Protein folding is remarkable as a chemical reaction with a highly non-Arrhenius temperature dependence. Although for real proteins some of this behavior is doubtless due to the entropic nature of hydrophobic forces, we see here that this unusual temperature dependence also arises from the competition between trapping in misfolded states and drift down the folding funnel which is itself a competition between entropy and energy. These effects can be described through collective coordinates, which make quantitative the nature of the funnel. Our results establish that a description in terms of a single collective coordinate suffices to explain folding kinetics of small lattice models over a wide range of temperatures. We believe that a similar set of concepts can be used for the smaller real proteins because of their correspondence with the lattice models. The diffusive dynamics should be contrasted to the more traditional transition state theory. These simulations suggest that if transition state theory is used a very complex reaction coordinate, reflecting the detailed trapping dynamics, would have to be constructed. The single collective diffusive coordinate picture on the other hand is much more robust and can make use of experimental information about the dynamics of fluctuations in the molten globule, as well as simple thermodynamic measurements and theories about the molten globule state.
The lattice model studied here are described quite well by a single order parameter or reaction coordinate. Various scenarios for protein folding require the introduction of a few more collective coordinates to describe the funnel. Even the small helical proteins, that corresponding in a thermodynamic sense with the 27-mer lattice, require the introduction of coordinates describing the amount of helicity, degree of collapse and liquid crystallinity of the dynamical molten globules. If these order parameters are constant or their dynamics is ``slaved'' to the tertiary ordering parameter the one-dimensional diffusive theory for funnel dynamics will suffice for real proteins just as in the model systems. If the collective diffusion coefficients are wildly different for these extra degrees of freedom or when there is a free energy profile for the coupled order parameters with several minima, one must generalize the Kramer's expression to a few more dimensions. We believe such a few-dimensional generalization of the current diffusive dynamical description should suffice for the smaller fast folding protein molecules. On the other hand, sufficiently large proteins doubtless require a description in terms of geometrically localized order parameters not just a globally defined coordinate. One example of geometrically localized phenomena is provided by foldons.[22] Foldons are kinetically autonomous folding subunits, that are expected to exist in the larger proteins. Each foldon will have its own funnel. A related description involving local collective coordinates necessary for describing critical nuclei may also be helpful. Some arguments suggest critical nuclei may be large[23] so the system would be well treated by global reaction coordinate while others suggest nuclei may small[24] or indeed possibly unique.[25] Any of these cases can be accommodated by the diffusive funnel dynamics picture once the coordinates are specified. We expect the diffusive funnel dynamics picture (discussed here and previously by us[3]) will provide a convenient quantitative framework to analyze both simulation and laboratory experiments.