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The theory of receptor-ligand binding equilibria has long been well-established in biochemistry, and was primarily constructed to describe dilute aqueous solutions. Accordingly, few computational approaches have been developed for making quantitative predictions of binding probabilities in environments other than dilute isotropic solution. Existing techniques, ranging from simple automated docking procedures to sophisticated thermodynamics-based methods, have been developed with soluble proteins in mind. Biologically and pharmacologically relevant protein-ligand interactions often occur in complex environments, including lamellar phases like membranes and crowded, nondilute solutions. Here, we revisit the theoretical bases of ligand binding equilibria, avoiding overly specific assumptions that are nearly always made when describing receptor-ligand binding. Building on this formalism, we extend the asymptotically exact Alchemical Free Energy Perturbation technique to quantifying occupancies of sites on proteins in a complex bulk, including phase-separated, anisotropic, or nondilute solutions, using a thermodynamically consistent and easily generalized approach that resolves several ambiguities of current frameworks. To incorporate the complex bulk without overcomplicating the overall thermodynamic cycle, we simplify the common approach for ligand restraints by using a single distance-from-bound-configuration (DBC) ligand restraint during AFEP decoupling from protein. DBC restraints should be generalizable to binding modes of most small molecules, even those with strong orientational dependence. We apply this approach to compute the likelihood that membrane cholesterol binds to known crystallographic sites on three GPCRs (β 2 -adrenergic, 5HT-2B, and μ-opioid) at a range of concentrations. Nonideality of cholesterol in a binary cholesterol:phosphatidylcholine (POPC) bilayer is characterized and consistently incorporated into the interpretation. We find that the three sites exhibit very different affinities for cholesterol: The site on the adrenergic receptor is predicted to be high affinity, with 50% occupancy for 1:10 9 CHOL:POPC mixtures. The sites on the 5HT-2B and μ-opioid receptor are predicted to be lower affinity, with 50% occupancy for 1:10 3 CHOL:POPC and 1:10 2 CHOL:POPC, respectively. These results could not have been predicted from the crystal structures alone.

A Streamlined, General Approach for Computing Ligand Binding Free Energies and Its Application to GPCR-Bound Cholesterol