Abstract-In the realm of discrete-time modeling for gene regulatory networks, significant focus has been placed on addressing the time lags inherent in the process of DNA transcription to RNA and the subsequent translation of mRNA to proteins. These temporal delays have been consistently incorporated into discrete gene regulatory network models. However, true gene regulatory networks are also subject to spatial variables, due to the uneven distribution of protein and mRNA concentrations. The integration of reaction-diffusion terms is thus essential to fully represent the impact of spatial dynamics on gene regulatory networks. In such networks, reaction-diffusion dynamics highlight the complex interactions between neighboring spatial regions, where closeness leads to mutual influences on their functional activities. Based on this conceptual groundwork, this study introduces a discrete-time gene regulatory network model that includes the mutual interconnections between spatial areas. To guarantee the model's robust stability, we have established delay-dependent stability criteria using carefully designed Lyapunov-Krasovskii functions, framed within the context of linear matrix inequalities. The robustness and effectiveness of our approach are demonstrated through a numerical example presented in this work.
Index Terms-Discrete-time genetic regulatory networks; Robust stability; Reaction-diffusion; Linear matrix inequalities
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I. INTRODUCTION
Historically, scientific inquiry predominantly centered on elucidating the functions of select genes, proteins, and molecules. However, contemporary understanding underscores that the orchestration of most biological functions transcends the control exerted by individual molecules or genes, instead arising from the intricate interplay among myriad components. This interaction entails the regulation of gene expression or function by the influence or activity of other genes, and conversely. With the progressive evolution of biological science, particularly the nuanced exploration of molecular mechanisms witnessed in recent years, there has emerged an increasingly profound appreciation among experts for the fundamental significance of genetic networks. Given the inherent complexity of authentic genetic networks, it becomes imperative to employ simplification strategies for effective comprehension. Genetic regulatory networks (GRNs) have emerged as potent tools in this regard, offering valuable insights into intricate biochemical processes such as gene transcription, translation, and protein diffusion within cellular microenvironments such as the cytosol and nucleus.
Researchers have made significant strides in the field of Genetic Regulatory Networks (GRNs), leading to the development of a diverse array of models. These include Bayesian models [1-3], Boolean models [4], and differential equations models [5-9]. Among them, the differential equations models stand out for their ability to capture the essence of GRNs through the representation of continuous values. However, the integration of delayed discrete-time GRNs is crucial for their practical deployment and use in simulation scenarios [10]. Scholars such as Sakthivel [11] have contributed significantly to the robustness evaluation of discrete-time GRNs, especially in dealing with various forms of delays.
Time delays are a fundamental aspect of numerous scientific disciplines, encompassing areas such as chemistry, physics, neural networks [12, 13], and GRNs. Within GRNs, these delays originate from the inherently slow pace of biochemical processes, including gene transcription and translation, or from the finite switching capabilities of amplifiers. Concurrently, uncertainties arise due to imperfections in modeling, the influence of external factors, and variations in parameters. In response to these complexities, a multitude of research efforts have been dedicated to performing robust stability analyses of GRNs, specifically addressing the intricacies of time delays and uncertainties.
As mentioned above, genetic regulatory processes require the migration of regulatory proteins or metabolites within cellular compartments such as the cytoplasm and nucleus. Recognition of the importance of protein diffusion has highlighted the need to incorporate reaction-diffusion phenomena into genetic regulatory network (GRN) models, rather than assuming spatial homogeneity. Several studies [14–16] have highlighted the key role of reaction-diffusion dynamics in shaping GRN behavior, emphasizing that models that ignore these effects may produce inaccurate predictions of protein and mRNA concentrations. Therefore, the integration of reaction-diffusion mechanisms is essential in GRN modeling. However, the literature on this aspect remains limited, especially regarding discrete-time GRNs within plant systems.
In this study, we aim to incorporate reaction-diffusion dynamics into discrete-time GRNs and analyze the stability of discrete GRNs with reaction-diffusion terms, considering scenarios both with and without Brownian motion. Through computer simulations, we demonstrate the effectiveness of our theoretical findings and the impact of integrating reaction-diffusion dynamics into GRN modeling.
The main contributions of this work are summarized as follows:
(1) Integration of spatial diffusion: this study introduces the integration of spatial diffusion mechanisms into discrete GRNs, thus establishing a novel discrete coupled GRN incorporating reaction-diffusion dynamics.
(2) Stability theorem for Brownian motion and time delay: Taking into account Brownian motion and time delay, this study introduces a theorem aimed at determining the stability of discrete modeling GRNs incorporating reaction-diffusion processes. This theorem serves as an important tool for understanding the dynamic behavior of GRNs under the influence of these complex factors.
II. PROBLEM FORMULATION Following nonlinear delayed genetic regulatory network [17]:
...
where
...
In Eq. ... represent the concentrations of mRNA and protein of the i node at time m , respectively. The parameters ai and i c denote the degradation rates of mRNA and protein, and i d is the translation rate. The term Li signifies the basal transcriptional rate of the repressor of gene i , and (*) i f * is the Hill form regulatory function. This function represents the feedback regulation of the protein on transcription and is expressed as follows:
...
where Hi is the Hill coefficient, i v is a positive constant,*(m) and *(m) are time-varying delays satisfying
...
... is described as ...
where nn * * represent transcription factor n N ***1,2,3 , * is an activator of gene n , 0 represent that there is no link from node n * to n , nn *− * represent transcription factor n * is an repressor of gene n .
Now, assuming that X * and Y * are the equilibrium point vectors of Eq. (1), let ... , and ...
Eq. (1) can be expressed as follows:
...
Based on the fundamental discrete GRN model outlined in Eq. (1), we now advance to develop an enhanced discrete GRN model that integrates reaction-diffusion terms to capture the spatial-temporal dynamics more comprehensively.
...
Where
...
In Eq. (4), ( , ) i x m n and ( , ) i y m n are the concentrations of mRNA and protein of the i th node at the time m and space n .
A typical continuous-time genetic regulatory networks with reaction-diffusion terms [15] is described as:
...
where ... is constant, ... denote the diffusion rate matrices; ... demonstrate the concentrations of mRNA and protein at the ith node respectively; ˆ i a and ˆ i c represent degradation rates of the mRNA and protein, respectively; ˆ i b is a constant; g j (*) is the activation function; ... is the set of all the nodes which are repressors of gene I; ... is coupling matrix, which is defined as
...
where * ij represent transcription factor j is an activator of gene i, 0 represent that there is no link from node j to i, −*ij represent transcription factor j is a repressor of gene i.
According to the form of the reaction-diffusion terms in Eq. (5), *1 and *2 are defined to represent the reaction diffusion terms of concentrations of mRNA and protein respectively as follows:
...
where * and * are positive coupling coefficient. After substituting Eq. (6) into Eq. (4) and simplifying, we obtain
...
where ...
Assuming that X ˆ and Y ˆ are the equilibrium point vectors of Eq.
...
...
Considering that the concentrations of mRNA and proteins are influenced by molecular Brownian motion, Eq. (8) is rewritten as
...
where ... is a vector-form scalar Brownian motion with
...
and ... is the noise intensity matrix satisfying
...
where H1 and H2 are known constant matrices with appropriate dimensions.
Nonlinear function ... satisfies the Ineq. (11), because ... is a monotonically increase function with saturation
...
Ineq. (11) is reformulated in matrix notation, thereby engendering
...
where ...
In this study, we contemplate the imposition of Dirichlet boundary conditions, articulated as follows:
...
where ... represents boundary
Lemma 1 [16]. For any constant matrix 0 T W W= * , scalar r ** 0 , exist
...
Lemma 2 [18]. For any vectors X , ... is any positive definite matrix, exist following Ineq.
In this section, we will investigate a stability criterion for Eq. (8).
Theorem 1 For given scalars * min , * max , *min and *max satisfying Ineq. (2), * ( , ) 0 m n = , the trivial solution of Eq. (9) is stability if there exist scalars * * 0 , matrices ... and ... and ... such that the following linear matrix Ineq. (LMI) holds:
...
Where
...
Considering Ineq. (12), it follows that for diagonal matrices
...
Define a Lyapunov-Krasovskii functional candidate for Eq.
...
Where
...
...
...
Define
...
...
...
Considering Ineq. (12), it follows that for diagonal matrices ... there exist
Utilizing Lemma 1, we ascertain
...
...
...
...
Considering the spatial continuity and limitary of GRNs, we can obtain
...
...
Then, combing (17)-(36), we can obtain
...
The analysis above implies that Eq. (9), in the absence of Brownian motion, demonstrates mean square asymptotic stability, thereby completing the proof.
Theorem 2 For given scalars * min , * max , *min and *max satisfying Ineq. (2), the trivial solution of Eq. (9) is stability if there exist scalars * * 0 , matrices ... such that the following LMIs hold:
...
...
...
Where
...
...
Proof
Employing Lemma 2, we ascertain
...
...
...
...
Observing Ineq. (10), it is evident that
...
Subsequently, by consolidating Eqs. (17)-(36) and Eqs. (41)-(44), we obtain
...
Where
...
This is complete the proof.
IV. SIMULATION EXAMPLES
In this section, we present two examples to demonstrate the validity of Theorems 1 and 2. Specifically, Example 1 highlights the passivity properties of the proposed GRNs model without Brownian motion, while Example 2 exhibits the passivity characteristics of the same model when Brownian motion is taken into account.
A. Robust stability of proposed discrete-time RGNs without molecular Brownian motion
In this section, we consider a GRNs (9) with 5 nodes, where L =100 . The parameters are assumed to be [17]
...
where ... the time delays * * ( ) 4 2sin( / 2) m m = + and * * ( ) 4 sin( / 2) m m = + , so that min * = 2 , max * = 6 , min * = 3 , max * = 5 , coupling coefficient 1 2 * * = 0.2 . The simulation results for the trajectories of mRNA and protein concentrations in Example 1 are presented in Figs. 1-5.
Based on the references [15, Theorem 1], [18, Remark 2], [19, Theorem 1], and Theorem 1 of this paper, the maximum delay * * max max = is derived when ... 0.1, 0.3, 0.5,0.7,0.9,1.0* and ... . The following illustrations are provided for Tab. 1:
(1) If * * 0.7 , it is deemed achievable to satisfy the LMI conditions outlined in the aforementioned references and Theorem 1.
(2) When * = 0.9 , the LMIs conditions specified in [19, Theorem 1] are feasible, whereas the LMI conditions presented in [15, Theorem 1], [18, Remark 2], and Theorem 1 of this paper are infinite.
(3) When * =1 , the LMIs conditions discussed in [15, Theorem 1] and [18, Remark 2] are feasible, while the LMI conditions in [19, Theorem 1] are not. Only the LMI conditions from Theorem 1 of this paper remain infinite.
Therefore, within the specified range * *1 , Theorem 1 of this paper exhibits a reduced level of conservativeness compared to the other theorems mentioned.
B. Robust stability of proposed discrete-time RGNs with molecular Brownian motion
Considering the Brownian motion for Eq. (9), ..., by using the Toolbox YALMIP in MATLAB to solve the Eqs. (38)-(40), we can obtain the following feasible solution:
... *=2.0809 .
From Fig. 6, it can be observed that when the values of * and * fall within the range of region I, the system is robustly stable. Nevertheless, should the values of * and * reside within the confines of region II, the proposed GRN model ceases to exhibit stability.
V. CONCLUSION
In this seminal investigation, we introduce an innovative methodology aimed at integrating reaction-diffusion mechanisms into discrete-time GRNs. Our principal objective revolves around scrutinizing the robust stability of these networks in the presence of time-varying delays and Dirichlet boundary conditions, leveraging advanced Lyapunov-Krasovskii functions. Furthermore, we undertake a comprehensive analysis of asymptotic stability concerning GRNs incorporating reaction-diffusion terms alongside Brownian motion. This unprecedented inquiry marks the initial attempt to incorporate reaction-diffusion phenomena into discrete-time GRNs. To substantiate the efficacy and validity of our novel approach, we furnish detailed numerical examples and simulation outcomes. These findings serve to corroborate the accuracy and efficacy of our methodology, thus emphasizing its potential to significantly impact the realm of gene regulatory network modeling and analysis.
Manuscript received February 19, 2024; revised July 8, 2024. This research is supported by China Postdoctoral Science Foundation under Grant Nos: 2022M720650, the Project of Hebei Key Laboratory of Software Engineering (22567637H), the S&T Program of Qinhuangdao (No: 202301A315), ``Kunlun Elite" Talent Recruitment Research Project (No: 2023-QLGKLYCZX-028), and New Faculty (Ph.D.) Extended Research and Cultivation Program under (No: 202302lwys018).
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Abstract
In the realm of discrete-time modeling for gene regulatory networks, significant focus has been placed on addressing the time lags inherent in the process of DNA transcription to RNA and the subsequent translation of mRNA to proteins. These temporal delays have been consistently incorporated into discrete gene regulatory network models. However, true gene regulatory networks are also subject to spatial variables, due to the uneven distribution of protein and mRNA concentrations. The integration of reaction-diffusion terms is thus essential to fully represent the impact of spatial dynamics on gene regulatory networks. In such networks, reaction-diffusion dynamics highlight the complex interactions between neighboring spatial regions, where closeness leads to mutual influences on their functional activities. Based on this conceptual groundwork, this study introduces a discrete-time gene regulatory network model that includes the mutual interconnections between spatial areas. To guarantee the model's robust stability, we have established delay-dependent stability criteria using carefully designed Lyapunov-Krasovskii functions, framed within the context of linear matrix inequalities. The robustness and effectiveness of our approach are demonstrated through a numerical example presented in this work.
You have requested "on-the-fly" machine translation of selected content from our databases. This functionality is provided solely for your convenience and is in no way intended to replace human translation. Show full disclaimer
Neither ProQuest nor its licensors make any representations or warranties with respect to the translations. The translations are automatically generated "AS IS" and "AS AVAILABLE" and are not retained in our systems. PROQUEST AND ITS LICENSORS SPECIFICALLY DISCLAIM ANY AND ALL EXPRESS OR IMPLIED WARRANTIES, INCLUDING WITHOUT LIMITATION, ANY WARRANTIES FOR AVAILABILITY, ACCURACY, TIMELINESS, COMPLETENESS, NON-INFRINGMENT, MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Your use of the translations is subject to all use restrictions contained in your Electronic Products License Agreement and by using the translation functionality you agree to forgo any and all claims against ProQuest or its licensors for your use of the translation functionality and any output derived there from. Hide full disclaimer
Details
1 Yanshan University, Qinhuangdao 066004, China
2 Tongling University, Tongling, Anhui 244000, China
3 Qinghai Institute of Technology, Xining, 810016, China