Entropic Cost of Protein−Ligand Binding and Its Dependence on the Entropy in Solution

Abstract

Two theoretical formulations are proposed and compared for the loss of translational and rotational entropy upon protein-ligand binding in water. The two theories share the same approach to evaluate the translational and rotational entropy of the ligand when bound. The potential of the bound ligand is modeled by six harmonic oscillators that are parametrized from the force and torque magnitudes measured in a molecular dynamics simulation, yielding vibrational and librational entropies. In the aqueous phase, the theories differ because there is no unique way to assign the total entropy to molecules in solution. In one approach, the ligand is allowed unrestricted access to the full solution volume at the standard concentration and is assigned the same translational and rotational entropy as if it were an ideal gas. We term this a "molecule-frame" (MF) theory because it considers configurational space in the reference frame of the molecule of interest. The entropy of the solvent is penalized because it is excluded from the molecule's volume. In the second theory, all molecules including the solvent are confined by their neighbors in mean-field configurational volumes. This we term a "system-frame" (SF) theory because the configurational space available to all molecules is considered in the reference frame of the whole system. Molecules have vibrational and librational entropy in the same way as they do when bound. In addition, the discrete size of the solvent molecules quantizes the configurational space into an effective number of minima according to the solute molecule's standard concentration and the mean volume of a solvent molecule. This leads to the cratic entropy expressed in terms of the solute molecule's mole fraction. The equivalent number of minima in rotational space depends on both the solute molecule's volume and the solvent molecule's volume. This leads to an equation for the orientational entropy based on the proposed concept of "angle fraction". The MF and SF theories are applied to calculate the translational and rotational entropy losses involved in the formation of six different protein-ligand complexes, in two of which the ligand is water. The MF entropy losses range from -80 to -142 J K(-1) mol(-1) for ligands at the 1 M standard-state concentration and from -52 to -63 J K(-1) mol(-1) for water at the 55.6 M standard-state concentration. They depend logarithmically on both the number and strength of interactions between the ligand and protein through the forces and torques. This is observed to lead to moderate dependencies on the ligands' moments of inertia and masses. The SF entropy losses are smaller and range from -50 to -75 J K(-1) mol(-1) for ligands at the 1 M standard-state concentration and from 0 to -12 J K(-1) mol(-1) for water. They depend logarithmically on the ligand solvent's molecular volume and weakly on the relative strengths of the ligand's interactions with the protein and water. The cratic entropy loss in water at the standard concentration is constant and is also demonstrated to be implicit in MF theories. Entropy losses from the two approaches are also compared with those from other computational approaches and with experiment. The use of the force and torque magnitudes leads to smaller bound volumes than are obtained from ligand-displacement approaches. The general agreement of the SF entropy losses with those from experiment suggests that the SF theory is more consistent with the assumptions made in experimental measurements than the MF solvation theories, which would require a compensating entropy gain in the solvent in order to agree.