Free energy perturbation calculations with combined QM/MM Potentials complications, simplifications, and applications to redox potential calculations

The present work deals with a formal discussion on complications associated with using combined quantum mechanical and molecular mechanical (QM/MM) potentials in free energy perturbation simulations. Because quantum mechanical potentials are not trivially separable and because of the difficulty associated with QM calculations for "fractional" electronic Hamiltonians, conventional computational strategies cannot be straightforwardly applied to free energy simulations with QM/MM potentials. Here, we propose a unique coupling scheme, termed the "dual-topology-single-coordinate" scheme, in which the two chemical states are forced to adopt the same set of Cartesian coordinates during the perturbation simulation; the method is exact because the free energy difference is independent of the coupling path, although a formal proof is also given. The scheme combines the merits of the conventional dual- and single-topology approaches: both coupling potential and free energy derivative are straightforward to compute for any form of QM method (even if nonvariational) and QM/MM coupling scheme, calculations can be performed at end points without numerical/ sampling instabilities, no corrections related to artifactual reference states or Jacobian factors have to be considered, and numerical convergence is rapid. The method is illustrated with an application to the redox potential of FAD in cholesterol oxidase. It was found that the protein and solvent respond in approximately a linear fashion to the reduction of FAD with a small reorganization energy; the bulk solvent makes an essential contribution to compensate for the reorganization of the protein. The contribution of Asn 485 was found to be around 1.3 kcal/mol with several different simulations, which is close to the experimental estimate based on mutation studies.