Inclusion of Loss of Translational and Rotational Freedom in Theoretical Estimates of Free Energies of Binding. Application to a Complex of Benzene and Mutant T4 Lysozyme

Abstract

We present the first complete treatment for calculating theoretical estimates of free energies of formation of macromolecule−ligand complexes with molecular dynamics simulations, as the free energy for transforming the ligand into a non-interacting state by gradually diminishing the forces between macromolecule (plus solvent) and ligand. The calculations become possible due to the introduction of a specially designed potential (“molecular tweezers”) which restrains the spatial position and orientation of the ligand molecule and is gradually applied as the transformation proceeds from complexed to non-interacting components. The binding of benzene to a mutant T4 lysozyme (Morton et al. Biochemistry 1995, 34, 8564−8575) has been used as a test case. The simulations reproduce the value of the free energy of binding (−5.19 kcal/mol if the standard state of benzene is a 1 M aqueous solution) within the sum of experimental and statistical error. Another series of such simulations with rigid protein models provides an estimate of the dependence of the free energy of binding on the protein conformation. The free energy of binding is found to decrease in the series: energy-minimized ligand-free protein (−3.5 kcal/mol), energy-minimized ligand-containing protein (−6.3 kcal/mol), and crystal structure (−8.5 kcal/mol). The free energy of binding to a series of snapshots from a protein−ligand dynamics trajectory varies between −7 and −9 kcal/mol. The “cratic” free energy contribution, which corresponds to the loss of translational and rotational freedom of the ligand molecule, was estimated at 7 kcal/mol. It has proved possible to decompose this into translational and rotational components and, from these free energies, estimate the remaining freedom of the benzene in the binding pocket, at 0.6 Å for positional range and 10−15° for angular range, in excellent agreement with the motion observed in a dynamics trajectory.