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On Wed, Feb 09, 2005 at 08:48:54PM +0000, Darren Wilkinson wrote:

> Now, if the numbers of molecules involved are quite large (but not large
> enough to justify a deterministic simulation), then the continuous
> diffusion approximation to the discrete-state Markov process might be
> acceptable.

That's one approximation. Would the following be another for the case
of a reaction corresponding to biomass formation? I'm going to treat
the biomass formation process essentially as something draining
species from the model. I keep a _fractional_ tally per involved
species, aside from the "real" species tally. The former is
fractional (to be explained) and the latter is always integral.

If know a rate for this biomass accumulation reaction, then I know its
probability per unit time. On each occurance, I bump up the
fractional tallies of all involved species by the fractional
stoichiometry. If that causes any to surpass an integer, I decrement
the "real" tally by the integral amount and keep the remainder in the
fractional tally. Meanwhile, all the normal steps in the simulation
just work off of the integral, "real" tallies.

The approximation here is gentler than the one above. If the number
of molecules is large enough that the ratio of the fractional
accumulator to the number of molecules is small enough, then the
impact on any computed probability based upon the integral number is
small. This is perhaps a weaker constraint than that needed for
continuous diffussion?

The price you pay is that mass is not balanced instantaneously, but
only on average and with a precision corresponding to the ratio
of fractional to "real" tallies. I'm willing to say that the
biomass comosition is fluctuating and the rest of the system
remains balanced.

In a sense, I'm putting a parasitic element into the sim to drain off
species when the impact is acceptable. That element of the sim is
driven by a probability-per-unit time associated with the biomass
reaction. Meanwhile, the normal stochastic sim is clunking along.

I'm making this up off the top of my head as I type. It shows.

-Ed

--
Ed Frank 630-252-4548 (-5986, fax)
Division of Math & Computer Science Argonne National Laboratory