Application of Shannon-like diversity measures to cell-based chemistry spaces

The use of multi-dimensional “chemistry spaces” to represent large compound collections has become widespread in pharmaceutical research. In such spaces compounds are treated as points. Points in close proximity represent similar compounds, while distant points represent dissimilar compounds. Assessing the diversity of a compound collection, thus, is tantamount to characterizing the distribution of points in chemistry space. To facilitate many procedures such as selecting subsets of compounds for screening, for compound acquisition and designing combinatorial libraries, chemistry spaces have been partitioned into sets of non-overlapping, multi-dimensional cells, which are generated by dividing each axis into a number of equal-sized bins. This leads to a lattice of (N_bins)^N_dim{(N_{bins})^{N_{\rm dim}}} cells, where N _bins is the number of bins on each axis and N _dim is the dimensionality of the space. One diversity measure that is typically used in cell-based chemistry spaces is identical in form to Shannon entropy, D_Ncpd^cpd{D_{N_{cpd}}^{cpd}} A normalized measure of this Shannon entropy given by, D_rel^cpd{D_{rel}^{cpd}} enables comparison between compound collections that occupy different number of occupied cells. Although D_rel^cpd{D_{rel}^{cpd}} characterizes the uniformity and “spreadout” of the corresponding compound collection, it treats cells as positionally independent. Some of the positional information lost can be recaptured by another diversity measure, which is also related in form to Shannon entropy. This new measure D_Nbin^cell (l){D_{N_{bin}}^{cell} (\lambda)} characterizes the distribution of occupied cells along each axis of chemistry space. The normalized measure á D_rel^cell ñ{\left\langle {D_{rel}^{cell}}\right\rangle} over all axes is given then by the average. Examples illustrating the applicability of these two Shannon-like measures to compound collections are presented.