On the Use of Enveloping Distribution Sampling (EDS) to Compute Free Enthalpy Differences between Different Conformational States of Molecules: Application to 310-, α-, and π-Helices.

Enveloping distribution sampling (EDS) is a powerful method to compute relative free energies from simulation. So far, the EDS method has only been applied to alchemical free energy differences, i.e., between different Hamiltonians defining different systems, and not yet to obtain free energy differences between different conformations or conformational states of a system. In this article, we extend the EDS formalism such that it can be applied to compute free energy differences of different conformations and apply it to compute the relative free enthalpy ΔG of 310-, α-, and π-helices of an alanine deca-peptide in explicit water solvent. The resulting ΔG values are compared to those obtained by standard thermodynamic integration (TI) and from so-called end-state simulations. A TI simulation requires the definition of a λ-dependent pathway which in the present case is based on hydrogen bonds of the different helical conformations. The values of ⟨(∂VTI)/(∂λ)⟩λ show a sharp change for a particular range of λ values, which is indicative of an energy barrier along the pathway, which lowers the accuracy of the resulting ΔG value. In contrast, in a two-state EDS simulation, an unphysical reference-state Hamiltonian which connects the parts of conformational space that are relevant to the different end states is constructed automatically; that is, no pathway needs to be defined. In the simulation using this reference state, both helices were sampled, and many transitions between them occurred, thus ensuring the accuracy of the resulting free enthalpy difference. According to the EDS simulations, the free enthalpy differences of the π-helix and the 310-helix versus the α-helix are 5 kJ mol(-1) and 47 kJ mol(-1), respectively, for an alanine deca-peptide in explicit SPC water solvent using the GROMOS 53A6 force field. The EDS method, which is a particular form of umbrella sampling, is thus applicable to compute free energy differences between conformational states as well as between systems and has definite advantages over the traditional TI and umbrella sampling methods to compute relative free energies.