Frustration and packing in curved-filament assemblies: from isometric to isomorphic bundles

Densely packed bundles of 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. Here, we study the non-linear interplay between the 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 stability depend both on condensation mechanism, as well as the helical geometry of filaments.