A summary of the Bryngelson and Wolynes Energy Landscape Picture

The energy landscape of any heteropolymer is complex. Because of the possibility of making many contacts involving residues that are distant in sequence, the energy landscape is rough. This is an effect of frustration. Protein-like heteropolymers obey a principle of minimal frustration, leading to an additional funnel-like aspect of the landscape. To quantify this, we recognize that the energy landscape is stratified, that is to say the statistical characteristics of the landscape depend on the distance from, or equivalently the similarity to, the ideal native structure. This similarity measure is known as an order parameter in the physics of phase transitions, and for small systems such as proteins can also be used as a reaction coordinate for computing the folding rate. (Here this the reaction coordinate will be called n.) Clearly such a reaction coordinate is not unique. A picture of such a stratified landscape with kinetic connections is shown in figure 1.

  
Figure 1: The top figure shows the kinetic landscape versus an arbitrary reaction coordinate (n). The vertical line shows the glass transition point ( ) in which the behavior changes from many paths to few paths. In addition to the native state there are a number of local minimum. The bottom figure shows the free energy plotted as a function of the same coordinate. The line at denotes the transition region.

BW show that to a first approximation, the folding time can be computed by first grouping together states with in a stratum having a common value of the reaction coordinate. A diffusion equation for the probability flow between strata is derived under the assumption that the reaction coordinate can only change by relatively small steps, and it is written as

The average direction of the flow is given by the gradient of the free energy. The diffusion coefficient, D(n), depends on the trapping in local minima and reflects the ruggedness of the energy landscape in the system proximity of the glass transition.

The free energy F(n) incorporates the balance between two terms, the energy that is decreased as the native state is approached and the entropic term -T S(n) which decreases with unfolding. The shape of the free energy profile is therefore strongly temperature dependent as illustrated in figure 1b. At high temperatures, folding is an uphill process thermodynamically so folding is exponentially suppressed. At the folding temperature, the free energy profile is typically bistable with a small thermodynamic barrier which arises from the incomplete cancellation of the entropy by the energy as the systems descends through the funnel. At low temperatures, folding becomes a downhill process. Thus, if the diffusion coefficient were temperature independent, the rate of folding would have a dependence on the thermodynamic driving force (which depends on 1/T) just as a solid state rectifier depends on the applied voltage (as in the famous example of Feynman). The folding time can be written as a double integral

 

When the barrier exists, as at , F(n) has a double-well with the bottom of one well close to and the bottom of the other well at . In this situation this double integral can be approximated by a Kramer's like law,[16, 17]

 

where

and

and is the curvature around and is the curvature in the top of the barrier. is the effective diffusion coefficient on an equivalent flat landscape.

The prefactor of Kramer's expression reflects the multiple recrossings of the barrier through diffusion that is controlled by trapping. When the process is entirely downhill, the double integral becomes simply proportional to the diffusion coefficient, and only weakly depends on the slope of the free energy gradient.

The configurational diffusion coefficient depends both on the local moves allowed to the protein and the energy landscape topography. BW analyzes configurational diffusion by assuming the limit of an uncorrelated rugged energy landscape and Metropolis rules. This is, of course, a caricature of the dynamics in which the landscape will have correlations and the barriers to escape from traps may be surmounted by successively melting out local clusters rather than globally changing the protein conformation. According to the BW analysis, the diffusion coefficient has a strong temperature dependence that arises from the necessity to escape from local minima on the rough energy landscape. At high temperatures, D follows a Ferry law typical of glasses,

 

where is the local mean square fluctuation in energy. At intermediate temperatures, the strongly non-Arrhenius temperature dependence becomes itself moderated

This equation is valid for temperatures between and . This form arises because of the assumed Metropolis dynamics. The characteristic temperature is the ideal glass transition where the configurational entropy would vanish at equilibrium. This is given by the condition

where is configurational entropy at n. We see that at , the diffusion coefficient is diminished from its bare value by a factor of the total number configuration states, thus giving rise to a possibility of a Levinthal paradox. Below , we expect both that the diffusion coefficient will still possess an activation energy but the activation energy will not be self-averaging and that the free energy itself can fluctuate considerably and will be very sensitive to details of the model and to the sequence of the heteropolymer. In the theory of disordered systems this is known as the emergence of non self-averaging behavior, the hallmark of the ideal thermodynamic spin glass transition in mesoscopic systems. Below , the slow folding process can be better described by a few kinetic pathways than by the statistical average laws appropriate for the faster events. Both the temperature dependence of D and the interplay of entropy and energetic ruggedness in the driving force lead to a parabolic dependence of the folding rate on the inverse temperature as shown in figure 2.

   
Figure 2: Mean first passage time ( ) is plotted as a function of the inverse temperature ( ). Note, parabolic dependence of the folding time on .

This rough form has been confirmed in many lattice simulations. Leite and Onuchic[18] have shown that for an energy landscape model for solvent polarization, analogous to the BW landscape, there is a gradual transition into a slow non-self averaging dynamics. Fluctuations gradually dominates the kinetics below this temperature.

Folding rates depends on both diffusion and free energy profile in a way that is significantly different from the standard transition state theory (TST) The extent to which the rate is determined by the free energy profile versus the diffusion coefficient, that incorporates the multiple barrier crossing, depends on the choice for the reaction coordinate. Notice however, that while it may be impossible or at least difficult to find a reaction coordinate for which TST is exact, the diffusion formula (eq. 2) is essentially invariant to this choice as long as the elementary moves are reasonably local for this coordinate.

In the original BW treatment, this reaction coordinate was represented by an order parameter that measured the similarity between any given configuration and the native state of the protein in terms of the fraction of correct dihedral angles, a coordinate which is manifestly local because of the elementary moves of the protein at the microscopic level are controlled by isomerization of the peptide bond. Another possible reaction coordinate is the number of correct contacts, a coordinate that is only partially local.