I would like to run a discrete simulation of a biological system as it can be done, e.g., using the Gillespie algorithm. However, the Gillespie algorithm requires you to know the reaction rate constants of each involved reaction whereas I only know the Michaelis constant $K_M$ and the maximum rate $V_{max}$ of each reaction.
Since the Gillespie algorithm only uses the reaction rate constants to calculate the reaction rate $v$ (which, again, is used to determine the probability of a given reaction), I wonder wheter it is justifiable to simply replace this calculation of $v = k * [S]$ by $v = \frac{V_{max}\cdot[S]}{K_M + [S]}$ for each reaction $S \rightarrow P$ and, apart from that, use the Gillespie algorithm as is? If not, is there any better way to run a discrete simulation of the system with the given parameters?
Note: I've already asked in a different question whether it is possible to get (an estimation of) the reaction rates knowing only the parameters $K_M$ and $V_{max}$. I hope my both questions are sufficiently distinct to justify two different posts.