Hello,
I am studying a basic model that will aid in understanding more complex models. The basic model is the Michaelis-Menten Reaction with Substrate Input with No Product. The enzymatic reaction scheme is below:
Null -> S + E <-> SE -> E,
with production rate ‘Q’, complex association rate ‘a’, complex dissociation rate ‘d’, and catalytic rate ‘k’.
I wrote down the Reaction Rate equations (ODEs) for the system and solved for the steady-state concentration of S. It turns out that it is defined only when Q < V_{max,E}. There is a “protein catastrophe” for large enough Q.
This is an open system that does not satisfy the property of detailed balance. I tried investigating with an analytical treatment of the Chemical Master Equation. I also tried Moment Closure Approximation methods. After much effort and literature survey, I found it is highly unlikely that one can derive the stationary distribution for S, let alone its limiting distribution.
So I sampled trajectories of S from the (exact) CME using the Gillespie algorithm. I began the stochastic simulation at the steady-state concentrations given by numerical integration of the ODEs. I chose parameter values such that the system was very close to the protein catastrophe (i.e. large enough Q to yield S ~ 10^7). However, even after running the simulation for a very long time, I found that both mean(S) and variance(S) had not settled to a steady-state. Instead, they were fluctuating with respect to simulation time.
My questions:
- Why could this be happening for the Mean? I thought that the large species numbers in the system would remedy any discrepancy between the deterministic average and the stochastic mean.
- Why for the Variance?
You see, this poses some issues because my end goal is to calculate the Coefficient of Variation for S, as a function of Q.
Thanks for any help.
