On the calculation of absolute macromolecular binding free energies

The standard framework for calculating the absolute binding free energy of a macromolecular association reaction A + B --> AB with an association constant K(AB) is to equate chemical potentials of the species on the left- and right-hand sides of this reaction and evaluate the chemical potentials from theory. This theory involves (usually hidden) assumptions about what constitutes the bound species, AB, and where the contribution of the solvent appears. We present here an alternative derivation that can be traced back to Bjerrum, in which the expectation value of K(AB) is obtained directly through the statistical mechanical method of evaluating its ensemble (Boltzmann-weighted) average. The generalized Bjerrum approach more clearly delineates: (i) the different contributions to binding; (ii) the origin of the much-discussed and somewhat controversial association entropy term; and (iii) where the solvent contribution appears. This approach also allows approximations required for practical evaluation of the binding constant in complex macromolecular systems, to be introduced in a well defined way. We provide an example, with application to test cases that illustrate a range of binding behavior.