Conformational substates and uncertainty in macromolecular free energy calculations

The role of conformational substates in free energy simulations of a small polypeptide is studied by molecular dynamics and multiwindow thermodynamic integration. The peptide is a model of a surface loop in staphyloccocal nuclease. An Ala-Pro segment of the polypeptide is changed to Gly-Pro and the effect on the cis-trans equilibrium of the peptide bond considered. In the simulations, many local minima, or conformational substates, are detected. Their relative probabilities cannot in general be derived reliably from the simulations. The resulting uncertainty in the calculated free energy differences between cis and trans isomers is analyzed and compared to other sources of error. With simulations of 20-200 ps, random statistical error was negligible. Sample-size hysteresis was also negligible. Changing the nonbonded parameters and the dielectric constant also had a small effect. The derivative of the Hamiltonian, with respect to the coupling parameter, displays a fast (1 ps) relaxation, with a slower (10 ps) relaxation superimposed in some cases; thus equilibration periods of a few picoseconds, after changing to a new value of the coupling parameter, only approximately eliminate time-lag hysteresis. We compare several methods of averaging over the conformational substates. The first uses probabilities taken directly from the simulations. The second uses equal probabilities. The third uses a Boltzmann weighting of the substates based on their mean potential energies. The Ala → Gly free energies from the three calculations differ by 1 kcal/mol, 50% of the total. Thus, the largest source of uncertainty by far is the presence of multiple substates. The free energy barriers and connectivity between substates are discussed, as well as the structural features that affect isomerization of the X-Pro peptide bond.