Biological macromolecules display intricate geometric and topological organization that defies traditional descriptors based solely on atom-level coordinates or sequence information. This dissertation introduces an integrated framework that advances both computational algebraic and geometric topology to capture multiscale structure–function relationships in biomolecular data. In the algebraic domain, we expand persistent homology to higher-order N-chain complexes, producing generalized, efficiently computable descriptors; in the geometric domain, we develop a suite of multiscale invariants—including the multiscale Gauss linking integral, evolutionary Khovanov homology, and persistent Khovanov homology—to quantify entanglement in knot-type data. Applied to protein–ligand affinity prediction, DNA/RNA topological analysis, and macromolecular flexibility assessment, these tools yield interpretable features with competitive accuracy, underscoring the promise of topological approaches in contemporary biological research.
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