The Weighted Mean Curvature Derivative of a Space-Filling Diagram

• Published in: Computational and Mathematical Biophysics Vol. 8; no. 1; pp. 51 - 67
• Main Authors: Akopyan, Arsenyi, Edelsbrunner, Herbert
• Format: Journal Article
• Language: English
• Published: De Gruyter 27.07.2020
• Subjects: 52A38; 92E10; alpha shapes; computer implementation; derivatives; discontinuities; inclusion-exclusion; intrinsic volume; Molecular dynamics; proteins; space-filling diagrams
• DOI: 10.1515/cmb-2020-0100
• ISSN: 2544-7297

Representing an atom by a solid sphere in 3-dimensional Euclidean space, we get the space-filling diagram of a molecule by taking the union. Molecular dynamics simulates its motion subject to bonds and other forces, including the solvation free energy. The morphometric approach [12, 17] writes the latter as a linear combination of weighted versions of the volume, area, mean curvature, and Gaussian curvature of the space-filling diagram. We give a formula for the derivative of the weighted mean curvature. Together with the derivatives of the weighted volume in [7], the weighted area in [3], and the weighted Gaussian curvature [1], this yields the derivative of the morphometric expression of the solvation free energy.