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Theoretical model-based equations for the linear free energy relationships of the biological activity of ionizable substances. 1. Equilibrium-controlled potency.

Because of the ambiguities of how to treat ionization in empirical equations which relate biological activity to partition coefficient by use of a (log P)2 term, a theoretical approach to the problem is proposed. Based on a simplified view of assays of potency following in vitro or continuous infusion administration of drugs, equations have been derived from a combination of mass law, equilibrium, and extrathermodynamic assumptions. In general form the equations which relate potency to partition coefficient (P) and degree of ionization (alpha) are the following. If the neutral form reacts with the receptor, log (1/C) =-log [1 + SIGMAM(DIPci) + sigman[aj/Pb(1-alpha4y]] + X. If the ionic form reacts with the receptor, log (1/C) =-log [1 + (1 - Alphan)/(alphan)[sigmam(diPci) + sigman[aj/Pb(1-alphaj)]]] + X. In this generalized model there are m nonaqueous compartments and n aqueous compartments of different pH. The parameters a, b, c, and d can be interpreted in terms of the model. The shape of the log (1/C) vs. log P curve may be asymptotic, linear, or composed of two portions of unequal slope which meet at an optimum or a bend. With the use of these equations it is possible to examine whether the ion or the neutral form is the active species and whether there is hydrophobic bonding to the receptor and/or an inert compartment. The models may be further extended to include terms other than log P and alpha.