Theory on the rate equations of Michaelis-Menten type enzyme kinetics with competitive inhibition

Abstract

We derive approximate expressions under various conditions of validity over both pre- and post-steady state regimes of the velocity-substrate-inhibitor phase-spaces of fully and partially competitive enzyme kinetic schemes. Our refinement over the currently available standard quasi steady state approximations (QSSA) seems to be valid for wide range of enzyme to substrate and inhibitor ratios. Further, we show that, the enzyme-inhibitor-substrate system can exhibit a complicated dynamical behavior with non-monotonic trend in the temporal depletion of the substrate and there is also a possibility of the occurrence of two steady states with respect to enzyme-substrate and enzyme-inhibitor complexes. We define the ratios 〖f_S=v〗_max⁄((K_MS+e_0 ) ) and 〖f_S=u〗_max⁄((K_MI+e_0 ) ) as the acceleration factors with respect to the conversion dynamics of substrate and inhibitor into their respective products. Here K_MS and K_MI are the Michaelis-Menten parameters associated with the binding of substrate and inhibitor with the enzyme, vmax and umax are the respective maximum reaction velocities. When f_S⁄f_I <1, then the enzyme-substrate complex can reach the full-fledged steady state only after the depletion of the enzyme-inhibitor complex. On the other hand, when f_S⁄f_I >1 then the enzyme-inhibitor complex can reach the full-fledged steady state only after the depletion of the enzyme-substrate complex. This complicated behavior of the enzyme-substrate-inhibitor system especially when f_S⁄f_I ≠1 is the root cause of large amount of error in the estimation of various kinetic parameters both in the cases of fully and partially competitive inhibition schemes using the standard QSSA methods. Remarkably, we show that our refined expressions for the reaction velocities over enzyme-substrate-inhibitor space can control this error more significantly than the currently available standard QSSA velocity expressions.