Bioinformatics Advance Access originally published online on March 17, 2005
Bioinformatics 2005 21(11):2722-2729; doi:10.1093/bioinformatics/bti392
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Noise-induced cooperative behavior in a multicell system
1Department of Electrical Engineering and Electronics, Osaka Sangyo University Daito, Osaka 574-8530, Japan
2Department of Mathematical Sciences, Tsinghua University Beijing 100084, China
3Institute of Industrial Science, The University of Tokyo Komaba 4-6-1, Meguro, Tokyo 153-8505, Japan
4ERATO Aihara Complexity Modelling Project, JST Oyama 45-18, Shibuya-ku, Tokyo 151-0065, Japan
*To whom correspondence should be addressed.
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Abstract |
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 TOP Abstract 1 INTRODUCTION2 GENE REGULATORY NETWORK...3 IMPLEMENTATION BY A...4 DISCUSSIONREFERENCES |
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Motivation: All cell components exhibit intracellular noise on account of random births and deaths of individual molecules, and extracellular noise because of environment perturbations. Gene regulation in particular, is an inherently noisy process with transcriptional control, alternative splicing, translation, diffusion and chemical modification reactions, all of which involve stochastic fluctuations. Such stochastic noises may not only affect the dynamics of the entire system but may also be exploited by living organisms to actively facilitate certain functions, such as cooperative behavior and communication.
Results: We have provided a general model and an analytic tool to examine the cooperative behavior of a multicell system with both intracellular and extracellular stochastic fluctuations. A multicell system with a synthetic gene network is adopted to demonstrate the effects of noises and coupling on collective dynamics. These results establish not only a theoretical foundation but also a quantitative basis for understanding essential roles of noises on cooperative dynamics, such as synchronization and communication among cells.
Availability:
Contact: chen{at}elec.osaka-sandai.ac.jp
Supplementary information: http://www.sat.t.u-tokyo.ac.jp/research.html or http://www.elec.osaka-sandai.ac.jp/lab/chen/support.html
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1 INTRODUCTION |
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TOPAbstract1 INTRODUCTION2 GENE REGULATORY NETWORK...3 IMPLEMENTATION BY A...4 DISCUSSIONREFERENCES |
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Cooperative behaviors are essential coordinated responses resulting from an integrated exchange of information by cell communication in both prokaryotes and eukaryotes (Perbal, 2003). For multicellular organisms, complex patterned structures can be created through cell to cell communication from identical and unreliable components, whereas bacteria display various social behaviors and cellular differentiations, such as quorum sensing in Gram-positive and Gram-negative strains, owing to intercellular communications (Weiss and Knight, 2000; Taga and Bassler, 2003). Generally, cooperative behavior, such as intercellular communication is accomplished by transmitting individual cell reactions via intercellular signals to neighboring cells and further integrating to generate a global cellular response at the level of molecules, tissues, organs and a body. The ability to communicate between cells is an absolute requisite to ensure appropriate and robust coordination of the cell activities at all levels of organisms under an uncertain environment. However, all cell components exhibit intracellular noises owing to random births and deaths of individual molecules, and extracellular noises owing to environment fluctuations (Elowitz et al., 2002; Paulsson, 2004). Gene regulation in particular, is an inherently noisy process with transcriptional control, alternative splicing, translation, diffusion and chemical modification reactions of transcriptional factors, all of which involve stochastic fluctuations owing to low copy numbers of many molecules per cell and uncertainty of an external environment. Such stochastic noises may not only affect the dynamics of the entire system but may also be exploited by living organisms to actively facilitate certain functions, such as synchronization and communication.
To understand the basic mechanism of cell–cell cooperative or collective dynamics, a few theoretical models have been successfully developed so far by studying both natural and synthetic gene networks (Weiss and Knight, 2000; Taga and Bassler, 2003; McMillen et al., 2002; Bulter et al., 2004). By ignoring the effects of noises and diffusion in particular, McMillen et al. (2002) proposed a synchronization scheme based on the mechanism of relaxation oscillation or ‘fast threshold modulation’ in a synthetic system, which fulfills the communication by producing and responding to a small intercell signaling molecule known as autoinducer (AI) in the quorum sensing apparatus of the marine bacterium, Vibrio fischeri.
In this paper, we aim to develop a general model on collective dynamics in a multicellular system, by further considering the effects of stochastic fluctuations and signal diffusion processes in gene regulatory networks. Specifically, we first present a theoretical framework to equivalently transform a stochastic multicell system into a set of coupled mean field equations, which in turn are converted to a deterministic system with the assumption of the Gaussian distributions. Sufficient conditions for stochastic synchronization are also obtained based on ‘Hopf bifurcation’. Then we design and construct a synthetic gene regulatory network by using an operon with genes luxR and luxI and promoter placLux0. Intercellular communication is facilitated between cells by diffusing a small protein, AI produced by protein LuxI, that results in cooperative behaviors of all cells with stochastic perturbations. From the viewpoint of stochastic dynamics, such cooperative dynamics or synchronization is an in-phase behavior of the probability distributions for all cells. In contrast to the non-cooperative effect of intracellular noises, as shown both analytically and numerically in this paper, extracellular noises, common to all cells, actually enhance signaling interchange and can induce a collective behavior. Different from coupling that play a dominant role to coordinate the synchronization, noises act rather as a compensating force to induce the synchronous oscillation and also constantly perturb the system in an unpredicted manner. The results of this paper establish a theoretical foundation that can be directly applied to model, analyze or even predict cooperative behaviors of coupled biosystems at the level of molecules. The numerical simulations also confirm that interplay between noises and coupling induces regular or cooperative dynamics and actively mediates synchrony of the multicell system. In the Appendices of the Supplementary material, we give detailed derivation of both the master and cumulant equations by describing the slow and spatially well-mixed diffusive signals with the time-delayed mean field. In addition, a modified Gillespie algorithm is also provided for the multicell system with mean field variables and time delays.
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2 GENE REGULATORY NETWORK WITH STOCHASTIC NOISES |
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TOPAbstract1 INTRODUCTION2 GENE REGULATORY NETWORK...3 IMPLEMENTATION BY A...4 DISCUSSIONREFERENCES |
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On account of the low copy numbers for many molecules in living cells, the origin of stochasticity in a cell can be traced to random transitions among the discrete chemical states. Generally, the master equation is adopted to represent such random and discrete nature of biochemical reactions (Van Kampen, 1992), which inherently exists in all life processes. As derived in Appendix A of Supplementary material, with appropriate approximation to the master equation, the gene regulatory network in a cell can be expressed by a set of Langevin equations or stochastic differential equations as follows:
 | (1) |
= (1,...,m) is Gaussian noises with zero means and covariances Kij, i.e. 
i(t)
= 0 and i(t)j(t') = Kij(x(t))
(t – t'). f = (f1,...,fm) is the reaction rates of molecules and {Kij} is a m x m matrix. Gaussian noises are called intracellular noises inside the cell and derived from the random and discrete chemical reactions.
Assume that there are n identical cells, which are coupled with a common environment. Then, the corresponding Langevin equations of the entire system are for k = 1,...,n,
 | (2) |
called cellular noises includes both intracellular and extracellular noises, and are assumed as independent Gaussian white noises with zero means and covariances 
, i.e. 
and 
, where Kij and 
are the intracellular and the coupling covariances, respectively, and 
represents the extracellular covariance with dij = 
ij = 0 for i 
j. In particular, extracellular noises that are common to all cells represent external noises originating from outside of the cells owing to environment perturbations, such as temperature and culture fluctuations. The m x m cellular noise covariance matrix {Kij} and Gaussian noises 
k can be derived from the random and discrete chemical reactions (Van Kampen, 1992; Risken, 1989), as shown in Appendix A of Supplementary material. Notice that dii = 0 if 
is not a coupling variable. Obviously, there are three different types of noises in our model, i.e. intracellular noises Kij derived from the master equation on account of the low copy numbers of molecules inside a cell, coupling noises on account of the random fluctuations of coupling, and extracellular noises originating from the environment perturbations, such as temperature fluctuations and imperfect culture mixing.
Assuming that the system is well mixed but with the delayed effect of molecular diffusion, by ignoring degradation processes and transient mixing dynamics of y in the extracellular environment (Dockery and Keener, 2001; Garcia-Ojalvo et al., 2004; McMillen et al., 2002; Wang and Chen, 2005), y can be approximated by 
with the consideration of the accumulative effect for time lag of signal molecules between the environment and each cell (Rosenblum and Pikovsky, 2004; Tsimring and Pikovsky, 2001; Huber and Tsimring, 2003; Wang and Chen, 2005), where 
and 
j represents the extracellular feedback time delay of xj owing to diffusion or transport processes of the molecule. When there are a sufficiently large number of cells, i.e. n 
, y approaches the mean field of x, i.e. y = x(t – ), where x(t – ) is the average of x(t – ) over time (Rosenblum and Pikovsky, 2004; Tsimring and Pikovsky, 2001; Hubor and Tsimring, 2003) and represents the time-delayed feedback effects. Next, we define N(t) 
x(t). Therefore, in the case of n , systems for all the cells have an identical evolution given by the non-linear stochastic equations. According to Equation (2),
 | (3) |
Although we assume that the environment is well mixed, the time lag for the diffusion and the transportation is also approximately considered in this paper, i.e. the effect of the diffusion and the coupling processes for a signal molecule is actually expressed by a mean field variable with a time delay. Specifically, the time delay is adopted to approximate the equivalent time lag of the diffusion process, whereas the mean field variable approximates the equilibrium dynamics of the coupling or mixing process. In other words, the whole process for diffusion and coupling is approximated by two separated processes in this paper, i.e. a diffusion process by a time delay and a coupling (mixing) process by a mean field variable. Such an approximation is also justified in many physical systems and biological networks (Tsimring and Pikovsky, 2001; Huber and Tsimring, 2003; Paulsson and Ehrenberg, 2001). Different from the interconnected coupling of neural networks, no two cells in Equation (3) were directly coupled but interacted indirectly through the mixing process with mean field N, which is more biologically plausible.
When the system is sufficiently large, we assume that the stochastic variables obey Gaussian distribution. Then, it can be proven that Equation (2) is equivalently expressed by the first and second cumulant evolution equations, which means that we can actually examine the dynamics by deterministic cumulants instead of the complicated stochastic variables. Let the first cumulants or means of x be N with each element Ni = xi, and the second cumulants or covariances of x be M with each element Mij = (xi – Ni)(xj – Nj). Then, as indicated in Appendix A of Supplementary material, by integrating overall x, the cumulant evolution equations of Equation (2) are
 | (4) |
fi(x(t)) and 
. The vector N clearly has m elements. However, the non-zero elements of the covariance vector M are at most m(m + 1)/2 but >m because no two molecules in a cell are generally independent of each other. Notice that the element dii of the diagonal matrix D is 0 if xi is not a coupling variable among cells. Since molecules among cells are not necessarily all coupled, many of diagonal elements of D are generally zero.
By examining deterministic Equation (4), we can figure out the qualitative dynamics of the original stochastic Equation (3), including the stochastic synchronization from the viewpoint of probability distribution. The synchrony of Equation (4) in particular, corresponds to that of Equation (3), because the original stochastic dynamics of x can be fully reconstructed from the deterministic mean N and covariance M owing to the assumption of Gaussian distributions. Specifically, if there is no communication between each cell and the environment, i.e. all dij = 0, and no extracellular noise in the environment, i.e. all ij = 0, then the stable equilibrium of Equation (4) corresponds to an unsynchronized random behavior owing to independently unpredictable fluctuations of each cell. However, with the increase of dij or ij, there may exist a bifurcation point that qualitatively synchronizes cells by transiting the system from the equilibrium to oscillation. Such a periodic oscillation of Equation (4) corresponds to the synchrony of cells owing to coupling or common fluctuations. Appendix C of Supplementary material gives general conditions of the bifurcation for Equation (4).
Instead of the mean field, when explicitly considering interconnection among cells in Equation (4) by replacing Ni(t – i) with 
according to y of Equation (2), we can also derive the synchronization conditions. However, to study such cooperative behaviors or phase-locked solutions of Equation (4) is a challenging problem. The pioneering work in this area was done by Winfree (1967, 1980, 1987) who simplified the problem by assuming that the oscillators are strongly attracted to their limit cycles, such that the amplitude variations can be neglected and only phase variations need to be considered. Moreover, Winfree discovered that mutual synchronization is a cooperative phenomenon, by a temporal analogue of the phase transitions encountered in statistical physics. In Appendix D of Supplementary material, we extend the previous works and further derive the sufficient conditions of synchronization. Clearly, if coupling between cells is sufficiently strong, the dynamics of all cells are synchronized in an identical phase, called in-phase synchronization. However, for a general coupled system, a phase-locked solution with different clusters is usually expected to exist, and is of greater interest in biology. Divide n cells into 
groups, where all the cells in each group have identical dynamics. Without loss of generality, let the phase of the first group be zero, i.e. 
1 = 0, and the phase of the k-th group be kT where T is the least period of the oscillation. From the analysis of Appendix D of Supplementary material based on global Hopf bifurcation theory (Alexander and Yorke, 1978; Alexander and Auchmuty, 1986; Chow and Mallet-Paret, 1978; Nussbaum, 1978), we can prove that there is a phase-locked solution bifurcating from the steady state for Equation (4) under the conditions of Theorem D.1, and that the corresponding phase of the k-th group is determined by
 | (5) |
= 1 means the in-phase synchronization. On account of the symmetry of Equation (2) or (4) for identical cells, there are formally solutions with the phase in the same form of Equation (5). When there are time delays in f, we can still prove the existence of a phase-locked solution, in which the phases have similar expression as Equation (5) but in an implicit function form of time delays. Since the synchronization mechanism is based on global Hopf bifurcation which is a restricted Hopf bifurcation, it is also easy to use the method for designing the biocircuits or predicting synchronous dynamics by disturbing a set of parameters according to the conditions of Theorem D.1 of Appendix D of Supplementary material. From Equations (1) and (4), apparently noises exert their effects by the second cumulants or covariances. In other words, without the second cumulants, the cumulant evolution equations (4) simply degenerate to a deterministic system or an averaging dynamics. Next, we show that extracellular noises actually play an active role to induce the cellular synchronization.
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3 IMPLEMENTATION BY A SYNTHETIC GENE REGULATORY NETWORK |
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TOPAbstract1 INTRODUCTION2 GENE REGULATORY NETWORK...3 IMPLEMENTATION BY A...4 DISCUSSIONREFERENCES |
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Recent progress in genetic engineering has made the design and implementation of de novo synthetic gene networks realistic from both theoretical and experimental viewpoints (Bulter et al., 2004; Hasty et al., 2001; Chen and Aihara, 2002a,b; Kobayashi et al., 2003; Wang et al., 2004; Chen et al., 2004), in particular for simple organisms, such as Escherichia coli and yeast. Designing and implementing synthetic gene networks provide a natural framework for reducing the complexity of gene regulation. Actually, from the theoretical prediction, several simple gene networks have been experimentally constructed, e.g. genetic toggle switch (Gardner et al., 2000), repressilator (Elowitz and Leibler, 2000), and other biocircuits (Paulsson, 2004; Bulter et al., 2004; Becskei and Serrano, 2000; Ozbudak et al., 2004). The data in these experiments agreed well with the theoretical predictions, which implies that mathematical model is a powerful tool for designing synthetic gene regulatory networks. Such simple models clearly represent a first step toward logical cellular control by manipulating and monitoring biological processes at the DNA level (Hasty et al., 2001; Kobayashi et al., 2003). Next, we adopt an artificial gene network as an implementation example to demonstrate noise-mediated communication.
3.1 Model
In this work, we design a synthetic gene network by using an extended version two-gene model (Zhou et al., 2004), i.e. luxI and luxR with a promoter placLux0, as shown in Figure 1. Genes luxI and luxR coordinating the behavior of bacteria, such as quorum sensing, were initially discovered in the marine bacterium, V. fischeri. In Figure 1, genes luxI and luxR are constructed as an operon and are both under the control of the promoter placLux0. Cell–cell coupling is accomplished by diffusing a small signal molecule into the extracellular environment, i.e. AI that plays a major role in the cell–cell communication or quorum sensing in V.fischeri. The protein LuxI is an AI synthase that produces AI. Both proteins LuxR and AI are first dimerized and then form a complex, i.e. a hetero-tetramer, which inhibits the activity of the promoter placLux0. As a signal molecule, AI freely diffuses into the environment to exchange information with other cells, and then enters a cell to alter gene expression. Such a circuit can be engineered on plasmids (Weiss and Knight, 2000), and then be cloned to multiple copies, e.g. by PCR. The engineered plasmids are assumed to grow further in E.coli. In order to measure the behavior of the genetic network, a gene for green fluorescent protein or yellow fluorescent protein is incorporated in each plasmid under the control of the targeted promoter to represent the targeted gene in experiments (Weiss and Knight, 2000).
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Let AI2 and LuxR2 indicate AI and LuxR dimers, and AL and ALD represent AI2–LuxR2 and AI2–LuxR2–DNA complexes, respectively. Then, the AI synthesis, the multimerization reactions of proteins and the binding reaction on the regulatory region of DNA in a cell are described as
 | (6) |
 | (7) |
 | (8) |
 | (9) |
 | (10) |
However, the reactions involving transcription and translation, and degradation in a cell are expressed as
 | (11) |
 | (12) |
 | (13) |
 | (14) |
(0 < < 1) is a repression coefficient. As shown in Equations (11) and (12), mRNALuxI and mRNALuxR are produced by the same reactions owing to the operon. Molecules LuxI, LuxR, mRNALuxI, mRNALuxR and AI degrade, at rates ei, er, emi, emr and ea, respectively, which are also considered in the model.
Among cells, AI freely diffuses between cell membranes, and such a process is modeled by linear coupling with a time delay that is mainly on account of the diffusion process. We assume that the system is a well mixed homogenous culture, i.e. diffusive global coupling, and all the transportation and diffusion processes of chemical complexes are expressed by the corresponding means with feedback time delays . Thus, the equivalent chemical reaction of AI diffusion is expressed as
 | (15) |
A and V = 
A where and are the individual cell volume and total culture or environment volume respectively, and A is the Avogadro number. However, the effects of extracellular noises with Gaussian distributions are also assumed to associate with the diffusion process of AI and can be equivalently expressed in the form of the following chemical reaction
 | (16) |
2 is the noise intensity or variance, which affects the cellular dynamics through cellular signals AI and Y. From Equations (6) to (16), according to Appendix A of Supplementary material we first derive the master equation (A1), and then transform it to the Langevin equation (2), i.e. Equation (A6) or (A7). Appendix A of Supplementary material shows the detailed derivation process and the equations. By considering cell coupling or the diffusion of AI, the system is finally expressed by the first and second cumulant evolution equations with time delays. When the system is sufficiently large, the fluctuations of variables for the system are assumed to approach the Gaussian distribution, which can be exactly described by the first and second cumulants (means and covariances) of Equation (4), i.e. Equations (A10) and (A11) in Appendix A of Supplementary material.
Although intracellular dynamics can be derived from the reaction Equations (6)–(14), to compare the simulation results between the derived evolution equations (3) and (4) and the original master equation (A1), coupling and extracellular noises should be equivalently expressed in the master equation or in chemical reaction equations. In this paper, we use the chemical reactions Equations (15) and (16) to equivalently express the effects of coupling and noises on the master equation. Specifically, the chemical reaction equation (15) represents the linear coupling of AI between each cell and the environment. Clearly, the contribution of the reaction (15) to the dynamics of AI is Ydv/V – AId = vd(Y/V – AI/v) in terms of the number, and is d([Y] – [AI]) in terms of the concentration, which is consistent with the linear coupling of Equation (3). However, the chemical reaction (16) describes the effect of extracellular noises on the dynamics of AI. That is, the contribution of the reaction (16) to the dynamics of AI is 2v Y/(2Y) – 2vAI/(2AI) = 0 in terms of the mean value, but is 2v Y/(2Y) + 2v AI/(2AI) = v2 in terms of the variance of the molecule number or 2 in terms of the variance of the concentration, which is also consistent with the settings (zero mean and variance 2) of the extracellular noises in Equation (3).
The parameter values are set as follows: ka = 3.0 min–1, ei = er = 3.0 min–1, ea = 
min–1, emi = emr = 10 min–1, k1 = 
(nM · min), k2 = k3 = k4 = 0.3(nM · min), k–1 = k–2 = 1 min–1, k–3 = k–4 = 2 min–1, km = 3.0 min–1, kpi = 10 min–1, kpr = 20 min–1, = 0.1, nD = 10, the individual E.coli cell volume = 1 x 10–15l and the total culture volume = 2 x 10–3l, which are principally from Bulter et al. (2004) and Belta et al. (2001) with slight modification. Define the cell volume and culture volume factors to be v = A and V = A respectively, where A = 6.022 x 1023 is the Avogadro number to change the unit of volume from l to M. In this paper, d is in min–1 and is in nM.
x and y represent the concentrations of molecules in this paper whereas X and Y describe the number of molecules. In other words, all variables in this paper are the concentrations except those in Appendices (Supplementary material) where X and Y are the number of molecules.
3.2 Implementation results
We now analyze how the noises, coupling and time delays affect the dynamics of cells, leading to cooperative behaviors, by studying and comparing stochastic master equation (A1) and deterministic evolution equations (A10) and (A11) numerically.
We first examine the effects of extracellular noise intensity on cooperative behaviors among cells. The bifurcation diagram of the AI mean value as a function of the noise deviation in the evolution equations (A10) and (A11) is indicated in Figure 2, which shows that the noise can actually induce the oscillation by the Hopf bifurcation. Clearly, without extracellular noises, the evolution equations for cumulants and covariances converge to a stable steady state or a stationary probability distribution with constant means and covariances, which corresponds to asynchronously fluctuating behaviors of cells. As the noise intensity increases, the steady state becomes unstable and the multicell system becomes oscillatory, which implies that the synchronized oscillation is induced by extracellular noises. In other words, for such a system, noises provide extra dynamics, e.g. the dynamics of the second cumulants that are originated from fluctuations or other unknown energy sources beyond the coupled system, so that a cooperative but oscillatory behavior is stimulated among cells. The cooperative behavior, which is also confirmed by the simulation of the stochastic master equation, is shown in Figure 3A. The detailed algorithm of the modified Gillespie method for the master equation with delays is given in Appendix B of Supplementary material. Such behavior corresponds to the in-phase synchronization induced by extracellular noises in the evolution equations. Although the numerical simulation by deterministic evolution equations can be done within a reasonable CPU time even for a large number of cells, the stochastic simulation is very time consuming owing to the master equation and generally requires >10 h for one-case study of three cells (Dell Latitude D400, Pentium M 1.40 GHz CPU). Therefore, owing to time-consuming computation of stochastic simulation, we only use three cells to show the results. As seen in Figure 3A, clearly all cells oscillate almost in the same way with the synchronous fluctuations, where the noise deviation is 5 nM. However, in the absence of the extracellular noises, the mean values of AIs evolve to a steady state, as indicated by the results of both the evolution equations (A10) and (A11) and the stochastic master equation (A1) shown in of Figure 3B.
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Generally, the coupling enhances the synchronization among cells, whereas the time delay
and the extracellular noise tend to induce the oscillation in each cell. Next, we numerically examine the effects of coupling, extracellular noises and time delay of the diffusion process, on cooperative behaviors among cells. The bifurcation diagram of the mean value for AI in the evolution equations (A10) and (A11) as a function of the time delay is shown in Figure 2B. When there is no time delay or the time delay is small, the evolution equations for cumulants and covariances converge to a stable steady state or a stationary probability distribution with constant means and covariances, for which each cell does not fluctuate. However, with a large time delay, the multicell system becomes synchronously oscillatory. The extracellular noise has the effect on oscillation similar to the time delay, as indicated in Figure 2A. However, the oscillatory and steady state regions as functions of the noise deviation and the coupling strength d are shown in Figure 2C. Clearly, the oscillatory region grows with the noise deviation and is also related to the coupling strength. In particular, for a small noise, the multicell system converges to a steady state, which is also a trivial synchronous state. Therefore, for such a system, coupling and time delay as well as noises can also significantly affect cooperative dynamics. The cooperative dynamics induced by the time delay are confirmed in Figure 4A, whereas the convergence to a steady state of cumulants with a small time delay is shown in Figure 4B by both the evolution equations (A10) and (A11) and the master equation (A1). On the other hand, the effect of coupling on cooperative behavior is demonstrated in Figure 3A and C. All cells almost synchronously oscillate for a large coupling d (Fig. 3A). However, when the coupling d is changed to a small value, the multicell system still oscillates but in an asynchronous manner (Fig. 3C). Such facts imply that the coupling generally enhances the cooperative dynamics by controlling the stability of synchronization. All these figures are obtained with random initial phases. In other words, the simulation for all cells starts from asynchronously initial conditions, to demonstrate the effect of active synchronization.
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Now we compare the results of the stochastic and deterministic approaches in Figures 2–4. The deterministic results in Figure 2 were obtained by numerical integration of Equations (A10) and (A11), whereas the stochastic results in Figures 3 and 4 were obtained by the modified Gillespie algorithm in Appendix B of Supplementary material. The results in this paper indicate, nevertheless, that when the system is oscillatory, the cooperative behaviors among cells can be observed clearly from the evolution equations (Fig. 3C). All the figures show good agreement between stochastic and deterministic approaches, except for quantitative differences in peak levels. Such differences arise possibly because of a small number of cells in the simulation, and are expected to be further reduced if more cells are considered.
In this model, we examine the noise or coupling-induced cooperative behaviors by using one signal molecule AI in identical cells. Recent research shows that chemical communication among living organisms is widespread, and many bacteria have two or more sensing systems to monitor their environment and respond to fluctuations for the presence of other bacteria or even other species (Taga and Bassler, 2003). Therefore, by adding multiple species-specific AIs in the model, we may also expect diverse and complex cooperative behaviors among different species similar to those shown in this work.
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4 DISCUSSION |
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TOPAbstract1 INTRODUCTION2 GENE REGULATORY NETWORK...3 IMPLEMENTATION BY A...4 DISCUSSIONREFERENCES |
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In this paper we investigated cooperative behaviors in a general coupled noisy system for microbial cells, which is applied to a cell–cell communication with a biologically plausible model. The essential cooperative mechanism is analyzed by Hopf bifurcation theory. We examined both analytically and numerically the effects of noises and coupling on cooperative behaviors with the consideration of diffusion processes. We show that both noises and time delays enhance the cooperative behaviors and play a major role in the dynamic behaviors of the coupled system. Our analytical results enable us to predict, by assessing a set of parameter values including noises, whether or not the intercellular coupling and the noises synchronize the cells so as to fulfill the intercellular communication or attain a concerted biological behavior in a population. Such dynamic analysis can also be used to design a robust experimental implementation with potential implications for biotechnological applications.
We aim to treat cellular dynamics in a more exact manner, e.g. from the master equation to Langevin equations and cumulant evolution equations including the derivation of intracellular and extracellular noises from the chemical reactions. Although there are still several approximations in our model, such as Langevin equations that approximate discrete numbers of molecules by continuous concentrations, cumulant equations that require the assumption of Gaussian distributions for the random variables and time delays for diffusion processes, we find that the evolution equations approximate the stochastic master equations very well, even for a small number of cells. Since the first and second cumulants represent the Gaussian distributions exactly, the oscillation of cumulants implies the cooperative behaviors among cells, as indicated in the simulations.
Generally, noises promote oscillation by introducing extra dynamics, which are originated from random fluctuations or the second cumulants. Such extra dynamics under certain conditions may play a crucial role as an energy source to excite a cooperative behavior or lead to ordered states among cells, as indicated in this paper. Actually, it is well known that noise plays a significant role in stochastic resonance of a non-autonomous system (with a periodic driving force) and coherence resonance of an autonomous system. In particular, coherence resonance phenomena show that noises enhance temporal regularity of a dynamical system, which was first noticed by Sigeti and Horsthemke (1989) and then theoretically analyzed by Pikovsky and Kurths (1997) with the activation and the excursion times. Since then, many theoretical and experimental works including coupled coherence resonance oscillators (Postnov et al., 2000; Han et al., 1999) and coupled chaotic systems (Kiss et al., 2003; Zhan et al., 2002) have appeared mainly based on the analysis of resonance behaviors. All these studies indicate that noises can play a stabilizing role in coupled oscillators and maps, and have an effect to drive the stochastic system toward rather a regular dynamics.
The intracellular and extracellular noises play different roles in cell communication. Since the intracellular noises of each cell are independent, they generally tend to disturb a cooperative behavior between cells. However, the extracellular noises are nearly common to all cells due to their common environment, and this has the effect of synchronizing the dynamics of all cells by exerting the same fluctuations on each cell through signal molecules, as we have shown in this paper.
From the mathematical viewpoint, there is another question for stability of dynamics, which is probably due to Hopf bifurcation in our model. Actually, Hopf bifurcation is generic for generating oscillation of a nonlinear dynamical system. For example, coupled relaxation oscillators considered by McMillen et al. (2002) are capable of producing Hopf bifurcation through which the coupled system is synchronized. Such a solution can be mathematically proved to be stable by evaluating the corresponding Floquet multipliers. Technically speaking, if the oscillation is hyperbolically and asymptotically stable, then the cooperative behaviors can be observed, because mathematically they are stable for appropriate noise deviation (see Zhou, 2003 for details).
The effects of coupling on cooperative behaviors across a population of oscillators have not been extensively studied and the roles of noises are not well understood. We showed that noises and time delay can induce cooperative behaviors or ‘order’, which seems to contradict to our intuitive predictions based on a ‘negative’ meaning of the word ‘noise’ or randomness. Our theoretical and numerical results suggest that such an essential and constructive role played by noises and coupling may make living organisms organize their various apparatuses harmoniously and actively accomplish mutual communications.
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Acknowledgments |
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This study is partially supported by a Grant-in-Aid No. 12208004 from the Ministry of Education, Culture, Sports, Science and Technology, the Japanese Government.
Received on December 5, 2004; revised on February 23, 2005; accepted on March 15, 2005
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TOPAbstract1 INTRODUCTION2 GENE REGULATORY NETWORK...3 IMPLEMENTATION BY A...4 DISCUSSIONREFERENCES |
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