Derivation of Michaelis-Menten kinetics for discrete stochastic simulations
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
$$
a(S) = \frac{V_{max}\cdot [S]}{K_M + [S]} \cdot \Omega = \frac{V_{max}\cdot S / \Omega \cdot \Omega}{K_M + S / \Omega} = \frac{V_{max}\cdot S}{K_M + S/\Omega}.
$$