From ODE Hill kinetics to stochastic mass action

Question

I am facing a problem translating a model with ODE to a stochastic model.

The original model contains two expressions:

1) k6f2*PKB_S473P^n6/(km6^n6+PKB_S473P^n6))*AS160

2) k9f1*S6K*mTORC1a^n9/(km9^n9+mTORC1a^n9)

that are described with Hill kinetics.

I was wondering how to tranlate the previous two expressions into a mass action law, to be able to simulate them with a stochastic simulator.

Thanks all in advance!

Answer 1

Transforming Hill kinetics into mass action kinetics cannot be done in a satisfying way, because for large concentrations, the former lead to bounded rates, while the latter does not. Mass action law will not show the saturation effect that can be seen with Hill kinetics. At best, you can approximate the Hill function locally with mass action, but then your simulation should be done around a stable steady state. Alternatively, if you introduce intermediate species and reactions with appropriate coefficients, you can obtain Michaelis-Mentens kinetics (Hill kinetics with unit Hill coefficient). In principle, you could even regain cooperativity using more complex intermediate species, but this is probably rather involved.