Enzyme Kinetics: The Michaelis-Menten Curve and Its Inhibitors

The Michaelis-Menten model

An enzyme E binds a substrate S to form a complex ES, which either falls back apart or proceeds to product P, releasing the enzyme unchanged:

E + S  ⇌(k1,k-1)  ES  →(k2)  E + P

Leonor Michaelis and Maud Menten (1913), building on earlier work by Victor Henri, applied the steady-state approximation — assuming ES forms and breaks down at equal rates once the reaction settles — to derive the rate of product formation as a function of substrate concentration:

v = Vmax [S] / (Km + [S])

Vmax is the maximum rate, reached only as [S] → ∞, when every enzyme molecule is saturated with substrate: Vmax = k2·[E]total. Km, the Michaelis constant, is the substrate concentration at which v = Vmax/2 — a practical measure of the enzyme's apparent affinity for its substrate (a low Km means the enzyme reaches half-saturation at low [S], i.e. binds tightly; formally Km = (k-1 + k2)/k1).

Why the curve is a hyperbola, not a line

At low [S], v is roughly proportional to [S] (first-order in substrate) because most enzyme sits unbound and free to react with whatever substrate arrives. At high [S], nearly every enzyme molecule is already occupied, so v flattens toward Vmax regardless of how much more substrate you add (zero-order in substrate) — the reaction is now limited by how fast ES can turn over to product, not by how much S is around. That transition from linear to flat is exactly a rectangular hyperbola.

Linearising it: Lineweaver-Burk

Fitting a hyperbola by eye is unreliable, so before nonlinear regression was routine, biochemists inverted the equation into a straight line — the Lineweaver-Burk double-reciprocal plot:

1/v = (Km/Vmax)·(1/[S]) + 1/Vmax

Plotting 1/v against 1/[S] gives a line whose slope is Km/Vmax, whose y-intercept is 1/Vmax and whose x-intercept is −1/Km. It's still taught because it makes the type of inhibition visually obvious, even though it distorts experimental error at low [S] (small errors in v become huge errors in 1/v) and modern kinetics work fits the untransformed hyperbola numerically instead.

Three ways to jam the machine

Competitive inhibition: an inhibitor structurally resembles the substrate and competes for the same active site. It can always be out-competed by enough substrate, so Vmax is unchanged but the apparent Km rises (more substrate is needed to reach half-saturation). On a Lineweaver-Burk plot, competitive inhibitor lines share the same y-intercept (Vmax unchanged) but the slope steepens.

Non-competitive inhibition: the inhibitor binds a separate allosteric site, on either free enzyme or the ES complex, distorting the active site's geometry without blocking substrate binding. Adding more substrate can't rescue this, so Vmax drops; Km is unchanged in the pure (symmetric) case because binding affinity for S itself isn't affected, only turnover is impaired.

Uncompetitive inhibition: the inhibitor binds only the ES complex, not free enzyme. Both Vmax and the apparent Km decrease by the same factor — counterintuitively, the enzyme appears to bind substrate more tightly, because pulling ES out of the equilibrium (by locking it as ESI) shifts the E+S ⇌ ES equilibrium toward more ES formation, even as fewer ES complexes ever reach product.

Why it matters outside the classroom

Michaelis-Menten kinetics and inhibition patterns are the language of pharmacology (most drugs are competitive or non-competitive enzyme or receptor inhibitors), of metabolic engineering (Km and Vmax values tell you which enzyme in a pathway is the bottleneck), and of diagnostics (many clinical assays quantify a target by how it perturbs a well-characterised enzyme's kinetics).

Frequently asked questions

Try it live

Everything above runs in your browser — open Enzyme Kinetics and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.