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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 175068, 10 pages
Entrained Collective Rhythms of Multicellular Systems: Partial Impulsive Control Strategy
1School of Economics and Management, Nanjing University of Information Science ####^~^~^~^~^~^####x26; Technology, Nanjing 210044, China
2College of Energy and Electrical Engineering, Hohai University, Nanjing 210098, China
Received 26 September 2013; Accepted 13 November 2013
Academic Editor: Jinde Cao
Copyright © 2013 Lifei Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper is concerned with the study of entrained collective rhythms of multicellular systems by using partial impulsive control strategy. The objective is to design an impulsive controller based on only those partially available cell states, so that the entrained collective rhythms are guaranteed for the multicellular systems with cell-to-cell communication mechanism. By using the newly developed impulsive integrodifferential inequality, the sufficient conditions are derived to achieve the entrained collective rhythms of multicellular systems. A synthetic multicellular system with simulation results is finally given to illustrate the usefulness of the developed results.
Complex physiological rhythms are ubiquitous in living organisms, which are central to life, such as our daily cycle of waking and sleeping and the beating of our hearts. Collective rhythms are normally generated by thousands of divers clock cells which manage to function in a coherent oscillatory state [1, 2]. In fields ranging from circadian biology to endocrinology, however, it remains an exciting challenge to understand how collective rhythms emerge in multicellular structures [3####^~^~^~^~^~^####x2013;7].
Elucidating the collective dynamics of multicellular systems not only is essential for the understanding of the rhythmic phenomena of living organisms at both molecular and cellular levels but also has many potential applications in bioengineering areas. For example, in cancer chemotherapy, treatments could be based on the circadian rhythm of cell division . Over the past decade or so, many researchers have paid a great deal of attention to study the collective dynamics of multicellular systems. For instance, in [9####^~^~^~^~^~^####x2013;12], the authors considered stability of genetic networks and neural networks. In , the authors pointed out that intercell signaling mechanism does lead to synchronous behavior across a population of cells. In , after making real-time analysis of the gene expression, the authors showed the synchronized rhythms of clock gene transcription across hundreds of neurons within the mammalian suprachiasmatic nucleus (SCN) in organotypic slice culture. In addition, based on the Lyapunov stability theory, the collective rhythms of multicellular systems were further studied in . For the other relevant results, please see [15####^~^~^~^~^~^####x2013;17].
Although there are significant advances on elucidating the collective behaviors of biological organisms in recent years, the essential mechanisms from which the collective rhythms arise remain to be fully understood. It is well known that coupling among cells is not sufficient to achieve collective rhythms. In fact, the collective rhythms of multicellular systems are far away from being well understood and warrant further and insightful study.
On the other hand, experimental results have already shown that external stimuli play an important role in achieving the collective rhythms. In , physiological rhythms were induced by regular or periodic inputs occurring in the context of medical devices. In , an external voltage was applied to enhance the synchronization of electronic synthetic genetic networks. In , it was shown that a specific collective behavior could be realized by changing the frequency and amplitude of the periodic stimuli. Another well-known example is that organisms usually display a circadian rhythm, where the key processes show a 24-hour periodicity entrained to the light-dark cycle . In , the authors studied the rhythmic process of the circadian oscillators under the effect of the daily light-dark cycle. Furthermore, from the view of impulsive control systems, collective behaviors of coupled systems were investigated and some interesting results have been obtained in [23####^~^~^~^~^~^####x2013;27], and, for the other relevant results, please refer to [28####^~^~^~^~^~^####x2013;32] and the references therein.
However, in the above-mentioned results, one basic assumption is that the external stimuli are applied to all the cells in the community, that is very expensive or unrealistic in practice. Actually, in many practical medical cases, only partial specific cells could be detected and utilized. In these situations, the external stimuli are applied to only those cells in the community. To the best of our knowledge, there are few results in the open literature on the entrained collective rhythms of multicellular systems by applying impulsive control based on the partially available cell states.
This paper is to study the entrained collective rhythms of multicellular systems with only partially available cell states. By using the newly developed impulsive integrodifferential inequality, a new criterion is derived to ensure the entrained collective rhythms of multicellular systems. It is shown that when the spontaneous synchrony cannot be achieved, an appropriate periodic stimulus could achieve a collective rhythm even only with partially available cell states. It is noted that the proposed partial impulsive control method can be also easily extended to study other complex systems.
The rest of the paper is organized as follows. Section 2 formulates the problem of the entrained collective rhythms and provides some useful lemmas. Section 3 presents the main results for entrained collective rhythms of multicellular systems. A synthetic multicellular system will be employed to illustrate the effectiveness of the developed results in Section 4, which is followed by conclusions in Section 5.
2. Model Description and Problem Formulation
To make it easy for the readers, let us start from a single cell model of the form where represents the concentrations of proteins, RNAs, and other chemical complexes, is the positive diagonal matrix denoting the degradation and dilution rate, and is the complex regulatory function, which usually is of the Michaelis-Menten or Hill form.
Remark 1. It is known that many biological models can be represented by (1), such as the Goodwin model  and the toggle switch . Furthermore, the regulatory function in model (1) is usually monotonically increasing or decreasing.
Without loss of generality, the regulatory function is always assumed to satisfy the following assumption.
Assumption 2. The regulatory function in (1) satisfies
for all , and .
Consider multicellular systems with cell-to-cell communication mechanism described as follows: where is the state of the th cell, denoting the concentrations of chemical complexes in this cell, and is the total cell number of the entire community. The third term in model (3) describes the capability of cells to communicate with each other in order to coordinate the behavior of the entire community. is the coupling structure matrix that represents the communications between different cells, and is the inner coupling structure that represents the connections of different chemical complexes in one cell. satisfies the diffusive coupling condition It can be noted that such coupling is biologically plausible in many biological systems, such as the quorum sensing mechanism in bacteria [2, 35].
Suppose only cell states in the community are measurable for the multicellular systems (3). Consider the following linear impulsive controller based on those measurable cell states: where is the state of the isolated cell described in (1), is the number of the measurable cell states, is the gain matrix, and is the Dirac impulse function with discontinuity points .
Then the impulsive-controlled multicellular systems with partial states can be described by the following impulsive differential equation:
Defining , one can obtain the following error system:
Then the problem of entrained collective rhythms is to design the partial impulsive controller (5) such that the stability of the error system (7) is guaranteed. Before presenting the main results, some useful lemmas are introduced in advance.
Lemma 3 (see ). If is a positive definite matrix and is a symmetric matrix, then where and are the minimum and maximum eigenvalues of the matrix, respectively.
Lemma 4. For positive scalars ,####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;,####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;and####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;,####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;if####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;satisfies where is a continuous function, then one has for , where is the solution to the following impulsive integrodifferential equation:
Proof. Firstly, we prove
If argument (11) is not right, then there exists such that
Considering the continuity of , on , there must exist such that
then it yields
which contradicts the condition , so (11) holds.
Suppose , for all ; then by , similarly, one has , for all . By using the mathematical induction method, one can conclude , for all for any positive integer . The proof is thus complete.
Lemma 5 (Grownwall-Bellman Inequality ). Let be a real value continuous function and a nonnegative continuous function on . If a continuous function has the property that then on one has
Definition 6. The multicellular system (6) is said to achieve collective rhythms with the designed partial impulsive controller, if there exist scalars and such that where and is the initial condition.
3. Main Results
In this section, by using the proposed impulsive integrodifferential inequality, a sufficient condition guaranteeing the entrained collective rhythms of multicellular systems is derived.
Theorem 7. For a given scalar , if there exist matrices , , , scalars , , , , and positive scalars , , , and such that for , and for , and, for any impulsive time sequence satisfying for , where , , , and , , then the entrained collective rhythms of multicellular systems (6) are achieved.
Proof. Consider the following Lyapunov function:
where and are positive definite matrices to be determined.
For any , , taking the Dini derivative along the trajectories of (7), we have It follows from Assumption 2 that where , and . One also has Substituting (25)####^~^~^~^~^~^####x2013;(26) into (24) yields Define . Then it follows from condition (18) and Lemma 3 that where and .
Furthermore, for any , one can also get where , .
Defining it follows from (20) and (29) that where and , which implies Then substituting (31) into (28) yields On the other hand, when , it follows from (19) that where , .
For any scalar , define the following impulsive integrodifferential equation: It then follows from Lemma 4 that , for all .
The solution to (34) can be expressed as follows: where is the Cauchy matrix of the linear impulsive system
Furthermore, noting that and , it follows from condition (21) that could be estimated as Defining , one has Noting from condition (21) that , then one gets Defining , and , one obtains Then by using Lemma 5, it is easy to get which implies Let , one can get Then it follows from (31) that Furthermore, it follows from condition (21) that . Therefore together with (43) and (44), one can conclude that condition (17) is satisfied; that is, the entrained collective rhythms of multicellular systems (6) are achieved. The proof is thus completed.
Remark 8. The obtained result not only provides a new prospective to understand the interactions between the external stimuli and intrinsic physiological rhythms but also is potentially useful for the development of some medical devices. The result presented here is more effective in comparison with those in [38####^~^~^~^~^~^####x2013;41], where it is assumed that all the states are available for feedback purpose.
As a special case, if the positive matrices and in the Lyapunov function (22) are chosen as the identity matrix, the following simplified result could be readily obtained.
Corollary 9. For a given scalar , if there exist scalars ,####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;,####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;,####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;, and positive scalars ,####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;,####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;,####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;and such that for and for , and, for any impulsive time sequence satisfying for , where ,####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;,####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;, and ,####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;, then the entrained collective rhythms of multicellular systems (6) are achieved.
4. Numerical Example
In this section, a synthetic multicellular system composed of Goodwin oscillators  is employed to illustrate the effectiveness of the proposed control strategy.
4.1. Goodwin Oscillators
The Goodwin oscillator is a genetic network with negative feedback formed in a cyclic way , where metabolites repress the enzymes which are essential for their own synthesis by inhibiting the transcription of the molecule DNA to messenger RNA (mRNA) (see  for more details). The regulatory scheme of the Goodwin oscillator can be shown in Figure 1.
A modified model reflecting the above regulation process is given by where , , and are the concentrations of mRNA, the enzyme, and the product of the reaction of the enzyme and a substrate, respectively. , , and are the degradation rates of each component, respectively. , , and denote the rates of transcription, translation, and catalysis, respectively. and are two positive constants. is the Hill coefficient denoting the cooperativity of the end product repression. It should be pointed out that model (46) is slightly different from the model in . By changing the linear degradation form of the product to Michaelis-Menten form, the limit cycle oscillations can occur for a lower value of the Hill coefficient (see  for details).
The parameters are chosen as , , and , and the Hill coefficient is taken as . Given 4 random initial conditions, the simulation results are recorded in Figures 2 and 3. Figure 2 shows the time response of every products, and Figure 3 shows the limit cycle of the oscillators in phase space.
4.2. Multicellular System Model
Without loss of generality, we perform the simulation study on the scale-free network structure, which is assumed to obey the scale-free distribution of the Barab####^~^~^~^~^~^####xe1;si-Albert (BA) model . The parameters are given as and . Figure 4 is the generated BA network graph.
The inner coupling matrix is given by , and the regulation function in Michaelis-Menten form satisfies condition (2). Assume that only the first nodes are able to accept the external inputs. The parameters are chosen as , , and . The impulse magnitude is given as , and impulse interval is chosen as . It follows from Corollary 9 that the entrained collective behaviors can be achieved. Figures 5, 6, and 7 show the rhythmic errors of the multicellular systems.
It is noted that since only 20 measurable cells in the network are subject to the external stimuli, the approaches in [38####^~^~^~^~^~^####x2013;41], which need the information of all the states, cannot be applied in this case.
In this paper, the entrained collective rhythms of multicellular systems have been investigated. It is shown that the entrained collective behavior can be achieved via impulsive control even when only partial states of multicellular systems are available. With the help of the newly developed impulsive integrodifferential inequality, the sufficient conditions are derived to ensure the entrained collective rhythms of multicellular systems. A synthetic multicellular system is finally used to illustrate the effectiveness of the developed impulsive control strategy.
The work was supported in part by the National Natural Science Foundation of China under Grant 61104045, in part by Technology Foundation for Selected Overseas Chinese Scholar, Ministry of Personnel of China, and in part by the Fundamental Research Funds for the Central Universities of China under Grant 2012B03514.
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