| Description |
We analyze different models of several chemical reactions. We find that, for some reactions, the steady state behavior of the chemical master equation, which describes the continuous time, discrete state Markov process, is poorly approximated by the deterministic model derived from the law of mass action or a mean field model derived in a similar way. We show that certain simple enzymatic reactions have bimodal stationary distributions in appropriate parameter ranges, though the deterministic and mean field models for these reactions do not have the capacity to admit multiple equilibrium points no matter the choice of rate constants. We provide power series expansions for these bimodal distributions. We also consider several variants of an autocatalytic reaction. This reaction's deterministic model predicts a unique positive stable equilibrium, but the only stationary distribution of its chemical master equation predicts extinction of the autocatalytic chemical species with probability 1. We show that the transient distribution of this chemical master equation is centered near the deterministic equilibrium and that the stationary distribution is only reached on a much longer time scale. Finally, we consider a model for the rotational direction switching of the bacterial rotary motor and propose two possible reductions for the state space of the corresponding Markov chain. One reduction, a mean field approximation, is unable to produce physically realistic phenomena. The other reduction retains the properties of interest in the system while significantly decreasing the computation required for analysis. We use this second reduction to fit parameters for the full stochastic system and suggest a mechanism for the sensitivity of the switch. |
