  Eigenvalue disparity, also known as timescale separation, permits the
systematic reduction of deterministic models of enzyme kinetics. Geometric
singular perturbation theory, of which eigenvalue disparity is central,
provides a coordinate-free framework for deriving reduced mass action models in
the deterministic realm. Moreover, homologous deterministic reductions are
often employed in stochastic models to reduce the computational complexity
required to simulate reactions with the Gillespie algorithm. Interestingly,
several detailed studies indicate that timescale separation does not always
guarantee the accuracy of reduced stochastic models. In this work, we examine
the roles of timescale separation and coordinate transformations in the
reduction of the Linear Noise Approximation (LNA) and, unlike previous studies,
we do not require the system to be comprised of distinct fast and slow
variables. Instead, we adopt a coordinate-free approach. We demonstrate that
eigenvalue disparity does not guarantee the accuracy of the reduced LNA, known
as the slow scale LNA (ssLNA). However, the inaccuracy of the ssLNA can often
be eliminated with a proper coordinate transformation. For planar systems in
separated (standard) form, we prove that the error between the variances of the
slow 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 to
construct a reduced equation that is accurate near the singular limit in the
deterministic regime. However, this reduction in the stochastic regime is far
less accurate, which illustrates that eigenvalue disparity plays a central role
in stochastic model reduction.
