Solvation free energies of alanine peptides: the effect of flexibility

Abstract

The electrostatic (ΔGel), van der Waals cavity-formation (ΔGvdw), and total (ΔG) solvation free energies for 10 alanine peptides ranging in length (n) from 1 to 10 monomers were calculated. The free energies were computed both with fixed, extended conformations of the peptides and again for some of the peptides without constraints. The solvation free energies, ΔGel, and components ΔGvdw, and ΔG, were found to be linear in n, with the slopes of the best-fit lines being γel, γvdw, and γ, respectively. Both γel and γ were negative for fixed and flexible peptides, and γvdw was negative for fixed peptides. That γvdw was negative was surprising, as experimental data on alkanes, theoretical models, and MD computations on small molecules and model systems generally suggest that γvdw should be positive. A negative γvdw seemingly contradicts the notion that ΔGvdw drives the initial collapse of the protein when it folds by favoring conformations with small surface areas. When we computed ΔGvdw for the flexible peptides, thereby allowing the peptides to assume natural ensembles of more compact conformations, γvdw was positive. Because most proteins do not assume extended conformations, a ΔGvdw that increases with increasing surface area may be typical for globular proteins. An alternative hypothesis is that the collapse is driven by intramolecular interactions. We find few intramolecular H-bonds but show that the intramolecular van der Waals interaction energy is more favorable for the flexible than for the extended peptides, seemingly favoring this hypothesis. The large fluctuations in the vdw energy may make attributing the collapse of the peptide to this intramolecular energy difficult.

Figure 1

The 10 alanine peptide oligomers examined in this study in their equilibrated extended conformations created in PyMOL.

Figure 2

A free energy cycle that could be used to compute the change (ΔΔG) in solvation energy between a peptide of length N and one of length N − 1. A blue background indicates that a molecule is in explicit solvent, while a white background indicates that it is in vacuum. The large circles represent the first N − 1 residues of an alanine peptide, the small circles represent the cap on the peptide of length N − 1, and the medium-sized circle represents the N’th residue of the peptide of length N. A black circle means that the atoms in question are charged, and a white circle means that they are uncharged.

Figure 3

The cumulative running estimates of the integrands (a. dU (0.2)/dλ for the van der Waal’s (cavity-formation) and b. dU (0.5)/dλ for the electrostatic solvation free energy calculations) of the thermodynamic integration free energy calculations as functions of simulation time for the extended peptide with 10 alanine residues.

Figure 4

The integrands (〈dU (λ)/dλ〉 for a. the van der Waal’s (cavity-formation) and b. the electrostatic solvation free energy calculations) of the thermodynamic integration free energy calculations as functions of λ for the extended peptide with 10 alanine residues.

Figure 5

The differences between the solvation free energies of peptides of length n and those of length n − 1 plotted as functions of n. Squares represent the results of alchemical free energy calculations, and diamonds represent the differences between the free energy estimates computed by thermodynamic integration. a. The total solvation free energy. b. The van der Waal’s (cavity-formation) component of the solvation free energy. c. The electrostatic component of the solvation free energy.

Figure 6

Solvation free energies of the extended alanine peptides plotted as functions of the number (N) of residues in the peptide. a. The total solvation free energy. b. The van der Waal’s (cavity-formation) component of the solvation free energy. c. The electrostatic component of the solvation free energy.

Figure 7

Solvation free energies of the flexible alanine peptides plotted as functions of the number (n) of residues in the peptide. a. The total solvation free energy. b. The van der Waal’s (cavity-formation) component of the solvation free energy. c. The electrostatic component of the solvation free energy.