convert between deterministic to stochastic rates?

Question

I have a bit of confusion, if somebody could shed a light that would be appreciated.

Suppose a reaction of the form (reaction 4) in the ref. paper

R4:= S1+S2->> S3 at some rate $k_1.$

S1 and S2 are two species with population number $X_1$ and $X_2.$

Then the reaction rate in general is:

$$k_1 X_1X_2.$$

If we want to convert between deterministic to stochastic rates, we need to know the units.

For example the rate above becomes $$\frac{k_1 X_1 X_2}{n_a V}.$$

where $n_a$ is the Avogadro's number.

The stochastic rate or probability per unit time is

$$c_1 X_1 X_2.$$

Equating the equations of the two rates become

$$\frac{k_1 X_1 X_2}{n_a V}=c_1 X_1 X_2.$$

This leads to $$K_1=c_1 n_a V$$

Is this what is written in Gillespie's paper (highlighted in red) or I am missed something out. It confuses me when he mentioned the average reaction rate and products of the average densities.

Thank you

please see image below.

it is from Gillespie 1976 stochastic simulation algorithm

ref: http://web.mit.edu/endy/www/scraps/signal/JCompPhys%2822%29403.pdf

Answer 1

While it is interesting to go back to the basics sometimes, it may not be the most helpful as more recent resources may present that in a clearer way. I am notably recommending to look at the works of Thomas Kurtz and David Anderson on the topic.

That said, assume first that the propensity functions are amass-action and that the volume is given by $V$. Define further $d$ to be the reaction rate in the deterministic setting and $s$ to be the reaction rate in the stochastic setting.

Then, we get the following correspondence:

• zeroth order: $s=dV$,
• first order (i.e. monomolecular): $s=d$,
• second-order (i.e. bimolecular, same species, homodimer): $s=d/V$, and
• second-order (i.e. bimolecular, different species, heterodimer): $s=2d/V$.

This can be be simply obtained by looking at the propensities and rescaling appropriately by the volume. For, instance, if you have $sX$ in the stochastic setting, and rescale that with $V$ as $sX/V$. Since $[x]=X/V$ (where $[x]$ denotes concentration), then we get that $s[x]$ and, hence, $s=d$.

Similarly, pick $sX_1X_2$ and rescale with the volume to get $sX_1X_2/V$. In order to get everything in terms of concentrations, we need to pick $s=d/V$ to get $dX_1X_2/V^2=d[x_1][x_2]$.