Implicit ligand theory for relative binding free energies

Abstract

Implicit ligand theory enables noncovalent binding free energies to be calculated based on an exponential average of the binding potential of mean force (BPMF)-the binding free energy between a flexible ligand and rigid receptor-over a precomputed ensemble of receptor configurations. In the original formalism, receptor configurations were drawn from or reweighted to the apo ensemble. Here we show that BPMFs averaged over a holo ensemble yield binding free energies relative to the reference ligand that specifies the ensemble. When using receptor snapshots from an alchemical simulation with a single ligand, the new statistical estimator outperforms the original.

FIG. 1.

Schematic representation of apo and holo ensembles in configuration space. The circles denote regions in configuration space that are important to various thermodynamic ensembles. Solid blue and dashed red circles indicate two different ligands. In the left panel, a series of circles of the same color indicate one alchemical pathway from the apo to a holo ensemble. The left panel describes induced-fit binding processes in which apo and holo ensembles are distinct. The right panel describes conformational selection processes in which the holo ensembles are subsets of the apo ensemble.

FIG. 2.

Binding free energies for the benchmark ligands estimated by YANK (x-axis) and AlGDock (y-axis) using the apo estimator (left column) or holo estimator (right column). Active ligands are shown as small dots and inactive ones as diamonds. Error bars denote the standard deviation from three independent YANK calculations (x-axis) or from bootstrapping BPMFs (y-axis), with the range of error bars representing a single standard deviation. A least-squares linear regression is shown as a dashed line. Each row corresponds to a group. Results for other ligands are shown in Fig. 3.

FIG. 3.

Binding free energies for the benchmark ligands, continued from Fig. 2.

FIG. 4.

Convergence of binding free energies. RMSE with respect to the final results of binding free energies estimated using apo (blue solid lines) and holo (green dashed lines) estimators for single groups. Error bars denote the standard deviation from bootstrapping. Different panels correspond to different reference ligands.

FIG. 5.

Convergence of binding free energies. Pearson’s R with respect to the final results of binding free energies estimated using apo (blue solid lines) and holo (green dashed lines) estimators for single groups. Error bars denote the standard deviation from bootstrapping. Different panels correspond to different reference ligands.

FIG. 6.

Binding free energies for 24 ligands estimated by YANK (x-axis) and AlGDock (y-axis) using the apo estimator (left) or holo estimator (right). Results from 6 groups were combined using the exponential average across BPMFs from all groups for the apo estimator or the minimum of six estimates for the holo estimator. Active ligands are shown as dots and inactive ones as diamonds. Error bars denote the standard deviation from three independent YANK calculations (x-axis) or from bootstrapping BPMFs (y-axis), with the range of error bars representing a single standard deviation. The linear regression is shown as a dashed line.

FIG. 7.

Convergence of binding free energies. RMSE (left) and Pearson’s R (right) with respect to the final results of binding free energies estimated using apo (blue solid lines) and holo (green dashed lines) estimators for receptor snapshots combined from 6 YANK simulations. Error bars denote the standard deviation from bootstrapping.

FIG. 8.

The Jensen-Shannon divergence between the benchmark holo ensembles (rows) and the apo and reference holo ensembles (columns).

FIG. 9.

Jensen-Shannon divergence versus deviation of ILT estimates from YANK. Absolute binding free energy estimates using the apo estimator based on 36 BPMFs to receptor snapshots drawn from the apo state of six YANK simulations are shown as red circles. Relative binding free energy estimates using the holo estimator based on 6 BPMFs to receptor snapshots drawn from the holo state of the YANK simulation for the reference ligand are shown as follows: methylpyrrole (blue downward triangles), benzene (green upward triangles), p-xylene (cyan leftward triangles), phenol (magenta rightward triangles), DL-camphor (yellow squares), and n-hexylbenzene (black diamonds).