  Ligand-based virtual screening aims to reduce the cost and duration of drug
discovery campaigns. Shape similarity can be used to screen large databases,
with the goal of predicting potential new hits by comparing to molecules with
known favourable properties. This paper presents the theory underpinning
RGMolSA, a new alignment-free and mesh-free surface-based molecular shape
descriptor derived from the mathematical theory of Riemannian geometry. The
treatment of a molecule as a series of intersecting spheres allows the
description of its surface geometry using the Riemannian metric, obtained by
considering the spectrum of the Laplacian. This gives a simple vector
descriptor constructed of the weighted surface area and eight non-zero
eigenvalues, which capture the surface shape. We demonstrate the potential of
our method by considering a series of PDE5 inhibitors that are known to have
similar shape as an initial test case. RGMolSA displays promise when compared
to existing shape descriptors and in its capability to handle different
molecular conformers. The code and data used to produce the results are
available via GitHub: https://github.com/RPirie96/RGMolSA.
