We analyzed folding routes predicted by a variational model in terms of a generalized formalism of the capillarity scaling theory for 28 two-state proteins. The scaling exponent ranged from 0.2 to 0.45 with an average of 0.33. This average value corresponds to packing of rigid objects.That is, on average the folded core of the nucleus is found to be relatively diffuse. We also studied the growth of the folding nucleus and interface along the folding route in terms of the density or packing fraction. The evolution of the folded core and interface regions can be classified into three patterns of growth depending on how the growth of the folded core is balanced by changes in density of the interface. Finally, we quantified the diffuse versus polarized structure of the critical nucleus through direct calculation of the packing fraction of the folded core and interface regions. Our results support the general picture of describing protein folding as the capillarity-like growth of folding nuclei.
Deep Dive into Capillarity-like growth of protein folding nuclei.
We analyzed folding routes predicted by a variational model in terms of a generalized formalism of the capillarity scaling theory for 28 two-state proteins. The scaling exponent ranged from 0.2 to 0.45 with an average of 0.33. This average value corresponds to packing of rigid objects.That is, on average the folded core of the nucleus is found to be relatively diffuse. We also studied the growth of the folding nucleus and interface along the folding route in terms of the density or packing fraction. The evolution of the folded core and interface regions can be classified into three patterns of growth depending on how the growth of the folded core is balanced by changes in density of the interface. Finally, we quantified the diffuse versus polarized structure of the critical nucleus through direct calculation of the packing fraction of the folded core and interface regions. Our results support the general picture of describing protein folding as the capillarity-like growth of folding nu
The modern theory of protein folding describes the mechanism for folding as an entropic bottleneck arising from the decreasing number of accessible pathways available to a protein as it becomes ordered. 1,2 The collection of partially ordered conformations corresponding to this bottleneck region is known as the transition state ensemble or critical folding nucleus. 3 Although it is common to focus on the degree of native-like order of specific residues, a complete description of the protein folding mechanism also includes the spatial properties such as size or density of the transition state ensemble. Indeed, shortly after characterizing the transition state ensemble of CI2, Fersht summarized the structure of the critical nucleus by a spatial description through the proposal of the nucleation-condensation mechanism. 4 This critical nucleus can be thought of as an expanded, partially ordered version of the native state ensemble with concomitant longranged tertiary and local secondary structure.
It is now clear that while diffuse nuclei appear to be the general rule, some nuclei are less diffuse than others. 5 Polarized nuclei have highly structured residues which are spatially clustered in the native structure, while the rest of the residues show little definite order. 6,7,8,9 Such nuclei are similar to the capillarity approximation in homogeneous nucleation in which the free energy of a stable phase droplet is separated from the metastable phase by a very sharp interface. 10,11 Exploiting this analogy, Wolynes describes a nucleus with capillarity-like order in which the interface surrounding a relatively folded core is broadened by wetting of partially ordered residues. 11 In this picture, folding can be described as the growth of the folding nucleus: a wave of order moving across the protein as the edge of the nucleus expands to ultimately consume the entire molecule. 11,12 The extended partially ordered interface of a capillarity-like ordered nucleus separates space into three regions: a folded core, a partially ordered interface region, and unfolded halo (see Fig. 1). In this paper, we monitor the structural development of the nucleus along the folding route through the evolution of the packing fraction of the folded core and the interface. As shown in Fig. 1, growth of the nucleus can be described by fluxes of residues passing through two moving surfaces: one surface separates the folded core and interface, and the other surface separates the interface region and the unfolded halo. As the protein folds, the evolution of the interface is determined by the interfacial volume and the net flux of residues entering the interface.
Our analysis is based on folding routes calculated for 28 two-state proteins from a cooperative variational model described in 13 . We note this model includes neutral cooperativity due to repulsive excluded volume interactions. This form of cooperativity has been shown to broaden the range of barrier heights allowing direct comparison between calculated and measured folding rates. 13 Not surprisingly, cooperativity tends to sharpen the interface between folded and unfolded regions. Nevertheless, the interface from this model is generally not nearly as sharp as a strict capillarity description in which a residue can be clearly identified as being either completely folded or completely unfolded as some other analytic models assume. 14,15,16 In fact, an unbiased analysis of the spatial properties of the folding nucleus fundamentally depends the model’s ability to describe partial order.
Capillarity picture of folding nuclei. The capillarity approximation of folding nuclei is based on classical nucleation theory of first order phase transition kinetics. 11,17 . Within the capillarity approximation, the free energy of a nucleus with volume V f and surface area A f can be written as a sum of two terms
where ∆f denotes the bulk free energy difference per unit volume between the unfolded and folded ensembles, and γ is the surface tension between the folded and unfolded regions.
A folded core with native-like density has a volume per monomer independent of its size.
Relaxing this assumption, we assume that the number of residues in the folded core, N f , scales with and its volume, V f , according to
Here, ν is the scaling exponent associated with the lengthscale of the folded core R ∼ b 0 N ν f , and b 3 is a geometry dependent elementary volume proportional to the monomer volume, b 3 0 . The free energy of a folded nucleus with N f residues then has the form: 11
At the folding transition temperature, T f , finite size depression of the surface energy suggests that γ ∼ ∆f bN ν where N is the number of monomers in the protein. The maximum of the free energy occurs at N † f = (2/3) 1/ν N, giving the size of the critical nucleus, and the associated free energy barrier scales as ∆F † ∼ N 2ν . If we assume that the folded core has native like packing, ν = 1/3 and b 3 is the native-like
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