Cooperative Effects of Noise and Coupling on Stochastic Dynamics of a Membrane-Bulk Coupling Model

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1 Commun. Theor. Phys. (Beijing, China) 51 (2009) pp c Chinese Physical Society and IOP Publishing Ltd Vol. 51, No. 3, March 15, 2009 Cooperative Effects of Noise and Coupling on Stochastic Dynamics of a Membrane-Bulk Coupling Model TANG Jun, 1,2, JIA Ya, 2, and YI Ming 2 1 College of Science, China University of Mining Technology, Xuzhou , China 2 Department of Physics and Institute of Biophysics, Huazhong Normal University, Wuhan , China (Received April 25, 2008) Abstract Based on a membrane-bulk coupling cell model proposed by Gomez-Marin et al. [ Phys. Rev. Lett. 98 (2007) ], the cooperative effects of noise and coupling on the stochastic dynamical behavior are investigated. For parameters in a certain region, the oscillation can be induced by the cooperative effect of noise and coupling. Whether considering the coupling or not, corresponding coherence resonance phenomena are observed. Furthermore, the effects of two coupling parameters, cell size L and coupling intensity k, on the noise-induced oscillation of membranes are studied. Contrary effects of noise are found in and out of the deterministic oscillatory regions. PACS numbers: Ca, Fh, Bb Key words: membrane-bulk cell model, coupling, noise, coherence resonance 1 Introduction In nonlinear biological systems, the multi-system coupling is ubiquitous, and essential for specific biological function. Many biological behaviors, such as calcium signals propagation, [1] optimal signaling in coupled hormone systems, [2] cell-cell communication in multicellular systems, [3] result from the inter-system coupling. In coupled hormone systems, cells are coupled to each other by mutual feedback, [2] so this coupling is defined as direct coupling between cells. However, in the multicellular systems of Ref. [3], signal molecules diffuse between cells and extracellular environment. [3] Multi-cell coupling is indirectly accomplished through extracellular environment, it is accordingly defined as indirect coupling. Noises can be introduced into the biochemical systems by diverse sources. The external noise originates from the random variations of one or more of the externally set control parameters, such as the rate constants associated with a given set of reactions. The internal noise comes from discrete nature of biochemical events such as transcription, translation, multimerization, and protein/mrna decay processes. The effects of noise on biochemical systems have been extensively studied from both theoretical and experimental points of view. For instance, Liu et al. studied a positive feedback gene regulation system, their results indicated that different source of noise could induce state transition between multi-steady states. [4] Based on a Ca 2+ model presented by Li and Rinzel, [5] Shuai et al. found internal noise play an important role in inducing sub- or super-threshold oscillation. [6] Common noise may synchronize oscillators in different level of biological systems. [7,8] The response of nonlinear system with or without external periodic signal may be enhanced through an optimal amount of noise, this phenomenon is called coherence resonance (CR). [9] CR is originally found in a simple dynamical system in the vicinity of a saddle-node bifurcation [10] and is due to non-uniformity of the noiseinduced limit cycle. [11] CR has been extensively studied in many nonlinear systems. [9,12 15] The cooperative effects of noise and coupling, which produce synchronization between systems, [16] on biological systems are intensively studied. [17,18] Recently, Gomez- Marin et al. presented a Membrane-Bulk coupling model, which involved a one-dimensional bulk and two pointlike boundaries. [19] The bulk couples with two boundaries through diffusion. In another viewpoint, it can be seen as indirect coupling between two boundaries through bulk diffusion. Their study indicates the coupling can induce spatiotemporal oscillation. However, the noise effect has not been considered by Gomez-Marin et al. This paper introduces noise into the Gomez-Marin model, and focuses on the cooperative effect of noise and coupling on the dynamical behavior. Results show the oscillation can be induced by the cooperative effect of noise and coupling. With or without the coupling considered, corresponding coherence resonance phenomena are observed. The effects of coupling on noise-induced oscillation of two membranes are studied, and contrary effects of noise are found in and out of the deterministic oscillatory regions. 2 Model The simple cell model discussed in this paper is composed of a one-dimensional bulk and two boundaries (membranes). The boundaries is active whereas the bulk is passive, i.e., the biochemical reactions only occur in the The project supported by the National Natural Science Foundation of China under Grant No tjuns1979@yahoo.com.cn jiay@phy.ccnu.edu.cn

2 456 TANG Jun, JIA Ya, and YI Ming Vol. 51 boundaries. The reactions involve two proteins, and we respectively define u and v as concentrations of activator and inhibitor protein. The dimensionless membrane dynamics is given by du dt = u q(u 2)3 + 4 v, (1) dv dt = ε(zu v) + k u x, (2) which is a modified Fitzhugh Nagumo (FHN) model. In the model, q and v are constant parameters, ε determines a different time scale for two species, which is invariably set as in this paper. The x stands for spatial position. The positions of membranes are set as 0 and L, where L is the cell length. The term k u/ x in Eq. (2) accounts for the exchange between the membranes and bulk, while k determines the coupling strength. For simplification, we consider exchange of only one protein. In the passive bulk, no reaction occurs but the degeneration of inhibitor protein. Considering the diffusion and degeneration of inhibitor protein, the bulk dynamics is given by v(x, t) t = D v σv, (3) x where D is the diffusion coefficient, and σ is degeneration rate. To study the noise effects on the dynamical behaviors, a stochastic version of the presented model will be introduced. Equations (1) and (2) are a modified FHN model to describe Min system in E. coli, which involves the cell division regulators. [20] Herein, the activator u would correspond to the protein MinD and the inhibitor v to the protein MinE. [19] The noises in Min system are due to the low copy number of proteins, the discrete nature of biochemical reactions, or the environmental fluctuations. This FHN-type model is used to describe Min system due to their similar dynamics. In fact, the Min system involves additional reactors, such as MinC protein, and more detailed biochemical reactions. Thus, simple additive noises, which are enough for issues we are interested in, are considered to account for all possible noise sources. The stochastic version of the model is given by du dt = u q(u 2)3 + 4 v + ξ(t), (4) dv dt = ε(zu v) + k u + η(t), (5) x where ξ(t) and η(t) are uncorrelated Gaussian white noises, and the statistical properties of them are given by ξ(t) = η(t) = 0, (6) ξ(t)ξ(t ) = η(t)η(t ) = 2αδ(t t ). (7) α is the noise intensity. All model parameters are obtained from Ref. [19] with slightly modification. 3 Results and Discussion Based on the deterministic model, we first investigate the effect of coupling on dynamics of one of the two boundaries. Set z as the bifurcation parameter, the bifurcation diagrams of the deterministic model are compared in Fig. 1(a). In Fig. 1(b), the position of left bifurcation point is plotted against coupling strength k. Obviously, the oscillatory region is drastically shifted by indirect coupling of two boundaries, moreover, the coupling can slightly broaden the oscillatory region [see inset of Fig. 1(b)]. Fig. 1 Comparison of the bifurcation diagrams of the presented model with and without coupling by setting q = 5.0, D = 0.7, and length of bulk L = 10. k is variable and z is the bifurcation parameter. v is the inhibitor concentration of one boundary. (a) Two bifurcation diagram comparison: without coupling by setting k = 0; with coupling and k = 7ε. (b) Left bifurcation point z L is plotted against coupling strength k, inset: width of oscillatory region z against k. To illuminate the cooperative effect of coupling and noise, the dynamics of the system with or without them are simulated. As an example, parameter z is set as 4.3, for which both the systems with or without coupling is monostable [see Fig. 1(a)]. Obviously, regardless of the noise effect, the boundary will be in a steady state either with or without coupling. Simulation results indicate that, with coupling, small noise can induce oscillation. But when we turn our steps to the condition of without coupling, it is found that even for larger strength, noise only produces small fluctuation right around the steady

3 No. 3 Cooperative Effects of Noise and Coupling on Stochastic Dynamics of a Membrane-Bulk Coupling Model 457 state [see Fig. 2]. The comparison of Fig. 2(b) and 2(c) is more obviously exhibited in Fig. 3. which is relevant power spetra. When noise and coupling effect are both involved, a distinct peak with frequency of about is found in the power spetra. But there is no peak without coupling effect. In one word, the noise and coupling cooperatively induce oscillation. with or without coupling, it is clearly seen that τ c first increases to some maximum and then decreases. This is the coherence resonance phenomenon, and the maximal τ c corresponds to most regular oscillation of the system. The maximal values of τ c are observed at almost the same noise intensity for both curves. On the other hand, the coupling reduces the regularity of the oscillation, and when coupling is involved, larger noise is needed to induce oscillation. Fig. 2 Time series of inhibitor concentration v. (a) Without noise, i.e., α = 0; with coupling and k = 7ε; (b) Without coupling, i.e., k = 0; with noise and α = ; (c) With coupling and k = 7ε; with noise and α = All other parameters are set as Fig. 1. Fig. 3 Power spectra of time series (b) and (c) in Fig. 2. CR is an important phenomena in noise effect. Several quantities, such as signal-to-noise ratio R, correlation time τ c, are presented to characterize this phenomena. To obtain τ c, we should first compute the normalized autocorrelation function [10] ṽ(t)ṽ(t + τ) C(τ) = ṽ 2, ṽ = v v. (8) Then the characteristic correlation time can be integrated as follows: τ c = 0 C 2 (τ)dτ. (9) Obviously, lager τ c corresponds to more regular oscillation. As an example, the dependence of τ c on the noise intensity is presented in Fig. 4, where z (z = 3.4) is set between two oscillatory region [see Fig. 1(a)]. Whether Fig. 4 Characteristic correlation time τ c vs. noise intensity for different coupling condition. The parameters are set as Fig. 1, and z = 3.4. The characters of the bulk, which is an indirect coupling media of two boundaries, are also important factors affecting the dynamics of the system as well as coupling intensity k. Our simulation results indicate that the length of bulk L affects the dynamics of the system very similarly to k, so the effects of them will be described comparatively. In what follows, the effects of L and k on the system will be studied, and parameter z is invariably set 3.5. As shown in Fig. 5, the system exhibits bifurcation behavior when L and k are varied. When k is fixed at 7ε, the system is oscillatory if L is in a region ( ), and the system will be monostable elsewhere. When L is fixed at 10.0, the system is oscillatory if k is in a region ( ), and the system will be monostable elsewhere. Characteristic correlation time τ c is plotted against L and k for varied noise intensity α in Fig. 6. Obviously, τ c increases when the parameter (k or L) is close to the deterministic oscillatory region (indicated by the dashed lines), i.e., the intrinsic oscillation of the system is more regular than that induced by the noises. Moreover, in the deterministic oscillatory region, τ c reduces with the increase of noise intensity. However, when the system goes out of the oscillatory region, the result is totally reversed. We can infer from the simulation results that the noises destroy the regularity of the oscillation when the system is in the oscillatory region, thus, the lager the noise intensity is, the worse regular the oscillation is. On the contrary, the effect of noise is inducing oscillation out of the oscillatory region. So, lager the noise intensity is, more regular oscillation is induced.

4 458 TANG Jun, JIA Ya, and YI Ming Vol. 51 Fig. 5 Bifurcation diagram of the model, where the length of bulk L and coupling intensity k are the bifurcation parameters, v is the inhibitor concentration of one boundary. The parameters are set as the same as Fig. 1 except z = 3.5. (a) k = 7ε; (b) L = 10. Fig. 6 Characteristic correlation time τ c vs. bulk length L and coupling intensity k for varied noise intensity α. All constant parameters are set as Fig. 5. Through comparison in Figs. 5 and 6, we can conclude that L and k affect the dynamics of the system similarly. It is reasonable because L and k are both parameters related to coupling, and their effects on the system are both coupling effects. 4 Conclusion In conclusion, the cooperative effects of coupling and noise on a nonlinear system, which is used to describe the interaction between bulk and boundaries of E. coli, are studied. Our simulation results indicate that coupling can change the bifurcation diagram of the boundaries, including the position and width of the oscillatory region. When the system is in a stable steady state, noise and coupling can cooperatively induced oscillation. Whether involving coupling or not, CR phenomena are observed for given value of z, i.e., there is an optimal noise intensity for inducing most regular oscillation. Moreover, the length of bulk L and coupling intensity k can affect the regularity of oscillation very similarly. Out of the oscillatory region, the lager the noise intensity is, the more regular oscillation is induce. Through comparison, it is found that noises act as totally different roles in and out of oscillatory region. It should be pointed out that stochastic FHN model has been intensively studied, the methods of adding noise term vary according to special biological system. [21 24] Only simple Gaussion white additive noises are considered in this paper. When more detailed reactions are involved, instead of simply additive Gaussion white type, other noise sources will be considered. It is a motivation of our further work. References [1] W.D. Kepseu and P. Woafo, Phys. Rev. E 73 (2006) [2] Q.S. Li and H.Y. He, J. Chem. Phys. 123 (2005) [3] T.S. Zhou, et al., Phys. Rev. Lett. 95 (2005) [4] Quan Liu and Ya Jia, Phys. Rev. E. 70(2004) [5] Y.X. Li and J. Rinzel, J. Theor. Biol. 166 (1994) 461.

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