Converting IC₅₀ to \(K_d\)#

An IC₅₀ is an assay-dependent midpoint, whereas \(K_d\) is an equilibrium dissociation constant. bindcurve implements three conversions for the specific case of one-site, mutually exclusive, competitive equilibrium between a labeled tracer and an unlabeled competitor.

The notation is:

Symbol

Code

Meaning

\([R_T]\)

RT

total receptor concentration

\([L_T^*]\)

LsT

total tracer concentration

\(K_d^*\)

Kds

tracer dissociation constant

\(K_d\)

Kd

competitor dissociation constant

\(\mathrm{IC}_{50}\)

IC50

total competitor giving 50% tracer displacement

Here, 50% displacement means \([RL^*]_{50}=[RL^*]_0/2\), where the subscript 0 denotes the competitor-free state. These conversions do not apply to noncompetitive or irreversible mechanisms, nonequilibrium measurements, functional-response IC₅₀ values, heterogeneous sites, or the four-state incomplete-competition model.

Cheng-Prusoff approximation#

Function: cheng_prusoff

The familiar conversion used by the API is

\[ K_d=\frac{\mathrm{IC}_{50}} {1+\dfrac{[L_T^*]}{K_d^*}}. \]

The classical relationship is written in terms of free tracer. Substitution of total tracer is accurate when tracer depletion by receptor is negligible, so \([L^*]\approx[L_T^*]\), and bound competitor contributes negligibly to its total concentration. In bindcurve, this approximation is the low-receptor limit of the finite-concentration conversions below.

Munson-Rodbard finite-concentration correction#

Function: cheng_prusoff_corrected

Let

\[ y_0=\frac{[RL^*]_0}{[L^*]_0} \]

be the bound-to-free tracer ratio before competitor is added. Under the same one-site competitive-equilibrium model, the exact finite-concentration correction is

\[ K_d= \frac{\mathrm{IC}_{50}} {1+ \dfrac{[L_T^*](y_0+2)}{2K_d^*(y_0+1)} +y_0} -K_d^*\frac{y_0}{y_0+2}. \]

The second term is subtracted. The original article printed a plus sign; the authors’ erratum corrected it to a minus sign. When \(y_0\to0\), this expression reduces to the Cheng-Prusoff approximation.

Some combinations of \(\mathrm{IC}_{50}\), \([L_T^*]\), \(K_d^*\), and \(y_0\) are incompatible with the assumed equilibrium and produce a non-positive result. bindcurve rejects such inputs rather than reporting them as affinities.

Nikolovska-Coleska finite-concentration correction#

Function: coleska

This form derives the competitor-free state from \([R_T]\), \([L_T^*]\), and \(K_d^*\) instead of requiring \(y_0\).

Before competitor is added, free receptor obeys

\[ [R_0]^2+a[R_0]+b=0, \]

where

\[ a=[L_T^*]+K_d^*-[R_T], \qquad b=-K_d^*[R_T]. \]

The physical root is

\[ [R_0]=\frac{-a+\sqrt{a^2-4b}}{2}. \]

The remaining competitor-free quantities are

\[ [L_0^*]=\frac{[L_T^*]}{1+[R_0]/K_d^*}, \qquad [RL_0^*]=\frac{[R_T]}{1+K_d^*/[L_0^*]}. \]

At 50% displacement,

\[ [RL_{50}^*]=\frac{[RL_0^*]}{2}, \qquad [L_{50}^*]=[L_T^*]-[RL_{50}^*], \]

and tracer equilibrium gives

\[ [R_{50}]=K_d^*\frac{[RL_{50}^*]}{[L_{50}^*]}. \]

Receptor conservation then requires

\[ [RL_{50}]=[R_T]-[R_{50}]-[RL_{50}^*]. \]

Both terms on the right are subtracted. The free competitor concentration is

\[ [L_{50}]=\mathrm{IC}_{50}-[RL_{50}], \]

and the competitor dissociation constant follows directly from its equilibrium:

\[ K_d=\frac{[R_{50}][L_{50}]}{[RL_{50}]}. \]

An algebraically equivalent final expression used by the implementation is

\[ K_d=\frac{[L_{50}]} {1+\dfrac{[L_{50}^*]+[R_0]}{K_d^*}}. \]

Under consistent inputs, the Munson-Rodbard and Nikolovska-Coleska corrections recover the same one-site equilibrium \(K_d\). The former requires \(y_0\); the latter derives the initial state from total receptor and tracer concentrations.

References#