  This work addresses the question of whether it is possible to define simple
pair-wise interaction terms to approximate free energies of proteins or
polymers. Rather than ask how reliable a potential of mean force is, one can
ask how reliable it could possibly be. In a two-dimensional, infinite lattice
model system one can calculate exact free energies by exhaustive enumeration. A
series of approximations were fitted to exact results to assess the feasibility
and utility of pair-wise free energy terms. Approximating the true free energy
with pair-wise interactions gives a poor fit with little transferability
between systems of different size. Adding extra artificial terms to the
approximation yields better fits, but does not improve the ability to
generalise from one system size to another. Further, one cannot distinguish
folding from non-folding sequences via the approximated free energies. Most
usefully, the methodology shows how one can assess the utility of various terms
in lattice protein/polymer models.
