  In an experimental study of single enzyme reactions, it has been proposed
that the rate constants of the enzymatic reactions fluctuate randomly,
according to a given distribution. To quantify the uncertainty arising from
random rate constants, it is necessary to investigate how one can simulate such
a biochemical system. To do this, we will take the Gillespie's stochastic
simulation algorithm for simulating the evolution of the state of a chemical
system, and study a modification of the algorithm that incorporates the random
rate constants. In addition to simulating the waiting time of each reaction
step, the modified algorithm also involves simulating the random fluctuation of
the rate constant at each reaction time. We consider the modified algorithm in
a general framework, then specialize it to two contrasting physical models, one
in which the fluctuations occur on a much faster time scale than the reaction
step, and the other in which the fluctuations occur much more slowly. The
latter case was applied to the single enzyme reaction system, using in part the
Metropolis-Hastings algorithm to enact the given distribution on the random
rate constants. The modified algorithm is shown to produce simulation outputs
that are corroborated by the experimental results. It is hoped that this
modified algorithm can subsequently be used as a tool for the estimation or
calibration of parameters in the system using experimental data.
