Two-state folders

This correlation function is shown under different conditions in Fig. 5. In the folded state, the decay of the correlation function is characterized by a relatively narrow dynamical range and short time scales, on the order of 1. The rapid decay of C(t) in this case can be attributed to fluctuations around the native conformation, within the corresponding basin of attraction. The fact that those fluctuations are fast is consistent with the fact that this is a rapid two-state folder. More specifically, the native conformation is expected... 

Fig. 3. Schematic free energy landscapes of two-state folders for increasing denat-urant concentration (or increasing temperatures). Note that the unfolded state (U) becomes more stable with respect to the native state (N). The folding rate constant is inversely proportional to the exponent of the barrier height...

Figure 2 Free energy surfaces for a two-state folder as a function of a reaction coordinate q, which is plotted at different temperatures. The folded (F) and denatured (D) states are separated by a free energy barrier located at (a) Above the folding transition temperature (T > Tf), the entropic contribution of the free energy dominates and the unfolded state is favored, (b) At the folding transition temperature (T = Tf), both the unfolded and the folded states are equally populated, (c) Below the transition temperature (T < Tf), the favorable energetic contributions to folding dominate and the native basin is favored.

Figure 3 Two-state (D = F) versus three-state (D = I = F) folding, (a) Free energy surface for a two-state folder as a function of the reaction coordinate q. (b) Free energy surface for a three- state folder as a function of the reaction coordinate q. An additional minimum corresponding to an intermediate state I is present, (c) Single exponential kinetics of folding for a two-state folder, (d) Nonexponential kinetics of folding for a three-state protein, (e) Linear chevron plot for a two state folder, (f) Chevron plot with rollover for a three-state folder.

Figure 6 (a) The native state of the HT model. Hydrophobic residues are black, hydrophilic residues are gray, and neutral residues are white, (b) Specific heat (Cy) curve and plots of the average fraction of native contacts (Q) and total number of contacts (C) as a function of temperature for the original HT model, (c) Same curves as in (b) for the minimally frustrated Go-model. The original HT model shows a nonspecific transition, whereas the minimally frustrated model displays the signatures of a two-state folder. Adapted From Nymeyer et al. (Ref. 64). 

Most small, single-domain proteins display thermodynamic and kinetic signatures of two-state folders. The simplicity of this folding scenario, in which only unfolded (U) and native (N) states are populated to any significant degree, led to the development of kinetic models analogous to those first introduced in the context of chemical rate processes. 

The sequence, whose native state is shown in Fig. 4, displays two-state kinetics for the temperatures T > 0.87 that is, P (t) exp(— ), where Xf is the folding time. To probe the sequence of events en route to the native conformation, we computed Rg t)), which reveals two stages in collapse. Initial rapid burst phase is followed by a gradual chain compaction (Fig. 5). The overall collapse time Xc is associated with the second characteristic time. From the approach to the native conformation we draw the following general conclusions regarding two-state folders ... 

Figure 6. The sketch of the protein folding pathways. The fast (upper) folding pathway includes the formation of native-like collapsed states In, which rapidly convert into the native state N. The fraction of protein molecules, folding along this pathway, is >. For two-state folders, 1. The lower track (followed by 1 - molecules) represents slow pathway(s), which fold by a three-stage kinetic mechanism. At the first stage, nonspecific collapse species Insc form, which later convert into a collection of discrete native-like intermediates . The transition from Ij to the native state is slow and represents the rate-limiting step in the slow pathway. The degree of heterogeneity in the folding pathways depends on the sequence and external conditions.

A molecular folding process is much more complex and generally requires interactive cooperativity, i.e., the formation of a unique functional protein fold can take comparatively long (up to the order of seconds) and include weakly stable intermediate states. The collective folding is also necessary to avoid dislocations. Hence, the free-energy contour depicted in Fig, 2.1 will only apply to a subclass of proteins, so-called two-state folders. 

For the study of the folding transition, the introduction of an effective parameter that uniquely describes the macrostate of the ensemble of heteropolymer conformations is useful. A typical measure is the contact number q( X), which for a given conformation X is simply defined as the fraction of the already formed native contacts n(X) in conformation X and the total number of native contacts tot in the final fold, i.e., (X) = (X)/ tot- Then, the statistical ensemble average of this quantity q X)) at a given temperature characterizes its macrostate. Roughly, for a two-state folder, if q X)) > 0.5, native-like conformations are dominating the statistical ensemble. If less than half the total number of contacts is formed, the heteropolymer tends to reside in the pseudophase of denatured conformations. 

The classification of the heteropolymer with sequence 20.6 as a two-state folder arises from the analysis of the free-energy landscape. We assume that gr is a suitable parameter that describes the macrostate of the system adequately. Considering this parameter as a constraint, we can formally average out the conformational degrees of freedom, and the probability for a conformation in a macrostate with contact parameter q reads... 

Several proteins are known to be two-state folders, and their folding transitions exhibit the features we have also seen in the coarse-grained model study presented in this chapter. A famous example is chymotrypsin inhibitor 2 (CI2) [205], one of the first proteins, where two-state folding characteristics have been identified experimentally. Clear signals... 

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