  Curvature-sensing and curvature-remodeling proteins are known to reshape cell
membranes, and this remodeling event is essential for key biophysical processes
such as tubulation, exocytosis, and endocytosis. Curvature-inducing proteins
can act as curvature sensors as well as induce curvature in cell membranes to
stabilize emergent high curvature, non-spherical, structures such as tubules,
discs, and caveolae. A definitive understanding of the interplay between
protein recruitment and migration, the evolution of membrane curvature, and
membrane morphological transitions is emerging but remains incomplete. Here,
within a continuum framework and using the machinery of Monte Carlo
simulations, we introduce and compare three free-energy methods to delineate
the free-energy landscape of curvature-inducing proteins on bilayer membranes.
We demonstrate the utility of the Widom test-particle/field insertion
methodology in computing the excess chemical potentials associated with
curvature-inducing proteins on the membrane-- in particular, we use this method
to track the onset of morphological transitions in the membrane at elevated
protein densities. We validate this approach by comparing the results from the
Widom method with those of thermodynamic integration and Bennett acceptance
ratio methods. Furthermore, the predictions from the Widom method have been
tested against analytical calculations of the excess chemical potential at
infinite dilution. Our results are useful in precisely quantifying the
free-energy landscape, and also in determining the phase boundaries associated
with curvature-induction, curvature-sensing, and morphological transitions.
This approach can be extended to studies exploring the role of thermal
fluctuations and other external (control) variables, such as membrane excess
area, in shaping curvature-mediated interactions on bilayer membranes.
