Helical states of nonlocally interacting molecules and their linear stability: geometric approach

TL;DR

This paper develops a geometric framework on SE(3) to analyze the stability of helical conformations in nonlocally interacting molecules, revealing how non-local interactions can induce shape instabilities.

Contribution

It introduces an SE(3)-based method to analyze linear stability of helical molecules with non-local interactions, providing explicit dispersion relations and stability criteria.

Findings

Helical shapes are stable under certain conditions.

Non-local interactions can induce instabilities leading to helical configurations.

Explicit stability analysis for a specific charged rod model.

Abstract

The equations for strands of rigid charge configurations interacting nonlocally are formulated on the special Euclidean group, SE(3), which naturally generates helical conformations. Helical stationary shapes are found by minimizing the energy for rigid charge configurations positioned along an infinitely long molecule with charges that are off-axis. The classical energy landscape for such a molecule is complex with a large number of energy minima, even when limited to helical shapes. The question of linear stability and selection of stationary shapes is studied using an SE(3) method that naturally accounts for the helical geometry. We investigate the linear stability of a general helical polymer that possesses torque-inducing non-local self-interactions and find the exact dispersion relation for the stability of the helical shapes with an arbitrary interaction potential. We explicitly determine the linearization operators and compute the numerical stability for the particular example of a linear polymer comprising a flexible rod with a repeated configuration of two equal and opposite off-axis charges, thereby showing that even in this simple case the non-local terms can induce instability that leads to the rod assuming helical shapes.