Logistic model#

The ic50 model is an empirical four-parameter inhibitory dose-response model. It describes a monotonic decrease from an upper response to a lower response as the concentration increases:

\[ y(x) = y_{\min} + \frac{y_{\max}-y_{\min}} {1 + \left(\dfrac{x}{\mathrm{IC}_{50}}\right)^h}. \]

The parameters are:

  • \(x \ge 0\): the raw, untransformed concentration;

  • \(y_{\min}\): the response approached at high concentration;

  • \(y_{\max}\): the response approached at zero concentration;

  • \(\mathrm{IC}_{50} > 0\): the concentration at the midpoint;

  • \(h > 0\): the Hill slope, exposed as hill_slope.

\[ y(\mathrm{IC}_{50}) = \frac{y_{\min}+y_{\max}}{2}. \]

This corresponds to GraphPad Prism’s variable-slope inhibitory dose-response equation: ymax is Top and ymin is Bottom. bindcurve writes the inhibitory Hill slope as the positive quantity \(h\); it is the negative of Prism’s signed HillSlope. The curve decreases when \(y_{\max}>y_{\min}\).

Concentration and logarithms#

Concentrations supplied to bindcurve must remain on their original linear scale. The model is evaluated from the dimensionless ratio \(x/\mathrm{IC}_{50}\); it does not accept \(\log_{10}(x)\) as its input axis.

For numerical stability and to enforce positivity, bindcurve optimizes \(\mathrm{IC}_{50}\) internally on a base-10 logarithmic coordinate. This is an optimizer detail, not a separate logIC50 model. The public fitted value is returned on the same concentration scale as the input data, and

\[ \log\mathrm{IC}_{50}=\log_{10}(\mathrm{IC}_{50}). \]

Important

The conventional quantity \(\mathrm{pIC}_{50}\) is \(-\log_{10}(\mathrm{IC}_{50}/1\ \mathrm{M})\). The numerical concentration must therefore be expressed in molar units before taking the negative logarithm. For example, \(1\ \mathrm{\mu M}\) corresponds to \(\mathrm{pIC}_{50}=6\), not 0.

Interpretation#

\(\mathrm{IC}_{50}\) is a curve midpoint under the conditions of the assay. It is not generally a thermodynamic dissociation constant. Its relationship to a competitor \(K_d\) depends on the binding mechanism, tracer concentration, receptor concentration, depletion, and whether equilibrium was reached. See the IC₅₀-to-\(K_d\) conversions for the one-site competitive equilibrium case.