Frustration and Packing in Curved-Filament Assemblies: From Isometric to Isomorphic Bundles

TL;DR

This paper investigates how the intrinsic curvature of biological filaments influences the packing structure of filament bundles, revealing size-dependent transitions between isometric and isomorphic packing states and their impact on bundle stability.

Contribution

It introduces a theoretical framework for understanding the interplay between filament curvature, packing geometry, and thermodynamic factors in bundle formation.

Findings

Bundle packing transitions from isometric to isomorphic with increasing size.

Elastic constraints lead to smooth size-dependent packing evolution.

Osmotic compression can cause a singular transition at finite bundle radius.

Abstract

Densely-packed bundles of biological filaments (filamentous proteins) are common and critical structural elements in range of biological materials. While most bundles form from intrinsically straight filaments, there are notable examples of protein filaments possessing a natural, or intrinsic, curvature, such as the helical bacterial flagellum. We study the non-linear interplay between thermodynamic preference for dense and regular inter-filament packing and the mechanical preference for uniform filament shape in bundles of helically-curved filaments. Geometric constraints in bundles make perfect inter-filament (constant spacing, or isometric) packing incompatible with perfect intra-filament (constant shape, or isomorphic) packing. As a consequence, we predict that bundle packing exhibits a strong sensitivity to bundle size, evolving from the isometric packing at small radii to an isomorphic packing at large radii. The nature of the transition between these extremal states depends on thermodynamic costs of packing distortion, with packing in elastically-constrained bundles evolving smoothly with size, while packing in osmotically-compressed bundles may exhibit a singular transition from the isometric packing at a finite bundle radius. We consider the equilibrium assembly of bundles in a saturated solution of filaments and show that mechanical cost of isomorphic packing leads to self-limited equilibrium bundle diameters, whose size and range of thermodynamic stability depend both on condensation mechanism, as well as the helical geometry of filaments.