[Paper Review] A topological study of protein folding kinetics

Eleni Panagiotou, Kevin W. Plaxco|arXiv (Cornell University)|Dec 3, 2018

Protein Structure and DynamicsBiochemistry, Genetics and Molecular Biology22 references3 citations

TL;DR

This study investigates how protein folding kinetics relate to topological and geometric properties of the native state, using the Gauss linking integral, writhe, and torsion to analyze a set of two-state, non-knotted proteins. It finds that decreasing folding rates correlate with increasingly negative global writhe and torsion, driven by an increase in sequence-distant contacts and negative linking between secondary structures—particularly helices and coils—suggesting topology plays a key role beyond contact count alone.

ABSTRACT

Focusing on a small set of proteins that i) fold in a concerted, all-or-none fashion and ii) do not contain knots or slipknots, we show that the Gauss linking integral, the torsion and the number of sequence-distant contacts provide information regarding the folding rate. Our results suggest that the global topology/geometry of the proteins shifts from right-handed to left-handed with decreasing folding rate, and that this topological change is associated with an increase in the number of more sequence-distant contacts.

Motivation & Objective

• To understand how topological and geometric features of the native protein structure influence folding kinetics.
• To assess whether the Gauss linking integral and related topological invariants (writhe, torsion) provide predictive power for folding rates beyond existing measures like sequence-distant contacts.
• To investigate the role of secondary structure elements (α-helices, β-strands, coils) in shaping the global topology of the protein and its impact on folding.
• To determine if negative linking or writhe in secondary structure pairs correlates with slower folding, even when contact count is held constant.

Proposed method

• Proteins were modeled as polygonal curves using Cα atom coordinates, enabling topological analysis.
• The Gauss linking integral was computed between all pairs of secondary structure elements to quantify entanglement.
• Writhe and torsion were calculated for the entire protein chain and for individual coils to assess geometric complexity.
• Sequence-distant contacts were defined as Cα–Cα distances <6 Å with sequence separation >12 residues, used as a benchmark for comparison.
• Statistical correlations were computed between folding rates and topological/geometric parameters, including relative proportions of negative linking, writhe, and torsion.
• The analysis focused on a curated set of two-state, non-knotted proteins to isolate topological effects from kinetic complexity.

Experimental results

Research questions

• RQ1 Does the global writhe and torsion of the protein chain correlate with its folding rate?
• RQ2 How do the linking numbers between secondary structure elements (e.g., α-helices, β-strands) relate to folding kinetics?
• RQ3 To what extent do negative writhe or torsion in coils contribute to slower folding, independent of contact count?
• RQ4 Is the number of sequence-distant contacts a sufficient proxy for topological complexity, or do other measures like linking number provide additional predictive power?

Key findings

• Folding rates decrease as the global writhe and torsion of the protein become more negative, indicating a shift from right-handed to left-handed topology with slower folding.
• Proteins with more sequence-distant contacts exhibit more negative linking numbers between α-helices and coils, suggesting a topological origin for contact complexity.
• The number of pairs of α-helices with negative linking number correlates with slower folding (R² = 0.195), and this correlation is stronger than that with contact count alone.
• Coils with negative writhe or torsion are associated with slower folding, even when contact count is low, indicating that geometric complexity in loops contributes independently to kinetic barriers.
• The presence of antiparallel β-strands contributes to a 0.5 ratio of negative to positive linking pairs, due to their alternating linking contributions, which may stabilize complex topologies.
• The Gauss linking integral and torsion provide complementary information to contact count, as folding rates are more sensitive to negative writhe/torsion than to contact number alone.