Minimal Cycle Representatives in Persistent Homology Using Linear Programming: An Empirical Study With User's Guide

Spatial and Sequential Topological Analysis of Molecular Dynamics Simulations of IgG1 Fc Domains

Monoclonal antibodies are utilized in a wide range of biomedical applications. The NIST monoclonal antibody is a resource for developing analysis methods for monoclonal antibody based biopharmaceutical platforms. Techniques from topological data analysis quantify structural features such as loops and tunnels which are not easily measured by classical data analysis methods. In this paper, we introduce the Gaussian CROCKER column differences (GCCD) matrix, which augments standard topological data analysis summaries with biological sequence information. We use GCCD matrices to successfully differentiate between glycosylated and aglycosylated conformations from molecular dynamics simulations of the NIST monoclonal antibody Fc domain. We are optimistic that other researchers will be able to utilize GCCD matrices to quantify multiscale spatial and sequential features.

Tight basis cycle representatives for persistent homology of large biological data sets

Persistent homology (PH) is a popular tool for topological data analysis that has found applications across diverse areas of research. It provides a rigorous method to compute robust topological features in discrete experimental observations that often contain various sources of uncertainties. Although powerful in theory, PH suffers from high computation cost that precludes its application to large data sets. Additionally, most analyses using PH are limited to computing the existence of nontrivial features. Precise localization of these features is not generally attempted because, by definition, localized representations are not unique and because of even higher computation cost. Such a precise location is a sine qua non for determining functional significance, especially in biological applications. Here, we provide a strategy and algorithms to compute tight representative boundaries around nontrivial robust features in large data sets. To showcase the efficiency of our algorithms and the precision of computed boundaries, we analyze the human genome and protein crystal structures. In the human genome, we found a surprising effect of the impairment of chromatin loop formation on loops through chromosome 13 and the sex chromosomes. We also found loops with long-range interactions between functionally related genes. In protein homologs with significantly different topology, we found voids attributable to ligand-interaction, mutation, and differences between species.

Homology of homologous knotted proteins

Quantification and classification of protein structures, such as knotted proteins, often requires noise-free and complete data. Here, we develop a mathematical pipeline that systematically analyses protein structures. We showcase this geometric framework on proteins forming open-ended trefoil knots, and we demonstrate that the mathematical tool, persistent homology, faithfully represents their structural homology. This topological pipeline identifies important geometric features of protein entanglement and clusters the space of trefoil proteins according to their depth. Persistence landscapes quantify the topological difference between a family of knotted and unknotted proteins in the same structural homology class. This difference is localized and interpreted geometrically with recent advancements in systematic computation of homology generators. The topological and geometric quantification we find is robust to noisy input data, which demonstrates the potential of this approach in contexts where standard knot theoretic tools fail.