  The conformational free energy landscape of a system is a fundamental
thermodynamic quantity of importance particularly in the study of soft matter
and biological systems, in which the entropic contributions play a dominant
role. While computational methods to delineate the free energy landscape are
routinely used to analyze the relative stability of conformational states, to
determine phase boundaries, and to compute ligand-receptor binding energies its
use in problems involving the cell membrane is limited. Here, we present an
overview of four different free energy methods to study morphological
transitions in bilayer membranes, induced either by the action of curvature
remodeling proteins or due to the application of external forces. Using a
triangulated surface as a model for the cell membrane and using the framework
of dynamical triangulation Monte Carlo, we have focused on the methods of Widom
insertion, thermodynamic integration, Bennett acceptance scheme, and umbrella
sampling and weighted histogram analysis. We have demonstrated how these
methods can be employed in a variety of problems involving the cell membrane.
Specifically, we have shown that the chemical potential, computed using Widom
insertion, and the relative free energies, computed using thermodynamic
integration and Bennett acceptance method, are excellent measures to study the
transition from curvature sensing to curvature inducing behavior of membrane
associated proteins. The umbrella sampling and WHAM analysis has been used to
study the thermodynamics of tether formation in cell membranes and the
quantitative predictions of the computational model are in excellent agreement
with experimental measurements. Furthermore, we also present a method based on
WHAM and thermodynamic integration to handle problems related to
end-point-catastrophe that are common in most free energy methods
