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

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 (β2-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 1:109 CHOL:POPC mixtures.
The sites on the 5HT-2B and μ-opioid receptor are predicted
to be lower affinity, with 50% occupancy for 1:103 CHOL:POPC
and 1:102 CHOL:POPC, respectively. These results could
not have been predicted from the crystal structures alone.