Cheng-Prusoff Equation: Converting IC50 to Ki

The Cheng-Prusoff equation converts an IC₅₀ — the inhibitor concentration that produces 50% inhibition in a specific assay — into a K_i, the inhibition constant that describes an inhibitor’s intrinsic, assay-independent binding affinity for its target. The two numbers answer different questions. IC₅₀ answers “how much inhibitor did it take to cut activity in half, under these exact assay conditions?” K_i answers “how tightly does this inhibitor actually bind the target, independent of what substrate concentration or radioligand concentration happened to be used?” Because IC₅₀ shifts with substrate concentration while K_i in principle does not, K_i is the number that should be comparable across papers, across assay formats, and across labs — but only when the conversion is applied to data for which its assumptions actually hold.

Why the Conversion Is Necessary

IC₅₀ is an operational, assay-bound number. Run the same competitive inhibitor against the same enzyme at a higher substrate concentration and the measured IC₅₀ increases, because more inhibitor is needed to out-compete the additional substrate for the active site — even though nothing about the inhibitor’s actual affinity for the enzyme has changed. Two labs reporting different IC₅₀ values for the same compound against the same target are not necessarily describing a different compound; they may simply have run the assay at different substrate concentrations relative to the enzyme’s Michaelis constant (K_m). This is the practical reason the conversion matters for anyone evaluating a pharmacology methods section: a bare IC₅₀, reported without the assay’s substrate concentration and K_m, is not directly comparable to another lab’s IC₅₀ for the same target. K_i, correctly derived, removes that assay-condition dependence.

The Equation and Its Terms

For classical competitive inhibition of an enzyme, the Cheng-Prusoff equation is:

K_i = IC₅₀ / (1 + [S]/K_m)

• IC₅₀ — the inhibitor concentration producing 50% inhibition of enzyme activity under the assay’s specific conditions, read off a dose-response curve.
• [S] — the substrate concentration actually used in the assay.
• K_m — the Michaelis constant of the enzyme for that substrate, independently determined (typically from a separate Michaelis-Menten kinetics experiment on the same enzyme/substrate pair).
• K_i — the resulting inhibition constant, interpreted as the dissociation constant of the enzyme-inhibitor complex.

The receptor-binding equivalent, used for radioligand competition-binding assays rather than enzyme activity assays, substitutes the radioligand concentration and its dissociation constant for substrate and K_m:

K_i = IC₅₀ / (1 + [L]/K_d)

where [L] is the concentration of labeled ligand used in the competition assay and K_d is that ligand’s own equilibrium dissociation constant for the receptor, determined by a separate saturation-binding experiment. The logic is identical; only the terms describing the labeled probe change. This distinction matters when reading a methods section: an enzyme-assay IC₅₀ is converted with K_m, a binding-assay IC₅₀ is converted with the tracer’s K_d, and mixing the two up produces a K_i that does not mean what it claims to.

Derivation Logic: Why the [S]/K_m Correction Term Exists

The correction factor (1 + [S]/K_m) exists because a competitive inhibitor and the substrate are racing for the same binding site. Under Michaelis-Menten kinetics, the apparent affinity an inhibitor needs to achieve a given fractional inhibition depends on how much substrate it has to out-compete. At the limit where [S] is negligible relative to K_m (very low substrate concentration), there is essentially no substrate competing for the site, so IC₅₀ approaches K_i directly — the correction term collapses toward 1. As [S] rises relative to K_m, progressively more inhibitor is required to achieve the same 50% inhibition, because substrate occupancy of the active site is itself higher and harder to displace; IC₅₀ rises linearly with [S]/K_m, while the underlying K_i — the inhibitor’s actual dissociation constant — stays constant. Cheng and Prusoff derived the relationship from steady-state competitive-inhibition kinetics precisely to back this substrate-dependence out of the observed IC₅₀, leaving a number that reflects only the inhibitor-enzyme interaction.

This is also why the equation requires an independently measured K_m (or K_d) for the same enzyme/substrate or receptor/ligand pair under matching conditions (temperature, pH, buffer) — the conversion is not self-contained within a single inhibition curve. A K_i calculated with a K_m value borrowed from a different assay format, a different species’ ortholog, or different buffer conditions is not a valid conversion, even though the arithmetic will still produce a number.

Worked Example (Illustrative)

The following is a constructed illustrative example, not data from a specific published study, used to demonstrate the calculation.

An enzyme inhibition assay is run at a substrate concentration [S] of 20 μM. The enzyme’s K_m for this substrate, determined separately, is 10 μM. A dose-response curve for the test inhibitor yields an IC₅₀ of 150 nM.

Applying the equation:

K_i = IC₅₀ / (1 + [S]/K_m) = 150 nM / (1 + 20/10) = 150 nM / 3 = 50 nM

Here, because the assay was run at twice the enzyme’s K_m, the observed IC₅₀ (150 nM) overstates how much inhibitor is actually needed to occupy the enzyme in the absence of substrate competition — the true binding affinity, K_i, is three times tighter than the raw IC₅₀ suggests. If a second lab ran the identical inhibitor and enzyme at [S] = K_m instead, their raw IC₅₀ would come out to 100 nM (IC₅₀ = K_i × (1 + [S]/K_m) = 50 × 2), a different number from the first lab’s 150 nM — yet both would converge on the same K_i of 50 nM once correctly converted. That convergence, not the raw IC₅₀, is what makes K_i the number worth comparing across studies.

When the Conversion Is Invalid

The Cheng-Prusoff equation is a specific mathematical consequence of classical competitive-inhibition kinetics. It does not generalize to every inhibition mechanism, and applying it outside its assumptions produces a K_i-labeled number that is not a true dissociation constant. The conditions below are the ones most likely to be violated in practice, and the ones worth checking when evaluating someone else’s reported K_i.

Non-Competitive and Uncompetitive Inhibition

The standard form of the equation assumes the inhibitor and substrate compete for the same site, so that raising [S] antagonizes inhibition. Non-competitive inhibitors bind a site distinct from the substrate-binding site and can inhibit regardless of substrate occupancy; uncompetitive inhibitors bind only the enzyme-substrate complex, and inhibition increases with [S] rather than decreasing. For both mechanisms, the (1 + [S]/K_m) correction term is the wrong correction — it was derived specifically for the competitive case, where higher substrate makes the inhibitor’s job harder in a particular, quantifiable way. Applying it to non-competitive or uncompetitive data either under- or over-corrects, and mixed-mechanism inhibition requires its own explicit kinetic model rather than the standard formula.

Allosteric Mechanisms

Allosteric inhibitors bind a site topologically distinct from the active site and act by conformational modulation rather than direct active-site occupancy competition. Even when an allosteric inhibitor happens to reduce activity in a substrate-concentration-dependent way, that dependence does not follow the same steady-state derivation Cheng and Prusoff used, and a K_i extracted with the standard equation is not interpretable as a simple binding dissociation constant. Allosteric inhibition generally needs a mechanism-specific model (and, where cooperativity is present, a Hill-type or allosteric ternary-complex analysis) rather than a direct IC₅₀-to-K_i conversion.

Substrate Concentration Not Properly Accounted For

The conversion requires both an accurately known [S] for the inhibition assay itself and an independently, correctly determined K_m under matching conditions. Two common failure points: the assay was run at an [S] that was not actually measured or was assumed rather than verified (common in high-throughput screening formats where [S] is nominally fixed but substrate depletion over the assay’s read window is not checked), or the K_m plugged into the equation was taken from a different source — a published value for a different species ortholog, a different assay temperature, or a different buffer — rather than measured in the same system. Either error propagates directly into the reported K_i, and because the arithmetic still executes cleanly, the resulting number carries a false precision: it looks like a rigorously derived constant while actually encoding an unverified or mismatched input.

Tight-Binding Inhibitors

The Cheng-Prusoff derivation assumes the free inhibitor concentration is not significantly depleted by binding to the enzyme — that is, enzyme concentration in the assay is low relative to K_i. When an inhibitor’s K_i approaches or falls below the enzyme concentration used in the assay (a “tight-binding” inhibitor), a substantial fraction of the inhibitor is sequestered by the enzyme itself, the simple algebraic relationship no longer holds, and the standard equation systematically overestimates K_i. This case requires a tight-binding-specific analysis (e.g., the Morrison equation) rather than the standard Cheng-Prusoff formula.

Radioligand-Binding Assays Using the Wrong Correction Term

For competition-binding assays, the correct correction uses the tracer’s own K_d and concentration, not an enzyme K_m/substrate pair — and, separately, using an IC₅₀ derived from a saturating or otherwise inappropriately high tracer concentration relative to its K_d without applying that correction at all is a well-documented and specifically named source of error in the pharmacology literature (see Sources below). A K_i reported from a binding assay without stating the tracer concentration and its K_d cannot be checked for validity by a reader, and should be treated as unverifiable as reported.

Reading a Reported K_i in a Methods Section

For anyone reviewing a pharmacology or drug-discovery methods section — a grant reviewer assessing a preclinical rationale, an IACUC or IBC member evaluating a proposed dosing justification, a journal reviewer, or a research administrator checking a sponsor’s investigator brochure — a reported K_i is only as trustworthy as the inputs behind it. A methods section that supports a converted K_i should state, at minimum: the assay type (enzyme activity vs. radioligand binding), the substrate or tracer concentration actually used, the K_m or K_d value applied and its source (measured in-house vs. cited from elsewhere), and the inhibition mechanism assumed. A K_i presented without any of these is not falsifiable by a reader and should be read as an IC₅₀ with an unverified label. This transparency expectation is consistent with the broader rigor-and-reproducibility principle that a quantitative claim should carry enough methodological detail for an independent reader to assess, not just reproduce, the result.

Frequently Asked Questions

Is Ki always smaller than IC50?

Not necessarily smaller in an absolute sense, but for competitive inhibition at [S] > 0, K_i is always less than or equal to the observed IC₅₀, since the correction factor (1 + [S]/K_m) is always ≥ 1. The two are equal only in the limiting case where [S] is negligible relative to K_m.

Can the Cheng-Prusoff equation be used without knowing Km?

No. K_m (or, for binding assays, the tracer’s K_d) is a required input, not something the equation derives. It must come from an independent kinetic experiment on the same enzyme-substrate or receptor-ligand pair under matching conditions; using a K_m from the literature for a different system introduces error that does not show up anywhere in the calculation itself.

Does a low IC50 always mean high potency?

Not on its own. A low IC₅₀ in one assay can be lower simply because that assay used a lower substrate or tracer concentration relative to K_m/K_d, not because the compound binds more tightly. This is exactly the comparability problem the Cheng-Prusoff conversion to K_i is meant to solve, and why comparing raw IC₅₀ values across studies without knowing each assay’s substrate conditions is unreliable.

What is the difference between the enzyme and receptor-binding forms of the equation?

They share the same mathematical structure but use different terms: the enzyme form uses substrate concentration [S] and the Michaelis constant K_m; the receptor-binding form, used for radioligand competition assays, uses labeled-ligand concentration [L] and the ligand’s own dissociation constant K_d. Applying the wrong pair of terms to the wrong assay type produces an incorrect K_i.

Why do published Ki values for the same compound sometimes disagree?

Common reasons include different assays being run at different substrate/tracer concentrations without correct K_m/K_d values applied, K_m/K_d values borrowed from a different experimental system, the inhibitor actually being tight-binding (violating the equation’s free-inhibitor assumption), or a genuinely non-competitive or allosteric mechanism being force-fit to a competitive-inhibition formula. Discrepancies are worth investigating at the level of assay conditions before concluding the compound behaves differently between labs.

Sources

Cheng Y, Prusoff WH. “Relationship between the inhibition constant (K_I) and the concentration of inhibitor which causes 50 per cent inhibition (I₅₀) of an enzymatic reaction.” Biochemical Pharmacology, 1973;22(23):3099-3108. DOI: 10.1016/0006-2952(73)90196-2 — the original derivation.

Craig DA. “The Cheng-Prusoff relationship: something lost in the translation.” Trends in Pharmacological Sciences, 1993;14(3):89-91 — a widely cited discussion of common misapplications, including the tight-binding and substrate-condition issues covered above.

Lazareno S, Birdsall NJM. “Estimation of antagonist K_b from inhibition curves in functional experiments: alternatives to the Cheng-Prusoff equation.” Trends in Pharmacological Sciences, 1993;14(6):237-239 — on cases (including some allosteric and functional-antagonism scenarios) where the standard equation does not apply and alternative estimation is needed.

For related methodology on measuring binding affinity directly rather than converting from an inhibition curve, see CASRAI’s guides to surface plasmon resonance (SPR) binding kinetics and K_D calculation and fluorescence polarization assay setup and validation, both of which report K_D directly from equilibrium or kinetic binding data rather than from a competition IC₅₀. Enzyme inhibition assays of the kind the classical Cheng-Prusoff equation applies to are typically read on a plate reader, and inhibitors intended for further development are frequently sourced from material generated during protein purification workflows. Assay-to-assay variability in a reported IC₅₀ or K_i is often summarized using the coefficient of variation. For the broader disciplinary context these assays sit within, see CASRAI’s overview of pharmacology as a research area.