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. 1997 Jun 10;94(12):6170-5.
doi: 10.1073/pnas.94.12.6170.

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

P G Wolynes  1
Affiliations

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

P G Wolynes. Proc Natl Acad Sci U S A. .

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 free-energy 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 fluctuational fine structure both to the free-energy profile and to the glassy dynamics is highlighted. The fluctuation 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.

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Figures

Figure 1
Figure 1
A schematic view, according to the capillarity picture of structure of a protein (lysozyme is shown for illustration) once it is advanced partially down the folding funnel. When several phases of the protein are possible, they can lead to a rather complex interface, as illustrated here. Thermal- and disorder-induced roughening of the interfaces can smooth out the transition region seriously. The figure shows several kinds of ordering that are possible: a region in which side chains are completely ordered (blue), a transition zone with topologically correct folding but no side-chain order (red), a molten-disordered phase (yellow), and the random coil (green). Having such a gradual progression of phases can lower the activation energy but broadens the interface between completely folded and unfolded regions. The front is shown, for clarity, as progressing from one end of the protein to the other. The most well ordered part may be buried in the core, depending on the relative surface tensions between the phases and the solvent.
Figure 2
Figure 2
The funnel free-energy profile in the capillarity approximation. The solid curve is the ideal profile of the free energy versus the folded fraction, assuming no heterogeneity of stabilization energy. An actual profile differs by a fluctuation that is random and Gaussianly distributed due to this heterogeneity. This is the “fine structure” of the folding funnel. A specific such profile is indicated by a dotted line. Notice the cusp-like minima occur at precisely Ñf = 0 and Ñf = 1, because the capillarity theory is an asymptotic result for large NTOT. Simulations and mean field theory show smooth minima at nonextreme values of the order parameters.
Figure 3
Figure 3
The various regimes of scaling behavior for the typical folding time with length are indicated for different values of TF/T, the folding temperature-to-ambient ratio, and TG/T, the glass temperature-to-ambient ratio. At high temperatures, folding is always thermodynamic uphill and the barrier grows like N. At low temperatures, the rate is trap-dominated, and the typical barrier behaves like N1/2, whereas the slowest escape (not indicated in the figure) will have bigger barriers, scaling like N. Fastest folding occurs for T ≪ TF but T ≫ TG where polynomial behavior of the folding time in N is to be found. This occurs for “well designed” proteins. The figure also shows the boundary between trap escape and quasi-free chain dynamics, as indicated by the mean field dynamic transition temperature TA. A realistic value of TF/TG is 1.6. In this figure we assume temperatures are all less than the collapse temperature TC. If T is bigger than TC, ruggedness effects can be neglected.
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