Connecting free energy surfaces in implicit and explicit solvent: an efficient method to compute conformational and solvation free energies

Abstract

The ability to accurately model solvent effects on free energy surfaces is important for understanding many biophysical processes including protein folding and misfolding, allosteric transitions, and protein–ligand binding. Although all-atom simulations in explicit solvent can provide an accurate model for biomolecules in solution, explicit solvent simulations are hampered by the slow equilibration on rugged landscapes containing multiple basins separated by barriers. In many cases, implicit solvent models can be used to significantly speed up the conformational sampling; however, implicit solvent simulations do not fully capture the effects of a molecular solvent, and this can lead to loss of accuracy in the estimated free energies. Here we introduce a new approach to compute free energy changes in which the molecular details of explicit solvent simulations are retained while also taking advantage of the speed of the implicit solvent simulations. In this approach, the slow equilibration in explicit solvent, due to the long waiting times before barrier crossing, is avoided by using a thermodynamic cycle which connects the free energy basins in implicit solvent and explicit solvent using a localized decoupling scheme. We test this method by computing conformational free energy differences and solvation free energies of the model system alanine dipeptide in water. The free energy changes between basins in explicit solvent calculated using fully explicit solvent paths agree with the corresponding free energy differences obtained using the implicit/explicit thermodynamic cycle to within 0.3 kcal/mol out of ∼3 kcal/mol at only ∼8% of the computational cost. We note that WHAM methods can be used to further improve the efficiency and accuracy of the implicit/explicit thermodynamic cycle.

Figure 1

(a) A schematic diagram illustrating the calculation of conformational free energy differences by connecting free energy surfaces in explicit and implicit solvents via localized decoupling. (b) Thermodynamic cycle. Note that ΔG_0→1,a1 is the transfer free energy for a cell a₁ in basin A from implicit solvent to explicit solvent, while ΔG_0→1,A is the transfer free energy for the whole basin A from implicit to explicit solvent which depends on both the localized transfer free energy ΔG_0→1,a1 and a curvature term, see text.

Figure 2

Dividing the free energy surface of a basin (e.g. α_L) into multiple cells and calculating the free energy of transferring a cell (a₁) of the α_L basin in implicit solvent to explicit solvent. The examples shown here are the free energy surfaces of alanine dipeptide in an implicit solvent (AGBNP2) and explicit solvent (TIP3P), projected onto the plane of φ–ψ dihedral angles. Also shown are the C5/β/α_R/C7_eq macrostate (−180°<φ<0°), and α_L/C7_ax macrostate (0°<φ<120°), each containing basins that interconvert rapidly.

Figure 3

Calculating the free energy difference between α_R and C7_ax basins by connecting the free energy surfaces of alanine dipeptide in vacuum and TIP3P water. The intramolecular hydrogen bond in the C7_ax conformer is shown in dashed line. Note that the α_L conformer has a very small population in vacuum. The C5/β/α_R/C7_eq and α_L/C7_ax macrostates are also indicated.