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In the model of reversible enzyme inhibition, competitive inhibition makes substrate binding and inhibitor binding mutually exclusive: $EI$ is allowed but $ESI$ is absent. Set ${\alpha }^{\prime }=1$, equivalently take $K_i^{\prime }\to \infty$. ThenFor the chosen $I=30\mu \mathrm{M}$, $\alpha =4$: the half-limiting substrate concentration rises from $20$ to $80\mu \mathrm{M}$, while the limiting reaction rate remains $100\mu \mathrm{M}\mathrm{mi}{\mathrm{n}}^{-1}$. At $s=200\mu \mathrm{M}$ the inhibited reaction rate is $500/7\approx 71.43\mu \mathrm{M}\mathrm{mi}{\mathrm{n}}^{-1}$, compared with the uninhibited $1000/11\approx 90.91\mu \mathrm{M}\mathrm{mi}{\mathrm{n}}^{-1}$. “Overcome by substrate” means that their ratio tends to one as $s\to \infty$, not that any finite addition removes the difference.

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.

In the branching model, mixed inhibition retains inhibitor binding to both free enzyme and the enzyme-substrate complex. Unequal $K_i$ and $K_i^{\prime }$ shift both apparent parameters; their equality gives pure noncompetitive inhibition. For the chosen unequal case, $I=30\mu \mathrm{M}$ gives $\alpha =4$ and ${\alpha }^{\prime }=2$, hence $V_{\max ,\mathrm{app}}=50\mu \mathrm{M}\mathrm{mi}{\mathrm{n}}^{-1}$ and $K_{M,\mathrm{app}}=40\mu \mathrm{M}$. Because $K_i<K_i^{\prime }$, binding favours free enzyme, and the apparent Michaelis constant increases. Reversing that inequality decreases it.

The limiting behaviour provides a useful diagnostic within this model:Low substrate concentration isolates $\alpha$ through the initial slope; high substrate concentration isolates ${\alpha }^{\prime }$ through the plateau. At $I=30\mu \mathrm{M}$, the chosen competitive inhibition and unequal mixed inhibition examples have the same low-$s$ slope, $1.25\mathrm{mi}{\mathrm{n}}^{-1}$, but different plateaus, $100$ and $50\mu \mathrm{M}\mathrm{mi}{\mathrm{n}}^{-1}$. Measuring only low substrate concentrations would miss that distinction.

Hans Adolf Krebs described the experimental reasoning in his 1953 Nobel lecture. Pigeon breast skeletal muscle retained strong oxidative activity after preparation in a Latapie mincing mill and suspension in aqueous medium. Small fumarate additions increased oxygen uptake by more than complete oxidation of the added fumarate could explain; added four-carbon acids could still be detected afterward. His Table 1 compares the increase in uptake with that calculated from complete consumption of the addition. These comparisons challenge the interpretation that the addition merely supplied extra fuel.^[ref]

The lecture identifies the preparation apparatus but does not describe the gas-measuring instrument behind Table 1. The result supports regeneration, not a unique complete reaction mechanism. Its later discussion combines citrate formation from oxaloacetate with breakdown back toward the four-carbon intermediates to close the cycle. Competitive inhibition supplies another comparison: malonate inhibits succinate dehydrogenase, and succinate accumulates. Such evidence links a blocked step to the surrounding metabolic pathway; damaged tissue and restricted inhibitor access can complicate interpretation.^[ref]

An increase above the added fuel's own oxygen demand is a particularly revealing control on the explanation: it compares the proposed cause with the largest effect that simple consumption could produce. The modern mitochondrial oxygen electrode assay asks a different question about a prepared mitochondrion, testing respiratory control through ADP additions. Neither an oxygen uptake signal nor an accumulating intermediate is by itself a direct measurement of ATP production.

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.