We show how stiffness manifests itself in the simulation of chemical reactions at both the continuous-deterministic level and the discrete-stochastic level. Existing discrete stochastic simulation methods, such as the stochastic simulation algorithm and the (explicit) tau-leaping method, are both exceedingly slow for such systems. We propose an implicit tau-leaping method that can take much larger time steps for many of these problems.
REFERENCES
1.
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.
5.
6.
8.
The Poisson random variable is the integer-valued random variable defined by Both the mean and the variance of are equal to 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 is
9.
We denote by the normal (or Gaussian) random variable with mean m and variance This random variable has the useful property that
10.
11.
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
, A variable stepsize implementation for stochastic differential equations
, SIAM J. Sci. Comput. (USA)
24
, 848
(2002
).14.
In the thermodynamic limit, the species populations and the system volume Ω diverge together, and proportionately. It turns out that, in this limit, all propensity functions diverge linearly with the system size, because the propensity function for an mth-order reaction will contain m factors along with a factor 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 typically become 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.
18.
19.
See Ref. 16, p. 385, Eqs. (6.1-29) and (6.1-30), and note that the functions and appearing in those equations are defined in Eqs. (6.1-13) to be the same as our functions and respectively.
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