Fitting a Concentration-Response Curve
Picture a curve fitted across two orders of magnitude when the transition it describes actually spans three. Software returns a midpoint. A confidence interval arrives with it, narrow enough to look authoritative. Yet the drawn line’s upper shoulder sits above every concentration anyone actually tested. Nothing warns you, because that interval reports how closely the model tracks the points it was handed, not whether those points were adequate. This single failure accounts for much of the concentration-response literature, and it is the right place to start.
Why an unobserved plateau contaminates everything downstream
A midpoint has no independent existence. It is defined by reference to the two plateaus, so its reliability is bounded by theirs. Push the upper asymptote into territory where no measurement was ever made and the midpoint follows it there. Both are then extrapolations wearing the costume of measurements, and the fitting software has no way to say so.
The diagnostic is unglamorous. Check whether real data points occupy the top and the bottom of the curve. Tidiness of the line tells you nothing: a smooth fit through inadequate coverage is smooth precisely because nothing out there exists to disagree with it.
Four parameters, four assertions
The standard model is the four-parameter logistic. Lower plateau, upper plateau, midpoint, slope, fitted against a logarithmic concentration axis, yielding the sigmoid everyone recognizes. Each parameter is an assumption before it is ever a result.
Together they assert three things about the system: that response moves in one direction only, that it flattens at both extremes, and that a single symmetric transition covers the whole range. Real systems break all three often enough that inspection should come before trust. Where saturation is not receptor-mediated at all, as with the concentration-dependent behavior noted for some mitochondrial compounds, the shape may never have been sigmoid to begin with.
Pinning a parameter down
Fix the lower plateau at zero, or the slope at unity, and the fit behaves better while the interval narrows. Whether this is honest depends entirely on where the constraint came from. A baseline measured independently is a genuine reason. Wanting a cleaner-looking result is not.
A constrained fit answers a conditional question: what is the midpoint, given that the curve looks like this? Where the given is carrying the argument, the answer belongs to the given rather than to the data. Whatever was constrained should be stated alongside the value.
Decisions inside the fit that move the answer
Several choices sit between raw points and reported number. None of them customarily appear in the write-up.
| Weighting | Leave a fit unweighted and the points with the largest absolute values dominate it. Where variability scales with response, weighting is the correct choice, and toggling between the two shifts the midpoint. |
| Pooling versus per-replicate | One curve through pooled points is not equivalent to fitting each replicate and then averaging the midpoints. The second route usually represents variability more honestly, and it turns on what counts as a replicate, which is the subject of technical and biological replicates. |
| Axis | This model is defined on the log axis. Fit it on a linear one and you have fitted a different model, which returns a different answer. |
| Exclusions | Drop a point near either extreme and a plateau moves, carrying the midpoint along with it. Drop one mid-transition and far less happens. That asymmetry is why the discipline described under outliers bites hardest here. |
Potency is not affinity, and neither travels well
What a functional assay yields at its midpoint is an EC50 or an IC50: a potency, in that system, under those conditions. Affinity is a separate quantity, and the difference is worked through in Ki, IC50 and EC50.
Those conditions do not travel with the number. Alter receptor expression, incubation time or serum content and the value moves, which is why a figure from one laboratory cannot simply be placed beside a figure from another. That model dependence is set out under cell line choice. The axis compounds it: what gets plotted is nominal concentration, and free concentration can diverge from it for the reasons laid out in why an in vitro concentration is not a dose.
Interrogating a value someone else reports
Five questions do most of the work. Over what concentration range was the experiment run? Were both plateaus reached inside that range? How many points actually defined the transition? Was any parameter constrained? What interval accompanies the estimate?
Strip away the range and the interval and what remains is a point estimate from an unspecified fit. Quoting it is possible. Comparing it with anything else is not, and recognizing that limitation before building an argument on the comparison saves effort later.
The line claims more than the points
Drawing a smooth curve through scattered data makes a forceful visual assertion: monotonic, saturating, continuous right across the axis, including stretches where nobody collected anything at all.
Usually the points underneath support something weaker. Response rose with concentration over the range tested. Where that is the honest limit of the evidence, that is the sentence worth writing, and the curve becomes a summary of what was seen rather than evidence about the regions it merely spans.
