Diffusive dynamics of the reaction coordinate for protein folding funnels

Abstract

The quantitative description of model protein folding kinetics using a diffusive collective reaction coordinate is examined. Direct folding kinetics, diffusional coefficients and free energy profiles are determined from Monte Carlo simulations of a 27-mer, 3 letter code lattice model, which corresponds roughly to a small helical protein. Analytic folding calculations, using simple diffusive rate theory, agree extremely well with the full simulation results. Folding in this system is best seen as a diffusive, funnel-like process. I IntroductionProtein 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. 1-7 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 ...

Dynamics, Energetics, and Structure in Protein Folding

“…Therefore, the apparent success of two-state analyses already suggests that even one-dimensional projections may suffice to describe coarse folding properties. The first direct evidence supporting the validity of one-dimensional folding surfaces has come from the analysis of simulations in the cubic lattice (28). Further support comes from the reasonable success in calculating relative folding rates from protein structures using one-dimensional free energy surfaces (29).…”

“…more compact)(3). Effective diffusion coefficients for folding on a one-dimensional energy surface are therefore expected to be complex functions of the order parameter, including both dynamic and energetic terms (28).…”

Structural correlations in protein folding funnels

“…The motion of this coordinate for small lattice models is diffusive, and rates are predicted well using a diffusional rate theory (25), since the free energy barrier is broad. It has also been shown that at the folding midpoint of the lattice protein the effects of site mutations faithfully reflect the structural correlations in the ensemble at the top of the barrier (11).…”

“…Since tertiary contacts are a dominant source of protein stabilization, a collective coordinate measuring the fraction of such contacts which are correct, Q, has been used in a number of off-lattice and lattice simulations (22)(23)(24)(25). The motion of this coordinate for small lattice models is diffusive, and rates are predicted well using a diffusional rate theory (25), since the free energy barrier is broad.…”

Diffusive reaction dynamics on invariant free energy profiles

“…Theoretical studies based on simulations of the reaction rate for complex systems, such as peptides and proteins, often show simple exponential kinetics (11)(12)(13). To be able to determine both the preexponential factor and the free energy barrier from simulations, it is necessary to have a method of constructing the onedimensional projected FES in terms of an appropriate reaction coordinate, if such exists.…”