Elsevier

Mathematical Biosciences

Volume 239, Issue 1, September 2012, Pages 97-105
Mathematical Biosciences

H filtering for discrete-time genetic regulatory networks with random delays

https://doi.org/10.1016/j.mbs.2012.05.002Get rights and content

Abstract

The H filtering problem is investigated in this paper for a class of discrete-time genetic regulatory networks (GRNs) with random delays. The addressed filtering problem is to estimate the concentrations of mRNA and protein, and the filtering error system is modeled as a Markovian switched system. By using a properly constructed Lyapunov function, a sufficient condition is derived in terms of linear matrix inequalities (LMIs), which can guarantee stochastic stabilization of the filtering error system. Then, an optimization problem with LMIs constraints is established to design an H filter which ensures an optimal H disturbance attenuation level. Finally, an illustrative example is given to demonstrate the effectiveness of the proposed results.

Highlights

► The H filtering problem is studied for a discrete-time GRN with random delays. ► The filtering error system is modeled as a Markovian switched system. ► The stochastic stability of the discrete-time GRN is analyzed. ► An optimization problem with LMIs constraints is established to design an H filter.

Introduction

Genetic regulatory networks (GRNs), structured by networks of regulatory interactions between DNA, RNA and proteins, have become an important new area of research in the biological and biomedical sciences, and a large number of results have been reported in the literature, such as [1], [2], [3], [4], [5], [6], [7] and the references therein.

It is revealed that time-delay is an important factor in dynamics of GRNs due to slow biochemical reaction such as actual regulation, transcription, translation, diffusion, and translocation, especially in that of a eukaryotic cell [8], [9]. A number of important results concerning GRNs with time delays have been reported, e.g., in [2], [6], [10], [11], [12], [13]. A couple of delay differential equations have been proposed to describe GRNs in [10], where the local stability and bifurcation are discussed to provide comprehensive information for understanding gene expression patterns and regulatory pathways. The robust asymptotic stability of GRNs with time-varying delays and parameter uncertainties was investigated in [11], [12], where the stability conditions were dependent on the bounds of the delays. In [13], the robust stability analysis problem for uncertain GRNs with interval time-varying delays was addressed, and the presented stability criteria were applicable to both fast and slow time-varying delays. Note that the models in [2], [6], [10], [11], [12], [13] were all described by differential equations. However, in implementing the continuous time network for computer simulation and experimental or computational purposes, it is common to discretize the continuous time network. Moreover, it is shown in [4], [14], [15], [16], [17] that some GRN models can be described by discrete-time dynamical systems, and these models are much more important than their continuous-time counterpart. In [14], [15], the discrete-time piecewise affine models have been proposed to find the explicit relations between the topological structure of the GRNs and the growth rate of the dynamical complexity, while the exponential stability analysis problem for discrete-time GRNs has been investigated in [16], [17].

Molecular noise has many roles in the biological functions of GRNs, including noise-induced amplification of signals, noise-drive divergence of cell fates, generation of errors in DNA replication leading to mutation and evolution, and maintenance of the quantitative individual of cells [18]. The nature of the noise and its value may not be fully known or measurable due to complexity of biological processes. Filtering is to filter out the intrinsic and extrinsic noises to make the system function properly [19]. The filtering technology has been introduced (see [20], [21], [22], [23], and the references therein). For GRNs model, the robust filtering problem has been investigated in [19], [24] for a class of gene expression model with stochastic disturbances and parameter uncertainties, where the time delay was not considered. The filter design approaches were presented in [25], [26] for a delayed GRNs with stochastic disturbance, and the filtering error system was exponentially mean square stable with a prescribed decay rate. By taking both the external noise and model switching into account, the H filtering problem was investigated in [27]. Recently, the extended Kalman filtering for the discrete-time GRNs was studied in [28]. Note that most existing results are concentrated on the filtering problem of GRNs by differential equations. However, the filtering problem for discrete-time GRNs structures still remains challenging. This motivates the present research.

In this paper, we are concerned with the H filtering problem for a class of discrete-time GRNs with random delays. The purpose of the addressed filtering problem is to estimate the true concentrations of the mRNA and protein. The time delays, existing in both the translation process and feedback regulation process, are described as a Markov chain. Then, the filtering error system is modeled as a Markovian switched system. By applying Lyapunov method, a sufficient condition for the H filter is derived in terms of LMIs. The designed controller guarantees that the filtering error system is stochastically stable and ensures a prescribed H performance. Moreover, an optimization problem with LMIs constraints is formulated to design the H filter. Finally, an illustrative example is presented to show the effectiveness of the proposed results.

Section snippets

Problem formulation

The considered discrete-time GRN with random delays, n mRNAs and n proteins can be described by the following difference equations [16], [17]Mi(k+1)=e-aihMi(k)+ϕi(h)j=1nbijfj(Pj(k-d(k)))+Wi,Pi(k+1)=e-cihPi(k)+φi(h)[diMi(k-d(k))],i=1,2,,nwhere Mi(k)RandPi(k)R are the concentrations of mRNA and protein of the ith gene, respectively; h is a fixed positive real number denoting a uniform discretionary step size; ai>0 and ci>0 are the degradation rates of mRNA and protein, respectively; di is the

H performance analysis and filter design

The stability analysis for the filtering error system (8) with w(k)=0 and v(k)=0 is presented in the following theorem.

Theorem 1

The filtering error system (8) with w(k)=0 and v(k)=0 is stochastically stable, if for each r  N, there exist matrices Pi(r) > 0, Qi > 0, i = 1, 2, Af, Bf, Cf and Df, such that the following matrix inequalityΩ=Ω1100Ω140Ω22Ω230Ω25Ω3300Ω440Ω55<0holds, whereΩ11=A¯TP¯1(r)A¯-P1(r)+μQ1,Ω14=A¯TP¯1(r)B¯,Ω22=C¯TP¯2(r)C¯-P2(r),Ω23=C¯TP¯2(r)D¯Z,Ω25=-ZTL,Ω33=ZTD¯TP¯2(r)D¯Z-Q1,Ω44=B¯TP¯1(

Illustrative example

In the following, we illustrate the application of the proposed results to an existing biological system. The dynamics of repressilator has been theoretically predicted and experimentally investigated in Escherichia coli [1]. This system is a cyclic negative-feedback loop comprising three repressor genes (lacl, tetR and cl) and their promoters. It is described as followsm˙i=-mi+αi1+pjHp˙i=-βi(pi-mi)where i=lacl,tetR,cl, j=cl,lacl,tetR, mi and pi are the concentrations of the three mRNA and

Conclusions

The H filtering problem has been studied in this paper for a class of discrete-time GRNs with random delays. The random delays were described as a Markov chain, and the filtering error system was modeled as a Markovian switched system. By utilizing the Lyapunov functional method and some stochastic analysis tools, the existence of a full order filter was presented in terms of LMIs. The designed filter guarantees that the filtering error system is stochastically stable and ensures an optimal H

Acknowledgements

The authors would like to thank the anonymous reviewers for their valuable suggestions to improve the quality of this article. This research work was supported in part by the National Natural Science Foundation of China under Grants 60904040, 60974017 and 61104063, and in part by the Zhejiang Provincial Natural Science Foundation of China under Grant Y1110484.

References (33)

  • M.C. de Oliveira et al.

    A new discrete-time robust stability condition

    Systems and Control Letters

    (1999)
  • J. Xiong et al.

    On robust stabilization of Markovian jump systems with uncertain switching probabilities

    Automatica

    (2005)
  • M.B. Elowitz et al.

    A synthetic oscillatory network of transcriptional regulators

    Nature

    (2000)
  • C. Li et al.

    Stability of genetic networks with sum regulatory logic: Lur’e system and LMI approach

    IEEE Transactions on Circuits and Systems-I: Regular Papers

    (2006)
  • I. Ivanov et al.

    Modeling genetic regulatory networks: continuous or discrete

    Journal of Biological Systems

    (2006)
  • B.S. Chen et al.

    Robust H stabilization design in gene networks under stochastic molecular noises: fuzzy-interpolation approach

    IEEE Transactions on System, Man and Cybernetics Part B – Cybernetics

    (2008)
  • Cited by (0)

    View full text