(i) the β-hairpin peptide exhibits two-state behavior in both its equilibrium and kinetics; (ii) the apparent activation energy for the folding rate calculated from the two-state analysis is negative; and (iii) the rate of β-hairpin formation is much (>10-fold) slower than that of the α-helices that have been studied up to now in short peptides.
To explain these results, we used a simple statistical mechanical model which was only briefly described (27). Here, we present a detailed description of the model, test one of its major approximations, and use the model to predict kinetic and equilibrium properties expected for other β-hairpinforming peptides. We shall see that analysis of β-hairpin thermodynamics and kinetics addresses many of the same issues that arise in considering the folding of a small protein.
Description of the Model. Our objective has been to develop a model for protein secondary structure kinetics, which can be used to analyze experimental data and to predict new experiments. In this work, the model is applied to a β-hairpin, but it also can be applied to helices and is readily adapted for more complex structures. We adopt a description, which uses pairs of ϕ,ψ, dihedral angles to define the conformation of each molecule; the complete native structure is formed when all of the residues have native values for these angles. Formation of the native structure is opposed by the loss of conformational entropy and favored by the formation of stabilizing interactions, i.e., hydrogen bonds and hydrophobic interactions (Fig. 1). The model postulates that two groups interact only when all of the dihedral angles of the sequence connecting them are native. This restriction considerably simplifies the model by identifying three-dimensional structures with sequences of peptide bond conformations.
A second simplifying step is to consider only two conformations for the backbone dihedral angles, native and nonnative (in a spirit similar to the “correct” and “incorrect”
parameter of the Zwanzig model; ref. 30). The nonnative conformation of a dihedral angle pair is not a unique conformation but is the set of all conformations that are incompatible with the native structure. An additional feature of the model is that pairs of ϕ,ψ dihedral angles are assumed to rotate between native and nonnative values simultaneously.† We chose the dihedral angles ψ of residue i and ϕ of residue i + 1 (Fig. 2) so that the peptide bond, rather than the residue, is the conformational unit. Formation of a backbone-backbone hydrogen bond is therefore associated with the transformation of one pair of ψi, ϕi+1 angles in each β strand from nonnative to native values.
In our thermodynamic description of the β-hairpin, we consider only three factors. These are the stabilizing effect of the hydrogen bonds between the backbone carbonyl and amide of the N- and C-terminal β strands, the stabilizing effect of the three hydrophobic interactions among the four side chains of the hydrophobic cluster (Fig. 1). and the destabilizing effect of the loss of conformational entropy when fixing pairs of dihedral angles in the native hairpin conformation. Nonnative interactions, such as wrong hydrogen bonds or hydrophobic interactions, are ignored. We also ignore electrostatic interactions among the charged side chains and chain termini (their importance could be assessed by experiments on the ionic strength dependence of the equilibrium and kinetics which have not yet been performed). Each thermodynamic factor is considered to be homogeneous, i.e., independent of side chain and position in the native structure. We assume that the free energies of formation for each of the three hydrophobic interactions, ΔGsc, are identical. Each of the backbonebackbone hydrogen bonds, including the one in the turn region, is assumed to have the same free energy, ΔGhb. The conformational entropy loss for the strand and turn regions also is assumed to be the same (ΔSconf). which is equivalent to assuming that the residues in the turn have a propensity for this conformation equal to the propensity of the strand residues to be in a strand conformation. To further reduce the number of parameters, we assume that the hydrogen bond is purely enthalpic, i.e., ΔGhb=ΔHhb and that the hydrophobic inter
When pairs of dihedral angles are used instead of single dihedral angles, the specification of a pair of angles produces a problem in phasing between the loss of entropy and the compensating decrease in interaction free energy. Either choice of ϕ,ψ, pairs represents a compromise. This can be illustrated by considering the formation of a six-residue β-hairpin with a side-chain interaction between residues two and five. To form the backbone-backbone hydrogen bond requires native values for four dihedral angles, ϕ3,ψ3,ϕ4,ψ4. If we were only concerned with hydrogen bond formation, as in helix-coil theory for homopolypeptides, then the natural choice for the dihedral angle pairs would be the ϕ and ψ associated with the same residue—in this case the two pairs ϕ3,ψ3, and ϕ 4,ψ4. With this choice, however, formation of the two- to five-side-chain interaction requires that eight dihedral angles assume native values—when only six. i.e., ψ2,ϕ 3,ψ3,ϕ4,ψ4, and ϕ 5, actually are required. So, in choosing ψi,ϕi+1 instead of ϕi,ψi, pairs, we overestimate the loss in entropy associated with formation of the first hydrogen bond, in favor of accurately representing the compensation between entropy loss and formation of side-chain interactions in subsequent steps.