Appendix 3

Calculation of IC50 and Apparent IC50

At equilibrium, the concentrations of all chemical species in the competitive binding system are governed by Equations (2)–(6), where \([\mathrm{P}]_0\), \([\mathrm{L}]_0\), and \([\mathrm{I}]_0\) represent the total concentrations of protein, ligand, and inhibitor, respectively. Assuming that \([\mathrm{P}]_0\), \([\mathrm{L}]_0\), \(K_\mathrm{d}\), and \(K_\mathrm{i}\) are known, Equations (7) and (8) can be derived from Equations (2) and (4), and from Equations (3) and (5), respectively.

When the protein–ligand complex is inhibited by 50%, the equilibrium complex concentration is defined as \([\mathrm{PL}]_{\mathrm{eq\text{-}50}} = 0.5 \times [\mathrm{PL}]_{\mathrm{eq\text{-}0}}\), where \([\mathrm{PL}]_{\mathrm{eq\text{-}0}}\) denotes the equilibrium concentration of the protein–ligand complex in the absence of inhibitor. \([\mathrm{PL}]_{\mathrm{eq\text{-}0}}\) can be calculated using Equation (1).

Given \([\mathrm{PL}]_{\mathrm{eq\text{-}50}}\), the equilibrium concentration of free protein \([\mathrm{P}]_{\mathrm{eq\text{-}50}}\) can be determined by Equation (7), and the equilibrium concentration of the protein–inhibitor complex \([\mathrm{PI}]_{\mathrm{eq\text{-}50}}\) can be determined by Equation (6). The apparent IC50 is the \([\mathrm{I}]_0\) that elicits a 50% decrease in \([\mathrm{PL}]_{\mathrm{eq\text{-}0}}\), and it can be calculated from Equation (8). Consequently, the IC50, defined as the free inhibitor concentration \([\mathrm{I}]\) when the protein–ligand complex is displaced by 50%, can be calculated using Equation (5).

By substituting Equation (8) into Equation (5), we obtain Equation (9). Since \([\mathrm{PI}]_{\mathrm{eq\text{-}50}}\) and \([\mathrm{P}]_{\mathrm{eq\text{-}50}}\) are fully determined by \([\mathrm{PL}]_{\mathrm{eq\text{-}50}}\), \([\mathrm{P}]_0\), \([\mathrm{L}]_0\), and \(K_\mathrm{d}\), the IC50 (i.e., \([\mathrm{I}]\)) is proportional to \(K_\mathrm{i}\). However, the apparent IC50 is not directly proportional to \(K_\mathrm{i}\).

(1)\[[\mathrm{PL}]_\mathrm{eq} = \frac{\left([\mathrm{P}]_0 + [\mathrm{L}]_0 + K_\mathrm{d}\right) - \sqrt{\left([\mathrm{P}]_0 + [\mathrm{L}]_0 + K_\mathrm{d}\right)^2 - 4[\mathrm{P}]_0[\mathrm{L}]_0}}{2}\]

(2)\[K_\mathrm{d} = \frac{[\mathrm{P}][\mathrm{L}]}{[\mathrm{PL}]}\]

(3)\[K_\mathrm{i} = \frac{[\mathrm{P}][\mathrm{I}]}{[\mathrm{PI}]}\]

(4)\[[\mathrm{L}] = [\mathrm{L}]_0 - [\mathrm{PL}]\]

(5)\[[\mathrm{I}] = [\mathrm{I}]_0 - [\mathrm{PI}]\]

(6)\[[\mathrm{PI}] = [\mathrm{P}]_0 - [\mathrm{P}] - [\mathrm{PL}]\]

(7)\[[\mathrm{P}] = \frac{K_\mathrm{d}\,[\mathrm{PL}]}{[\mathrm{L}]_0 - [\mathrm{PL}]}\]

(8)\[[\mathrm{I}]_0 = \frac{[\mathrm{PI}]\left(K_\mathrm{i} + [\mathrm{P}]\right)}{[\mathrm{P}]}\]

(9)\[[\mathrm{I}] = \frac{[\mathrm{PI}]\,K_\mathrm{i}}{[\mathrm{P}]}\]