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The paper investigates quasisynchronization in a communication system, which consists of cells communicating through quorum sensing. With the help of Lyapunov function method and Lur’e system approach, some sufficient conditions for quasisynchronization are presented, and a bound on the synchronization errors is derived. The obtained theoretical results show that the synchronization quality is influenced by two parameters detrimentally: the error bound depends almost linearly on the mismatches between cells and depends sensitively on the diffusion rates of the signals inward the cell membrane. Numerical experiments are carried out to verify the theoretical results.

In the past decades, increasing interest has been shown to study the dynamics of coupled oscillator networks, which describes many complex systems in the field of nature and science. Due to the couplings among these oscillators, different types of synchronization can be realized in such systems.

In particular, a coupled oscillator network is called to be quasisynchronized, or weak synchronized, if the synchronization errors will be in some neighborhood of zero but will not tend to zero eventually. In other words, quasisynchronization means that the dynamical trajectories of each cell are similar but different from each other [

To the best of our knowledge, most of researches on synchronization focused on two oscillators coupled directly [

Due to the biological diversity, there are usually some parameter mismatches between the coupled oscillators in biological systems. Therefore, complete synchronization is hard to be achieved. Instead, quasisynchronization, which implies a state of synchronization with an error level, is more common in the biological systems. Motivated by the complexity and similarity of biological organisms and the potential applications, the paper studies quasisynchronization in quorum sensing systems with parameter mismatches. Through a new method different from many previous researches [

The rest of the paper is organized as follows. In Section

Quorum sensing is a cell concentration dependent phenomenon in bacteria and fungi, which is mediated by small, diffusible signaling molecules that accumulate in the extracellular environment [

Scheme of the repressilator network coupled through signaling molecules, termed quorum sensing. The synchronization scheme of quorum sensing is based on the diffusion of a small molecule (autoinducer AI) to and from the cells.

Before we carry out the dynamics model for the

Then the dynamics model for the

We suppose that all the parameters mentioned above are time-varying in the vicinity of certain constants, which indicates that all the cells are similar but different from each other. For convenience, we decompose these parameters into two parts: a constant part that determines the values of the parameters and a time-varying part representing the parameter mismatches. For example, let the parameter

Denoting

Since the cells in realistic organisms are similar but different from each other, one can suppose that the mismatch matrices

Since there exist parameter mismatches between different cells, multicell system (

The multicell system (

The regulatory functions

If there exist symmetric matrices

Define a Lyapunov function with respect to multicell system (

Based on Lyapunov direct method, if the time derivative of

If matrix inequalities (

Theorem

In fact, the proof of the theorem shows that the mismatches that go against synchronization could be compensated by the linear function

As a special case, if there are no parameter mismatches between different cells, which implies that all the cells are identical, then it is easy to conclude that multicell system (

If there exist symmetric matrices

In order to demonstrate the effectiveness of our theoretical analysis, we give numerical examples based on multicell system (

Time evolutions of the synchronization errors

In order to verify the relationship between the synchronization errors and the mismatches between cells, the figure of the synchronization errors transition for increasing the mismatches

Synchronization errors transition for increasing the mismatch bound

Numerical simulations are also carried out to verify the relationship between the synchronization errors and the diffusion rate of the signals inward the cell membrane in the region

Synchronization errors transition for increasing the parameter

From Figures

Many previous researches studied the collective behavior of biological systems by using complete synchronization. However, in many real biological systems, individual organisms are similar but different from each other, and their behaviors are not completely identical either. Therefore, it is meaningful to carry out researches on quasisynchronization instead of complete synchronization. Our results on quasisynchronization in multicell systems coupled by quorum sensing indicate that mismatches between cells can lead to quasisynchronization, and the synchronization errors depend heavily on the parameter mismatches. Theoretical analysis shows that the synchronization errors will decrease if the coupling strength increases. All the results agree well with numerical simulations and biological phenomena in practice.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This research was supported by Zhejiang Provincial NSF of China (nos. LQ12A01003 and LQ12A01002), NNSF of China (nos. 61203155, and 11001069), Guangxi Key Laboratory of Trusted Software (no. kx201417), and the Science Foundation of Guangxi Province (no. 2013GXNSFAA019006).