Complete and Bi-Continuous Invariant of Protein Backbones under Rigid Motion

3^rd April 2025, 14:30
Will Jeffcott
Computer Science

Abstract

Proteins are large biomolecules that regulate all living organisms and consist of one or several chains. The primary structure of a protein chain is a sequence of amino acid residues whose three main atoms (alpha-carbon, nitrogen, and carbonyl carbon) form a protein backbone. The tertiary structure is the rigid shape of a protein chain represented by atomic positions in 3-dimensional space. Because different geometric structures often have distinct functional properties, it is important to continuously quantify differences in rigid shapes of protein backbones. This allows us to accurately measure whether two chains are similar. Unfortunately, many widely used similarities of proteins fail axioms of a distance metric and discontinuously change under tiny perturbations of atoms. The talk will introduce a complete invariant that identifies any protein backbone in 3-dimensional space uniquely under rigid motion. The new invariant is Lipschitz bi-continuous in the sense that it changes up to a constant multiple of a maximum perturbation of atoms, and vice versa. The invariant helped detect thousands of (near-)duplicates in the Protein Data Bank, whose presence skews machine learning predictions such as AlphaFold. The resulting invariant space allows us to produce low-dimensional maps with analytically defined coordinates that reveal substantial variability in the protein universe.