n\times n} $$, which makes the below linear matrix inequalities (LMIs) feasible:

where

where $$ \psi_{1a}=\psi_{1,\varepsilon(x)} $$, $$ \psi_{2a}=\psi_{2,\varepsilon(x)} $$, $$ \mathbb{N}=\text{diag}(\mathbb{N}_1,\mathbb{N}_2,\ldots,\mathbb{N}_n) $$, $$ \varpi=\nu_2-\nu_1 $$, and $$ \zeta_\upsilon=\left[0_{n,(\upsilon-1)n} \ I_{n} \ 0_{n,(7-\upsilon)n}\right] $$, $$ (\upsilon=1,\ldots,7) $$.

Proof. Construct the LKFs for the system (9) as follows:

where

The time derivative of $$ V(x) $$ is calculated as follows:

According to (9), the following is obviously true

According to the definition of the transfer probability matrix (4), for symmetric matrices $$ W_a\in\Re^{n\times n} $$ and $$ \bar{W}_a\in\Re^{n\times n} $$, one gets

Based on (17) and the boundary conditions (2), using the integration by parts, one can get

and

Following (19) and (20) and applying Lemma 1, we can derive that

From (17) and the boundary conditions (2), we have that

and

Based on (10), for arbitrary diagonal matrix $$ R_1>0 $$, one obtains

By combining (14)–(24), it can be proved that

where $$ \ell(x,y)=\text{col} (\phi_m,\phi_p,\phi_m(x-\beta_a(x),y),$$$$\phi_p(x-\alpha_a(x),y),f(\phi_p(x-\alpha_a(x),y)),$$$$\frac{\partial \phi_m}{\partial x},\frac{\partial \phi_p}{\partial x}) $$.

Therefore, combining the above analysis, one can conclude that the error system (9) is asymptotically stable. The proof is complete. $$ \blacksquare $$

Having performed the stability analysis in Theorem 1, we will deal with the design of a non-fragile estimator. The following theorem gives a solution for designing a non-fragile estimator.

Theorem 2. Given positive constants $$ \psi_{1a} $$, $$ \psi_{2a} $$, $$ \varpi $$, $$ \gamma_1 $$, $$ \gamma_2 $$, and matrix $$ \mathbb{N} $$, the error system (9) is asymptotically stable if there exist positive matrices $$ P_{l a} $$, $$ F_l $$, $$ Q_l $$, $$ \Gamma_{l a} $$, $$ (l=1, 2) $$ with suitable dimensions, symmetric matrices $$ W_a\in\Re^{n\times n} $$ and $$ \bar{W}_a\in\Re^{n\times n} $$, and the diagonal matrix $$ R_1>0\in\Re^{n\times n} $$, which makes the below LMIs feasible:

where $$ \hat{\Psi}=\Psi_1+\hat{\Psi}_1+\hat{\Psi}_2 $$, $$ \mathbb{A}_{1a}= \bigg[-A_{1a}^TQ_1^T\ \ \ \ \underbrace{0\ \ \ \ 0\ \ \ \ \ldots\ \ \ \ 0}_{6\ \text{times}} \bigg]^T $$, $$ \mathbb{C}_{1a}=\bigg[ C_{1a}N\ \ \ \ \underbrace{0\ \ \ \ 0\ \ \ \ \ldots\ \ \ \ 0}_{4\ \text{times}}\ \ \ \ C_{1a}N\ \ \ \ 0 \bigg] $$, $$ \mathbb{A}_{2a}=\bigg[ 0\ \ \ \ -A_{2a}^TQ_2^T\ \ \ \ \underbrace{0\ \ \ \ 0\ \ \ \ \ldots\ \ \ \ 0}_{5\ \text{times}}\bigg]^T $$, $$ \mathbb{C}_{2a}= \bigg[ 0\ \ \ \ C_{2a}\bar{N}\ \ \ \ \underbrace{0\ \ \ \ 0\ \ \ \ \ldots\ \ \ \ 0}_{4\ \text{times}}\ \ \ \ C_{2a}\bar{N}\bigg] $$, and

Furthermore, the other parameters are consistent with Theorem 1, and the estimator gain matrices are obtained by $$ K_{1a}=\bar{Q}_1^{-1}\Gamma_{1a} $$ and $$ K_{2a}=\bar{Q}_2^{-1}\Gamma_{2a} $$.

Proof. Based on (8) and the proof procedure of Theorem 1, one obtains that

Since $$ \Lambda_{1a}^{\text{T}}(x)\Lambda_{1a}(x)\leq I $$ and $$ \Lambda_{2a}^{\text{T}}(x)\Lambda_{2a}(x)\leq I $$, by using Lemma 2, one obtains

and

According to (27), (28), and (29), by using Lemma 3, we can get

where $$ \mathbb{A}_{1a} $$, $$ \mathbb{A}_{2a} $$, $$ \mathbb{C}_{1a} $$, and $$ \mathbb{C}_{2a} $$ are already given in the statement of Theorem 2.

By combining (14)–(24) and (27)–(30), it can be proved that

where $$ \ell(x,y)=\text{col} (\phi_m,\phi_p,\phi_m(x-\beta_a(x),y), $$$$ \phi_p(x-\alpha_a(x),y),f(\phi_p(x-\alpha_a(x),y)),$$$$ \frac{\partial \phi_m}{\partial x},\frac{\partial \phi_p}{\partial x}) $$, we employ Schur's complement, which can convert the inequality (30) into (26).

Combining the above analysis, we can conclude that the system (9) is asymptotically stable when the inequalities (12) and (26) hold. The proof is complete. $$ \blacksquare $$

4. SIMULATION EXAMPLE

In this section, a numerical simulation is provided to illustrate the validity of the theoretical results in this paper. The corresponding parameters of the system (1) are selected as:

The selection of the transfer matrix with partially unknown probabilities is shown below:

based on the above transfer matrix, the process of switching between modes is shown in Figure 1.

Figure 1. Switching Modes of the system

The mode-dependent time-varying delays and related parameters are shown in Table 1.

The parameters of the estimator gain perturbation are chosen as:

Let $$ \gamma_1=\gamma_2=0.5 $$, $$ \varpi=1 $$, $$ \mathbb{N}=0.65I_3 $$, and regulation function $$ f(\epsilon)=\frac{\epsilon^2}{1+\epsilon^2} $$, by solving the LMIs in Theorem 2, it can prove that the LMIs (12) and (26) of this paper are feasible; the estimator gain matrices are derived:

Based on the obtained estimator gain matrix, Figures 2-5 illustrate the simulation results we obtained. Figures 2 and 3 demonstrate the estimation errors for mRNA and protein, respectively. To observe the trend of the estimation errors more clearly, we take the spatial point at $$ y = 0.2 $$ as the example and display the trajectory plots of the estimation errors of mRNA and protein in Figures 4 and 5, respectively. The above simulation results show that the estimation errors are asymptotically stable, thus proving the effectiveness of the non-fragile state estimator designed in this paper.

Figure 2. The evolution of mRNA estimation errors.

Figure 3. The evolution of protein estimation errors.

Figure 4. The trajectories of mRNA estimation errors $$ \phi_{mh} $$, $$ h=1,2,3 $$.

Figure 5. The trajectories of protein estimation errors $$ \phi_{ph} $$, $$ h=1,2,3 $$.

5. CONCLUSIONS

This paper has investigated the non-fragile state estimation for Markovian jump RDGRNs with mode-dependent time-varying delays. Moreover, the switching topology of the system satisfies the Markov chain with partial unknown transfer probabilities. By utilizing the suitable LKFs, sufficient conditions that guarantee the asymptotic stability of the error system were established. Ultimately, a numerical example was used to demonstrate the validity of the designed non-fragile state estimator. Nonetheless, some limitations of this paper still exist. For example, the measurement methods adopted in this paper are costly and challenging to implement. Therefore, we will employ point measurements to reduce the measurement cost in future work.

DECLARATIONS