Hierarchical description of molecular structures based on promolecular electron density representations

In a recent work [L. Leherte, J. Math. Chem. 29, 47-83 (2001)], it was shown how low resolution graph representations of molecular structures could be used to easily compare different drug molecules. These graph representations were composed of vertices and edges that were generated using a critical point (CP) analysis of low resolution electron density (ED) maps, and used as pharmacophore models of biologically active/affine molecules.

The aim of the on-going work is to establish a hierarchical description of these CP graphs based on promolecular ED representations in order to assign peaks (or local maximum of the ED function) observed at a given resolution to a chemical function or group of atoms. In this context, every peak at a given resolution could be related to peaks obtained at a higher resolution level, and eventually to the constituting atoms (seen as the highest resolution peaks) of the molecular structure. It is thus expected that CPs of a low resolution graph will be characterized not only by their local density and ED curvature values, but also by a chemical meaning for further applications in molecular similarity search, for example.

In this purpose, the Atomic Shell Approximation (ASA) was selected [L. Amat and R. Carbó-Dorca, J. Comput. Chem. 19, 2023-2039 (1997)]. The ASA formalism allows the calculation of a promolecular ED distribution in terms of weighted summation over atomic ED distributions which are described in terms of series of squared 1s Gaussian functions fitted from atomic basis set representations. In order to calculate smoothed versions of the promolecular ED function, the ASA approach is coupled with the formalism developed by Kostrowicki et al. [J. Kostrowicki, L. Piela, B. J. Cherayil, H. A. Scheraga, J. Phys. Chem. 95, 4113-4119 (1991)] wherein deformed versions of the ED distribution are expressed as solutions of the diffusion equation.

For each of the so-calculated ED maps, the local maxima of the ED function are determined using a hierarchical clustering algorithm wherein peaks obtained at a given resolution are used as starting points for discovering peaks at the next lower resolution level [Y. Leung, J.-S. Zhang, Z.-B. Xu, IEEE Trans. on Pattern Analysis and Machine Intelligence 22, 1396-1410 (2000)]. In the so-obtained hierarchical description, peaks at a given resolution level are linked to those obtained at the next lower resolution through gradient trajectories of the ED function. Results can be presented in terms of dendrograms wherein each node corresponds to a well-defined molecular substructure (see illustrated example below):


Molecular structure of 7-chloro-1, 3-dihydro-1-methyl-5-(phenyl)-2H-1 (Diazepam) and substructures obtained at resolution levels 1.1 (green) and t = 1.6 (red). H atoms are not shown for clarity. Dendrogram resulting from a hierarchical description of a promolecular ED of Diazepam as a function of resolution. Resolution levels 1.1 and 1.6 are shown using horizontal lines.


We present results obtained for various molecular systems: benzodiazepine-related molecules, thrombin inhibitors, and a small polypeptide. Comparisons are made with other graphs of low resolution peaks, obtained from a critical point analysis of simulated X-ray diffraction ED maps and wavelet-based smoothed ED maps.