Stiffness in stochastic chemically reacting systems: The implicit tau-leaping method

Journal of Chemical Physics

Volumn 119, Issue 24, 2003, Pages 12784-12794

  Rathinam, Muruhan ^a   Petzold, Linda R ^b   Cao, Yang ^b   Gillespie, Daniel T ^c  

^a UNIVERSITY OF MARYLAND BALTIMORE COUNTY   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

^c Daniel T Gillespie Consulting   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ALGORITHMS; CHEMICAL REACTIONS; COMPUTER SIMULATION; DIFFERENTIAL EQUATIONS; RANDOM PROCESSES; STIFFNESS;

CONTINUOUS DETERMINISTIC LEVEL; DISCRETE STOCHASTIC LEVEL; TAU-LEAPING METHOD;

QUANTUM THEORY;

EID: 0942279178     PISSN: 00219606     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.1627296     Document Type: Article

References (19)

1. H. H. McAdams and A. Arkin, Trends Genet. 15, 65 (1999).
2. H. H. McAdams and A. Arkin, Proc. Natl. Acad. Sci. U.S.A. 94, 814 (1997).
3. A. Arkin, J. Ross, and H. H. McAdams, Genetics 149, 1633 (1998).
4. N. Fedoroff and W. Fontana, Science 297, 1129 (2002).
5. M. A. Gibson and J. Bruck, J. Phys. Chem. 104, 1876 (2000).
6. D. T. Gillespie, J. Chem. Phys. 115, 1716 (2001).
7. ji of Refs. 6, 10, 11. The present indexing corresponds to the commonly accepted definition of the stoichiometric matrix.
8. n]/n! (n=0.1,⋯). Both the mean and the variance of P(a, τ) are equal to aτ. P(a, τ) can be interpreted physically as the number of events that will occur in any finite time τ, given that the probability of an event occurring in any future infinitesimal time dt is adt.
9. 2)=m + σN(0,1).
10. D. T. Gillespie, J. Chem. Phys. 113, 297 (2000).
11. D. T. Gillespie, J. Phys. Chem. A 106, 5063 (2002).
12. P. F. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, 2nd ed. (Springer, Berlin, 1995).
13. P. M. Burrage and K. Burrage, SIAM J. Sci. Comput. (USA) 24, 848 (2002).
14. -(m-1). As a consequence, while the terms under the first summation sign in Eq. (4) are roughly proportional to the system size, the terms under the second summation sign are roughly proportional to the square root of the system size. So, in the thermodynamic limit, the latter terms negligibly small compared to the former terms. Of course, real systems, no matter how large, are necessarily finite, and in situations where the terms in the first summation in Eq. (4) add up to practically zero (for instance at equilibrium), the fluctuating second sum can become important.
15. U. M. Ascher and L. R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations (SIAM, 1998).
16. D. T. Gillespie, Markov Processes: An Introduction for Physical Scientists (Academic, Philadelphia, PA, 1992).
17. D. T. Gillespie, J. Comput. Phys. 22, 403 (1976).
18. I. G. Curtiss and P. J. Staff, J. Chem. Phys. 44, 990 (1976).
19. See Ref. 16, p. 385, Eqs. (6.1-29) and (6.1-30), and note that the functions v(x) and a(x) appearing in those equation are defined in Eqs. (6.1-13) to be the same as our functions A(x) and D(x), respectively.