Absolute Binding Free Energies:  A Quantitative Approach for Their Calculation

The computation of absolute binding affinities by molecular dynamics (MD) based free energy simulations is analyzed, and an exact method to carry out such a computation is presented. The key to obtaining converged results is the introduction of suitable, auxiliary restraints to prevent the ligand from leaving the binding site when the native ligand−receptor interactions are turned off alchemically. We describe a versatile set of restraints that (i) can be used in MD simulations, that (ii) restricts both the position and the orientation of the ligand, and that (iii) is defined relative to the receptor rather than relative to a fixed point in space. The free energy cost, ΔAr, for this set of restraints can be evaluated analytically. Although the techniques were originally developed for the gas phase, the resulting expression is exact, since all contributions from solute−solvent interactions cancel from the final result. The value of ΔAr depends only on the equilibrium values and force constants of the chosen harmonic restraint terms and, therefore, can be easily calculated. The standard state dependence of binding free energies is also investigated, and it is shown that the present approach takes this into account correctly. The analytical expression for ΔAr is verified numerically by calculations on the complex formed by benzene with the L99A mutant of T4 lysozyme. The overall approach is illustrated by a complete binding free energy calculation for a complex based on a simplified model for tyrosine bound to tyrosyl-tRNA-synthetase. The results demonstrate the usefulness of the proposed set of restraints and confirm that the calculated binding free energy is independent of the details of the restraints. Comparisons are made with earlier formulations for the calculation of binding free energies, and certain limitations of that work are described. The relationship between ΔAr and the loss of translational and rotational entropy during a binding process is analyzed.