Introduction

Protein folding is a collective self-organization process, conventionally described as a chemical reaction. However, this process generally does not occur by an obligate series of discrete intermediates, a ``pathway,'' but by a multiplicity of routes down a folding funnel.[1, 2, 3, 4, 5, 6, 7] Dynamics within a folding funnel involves the progressive formation of an ensemble of partially ordered structures, which is transiently impeded in its flow toward the folded structure by trapping in local minima on the energy landscape. As one proceeds down the folding funnel, the different ensembles of partially ordered structures can be conveniently described by one or more collective reaction coordinates or order parameters. Thermodynamically this funnel is characterized by a free energy that is a function of the reaction coordinate which is determined by the competition between the energy and entropy. A crucial feature of the funnel description is the concerted change in both the energy and the entropy as one moves along the reaction coordinate. As the entropy decreases so does the energy. The gradient of the free energy determines the average drift up or down the funnel. Superimposed on this drift is a stochastic motion whose statistics depends on the jumps between local minima. To first approximation this process can be described as diffusion. Folding rates are determined both by the free energy profile of the funnel and the stochastic dynamics of the reaction coordinates. In this paper we study the dynamics of a reaction coordinate describing the folding funnel of a lattice model with thermodynamic parameters which theoretical arguments suggest realistically correspond with fast folding small helical proteins.[3] We determine from simulation both the free energy profiles and effective configurational diffusion coefficients for a folding reaction coordinate as a function of temperature. Folding times, also determined by simulations, are compared to analytical calculations based on these quantities.

The kinetic analysis of a single folding funnel is most appropriate for the fast folding proteins that mainly flow downhill progressively in energy to the folded state. During folding, even for this case, trapping in local minima will occur for times very short compared the overall folding time. Alternatively, when the residence time in these traps becomes too long leading to a substantial slow down of the folding time, the traps can be thought of as creating additional funnels. This occurs near the glass transition of the heteropolymer. Good folding sequences will fold at a temperature , which is above the glass transition temperature, , and have a single dominant folding funnel. As the chain moves downhill energetically in its dominant funnel and becomes more similar to the native structure , the configurational entropy of the chain (number of available states) is reduced. For the model discussed here, a single order parameter suffices to measure similarity. For real proteins more order parameters may be necessary to quantify the similarity of the configuration to the native structure for example: degree of collapse, helicity as well as fraction of correct contacts and dihedral angles in the general case. A free energy surface as a function of these order parameters can then be calculated. For the case of a single dominant funnel, these order parameters may also be associated with reaction coordinates. The motion of these reaction coordinates will be largely diffusive due to the transient trapping. For the single funnel case, the folding time is generally determined both by the difficulty to overcome the free energy barrier and a prefactor that depends on the ruggedness of the landscape that enters via the diffusion coefficients. This general quantitative description of folding using diffusive coordinates along with energy landscape ideas to account for trapping was introduced by Bryngelson and Wolynes[8, 9] (BW). A short overview of their ideas is given in the next section. Many of the qualitative ideas from that description (e.g. the importance of the relationship between and for kinetics) have been qualitatively confirmed in lattice studies of model proteins[10, 11] and others have studied the dynamical properties of these model systems.[12, 13, 14, 15, 6] The main goal of the present paper is to quantitatively compare the BW expressions for the folding time and effective diffusion coefficients with the simulation results over a range of thermodynamics conditions for a model that theory suggests realistically corresponds with the faster folding small proteins.

In a previous paper,[3] we started this analysis. There it was shown that at the folding temperature ( ) the trajectories of appropriate collective reaction coordinates are Brownian and must surmount a modest thermodynamic free energy barrier. The folding time at was well described by the diffusive rate formula. This work shows how, using a combination of simulation results and analytical theories, to formulate a law of corresponding states relating simple lattice models to small helical proteins. Since free energy surfaces are temperature dependent, various scenarios for the folding mechanism apply at different temperatures.[2] By testing the validity of analytical formulas over a wide temperature range, we hope to provide a route to use theoretical descriptions to design and understand quantitative experiments.