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From Euclidean distance, to polymer uncrossing and knotting, to protein folding rate prediction

Banff International Research Station for Mathematical Innovation and Discovery

A fundamental problem of relevance 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 distance between non-crossing space curves to uniquely define the Euclidean distance between two biopolymer conformations. We apply this order parameter to the problem of protein folding rates and reaction coordinates. To do so, we develop a method for generating a diverse conformational ensemble, to characterize properties of the unfolded states of intrinsically disordered or intrinsically folded proteins. We find that for a randomly selected dataset of 15 non-homologous 2- and 3- state proteins, quantities such as the average root mean squared deviation between the folded structure and unfolded ensemble correlate with folding rates as strongly as absolute contact order. In an all-atom representation that respects steric constraints, the distance travelled per residue naturally partitions into a smooth “laminar” and subsequent “turbulent” part of the trajectory. This latter conceptually simple measure with no fitting parameters predicts folding rates in 0 M denaturant with remarkable accuracy (r = -0.95, p = 1e-7). The high correlation between folding times and sterically modulated,\\r\\nreconfigurational motion supports the rapid collapse of proteins prior to the transition state as a generic feature in the folding of both two-state and multi-state proteins. This method for generating unfolded ensembles provides a powerful approach to address various questions in protein evolution, misfolding and aggregation, transient structures, and molten globule and disordered protein phases.

Mathematics; Biology and other natural sciences; Manifolds and cell complexes; Mathematical biology

video/mp4

Author affiliation: University of British Columbia

2014-08-07

Attribution-NonCommercial-NoDerivs 2.5 Canada

http://hdl.handle.net/2429/49722

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/2.5/ca/