H∞ mode-independent filter design for Markovian jump genetic regulatory networks with time-varying delays
Introduction
In recent years, the interaction between genes, proteins and molecules forming cellular systems have been one of the most important aspects of post-genomic biology. Specifically, stability and modeling problem of gene regulatory networks (GRNs), as a significant area of research in the biological and biomedical sciences, have attracted much attention [1], [2], [3], [4], [5]. GRNs are the mechanisms that regulate the expression of genes. The change in expression of genes is regulated negatively or positively by their own produced proteins. The main mathematical models proposed to model the genetic networks are Boolean networks [6], [7] and differential equation models [2], [8]. Examining gene expression data, it seems that the gene expression levels tend to be more continuous rather than binary.
In differential equation modeling of GRNs, similar to modeling other dynamical systems, the exact model can hardly be obtained. It is mostly because of the modeling errors, external perturbation and parameter fluctuations. So it is important to study the robust problem of such networks with noise, errors and perturbations. Also, It is shown in [9], [10], [11] that the time taken for the gene transcription and translation are not negligible in the dynamics of GRNs. The mathematical models without addressing the delay effects may provide wrong predictions of the mRNA and protein concentrations. The delays are usually considered as time-varying delays [9] and [10], [11], [12].
The existence of switching mechanisms in gene networks is a well known fact [13], [14]. To address the switching nature, authors in [15], [16] have proposed Markovian jump GRNs with the emphasis on quantitatively describing of gene regulation. Markovian jump GRNs are hybrid systems with their discrete state varying as a continuous-time finite state Markov process. So, GRNs can be assumed as a type of Markovian jump nonlinear systems with noise and delays. The stochastic stability and control problem of delayed nonlinear systems with and without Markovian jumps are investigated in [17], [18], [19].
To achieve some biological objectives such as identifying genes of interest and drugs extraction, biologists are interested in knowing the concentration values of mRNA and protein in gene networks. To this end, the problem of filtering has been investigated for nonlinear genetic regulatory networks in many recent works such as in [20], [21], [22], [23]. In [20], authors have designed a filter for stochastic GRNs including noise and fixed transcription and translation delays. But the switching mechanism in GRNs has been disregarded. In [21], the problem of state estimation for Markovian jump GRNs with time-varying delays has been investigated but the noise in GRNs dynamics has not been considered. [22], [23] present the investigations on the filtering for the GRNs with distributed time delays.
In all above works on Markovian jump GRNs filtering, the complete access to present system mode is presumed. In other words, the filter gains are all obtained as mode-dependent values. Such an assumption may not hold in many real world situations. There are many situations in which the state of the Markov process is not available for observation. For instance, in a cancer treatment application, it is not possible to track the expression status of all genes in the network. The availability of the Markov states may be limited by cost, physical accessibility or other considerations. Clearly, in such a case the filtering techniques of previous works cannot be implemented. The control problem of GRNs with non-accessible Markov states in Probabilistic Boolean Networks is investigated in [24]. However, to the best knowledge of authors, all dynamical filters designed to monitor the states of GRNs need the switching status to be available. In the literature of Markovian Jump Systems (MJSs), the problem of mode-independent filtering is addressed in [25], [26], [27]. To design mode-independent and deterministic filters or controllers for MJSs, It is important to use mode-dependent Lyapunov functions. Using mode-independent Lyapunov functions, introduces very strong conservatism on the filter performance [26].
Even in the area of MJSs, there exist few numbers of studies on the mode-independent filtering in presence of noise and time-varying delays. According to that mentioned above, there is a strong motivation to design a mode independent filter based on mode dependent Lyapunov functions. Such filters have been neither investigated in the literature of MJLs and gene regulatory networks with time delays.
The main idea in this paper is to design a mode-independent filter for dynamic GRN model including noise and time-varying delays while using mode-dependent Lyapunov functions. The designed asymptotically stable filter ensures mean square stability for the estimation error dynamics and a prescribed upper-bound on the ℒ2 -induced gain from the disturbance signals to the estimation error. The design method is based on assuming a special form for Lyapunov matrices. To synthesize the filter gains from the stochastic stability conditions some techniques and transformations are applied. The filter designs are given in terms of linear matrix inequalities (LMIs). The effectiveness of results are tested via a simulation example.
This paper is organized as follows: Section 2 describes the model of Markovian jump Genetic regulatory networks and the filter structure and gives some definitions and preliminaries on stochastic stability of GRNs; Section 3 presents the main results on filter design, Section 4 gives a simulation example and finally Section 5 concludes the paper.
Section snippets
System description and preliminaries
In this paper, the following genetic regulatory networks are considered [2]:in which m(t)=[m1(t),m2(t),…,mn(t)]T, p(t)=[p1(t),p2(t),…,pn(t)]T and mi(t),pi(t)∈R are the concentrations of mRNA and protein of ith node. The parameters in (1) are considered as follows:in which, ami’s and api’s are the degradation rates of mRNA and
Main results
In this section, we introduce the main results which ensure the stochastic stability of augmented filter/GRN system with Markovian jumping parameters. Then, an approach for the synthesis of the deterministic filter gains from LMIs is presented. The stochastic stability conditions of GRN system (11) is given in the following lemma which is a modified version of the work presented in [15], [30]. The results of the previous works are adapted here to express the conditions of stochastic stability
Simulation results
In this section, we examine our results to show the effectiveness of the proposed method. As the first example, consider a Markovian jump GRN system like (11) with two operating modes and the following parameters:
Conclusion
The problem of H∞ mode-independent filter design for Markovian jump gene regulatory networks with disturbances, noise and time-varying delays was considered. Based on mode-dependent Lyapanuv–Krasovski functionals and a rectangular form consideration for Lyapunov matrix and applying some techniques and congruence transformations, the deterministic filter gains were synthesized from LMI conditions. The results show that the proposed method is effective and can stabilize the augmented GRNs/filter
Mohammad Mohammadian received the B.S. degree in Electrical Engineering from Amirkabir University of Technology in 2008, the M. Sc. degree from Tarbiat Modares University in 2011. He is currently PhD candidate in control engineering department, Tarbiat Modares University, Tehran, Iran. His research interests include stochastic switching systems, stochastic control and Systems Biology.
References (31)
- et al.
Mathematical modeling of gene networks
Neuron
(2000) - et al.
Stability analysis of uncertain genetic sum regulatory networks
Automatica
(2008) - et al.
Robustness and fragility of Boolean models for genetic regulatory networks
J. Theor. Biol.
(2005) Boolean formalization of genetic control circuits
J. Theor. Biol
(1973)Autoinhibition with transcriptional delay: A simple mechanism for the Zebrafish Somitogenesis Oscillator
Curr. Biol.: CB
(2003)Oscillatory expression of Hes1, p53, and NF-؛B driven by transcriptional time delays
Curr. Biol.: CB
(2003)Stochastic and delayed stochastic models of gene expression and regulation
Math. Biosci.
(2010)- et al.
Stochastic stability of Markovian switching genetic regulatory networks
Phys. Lett. A
(2009) - et al.
Robust stochastic stability analysis of Markovian switching genetic regulatory networks with discrete and distributed delays
Neurocomputing
(Dec. 2010) - et al.
Robust H∞ control for nonlinear uncertain stochastic T–S fuzzy systems with time-delays
Appl. Math. Lett.
(2011)
State estimation for Markov-type genetic regulatory networks with delays and uncertain mode transition rates
Phys. Lett. A
Robust H∞ filter design for uncertain nonlinear singularly perturbed system with Markovian jumps: An LMI approach
Inf. Sci.
Passivity analysis for stochastic Markovian switching genetic regulatory networks with time-varying delays
Commun. Nonlinear Sci. Numer. Simul.
Stability of genetic networks with SUM regulatory logic: Lur'e system and LMI approach
IEEE Trans. Circuits Syst. Regul. Pap.
Modeling and simulation of genetic regulatory systems: a literature review
J. Comput. Biol.
Cited by (39)
Dynamic output feedback control of discrete-time switched GRNs with time-varying delays
2020, Journal of the Franklin InstituteCitation Excerpt :Switched systems, composed of a class of subsystems and a switching law that orchestrates the switching among the subsystems, assuring stability and a certain performance. The last decades have witnessed a growing interest from the scientific community on the stability analysis [12–15], state estimation [16–18], filter design [19,20] and controller design [21] for switched GRNs with time delays. More specifically, the authors in [12] investigated the problem of decentralized event triggered exponential stability for uncertain GRNs with distributed delays and Markov jump parameters.
A state estimation H<inf>∞</inf> issue for discrete-time stochastic impulsive genetic regulatory networks in the presence of leakage, multiple delays and Markovian jumping parameters
2018, Journal of the Franklin InstituteCitation Excerpt :Until now, the Boolean model and the differential equation model [1,5,31,46–48] are the basic models related to GRNs. In the first highlighted model, the genes that express from the network are simulated to be either ON or OFF and there is no interposed action levels are usually taken into deliberation, and gene state is decided by the Boolean function of the other related gene states [5,19–29,32]. Whereas, the variables involve, which defines the change rate of the concentration of mRNAs and proteins, are continuous values in the differential equation model.
Non-uniform sampled-data control for stochastic passivity and passification of Markov jump genetic regulatory networks with time-varying delays
2016, NeurocomputingCitation Excerpt :The study of Markov jump GRNs has received much attention due to their theoretical importance and potential applications. It is worth mentioning that most of the researches mainly focus on the stability problem or the state estimation problem of GRNs and have achieved remarkable results [6,12,32,33]. In this paper, we investigate the non-uniform sampled-data control scheme to solve the passivity and passification problems of Markov jump GRNs.
Delay-dependent robust H<inf>∞</inf> filtering of uncertain stochastic genetic regulatory networks with mixed time-varying delays
2015, NeurocomputingCitation Excerpt :To obtain the right amount of proper drugs as artificial input control, it is important to estimate the network states such that the overall error could asymptotically converge to zero for a GRN including noise perturbations, delays, and parameter uncertainties. Therefore, the estimation problem for the kinds of GRNs is important from an application point of view, and has been paid attention by some scholars [19–40]. In [34], the filtering problem of uncertain stochastic GRNs with Itô-type stochastic disturbance, norm-bounded parameter uncertainties, and time-varying discrete delays without considering distributed delays has been studied.
Mohammad Mohammadian received the B.S. degree in Electrical Engineering from Amirkabir University of Technology in 2008, the M. Sc. degree from Tarbiat Modares University in 2011. He is currently PhD candidate in control engineering department, Tarbiat Modares University, Tehran, Iran. His research interests include stochastic switching systems, stochastic control and Systems Biology.
Amir Hossein Abolmasoumi received the B.S. degree in Electrical Engineering from University of Tehran in 2004, the M. Sc. degree from Tarbiat Modares University in 2007 and his PhD in control engineering department, Tarbiat Modares University, Tehran, Iran. He is currently with the Electrical Engineering department of Arak University. His research interests include stochastic switching systems, delayed dynamical systems and congestion control in communication networks.
Hamid Reza Momeni was born in Khomain, Iran. He received his B. Sc., M. Sc. and Ph. D. degrees in electrical engineering from Sharif university of technology, in 1977, university of Wisconsin at Madison, USA, in 1979 and Imperial college of London, England, in 1987, respectively. His research interests included Adaptive control, robust control, Teleoperation systems, Industrial control, Instrumentation, Automation, Navigation and Guidance. He is associate professor in Electrical Engineering department, university of Tarbiat Modares, Tehran, Iran.