On the calculation of absolute macromolecular binding free energies

Abstract

The standard framework for calculating the absolute binding free energy of a macromolecular association reaction A ؉ B 3 AB with an association constant KAB 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 KAB is obtained directly through the statistical mechanical method of evaluating its ensemble (Boltzmannweighted) 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.T he importance to biochemistry of being able to calculate absolute and relative binding affinities from the structures of macromolecular complexes is clear. A large body of literature on the theory, practice, and application exists. See, for example, refs. 1-8 for recent papers that discuss or review the theory and methods. Nevertheless, there is still controversy over some basic theoretical issues. There is agreement that association is accompanied by a loss of translational and rotational (association) entropy but disagreement over the magnitude, whether it varies significantly from system to system or can be approximated by a single value, what theoretical model should be used to estimate it, and how to include solvent effects. This controversy has motivated recent experimental studies using tethered dimerization to estimate translational entropy loss (9, 10). Accurate accounting of the association term is essential for the calculation of absolute binding free energies. Alternatively, if it could be reliably established by experiment or theory that this term is constant for some set of binding reactions, this would greatly simplify the calculation of relative binding free energies for those systems.The traditional theoretical framework for calculating the absolute binding free energy for the reaction A ϩ B 3 AB is to equate the chemical potentials of the species on the left-and right-hand sides of this reaction and to evaluate the chemical potentials of the three species, A, B, and AB by using some theory, as described in Hill's classic statistical mechanics treatment of association (11). Recent work describes the extension of this approach to macromolecular association and to ligand-membrane association, along with a detailed discussion of other theoretical issues (1, 2). However, the chemical potential approach is not the only ...

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