1887
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.
oa
Enhanced identification and exploitation of time scales for model reduction in stochastic chemical kinetics
Rent:
Rent this article for
Access full text Article
/content/aip/journal/jcp/129/24/10.1063/1.3050350
1.
1.V. Henri, Lois Générales de l’Action des Diastases (Hermann, Paris, 1903).
2.
2.L. Michaelis and M. Menten, Biochem. Z. 49, 333 (1913).
3.
3.A. Ciliberto, F. Capuani, and J. J. Tyson, PLOS Comput. Biol. 3, e45 (2007).
4.
4.C. Gomez-Uribe, G. C. Verghese, and L. A. Mirny, PLOS Comput. Biol. 3, e246 (2007).
5.
5.A. R. Tzafriri and E. R. Edelman, Biochem. J. 402, 537 (2007).
6.
6.M. G. Pedersena, A. M. Bersani, and E. Bersani, Bull. Math. Biol. 69, 433 (2007).
http://dx.doi.org/10.1007/s11538-006-9136-2
7.
7.D. Del Vecchio, A. J. Ninfa, and E. D. Sontag, Mol. Syst. Biol. 4, 161 (2008).
8.
8.M. Frankowicz, M. Moreau, P. Szczȩsny, J. Tòth, and L. Vicente, J. Phys. Chem. 97, 1891 (1993).
http://dx.doi.org/10.1021/j100111a029
9.
9.E. L. Haseltine and J. B. Rawlings, J. Chem. Phys. 117, 6959 (2002).
10.
10.C. Rao and A. Arkin, J. Chem. Phys. 118, 4999 (2003).
11.
11.M. R. Roussel and R. Zhu, J. Chem. Phys. 121, 8716 (2004).
http://dx.doi.org/10.1063/1.1802495
12.
12.Y. Cao, D. T. Gillespie, and L. R. Petzold, J. Chem. Phys. 122, 14116 (2005).
13.
13.J. Goutsias, J. Chem. Phys. 122, 184102 (2005).
http://dx.doi.org/10.1063/1.1889434
14.
14.E. L. Haseltine and J. B. Rawlings, J. Chem. Phys. 123, 164511 (2005).
15.
15.S. Peleš, B. Munsky, and M. Khammash, J. Chem. Phys. 125, 204104 (2006).
http://dx.doi.org/10.1063/1.2397685
16.
16.E. A. Mastny, E. L. Haseltine, and J. B. Rawlings, J. Chem. Phys. 127, 094106 (2007).
http://dx.doi.org/10.1063/1.2764480
17.
17.D. T. Gillespie, J. Phys. Chem. 81, 2340 (1977).
http://dx.doi.org/10.1021/j100540a008
18.
18.M. Gibson and J. Bruck, J. Phys. Chem. A 104, 1876 (2000).
http://dx.doi.org/10.1021/jp993732q
19.
19.S. MacNamara, A. Bersani, K. Burrage, and R. Sidje, J. Chem. Phys. 129, 95105 (2008).
20.
20.C. Gomez-Uribe, Ph.D. thesis, Massachusetts Institute of Technology, 2008.
21.
21.R. Bundschuh, F. Hayot, and C. Jayaprakash, Biophys. J. 84, 1606 (2003).
22.
22.P. B. Warren, S. Tanase-Nicola, and P. R. ten Wolde, J. Chem. Phys. 125, 144904 (2006).
http://dx.doi.org/10.1063/1.2356472
23.
23.M. J. Morelli, R. J. Allen, S. Tanase-Nicola, and P. R. ten Wolde, J. Chem. Phys. 128, 045105 (2008).
http://dx.doi.org/10.1063/1.2821957
24.
24.R. Heinrich and S. Schuster, The Regulation of Cellular Systems, 1st ed. (Chapman and Hall, London, 1996), pp. 112134.
25.
25.C. A. Gómez-Uribe and G. C. Verghese, J. Chem. Phys. 126, 024109 (2007).
26.
26.J. Goutsias, Biophys. J. 92, 2350 (2007).
27.
27.A. Singh and J. Hespanha, in Proceedings of the 45th IEEE Conference on Decision and Control, 2006 (unpublished).
28.
28.A. R. Tzafriri, Bull. Math. Biol. 65, 1111 (2003).
http://dx.doi.org/10.1016/S0092-8240(03)00059-4
29.
29.D. Barik, M. R. Paul, W. T. Baumann, Y. Cao, and J. J. Tyson, Biophys. J. 95, 3563 (2008).
30.
30.R. Gallager, Discrete Stochastic Processes (Springer, New York, 2001).
31.
31.D. T. Gillespie, Physica A 188, 404 (1992).
http://dx.doi.org/10.1016/0378-4371(92)90283-V
32.
32.H. Khalil, Nonlinear Systems (Prentice-Hall, Englewood Cliffs, NJ, 2002).
33.
33.J. A. Borghans, R. J. de Boer, and L. A. Segel, Bull. Math. Biol. 58, 43 (1996).
http://dx.doi.org/10.1007/BF02458281
34.
34.W. E, D. Liu, and E. Vanden-Eijnden, J. Chem. Phys. 123, 194107 (2005).
http://dx.doi.org/10.1063/1.2109987
35.
35.D. T. Gillespie, L. R. Petzold, and Y. Cao, J. Chem. Phys. 126, 137101 (2007);
http://dx.doi.org/10.1063/1.2567036
35.D. T. Gillespie, L. R. Petzold, and Y. Cao, J. Chem. Phys.126, 137102 (2007).
http://dx.doi.org/10.1063/1.2567071
36.
36.Y. Cao, D. T. Gillespie, and L. R. Petzold, J. Chem. Phys. 123, 144917 (2005).
http://dx.doi.org/10.1063/1.2052596
37.
37.G. E. Briggs and J. B. S. Haldane, Biochem. J. 19, 339 (1925).
38.
38.L. A. Segel, Bull. Math. Biol. 50, 579 (1988).
http://dx.doi.org/10.1016/S0092-8240(88)80057-0
39.
39.K. Ball, T. G. Kurtz, L. Popovic, and G. Rempala, Ann. Appl. Probab. 16, 1925 (2006).
http://aip.metastore.ingenta.com/content/aip/journal/jcp/129/24/10.1063/1.3050350
Loading
View: Figures

Figures

Image of FIG. 1.

Click to view

FIG. 1.

Empirical probability distribution obtained from 20 000 realizations of the full CME in Eq. (B1) via the Gillespie algorithm (Ref. 17) at six different time points. We used , and initial enzyme and total substrate equal to 10 and , respectively.

Image of FIG. 2.

Click to view

FIG. 2.

Mean and standard deviation of the total substrate computed from 10 000 realizations for each of the stochastic descriptions considered. The realizations were all obtained from Monte Carlo simulation using the Gillespie algorithm. All parameters are the same as those in Fig. 1. In this case, all approximate methods work well but require only of computation, vs for simulation of the full (“exact”) model.

Image of FIG. 3.

Click to view

FIG. 3.

Errors of the mean and standard deviation of the total substrate vs for the different approximations discussed here. Except for , we used the same parameters as in Fig. 2. We used 2000 Monte Carlo realizations for the smallest value of and increased the number of realizations linearly, working with 20 000 realizations for the largest value of shown; this was done to maximize accuracy for the available computational power. Our method is uniformly more accurate as is varied.

Image of FIG. 4.

Click to view

FIG. 4.

Empirical probability distribution obtained from 120 000 realizations of the full (nonapproximate) CME in Eq. (B1) via the Gillespie algorithm at six different time points. The system parameters are , , and initial enzyme and total substrate both equal to 100. Note that the separation of time scales in this example is not as extreme as that of the example in Fig. 1.

Image of FIG. 5.

Click to view

FIG. 5.

Mean and standard deviation of the total substrate computed from 10 000 realizations for each of the stochastic descriptions considered for the system parameters of Fig. 4. The realizations were all obtained from Monte Carlo simulation using the Gillespie algorithm. Our method produces more accurate solutions than the other approximate methods shown.

Image of FIG. 6.

Click to view

FIG. 6.

Errors of the mean and standard deviation of the total substrate vs for the different approximations discussed here. Except for , we used the same parameters as in Fig. 5. We used 1000 Monte Carlo realizations for the smallest value of and increased the number of realizations linearly, working with 10 000 realizations for the largest value of shown; this was done in order to maximize accuracy for the available computational power. Our method is uniformly better as is varied.

Loading

Article metrics loading...

/content/aip/journal/jcp/129/24/10.1063/1.3050350
2008-12-31
2014-04-21

Abstract

Widely different time scales are common in systems of chemical reactions and can be exploited to obtain reduced models applicable to the time scales of interest. These reduced models enable more efficient computation and simplify analysis. A classic example is the irreversible enzymatic reaction, for which separation of time scales in a deterministic mass action kinetics model results in approximate rate laws for the slow dynamics, such as that of Michaelis–Menten. Recently, several methods have been developed for separation of slow and fast time scales in chemical master equation (CME) descriptions of stochastic chemical kinetics, yielding separate reduced CMEs for the slow variables and the fast variables. The paper begins by systematizing the preliminary step of identifying slow and fast variables in a chemical system from a specification of the slow and fast reactions in the system. The authors then present an enhanced time-scale-separation method that can extend the validity and improve the accuracy of existing methods by better accounting for slow reactions when equilibrating the fast subsystem. The resulting method is particularly accurate in systems such as enzymatic and protein interaction networks, where the rates of the slow reactions that modify the slow variables are not a function of the slow variables. The authors apply their methodology to the case of an irreversible enzymatic reaction and show that the resulting improvements in accuracy and validity are analogous to those obtained in the deterministic case by using the total quasi-steady-state approximation rather than the classical Michaelis–Menten. The other main contribution of this paper is to show how mass fluctuation kinetics models, which give approximate evolution equations for the means, variances, and covariances of the concentrations in a chemical system, can feed into time-scale-separation methods at a variety of stages.

Loading

Full text loading...

/deliver/fulltext/aip/journal/jcp/129/24/1.3050350.html;jsessionid=2ngp4c6b5ookr.x-aip-live-01?itemId=/content/aip/journal/jcp/129/24/10.1063/1.3050350&mimeType=html&fmt=ahah&containerItemId=content/aip/journal/jcp
true
true
This is a required field
Please enter a valid email address
752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Enhanced identification and exploitation of time scales for model reduction in stochastic chemical kinetics
http://aip.metastore.ingenta.com/content/aip/journal/jcp/129/24/10.1063/1.3050350
10.1063/1.3050350
SEARCH_EXPAND_ITEM