## Generalizing Euclidean distance to understand polymer uncrossing and knotting : A physicist’s foray into protein folding

Physics often deals quite rightfully with symmetries and conservation laws, while molecular biology has historically retained little of this distinguished standard. Even the simplest biological macromolecules are aperiodic, have disordered energetics, and have enormously vast phase space relevant at biological temperatures. A central biophysical problem capturing all of these aspects is the question of how a protein, once synthesized by a ribosome in the cell, spontaneously folds up to it’s biologically functional structure. In this context, equilibrium and non-equilibrium statistical mechanics, as formulated in what has been called the energy landscape theory, has been essential in understanding protein folding, function, and evolution. Unfortunately however, geometry, structure, and transformation often fade away into the ensemble of a statistical mechanical description. A fundamental problem which is nevertheless central to protein folding and structural comparison of biomolecules is the notion of what *distance* means for higher-dimensional objects such as a polymer. Here we generalize the notion of distance between points to the distance between non-crossing space curves to uniquely define the Euclidean distance between two biopolymer conformations. We then tackle the conceptual and practical hurdles required to apply this quantity to the problem of protein folding ; we find that a simple measure of distance, with no adjustable parameters, predicts folding rates with remarkable accuracy (a correlation of 0.95). We discuss the implications of this finding.