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Available online at www.sciencedirect.com

Journal of the Franklin Institute 351 (2014) 4395–4414

Exponential synchronization of switched genetic

oscillators with time-varying delays

Xiongbo Wan

a,b,

,LiXu

, Huajing Fang

, Fang Yang

, Xuan Li

College of Engineering, Huazhong Agricultural University, Wuhan 430070, China

Department of Electronics and Information Systems, Akita Prefectural University, 84-4 Tsuchiya-Ebinokuchi,

Honjo, Akita 015-0055, Japan

Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074,

China

Received 13 June 2013; received in revised form 2 April 2014; accepted 3 June 2014

Available online 9 June 2014

Abstract

This paper addresses the problem of exponential synchronization of switched genetic oscillators with time-

varying delays. Switching parameters and three types of nonidentical time-varying delays, that is, the self-delay, the

intercellular coupling delay, and the regulatory delay are taken into consideration in genetic oscillators. By utilizing

the Kronecker product techniques and ‘delay-partition’ approach, a new Lyapunov–Krasovskii func tional is

proposed. Then, based on the average dwell time approach, Jensen's integral inequality, and free-weighting matrix

method, delay-dependent sufﬁcient conditions are derived in terms of linear matrix inequalities (LMIs). These

conditions guarantee the exponential synchronization of switched genetic oscillators with time-varying delays

whose upper bounds of derivatives are known and unknown, respectively. A numerical example is presented to

demonstrate the effectiveness of our results.

www.elsevier.com/locate/jfranklin

http://dx.doi.org/10.1016/j.jfranklin.2014.06.001

☆

This work was supported by the National Natural Science Foundation of China under Grant nos. 61203085 and

61034006., and the Fundamental Research Funds for the Central Universities under Grant 2013JC003.

Corresponding author at: College of Engineering, Huazhong Agricultural University, Wuhan 430070, China.

E-mail addresses: [email protected] (X. Wan), [email protected] (L. Xu), [email protected] (H. Fang),

[email protected] (F. Yang), [email protected] (X. Li).

1. Introduction

Over the past decades, genetic networks have received considerable attention due to their wide

application in biological and biomedical sciences. Modeling genetic networks as dynamic

systems have p rovided a powerful tool for elucidati ng genetic regulation process in living

organisms. The existing genetic network models include Boolean models [1], different ial

equation model s [2,3], Petri net models [4], and discrete-time piecewise afﬁne model [5,6].

Among them, differential equation models are the most popular ones in which the state variables

usually denote the concentrations of proteins, messenger ribonucleic acids (mRNAs) and

other small molecules, and the regulatory functions are usually in Michaelis–Menten or Hill form

[7–13].

The oscillatory behavior of genetic networks is a fundamental challenge in the research ﬁeld of

systems biology. There are two typical examples of genetic oscillators, that is, the cell cycle

oscillators [14] and circadian clocks [15]. Understanding the molecular mechanisms of

oscillations and their collective behaviors is important for clarifying the dynamics of cellular life

and for diagnosing and treating diseases. Synchronization, as a type of typical collective behavior

and a basic motion in nature, can be used to explain many natural phenomena [16–18]. Study on

synchronization of coupled genetic oscillators has important biological implications and potential

engineering applications [19,20]. For example, it is essential for the understanding of the

rhythmic phenomena and the inherent mecha nisms of living organisms at both molecular and

cellular levels [16,19,20]. Moreover, the synchronization criteria of genetic oscillators can

provide biologists with the opportunity to test mathematical models for the dynamics of gene

regulation. In addition, the mechanism of synchronization is helpful for medical diagnosis and

treatment and for the creation of synthetic gene circuits.

So far, the synchronization problem of genetic oscillators has been intensively investigated

during the past few years [19–23].In[19], several sufﬁcient conditions for synchronization of

coupled nonidentical genetic oscillators have been presented. Moreover, the estimation of the

bound of the synchronization error has been given simultaneously. In [20], based on the Lur'e

system approach, an easy-veriﬁed sufﬁcient condition for the stochastic synchronization of

genetic oscillators with random noise perturbations has been established. In [21], synchronization

of cellular clock in the suprachiasmatic nucleus (SCN) has been experimentally investigated and

fruitful ﬁndings which indicate the vital factors affecting this synchronization have been

reported. For example, it has been pointed out that Na

-dependent action potentials contribute to

establishing cellular synchrony and maintaining spontaneous oscillation across the SCN. In [22],

the effect of coupling on synchronization through intercellular signaling in a population of

Escherichia coli cells has been analyzed. The authors have given many results which establish

not only a theoretical foundation but also a quantitative basis for understanding the essential

cooperative dynamics in a population of cells. For example, they have shown that an appropriate

design of the coupling and the inner-linking matrix can ensure global synchronization of the

coupled synthetic biological system. In [23], robust synchronization control of synthetic genetic

oscillators under intr insic and extrinsic molecular noises has been studied.

Notice that time delay has not been taking into account in the above genetic oscillators. In fact,

due to the slow processes of transcription, translation, and translocation or the ﬁnite switching

speed of ampliﬁers, time delays inherently exist in genetic networks. Therefore, incorporation of

time delays is necessary when studying the synchronization problem of genetic oscillators.

In [24], Qiu and Cao have studied the global exponential synchronization problem of delay-

coupled identical genetic oscillators with time delays. In [25], genetic oscillators with delayed

X. Wan et al. / Journal of the Franklin Institute 351 (2014) 4395–44144396

intercellular coupling have been studied by noticing that a signal transmitted from one cell to

neighboring cells shall take a period of time. In [26], synchronization of a class of genetic

oscillators with Markovian jumping parameters, time delays, and stochastic ﬂuctuation has been

studied. Zhang et al. [27] have further investigated the synchronization problem for Markovian

jump genetic oscillators with partially known transition rates and nonidentical feedback delay.

As pointed out in [9], the hybrid system involving some kind of switching is natural and

promising in mathematically modeling the complex gene regulations. For Markovian switching

genetic networks, fruitful results can be found in the literature (see, e.g., [9,10,26,27]). In [11],

the stability problem of switched genetic regulatory networks (GRNs) with time-var ying delays

has also been investigated. In [12], Zhang et al. have further studied the exponential stability of

stochastic delayed switched GRNs with both stable and unstable subsystems. However, till now,

little attention has been paid on the analysis of exponential synchronization of switched genetic

oscillators. Furthermore, when studying the synchronization problem of delayed genetic

oscillators, constant delay has been studied in almost all of the existing literature (e.g. [24–27]),

while time-varying delay has seldom been discussed. In addition, in coupled genetic oscillators,

three types of delays, that is, the self-delay, the intercellular coupling delay, and the regulatory

delay, may be nonidentical. Although synchronization of genetic oscillators with nonidentical

intercellular coupling delay and self-delay has been investigated in [27], synchronization for

genetic oscillators with the above three types of nonidentical delays has not been discussed yet,

possibly due to its mathematical complexity.

Motivated by the above discussions, in this paper, we are concerned with the exponential

synchronization of switch ed genetic oscillators with three types of nonidentical time-varying

delays. By utilizing the Kronecker product techniques and ‘delay-partition’ approach, a new

Lyapunov–Krasovskii functional is proposed. Then, based on the average dwell time approach,

Jensen's integral inequality, and free-weighting matrix method, d elay-dependent sufﬁcient

conditions are derived in terms of linear matrix inequalities (LMIs) to guarantee the exponential

synchronization of genetic oscillators with known and unknown upper bounds of derivatives of

time delays, respectively. A numerical example is presented to demonstrate the effectiveness of

our results.

The main contributions of this paper include: (1) A class of switched genetic oscillators with

unavailable transition rates is proposed. Although it is well known that switching is natural and

promising for modeling the complex gene regulations, little attention has been paid to the

switched genetic oscillators. (2) Time-varying delays are assumed to be included in the proposed

switched geneti c oscillators. As we know, for the synchronization problem of delayed genetic

oscillators, constant delay has been widely studied in the existing literature, but time-varying

delay has seldom been discussed. (3) Three types of nonidentical time-varying delays in genetic

oscillators, that is, the self -delay, the intercellular coupling delay, and the regulatory delay, are

distinguished and carefully investigated. (4) New techniques including average dwell time

approach, ‘delay-partition’ approach, Jensen's integral inequality, and free-weighting matrix

method are employed to establish the main results.

Notations: Throughout this paper, R

denotes the n-dimensional Euclidean space. R

nm

is the

set of real n m matrices. For vector xA R

, its norm is deﬁned as J x J ¼

ﬃﬃﬃﬃﬃﬃﬃ

. The supers cript

T repres ents the matrix transposition. I and 0 denote the identity matrix and the zero matrix with

compatible dimensions, respectively. diagfg denotes a diagonal matrix. The notation A4B

(respectively, A Z B) means that the matrix A B is symmetric positive deﬁnite (respectively,

positive semi-deﬁnite), where A and B are matrices with the same dimensions. A  B is the

Kronecker product of matrices A and B. In symmetric block matrices or long matrix expressions,

X. Wan et al. / Journal of the Franklin Institute 351 (2014) 4395–4414 4397

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