Accuracy of the Michaelis-Menten approximation when analysing e↵ects of molecular noise

Quantitative biology relies on the construction of accurate mathematical models, yet the e↵ectiveness of these models is often predicated on making simplifying approximations that allow for direct comparisons with available experimental data. The Michaelis-Menten approximation is widely used in both deterministic and discrete stochastic models of intracellular reaction networks, due to the ubiquity of enzymatic activity in cellular processes and the clear biochemical interpretation of its parameters. However, it is not well understood how the approximation applies to the discrete stochastic case or how it extends to spatially inhomogeneous systems. We study the behaviour of the discrete stochastic Michaelis-Menten approximation as a function of system size and show that significant errors can occur for small volumes, in comparison with a corresponding mass action system. We then explore some consequences of these results for quantitative modelling. One consequence is that fluctuation-induced sensitivity, or stochastic focusing, can become highly exaggerated in models that make use of Michaelis-Menten kinetics even if the approximations are excellent in a deterministic model. Another consequence is that spatial stochastic simulations based on the reaction-di↵usion master equation can become highly inaccurate if the model contains Michaelis-Menten terms.