  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.
