Use of the harmonic mean to the determination of dissociation constants of stereoisomeric mixtures of biologically active compounds

Herein we introduce the derivation of a mathematical expression to evaluate the dissociation constant of a mixture of stereoisomers in equal amounts (KdMIX), when the corresponding dissociation constants (Kd) or medium response (MR50) of the pure stereoisomers are known; the final equation takes the form of the harmonic mean. In order to validate the equation, we carried out a bibliographic search of experimental data of enantiomeric molecules with biological activity, considering the Kd's or MR50's of the isolated enantiomers as well as that of the racemate. The comparisons between the experimental dissociation constants of the mixtures (KdEXP or MR50EXP) and the calculated values (KdMIX or MR50MIX) were consistent; the similarity between these values is supported through statistical analyses of group comparison and simple linear correlation. The equation we obtained, which corresponds to the harmonic mean, was used to predict the values of KdMIX (or MR50MIX) or Kd (or MR50) in systems when only two of the experimental values are known: either the dissociation constants of both enantiomers or the Kd (or MR50) of one of the enantiomers and dissociation constant of the racemate.