  Cell membranes interact with a myriad of curvature-active proteins that
control membrane morphology and are responsible for mechanosensation and
mechanotransduction. Some of these proteins, such as those containing BAR
domains, are curved and elongated, and hence may adopt different states of
orientational order, from isotropic to maximize entropy to nematic as a result
of crowding or to adapt to the curvature of the underlying membrane. Here,
extending the work of [Nascimento et. al, Phys. Rev. E, 2017, 96, 022704], we
develop a mean-field density functional theory to predict the orientational
order and evaluate the free-energy of ensembles of elongated and curved objects
on curved membranes. This theory depends on the microscopic properties of the
particles and explains how a density-dependent isotropic-to-nematic transition
is modified by anisotropic curvature. We also examine the coexistence of
isotropic and nematic phases. This theory lays the ground to understand the
interplay between membrane reshaping by BAR proteins and molecular order,
examined in [Le Roux et. al, Submitted, 2020].
