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.
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