# Efficient sampling of knotting-unknotting pathways for semiflexible Gaussian chains

## TL;DR

This paper introduces a stochastic method for efficiently generating conditioned overdamped Langevin paths of semi-flexible Gaussian chains, enabling detailed analysis of knotting and unknotting transitions in soft matter and DNA systems.

## Contribution

The authors develop an exact local stochastic differential equation approach to generate independent, conditioned paths between arbitrary conformations, including knotted states.

## Key findings

- Transition routes can involve more complex topologies than initial and final states.
- Average crossings, writhe, and unknotting number vary non-monotonically over time.
- The method enables analysis of topological reconnections in soft matter and DNA.

## Abstract

We propose a stochastic method to generate exactly the overdamped Langevin dynamics of semi-flexible Gaussian chains, conditioned to evolve between given initial and final conformations in a preassigned time. The initial and final conformations have no restrictions, and hence can be in any knotted state. Our method allows the generation of statistically independent paths in a computationally efficient manner. We show that these conditioned paths can be exactly generated by a set of local stochastic differential equations. The method is used to analyze the transition routes between various knots in crossable filamentous structures, thus mimicking topological reconnections occurring in soft matter systems or those introduced in DNA by topoisomerase enzymes. We find that the average number of crossings, writhe and unknotting number are not necessarily monotonic in time and that more complex topologies than the initial and final ones can be visited along the route.