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 ...

The conformational space of a 20-residue three-stranded antiparallel β-sheet peptide (double hairpin) was sampled by equilibrium folding/unfolding molecular dynamics simulations for a total of 20 µs. The resulting one-dimensional free-energy profiles (FEPs) provide a detailed description of the free-energy basins and barriers for the folding reaction. The similarity of the FEPs obtained using the probability of folding before unfolding (p fold) or the mean first passage time supports the robustness of the procedure. The folded state and the most populated free-energy basins in the denatured state are described by the one-dimensional FEPs, which avoid the overlap of states present in the usual one- or two-dimensional projections. Within the denatured state, a basin with fluctuating helical conformations and a heterogeneous entropic state are populated near the melting temperature at about 11% and 33%, respectively. Folding pathways from the helical basin or enthalpic traps (with only one of the two hairpins formed) reach the native state through the entropic state, which is on-pathway and is separated by a low barrier from the folded state. A simplified equilibrium kinetic network based on the FEPs shows the complexity of the folding reaction and indicates, as augmented by additional analyses, that the basins in the denatured state are connected primarily by the native state. The overall folding kinetics shows single-exponential behavior because barriers between the non-native basins and the folded state have similar heights.

Disconnectivity graphs are widely used for understanding the multidimensional potential energy surfaces (PES) of complex systems. Since entropic contributions to the free energy can be important, particularly for polypeptide chains and other polymers, conclusions concerning the equilibrium properties and kinetics of the system based on potential energy disconnectivity graphs (PE DG) can be misleading. We present an approach for constructing free energy surfaces (FES) and free energy disconnectivity graphs (FE DG) and give examples of their applications to peptides. They show that the FES and FE DG can differ significantly from the PES and PE DG.

We show that the balanced minimum-cut procedure introduced in PNAS 2004, 101, 14766 can be reinterpreted as a method for solving the constrained optimization problem of finding the minimum cut among the cuts with a particular value of an additive function of the nodes on either side of the cut. Such an additive function (e.g., the partition function of the reactant region) can be used as a progress coordinate to determine a one-dimensional profile (FEP) of the free-energy surface of the protein-folding reaction as well as other complex reactions. The algorithm is based on the network (obtained from an equilibrium molecular dynamics simulation) that represents the calculated reaction behavior. The resulting FEP gives the exact values of the free energy as a function of the progress coordinate; i.e., at each value of the progress coordinate, the profile is obtained from the surface with the minimal partition function among the surfaces that divide the full free-energy surface between two chosen end points. In many cases, the balanced minimum-cut procedure gives results for only a limited set of points. An approximate method based on p(fold) is shown to provide the profile for a more complete set of values of the progress coordinate. Applications of the approach to model problems and to realistic systems (beta-hairpin of protein G, LJ38 cluster) are presented.