Folding funnels and energy landscapes of larger proteins within the capillarity approximation

Abstract

The characterization of protein-folding kinetics with increasing chain length under various thermodynamic conditions is addressed using the capillarity picture in which distinct spatial regions of the protein are imagined to be folded or trapped and separated by interfaces. The quantitative capillarity theory is based on the nucleation theory of first-order transitions and the droplet analysis of glasses and random magnets. The concepts of folding funnels and rugged energy landscapes are shown to be applicable in the large size limit just as for smaller proteins. An ideal asymptotic freeenergy profile as a function of a reaction coordinate measuring progress down the funnel is shown to be quite broad. This renders traditional transition state theory generally inapplicable but allows a diffusive picture with a transition-state region to be used. The analysis unifies several scaling arguments proposed earlier. The importance of f luctuational fine structure both to the free-energy profile and to the glassy dynamics is highlighted. The f luctuation effects lead to a very broad trapping-time distribution. Considerations necessary for understanding the crossover between the mean field and capillarity pictures of the energy landscapes are discussed. A variety of mechanisms that may roughen the interfaces and may lead to a complex structure of the transition-state ensemble are proposed.Proteins can be thought of as mesoscopic systems; that is, they are large in atomistic terms but are too small to be completely analyzed by macroscopic reasoning alone. The energy landscape theory of protein folding describes folding kinetics through a statistical characterization of the energies of different conformations (1, 2). This makes a bridge with the theory of phase transitions in disordered systems. The theory suggests that rapidly foldable proteins have an energy landscape dominated by a funnel to a large basin of attraction, the native state, and many small rugged features that give rise to trapping in local minima (3-5). Reliable folding requires the guiding forces of the funnel to be strong enough to overcome the entropy of the unfolded states. This must occur at a temperature such that trapping (also favored at low temperature) is not so strong that flow down the funnel is too slow. Much of the qualitative content of this picture has been confirmed through the computer simulation of small lattice models of proteins (5-8). Quantitatively, the analytical theories of folding landscapes have often made use of mean field descriptions that should be most accurate when the protein is small enough that we can think of each rearranging subunit of the chain as being able to interact with a fair fraction of the others at any time (1, 2, 9-13). This is true of the smallest lattice models. If the rearranging units are appropriately taken to be small segments of helices rather than individual residues, it is also true for the smallest globular proteins in nature, which have about 70 residues (4). This paper discusses how the fol...

Capillarity theory for the fly-casting mechanism

“…The latter effect is very analogous to the surface tension of a liquid condensing from a gas. The capillarity picture of folding takes over the picture of nucleation at a first-order transition more or less intact to describe the change from folded to unfolded states (23,24). This picture might be called the "spherical cow" approximation to folding because it ignores, to zeroth order, the specifics of protein topology and connectivity.…”

Capillarity-like growth of protein folding nuclei

“…Such nuclei are similar to the capillarity approximation in homogeneous nucleation, where a stable phase droplet is separated from the metastable phase by a sharp interface. Several theoretical studies have appealed to this analogy to describe the mechanism for folding (10)(11)(12)(13)(14). Wolynes (14) describes a nucleus with capillarity-like order in which the interface surrounding a relatively folded core is broadened by wetting of partially ordered residues.…”

“…Wolynes (14) describes a nucleus with capillarity-like order in which the interface surrounding a relatively folded core is broadened by wetting of partially ordered residues. In this picture, folding can be described as a wave of order moving across the protein as the edge of the nucleus expands to ultimately consume the entire molecule (14,15) The extended partially ordered interface of a capillarity-like ordered nucleus separates space into three regions: a folded core, a partially ordered interface region, and unfolded halo (see Fig. 1).…”

“…If we assume that the folded core has native-like packing, ϭ 1/3, and b 3 is the native-like volume per monomer, so that N f † ϭ (2/3) 3 N, and ⌬F † Ϸ N 2/3 (13,14). Simulations and alternative theoretical considerations also suggest that barrier height (logarithm of the folding time) scales sublinearly on chain length, ⌬F † Ϸ N p , with 0 Յ p Յ 1 (13,14,20,21).…”

“…If we assume that the folded core has native-like packing, ϭ 1/3, and b 3 is the native-like volume per monomer, so that N f † ϭ (2/3) 3 N, and ⌬F † Ϸ N 2/3 (13,14). Simulations and alternative theoretical considerations also suggest that barrier height (logarithm of the folding time) scales sublinearly on chain length, ⌬F † Ϸ N p , with 0 Յ p Յ 1 (13,14,20,21). Direct analysis of folding rate data to determine the scaling exponent p encounters the difficulty that the range of N is too small to distinguish between different values of p (22)(23)(24)(25)(26).…”