Topologically Directed Simulations Reveal the Impact of Geometric Constraints on Knotted Proteins

TL;DR

This paper addresses how knots form and untie in biopolymers by introducing topological steering based on knotoid projections of open curves onto the sphere $S^2$ and by defining global complexity functionals $\text{AUN}$ and $\text{TUN}$. It develops a gradient-like optimization over perturbations to direct configurations toward higher or lower knotoid complexity, and demonstrates the approach with Langevin dynamics of semiflexible polymers and with a growing self-avoiding walk (GSAW) model. The results show that without protein-like geometric constraints, topologically guided evolution favors alternating torus knots, while imposing protein-like angular constraints shifts the knot spectrum toward twist knots, aligning with observations in knotted proteins. The work provides a principled method to generate large samples of knotted protein-like polymers and highlights local geometry as a key determinant of knotting spectra, with implications for understanding folding pathways and protein knotting.

Abstract

Simulations of knotting and unknotting in polymers or other filaments rely on random processes to facilitate topological changes. Here we introduce a method of \textit{topological steering} to determine the optimal pathway by which a filament may knot or unknot while subject to a given set of physics. The method involves measuring the knotoid spectrum of a space curve projected onto many surfaces and computing the mean unravelling number of those projections. Several perturbations of a curve can be generated stochastically, e.g. using the Langevin equation or crankshaft moves, and a gradient can be followed that maximises or minimises the topological complexity. We apply this method to a polymer model based on a growing self-avoiding tangent-sphere chain, which can be made to model proteins by imposing a constraint that the bending and twisting angles between successive spheres must maintain the distribution found in naturally occurring protein structures. We show that without these protein-like geometric constraints, topologically optimised polymers typically form alternating torus knots and composites thereof, similar to the stochastic knots predicted for long DNA. However, when the geometric constraints are imposed on the system, the frequency of twist knots increases, similar to the observed abundance of twist knots in protein structures.

Figures (8)

• Figure 1: a. Projection of an open 3D curve onto three surfaces. Each projection corresponds to a knotoid type. b. A schematic for the $S^2$-distribution of knotoids of the curve in a. Each colour indicates a specific knotoid type, with the corresponding unravelling number added as a label. c. An example of a diagram unravelling into simpler knotoids through forbidden moves. The unravelling number of the left-most knotoid is equal to 2, as the penultimate diagram in the sequence differ by the trivial knotoid also by a single Reidemeister I move.
• Figure 2: A 3D curve creating a slipknot (left), and an open-ended trefoil (right). For the slipknot, while the $\text{TUN}$ adequately captures the local entanglement of the curve, the $\text{AUN}$ is very close to $0$.
• Figure 3: a. AUN of a semiflexible polymer initially in a deep $3_1$ knotted configuration as it unties. Shades of purple show the topologically optimal untying, while blue shows an ensemble average of several untying events driven by thermal fluctuations. b. Same as a., with TUN instead of AUN. Shades of green show the topologically optimal untying, while blue shows an ensemble average of several untying events driven by thermal fluctuations. c. Comparison between AUN and TUN-driven unknotting.
• Figure 4: Topologically optimal knotting of an initially unknotted open semiflexible chain, forming alternating torus knots of increasing complexity.
• Figure 5: Schematic of the growing tangent-sphere chain, in which successive spheres are added according to a bending angle $\theta$ and a dihedral angle $\phi$.