# Derivation of Michaelis-Menten kinetics for discrete stochastic simulations

[...] 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?
• I guess the article is incorrect. I checked different possibilities but none of them seem to give this relation. The papers cited by this article also do not have this formula. Their supplementary data also does not explain anything. If $K_M$ is defined differently then this formula may possibly be right but since the article does not mention anything like that, we should not assume so. Oct 14, 2016 at 7:16
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 $$a(S) = \frac{V_{max}\cdot [S]}{K_M + [S]} \cdot \Omega = \frac{V_{max}\cdot S / \Omega \cdot \Omega}{K_M + S / \Omega} = \frac{V_{max}\cdot S}{K_M + S/\Omega}.$$