The Quasi-Steady State Assumption in an Enzymatically Open System

TL;DR

This paper extends the quasi-steady state assumption (QSSA) to enzymatically open systems with enzyme input and removal, providing new approximations, validity conditions, and analyzing oscillatory behavior in such systems.

Contribution

The authors develop a modified QSSA for open enzymatic systems, derive validity conditions, and compare approximations with full system dynamics across various parameters.

Findings

Approximations remain accurate even outside strict validity conditions.

Derived conditions ensure the validity of the new QSSA.

Numerical simulations reveal potential for damped oscillations.

Abstract

The quasi-steady state assumption (QSSA) forms the basis for rigorous mathematical justification of the Michaelis-Menten formalism commonly used in modeling a broad range of intracellular phenomena. A critical supposition of QSSA-based analyses is that the underlying biochemical reaction is enzymatically "closed," so that free enzyme is neither added to nor removed from the reaction over the relevant time period. Yet there are multiple circumstances in living cells under which this assumption may not hold, e.g. during translation of genetic elements or metabolic regulatory events. Here we consider a modified version of the most basic enzyme-catalyzed reaction which incorporates enzyme input and removal. We extend the QSSA to this enzymatically "open" system, computing inner approximations to its dynamics, and we compare the behavior of the full open system, our approximations, and the closed system under broad range of kinetic parameters. We also derive conditions under which our new approximations are provably valid; numerical simulations demonstrate that our approximations remain quite accurate even when these conditions are not satisfied. Finally, we investigate the possibility of damped oscillatory behavior in the enzymatically open reaction.