Hidden complexity of free energy surfaces for peptide (protein) folding

Abstract

An understanding of the thermodynamics and kinetics of protein folding requires a knowledge of the free energy surface governing the motion of the polypeptide chain. Because of the many degrees of freedom involved, surfaces projected on only one or two progress variables are generally used in descriptions of the folding reaction. Such projections result in relatively smooth surfaces, but they could mask the complexity of the unprojected surface. Here we introduce an approach to determine the actual (unprojected) free energy surface and apply it to the second beta-hairpin of protein G, which has been used as a model system for protein folding. The surface is represented by a disconnectivity graph calculated from a long equilibrium folding-unfolding trajectory. The denatured state is found to have multiple low free energy basins. Nevertheless, the peptide shows exponential kinetics in folding to the native basin. Projected surfaces obtained from the present analysis have a simple form in agreement with other studies of the beta-hairpin. The hidden complexity found for the beta-hairpin surface suggests that the standard funnel picture of protein folding should be revisited.

Fig. 1.

Schematic model system showing a trajectory and the corresponding TRDG. (a) Three FE basins (A–C) are shown with the transitions between them. Dots show clusters that were visited more the once and the connecting line corresponds to a single transition found in the trajectory. Dashed lines show the minimum cuts that separates A from B and C and B from C. (b) The TRDG was constructed as described and shows that the FES consists of three basins; the lowest (representative) nodes in each basin are labeled as in a.

Fig. 2.

TRDG of β-hairpin calculated with EEF1 solvation model (16) at 360 K. Representative structures for the deepest FE minima are shown. The left vertical axis shows the Z_i for the minima and the Z_ij for the barriers. The right vertical axis shows F_i =–kT ln(Z_i) and –kT ln(Z_ij) in units of kcal/mol for the minima and barriers, respectively. The free energy barrier is F_ij =–kT ln(Z_ij) –kT ln(Z_ij × h/kT × 1/t_q) =–kT ln(Z_ij) + 3.61 (4, 33); h is Plank constant and t_q = 20 ps is the sampling interval. For example, the height of the free energy barrier between the native basin and the denatured basin is about –kT ln (8 × 6.29.10^–3) + kT ln(69,065) = 10 kcal/mol, and the mean unfolding time is 69,065/8 × 20 ps ≈ 173 ns, where 69,065 is the partition function of the native basin (Fig. 5).

Fig. 3.

Structures of the most-visited clusters in the entropic basin of the denatured state.

Fig. 4.

FESs (kcal/mol) projected on two dimensions. (a) The rmsd from NMR native structure and radius of gyration in Å.(b) The two most important principal components in Å.

Fig. 5.

Transition matrix between major basins represented as a simplified network. The numbers in brackets show the number of times system visited the basin (which corresponds to the partition function of the basin) and the numbers on the edges show the number of direct transitions in each direction between the basins. We note that the number of direct transitions is different from those appearing in Fig. 2, which is based on all transitions (direct and indirect) between the basins; the latter measures the overall flow, and therefore the barriers, between the basins (4).

Fig. 6.

Cumulative distribution (f) of the first passage times for reaching the native structure, starting form the completely extended structure; , where p is the probability distribution of the first passage time. Crosses are for the MD simulation (35 events, the mean first passage time is ≈350 ns) and the line is for MC simulation with {T} transition matrix (10,000 events, the mean first passage time is ≈422 ns) (see Methods).