Two different types of kinetics, where the initial rate increases faster or slower than the reactant concentration, can coexist on bell-shaped kinetic dependencies

We have shown the possibility of mathematical analysis of complex biochemical reactions without simplifying assumptions regarding their parameters by investigating the mechanism of adenosine triphosphate (ATP) hydrolysis by the “basal” Mg 2+ -ATPase in which ATP (substrate), Mg 2+ (essential activator) and MgATP (true substrate) compete for binding with the enzyme catalytic site. For this mechanism, we proved, using the quasi-steady state principle by Bodenstein, the existence and uniqueness of a quasi-steady state at any fixed parameters (initial reactant concentrations, positive reaction rate constants). On this basis, the definition of the initial reaction rate v as a function of the initial concentrations of ATP ( x 1 ), Mg 2+ ( x 2 ), ATPase ( x 3 ) is given and the dependencies v vs. x i are investigated. We proved that at fixed initial concentrations of any two reactants, the kinetic curve v vs. x i has a unique maximum point. In different regions of the kinetic curve located before and after the maximum point, the initial rate may vary in different ways: in one region it can change more slowly than the reactant concentration, as in the case of hyperbolic dependencies, and in the other it can change much faster than the reactant concentration, as in the case of sigmoid dependencies, which is typical for regulatory enzymes. This initial rate behaviour causes the existence of four different types of bell-shaped kinetic curves. To quantify the degree of sigmoidity of kinetics, we used derivatives ∂ln v/∂ln x_i which allowed us to estimate the maximal velocity of the initial rate change.