Substrate-Dependent Reversal of Inhibition and Activation by Hyperbolic Enzyme Modifiers

A compound's effect on enzyme activity is frequently assigned as fixed inhibition or activation, independent of the substrate concentration used to measure it. Within the general modifier (Botts–Morales) framework, this characterization becomes conditional whenever a modifier's two coefficients — α, governing substrate binding, and β, governing the catalytic step — act in opposing directions relative to unity. For such a modifier, v/v₀ = (1 + βx/α)(1 + σ)/(1 + σ + x + σx/α), with σ = [S]/K_M and x = [X]/K_x. Setting v/v₀ = 1 yields a crossover substrate concentration σ* = (α − β)/(β − 1), which is positive and physically accessible when the condition for sign reversal is met. A critical and underappreciated consequence follows directly from the form of σ*: because it depends only on α and β, and not on modifier concentration, a complete modifier titration at a fixed substrate concentration produces a convincing dose–response curve while leaving the substrate-dependent inversion entirely undetected. Using published α and β values from three characterized enzyme–modifier systems, we show that σ* falls within accessible substrate ranges, so the same compound is correctly described as a partial inhibitor at one substrate concentration and as an activator at another. This reversal, an exact consequence of the general modifier scheme, has received little attention in the enzyme-activation literature. We propose that a minimal substrate series should be included in the characterization of any modifier assigned a fixed mechanistic classification