The chemical Langevin and Fokker-Planck equations for the reversible isomerization reaction

Journal of Physical Chemistry A

Volumn 106, Issue 20, 2002, Pages 5063-5071

  Gillespie, Daniel T ^a  

^a Dan T Gillespie Consulting   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

CHEMICAL FOKKER-PLANCK EQUATION; CHEMICAL LANGEVIN EQUATION; CHEMICAL MASTER EQUATION; REACTION RATE EQUATION;

DIFFERENTIAL EQUATIONS; HEAVY IONS; KINETIC THEORY; MARKOV PROCESSES; MOLECULES; RATE CONSTANTS;

ISOMERIZATION;

EID: 0037162152     PISSN: 10895639     EISSN: None     Source Type: Journal    
DOI: 10.1021/jp0128832     Document Type: Article

References (14)

1. Gillespie, D. T. Physica A 1992, 188, 404-425.
2. McQuarrie, D. A. J. Appl. Prob. 1967, 4, 413-478.
3. McAdams, H. H. and Arkin, A. P. Proc. Natl. Acad. Sci. U.S.A. 1997, 94, 814-819.
4. Gillespie, D. T. J. Chem. Phys. 2000, 113, 297-306.
5. j(t′)) = δ(j,j′) δ(t-t′), where the first delta function is Kronecker's and the second is Dirac's. But this equation contains no information that is not already contained, arguably more clearly, in eq 6; also, it tends to obscure the fact that the "derivative" on the left side does not really exist by ordinary mathematical standards. In any case, we shall not be needing this alternate version of the chemical Langevin equation in our work here.
6. Equation 7 can be formally obtained by Taylor expanding the right side of the CME (4) and then summarily dropping all terms with derivatives of order >2. But that is not the way eq 7 is deduced in ref 4: In ref 4, eq 7 is inferred directly from eq 6 by appealing to some general results in continuous Markov process theory [see, e.g.: Gillespie, D. T. Am. J. Phys. 1996, 64, 1246-1257], and eq 6 in turn is shown to be a direct approximate consequence of the fundamental premise (2) whenever conditions i and ii are satisfied.
7. Zwanzig, R. J. Phys. Chem. B 2001, 105, 6472-6473.
8. 1 a finite constant. But we have no need to invoke that special limit here.
9. 1/2 in the Langevin equation will contain the product of a normal with the square root of a normal, which is not a normal; therefore, X(t+dt) will not be normal.
10. 1(x)dt the expression in eq 19b will be -1, again in agreement with eq 19a. If neither of these two things happen, both Θ functions in eq 19b will vanish, and we have the third eventuality in eq 19a.
11. The stochastic simulation algorithm (SSA) is described in: Gillespie, D. T. J. Comput. Phys. 1976, 22, 403-434
12. 1(x)Delta;t, or no reaction otherwise. This is a conceptually straightforward implementation of the fundamental premise (2), but it has the practical drawback that is approximate; it becomes exact only in the limit that Delta;t becomes infinitesimally small, in which limit the procedure becomes infinitely slow and infinitely consuming of random numbers r. The SSA by contrast is exact with respect to premise (2), and hence also with respect to the CME (4). It requires two random numbers for each reaction event; one of those random numbers determines the time to the next reaction event, and the other determines the identity (index) of that reaction. The SSA does not require one to choose a time step size, nor does it entail approximating an infinitesimal time interval dt by a finite time interval Delta;t. The main limitation of the SSA derives from the fact that it does dutifully simulate every reaction event that occurs in the system: If the molecular population level of any reactant species happens to be so large that an enormous number of reaction events actually occur per unit of real time, the progress of the SSA in real time will be extremely slow. Of course, this limitation also applies to the other simulation algorithm.
13. For an extensive discussion of the Langevin and related approximation strategies for accelerating the stochastic simulation algorithm, see: Gillespie, D. T. J. Chem. Phys. 2001, 115, 1716-1733. Work in this area is ongoing.
14. -(m-1) (see ref 1), which effectively "cancels" all but one of the population factors in the thermodynamic limit. The CLE (6) thus implies quite generally the well known rule-of-thumb that "fluctuations scale like the square root of the molecular population."