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Stochastic Stability of Switched Genetic Regulatory Networks With Time-Varying Delays Wenbing Zhang, Yang Tang , Member, IEEE, Xiaotai Wu, and Jian-An Fang
Abstract—This paper investigates the exponential stability problem of switched stochastic genetic regulatory networks (GRNs) with time-varying delays. Two types of switched systems are studied respectively: one is the stochastic switched delayed GRNs with only stable subsystems and the other is the stochastic switched delayed GRNs with both stable and unstable subsystems. By using switching analysis techniques and the modified Halanay differential inequality, new criteria are developed for the exponential stability of switched stochastic GRNs with time-varying delays. Finally, an example is given to illustrate the main results. Index Terms—Exponential stability, stable subsystems, switched genetic regulatory networks, unstable subsystems.
I. INTRODUCTION ECENTLY, DUE to the potential significance in biological engineering applications, genetic regulatory networks (GRNs) have received increasing attention from biological and biomedical sciences [1]–[3], when modeling GRNs, time delays are ubiquitous due to the finite speed in the slow process of transcription, translation and translocation. Time delays in GRNs may cause instability and oscillation [4]. Therefore, It is of great importance to model GRNs with time delays. On the other hand, gene regulation process is an intrinsically noisy process due to intracellular and extracellular noise perturbations and environmental fluctuations [5]. Hence, it is necessary to investigate the effects of time delays and stochastic disturbances on the stability of GRNs. Many real world genetic regulatory networks intrinsically display unstable [6] or chaotic behavior [7]. For instance, though microarrays allow measurement of thousands of genes simultaneous, gene expressions in practice can be gathered only over a few time points due to high cost and time involved, and limitations of the experiments, which makes building GRNs inherently an ill-posed problem in practice, leading such networks unstable and irreproducible [6], and highly
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Manuscript received December 04, 2012; revised June 22, 2013; accepted May 23, 2014. Date of publication June 02, 2014; date of current version September 23, 2014. This paper was supported by the National Natural Science Foundation of China (61203235), the Key Creative Project of Shanghai Education Community (Grant No. 13ZZ050), the Key Foundation Project of Shanghai (Grant No. 12JC1400400), the Natural Science Foundation of the Higher Education Institutions of Jiangsu Province, China (Grant No.14KJB120014), and the Alexander von Humboldt Foundation of Germany. Asterisk indicates corresponding author. W. Zhang is with the Department of Mathematics, Yangzhou University, Jiangsu, China (e-mail: [email protected]). *Y. Tang is with the Potsdam Institute for Climate Impact Research, Potsdam 14473, Germany, and also with the Institute of Physics, Humboldt University of Berlin, Berlin 12489, Germany (e-mail: [email protected]). X. Wu is with the Department of Mathematics, Anhui Polytechnic University, Wuhu, Anhui 241000, China. J. Fang is with the School of Information Science and Technology, Donghua University, Shanghai 201620, China. Digital Object Identifier 10.1109/TNB.2014.2327582
nonlinear dynamics exhibited by genetic regulatory systems can be characterized into two broad regimes, to wit, an ordered regime where the system is robust against perturbations, and a chaotic regime where the system is extremely sensitive to perturbations [7]. Moreover, the existence of time delay could be pivotal in inducing oscillations in gene regulation [8], which means that the systems are unstable as well. Obviously, the model constructed by both stable and unstable subsystems can generalize the model constructed by only stable subsystems. Therefore, the model constructed by both stable and unstable subsystems can approximate the real-world genetic regulatory networks more accurately than that constructed by only stable subsystems. Overall, when modeling the real-world delayed genetic regulatory networks, it is of great importance to take into account both stable and unstable subsystems. Recently, the stability of switched systems has received increasing attention, such as switched GRNs [9] and switched linear systems [10], [11]. However, in [9], the authors only considered switched GRNs with stable subsystems, and the unstable subsystems were overlooked. Therefore the method used in [9] failed to deal with the stability of GRNs with both stable and unstable subsystems. Moreover, the results from switched systems [10]–[13] failed to analyze the stability of switched delayed systems due to the existence of delay. Hence, it is of practical importance to model GRNs with both stable and unstable subsystems, which are more general than GRNs with only stable subsystems. In addition to the difficulty of dealing with the stability problem of GRNs with stable and unstable subsystems, time-delays should be also taken into account to model switched GRNs, which will inevitably render difficulty to analyze such comprehensive systems. However, to our best knowledge, the stochastic stability problem of switched GRNs with both stable and unstable subsystems has received little research attention, not to mention the case of time delays are included, primarily due to the mathematical complexity in derivation, which motivates us to shorten such a gap. Motivated by the above discussion, we investigate the exponential stability of switched delayed GRNs with both stable and unstable subsystems. The contributions of this paper are listed as follows: 1) the stability problem is investigated for stochastic switched delayed GRNs with both stable and unstable subsystems, which encompasses some recently GRNs models as special cases; 2) the obtained results are developed through the modified Halanay differential inequality and piecewise Lyapunov function, which are easy to verify. Notations: Throughout this paper, and denote, respectively the set of nonnegative integers and positive integers. The vector norm is defined as . For matrix , where represents the largest eigenvalue. denotes the matrix inverse.
1536-1241 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
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Moreover, for real symmetric matrices and , the notation (respectively, ) means that the matrix is negative semidefinite (respectively, negative definite). For , we say that a function from to is piecewise continuous if the function has at and most a finite number of jumps discontinuous on are continuous from the right for all points in . Given denotes the family of piecewise to with the norm continuous functions from . Let be the family of
-adapted and -valued random variables such that , where denotes the expectation operator. is infinitesimal operator which is defined as in [14]. II. MODEL FORMULATION AND SOME PRELIMINARIES
where satisfies the following inequality:
(5) for all . Thus, we know that sector condition
where and denote respectively, the concentrations of mRNA and the proteins of the th gene at time and are the degradation rates of mRNA and prorepresents the translation tein of the th gene, respectively; rate. is a monotonically increasing function denoting the feedback regulation of the protein on the transcription which is generally expressed by a monotonic function of the Hill form: , where is the Hill coefficient and is a positive constant, is the base tranis the scriptional rates of the repressor of gene and coupling matrix of the GRNs (1), which is defined as follows:
The two positive functions and denote respectively, the translation time delays and the feedback regulation delays and satisfy the following conditions: (2) The GRNs model in (1) can be rewritten in the following compact matrix form: (3) where and
satisfies the following
(6) where . Taking into account the parameter fluctuations and the intracellular noises perturbations [17], [18], the stochastic switched GRNs model is considered as follows:
In this section, some preliminaries including model formulation, lemmas and definitions are given. Consider the following delayed GRNs with SUM regulatory logic consisting of mRNAs and proteins [15], [16]: (1)
. Here, we assume
(7)
where and are mutually uncorrelated one-dimensional Brownian motions, and are the noise intensity funcis a piecewise contions. stant function, continuous from the right, specifying the index of the active subsystem. For simplicity, can be written for . The time seas quence satisfies and , where is the initial time and we assume that there is no switching at . For the purpose of simplicity, we mark . The initial conditions of the GRNs (7) are assumed to be
where . Remark 1: Generally, two kinds of switched systems have been investigated in the literature, i.e., the switched system with stable subsystems [10] and the switched systems with both stable and unstable subsystems [19]. Obviously, the switched systems with both stable and unstable subsystems are more practical and general than the ones with only stable subsystems. Therefore, recent works have been devoted to the stability problem of switched systems with both stable and unstable subsystems [19]. Unfortunately, time-delays are neglected in the above results due to the mathematical difficulty. It should be mentioned that our results are not only restricted to GRNs, but also can be extended to usual switched delayed systems with both stable and unstable subsystems. The following assumptions, definitions and lemmas will be used in the proof of the main results. Assumption 1: The noise intensity functions in (7) satisfy the following condition
. Let and be an equilibrium of (3). Shifting the equilibrium point of the and system (3) to the origin by letting , we have (4)
for all
, where
are positive constants.
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Assumption 2: There exists a constant such that , for all . Definition 1: The stochastic GRNs (7) are said to be exponentially stable in mean square if there exist positive constants and , such that (8) holds for all
and any initial data . Then and are called decay coefficient and decay rate respectively. Lemma 1: Let is a diagonal positive definite matrix with appropriate dimensional, then the following inequality holds
. Then we have . Thus, we have (16) holds. And
When when holds. Therefore, we have and
from these inequalities. In view of (11) , we have
(9) Lemma 2 ([20]): If nite matrix and
is a symmetric positive defiis a symmetric matrix, then (17) (10)
which contracts (16), and hence (15) holds. Let get
, we can (18)
III. MAIN RESULTS In this section, we develop some criteria for the stability of the switched GRNs (7) by utilizing the modified Halanay differential inequality and switched analysis techniques. Before presenting the main results, we first establish two theorems, which will be used to derive the main results. Theorem 1: Let be a continuous, and nonnegative func. Suppose and satisfies tion on the following inequality (11) Then
Thus, the proof is completed. Remark 2: Recently, the Halanay differential inequality [21] has been widely used to investigated the stability or synchronization problem of various network systems, e.g., GRNs [22], impulsive coupled neural networks [23] and impulsive neural networks [24]. Unfortunately, the famous Halanay differential inequality [21] cannot be directly applied to stochastic systems. Theorem 1 extends the Halanay differential inequality for stochastic systems and therefore it will be used to prove the stability of stochastic switched delayed GRNs. Theorem 2: Assume that there exist positive constants and such that
(12)
(19)
, , there
(20) (21)
where
is the unique solution of the equation . . Note that Proof: Let we have . Since and exists a unique such that
(22)
(13) When
where
, it is clear that (14)
Denote for any
. In the following, we will prove that
and . is the unique solu. tion of equation Then the trivial solution of system (7) is exponentially stable in mean square. Proof: Note that . By (20), we have for
(15) Suppose that (15) is not true, then there exists some such that . Set
, (23)
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where
From Theorem 1, we have for (24) Since
and from (24), we have
is the unique positive solution of the equation . Then the GRNs (7) are exponentially stable in mean square. Proof: Consider the following Lyapunov functional:
(25)
(26) For any Then, we have
, it is true that
Taking the derivative of we have for
along the trajectory of system (7),
. (27)
Hence, for any
(37)
, one can get from (26) that In view of Lemma 1 and Lemma 2, one can obtain (28)
Then, from (25) and (28), we can get (38)
(29) Then, for
Form Assumption 1, we have
, we have
(30) By a simple induction, we can obtain for any
(39) From (37)–(39), we have (31)
where . Then, we can conclude from (8) that the GRNs (7) are exponentially stable in mean square. This completes the proof. Now we are in a position to investigate the stability problem of switched GRNs in (7) with time-varying delays and stochastic perturbations. We will derive the stability criteria for the stochastic switched GRNs (7) based on Theorem 2. Theorem 3: Consider the stochastic switched GRNs (7), if for any , there exist positive definite matrices , positive constants and diagonal positive definite matrices such that
(32) (33) (34) (35) (36)
(40) From Theorem 2, we can conclude that the switched delayed GRNs (7) are exponentially stable in mean square. This completes the proof of Theorem 3. Remark 3: Inequalities (32) and (33) in Theorem 3 imply that all the subsystems of the switched GRNs (7) are exponentially stable. Therefore, Theorem 3 is only suitable to the case that (7) is composed of stable subsystems. However, as shown in theoretical and biological results [6], [11], [25], [26], the unstable subsystems cannot be avoided in many applications. Hence it is desirable to consider a more general case that both unstable and stable subsystems exist in the GRNs model simultaneously. In the following theorem, we deal with the case that the switched GRNs (7) include both stable and unstable subsystems.
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In order to derive the stability conditions of switched GRNs with both stable and unstable subsystems, we need the following two lemmas. Lemma 3: Let be a continuous and nonnegative function on . Suppose that
tems. For the sake of simplicity, we denote as the set of the index of stable subsystems and denotes the set of index denote the total of unstable subsystems. Let activation time of the stable subsystems (unstable subsystems) during . Then, we need the following assumption. , such that Assumption 3: There exists a scalar the following assumption holds
(41) where
(47)
. Then (42)
, let Proof: For any , where . Then we will prove that for . Similar to the proof of Theorem 1, suppose that it is not true, then there exists some , such that
where will be defined latter. Then the following theorem can be obtained based on Theorem 1, Lemma 3 and Lemma 4. Theorem 4: Consider the stochastic delayed switched GRNs (7) with both stable and unstable subsystems, if for any , there exist positive definite matrices , positive constants and diagonal positive definite matrices such that
, we have
If
(48) where the last two inequalities are followed by the definition of . If , one gets (49) (50) Then, we always have
(51)
. Thus, from (41), we have where
where the last inequality holds due to the definition of . This . Thus , let , contradicts then the proof is completed. Then, similar to Theorem 2, the following lemma is readily available. Lemma 4: Assume that there exist positive constants and such that (43)
is the unique positive solution of the equation . Then the GRNs (7) with both stable and unstable subsystems are exponentially stable in mean square. Proof: Choosing the same Lyapunov functional as Theorem 3 and from (48), we have if , (52) From Theorem 1, we have (53) for
and
, it yields (54)
(44) (45) then for any
Then, from Lemma 3, we have (55)
, (46)
where . In the following, we will consider the stability of stochastic delayed switched GRNs with both stable and unstable subsys-
Without loss of generality, we assume that the evolution of stable subsystems and unstable switched GRNs start with subsystems, and then goes into stable subsystems, subsystems, and so on, where and are positive constants. For , we assume that there are
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stable subsystems and tween the interval . Then, if we have from Lemma 3
unstable subsystems be,
stochastic systems with both stable and unstable subsystems as well as time delays and will have wider applications than the results in [19], [25], [27]. IV. ILLUSTRATIVE EXAMPLE In this section, a simulation example is presented to illustrate the usefulness and flexibility of the developed results in this paper. It is assumed that each hybrid system here has only two subsystems, and the switching sequence is . Example 1: Reconsider (7) with
Considering Lemma 4, we have
(56) Then, using Theorem 3, we can get is taken as
(57)
which means that intensity functions are given as follows:
, . The noise
By repeating the same steps as (56) and (57), it is true that for
(58) Similarly if holds for
, we can also get (58) . Thus, in view of Assumption 3, it yields
with transcriptional time delays and . According to Theorem 4, we can get
. Let
(59) Thus we can get that , and solving (51) we have Let
Thus, it can be checked that
. .
(60) V. CONCLUSION
From (58)–(60), we have for
(61) where
. Then it yields
(62) where . From Definition 1, the switched delayed GRNs (7) with both stable and unstable subsystems are exponentially stable in mean square. This completes the proof. Remark 4: Recently, the stability of switched systems with stable and unstable subsystems has received increasing attention [19], [25], [27]. It is worth pointing out that time delays in [19], [25], [27] are ignored due to the mathematical complexity. However, in practice, time delays widely exist in various systems. Hence, Theorem 4 gives a unified framework for
In this paper, the exponential stability is investigated for a class of switched stochastic GRNs with time delays. By utilizing the dwell time approach and the modified Halanay differential inequality, the stability of stochastic switched delayed GRNs with only stable subsystems or both stable and unstable subsystems are investigated, respectively. When only stable subsystems exist in the switched GRNs, the stability of switched GRNs can be guarantied if the dwell time is larger than a derived lower bound. If both stable and unstable subsystems exist in switched GRNs, the stability condition is related to the total activation time of unstable subsystems and stable subsystems. REFERENCES [1] A. Becskei and L. Serrano, “Engineering stability in gene networks by autoregulation,” Nature, vol. 405, no. 6786, pp. 590–593, 2000. [2] T. Schlitt and A. Brazma, “Current approaches to gene regulatory network modelling,” BMC Bioinformat., vol. 8, p. s9, 2007. [3] Y. Tang, Z. Wang, H. Gao, S. Swift, and J. Kurths, “A constrained evolutionary computation method for detecting controlling regions of cortical networks,” IEEE/ACM Trans. Comput. Biol. Bioinformat., vol. 9, pp. 1569–1581, 2012. [4] L. Chen and K. Aihara, “Stability of genetic regulatory networks with time delay,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 49, no. 5, pp. 602–608, May 2002. [5] N. Maheshri and E. K. O’Shea, “Living with noisy genes: How cells function reliably with inherent variability in gene expression,” Annu. Rev. Biophys. Biomol. Struct., vol. 36, pp. 413–434, 2007.
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[6] J. C. Rajapakse and P. A. Mundra, “Stability of building gene regulatory networks with sparse autoregressive models,” BMC Bioinformat., vol. 12, p. S17, 2011. [7] I. Shmulevich, S. A. Kauffman, and M. Aldana, “Eukaryotic cells are dynamically ordered or critical but not chaotic,” PNAS, vol. 102, no. 38, pp. 13439–13444, 2005. [8] D. Bratsun, D. Volfson, L. S. Tsimring, and J. Hasty, “Delay-induced stochastic oscillations in gene regulation,” PNAS, vol. 102, no. 41, pp. 14593–14598, 2005. [9] Y. Yao, J. Liang, and J. Cao, “Stability analysis for switched genetic regulatory networks: An average dwell time approach,” J. Franklin Inst., vol. 348, no. 10, pp. 2718–2733, 2011. [10] J. Hespanha and A. S. Morse, “Stability of switched systems with average dwell time,” in Proc. 38th IEEE Conf. Decision Control, 1999, vol. 3, pp. 2655–2660. [11] G. Zhai, B. Hu, K. Yasuda, and A. N. Michel, “Stability analysis of switched systems with stable and unstable subsystems: An average dwell time approach,” Int. J. Syst. Sci., vol. 32, no. 8, pp. 1055–1061, 2001. [12] Y. Tang, H. Gao, and J. Kurths, “Distributed robust synchronization of dynamical networks with stochastic coupling,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 61, no. 5, pp. 1508–1519, 2014. [13] Y. Tang, H. Gao, W. Zou, and J. Kurths, “Distributed synchronization in networks of agent systems with nonlinearities and random switchings,” IEEE Trans. Cybern., vol. 43, pp. 358–370, 2013. [14] Y. Sun, G. Feng, and J. Cao, “Stochastic stability of Markovian switching genetic regulatory networks,” Phys. Lett. A, vol. 373, no. 18–19, pp. 1646–1652, 2009. [15] C. Li, L. Chen, and K. Aihara, “Stability of genetic networks with sum regulatory logic: Lure system and LMI approach,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 53, no. 11, pp. 2451–2458, Nov. 2006. [16] Z. Wang, H. Gao, J. Cao, and X. Liu, “On delayed genetic regulatory networks with polytopic uncertainties: Robust stability analysis,” IEEE Trans. NanoBiosci., vol. 7, no. 2, pp. 154–163, Jun. 2008. [17] W. Zhang, Y. Tang, J. Fang, and X. Wu, “Stochastic stability of genetic regulatory networks with a finite set delay characterization,” Chaos, vol. 22, p. 023106, 2012. [18] Z. Wang, J. Lam, G. Wei, K. Fraser, and X. Liu, “Filtering for nonlinear genetic regulatory networks with stochastic disturbances,” IEEE Trans. Autom. Control, vol. 23, no. 10, pp. 2448–2457, Nov. 2008. [19] F. Wei, T. Jie, and Z. Ping, “Stability analysis of switched stochastic systems,” Automatica, vol. 47, no. 1, pp. 148–157, 2011. [20] Z.-H. Guan, D. J. Hill, and X. Shen, “On hybrid impulsive and switching systems and application to nonlinear control,” IEEE Trans. Autom. Control, vol. 50, no. 7, pp. 1058–1062, Jul. 2005. [21] A. Halaney, Differential Equations: Stability, Oscillations, Time Lags. New York: Academic, 1966. [22] Z. Wang, X. Liao, S. Guo, and G. Liu, “Stability analysis of genetic regulatory network with time delays and parameter uncertainties,” IET Control Theory Appl., vol. 4, no. 10, pp. 2018–2028, Oct. 2010. [23] J. Lu, D. W. C. Ho, J. Cao, and J. Kurths, “Exponential synchronization of linearly coupled neural networks with impulsive disturbances,” IEEE Trans. Neural Netw., vol. 22, no. 2, pp. 169–175, Feb. 2011. [24] Z. Yang and D. Xu, “Stability analysis of delay neural networks with impulsive effects,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 22, no. 8, pp. 517–521, Aug. 2005. [25] L. Zhang and H. Gao, “Asynchronously switched control of switched linear system with average dwell time,” Automatica, vol. 46, no. 5, pp. 953–958, 2010. [26] H. Lin and P. J. Antsaklis, “Stability and stabilizability of switched linear system: A survey of recent results,” IEEE Trans. Autom. Control, vol. 54, no. 2, pp. 308–322, Feb. 2009. [27] F. Wei and J. Zhang, “Stability analysis and stabilization control of multi-variable switched stochastic systems,” Automatica, vol. 42, no. 1, pp. 169–176, 2006.
Wenbing Zhang received the M.S. degree in applied mathematics from the YangZhou University, JiangSu, China, and the Ph.D. degree on pattern recognition and intelligence systems from the DongHua University, China, in 2009 and 2012, respectively. From July 2012 to June 2013, he was a Research Associate in The Hong Kong Polytechnic University, Hung Hom Kowloon, Hong Kong, China. His main research interests are synchronization/consensus, networked control system, genetic regulatory networks. He is a very active reviewer for many international journals.
Yang Tang (M’11) received the B.S. and the Ph.D. degrees in electrical engineering from Donghua University, Shanghai, China, in 2006 and 2010, respectively. He was a Research Associate with the Hong Kong Polytechnic University, Kowloon, Hong Kong, from 2008 to 2010. He was an Alexander von Humboldt Research Fellow with the Humboldt University of Berlin, Berlin, Germany, from 2011 to 2013. He was a Visiting Research Fellow with Brunel University, London, U.K., in 2012. He has been a Research Scientist with the Potsdam Institute for Climate Impact Research, Potsdam, Germany, and a Visiting Research Scientist with the Humboldt University of Berlin, since 2013. He has published more than 30 refereed papers in international journals. His current research interests include synchronization/consensus, networked control systems, evolutionary computation, and bioinformatics and their applications. Dr. Tang is an Associate Editor of Neurocomputing and a Leader Guest Editor of the Journal of the Franklin Institute on “Synchronizability, Controllability and Observability of Networked Multi-Agent System.”
Xiaotai Wu received the B.S. degree in 2003 from Fuyang Normal College, Anhui, China, and the M.S. degree in 2006 from Jiangsu University, Zhenjiang, China, both in mathematics, and the Ph.D. degree in 2012 from Donghua University, Shanghai, China. His research interests include stability of stochastic systems, and control of stochastic systems.
Jian-an Fang received the B.S., M.S., and Ph.D. degrees in electrical engineering from Donghua University, Shanghai, China, in 1988, 1991, and 1994, respectively. He has been a Professor with Donghua University, Shanghai, China, since 2001. Prof. Fang was elected a Council Member of Shanghai Automation Association and a Council Member of Shanghai Microcomputer Applications, in 2005 and 2006, respectively.
IEEE TRANSACTIONS ON NANOBIOSCIENCE, VOL. 13, NO. 3, SEPTEMBER 2014
Stochastic Stability of Switched Genetic Regulatory Networks With Time-Varying Delays Wenbing Zhang, Yang Tang , Member, IEEE, Xiaotai Wu, and Jian-An Fang
Abstract—This paper investigates the exponential stability problem of switched stochastic genetic regulatory networks (GRNs) with time-varying delays. Two types of switched systems are studied respectively: one is the stochastic switched delayed GRNs with only stable subsystems and the other is the stochastic switched delayed GRNs with both stable and unstable subsystems. By using switching analysis techniques and the modified Halanay differential inequality, new criteria are developed for the exponential stability of switched stochastic GRNs with time-varying delays. Finally, an example is given to illustrate the main results. Index Terms—Exponential stability, stable subsystems, switched genetic regulatory networks, unstable subsystems.
I. INTRODUCTION ECENTLY, DUE to the potential significance in biological engineering applications, genetic regulatory networks (GRNs) have received increasing attention from biological and biomedical sciences [1]–[3], when modeling GRNs, time delays are ubiquitous due to the finite speed in the slow process of transcription, translation and translocation. Time delays in GRNs may cause instability and oscillation [4]. Therefore, It is of great importance to model GRNs with time delays. On the other hand, gene regulation process is an intrinsically noisy process due to intracellular and extracellular noise perturbations and environmental fluctuations [5]. Hence, it is necessary to investigate the effects of time delays and stochastic disturbances on the stability of GRNs. Many real world genetic regulatory networks intrinsically display unstable [6] or chaotic behavior [7]. For instance, though microarrays allow measurement of thousands of genes simultaneous, gene expressions in practice can be gathered only over a few time points due to high cost and time involved, and limitations of the experiments, which makes building GRNs inherently an ill-posed problem in practice, leading such networks unstable and irreproducible [6], and highly
R
Manuscript received December 04, 2012; revised June 22, 2013; accepted May 23, 2014. Date of publication June 02, 2014; date of current version September 23, 2014. This paper was supported by the National Natural Science Foundation of China (61203235), the Key Creative Project of Shanghai Education Community (Grant No. 13ZZ050), the Key Foundation Project of Shanghai (Grant No. 12JC1400400), the Natural Science Foundation of the Higher Education Institutions of Jiangsu Province, China (Grant No.14KJB120014), and the Alexander von Humboldt Foundation of Germany. Asterisk indicates corresponding author. W. Zhang is with the Department of Mathematics, Yangzhou University, Jiangsu, China (e-mail: [email protected]). *Y. Tang is with the Potsdam Institute for Climate Impact Research, Potsdam 14473, Germany, and also with the Institute of Physics, Humboldt University of Berlin, Berlin 12489, Germany (e-mail: [email protected]). X. Wu is with the Department of Mathematics, Anhui Polytechnic University, Wuhu, Anhui 241000, China. J. Fang is with the School of Information Science and Technology, Donghua University, Shanghai 201620, China. Digital Object Identifier 10.1109/TNB.2014.2327582
nonlinear dynamics exhibited by genetic regulatory systems can be characterized into two broad regimes, to wit, an ordered regime where the system is robust against perturbations, and a chaotic regime where the system is extremely sensitive to perturbations [7]. Moreover, the existence of time delay could be pivotal in inducing oscillations in gene regulation [8], which means that the systems are unstable as well. Obviously, the model constructed by both stable and unstable subsystems can generalize the model constructed by only stable subsystems. Therefore, the model constructed by both stable and unstable subsystems can approximate the real-world genetic regulatory networks more accurately than that constructed by only stable subsystems. Overall, when modeling the real-world delayed genetic regulatory networks, it is of great importance to take into account both stable and unstable subsystems. Recently, the stability of switched systems has received increasing attention, such as switched GRNs [9] and switched linear systems [10], [11]. However, in [9], the authors only considered switched GRNs with stable subsystems, and the unstable subsystems were overlooked. Therefore the method used in [9] failed to deal with the stability of GRNs with both stable and unstable subsystems. Moreover, the results from switched systems [10]–[13] failed to analyze the stability of switched delayed systems due to the existence of delay. Hence, it is of practical importance to model GRNs with both stable and unstable subsystems, which are more general than GRNs with only stable subsystems. In addition to the difficulty of dealing with the stability problem of GRNs with stable and unstable subsystems, time-delays should be also taken into account to model switched GRNs, which will inevitably render difficulty to analyze such comprehensive systems. However, to our best knowledge, the stochastic stability problem of switched GRNs with both stable and unstable subsystems has received little research attention, not to mention the case of time delays are included, primarily due to the mathematical complexity in derivation, which motivates us to shorten such a gap. Motivated by the above discussion, we investigate the exponential stability of switched delayed GRNs with both stable and unstable subsystems. The contributions of this paper are listed as follows: 1) the stability problem is investigated for stochastic switched delayed GRNs with both stable and unstable subsystems, which encompasses some recently GRNs models as special cases; 2) the obtained results are developed through the modified Halanay differential inequality and piecewise Lyapunov function, which are easy to verify. Notations: Throughout this paper, and denote, respectively the set of nonnegative integers and positive integers. The vector norm is defined as . For matrix , where represents the largest eigenvalue. denotes the matrix inverse.
1536-1241 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
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Moreover, for real symmetric matrices and , the notation (respectively, ) means that the matrix is negative semidefinite (respectively, negative definite). For , we say that a function from to is piecewise continuous if the function has at and most a finite number of jumps discontinuous on are continuous from the right for all points in . Given denotes the family of piecewise to with the norm continuous functions from . Let be the family of
-adapted and -valued random variables such that , where denotes the expectation operator. is infinitesimal operator which is defined as in [14]. II. MODEL FORMULATION AND SOME PRELIMINARIES
where satisfies the following inequality:
(5) for all . Thus, we know that sector condition
where and denote respectively, the concentrations of mRNA and the proteins of the th gene at time and are the degradation rates of mRNA and prorepresents the translation tein of the th gene, respectively; rate. is a monotonically increasing function denoting the feedback regulation of the protein on the transcription which is generally expressed by a monotonic function of the Hill form: , where is the Hill coefficient and is a positive constant, is the base tranis the scriptional rates of the repressor of gene and coupling matrix of the GRNs (1), which is defined as follows:
The two positive functions and denote respectively, the translation time delays and the feedback regulation delays and satisfy the following conditions: (2) The GRNs model in (1) can be rewritten in the following compact matrix form: (3) where and
satisfies the following
(6) where . Taking into account the parameter fluctuations and the intracellular noises perturbations [17], [18], the stochastic switched GRNs model is considered as follows:
In this section, some preliminaries including model formulation, lemmas and definitions are given. Consider the following delayed GRNs with SUM regulatory logic consisting of mRNAs and proteins [15], [16]: (1)
. Here, we assume
(7)
where and are mutually uncorrelated one-dimensional Brownian motions, and are the noise intensity funcis a piecewise contions. stant function, continuous from the right, specifying the index of the active subsystem. For simplicity, can be written for . The time seas quence satisfies and , where is the initial time and we assume that there is no switching at . For the purpose of simplicity, we mark . The initial conditions of the GRNs (7) are assumed to be
where . Remark 1: Generally, two kinds of switched systems have been investigated in the literature, i.e., the switched system with stable subsystems [10] and the switched systems with both stable and unstable subsystems [19]. Obviously, the switched systems with both stable and unstable subsystems are more practical and general than the ones with only stable subsystems. Therefore, recent works have been devoted to the stability problem of switched systems with both stable and unstable subsystems [19]. Unfortunately, time-delays are neglected in the above results due to the mathematical difficulty. It should be mentioned that our results are not only restricted to GRNs, but also can be extended to usual switched delayed systems with both stable and unstable subsystems. The following assumptions, definitions and lemmas will be used in the proof of the main results. Assumption 1: The noise intensity functions in (7) satisfy the following condition
. Let and be an equilibrium of (3). Shifting the equilibrium point of the and system (3) to the origin by letting , we have (4)
for all
, where
are positive constants.
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Assumption 2: There exists a constant such that , for all . Definition 1: The stochastic GRNs (7) are said to be exponentially stable in mean square if there exist positive constants and , such that (8) holds for all
and any initial data . Then and are called decay coefficient and decay rate respectively. Lemma 1: Let is a diagonal positive definite matrix with appropriate dimensional, then the following inequality holds
. Then we have . Thus, we have (16) holds. And
When when holds. Therefore, we have and
from these inequalities. In view of (11) , we have
(9) Lemma 2 ([20]): If nite matrix and
is a symmetric positive defiis a symmetric matrix, then (17) (10)
which contracts (16), and hence (15) holds. Let get
, we can (18)
III. MAIN RESULTS In this section, we develop some criteria for the stability of the switched GRNs (7) by utilizing the modified Halanay differential inequality and switched analysis techniques. Before presenting the main results, we first establish two theorems, which will be used to derive the main results. Theorem 1: Let be a continuous, and nonnegative func. Suppose and satisfies tion on the following inequality (11) Then
Thus, the proof is completed. Remark 2: Recently, the Halanay differential inequality [21] has been widely used to investigated the stability or synchronization problem of various network systems, e.g., GRNs [22], impulsive coupled neural networks [23] and impulsive neural networks [24]. Unfortunately, the famous Halanay differential inequality [21] cannot be directly applied to stochastic systems. Theorem 1 extends the Halanay differential inequality for stochastic systems and therefore it will be used to prove the stability of stochastic switched delayed GRNs. Theorem 2: Assume that there exist positive constants and such that
(12)
(19)
, , there
(20) (21)
where
is the unique solution of the equation . . Note that Proof: Let we have . Since and exists a unique such that
(22)
(13) When
where
, it is clear that (14)
Denote for any
. In the following, we will prove that
and . is the unique solu. tion of equation Then the trivial solution of system (7) is exponentially stable in mean square. Proof: Note that . By (20), we have for
(15) Suppose that (15) is not true, then there exists some such that . Set
, (23)
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where
From Theorem 1, we have for (24) Since
and from (24), we have
is the unique positive solution of the equation . Then the GRNs (7) are exponentially stable in mean square. Proof: Consider the following Lyapunov functional:
(25)
(26) For any Then, we have
, it is true that
Taking the derivative of we have for
along the trajectory of system (7),
. (27)
Hence, for any
(37)
, one can get from (26) that In view of Lemma 1 and Lemma 2, one can obtain (28)
Then, from (25) and (28), we can get (38)
(29) Then, for
Form Assumption 1, we have
, we have
(30) By a simple induction, we can obtain for any
(39) From (37)–(39), we have (31)
where . Then, we can conclude from (8) that the GRNs (7) are exponentially stable in mean square. This completes the proof. Now we are in a position to investigate the stability problem of switched GRNs in (7) with time-varying delays and stochastic perturbations. We will derive the stability criteria for the stochastic switched GRNs (7) based on Theorem 2. Theorem 3: Consider the stochastic switched GRNs (7), if for any , there exist positive definite matrices , positive constants and diagonal positive definite matrices such that
(32) (33) (34) (35) (36)
(40) From Theorem 2, we can conclude that the switched delayed GRNs (7) are exponentially stable in mean square. This completes the proof of Theorem 3. Remark 3: Inequalities (32) and (33) in Theorem 3 imply that all the subsystems of the switched GRNs (7) are exponentially stable. Therefore, Theorem 3 is only suitable to the case that (7) is composed of stable subsystems. However, as shown in theoretical and biological results [6], [11], [25], [26], the unstable subsystems cannot be avoided in many applications. Hence it is desirable to consider a more general case that both unstable and stable subsystems exist in the GRNs model simultaneously. In the following theorem, we deal with the case that the switched GRNs (7) include both stable and unstable subsystems.
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In order to derive the stability conditions of switched GRNs with both stable and unstable subsystems, we need the following two lemmas. Lemma 3: Let be a continuous and nonnegative function on . Suppose that
tems. For the sake of simplicity, we denote as the set of the index of stable subsystems and denotes the set of index denote the total of unstable subsystems. Let activation time of the stable subsystems (unstable subsystems) during . Then, we need the following assumption. , such that Assumption 3: There exists a scalar the following assumption holds
(41) where
(47)
. Then (42)
, let Proof: For any , where . Then we will prove that for . Similar to the proof of Theorem 1, suppose that it is not true, then there exists some , such that
where will be defined latter. Then the following theorem can be obtained based on Theorem 1, Lemma 3 and Lemma 4. Theorem 4: Consider the stochastic delayed switched GRNs (7) with both stable and unstable subsystems, if for any , there exist positive definite matrices , positive constants and diagonal positive definite matrices such that
, we have
If
(48) where the last two inequalities are followed by the definition of . If , one gets (49) (50) Then, we always have
(51)
. Thus, from (41), we have where
where the last inequality holds due to the definition of . This . Thus , let , contradicts then the proof is completed. Then, similar to Theorem 2, the following lemma is readily available. Lemma 4: Assume that there exist positive constants and such that (43)
is the unique positive solution of the equation . Then the GRNs (7) with both stable and unstable subsystems are exponentially stable in mean square. Proof: Choosing the same Lyapunov functional as Theorem 3 and from (48), we have if , (52) From Theorem 1, we have (53) for
and
, it yields (54)
(44) (45) then for any
Then, from Lemma 3, we have (55)
, (46)
where . In the following, we will consider the stability of stochastic delayed switched GRNs with both stable and unstable subsys-
Without loss of generality, we assume that the evolution of stable subsystems and unstable switched GRNs start with subsystems, and then goes into stable subsystems, subsystems, and so on, where and are positive constants. For , we assume that there are
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stable subsystems and tween the interval . Then, if we have from Lemma 3
unstable subsystems be,
stochastic systems with both stable and unstable subsystems as well as time delays and will have wider applications than the results in [19], [25], [27]. IV. ILLUSTRATIVE EXAMPLE In this section, a simulation example is presented to illustrate the usefulness and flexibility of the developed results in this paper. It is assumed that each hybrid system here has only two subsystems, and the switching sequence is . Example 1: Reconsider (7) with
Considering Lemma 4, we have
(56) Then, using Theorem 3, we can get is taken as
(57)
which means that intensity functions are given as follows:
, . The noise
By repeating the same steps as (56) and (57), it is true that for
(58) Similarly if holds for
, we can also get (58) . Thus, in view of Assumption 3, it yields
with transcriptional time delays and . According to Theorem 4, we can get
. Let
(59) Thus we can get that , and solving (51) we have Let
Thus, it can be checked that
. .
(60) V. CONCLUSION
From (58)–(60), we have for
(61) where
. Then it yields
(62) where . From Definition 1, the switched delayed GRNs (7) with both stable and unstable subsystems are exponentially stable in mean square. This completes the proof. Remark 4: Recently, the stability of switched systems with stable and unstable subsystems has received increasing attention [19], [25], [27]. It is worth pointing out that time delays in [19], [25], [27] are ignored due to the mathematical complexity. However, in practice, time delays widely exist in various systems. Hence, Theorem 4 gives a unified framework for
In this paper, the exponential stability is investigated for a class of switched stochastic GRNs with time delays. By utilizing the dwell time approach and the modified Halanay differential inequality, the stability of stochastic switched delayed GRNs with only stable subsystems or both stable and unstable subsystems are investigated, respectively. When only stable subsystems exist in the switched GRNs, the stability of switched GRNs can be guarantied if the dwell time is larger than a derived lower bound. If both stable and unstable subsystems exist in switched GRNs, the stability condition is related to the total activation time of unstable subsystems and stable subsystems. REFERENCES [1] A. Becskei and L. Serrano, “Engineering stability in gene networks by autoregulation,” Nature, vol. 405, no. 6786, pp. 590–593, 2000. [2] T. Schlitt and A. Brazma, “Current approaches to gene regulatory network modelling,” BMC Bioinformat., vol. 8, p. s9, 2007. [3] Y. Tang, Z. Wang, H. Gao, S. Swift, and J. Kurths, “A constrained evolutionary computation method for detecting controlling regions of cortical networks,” IEEE/ACM Trans. Comput. Biol. Bioinformat., vol. 9, pp. 1569–1581, 2012. [4] L. Chen and K. Aihara, “Stability of genetic regulatory networks with time delay,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 49, no. 5, pp. 602–608, May 2002. [5] N. Maheshri and E. K. O’Shea, “Living with noisy genes: How cells function reliably with inherent variability in gene expression,” Annu. Rev. Biophys. Biomol. Struct., vol. 36, pp. 413–434, 2007.
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[6] J. C. Rajapakse and P. A. Mundra, “Stability of building gene regulatory networks with sparse autoregressive models,” BMC Bioinformat., vol. 12, p. S17, 2011. [7] I. Shmulevich, S. A. Kauffman, and M. Aldana, “Eukaryotic cells are dynamically ordered or critical but not chaotic,” PNAS, vol. 102, no. 38, pp. 13439–13444, 2005. [8] D. Bratsun, D. Volfson, L. S. Tsimring, and J. Hasty, “Delay-induced stochastic oscillations in gene regulation,” PNAS, vol. 102, no. 41, pp. 14593–14598, 2005. [9] Y. Yao, J. Liang, and J. Cao, “Stability analysis for switched genetic regulatory networks: An average dwell time approach,” J. Franklin Inst., vol. 348, no. 10, pp. 2718–2733, 2011. [10] J. Hespanha and A. S. Morse, “Stability of switched systems with average dwell time,” in Proc. 38th IEEE Conf. Decision Control, 1999, vol. 3, pp. 2655–2660. [11] G. Zhai, B. Hu, K. Yasuda, and A. N. Michel, “Stability analysis of switched systems with stable and unstable subsystems: An average dwell time approach,” Int. J. Syst. Sci., vol. 32, no. 8, pp. 1055–1061, 2001. [12] Y. Tang, H. Gao, and J. Kurths, “Distributed robust synchronization of dynamical networks with stochastic coupling,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 61, no. 5, pp. 1508–1519, 2014. [13] Y. Tang, H. Gao, W. Zou, and J. Kurths, “Distributed synchronization in networks of agent systems with nonlinearities and random switchings,” IEEE Trans. Cybern., vol. 43, pp. 358–370, 2013. [14] Y. Sun, G. Feng, and J. Cao, “Stochastic stability of Markovian switching genetic regulatory networks,” Phys. Lett. A, vol. 373, no. 18–19, pp. 1646–1652, 2009. [15] C. Li, L. Chen, and K. Aihara, “Stability of genetic networks with sum regulatory logic: Lure system and LMI approach,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 53, no. 11, pp. 2451–2458, Nov. 2006. [16] Z. Wang, H. Gao, J. Cao, and X. Liu, “On delayed genetic regulatory networks with polytopic uncertainties: Robust stability analysis,” IEEE Trans. NanoBiosci., vol. 7, no. 2, pp. 154–163, Jun. 2008. [17] W. Zhang, Y. Tang, J. Fang, and X. Wu, “Stochastic stability of genetic regulatory networks with a finite set delay characterization,” Chaos, vol. 22, p. 023106, 2012. [18] Z. Wang, J. Lam, G. Wei, K. Fraser, and X. Liu, “Filtering for nonlinear genetic regulatory networks with stochastic disturbances,” IEEE Trans. Autom. Control, vol. 23, no. 10, pp. 2448–2457, Nov. 2008. [19] F. Wei, T. Jie, and Z. Ping, “Stability analysis of switched stochastic systems,” Automatica, vol. 47, no. 1, pp. 148–157, 2011. [20] Z.-H. Guan, D. J. Hill, and X. Shen, “On hybrid impulsive and switching systems and application to nonlinear control,” IEEE Trans. Autom. Control, vol. 50, no. 7, pp. 1058–1062, Jul. 2005. [21] A. Halaney, Differential Equations: Stability, Oscillations, Time Lags. New York: Academic, 1966. [22] Z. Wang, X. Liao, S. Guo, and G. Liu, “Stability analysis of genetic regulatory network with time delays and parameter uncertainties,” IET Control Theory Appl., vol. 4, no. 10, pp. 2018–2028, Oct. 2010. [23] J. Lu, D. W. C. Ho, J. Cao, and J. Kurths, “Exponential synchronization of linearly coupled neural networks with impulsive disturbances,” IEEE Trans. Neural Netw., vol. 22, no. 2, pp. 169–175, Feb. 2011. [24] Z. Yang and D. Xu, “Stability analysis of delay neural networks with impulsive effects,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 22, no. 8, pp. 517–521, Aug. 2005. [25] L. Zhang and H. Gao, “Asynchronously switched control of switched linear system with average dwell time,” Automatica, vol. 46, no. 5, pp. 953–958, 2010. [26] H. Lin and P. J. Antsaklis, “Stability and stabilizability of switched linear system: A survey of recent results,” IEEE Trans. Autom. Control, vol. 54, no. 2, pp. 308–322, Feb. 2009. [27] F. Wei and J. Zhang, “Stability analysis and stabilization control of multi-variable switched stochastic systems,” Automatica, vol. 42, no. 1, pp. 169–176, 2006.
Wenbing Zhang received the M.S. degree in applied mathematics from the YangZhou University, JiangSu, China, and the Ph.D. degree on pattern recognition and intelligence systems from the DongHua University, China, in 2009 and 2012, respectively. From July 2012 to June 2013, he was a Research Associate in The Hong Kong Polytechnic University, Hung Hom Kowloon, Hong Kong, China. His main research interests are synchronization/consensus, networked control system, genetic regulatory networks. He is a very active reviewer for many international journals.
Yang Tang (M’11) received the B.S. and the Ph.D. degrees in electrical engineering from Donghua University, Shanghai, China, in 2006 and 2010, respectively. He was a Research Associate with the Hong Kong Polytechnic University, Kowloon, Hong Kong, from 2008 to 2010. He was an Alexander von Humboldt Research Fellow with the Humboldt University of Berlin, Berlin, Germany, from 2011 to 2013. He was a Visiting Research Fellow with Brunel University, London, U.K., in 2012. He has been a Research Scientist with the Potsdam Institute for Climate Impact Research, Potsdam, Germany, and a Visiting Research Scientist with the Humboldt University of Berlin, since 2013. He has published more than 30 refereed papers in international journals. His current research interests include synchronization/consensus, networked control systems, evolutionary computation, and bioinformatics and their applications. Dr. Tang is an Associate Editor of Neurocomputing and a Leader Guest Editor of the Journal of the Franklin Institute on “Synchronizability, Controllability and Observability of Networked Multi-Agent System.”
Xiaotai Wu received the B.S. degree in 2003 from Fuyang Normal College, Anhui, China, and the M.S. degree in 2006 from Jiangsu University, Zhenjiang, China, both in mathematics, and the Ph.D. degree in 2012 from Donghua University, Shanghai, China. His research interests include stability of stochastic systems, and control of stochastic systems.
Jian-an Fang received the B.S., M.S., and Ph.D. degrees in electrical engineering from Donghua University, Shanghai, China, in 1988, 1991, and 1994, respectively. He has been a Professor with Donghua University, Shanghai, China, since 2001. Prof. Fang was elected a Council Member of Shanghai Automation Association and a Council Member of Shanghai Microcomputer Applications, in 2005 and 2006, respectively.
