Geometrical picture of reaction in enzyme kinetics

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Abstract
A generalization of the steady‐state and equilibrium approximations for solving the coupled differential equations of a chemical reaction is presented. This method determines the (steady‐state) reaction velocity in closed form. Decay from rapidly equilibrating networks is considered, since many enzyme mechanisms belong to this category. In such systems, after transients have died away, the phase‐space flow lies close to a unique trajectory, the slow manifoldM. Locating M reduces the description of the system’s progress to a one‐dimensional integration. This manifold is a solution of a functional equation, derived from the differential equations for the reaction. As an example, M for Michaelis–Menten–Henri mechanism is found by direct iteration. The solution is very accurate and the appropriate boundary conditions are obeyed automatically. I n a n a r b i t r a r y m e c h a n i s m, a t v a n i s h i n g d e c a y r a t e, t h e s l o w m a n i f o l d b e c o m e s a l i n e o f e q u i l i b r i u m s t a t e s, w h i c h s o l v e s t h e f u n c t i o n a l e q u a t i o n e x a c t l y; it is thus a good initial approximation at finite decay rate. These ideas are applied to a mechanism with both enzyme–substrate and enzyme–product complexes.