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Eigenvalue disparity, also known as timescale separation, permits thesystematic reduction of deterministic models of enzyme kinetics. Geometricsingular perturbation theory, of which eigenvalue disparity is central,provides a coordinate-free framework for deriving reduced mass action models inthe deterministic realm. Moreover, homologous deterministic reductions areoften employed in stochastic models to reduce the computational complexityrequired to simulate reactions with the Gillespie algorithm. Interestingly,several detailed studies indicate that timescale separation does not alwaysguarantee the accuracy of reduced stochastic models. In this work, we examinethe roles of timescale separation and coordinate transformations in thereduction of the Linear Noise Approximation (LNA) and, unlike previous studies,we do not require the system to be comprised of distinct fast and slowvariables. Instead, we adopt a coordinate-free approach. We demonstrate thateigenvalue disparity does not guarantee the accuracy of the reduced LNA, knownas the slow scale LNA (ssLNA). However, the inaccuracy of the ssLNA can oftenbe eliminated with a proper coordinate transformation. For planar systems inseparated (standard) form, we prove that the error between the variances of theslow variable generated by the LNA and the ssLNA is $\mathcal{O}(\varepsilon)$.We also address a nilpotent Jacobian scenario and use the blow-up method toconstruct a reduced equation that is accurate near the singular limit in thedeterministic regime. However, this reduction in the stochastic regime is farless accurate, which illustrates that eigenvalue disparity plays a central rolein stochastic model reduction.

On the role of eigenvalue disparity and coordinate transformations in the reduction of the linear noise approximation