Q-bio:Numerical simulation of a complex biochemical reaction

Brief description

A standard algorithm for simulating biochemical reactions is that due to Gillespie. Briefly, since elementary reactions have exponential wait times, one can add the rates for all reactions that can happen in a network, obtaining a rate for (an exponentially distributed) time for the next reaction of any time happening. Such exponential random number is easy to generate, and then, knowing that a reaction will happen at a particular time, one can sample which particular reaction will happen then by considering the relative values of all involved single-reaction rates. But if a reaction is not elementary (e.g., a Michaelis-Menten reaction), then the wait time is not exponentially distributed, and the probability of the next reaction happening per unit time depends on when the previous reaction happened (that is, the system becomes non-Markovian). Can we derive a generalization of the Gillespie algorithm for this setup? Gibson and Bruck have solved this problem (see reference below). That is, if a wait-time distribution is known, they provide an algorithm for simulations using it. However, how do we calculate the wait time distributions for nontrivial reactions? For linear directional pathways, this is easy to do. But even a simple Michaleis-Menten reaction, which involves a reversible binding step, presents problems. Can we calulate the wait time for an MM reaction? For other complex reactions?

Contact instructor

Ilya Nemenman and Anton Zilman

References

1. M Gibson and J Bruck. Efficient Exact Stochastic Simulation of Chemical Systems with Many Species and Many Channels. J Phys Chem A, Vol. 104, No. 9, pages 1876-1889, 2000. PDF.
2. Lecture by Gillespie
3. D Gillespie. Stochastic Simulation of Chemical Kinetics. "Ann Rev Phys Chem" 58, 35-55, 2007. PDF.

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