  The free energetics of water density fluctuations near a surface, and the
rare low-density fluctuations in particular, serve as reliable indicators of
surface hydrophobicity; the easier it is to displace the interfacial waters,
the more hydrophobic the underlying surface. However, characterizing the free
energetics of such rare fluctuations requires computationally expensive,
non-Boltzmann sampling methods like umbrella sampling. This inherent
computational expense associated with umbrella sampling makes it challenging to
investigate the role of polarizability or electronic structure effects in
influencing interfacial fluctuations. Importantly, it also limits the size of
the volume, which can be used to probe interfacial fluctuations. The latter can
be particularly important in characterizing the hydrophobicity of large
surfaces with molecular-level heterogeneities, such as those presented by
proteins. To overcome these challenges, here we present a method for the sparse
sampling of water density fluctuations, which is roughly two orders of
magnitude more efficient than umbrella sampling. We employ thermodynamic
integration to estimate the free energy differences between biased ensembles,
thereby circumventing the umbrella sampling requirement of overlap between
adjacent biased distributions. Further, a judicious choice of the biasing
potential allows such free energy differences to be estimated using short
simulations, so that the free energetics of water density fluctuations are
obtained using only a few, short simulations. Leveraging the efficiency of the
method, we characterize water density fluctuations in the entire hydration
shell of the protein, ubiquitin; a large volume containing an average of more
than six hundred waters.
