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 ␤-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 ␤-hairpin. The hidden complexity found for the ␤-hairpin surface suggests that the standard funnel picture of protein folding should be revisited. For relatively rigid systems (e.g., many organic molecules), and for flexible systems with a small number of significant degrees of freedom (e.g., short peptides), it is now possible to determine the potential energy surface and free energy surface (FES) by sampling the minima and saddles and constructing disconnectivity graphs to describe them (1-5). Direct extension of these methods to proteins (polypeptide chains with many degrees of freedom, a well defined native state, and a denatured state with a large number of conformations) has not been possible. Thus, most approaches used for such systems have resorted to essential simplifications in their description of the potential energy surface and FES. Progress variables, usually involving one or two degrees of freedom [e.g., number of native contacts, number of H bonds, number of native dihedral angles, rms deviation (rmsd), radius of gyration], have been selected and the FES was determined as a function of these variables. The projected FESs have generally been found to be relatively simple with a single or several low free energy barriers (a few kT or less). These results support the concept that a smooth FES provides the bias necessary for folding. Although the popular ''funnel picture'' is phrased in terms of the energy (6, 7), it is the FES that determines the folding behavior. In many proteins (8) the loss of entropy on folding nearly counterbalances the effective energy, and calculations for some model systems suggest that the entropy loss introduces an activation barrier, that leads to two-state folding (9). This result makes it all of the more important to determine the complexity of the FES of peptides and proteins. The well characterized ␤-hairpin of protein G (10) is studied here by a recently developed approach (4) based on disconnectivity graphs (1).An unprojected representation of the FES ...

Diffusive reaction dynamics on invariant free energy profiles

“…Theoretical studies based on simulations of the reaction rate for complex systems, such as peptides and proteins, often show simple exponential kinetics (11)(12)(13). To be able to determine both the preexponential factor and the free energy barrier from simulations, it is necessary to have a method of constructing the onedimensional projected FES in terms of an appropriate reaction coordinate, if such exists.…”

“…Given this projected surface and the calculated rate from simulations one can extract the rate coefficient and the free energy barrier. In a previous article (14) we showed how to use the minimum-cut procedure (11,15) for finding free energy barriers and constructing one-dimensional free energy profiles (FEPs). In that article, we considered the ballistic regime (i.e., the quenching interval was large enough that the number of recrossings of the transition state was negligible).…”

“…We then outline how the results are generalized to the multidimensional case; for the practically important case of clustered equilibrium trajectories, the method corresponds to the minimum-cut procedure for the corresponding network (11,14,15). Applications are made to a transition of the alanine dipeptide and to the folding reaction of a double ␤-hairpin miniprotein, both simulated by MD with implicit solvent.…”

“…In many cases (11,14), however, the standard progress variables (e.g., number of native contacts, radius of gyration) are not good reaction coordinates, because they do not preserve the barriers on the FES. Moreover, the diffusion coefficient is likely to vary in a complex way.…”

The OH stretch vibration of liquid water reveals hydrogen-bond clusters

“…In the case of protein folding, the archetypal complex system, a new arsenal of graph-based theoretical tools have been recently developed to circumvent this limitation [13][14][15]. Network approaches have pointed out the presence of multiple stable structures in the unfolded state of a protein, which is a somewhat surprising result, giving the fact that folding often appears to be a two-state process [13,16].…”