Dose-Response Curves: EC50, IC50 and the Hill Slope

An EC50 is not a measurement. It is a parameter of a curve fitted to measurements, and it inherits every assumption made during the fitting. Two numbers produced by the same equation, from data collected on different endpoints, over different concentration ranges, with different plateaus held fixed, are not comparable merely because both are labeled "EC50, nM".

This guide covers the four-parameter logistic model behind most published potency figures, what a Hill slope away from unity indicates, how to read a confidence interval on a potency estimate, and why a table assembled from EC50 values in several papers usually describes the assays rather than the compounds.

What the parameter actually names

EC50 is the concentration at which a fitted concentration-response curve sits halfway between its two plateaus; IC50 is the same midpoint on a descending curve. Neither is a binding constant, and neither is read off the data directly. Both are defined by the model, so changing the model changes the number with no new experiment.

The equation underneath is the Hill equation, introduced by A. V. Hill to describe the equilibrium relationship between oxygen tension and the saturation of hemoglobin, and since used extensively in pharmacology to analyze quantitative drug-receptor relationships ^[1]. A review of it notes that although the equation is widely used, its properties are not all well known, and sets out its mechanistic connection to the law of mass action alongside its descriptive uses ^[1].

The four-parameter logistic model

For an ascending response the model is:

Response = Bottom + (Top − Bottom) / (1 + (EC50 / [A])^h)

The parameters are the two plateaus, the midpoint concentration EC50, and the slope factor h, usually called the Hill slope. The midpoint algebra is worth checking once: at [A] = EC50 the ratio EC50 / [A] equals 1, so the denominator is 1 + 1 = 2 and the response is Bottom + (Top − Bottom) / 2. EC50 is the half-maximal point of the fitted curve, not of the observed data, and it moves whenever the fitted plateaus move.

The model has a long history in quantitative bioassay. In an early treatment applying it to a free fat cell bioassay of insulin, the author showed that the standard slope-ratio and parallel-line models can be considered approximations to the logistic in the extreme dose regions, with the parallel-line model expected to fit in the middle region ^[2].

What is fixed determines what is fitted

Constraining the plateaus

Analysts routinely fix Top at the mean of a positive control and Bottom at zero. This is a strong assumption, and it transfers the uncertainty in those controls into the EC50 without leaving a trace in the reported standard error. Evidence on the practice is assay-specific. Working on baroreflex curves from conscious rabbits, one group tested forcing the fits through resting blood pressure and heart rate values; for renal sympathetic nerve activity this produced a worse fit by increasing the unexplained variation and altered most of the estimated parameters, which led the authors to call the practice statistically questionable for that measurement ^[4]. The same study proposed a five-parameter asymmetric model that reduces to the four-parameter model when the data are symmetric ^[4], so allowing asymmetry costs a degree of freedom rather than a change of framework.

Which quantity you make a parameter

Nonlinear least squares gives standard errors that assume each parameter behaves close to linearly near the optimum, and that assumption depends on how the model is written. Examining ten parameters drawn from six published parameterizations of the four-parameter logistic, plus one new proposal, against ELISA and radioimmunoassay data, one analysis identified which are adequately near-linear on the Lowry-Morton lambda statistic and recommended those for practical use ^[5]. The bench consequence is familiar: fit log EC50 rather than EC50 — convention supported by this kind of analysis rather than a rule with a single citation, and why potency is reported and averaged in log units.

Outliers

Four parameters are estimated from perhaps eight to twelve concentrations, so one aberrant point carries real leverage. Simulating curves generated from four-parameter logistic models at constant relative potency, one group found that single, multiple or whole-curve outliers may cause a failure to declare similarity between test and reference curves, or yield a biased relative potency estimate; some situations needed no removal, but the results generally supported screening for outliers before fitting ^[6].

The Hill slope

The slope factor controls how many concentrations separate a barely-detectable response from a near-maximal one. Rearranging the model, the fractional response f satisfies f / (1 − f) = ([A] / EC50)^h. Setting f to 0.9 and 0.1 gives EC50 × 9^(1/h) and EC50 × 9^(−1/h), so the ratio between them is 9^(2/h) = 81^(1/h):

• h = 1: the 10% to 90% transition spans 81-fold, which is log10(81) = 1.91 orders of magnitude.
• h = 2: 81^(1/2) = 9-fold, or 0.95 orders of magnitude.
• h = 0.5: 81^2 = 6,561-fold, or 3.82 orders of magnitude.

This constrains design. An eight-point series at three-fold dilutions spans 3^7 = 2,187-fold, or 3.34 orders of magnitude. If the true slope is 0.5, the transition alone is wider than the tested range, and both plateaus become extrapolations. There is a real tension here: efficient estimation of potency calls for concentrating the doses where the slope is steepest, but to allow for assay-to-assay variability and unpredictable potencies, and to test parallelism between curves, it may be preferable to spread doses over a wide range and use a model such as the four-parameter logistic that can fit across the whole feasible range ^[2]. A paper that does not state its concentration range has withheld what you need to judge which side of that trade-off it landed on.

A slope away from unity is a question, not an answer

The Hill coefficient is the most commonly employed marker of cooperativity, but it originates from a structural hypothesis long since disproved; a critical survey of attempts to give it physical meaning concludes that it provides a minimum estimate of the free energy of interaction and correlates robustly with the population of incompletely saturated ligation intermediates ^[7]. It is not the number of binding sites. Reading a fitted h of 2.3 as "two sites" is reading the disproved hypothesis, not the data.

Steep slopes have mundane explanations that should be excluded first. Tight binding is the clearest documented case. Studying serum angiotensin converting enzyme inhibition by enalaprilat across substrate concentrations of 4 to 200 mM, one group found a competitive tight-binding relationship and reported apparent Hill coefficients greater than 1 when the conventional inhibitory sigmoid Emax model was applied; they attributed those slopes to discrepancies between free and total inhibitor concentration, concluded that such slopes could therefore be indicators of tight-binding conditions, and reported that the model overestimates Ki under those conditions ^[8]. Note the modal: offered as an indicator, not a diagnostic.

Steep slopes also arise in intact-cell functional assays where signal amplification lies between occupancy and readout. A meta-analysis of published concentration-response characterizations of the P2Y13 receptor pooled a Hill coefficient of 4.4 (95% CI 3.3 to 5.4) for ADP-like agonists on heterologously expressed receptor ^[9] — a statement about the pooled assay systems, presented as such.

Shallow slopes are, if anything, harder to interpret. A slope below one is consistent with heterogeneity of any kind: a mixed population of binding states, compound loss over a long incubation, or a readout summing two processes with different midpoints. Telling these apart takes an experiment designed to tell them apart, not a better fit to the same curve — standard bench reasoning rather than a finding from any one study.

Confidence intervals decide whether the figure means anything

A potency estimate without an interval is not interpretable, and the interval has to be read on the log scale. Take a fit returning log10 EC50 = −7.30 with a standard error of 0.12 log units. The point estimate is 10^(−7.30) = 5.01 × 10^(−8) M, or 50.1 nM. The 95% interval is −7.30 ± 1.96 × 0.12 = −7.30 ± 0.235, running from −7.535 to −7.065, which back-transforms to 29.2 nM and 86.1 nM. Two things follow. The interval is 20.9 nM below the estimate and 36.0 nM above it, so "50.1 ± 30 nM" is wrong in both directions. And its width is a ratio: 86.1 / 29.2 = 2.95, equivalently 10^(2 × 0.235) = 2.95. A three-fold interval cannot support a claim that two compounds differ two-fold.

Published intervals can be checked the same way. The P2Y13 meta-analysis reports an EC50 of 17.2 nM with a 95% CI of 7.7 to 38.5 nM for ADP-like agonists on heterologously expressed receptor ^[9]. The geometric center of those bounds is the square root of 7.7 × 38.5 = 296.45, which is 17.22 nM: the interval is symmetric in log space and the point estimate is a geometric mean. It spans 38.5 / 7.7 = 5.0-fold. For ATP-like agonists the same analysis reports 0.45 µM with a CI of 0.06 to 3.15 µM ^[9] — a 52.5-fold span, whose geometric center recomputes to the square root of 0.189 = 0.43 µM, close to the published 0.45 and consistent with rounding of the stated limits. An interval that wide cannot distinguish 0.1 µM from 3 µM, and quoting "0.45 µM" alone hides exactly that.

Whether two fitted EC50s differ is a model-comparison question, not an eyeball-the-error-bars one; dose-response software such as the drc package for R exposes inference on derived parameters for this reason ^[3]. Intervals from the asymptotic covariance matrix also rest on the near-linearity assumption above ^[5]; where that is doubtful, profile-likelihood or resampling intervals are the usual alternatives — convention in the field rather than a settled rule.

Why comparing EC50s across papers is usually invalid

The first reason is structural. The operational model of agonism describes agonist action with three parameters: an agonist-receptor dissociation constant KA, the total receptor concentration [R0], and a parameter KE defining transduction of the agonist-receptor complex into effect, with the ratio [R0]/KE described by the authors as a logical definition of an agonist's efficacy in a system ^[10]. A functional EC50 is a joint consequence of ligand and system. It is not KA, and it is not portable between preparations differing in receptor number or transduction.

The second reason is that inhibition constants are conditional on assay composition. For a competitive inhibitor the conventional Cheng-Prusoff correction is Ki = IC50 / (1 + [S]/Km), textbook practice rather than an empirical result. Its arithmetic is unforgiving: an IC50 of 40 nM measured at 4 × Km substrate gives Ki = 40 / (1 + 4) = 8 nM, and that same Ki measured at 0.25 × Km would present as an IC50 of 8 × 1.25 = 10 nM. Same inhibitor, same enzyme, IC50 values four-fold apart from substrate concentration alone. The enalaprilat study concluded that inhibition measured in vitro is in fact dependent on the substrate and enzyme concentrations present in the assay ^[8]. The correction carries an assumption too, documented most clearly for its receptor-pharmacology counterpart: a closer analysis of the Cheng-Prusoff equation and related power equations shows that the general power equation for a competitive antagonist reduces to the Cheng-Prusoff form only when the slope function of the agonist concentration-response curve is exactly unity, and that errors are inherent in using the Cheng-Prusoff equation when that slope function deviates from unity ^[11].

The third reason is that a time-dependent assay measures something that depends on how long it ran. Conventional IC50 and Emax values are highly sensitive to the number of cell divisions taking place over the course of a response assay, and that dependency has been shown to create artifactual correlations between genotype and drug sensitivity; alternative growth-rate metrics were derived to be insensitive to division number ^[12]. Two laboratories running 48-hour and 72-hour endpoints on cells with different doubling times are not measuring the same quantity.

The fourth reason is empirical. A comparison of two large pharmacogenomic studies found their genomic data well correlated while the drug response data were highly discordant; the authors stated that the source of the inconsistencies remained uncertain ^[13]. When two well-resourced groups measuring the same compounds on the same cell lines disagree, an informal cross-paper comparison of single figures has no better prospect.

The P2Y13 meta-analysis quantifies the same effect within one receptor: an EC50 for ADP-like agonists of 17.2 nM for signaling events induced by heterologously expressed receptor, against 1.76 µM for functional responses of endogenously expressed receptor ^[9]. That is a ratio of 1,760 / 17.2 = 102-fold, or 2.01 orders of magnitude, between two sets of published numbers for the same agonist class at the same receptor. The same analysis reports the EC50 of ADP-like agonists to be lower for brain than for blood P2Y13, and ADP-like agonists to be more potent for human than for rodent P2Y13 ^[9].

What is convention, and what is not known

Much of this territory is convention with thin primary support beneath it. The four-parameter logistic is the default because it is tractable and has an interpretable midpoint, not because concentration-response data have been shown to be symmetric on the log scale; the asymmetric five-parameter form exists precisely because they sometimes are not, and the evidence for choosing between them is assay-specific ^[4]. Fitting log EC50, reporting geometric rather than arithmetic mean potency, and reaching for profile or resampling intervals when the asymptotic ones look implausible are conventions with reasoning behind them ^[5] rather than results.

Several things are genuinely unsettled. We are not aware of a primary source establishing how far a fitted Hill slope must sit from unity, at a given number of replicates, before the departure is worth interpreting; because the slope is determined largely by the shoulders of the curve, the sparsest region of a typical dilution series, its own interval is usually the widest of the four and is rarely reported at all. There is no accepted general rule for when plateaus may be constrained. And no correction makes an EC50 from one system comparable to an EC50 from another, because the parameter is not defined independently of the system ^[10].

The practical reading is narrow but reliable. A potency figure is believable to the extent that the paper states the endpoint, the incubation time, the concentration range and spacing, whether the plateaus were fitted or fixed, the fitted slope, and an interval on the estimate. Absent those, it summarizes an experiment you cannot reconstruct, and its right use is as a starting concentration for your own curve rather than as a value to cite.