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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!

Nadir
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1 Answers1

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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.

pveber
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  • Thank you! What do you mean with "you can approximate the Hill function locally with mass action"? Must I write the Hill function as it is, or must do some changings? ... and, what do you mean with "locally"? – Nadir Jul 09 '13 at 13:12
  • Well, if you look close enough at a regular function, it looks like a straight line. Here, if you zoom at a particular portion of your Hill function, you can find a polynomial function (mass action kinetics) which is very close to it. But further from this portion, the approximation will get worse. This is what i meant by locally: you cannot find a new mass action kinetics expression that will correctly approximate the hill function for any concentration. – pveber Jul 09 '13 at 14:05