A general analysis of field-based molecular similarity indices

A formalism is presented that incorporates the entirety of all field-based molecular similarity indices of general form S-ij = Omega(ij) /h(Omega(ii), Omega(jj)), where the numerator is given by the inner product or "overlap" of field functions F-i and F-j corresponding to the ith and jth molecules, respectively, and the denominator is given by a suitable mean function of the self-similarities Omega(ii) and Omega(jj). This family of similarity indices includes the index initially introduced by Carbo nearly twenty years ago, where h(Omega(ii), Omega(jj)) is taken to be the geometric mean of Omega(ii) and Omega(jj), and the well-known indices due to Hodgkin and Richards, and Petke, where h(Omega(ii), Omega(jj)) is taken to be the arithmetic mean and maximum of Omega(ii) and Omega(jj), respectively. Two new indices based upon the harmonic mean and minimum of Omega(ii) and Omega(jj) are also defined, and it is demonstrated that the entire set of field-based similarity indices can be generated from a one-parameter family of functions, called generalized means, through proper choice of the parameter value and suitable limiting procedures. Ordering and rigorous bounds for all of the indices are described as well as a number of inter-relationships among the indices. The generalization of field-based similarity indices, coupled with the relationships among indices that have been developed in the present work, place the basic theory of these indices on a more unified and mathematically rigorous footing that provides a foundation for a better understanding of the quantitative aspects of field-based molecular similarity.