基于基因群体感应耦合模型的基因时钟同步
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摘要
生物体具有生物化学动力学属性。它们会受到时变因素的影响,例如来自环境的影响或生物体内细胞时钟产生的节律影响。这些节律发生器由许多时钟细胞组成,而这些时钟细胞虽然不同却能产生一致的节律行为。无论是在一个简单的种群中还是在一个庞大的网络中,同步时钟对于个体之间节律行为的调节都是至关重要的。因此这种群体行为产生的机制就成了人们研究的热点。
由于群体感应耦合机制的引入,虽然使得同步成为可能,但也给系统增加了噪声源。因此,分析噪声对同步时钟的影响也是十分必要的。
论文包含以下内容：
1.基本概念,如系统生物学,同步等,并指出这些概念的重要意义；同时回顾了同步研究的发展历程。
2.总结了复杂网络用到的一些理论基础。在非线性动力学方面,对连续时间系统的同步解做变分得到变分方程,然后对角化实现解耦,从而原系统可以简化为含有一个共同自治振子驱动的振子网络。在统计物理学中,连续相变在物质的各处同时发生,与单个分子周围的环境无关,由此引出平均场理论在振子模型中的应用。根据连续相变理论研究人员找到一个定量描述同步突然出现的量：序参量,该变量随一个控制参量变化而变化。
3.改进J-O模型。根据自然界中出现的同步时钟能对外界环境作出快速反应这一思想,本文作者改进了J-O模型,原模型和改进模型的数值仿真结果显示改进的模型能更快产生同步。同时运用序参量的原理定量刻画了耦合强度和序参量的关系。仿真结果验证了同步的产生是突变的而不是渐进的过程。
4.噪声分析。我们对比了两类随机微分方程仿真算法：slow-scale随机仿真算法(ssSSA)和Euler-Maruyama(EM)方法,由于EM方法概念上比较简单,本文作者采用该算法处理随机模型。首先利用平均值思想分析了噪声对系统整体的影响,结果显示耦合降低了噪声对系统的影响；其次,我们利用matlab三维做图方法刻画了每个细胞中噪声产生的影响。这样可以增加一维表示细胞个数的参量,我们可以看到耦合在降低细胞间差异方面的作用。
Organisms are of biochemical dynamics. They are continuously subjected to time-varying conditions in the form of both extrinsic driving from the environment and intrinsic rhythms generated by specialized cellular clocks within the organism itself. These rhythm generators are composed of thousands of clock cells that are intrinsically diverse but nevertheless manage to function in a coherent oscillatory state. Synchronized clocks are of fundamental importance in the coordination of rhythmic behaviour among individual elements in a community or a large complex sytem. Thus, there is considerable interest in investigating the mechanisms by which this collective behavior arises.
By virtue of the including of quorum sensing, which makes synchronization possible, it is unresistable to add a noise source to the system. Hence the analysis of noise effect to system is necessary.
This thesis consists of the following section:
1. Some element concepts, such as system biology, synchronization and so on, are presented of the article associated with the significance of their application. A review is also introduced.
2. Summarizing the theoretical principle concerning complex network, this article is focusing on explaining the applications of important ideas in nolinear dynamics and statistical physics. In the area of nonlinear dynamics, based on the variational principle, variation equation of the synchronous solution is presented for continuous-time system and then decouples by diagonalization. This leads to a networkof simplification driving by an autonomous oscillator; in the level of statistical physics, in the light of continuous phase transition researchers, to characterize quantitatively the transition to synchronization, define a quantity called order parameter with value that changes abruptly at the transition point as a certain control parameter varies.
3. Improved J-O model. Based the ideas that synchronous clocks in nautre can responde to environment fastly, we modify the J-O model. The numerical results of them reveal that the improved one can synchronize more rapidly; meanwhile, we characterize qunantitatively the relations between coupling parameter and order parameter. The result indicates the process of the transition to synchronization is abruptly, not gradually.
4. Noise analysis. We contrast two stochastic simulation algorithm:ssSSA and Euler-Maruyama and we employ the EM algorithm because it is a conceptional easy approach. We firstly address the noise effect averaged over a population of 3 repressilators. Simulation results manifest that coupling reduces the noise effect to system;furthermore, using 3-dimension picture in Matlab we associate the noise effect with the number of repressilators.
由于群体感应耦合机制的引入,虽然使得同步成为可能,但也给系统增加了噪声源。因此,分析噪声对同步时钟的影响也是十分必要的。
论文包含以下内容：
1.基本概念,如系统生物学,同步等,并指出这些概念的重要意义；同时回顾了同步研究的发展历程。
2.总结了复杂网络用到的一些理论基础。在非线性动力学方面,对连续时间系统的同步解做变分得到变分方程,然后对角化实现解耦,从而原系统可以简化为含有一个共同自治振子驱动的振子网络。在统计物理学中,连续相变在物质的各处同时发生,与单个分子周围的环境无关,由此引出平均场理论在振子模型中的应用。根据连续相变理论研究人员找到一个定量描述同步突然出现的量：序参量,该变量随一个控制参量变化而变化。
3.改进J-O模型。根据自然界中出现的同步时钟能对外界环境作出快速反应这一思想,本文作者改进了J-O模型,原模型和改进模型的数值仿真结果显示改进的模型能更快产生同步。同时运用序参量的原理定量刻画了耦合强度和序参量的关系。仿真结果验证了同步的产生是突变的而不是渐进的过程。
4.噪声分析。我们对比了两类随机微分方程仿真算法：slow-scale随机仿真算法(ssSSA)和Euler-Maruyama(EM)方法,由于EM方法概念上比较简单,本文作者采用该算法处理随机模型。首先利用平均值思想分析了噪声对系统整体的影响,结果显示耦合降低了噪声对系统的影响；其次,我们利用matlab三维做图方法刻画了每个细胞中噪声产生的影响。这样可以增加一维表示细胞个数的参量,我们可以看到耦合在降低细胞间差异方面的作用。
Organisms are of biochemical dynamics. They are continuously subjected to time-varying conditions in the form of both extrinsic driving from the environment and intrinsic rhythms generated by specialized cellular clocks within the organism itself. These rhythm generators are composed of thousands of clock cells that are intrinsically diverse but nevertheless manage to function in a coherent oscillatory state. Synchronized clocks are of fundamental importance in the coordination of rhythmic behaviour among individual elements in a community or a large complex sytem. Thus, there is considerable interest in investigating the mechanisms by which this collective behavior arises.
By virtue of the including of quorum sensing, which makes synchronization possible, it is unresistable to add a noise source to the system. Hence the analysis of noise effect to system is necessary.
This thesis consists of the following section:
1. Some element concepts, such as system biology, synchronization and so on, are presented of the article associated with the significance of their application. A review is also introduced.
2. Summarizing the theoretical principle concerning complex network, this article is focusing on explaining the applications of important ideas in nolinear dynamics and statistical physics. In the area of nonlinear dynamics, based on the variational principle, variation equation of the synchronous solution is presented for continuous-time system and then decouples by diagonalization. This leads to a networkof simplification driving by an autonomous oscillator; in the level of statistical physics, in the light of continuous phase transition researchers, to characterize quantitatively the transition to synchronization, define a quantity called order parameter with value that changes abruptly at the transition point as a certain control parameter varies.
3. Improved J-O model. Based the ideas that synchronous clocks in nautre can responde to environment fastly, we modify the J-O model. The numerical results of them reveal that the improved one can synchronize more rapidly; meanwhile, we characterize qunantitatively the relations between coupling parameter and order parameter. The result indicates the process of the transition to synchronization is abruptly, not gradually.
4. Noise analysis. We contrast two stochastic simulation algorithm:ssSSA and Euler-Maruyama and we employ the EM algorithm because it is a conceptional easy approach. We firstly address the noise effect averaged over a population of 3 repressilators. Simulation results manifest that coupling reduces the noise effect to system;furthermore, using 3-dimension picture in Matlab we associate the noise effect with the number of repressilators.
引文
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[2]Reppert S M, Weaver D R Forward genetic approach strikes gold:cloning of a mammalian clock gene. [J].Cell,1997,89:487-490.
[3]Liu, C., Weaver, D. R., Strogatz, S. H.& Reppert, S. M. Cellular construction of a circadian clock:period determination in the suprachiasmatic nuclei[J]. Cell,1997;91,855-860.
[4]Herzog, E. D., Takahashi, J. S.& Block, G. D. Clock controls circadian period in isolated suprachiasmatic nucleus neurons [J].Nat. Neurosci.1998,1:708-713.
[5]Honma, S., Shirakawa, T., Katsuno, Y., Namihira, M.& Honma, K. I. Circadian periods of single suprachiasmatic neurons in rats[J]. Neurosci. Lett.,1998,250:157-160.
[6]Nakamura, W., Honma, S., Shirakawa, T.& Honma, K.-I. Regional pacemakers composed of multiple oscillator neurons in the rat suprachiasmatic nucleus[J]. Eur. J.Neurosci.,2001,14: 1-10.
[7]Herzog, E. D., Aton, S. J., Numano, R., Sakaki, Y.& Tei, H. Temporal precision in the mammalian circadian system:a reliable clock from less reliable neurons[J]. J. Biol.Rhythms,2004,19:35-46.
[8]Goldbeter, A. Biochemical oscillations and cellular rhythms[M], Cambridge:CambridgeUniv. Press,1996.
[9]Hasty, J., McMillen, D.& Collins, J. J. Engineered gene circuits[J]. Nature,2002,420:224-230.
[10]Elowitz, M. B.& Leibler, S. A synthetic oscillatory network of transcription regulators [J].Nature,2000,403:335-338.
[11]Atkinson, M. R., Savageau, M. A., Myers, J. T.& Ninfa, A. J. Development of genetic circuitry exhibiting toggle switch or oscillatory behavior in Escherichia coli[J]. Cell,2003,113:597-607.
[12]Winfree, A. T. Biological rhythms and the behavior of populations of coupled oscillators [J].J. Theor. Biol,1967,16:15-42.
[13]Njus, D., Gooch, V. D.& Hastings. J. W. Precision of the gonyaulax circadian clock [J].Cell Biophys.,1981,3:223-231.
[14]Miller, M. B.& Bassler, B. L. Quorum sensing in bacteria [J].Annu. Rev. Microbiol.,2001,55:165-199.
[15]You, L., Cox, R. S., Ⅳ, Weiss. R.& Arnold. F. H. Programmed population control by cell-cell communication and regulated killing [J].Nature,2004,428:868-871.
[16]McMillen, D., Kopell, N., Hasty, J.& Collins, J. J. Synchronizing genetic relaxation oscillators by intercell signaling [J].PNAS,2002.99:679-684.
[17]Kuramoto, Y. Chemical oscillations, waves, and turbulence[M], Berlin:Springer,1984,pp. 68-77.
[18]Kiss, I. Z., Zhai, Y.& Hudson, J. L. Emerging coherence in a population of chemical oscillators [J].Science 2002; 296,1676-1678.
[19]Winfree, A. T. On Emerging Coherence[J]. Science,2002,298:2336-2337.
[20]Enright, J. T.Temporal precision in circadian systems:a reliable neuronal clock from unreliable components [J].Science,1980,209:1542-1545.
[21]Somers, D.& Kopell, N. Waves and synchrony in arrays of oscillators of relaxation and non-relaxation type[J]. Physica D,1995,89:169-183.
[22]Needleman, D. J., Tiesinga, P. H. E.& Sejnowski, T. J. Collective enhancement of precision in networks of coupled oscillators[J]. Physica D.2001,155:324-336.
[23]Barkai, N.& Leibler, S. Circadian clocks limited by noise [J].Nature,2000,403:267-268.
[24]Gonze, D., Halloy, J.& Goldbeter, A. Robustness of circadian rhythms with respect to molecular noise[J]. Proc. Natl. Acad. Sci. USA,2002.99:673-678.
[25]Vilar, J. M. G., Kueh, H. Y, Barkai, N. & Leibler, S. Mechanisms of noise-resistance in genetic oscillators[J]. Proc. Natl. Acad.Sci. USA,2002,99:5988-5992.
[26]Dockery, J. D.& Keener, J. P. A mathematical model for quorum sensing in Pseudomonas aeruginosa [J].Bull. Math. Biol.,2001,63:95-116.
[27]Elowitz, M. B., Levine, A. J., Siggia, R. D.& Swain, P. S. Stochastic gene expression in a single Cell [J].Science,2002,297:1183-1186.
[28]Ozbudak, E. M., Thattai, M., Kurtser, I., Grossman, A. D.& van Oudenaarden,A. Regulation of noise in the expression of a single gene[J]. Nat. Genet.,2002,31:69-73.
[29]Kaplan, H. B.& Greenberg. E. P. Diffusion of autoinducer is involved in regulation of the Vibrio fischeri luminescence system [J].J. Bacteriol.,1985,163:1210-1214.
[30]Strogatz, S. H. From Kuramoto to Crawford:exploring the onset of synchronization in populations of coupled oscillators[J]. Physica D,2000,143:1-20.
[31]Paulsson, J.& Ehrenberg, M. Q. Noise in a minimal regulatory network:plasmid copy number control [J].Rev. Biophys.,2001,34:1-59.
[32]Garcia-Ojalvo, J.& Sancho, J. M. Noise in spatially extended systems[M]. New York:Springer,1999.
[33]Welsh. D. K.& Reppert, S. M. Gap junctions couple astrocytes but not neurons in dissociated cultures of rat suprachiasmatic nucleus[J]. Brain Res.,1996,706:30-36.
[34]Schwartz, W. J, Gross, R. A.& Morton, M. T.The suprachiasmatic nuclei contain a tetrodotoxin-resistant circadian pacemaker[J]. PNAS,1987,84:1694-1698.
[35]Liu, C.& Reppert, S. M. GABA.synchronizes clock cells within the suprachiasmatic circadian clock [J].Neuron,2000,25:123-128.
[36]Yamaguchi, S., Isejima, H., Matsuo, T., Okura, R., Yagita, K., Kobayashi, M.& Okamura, H. Synchronization of Cellular clocks in the suprachiasmatic Nucleus[J]. Science, 2003,302:1408-1412.
[37]Richard, P., Teusink, B., Hemker, B. B., van Dam, K.& Westerhoff, H. V. Sustained oscillations in free-energy state and hexose phosphates in yeast [J].Yeast,1996,12:731-740.
[38]Dano, S., Sorensen, P. G.& Hynne, F. Sustained oscillations in living cells[J].Nature,1999, 402:320-322.
[39]Kuramoto, Y. Chemical Oscillations, waves, and turbulence Springer[M],Berlin,1984, pp:62-67.
[40]Ott, E., So, P., Barreto, E.& Antonsen, T. The onset of synchronism in systems of globally coupled chaotic and periodic oscillators[J].Physica D,2002,173:29-51.
[41]R. E. Mirollo and S. H. Strogatz. Synchronization of pulse-coupled biological oscillators[J]. SIAM J. Appl. Math,1990,50:1645-1662.
[42]S. H. Strogatz and I. Stewart.Coupled oscillators and biological synchronization[J]. Scientific American,1993,269 (6):102-09.
[43]Garcia-Ojalvo, M. B. Elowitz, and S. H. Strogatz. Modeling a synthetic multicellular clock: Repressilators coupled by quorum sensing[J]. PNAS,2004,101:10955-10960.
[44]D. J. Watts and S. H. Strogatz. Collective dynamics of 'small-world' networks[J]. Nature,1998, 393:440-42.
[45]S. H. Strogatz. Nonlinear dynamics and chaos:with applications to physics, biology, chemistry, and engineering[M]. Cambridge:Perseus Books,1994.
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