  Zero-order ultrasensitivity (ZOU) is a long known and interesting phenomenon
in enzyme networks. Here, a substrate is reversibly modified by two
antagonistic enzymes (a "push-pull" system) and the fraction in modified state
undergoes a sharp switching from near-zero to near-unity at a critical value of
the ratio of the enzyme concentrations, under saturation conditions. ZOU and
its extensions have been studied for several decades now, ever since the
seminal paper of Goldbeter and Koshland (1981); however, a complete
probabilistic treatment, important for the study of fluctuations in finite
populations, is still lacking. In this paper, we study ZOU using a modular
approach, akin to the total quasi-steady state approximation (tQSSA). This
approach leads to a set of Fokker-Planck (drift-diffusion) equations for the
probability distributions of the intermediate enzyme-bound complexes, as well
as the modified/unmodified fractions of substrate molecules. We obtain explicit
expressions for various average fractions and their fluctuations in the linear
noise approximation (LNA). The emergence of a "critical point" for the
switching transition is rigorously established. New analytical results are
derived for the average and variance of the fractional substrate concentration
in various chemical states in the near-critical regime. For the total fraction
in the modified state, the variance is shown to be a maximum near the critical
point and decays algebraically away from it, similar to a second-order phase
transition. The new analytical results are compared with existing ones as well
as detailed numerical simulations using a Gillespie algorithm.
