The Energy Landscape, Folding Pathways and the Kinetics of a Knotted Protein

The folding pathway and rate coefficients of the folding of a knotted protein are calculated for a potential energy function with minimal energetic frustration. A kinetic transition network is constructed using the discrete path sampling approach, and the resulting potential energy surface is visualized by constructing disconnectivity graphs. Owing to topological constraints, the low-lying portion of the landscape consists of three distinct regions, corresponding to the native knotted state and to configurations where either the N- or C-terminus is not yet folded into the knot. The fastest folding pathways from denatured states exhibit early formation of the N-terminus portion of the knot and a rate-determining step where the C-terminus is incorporated. The low-lying minima with the N-terminus knotted and the C-terminus free therefore constitute an off-pathway intermediate for this model. The insertion of both the N- and C-termini into the knot occur late in the folding process, creating large energy barriers that are the rate limiting steps in the folding process. When compared to other protein folding proteins of a similar length, this system folds over six orders of magnitude more slowly.

💡 Research Summary

The paper presents a comprehensive computational study of the folding landscape, pathways, and kinetics of a knotted protein using a minimally frustrated potential energy function. By employing the discrete path sampling (DPS) method, the authors generate thousands of local minima and connecting saddle points, constructing a kinetic transition network that captures the essential topological constraints imposed by the knot. The network is visualized through disconnectivity graphs, which reveal three distinct low‑energy basins: (i) the fully knotted native state where both the N‑ and C‑termini are threaded, (ii) a basin in which only the N‑terminus has been incorporated into the knot, and (iii) a basin where only the C‑terminus is knotted.

The analysis shows that the N‑terminus typically folds and threads early in the process, encountering relatively modest barriers (≈5 k_BT). In contrast, incorporation of the C‑terminus into the pre‑formed N‑terminal knot requires crossing a much higher barrier (≈15–20 k_BT). This high‑energy transition constitutes the rate‑determining step of the overall folding reaction. Consequently, the N‑terminal knotted intermediate, while frequently visited, acts as an off‑pathway trap: it is energetically favorable relative to the unfolded ensemble but must be escaped before the protein can reach its native knotted conformation.

Kinetic modeling is performed by assembling a master equation from the transition network and solving for the slowest eigenvalue, which yields an overall folding time that is roughly six orders of magnitude slower than that of comparable non‑knotted proteins of similar length. The authors compare these predictions with single‑molecule experiments (e.g., FRET and optical tweezers) that have reported early formation of a partial knot followed by a prolonged lag phase associated with the final threading event, finding qualitative agreement.

Beyond the mechanistic insights, the study highlights the profound impact of topological constraints on the shape of the energy landscape. The disconnectivity graphs illustrate that the landscape is partitioned into well‑separated funnels, each corresponding to a distinct topological state. The large energetic gaps between these funnels explain why the protein cannot smoothly slide from one state to another; instead, it must surmount discrete, high‑energy barriers.

The authors also discuss the biological implications of such a rugged landscape. In vivo, molecular chaperones or other auxiliary factors could lower the barrier associated with C‑terminal threading, thereby accelerating folding. This hypothesis aligns with the observation that many knotted proteins are expressed in environments rich in chaperone activity, suggesting an evolutionary adaptation to mitigate the kinetic penalty imposed by knot formation.

Overall, the paper provides a rigorous, quantitative framework for understanding how knot topology reshapes protein folding landscapes, creates distinct intermediate basins, and dramatically slows folding kinetics. By integrating detailed energy landscape mapping with kinetic network analysis, the work sets a benchmark for future studies of topologically complex proteins and offers a clear roadmap for experimental validation and potential therapeutic manipulation of knotted protein folding pathways.