Motivated to understand the behavior of biological filaments interacting with
membranes of various types, we study a theoretical model for the shape and
thermodynamics of intrinsically-helical filaments bound to curved membranes. We
show filament-surface interactions lead to a host of non-uniform shape
equilibria, in which filaments progressively unwind from their native twist
with increasing surface interaction and surface curvature, ultimately adopting
uniform-contact curved shapes. The latter effect is due to non-linear coupling
between elastic twist and bending of filaments on anisotropically-curved
surfaces, such as the cylindrical surfaces considered here. Via a combination
of numerical solutions and asymptotic analysis of shape equilibria we show that
filament conformations are critically sensitive to the surface curvature in
both the strong- and weak-binding limits. These results suggest that local
structure of membrane-bound chiral filaments is generically sensitive to the
curvature-radius of the surface to which it is bound, even when that radius is
much larger than the filament intrinsic pitch. Typical values of elastic
parameters and interaction energies for several prokaryotic and eukaryotic
filaments indicate that biopolymers are inherently very sensitive to the
coupling between twist, interactions and geometry and that this could be
exploited for regulation of a variety of processes such as the targeted
exertion of forces, signaling and self-assembly in response to geometric cues
including the local mean and Gaussian curvatures
