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BY 4.0 license Open Access Published by De Gruyter Open Access August 12, 2020

The Weighted Gaussian Curvature Derivative of a Space-Filling Diagram

  • Arsenyi Akopyan EMAIL logo and Herbert Edelsbrunner

Abstract

The morphometric approach [11, 14] writes the solvation free energy 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 Gaussian curvature. Together with the derivatives of the weighted volume in [7], the weighted area in [4], and the weighted mean curvature in [1], this yields the derivative of the morphometric expression of solvation free energy.

MSC 2010: 52A38; 92E10

References

[1] A. V. Akopyan and H. Edelsbrunner, The Weighted Mean Curvature Derivative of a Space-Filling Diagram, Computational and Mathematical Biophysics, 8(1):51–67, 2020.10.1515/cmb-2020-0100Search in Google Scholar

[2] U. Bauer and H. Edelsbrunner. The Morse theory of Cech and Delaunay complexes. Trans. Amer. Math. Soc. 369 (2017), 3741–3762.10.1090/tran/6991Search in Google Scholar

[3] P.O. Bonnet. Mémoire sur la théorie générale des surfaces. J. de l’École Polytechnique 32 (1848), 1–46.Search in Google Scholar

[4] R. Bryant, H. Edelsbrunner, P. Koehl and M. Levitt. The area derivative of a space-filling diagram. Discrete Comput. Geom. 32 (2004), 293–308.Search in Google Scholar

[5] W. Chauvenet. Treatise on Plane and Spherical Trigonometry. Ninth edition, Lippincott Company, Philadelphia, Pennsylvania, 1887.Search in Google Scholar

[6] H. Edelsbrunner. Geometry and Topology for Mesh Generation. Cambridge Univ. Press, Cambridge, England, 2001.10.1017/CBO9780511530067Search in Google Scholar

[7] H. Edelsbrunner and P. Koehl. The weighted-volume derivative of a space-filling diagram. Proc. Natl. Acad. Sci. 100 (2003), 2203–2208.10.1073/pnas.0537830100Search in Google Scholar PubMed PubMed Central

[8] H. Edelsbrunner and E.P. Mücke. Three-dimensional alpha shapes. ACM Trans. Graphics 13 (1994), 43–72.10.1145/174462.156635Search in Google Scholar

[9] H. Hadwiger. Beweis eines Funktionalsatzes für konvexe Körper. Abh. Math. Sem. Univ. Hamburg 17 (1951), 11–23.10.1007/BF02950740Search in Google Scholar

[10] H. Hansen-Goos, R. Roth, K.R. Mecke and S. Dietrich. Solvation of proteins: linking thermodynamics to geometry. Phys. Rev. Lett. 99 (2007), 128101:1–4.10.1103/PhysRevLett.99.128101Search in Google Scholar PubMed

[11] Y. Harano, R. Roth and S. Chiba. A morphometric approach for the accurate solvation thermodynamics of proteins and ligands. J. Comput. Chem. 34 (2013), 1969–1974.Search in Google Scholar

[12] P.-M. König, R. Roth and K.R. Mecke. Morphological thermodynamics of fluids: shape dependence of free energies. Phys. Rev. Lett. 93 (2004), 160601:1–4.10.1103/PhysRevLett.93.160601Search in Google Scholar PubMed

[13] K.R. Mecke. A morphological model for complex fluids. J. Phys.: Condens. Matter 8 (1996), 9663–9667.10.1088/0953-8984/8/47/080Search in Google Scholar

[14] R. Roth, Y. Harano and M. Kinoshita. Morphometric approach to the solvation free energy of complex molecules. Phys. Rev. Lett. 97 (2006), 078101.10.1103/PhysRevLett.97.078101Search in Google Scholar PubMed

Received: 2020-04-30
Accepted: 2020-07-21
Published Online: 2020-08-12

© 2020 Arsenyi Akopyan et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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