Efficient Approximation of Ligand Rotational and Translational Entropy Changes upon Binding for Use in MM-PBSA Calculations

A major uncertainty in binding free energy estimates for protein-ligand complexes by methods such as MM-PB(GB)SA or docking scores results from neglecting or approximating changes in the configurational entropies (ΔSconfig.) of the solutes. In MM/PB(GB)SA-type calculations, ΔSconfig. has usually been estimated in the rigid rotor, harmonic oscillator approximation. Here, we present the development of a computationally efficient method (termed BEERT) to approximate ΔSconfig. in terms of the reduction in translational and rotational freedom of the ligand upon protein-ligand binding (ΔSR/T), starting from the flexible molecule approach. We test the method successfully in binding affinity computations in connection with MM-PBSA effective energies describing changes in gas-phase interactions and solvation free energies. Compared to related work by Ruvinsky and co-workers, clustering bound ligand poses based on interactions with the protein rather than structural similarity of the poses, and an appropriate averaging over single entropies associated with an individual well of the energy landscape of the protein-ligand complex, were found to be crucial. Employing three data sets of protein-ligand complexes of pharmacologically relevant targets for validation, with up to 20, in part related ligands per data set, spanning binding free energies over a range of ≤7 kcal mol-1, reliable and predictive linear models to estimate binding affinities are obtained in all three cases (R2 = 0.54-0.72, p < 0.001, root mean squared error S = 0.78-1.44 kcal mol-1; q2 = 0.34-0.67, p < 0.05, root mean squared error sPRESS = 1.07-1.36 kcal mol-1). These models are markedly improved compared to considering MM-PBSA effective energies alone, scoring functions, and combinations with ΔSconfig. estimates based on the number of rotatable bonds, rigid rotor, harmonic oscillator approximation, or interaction entropy method. As a limitation, our method currently requires a target-specific training data set to identify appropriate scaling coefficients for the MM-PBSA effective energies and BEERT ΔSR/T. Still, our results suggest that the approach is a valuable, computationally more efficient complement to existing rigorous methods for estimating changes in binding free energy across structurally (weakly) related series of ligands binding to one target.