Geometric algebra generation of molecular surfaces

TL;DR

This paper introduces a novel geometric algebra method utilizing Clifford-Fourier transform for molecular surface generation, enabling improved PDE solutions and applications in biological sciences.

Contribution

It presents a new approach using Clifford-Fourier transform in geometric algebra for molecular surface generation and PDE solving, with validation on molecules and proteins.

Findings

Generated molecular surfaces match other definitions

Applicable to small molecules and proteins

Facilitates protein electrostatic analysis

Abstract

Geometric algebra is a powerful framework that unifies mathematics and physics. Since its revival in the middle of the 1960s by David Hestenes, it attracts great attention and has been exploited in many fields such as physics, computer science, and engineering. This work introduces a geometric algebra method for the molecular surface generation that utilizes the Clifford-Fourier transform which is a generalization of the classical Fourier transform. Notably, the classical Fourier transform and Clifford-Fourier transform differ in the derivative property in $R_k$ for k even. This distinction is due to the noncommutativity of geometric product of pseudoscalars with multivectors and has significant consequences in applications. We use the Clifford-Fourier transform in $R_3$ to benefit from the derivative property in solving partial differential equations (PDEs). The Clifford-Fourier transform is used to solve the mode decomposition process in PDE transform. Two different initial cases are proposed to make the initial shapes used in the present method. The proposed method is applied first to small molecules and proteins. To validate the method, the molecular surfaces generated are compared to surfaces of other definitions. Applications are considered to protein electrostatic analysis. This work opens the door for further applications of geometric algebra and Clifford-Fourier transform in biological sciences.