Free Energy Simulations:  Thermodynamic Reversibility and Variability

The energetic consequences of amino acid substitutions were investigated using molecular dynamics free energy simulations (MD/FES). A focus of the present study is how one can treat some of the intrinsic problems associated with the use of dummy atoms, which are introduced in standard MD / FES procedures working with a constant number of particles. Specifically, we show how one can introduce dummy atoms, which retain all the covalent interactions, in a “hybrid residue”, in such a way that the influence of the bonded interactions with dummy atoms do not influence the final free energy change. The simulations thus can be done using a transformation protocol in which all covalent bond contributions are maintained invariant throughout the calculations; only the nonbonded interactions are varied. That is, the number of atoms is maintained constant by introducing dummy atoms, which are covalently linked to the protein in question, but which have no nonbonded interactions at one or the other of the two end point reference states. The potential energy function describing the transformation was constructed such that all internal energy terms are invariant with respect to the thermodynamic coupling parameter λ (0 ≤ λ ≤ 1). This simulation procedure therefore has similarities with both the “single topology” and “dual topology” methods, which have been used in other investigations. This transformation and MD/FES strategy was evaluated using three different systems, in which an Ala is transformed into a Val (the italicized residues were transformed): formyl- Ala -ethanolamine; formyl-(Ala) 3 -Ala- Ala -Ala-(Ala) 3 -ethanolamine (as an α-helix); and formyl-(Ala) 3 -Val- Ala -Val-(Ala) 3 -ethanolamine (as an α-helix). The calculations examined how the free energy difference associated with an Ala-to-Val transformation depends on the choice of transformation path, the equilibration time, and the number windows used in the calculation. The results show that MD/FES provides for stable energy estimates for any intermediate state, as defined by the coupling parameter λ, including the end point reference states: λ = 0 and λ = 1. With an appropriate choice of sigmoid transformation path, free energy changes with minimal variability and good reversibility were obtained for all three systems using MD/FES lasting longer than 2.4 ns.

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