  Topological data analysis (TDA) is an emerging field in mathematics and data
science. Its central technique, persistent homology, has had tremendous success
in many science and engineering disciplines. However, persistent homology has
limitations, including its incapability of describing the homotopic shape
evolution of data during filtration. Persistent topological Laplacians (PTLs),
such as persistent Laplacian and persistent sheaf Laplacian, were proposed to
overcome the drawback of persistent homology. In this work, we examine the
modeling and analysis power of PTLs in the study of the protein structures of
the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) spike receptor
binding domain (RBD) and its variants, i.e., Alpha, Beta, Gamma, BA.1, and
BA.2. First, we employ PTLs to study how the RBD mutation-induced structural
changes of RBD-angiotensin-converting enzyme 2 (ACE2) binding complexes are
captured in the changes of spectra of the PTLs among SARS-CoV-2 variants.
Additionally, we use PTLs to analyze the binding of RBD and ACE2-induced
structural changes of various SARS-CoV-2 variants. Finally, we explore the
impacts of computationally generated RBD structures on PTL-based machine
learning, including deep learning, and predictions of deep mutational scanning
datasets for the SARS-CoV-2 Omicron BA.2 variant. Our results indicate that
PTLs have advantages over persistent homology in analyzing protein structural
changes and provide a powerful new TDA tool for data science.
