Deformable Protein Shape Classification Based on Deep Learning, and the Fractional Fokker–Planck and Kähler–Dirac Equations

The classification of deformable protein shapes, based solely on their macromolecular surfaces, is a challenging problem in protein–protein interaction prediction and protein design. Shape classification is made difficult by the fact that proteins are dynamic, flexible entities with high geometrical complexity. In this paper, we introduce a novel description for such deformable shapes. This description is based on the bifractional Fokker–Planck and Dirac–Kähler equations. These equations analyse and probe protein shapes in terms of a scalar, vectorial and non-commuting quaternionic field, allowing for a more comprehensive description of the protein shapes. An underlying non-Markovian Lévy random walk establishes geometrical relationships between distant regions while recalling previous analyses. Classification is performed with a multiobjective deep hierarchical pyramidal neural network, thus performing a multilevel analysis of the description. Our approach is applied to the SHREC'19 dataset for deformable protein shapes classification and to the SHREC'16 dataset for deformable partial shapes classification, demonstrating the effectiveness and generality of our approach.