Cylindrical confinement of semiflexible polymers

TL;DR

This paper analyzes the equilibrium configurations of semiflexible polymers confined to cylindrical surfaces, revealing stability conditions, topological invariants, and potential biological relevance to membrane fission.

Contribution

It introduces a detailed classification of bound states of semiflexible polymers on cylinders, including stability analysis and topological constraints, advancing understanding of polymer confinement physics.

Findings

Ground state with p=1 is an elliptic deformation of a parallel.

Long loops have energies approximately proportional to p or n.

Above a critical length, excited states become stable, preventing unfolding.

Abstract

Equilibrium states of a closed semiflexible polymer binding to a cylinder are described. This may be either by confinement or by constriction. Closed completely bound states are labeled by two integers: the number of oscillations, $n$, and the number of times it winds the cylinder, $p$, the latter a topological invariant. We examine the behavior of these states as the length of the loop is increased by evaluating the energy, the conserved axial torque and the contact force. The ground state for a given $p$ is the state with $n=1$; a short loop with $p=1$ is an elliptic deformation of a parallel; as its length increases it elongates along the cylinder axis, with two hairpin ends. Excited states with $n\geq2$ and $p=1$ possess $n$-fold axial symmetry. Short (long) loops possess energies $\approx p E_0$ ($nE_0$), with $E_0$ the energy of a circular loop with same radius as the cylinder; in long loops the axial torque vanishes. Confined bound excited states are initially unstable; however, above a critical length each $n$-fold state becomes stable: the folded hairpin cannot be unfolded. The ground state for each $p$ is also initially unstable with respect to deformations rotating the loop off the surface into the interior. A closed planar elastic curve aligned along the cylinder axis making contact with the cylinder on its two sides is identified as the ground state of a confined loop. Exterior bound states behave very differently, if free to unbind, as signaled by the reversal in the sign of the contact force. If $p=1$, all such states are unstable. If $p \geq 2$, however, a topological obstruction to complete unbinding exists. If the loop is short, the bound state with $p=2$ and $n=1$ provides a stable constriction of the cylinder, partially unbinding as the length is increased. This motif could be relevant to an understanding of the process of membrane fission mediated by dynamin rings.