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An enzyme inhibitor lowers an enzyme's reaction rate. In reversible enzyme inhibition, reducing free inhibitor concentration can restore activity; slow release can make this difficult to distinguish experimentally from irreversible inactivation. Competitive inhibition and mixed inhibition describe different changes in enzyme kinetics; pure noncompetitive inhibition is a special case of mixed inhibition, not a name for every inhibitor that binds away from an active site.^[ref]

The useful question is how adding substrate changes the inhibition. The model below connects binding to a measurable reaction rate, predicts when more substrate can overcome inhibition, and supplies examples for comparison with an enzyme inhibition initial-rate assay. Its inhibition constants are explicitly defined; neither a binding cartoon nor one reduced reaction rate by itself determines a mechanism.

An inhibition constant sets an inhibitor concentration scale in an enzyme kinetics model. In the rapid-binding construction under reversible enzyme inhibition, $K_i=[E]I/[EI]$ and $K_i^{\prime }=[ES]I/[ESI]$ are dissociation constants. Smaller values favour the corresponding bound population at fixed free inhibitor concentration. More generally, fitted inhibition constants need not be true binding dissociation constants; a mechanism must justify that interpretation.^[ref]

For example, choosing $I=K_i$ makes $[EI]=[E]$ in this model, but does not imply that half of all enzyme is $EI$: $ES$ and $ESI$ also contribute to the total population.

Construct a minimal model with one substrate and no appreciable accumulated reaction product. Only the enzyme-substrate complex $ES$ produces product, with catalytic rate constant $k_{\mathrm{cat}}$; $EI$ and $ESI$ are inactive. Write $s=[S]$ and $I=[\mathrm{I}]$ for free molar concentrations, and $E_T$ for total enzyme molar concentration. Assume a suitable quasi-steady-state approximation, rapid inhibitor binding and release relative to turnover, and negligible depletion of free inhibitor. In this construction the population relations areHere $K_M$ is the uninhibited Michaelis constant, and $K_i,K_i^{\prime }$ are the inhibition constants for the two indicated binding branches. The symbol $I$ denotes a free concentration, not an amount added to the tube. Define $V_{\max }=k_{\mathrm{cat}}{E}_T$. Conservation of enzyme gives an original short derivation:Thus fitting a Michaelis-Menten equation at a fixed $I$ would return the apparent parameters“Apparent” means conditional on that inhibitor concentration. The uninhibited Michaelis constant has not been redefined. Dimensional analysis checks the denominator: both $\alpha K_M$ and ${\alpha }^{\prime }s$ have units of molar concentration; the resulting $v$ has units of reaction rate.

The shared numerical choices make the mechanisms comparable. At $I=0$ and $s=20\mu \mathrm{M}$, all three models give $v=50\mu \mathrm{M}\mathrm{mi}{\mathrm{n}}^{-1}$. At $I=10\mu \mathrm{M}$, the respective reaction rates are $100/3$, $25$, and $30\mu \mathrm{M}\mathrm{mi}{\mathrm{n}}^{-1}$; at $I=30\mu \mathrm{M}$ they are $20$, $12.5$, and $50/3\mu \mathrm{M}\mathrm{mi}{\mathrm{n}}^{-1}$. These are independently chosen teaching models, not measured inhibition constants for a named enzyme.