Monday, December 10, 2012

Calculation of Host-Guest Binding Affinities Using a Quantum-Mechanical Energy Model

Hari S. Muddana and Michael K. Gilson J. Chem. Theory Comput. 2012, 8, 2023 (Paywall)
Contributed by Jan Jensen

This paper presents absolute binding energies for 29 ligands complexed with cucurbit[7]uril (CB7) computed using PM6-DH+ and the COSMO solvation model.  

This is a great model system for studying binding: CB7 is a macrocyclic molecule made of seven fused-ring monomers and has only one conformational minimum.  Many of the 29 ligands for which binding free energies have been measure experimentally are also conformationally restricted, and the binding free energies span a wide range: -5.3 to -21.5 kcal/mol. 

Because of this a fairly exhaustive conformational search was feasible and the number of conformations per complex ranged from 5 to 300 depending on the flexibility of the guest.  The conformational search was done using the OPLS-2005 all-atom force field and a low-mode conformational search algorithm implemented in the Schrodinger software suite. 

These structures where then used as a starting point for PM6-DH+/COSMO energy minimizations and subsequent vibrational analysis.  Given the number of conformations and ligands this paper represents a significant investment of CPU time even at a semiempirical level of theory.  The COSMO solvation energy is augmented by a non-polar solvation term that depends on the molecular surface area.  Though not explicitly stated this must have been done as a single point correction.

The computed binding energies correlate well ($R^2$=0.79) with the experimental results, but the root-mean-square error (RMSE) is high (11.4 kcal/mol) suggesting a systematic error.  To address this a three parameter fit was made involving the COSMO and non-polar solvation energy plus an off-set, which led to a $R^2$=0.91 and RMSE of 1.9 kcal/mol, excluding one outlier.  This analysis suggests that the COSMO solvation energies are overestimated by 4%, consistent with a similar analysis performed on 367 solvation energies for small neutral molecules.

The off-set was found to be -5.83 kcal/mol, i.e. the predicted binding energies are systematically underestimated by nearly 6 kcal/mol.  Interestingly, a recent analysis of explicit solvation thermodynamics of CB7, by Gilson and co-workers, "suggests that the water molecules in the CB7 cavity are unstable relative to bulk ...  If so, then treating the water in the cavity as a bulk dielectric might lead to a significant overestimation of the host’s solvation free energy and hence the underestimations of binding affinities observed here."  Furthermore, very recently  Rogers et al. was able to compute an absolute binding energy for one of the ligands to CB7 using thermodynamic integration and explicit solvent that was in excellent agreement with experiment.

The paper also provides a useful outline of how the thermodynamical aspects should be handled and analyzed, including a nice analysis of the conformational entropy change.  For example, the translational entropy must be computed using a volume of 1 L rather than that of an ideal gas a 1 bar, and that one should compute the Helmholtz, rather than the Gibbs, free energy change since the change in volume of the solution upon binding is negligible in the condensed phase.  There is also a reference to a real gem of a paper by Zhou and Gilson that reconciles the rigid rotor-harmonic oscillator approach to thermodynamics used here with that used in molecular dynamics studies.

While the agreement with experiment is impressive there is some room for improvement and this system provides a great model system for testing all aspects of the binding energy including ab initio calculations of the interaction energy and new continuum solvation methods.

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