A theoretical search for folding/unfolding nuclei in three-dimensional protein structures

Abstract

When a protein folds or unfolds, it has to pass through many half-folded microstates. Only a few of them can be seen experimentally. In a two-state transition proceeding with no accumulation of metastable intermediates [Fersht, A. R. (1995) Curr. Opin. Struct. Biol. 5, 79-84], only the semifolded microstates corresponding to the transition state can be outlined; they influence the folding/unfolding kinetics. Our aim is to calculate them, provided the three-dimensional protein structure is given. The presented approach follows from the capillarity theory of protein folding and unfolding [Wolynes, P. G. (1997) Proc. Natl. Acad. Sci. USA 94, 6170-6175]. The approach is based on a search for free-energy saddle point(s) on a network of protein unfolding pathways. Under some approximations, this search is rapidly performed by dynamic programming and, despite its relative simplicity, gives a good correlation with experiment. The computed folding nuclei look like ensembles of those compact and closely packed parts of the three-dimensional native folds that contain a small number of disordered protruding loops. Their estimated free energy is consistent with the rapid (within seconds) folding and unfolding of small proteins at the point of thermodynamic equilibrium between the native fold and the coil.

Figure 1

A pathway of sequential unfolding (and folding) of the native 3D structure S₀. S_U is the coil. The U–ν links in the intermediate S_ν keep their native positions and conformations (they are shown as a solid line against the background of a dotted cloud denoting the globule), whereas the other ν links (shown in dashed line) are unfolded.

Figure 2

Various unfolding intermediates (only a small number of them are shown) and a network of unfolding pathways. The arrows correspond to elementary unfolding steps, each of which is a transition of one link from the globular native-like part of the intermediate to the coil. Any continuous chain of arrows forms a possible unfolding pathway (see Fig. 1), and a molecule can go up and down along each of them.

Figure 3

Unfolding nuclei: correlation of theoretical and experimental results for CI2 (a), barnase (b), CheY (c), the src SH3 domain (d), and the α-spectrin SH3 domain (e). Hatched rectangles at the top of each plot show how the nucleus of the minimal free energy is located in the protein chain according to the calculations. The experimental Φ_f factors are shown with open circles (connected by dotted line for better presentation). The Φ factors calculated for the ensemble of the possible transition states are shown as a solid line with filled circles (the circles correspond to residues with experimentally determined Φ_f values). The experimental data do not include negative Φ_f values, because they have no clear structural interpretation (38); Φ_f values exceeding 1 are taken as 1; when various mutations give different Φ_f values, we take the highest ones. For barnase, we take Φ_f as 1-Φ_u (37) because its unfolding (u) at high denaturant concentration, as well as folding/unfolding at moderate concentration, is a two-state process, whereas its folding in water proceeds via a metastable intermediate (37). The rectangles and lines (at the bottom of each plot) show the native positions of the α-helices and the β-strands in the chain.