IC50 -> Kd conversion models

What function to use: convert

These models are used for converting \(\text{IC}_{50}\) to \(K_d\) using more or less approximate formulas.

Nikolovska-Coleska

Model name: coleska
References: Nikolovska-Coleska 2004

In the model of Nikolovska-Coleska, \(K_d\) is calculated by the equation:

\[ K_d = \frac{[L_{50}]}{\left(\frac{[L_{50}^*] + [R_0]}{K_d^*}\right) + 1} \]

where \([L_{50}^*]\) is the concentration of labeled ligand at 50% inhibition, \([L_{50}]\) is the concentration of unlabeled ligand at 50% inhibition, \(K_d^*\) is the dissociation constant of the labeled ligand, \(K_d\) is the dissociation constant of the unlabeled ligand, and \([R_0]\) is the concentration of receptor at 0% inhibition.

Following the equations for conservation of mass, \([R_0]\) could be expressed in a quadratic equation:
\[ [R_0]^2 + a[R_0] + b = 0 \]

where the coefficients are calculated as:
\[ a = [L^*]_T + K_d^* - [R]_T \]

\[ b = -K_d^* [R]_T \]

The physical solution for \([R_0]\) is then calculated by the quadratic formula:
\[ [R_0] = \frac{-a + \sqrt{a^2 - 4b}}{2} \]

The rest of the parameters are calculated as follows:
\[ [L_0^*] = \frac{[L^*]_T}{1 + \frac{[R_0]}{K_d^*}} \]

\[ [RL_0^*] = \frac{[R]_T}{1 + \frac{K_d^*}{[L_0^*]}} \]

\[ [RL_{50}^*] = \frac{[RL_0^*]}{2} \]

\[ [L_{50}^*] = [L^*]_T - [RL_{50}^*] \]

\[ [RL_{50}] = [R]_T + \frac{K_d^* [RL_{50}^*]}{[L_{50}^*]} + [RL_{50}^*] \]

\[ [L_{50}] = IC_{50} - [RL_{50}] \]

Cheng-Prusoff

Model name: cheng_prusoff

\[ K_d = \frac{\text{IC}_{50}}{1 + \frac{[L^*]_T}{K_d^*}} \]

Cheng-Prusoff Corrected

Model name: cheng_prusoff_corr
References: Munson 2008

\[ K_d = \frac{\text{IC}_{50}}{1 + \left(\frac{[L^*]_T(Y_0 + 2)}{2K_d^*(Y_0 + 1)}\right) + Y_0} + \frac{K_d^* Y_0}{Y_0 + 2} \]