Skip to main content
banner image
No data available.
Please log in to see this content.
You have no subscription access to this content.
No metrics data to plot.
The attempt to load metrics for this article has failed.
The attempt to plot a graph for these metrics has failed.
The full text of this article is not currently available.
/content/aip/journal/jcp/142/18/10.1063/1.4919834
1.
1.A. M. Lyapunov, “The general problem of the stability of motion,” Int. J. Control 55, 539-589 (1992).
http://dx.doi.org/10.1080/00207179208934253
2.
2.I. R. Epstein and A. R. Pojman, An Introduction to Nonlinear Chemical Dynamics (Oxford University Press, New York, 1998).
3.
3.R. E. Kalman and J. E. Bertram, “Control system analysis and design via the second method of Lyapunov: I–Continuous-time systems,” J. Basic Eng. 82, 371-393 (1960).
http://dx.doi.org/10.1115/1.3662604
4.
4.G. László and R. J. Field, “A three-variable model of deterministic chaos in the Belousov-Zhabotinsky reaction,” Nature 355, 808-810 (1992).
http://dx.doi.org/10.1038/355808a0
5.
5.R. B. Warden, R. Aris, and N. R. Amundson, “An analysis of chemical reactor stability and control–VIII: The direct method of Lyapunov,” Chem. Eng. Sci. 19, 149-172 (1964).
http://dx.doi.org/10.1016/0009-2509(64)85027-2
6.
6.D. T. Gillespie, “Stochastic simulation of chemical kinetics,” Annu. Rev. Phys. Chem. 58, 35-55 (2007).
http://dx.doi.org/10.1146/annurev.physchem.58.032806.104637
7.
7.C. W. Gardiner, Handbook of Stochastic Methods (Springer, Berlin, 1985), Vol. 3, pp. 2-20.
8.
8.H. H. McAdams and A. Arkin, “Stochastic mechanisms in gene expression,” Proc. Natl. Acad. Sci. U. S. A. 94, 814-819 (1997).
http://dx.doi.org/10.1073/pnas.94.3.814
9.
9.N. G. Van Kampen, Stochastic Processes in Physics and Chemistry (Elsevier, 1992), Vol. 1, pp. 86-97.
10.
10.D. T. Gillespie, “The chemical Langevin equation,” J. Chem. Phys. 113, 297-306 (2000).
http://dx.doi.org/10.1063/1.481811
11.
11.P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations (Springer, 1992), Vol. 23, pp. 34-40.
12.
12.R. Kubo, “The fluctuation-dissipation theorem,” Rep. Prog. Phys. 29, 255-284 (1966).
http://dx.doi.org/10.1088/0034-4885/29/1/306
13.
13.H. B. Callen and R. F. Greene, “On a theorem of irreversible thermodynamics,” Phys. Rev. 86, 702-710 (1952).
http://dx.doi.org/10.1103/PhysRev.86.702
14.
14.D. T. Gillespie, “A general method for numerically simulating the stochastic time evolution of coupled chemical reactions,” J. Comput. Phys. 22, 403-434 (1976).
http://dx.doi.org/10.1016/0021-9991(76)90041-3
15.
15.P. Smadbeck and Y. N. Kaznessis, “A closure scheme for chemical master equations,” Proc. Natl. Acad. Sci. U. S. A. 35, 14261-14265 (2013).
http://dx.doi.org/10.1073/pnas.1306481110
16.
16.B. Munsky and M. Khammash, “The finite state projection algorithm for the solution of the chemical master equation,” J. Chem. Phys. 124(4), 044104 (2006).
http://dx.doi.org/10.1063/1.2145882
17.
17.A. Ale, P. Kirk, and M. P. H. Stumpf, “A general moment expansion method for stochastic kinetic models,” J. Chem. Phys. 138(17), 174101 (2013).
http://dx.doi.org/10.1063/1.4802475
18.
18.A. Singh and J. P. Hespanha, “Approximate moment dynamics for chemically reacting systems,” IEEE Trans. Autom. Control 56(2), 414-418 (2011).
http://dx.doi.org/10.1109/TAC.2010.2088631
19.
19.V. Sotiropoulos and Y. N. Kaznessis, “Analytical derivation of moment equations in stochastic chemical kinetics,” Chem. Eng. Sci. 66(3), 268-277 (2011).
http://dx.doi.org/10.1016/j.ces.2010.10.024
20.
20.P. Smadbeck and Y. N. Kaznessis, “Efficient moment matrix generation for arbitrary chemical networks,” Chem. Eng. Sci. 84, 612-618 (2012).
http://dx.doi.org/10.1016/j.ces.2012.08.031
21.
21.C. E. Shannon, “A note on the concept of entropy,” Bell Syst. Tech. J. 27, 379-423 (1948).
http://dx.doi.org/10.1002/j.1538-7305.1948.tb01338.x
22.
22.E. T. Jaynes, Probability Theory: The Logic of Science (Cambridge University Press, 2003), pp. 343-372.
23.
23.P. B. Warren, S. Tănase-Nicola, and P. R. Ten Wolde, “Exact results for noise power spectra in linear biochemical reaction networks,” J. Chem. Phys. 125, 144904 (2006).
http://dx.doi.org/10.1063/1.2356472
24.
24.H. Salis, V. Sotiropoulos, and Y. N. Kaznessis, “Multiscale Hy3S: Hybrid stochastic simulation for supercomputers,” BMC Bioinf. 7, 93 (2006).
http://dx.doi.org/10.1186/1471-2105-7-93
25.
25.B. O. Palsson and E. N. Lightfoot, “Mathematical modelling of dynamics and control in metabolic networks. I. On Michaelis-Menten kinetics,” J. Theor. Biol. 111, 273-302 (1984).
http://dx.doi.org/10.1016/S0022-5193(84)80211-8
26.
26.M. Vellela and H. Qian, “Stochastic dynamics and non-equilibrium thermodynamics of a bistable chemical system: The Schlögl model revisited,” J. R. Soc., Interface 6, 925-940 (2009).
http://dx.doi.org/10.1098/rsif.2008.0476
27.
27.See supplementary material at http://dx.doi.org/10.1063/1.4919834 for a detailed derivation of the steady-state Jacobian of the probability distribution moments,JSS, and for an example. The derivation of correlation functions is also detailed in this material.[Supplementary Material]
http://aip.metastore.ingenta.com/content/aip/journal/jcp/142/18/10.1063/1.4919834
Loading
/content/aip/journal/jcp/142/18/10.1063/1.4919834
Loading

Data & Media loading...

Loading

Article metrics loading...

/content/aip/journal/jcp/142/18/10.1063/1.4919834
2015-05-08
2016-09-28

Abstract

We present elements of a stability theory for small, stochastic, nonlinear chemical reaction networks. Steady state probability distributions are computed with zero-information (ZI) closure, a closure algorithm that solves chemical master equations of small arbitrary nonlinear reactions. Stochastic models can be linearized around the steady state with ZI-closure, and the eigenvalues of the Jacobian matrix can be readily computed. Eigenvalues govern the relaxation of fluctuation autocorrelation functions at steady state. Autocorrelation functions reveal the time scales of phenomena underlying the dynamics of nonlinear reaction networks. In accord with the fluctuation-dissipation theorem, these functions are found to be congruent to response functions to small perturbations. Significant differences are observed in the stability of nonlinear reacting systems between deterministic and stochastic modeling formalisms.

Loading

Full text loading...

/deliver/fulltext/aip/journal/jcp/142/18/1.4919834.html;jsessionid=o_sjaE72giJiVloKVYYj6Yen.x-aip-live-03?itemId=/content/aip/journal/jcp/142/18/10.1063/1.4919834&mimeType=html&fmt=ahah&containerItemId=content/aip/journal/jcp
true
true

Access Key

  • FFree Content
  • OAOpen Access Content
  • SSubscribed Content
  • TFree Trial Content
752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
/content/realmedia?fmt=ahah&adPositionList=
&advertTargetUrl=//oascentral.aip.org/RealMedia/ads/&sitePageValue=jcp.aip.org/142/18/10.1063/1.4919834&pageURL=http://scitation.aip.org/content/aip/journal/jcp/142/18/10.1063/1.4919834'
Right1,Right2,Right3,