Geometric Algebra Modeling of Biomolecules

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. Its potential has yet to be leveraged in biology and biochemistry. This work introduces geometric algebra methods for protein-protein docking and molecular surface generation. The first method utilizes rotors for rotations instead of rotation matrices, exploiting the efficiency and compactness of geometric algebra operations. This approach reduces the number of operations needed significantly. The proposed method, GADOCK, was assessed on 36 enzyme-inhibitor pairs from the Protein Docking Benchmark and compared to the FMFT method. GADOCK outperforms FMFT in terms of execution time, accelerating the local search stage by 15\% on average, noting that this stage is by far the most expensive stage in the docking process, accounting for 90\%. On top of that, this efficiency enhancement was not in the trade of accuracy. Both methods produce very comparable case-by-case results, and GADOCK performs better on average, improving the accuracy by 8.42\% when averaged over all pairs and 19.82\% when averaged over good predictions only. The accuracy was measured using interface $C_{\alpha}$ RMSD On the other hand, the molecular surface generation method utilizes the Clifford-Fourier transform, a generalization of the classical Fourier transform. Notably, the classical Fourier transform and Clifford-Fourier transform differ in the derivative property in $\mathbb{R}_k$ for $k$ even. This distinction is due to the noncommutativity of the geometric product of pseudoscalars with multivectors and has significant consequences in applications. We use the Clifford-Fourier transform in $\mathbb{R}_3$ 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 the PDE transform. The present method proposes two different initial cases to make the initial shapes. The proposed method is applied first to small molecules and proteins. The molecular surfaces generated are compared to surfaces of other definitions to validate the method. Applications are considered to protein electrostatic surface potentials and solvation free energy. This work opens the door for further applications of geometric algebra and Clifford-Fourier transform in biological sciences.