General mathematical formula for near equilibrium relaxation kinetics of basic enzyme reactions and its applications to find conformational selection steps.

A general mathematical formula of basic enzyme reactions was derived with nearly no dependence on conditions nor assumptions on relaxation kinetic processes near equilibrium in a simple single-substrate-single-product enzyme reaction. The new formula gives precise relationships between the rate constants of the elementary reaction steps and the apparent relaxation rate constant, rather than the initial velocity that is generally used to determine enzymatic parameters according to the Michaelis-Menten theory. The present formula is shown to be complementary to the Michaelis-Menten formulae in a sense that the initial velocity and the relaxation rate constant data together could determine the enzyme-substrate dissociation constant K, which has been usually conditionally approximated by the Michaelis constant K within the framework of the Michaelis-Menten formulae. We also describe relaxation kinetics of enzyme reactions that include the conformational selection processes, in which only one enzymatic conformer among a conformational ensemble can bind with either the substrate or product. The present mathematical approaches, together with numerical computation analyses, suggested that the presence of conformational selection steps in enzymatic reactions can be experimentally detected simply by enzymatic assays with catalytic amounts of enzyme.