Reading enzyme inhibition patterns

An inhibitor is a molecule that slows down an enzyme. That is the definition. What makes inhibition interesting — and diagnostic — is that each mode of inhibition leaves a distinct fingerprint on the enzyme’s Michaelis-Menten kinetics. If you can identify the mode from a set of v-vs-[S] curves at different inhibitor concentrations, you know where on the enzyme the inhibitor binds and, often, whether it competes with substrate or with product. This guide extends the Michaelis-Menten intuition to inhibition patterns, so the diagnostic move stops feeling like memorization.

The four modes in one sentence each

• Competitive — inhibitor binds the free enzyme’s active site, blocks substrate binding, but has no effect once the enzyme is saturated with substrate.
• Uncompetitive — inhibitor binds only the enzyme-substrate complex, and does so at a site that is only formed after substrate has bound.
• Non-competitive (classical) — inhibitor binds a site distinct from the active site with equal affinity for E and ES, and once bound, catalysis stops.
• Mixed — inhibitor binds a site distinct from the active site, but with different affinities for E and ES. Classical non-competitive is the special case where those affinities are equal.

Each mode changes the apparent Km and the apparent Vmax in a distinct combination. The combination is your diagnostic.

What changes: apparent Km and apparent Vmax

Let α = 1 + [I]/Kᵢ (the factor by which free enzyme is depleted) and α’ = 1 + [I]/Kᵢ’ (the factor by which ES complex is depleted). Then the general Michaelis-Menten rate in the presence of inhibitor is (Cornish-Bowden Ch. 5):

v = (Vmax · [S]) / (α · Km + α’ · [S])

Which simplifies to

apparent Km = α · Km / α’ apparent Vmax = Vmax / α’

Now sort by mode:

• Competitive: only α > 1 (inhibitor binds E only). α’ = 1. Apparent Km increases (linearly with [I]/Kᵢ). Apparent Vmax unchanged.
• Uncompetitive: only α’ > 1 (inhibitor binds ES only). α = 1. Apparent Km decreases. Apparent Vmax decreases proportionally.
• Non-competitive (classical): α = α’ > 1 (inhibitor binds both, equally). Apparent Km unchanged. Apparent Vmax decreases.
• Mixed: both α > 1 and α’ > 1 but unequal. Apparent Km and apparent Vmax both shift; direction of Km shift depends on whether α > α’ (Km up) or α < α’ (Km down).

Every diagnostic move you make with a Lineweaver-Burk, Eadie-Hofstee, or direct-plot analysis is testing one of those four Km/Vmax patterns.

The Lineweaver-Burk fingerprint

The double-reciprocal plot (1/v vs 1/[S]) is a straight line with slope Km/Vmax and y-intercept 1/Vmax. When you add inhibitor and re-plot, the line moves in a specific way per mode:

• Competitive — lines intersect on the y-axis (1/Vmax unchanged, slopes fan out).
• Uncompetitive — lines are parallel (slope unchanged, y-intercept shifts up).
• Non-competitive — lines intersect on the x-axis (−1/Km unchanged, both intercepts shift).
• Mixed — lines intersect somewhere in the second quadrant, off both axes.

That single-picture summary is why Lineweaver-Burk plots refuse to die in biochemistry teaching even though every kinetics textbook since about 1990 has correctly pointed out that they weight low-[S] points too heavily and should not be used for numerical fitting (Cornish-Bowden §2.4; Segel Ch. 3). Diagnostic pattern good; numerical fitting bad.

For numerical fitting, non-linear regression of v vs [S] directly is the correct move — the Michaelis-Menten calculator uses that fit.

Why each mode makes physical sense

Competitive. Inhibitor and substrate are usually structural analogues (methotrexate for folate at dihydrofolate reductase, malonate for succinate at succinate dehydrogenase — Berg §17.3, Lehninger Fig. 6-15). Both compete for the active site. Add more substrate, you win the competition. Vmax is unchanged because at saturating [S], the inhibitor loses. Km shifts up because you need more substrate to reach half-saturation.

Uncompetitive. Inhibitor binds only after the substrate has bound, at a site that the ES conformation exposes but the free-E conformation does not. This is the rarest mode among small-molecule inhibitors of monomeric enzymes but shows up in multi-substrate enzymes and in some pharmacology (lithium’s action on inositol monophosphatase — Cornish-Bowden §5.9). Vmax decreases because you can no longer race through the catalytic step; Km appears to decrease because inhibitor traps ES, pulling the E + S ⇌ ES equilibrium to the right (Le Chatelier).

Non-competitive. Inhibitor binds an allosteric site with equal affinity in either enzyme state. Catalysis stops when bound. This makes Vmax fall (a fraction of the enzyme population is unusable at any moment), but Km is unchanged (the fraction that is usable still binds substrate exactly like normal).

Mixed. Non-competitive is the special case; mixed is the general one. Most real allosteric inhibitors are mixed — the inhibitor site is coupled to the active site through conformational communication, and the equilibrium constants for I binding are inevitably slightly different in E and ES.

Ki and how to extract it

Ki is the dissociation constant of the enzyme-inhibitor complex. For competitive inhibition,

apparent Km = Km · (1 + [I]/Kᵢ)

Rearranged: Ki = [I] / ((Km,app / Km) − 1). Measure Km at [I] = 0 and Km,app at some known [I], solve.

For non-competitive:

apparent Vmax = Vmax / (1 + [I]/Kᵢ)

Ki = [I] / ((Vmax / Vmax,app) − 1). Same shape, different equation.

For mixed, you get two K values (K_i for the E-I complex and K_i’ for the ES-I complex) from the intersecting-lines pattern on a Lineweaver-Burk plot. The formulas are ugly but tractable (Segel Ch. 4).

A canonical worked example

Notice how much diagnostic mileage came from two numbers (Km and Vmax with vs without inhibitor). You do not need to draw the Lineweaver-Burk plot to identify the mode if you already have the parameter estimates. The plot is a visualization aid; the numbers are the diagnostic.

Irreversible inhibitors — a different regime

Everything above assumes the enzyme-inhibitor complex is in rapid equilibrium — inhibitor comes off as easily as it goes on. Some inhibitors do not: they form a covalent bond with the enzyme and stay put (aspirin acetylating COX-1’s Ser530, penicillin covalently modifying the transpeptidase’s active-site serine, DFP inhibiting acetylcholinesterase — Berg §8.5). These are irreversible inhibitors, sometimes called mechanism-based inhibitors or “suicide substrates.”

Irreversible inhibitors do not fit the reversible framework. Instead, they show:

• Time-dependent loss of activity. The fraction of enzyme still active decays exponentially: [E]_t = [E]₀ · exp(−k_inact · [I] · t / (Ki + [I])).
• No recovery on dilution. Reversible inhibitors wash out; irreversible ones do not.
• A titration stoichiometry. One equivalent of inhibitor per active site permanently inactivates that enzyme copy.

If you see time-dependent inhibition that persists after gel-filtration, you are dealing with a covalent modifier, and you should switch from Ki analysis to k_inact/Ki reporting (Copeland’s Evaluation of Enzyme Inhibitors in Drug Discovery Ch. 8 is the standard reference).

What to measure, in what order

Practical protocol for characterising an unknown inhibitor:

1. Run a full [S]-titration at [I] = 0. Fit Km and Vmax.
2. Add inhibitor at 2-3 concentrations spanning ~0.5 Ki to 5 Ki. Repeat the [S]-titration at each.
3. Fit Km,app and Vmax,app at each [I]. Do this by non-linear regression, not Lineweaver-Burk.
4. Read the pattern: which of Km / Vmax shifted, and how did they scale with [I]?
5. Report Ki (and Ki’ if mixed) with its confidence interval. The Michaelis-Menten calculator will compute both.
6. Verify reversibility by dilution: if activity returns on 100× dilution, the inhibitor is reversible; if not, treat it as irreversible.

Skipping any step invites a wrong mode assignment, which invites a wrong Ki, which invites the wrong pharmacology in downstream cell work.

Practice problems

Where to go next

• The Michaelis-Menten calculator fits v-vs-[S] curves with or without inhibitor and reports Km,app, Vmax,app, and Ki with confidence intervals.
• Assays run at non-physiological pH will change the apparent Ki because ionization state of enzyme, substrate, or inhibitor may shift — pair with the pH calculator or the buffer calculator when setting up the assay buffer.

Read the plot in order: Vmax first, then Km, then match to one of four modes. The pattern is the diagnostic; Ki is the number that comes out after you know which formula to plug into.