  Recent years have witnessed a trend that advanced mathematical tools, such as
algebraic topology, differential geometry, graph theory, and partial
differential equations, have been developed for describing biological
macromolecules. These tools have considerably strengthened our ability to
understand the molecular mechanism of macromolecular function, dynamics and
transport from their structures. However, currently, there is no unified
mathematical theory to analyze, describe and characterize biological
macromolecular geometry, topology, flexibility and natural mode at a variety of
scales. We introduce the de Rham-Hodge theory, a landmark of 20th Century's
mathematics, as a unified paradigm for analyzing biological macromolecular
geometry and algebraic topology, for predicting macromolecular flexibility, and
for modeling macromolecular natural modes at a variety of scales. In this
paradigm, macromolecular geometric characteristic and topological invariants
are revealed by de Rham-Hodge spectral analysis. By using the Helmholtz-Hodge
decomposition, every macromolecular vector field is split into orthogonal
divergence-free, curl-free, and harmonic components with a distinct physical
interpretation. We organize the eigenvalues and eigenvectors of the 0-form
Laplace-de Rham operator into one of the most accurate protein B-factor
predictors. By combining the 1-form Laplace-de Rham operator and the
Helfrich-type curvature energy, we predict the natural modes of both X-ray
protein structures and cryo-EM maps. We construct accurate and efficient
three-dimensional discrete exterior calculus algorithms for the aforementioned
modeling and analysis of biological macromolecules. Using extensive
experiments, we validate that the proposed paradigm is one of the most
versatile and powerful tools for biological macromolecular studies.
