Chirality Effects in Molecular Chainmail

TL;DR

This study explores how linking chirality in molecular chainmail networks influences their Gaussian curvature, revealing that different linking patterns lead to positive, negative, or flat membrane geometries, with implications for biological and synthetic materials.

Contribution

It demonstrates the impact of linking chirality on membrane curvature in molecular networks through simulations and geometric analysis, a novel insight into topology-curvature relationships.

Findings

Alternating linking yields positive Gaussian curvature.

Non-alternating linking results in negative Gaussian curvature.

Partially non-alternating linking causes flat sheets with folding transitions.

Abstract

Motivated by the observation of positive Gaussian curvature in kinetoplast DNA networks, we consider the effect of linking chirality in square lattice molecular chainmail networks using Langevin dynamics simulations and constrained gradient optimization. Linking chirality here refers to ordering of over-under versus under-over linkages between a loop and its neighbors. We consider fully alternating linking, maximally non-alternating, and partially non-alternating linking chiralities. We find that in simulations of polymer chainmail networks, the linking chirality dictates the sign of the Gaussian curvature of the final state of the chainmail membranes. Alternating networks have positive Gaussian curvature, similar to what is observed in kinetoplast DNA networks. Maximally non-alternating networks form isotropic membranes with negative Gaussian curvature. Partially non-alternating networks form flat diamond-shaped sheets which undergo a thermal folding transition when sufficiently large, similar to the crumpling transition in tethered membranes. We further investigate this topology-curvature relationship on geometric grounds by considering the tightest possible configurations and the constraints that must be satisfied to achieve them.