Outer Synchronization of a Modified Quorum-Sensing Network via Adaptive Control

1School of Information Science and Engineering, Linyi University, Linyi 276005, China 2Key Laboratory of Complex Systems and Intelligent Computing in Universities of Shandong (Linyi University), Linyi 276005, China 3Department of Mathematics, Southeast University, Nanjing 210096, China 4College of Science and Technology, North China Electric Power University, Baoding 071000, China 5Department of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabia


Introduction
During the past decades, it has been discovered that bacteria, such as Escherichia coli, could communicate with each other through producing and monitoring one kind of signaling molecules [1,2].The signaling molecules could diffuse into different bacteria or the environment, and the bacteria could coordinate their gene expression and activities in response to the concentration of the signaling molecules.Then, the bacteria are coupled with each other by the intercellular signaling molecules [3] and display various social behaviors such as behaving synchronously [4].Obviously, the related researches have wide application prospects in biopharmaceutical industry and human health.Now, the mechanism of bacterial communication is widely known as quorum sensing, and more and more researchers began to study collective behavior caused by quorum sensing [5,6].In this paper, we modify one of the previous network models coupled through quorum sensing and discuss a typical kind of collective behavior based on several recent methods developed in the fields of complex networks and nonlinear dynamics.
Recent years have witnessed the great development in the study of complex networks and its collective dynamics [7,8].Synchronization is one of the most typical and most extensively studied kinds of collective dynamics, which implies the stability of zero solution of the synchronization error systems.Therefore, from the point of research methods, there are two main effective theoretical methods, i.e., the famous master stability function method [9,10] and Lyapunov function method [11][12][13].The former can be employed to discuss local stability of the synchronous state, and the latter can be used to explore global stability of the synchronous state.Up to now, dozens of different types of synchronization states have been proposed such as complete synchronization [14], cluster synchronization [15,16], lag synchronization [17,18], projection synchronization [19,20], and outer synchronization [21][22][23].Thereinto, outer synchronization has attracted many researchers' interest, which describes the synchronization 2 Mathematical Problems in Engineering between two or more networks.For instances, in a model of predator-prey interactions in ecological communities, all the predators form a network system and all the preys form another, and the two networks influence one another's evolution to keep the two species in check [24].Recently, outer synchronization of the fractional order node dynamics was considered in [21], and outer synchronization under aperiodically adaptive intermittent control was considered in [22].In many cases, networks can not realize a certain expected synchronization relying on just coupling interaction between different nodes [23].Therefore, many different kinds of output control methods have been introduced, such as pinning control [25], sliding mode control [26,27], adaptive control [28,29], and state feedback control [30].Thereinto, adaptive control could be used to design controllers for systems with uncertain parameters.Due to great demands from wide applications, many researches have been carried out to investigate synchronization induced by adaptive controllers [31].It is worth pointing out that there are few researches focused on outer synchronization of networks coupled through quorum sensing.
Motivated by the above discussions, this paper investigates outer synchronization induced by adaptive controllers in quorum-sensing network.At first, we present a modified model of previous quorum-sensing network by adding noise disturbances in case there exist three unknown parameter vectors and the network topology is also unknown.Then, effective adaptive controllers are designed to realize outer synchronization, parameter estimations are designed to identify the unknown parameter vectors, and topology estimations are designed to identify unknown network topology.Based on Lyapunov function method and matrix theory, this paper proves that adaptive outer synchronization is achieved in the quorum-sensing network.To the best of our knowledge, there are few researches focused on this subject by a similar method.In our opinion, there is a certain degree of values both in theory and in practice.
The rest of this paper is organized as follows.In Section 2, the synthetic gene network model coupled through quorum sensing is introduced.In Section 3, several criteria are derived for outer synchronization including the construction of adaptive controllers and parameter estimations.In Section 4, some numerical examples are provided to illustrate the effectiveness of the obtained results.Finally, conclusions are given to summarize the contributions of the paper in Section 5.

Problem Formulation
The synthetic gene network in Escherichia coli was first proposed by Garcia-Ojalvo J et al. [3].Consider the network consisting of  cells coupled through quorum sensing.Each cell consists of two basic parts illustrated in Figure 1.The first part is composed of three genes , ,  that express their respective proteins , , , which inhibit the transcription of the three genes , , , in a cyclic way.The second part of each cell is another gene regulated by protein A, which produces a protein and synthesizes a small molecule known as an autoinducer   .The autoinducer   can diffuse freely through the cell membrane, which activates the transcription of the genes in first part.For more detailed description, the reader is referred to previous articles [32,33].Now, we introduce the quorum-sensing network model.The dynamics of each node consists of the concentrations of three genes and their respective proteins, assume that the th cell is described by the following equations: where   (),   (), and   () are the concentrations of mRNA transcribed from genes , ,  in the th cell, respectively;   (),   (), and   () are the concentrations of the corresponding proteins, respectively;   () and   () are the concentrations of the autoinducer  inside the th cell and in the environment.The concentration dynamics of the autoinducer are governed by where  = 1, 2, . . ., .In the multicell system (1)-( 2), the parameters  1 ,  2 , . . .,  7 and   are the dimensionless degradation rates of the chemical molecules;  1 ,  2 ,  3 are the translation rates of the proteins from the mRNAs;  4 ,  5 ,  6 are the dimensionless transcription rates in the absence of repressor;  7 is the synthesis rate of AI;  = 4 is the Hill coefficient;  8 is the maximal contribution to the gene c transcription in the presence of saturating amounts of ;  and   measure the diffusion rate of  inward and outward the cell membrane.With the help of the quasi-steady state approximation Ṡ  () = 0, one gets that the extracellular  concentration can be approximated as which reduces (2) to the following form: Equations ( 1)-( 4) describe the concentration state of the th cell in the synthetic gene network model coupled through quorum sensing.Motivated by the quorum-sensing network ( 1)-( 4), we build up a network with three unknown parameter vectors and unknown network topology.The state dynamics are described by the following equations: where   () = (  (),   (),   (),   (),   (),   ()) ⊤ ,  1 = ( 1 ,  2 ,  3 ,  4 ,  5 ,  6 ) ⊤ ,  2 = ( 6 ,  4 ,  5 ,  1 ,  2 ,  3 ) ⊤ , and  3 =  7 + , and the diagonal matrix functions where  = 1, 2, . . ., .To meet the demands of broad applications, the matrix  = (  ) × is a coupling matrix denoting the network topology.The matrix element   is defined as follows: if there is a connection from node  to node ( ̸ = ), then define the coupling strength as   ̸ = 0; otherwise,   = 0. Let us assume that there are three unknown parameter vectors existing in the node dynamics,  1 ,  2 , and  3 , and the network topology matrix  = (  ) × is also unknown.

Adaptive Control Schemes for Outer Synchronization
In this section, several criteria are derived for outer synchronization induced by adaptive control schemes.At first, we need to introduce the following two assumptions.
If one or two of the unknown parameter vectors in the network (5) are given constants, Theorem 3 still holds after modifying the conditions slightly.For instance, supposing that the parameter vectors  1 and  2 are given, we modify network (7) as follows: then the synthetic gene network ( 5)- (21) with controllers (11) and estimations (22) can achieve outer synchronization.
If we do not consider the disturbances Δ 1 () and Δ 2 () in network (7), Theorem 3 still holds after modifying controllers (11) slightly.Then, we obtain the following corollary.

Conclusions
This paper builds a model of quorum-sensing network with disturbances, unknown parameter vector, and network topology and investigates the problem of outer synchronization between two quorum-sensing networks.In case that some systems' parameters are unknown in actual applications, adaptive parameter updating laws are designed to estimate the true values of those unknown parameters.Similarly, updating laws are also presented for the unknown elements of the network coupling matrix.Finally, some adaptive controllers are adopted to realize outer synchronization between two quorum-sensing networks.The validity of the proposed control schemes and updating laws is demonstrated by several numerical simulations.

Figure 1 :
Figure 1: Scheme of the repressilator network coupled through signaling molecules, termed quorum sensing.The synchronization scheme of quorum sensing is based on the diffusion of autoinducers (AI) to and from the cells.

Corollary 4 .
Suppose that Assumptions 1 and 2 hold.If the parameter estimation  3 () is designed as follows: