# The Weighted Mean Curvature Derivative of a Space-Filling Diagram

## TL;DR

This paper derives a formula for the derivative of the weighted mean curvature of a space-filling diagram, enabling more accurate calculations of solvation free energy in molecular simulations.

## Contribution

It provides a new formula for the derivative of weighted mean curvature, complementing existing derivatives for volume, area, and Gaussian curvature, to improve morphometric energy calculations.

## Key findings

- Derived the formula for the derivative of weighted mean curvature.
- Combined with existing derivatives for other geometric measures.
- Facilitates more precise computation of solvation free energy.

## Abstract

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 [HRC13,RHK06] 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 [EdKo03], the weighted area in [BEKL04], and the weighted Gaussian curvature [AkEd19], this yields the derivative of the morphometric expression of the solvation free energy.