Like the protein, the ligand is subject to harmonic TR and conformational restraints24. The TR restraints again comprise a distance, D1, two angles, A1 and A2, and three torsions, T1, T2, T3, defined relative to the three fixed dummy atoms N1, N2 and N3. The spring constant for D1 is set in the BAT.py input file using the lig_distance_force variable. For the angle and dihedral TR restraints, the spring constant is defined by the lig_angle_force parameter. The reference values of these restraints are taken from the initial coordinates. The ligand conformational restraints include harmonic potentials on the three distances between its anchor atoms L1, L2 and L3 (Fig. 2). In addition, essentially all dihedral angles are also restrained to make each ligand rigid and thereby accelerate convergence. For simplicity, torsions within rings are not excepted from the set of restraints, although they are not always necessary. The BAT.py script automatically assigns these restraints for each ligand. It uses the ligand’s AMBER parameter/topology (prmtop) file (Fig. 5) to identify all proper dihedral terms not involving a hydrogen atom, and assigns a restraint to one arbitrarily chosen dihedral term for each central bond. The spring constants for the ligand’s internal distance and dihedral restraints are set in the BAT.py input file via the lig_discf_force and lig_dihcf_force parameters, respectively, and their reference values are taken from the starting coordinates. Fig. 5 illustrates the assignment of 14 conformational dihedral restraints for the ligand from cocrystal structure 5uf0.

Example of ligand dihedral restraints, for PDB ID 5uf0. (Left) Section of AMBER parameter/topology file listing all the ligand dihedrals that do not include a hydrogen atom. Each row lists two torsions in terms of five indices; the first four map to specific atoms and the fifth maps to the associated force field parameters. Dihedrals restrained in the BAT procedure are highlighted in purple font, with redundant ones in black and improper dihedrals in red. (Right) Ligand from 5uf0 with restrained torsions highlighted with purple bonds. Cyan: carbon. White: hydrogen. Red: oxygen. Blue: nitrogen.

The free energies of attaching and releasing the ligand restraints may be separated into conformational and TR parts:

During the attachment stage (att), the ligand is in the binding site of the restrained protein. The conformational restraints are applied first, yielding the free energy change ΔGl,conf,att for making the ligand essentially rigid. The TR restraints are then applied, yielding the free energy change (ΔGl,TR,att) for restraining the ligand in the binding site. During the release stage (rel), ΔGl,conf,rel is computed with the ligand in a separate simulation box with no TR restraints present. The values of ΔGl,conf,att, ΔGl,TR,att, and ΔGl,conf,rel are calculated the same way as the protein conformational restraints, using MBAR (Eqs. (5)–(7)), with simulation windows having intermediate values of the harmonic spring constants, also defined by the attach_rest input array. The final term in Eq. (9), ΔGl,TR,rel, is calculated by numerical quadrature of the following integral, which is based on Euler angles and spherical coordinates:

Here C is the standard concentration, 1 M = 1/1661Å3, and r, θ and ϕ are the distance D1, angle A1, and dihedral T1, respectively24. In the last term on the right, which integrates over ligand orientation, Θ is the angle A2, Φ is the dihedral T2, and Ψ is the dihedral T3. These are three Euler angles which define the orientation of the ligand in space. The harmonic potential applied to the distance r has the form:

with kd the lig_distance_force spring constant and r0 its reference value. A similar expression is used for restrained angles and dihedrals:

with a a given angle/dihedral, ka the lig_angle_force spring constant, and a0 its reference value. The value of r0 is the reference distance, D1, from dummy atom N1 to ligand atom L1, in the bound state; this is always sets to 5.00 Å by construction (“Anchor atoms and dummy atoms”).

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