Geometric algebra generation of molecular surfaces.

Abstract

Geometric algebra is a powerful framework that unifies mathematics and physics. Since its revival in the 1960s, it has attracted great attention and has been exploited in fields like physics, computer science and engineering. This work introduces a geometric algebra method for the molecular surface generation that uses the Clifford-Fourier transform (CFT) which is a generalization of the classical Fourier transform. Notably, the classical Fourier transform and CFT differ in the derivative property in [Formula: see text] for k even. This distinction is due to the non-commutativity of geometric product of pseudoscalars with multivectors and has significant consequences in applications. We use the CFT in [Formula: see text] to benefit from the derivative property in solving partial differential equations (PDEs). The CFT is used to solve the mode decomposition process in PDE transform. Two different initial cases are proposed to make the initial shapes 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 surface potentials and solvation free energy. This work opens the door for further applications of geometric algebra and CFT in biological sciences.