The capillarity picture and the kinetics of one-dimensional protein folding.

E nergy landscape theory unites the study of protein folding with the theory of phase transitions. Dimensionality, whose key role in phase transitions is well known, comes to the fore in folding repeat proteins. Repeat proteins are made of near repetitions of 20–40 residues encoding recurring structural motifs. Each repeating element interacts only with its immediate neighbors, forming extended, globally one-dimensional structures that can be interrupted by thermally excited defects (1). The equilibrium properties of repeat proteins may be mapped onto a one-dimensional Ising model (2–4). Going beyond equilibrium, in this issue of PNAS, Werbeck et al. (5) delight us by studying the folding kinetics of a very long repeat protein, D34, a 12-ankyrin repeat fragment of AnkyrinR. Werbeck and Itzhaki (6) previously showed that, unlike short ankyrin repeat proteins, D34 populates an equilibrium folding intermediate at mild denaturing conditions, where the 12-repeat array is neither completely folded nor unfolded. They showed that this ‘‘trap’’ corresponds to an ensemble where the structured regions are polarized toward the C-terminal repeats. Nonuniform stability along the ‘‘superhelical’’ array yields two main cooperative folding ‘‘subdomains.’’ As in the classical helix–coil transition of secondary structures, the intrinsic stability of the individual folding elements is low compared with the free energy of stabilization gained by forming an ‘‘interface’’ between neighbors (1). The delicate balances of free energy in each element allows their folding to decouple and subdomains to emerge (7, 8). The present time-resolved experiments show how the one-dimensionality controls the dynamics. Monitoring the fluorescence, unfolding D34 shows a fast process yielding essentially the equilibrium intermediate. The barrier separating the native from the intermediate states appears ‘‘broad,’’ consistent with the picture of neighboring repeats unfolding sequentially (Fig. 1). A slower unfolding phase corresponds to unravelling the C-terminal subdomain. Mutations in this region affect both the rate and its urea dependence in a manner consistent with unfolding by two parallel routes. In principle, parallel routes are expected from the symmetric topology of repeat proteins (8). These have been experimentally traced in the shorter, fourankyrin-repeat protein Myotrophyn (9). In repeat proteins such distinct populated folding routes are not guaranteed to appear because the routes are selected based on the local energetics, and small perturbations easily reroute the transitions (8–11). For one-dimensional systems the details of the kinetic routes taken through the landscape crucially depend on inhomogeneities in the distribution of energies and entropy losses for folding along the array. Because intramolecular forces are short-ranged, repeat proteins fold through configurations in which residues contiguous in physical space will be either ordered or disordered, like droplets in a phase transition. The capillarity picture of folding (12) thus may be used to describe folding in a large protein. In the capillarity model, a front between the folded and unfolded parts can be defined. For elongated architectures this front orients orthogonal to the long axis and moves in a one-dimensional fashion. The nucleation of the capillarity front requires overcoming a free-energy barrier. The rate of the front’s propagation is reflected in a prefactor depending on the local landscape ruggedness and the friction from the solvent (12). The motion of the front is impeded by transient trapping in local minima, and when the residence time becomes too long, the traps can be thought as minifunnels within the main funnel (Fig. 1). Entropy is gained in stages and unfolding becomes intermittent (13), as often seen in mechanical stretching experiments (14). The measured rate of folding (1/ ) de-