Decomposition of interaction free energies in proteins and other complex systems

A recent analysis of Mark and van Gunsteren has questioned the validity of separating different free energy components in proteins, or indeed in any complex system. The separability of free energy terms is re-examined from both a theoretical and a numerical perspective. Using a power series expansion of the free energy, it is found that the leading terms are free energy components that arise from individual contributions to the Hamiltonian ("in situ" free energies). The energetic part of an in situ free energy component is given by the ensemble average of the corresponding Hamiltonian component, while the leading term in the entropic part, which was missing in the analysis of Mark and van Gunsteren, is given by the mean square fluctuation. In addition there are correlations between fluctuations in each Hamiltonian component, which give rise to a coupling, or correlation entropy. A simple system, whose configurational degrees of freedom can be completely sampled, was examined in order to determine the relative sizes of these different contributions to the free energy. Under certain conditions, the change in system free energy observed when a particular component of the Hamiltonian is removed or altered is well approximated by the change in the in situ free energy of that component. In practical terms, this allows one in these cases to separate out different free energy contributions.