# Is this adaption of the Gillespie algorithm using Michaelis constants justifiable?

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.

• The answer of @WYSIWYG is wrong in parts. (I can not comment because I have not enough reputation.) Gillespie does not make the assumption that "the reaction propensity (probability) in an infinitesimally small interval of time is same as the macroscopic reaction rate." He rather derives this statement from physical considerations about collisions of molecules. The derivation is given in Gillespies 1977 paper "Exact Stochastic Simulations of Coupled Chemical Reactions". The derivation holds for elementary reactions only. Michaelis-Menten rate obviously does not describe an elementary reaction. – PascalIv Dec 13 '17 at 12:36

The stochastic simulation algorithm by Gillespie makes the assumption that reaction propensity (probability) in an infinitesimally small interval of time is same as the macroscopic reaction rate. (For details, you can see my answer in Chemistry.SE).

You can model and simulate every step of enzyme catalysis or you can also use the Michaelis-Menten model. In any case, you can extend the same approximations to the stochastic model.

This is also applicable to other Michaelis-Menten like (or Hill) functions of the form $$\frac{x^n}{k^n+x^n} \qquad \text{or}\qquad \frac{k^n}{k^n+x^n}$$.

There are many papers that have done this. Note that the non-linear nature of these functions results in deviation of the stochastic behaviour from the deterministic behaviour. Read about stochastic focussing. Modelling each step exactly (binding, unbinding and catalysis) may result in different behaviour depending on your parameters. Whether you should use Michaelis-Menten function to describe reaction rate totally depends on how reasonable your approximation is.

Very interesting question. People use Gillespie's algorithm for several sort of kinetic functions such as Michaelis-Menten (MM), Hill etc., no matter what the original assumptions of Gillespie's algorithm are.

Concerning MM, There was a 2011 paper by Gillespie himself that shows that the approximation is applicable in discrete stochastic models and that the validity conditions are the same as in the deterministic regime.

https://www.ncbi.nlm.nih.gov/pubmed/21261403

Remember that, because MM is derived under a QSSA, you can break down the S->P transformation as its single-step components. See for instance a comparison here

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3578938/