Derivation of Michaelis-Menten kinetics for discrete stochastic simulations

Question

According to this article,

[...] the propensity function for the conversion reaction S → P in the well-mixed discrete stochastic case can be written $a(S) = \frac{V_{max}\cdot S}{K_m + S/\Omega}$ where $\Omega$ is the system volume.

I don't quite understand how this formula is derived from the non-discrete Michaelis-Menten kinetics $v = \frac{V_{max} \cdot [S]}{K_M + [S]}$ (see Wikipedia). According to my understanding, $[S] = S/\Omega$. If we apply this to the formula from Wikipedia we get $$ \frac{V_{max} \cdot[S]}{K_M + [S]} = \frac{V_{max} \cdot S/\Omega}{K_M + S/\Omega} = \frac{V_{max} \cdot S}{\Omega \cdot K_M + S} $$ which is not the same as $\frac{V_{max}\cdot S}{K_m + S/\Omega}$. So, how can one derive the formula from the quoted article (if it is correct)? If not, how can we correctly get to a discrete propensity function from the Michaelis-Menten kinetics?

Answer 1

I think I figured it out myself. Since $a(S)$ is in $\frac{mol}{s}$ and $v$ is in $\frac{\frac{mol}{l}}{s}$, we have that $a(S)= v * \Omega$. Therefore $$ a(S) = \frac{V_{max}\cdot [S]}{K_M + [S]} \cdot \Omega = \frac{V_{max}\cdot S / \Omega \cdot \Omega}{K_M + S / \Omega} = \frac{V_{max}\cdot S}{K_M + S/\Omega}. $$