Understanding the contagiousness of Covid-19 strains: A geometric approach

Protein-protein interaction occurs on surface patches with some degree of complementary geometric and chemical features. Building on this understanding, this study endeavors to characterize the spike protein of the SARS-CoV-2 virus at the morphological and geometrical levels in its Alpha, Delta, and Omicron variants. In particular, the affinity between different SARS-CoV-2 spike proteins and the ACE2 receptor present on the membrane of the human respiratory system cells is investigated. To achieve an adequate degree of geometrical accuracy, the 3D depth maps of the proteins in exam are filtered by developing an ad-hoc convolutional filter with a kernel implemented as a sphere of varying radius, simulating a ball rolling on the surface (similar to the 'rolling ball' filter). This ball ideally models a hypothetical molecule that could interface with the protein and is inspired by the geometric approach to macromolecule-ligand interactions proposed by Kuntz et al. in 1982. The aim is to mitigate the imperfections and to obtain a smoother surface that could be studied from a geometrical perspective for binding purposes. A set of geometric descriptors, borrowed from the 3D face analysis context is then mapped point-by-point onto protein depth maps. Following a feature extraction phase inspired by Histogram of Oriented Gradients and Local Binary Patterns, the final histogram features are used as input for a Support Vector Machine classifier to automatically classify the proteins according to their surface affinity, where a similarity in shape is observed between ACE2 and the spike protein of the SARS-CoV-2 Omicron variant. Finally, Root Mean Square Error analysis is used to quantify the geometrical affinity between the ACE2 receptor and the respective Receptor Binding Domains of the three SARS-CoV-2 variants, culminating in a geometrical explanation for the higher contagiousness of Omicron relative to the other variants under study.