Fitting a Concentration-Response Curve

Concentration-response data get a curve drawn through them and a number extracted from the curve. That number is frequently the only quantity carried forward, so what the fit assumed, how well it was constrained, and what the data actually covered all matter more than the figure suggests.

What the standard model assumes

The usual curve is a four-parameter logistic: a lower plateau, an upper plateau, a midpoint, and a slope. Fitted on a logarithmic concentration axis it produces the familiar sigmoid.

Each parameter is an assumption as much as a result. The model assumes the response is monotonic, that it saturates at both ends, and that a single symmetric transition describes the whole range. Real systems violate all three often enough that the fit should be inspected rather than trusted — and a system with no receptor-mediated saturation, such as the concentration-dependent behaviour noted for some mitochondrial compounds, may not be sigmoid at all.

The midpoint is only as good as the plateaus

The midpoint is defined relative to the two plateaus, so it cannot be determined more reliably than they are.

If the highest concentration tested has not reached a plateau, the upper asymptote is extrapolated rather than observed, and the midpoint is extrapolated with it. The fitting software will still return a value, often with a narrow-looking confidence interval, because the interval reflects the fit to the model rather than the adequacy of the data.

This is the most common defect in published concentration-response work: a curve fitted over two orders of magnitude when the transition spans three, with a reported midpoint that is an extrapolation presented as a measurement. The remedy is to look at whether the top and bottom of the curve contain real data points, not at how tidy the line looks.

Constraining parameters, and when it is honest

Fixing the lower plateau at zero or the slope at unity produces a better-behaved fit and a narrower interval. It is legitimate when there is an independent reason — a true baseline measured separately, for instance.

It is not legitimate as a way of rescuing a fit that the data do not support. A constrained fit answers the question “what is the midpoint, assuming the curve looks like this”, and where the assumption is doing the work, the answer belongs to the assumption. Any constraint applied should be stated.

What the extracted number is, and is not

A midpoint from a functional assay is an EC50 or IC50 — a potency in that system under those conditions. It is not an affinity, and the distinction is the subject of Ki, IC50 and EC50.

It is also system-dependent in ways the number does not carry. Receptor expression level, incubation time, and serum content all shift it, which is why values from different laboratories are not directly comparable — the model-dependence set out in cell line choice. And the concentration on the axis is the nominal concentration, which may differ from the free concentration for the reasons in why an in vitro concentration is not a dose.

Fitting decisions that quietly change the answer

• Weighting. An unweighted fit lets the points with the largest absolute values dominate. Where variability scales with the response, weighting is the correct choice, and switching between them moves the midpoint.
• Averaging before or after. Fitting one curve to pooled points and fitting each replicate separately then averaging the midpoints give different answers and different intervals. The second is usually more honest about the variability, and depends on what counts as a replicate — see technical and biological replicates.
• Log versus linear axis. The model is defined on the log axis. Fitting on a linear axis is a different model with a different answer.
• Excluded points. A dropped point at either extreme moves a plateau, and therefore the midpoint, more than a dropped point in the middle — which is why the discipline in outliers matters particularly here.

Reading a reported value

The useful questions are what concentration range was tested, whether both plateaus were reached within it, how many points defined the transition, whether any parameters were constrained, and what interval accompanies the estimate.

A midpoint reported without a range and an interval is a point estimate from an unspecified fit. It can be quoted but it cannot be compared with anything, which is a limitation worth recognising before building an argument on the comparison.

Why a curve looks more authoritative than it is

A fitted line drawn through scattered points is a strong visual claim: it asserts a smooth, saturating, monotonic relationship across the whole axis, including the regions where no data were collected.

The data underneath frequently support a weaker statement — that the response increased with concentration over the range tested. Where that is all the data show, it is the statement worth making, and the curve is a summary rather than evidence for the parts of the axis it spans without observations.