Analyzing the effect of homogeneous frustration in protein folding

The energy landscape theory has been an invaluable theoretical framework in the understanding of biological processes such as protein folding, oligomerization, and functional transitions. According to the theory, the energy landscape of protein folding is funneled toward the native state, a conformational state that is consistent with the principle of minimal frustration. It has been accepted that real proteins are selected through natural evolution, satisfying the minimum frustration criterion. However, there is evidence that a low degree of frustration accelerates folding. We examined the interplay between topological and energetic protein frustration. We employed a C_α structure‐based model for simulations with a controlled nonspecific energetic frustration added to the potential energy function. Thermodynamics and kinetics of a group of 19 proteins are completely characterized as a function of increasing level of energetic frustration. We observed two well‐separated groups of proteins: one group where a little frustration enhances folding rates to an optimal value and another where any energetic frustration slows down folding. Protein energetic frustration regimes and their mechanisms are explained by the role of non‐native contact interactions in different folding scenarios. These findings strongly correlate with the protein free‐energy folding barrier and the absolute contact order parameters. These computational results are corroborated by principal component analysis and partial least square techniques. One simple theoretical model is proposed as a useful tool for experimentalists to predict the limits of improvements in real proteins.Proteins 2013; 81:1727–1737. © 2013 Wiley Periodicals, Inc.     

First principles prediction of protein folding rates

Experimental studies have demonstrated that many small, single-domain proteins fold via simple two-state kinetics. We present a first principles approach for predicting these experimentally determined folding rates. Our approach is based on a nucleation-condensation folding mechanism, where the rate-limiting step is a random, diffusive search for the native tertiary topology. To estimate the rates of folding for various proteins via this mechanism, we first determine the probability of randomly sampling a conformation with the native fold topology. Next, we convert these probabilities into folding rates by estimating the rate that a protein samples different topologies during diffusive folding. This topology-sampling rate is calculated using the Einstein diffusion equation in conjunction with an experimentally determined intra-protein diffusion constant. We have applied our prediction method to the 21 topologically distinct small proteins for which two-state rate data is available. For the 18 beta-sheet and mixed alpha-beta native proteins, we predict folding rates within an average factor of 4, even though the experimental rates vary by a factor of approximately 4 x 10(4). Interestingly, the experimental folding rates for the three four-helix bundle proteins are significantly underestimated by this approach, suggesting that proteins with significant helical content may fold by a faster, alternative mechanism. This method can be applied to any protein for which the structure is known and hence can be used to predict the folding rates of many proteins prior to experiment.     

Importance of sequence specificity for predicting protein folding pathways: perturbed Gaussian chain model

Recent experimental and theoretical studies suggest that rates and pathways of protein folding are largely decided by topology of the native structures, at least for small proteins. However, some exceptions are known; for example, protein L and protein G have the same topology, but exhibit different characteristics of the TSE. Thus, folding pathways of some proteins are critically affected by detailed information on amino acid sequences. To investigate the sequence specificity, we calculate folding pathways of 20 small proteins using the perturbed Gaussian chain model developed by Portman et al. (Phys Rev Lett 1998;81:5237-5240; J Chem Phys 2001;114:5069-5081). Characteristics of the TSE predicted by the model are in good agreement with experimental phi-value data for many proteins at coarse-grained level. Especially, estimation of folding TSE for protein G and protein L based on both topology and additional sequence information are consistent with experimental phi-value data. With only topology information, however, the model predicts the TSE of protein G incorrectly. Moreover, the model that uses topology and sequence information describes free energy profiles of two-state and three-state folders consistently with experiment, whereas the topology only model predicts free energy profiles of some proteins incorrectly. This indicates that sequence specificity also has critical roles in determining the folding pathways for some proteins.     

Folding pathway of the b1 domain of protein G explored by multiscale modeling

The understanding of the folding mechanisms of single-domain proteins is an essential step in the understanding of protein folding in general. Recently, we developed a mesoscopic CA-CB side-chain protein model, which was successfully applied in protein structure prediction, studies of protein thermodynamics, and modeling of protein complexes. In this research, this model is employed in a detailed characterization of the folding process of a simple globular protein, the B1 domain of IgG-binding protein G (GB1). There is a vast body of experimental facts and theoretical findings for this protein. Performing unbiased, ab initio simulations, we demonstrated that the GB1 folding proceeds via the formation of an extended folding nucleus, followed by slow structure fine-tuning. Remarkably, a subset of native interactions drives the folding from the very beginning. The emerging comprehensive picture of GB1 folding perfectly matches and extends the previous experimental and theoretical studies.     

不同类型氨基酸网络参量与蛋白质折叠的关系

理论和实验研究表明,蛋白质天然拓扑结构对其折叠过程具有重要的影响．采用复杂网络的方法分析蛋白质天然结构的拓扑特征,并探索蛋白质结构特征与折叠速率之间的内在联系．分别构建了蛋白质氨基酸网络、疏水网、亲水网、亲水-疏水网以及相应的长程网络,研究了这些网络的匹配系数(assortativity coefficient)和聚集系数(clustering coefficient)的统计特性．结果表明,除了亲水-疏水网,上述各网络的匹配系数均为正值,并且氨基酸网和疏水网的匹配系数与折叠速率表现出明显的线性正相关,揭示了疏水残基间相互作用的协同性有助于蛋白质的快速折叠．同时,研究发现疏水网的聚集系数与折叠速率有明显的线性负相关关系,这表明疏水残基间三角结构(triangle construction)的形成不利于蛋白质快速折叠．还进一步构建了相应的长程网络,发现序列上间距较远的残基接触对的形成将使蛋白质折叠进程变慢．     

Energy barriers, cooperativity, and hidden intermediates in the folding of small proteins

Current theoretical views of the folding process of small proteins (< approximately 100 amino acids) postulate that the landscape of potential mean force (PMF) for the formation of the native state has a funnel shape and that the free energy barrier to folding arises from the chain configurational entropy only. However, recent theoretical studies on the formation of hydrophobic clusters with explicit water suggest that a barrier should exist on the PMF of folding, consistent with the fact that protein folding generally involves a large positive activation enthalpy at room temperature. In addition, high-resolution structural studies of the hidden partially unfolded intermediates have revealed the existence of non-native interactions, suggesting that the correction of the non-native interactions during folding should also lead to barriers on PMF. To explore the effect of a PMF barrier on the folding behavior of proteins, we modified Zwanzig's model for protein folding with an uphill landscape of PMF for the formation of transition states. We found that the modified model for short peptide segments can satisfy the thermodynamic and kinetic criteria for an apparently two-state folding. Since the Levinthal paradox can be solved by a stepwise folding of short peptide segments, a landscape of PMF with a locally uphill search for the transition state and cooperative stabilization of folding intermediates/native state is able to explain the available experimental results for small proteins. We speculate that the existence of cooperative hidden folding intermediates in small proteins could be the consequence of the highly specific structures of the native state, which are selected by evolution to perform specific functions and fold in a biologically meaningful time scale.     

Routes are trees: the parsing perspective on protein folding

An important puzzle in structural biology is the question of how proteins are able to fold so quickly into their unique native structures. There is much evidence that protein folding is hierarchic. In that case, folding routes are not linear, but have a tree structure. Trees are commonly used to represent the grammatical structure of natural language sentences, and chart parsing algorithms efficiently search the space of all possible trees for a given input string. Here we show that one such method, the CKY algorithm, can be useful both for providing novel insight into the physical protein folding process, and for computational protein structure prediction. As proof of concept, we apply this algorithm to the HP lattice model of proteins. Our algorithm identifies all direct folding route trees to the native state and allows us to construct a simple model of the folding process. Despite its simplicity, our model provides an account for the fact that folding rates depend only on the topology of the native state but not on sequence composition.     

Local and non-local native topologies reveal the underlying folding landscape of proteins

Due to Plaxco, Simons, Baker and others, it is now well known that the two-state single domain protein folding rate is fairly well predicted from knowledge of the topology of the native structure. Plaxco et al found that the folding rates of two-state proteins correlate with the average degree to which native contacts are 'local' within the chain sequence: fast-folders usually have mostly local structures. Here, we dissected the native topology further by focusing on non-local and local contacts using lower and upper bounds of allowable sequence separation in computing the average contact order. We analyzed non-local and local contacts of 82 two-state proteins whose experimental folding rates span over six orders of magnitude. We observed that both the number of non-local contacts and the average sequence separation of non-local contacts (non-local CO) are both negatively correlated with the folding rate, showing that the non-local contacts dominate the barrier-crossing process. Surprisingly, the local contact orders of the proteins also correlate with the folding rates. However, this correlation shows a strong positive trend indicating the role of a diffusive search in the denatured basin.     

Desolvation is a likely origin of robust enthalpic barriers to protein folding

Experimental data from global analyses of temperature (T) and denaturant dependence of the folding rates of small proteins led to an intrinsic enthalpic folding barrier hypothesis: to a good approximation, the T-dependence of folding rate under constant native stability conditions is Arrhenius. Furthermore, for a given protein, the slope of isostability folding rate versus 1/T is essentially independent of native stability. This hypothesis implies a simple relationship between chevron and Eyring plots of folding that is easily discernible when both sets of rates are expressed as functions of native stability. Using experimental data in the literature, we verify the predicted chevron-Eyring relationship for 14 proteins and determine their intrinsic enthalpic folding barriers, which vary approximately from 15 kcal/mol to 40 kcal/mol for different proteins. These enthalpic barriers do not appear to correlate with folding rates, but they exhibit correlation with equilibrium unfolding enthalpy at room temperature. Intrinsic enthalpic barriers with similarly high magnitudes apply as well to at least two cases of peptide-peptide and peptide-protein association, suggesting that these barriers are a hallmark of certain general and fundamental kinetic processes during folding and binding. Using a class of explicit-chain C(alpha) protein models with constant elementary enthalpic desolvation barriers between C(alpha) positions, we show that small microscopic pairwise desolvation barriers, which are a direct consequence of the particulate nature of water, can act cooperatively to give rise to a significant overall enthalpic barrier to folding. This theoretical finding provides a physical rationalization for the high intrinsic enthalpic barriers in protein folding energetics. Ramifications of entropy-enthalpy compensation in hydrophobic association for the height of enthalpic desolvation barrier are discussed.     

Use of quantitative structure-property relationships to predict the folding ability of model proteins

We investigate the folding of a 125-bead heteropolymer model for proteins subject to Monte Carlo dynamics on a simple cubic lattice. Detailed study of a few sequences revealed a folding mechanism consisting of a rapid collapse followed by a slow search for a stable core that served as the transition state for folding to a near-native intermediate. Rearrangement from the intermediate to the native state slowed folding further because it required breaking native-like local structure between surface monomers so that those residues could condense onto the core. We demonstrate here the generality of this mechanism by a statistical analysis of a 200 sequence database using a method that employs a genetic algorithm to pick the sequence attributes that are most important for folding and an artificial neural network to derive the corresponding functional dependence of folding ability on the chosen sequence attributes [quantitative structure-property relationships (QSPRs)]. QSPRs that use three sequence attributes yielded substantially more accurate predictions than those that use only one. The results suggest that efficient search for the core is dependent on both the native state's overall stability and its amount of kinetically accessible, cooperative structure, whereas rearrangement from the intermediate is facilitated by destabilization of contacts between surface monomers. Implications for folding and design are discussed. Proteins 33:177–203, 1998. © 1998 Wiley-Liss, Inc.     

Exploring zipping and assembly as a protein folding principle

It has been proposed that proteins fold by a process called "Zipping and Assembly" (Z&A). Zipping refers to the growth of local substructures within the chain, and assembly refers to the coming together of already-formed pieces. Our interest here is in whether Z&A is a general method that can fold most of sequence space, to global minima, efficiently. Using the HP model, we can address this question by enumerating full conformation and sequence spaces. We find that Z&A reaches the global energy minimum native states, even though it searches only a very small fraction of conformational space, for most sequences in the full sequence space. We find that Z&A, a mechanism-based search, is more efficient in our tests than the replica exchange search method. Folding efficiency is increased for chains having: (a) small loop-closure steps, consistent with observations by Plaxco et al. 1998;277;985-994 that folding rates correlate with contact order, (b) neither too few nor too many nucleation sites per chain, and (c) assembly steps that do not occur too early in the folding process. We find that the efficiency increases with chain length, although our range of chain lengths is limited. We believe these insights may be useful for developing faster protein conformational search algorithms.     

The building block folding model and the kinetics of protein folding.

Here we show that qualitatively, the building blocks folding model accounts for three-state versus the two-state protein folding. Additionally, it is consistent with the faster versus slower folding rates of the two-state proteins. Specifically, we illustrate that the building blocks size, their mode of associations in the native structure, the number of ways they can combinatorially assemble, their population times and the way they are split in the iterative, step-by-step structural dissection which yields the anatomy trees, explain a broad range of folding rates. We further show that proteins with similar general topologies may have different folding pathways, and hence different folding rates. On the other hand, the effect of mutations resembles that of changes in conditions, shifting the population times and hence the energy landscapes. Hence, together with the secondary structure type and the extent of local versus non-local interactions, a coherent, consistent rationale for folding kinetics can be outlined, in agreement with experimental results. Given the native structure of a protein, these guidelines enable a qualitative prediction of the folding kinetics. We further describe these in the context of the protein folding energy landscape. Quantitatively, in principle, the diffusion-collision model for the building block association can be used. However, the folding rates of the building blocks and traps in their formation and association, need to be considered.     

Statistical analyses of protein folding rates from the view of quantum transition

Understanding protein folding rate is the primary key to unlock the fundamental physics underlying protein structure and its folding mechanism.Especially,the temperature dependence of the folding rate remains unsolved in the literature.Starting from the assumption that protein folding is an event of quantum transition between molecular conformations,we calculated the folding rate for all two-state proteins in a database and studied their temperature dependencies.The non-Arrhenius temperature relation for 16 proteins,whose experimental data had previously been available,was successfully interpreted by comparing the Arrhenius plot with the first-principle calculation.A statistical formula for the prediction of two-state protein folding rate was proposed based on quantum folding theory.The statistical comparisons of the folding rates for 65 two-state proteins were carried out,and the theoretical vs.experimental correlation coefficient was 0.73.Moreover,the maximum and the minimum folding rates given by the theory were consistent with the experimental results.     

On prediction of folding nuclei in globular proteins

The approach described in this paper on the prediction of folding nuclei in globular proteins with known three dimensional structures is based on a search of the lowest saddle points through the barrier separating the unfolded state from the native structure on the free-energy landscape of protein chain. This search is performed by a dynamic programming method. Comparison of theoretical results with experimental data on the folding nuclei of two dozen of proteins shows that our model provides good phi value predictions for proteins whose structures have been determined by X-ray analysis, with a less limited success for proteins whose structures have been determined by NMR techniques only. Consideration of a full ensemble of transition states results in more successful prediction than consideration of only the transition states with the minimal free energy. In conclusion we have predicted the localization of folding nuclei for three dimensional protein structures for which kinetics of folding is studied now but the localization of folding nuclei is still unknown.     

Identification of cooperative folding units in a set of native proteins.

Cooperative unfolding penalties are calculated by statistically evaluating an ensemble of denatured states derived from native structures. The ensemble of denatured states is determined by dividing the native protein into short contiguous segments and defining all possible combinations of native, i.e., interacting, and non-native, i.e., non-interacting, segments. We use a novel knowledge-based scoring function, derived from a set of non-homologous proteins in the Protein Data Bank, to describe the interactions among residues. This procedure is used for the structural identification of cooperative folding cores for four globular proteins: bovine pancreatic trypsin inhibitor, horse heart cytochrome c, French bean plastocyanin, and staphylococcal nuclease. The theoretical folding units are shown to correspond to regions that exhibit enhanced stability against denaturation as determined from experimental hydrogen exchange protection factors. Using a sequence similarity score for related sequences, we show that, in addition to residues necessary for enzymatic function, those amino acids comprising structurally important folding cores are also preferentially conserved during evolution. This implies that the identified folding cores may be part of an array of fundamental structural folding units.     

Entropy capacity determines protein folding

Search and study of the general principles that govern kinetics and thermodynamics of protein folding generate a new insight into the factors controlling this process. Here, based on the known experimental data and using theoretical modeling of protein folding, we demonstrate that there exists an optimal relationship between the average conformational entropy and the average energy of contacts per residue-that is, an entropy capacity-for fast protein folding. Statistical analysis of conformational entropy and number of contacts per residue for 5829 protein structures from four general structural classes (all-alpha, all-beta, alpha/beta, alpha+beta) demonstrates that each class of proteins has its own class-specific average number of contacts (class alpha/beta has the largest number of contacts) and average conformational entropy per residue (class all-alpha has the largest number of rotatable angles phi, psi, and chi per residue). These class-specific features determine the folding rates: alpha proteins are the fastest folding proteins, then follow beta and alpha+beta proteins, and finally alpha/beta proteins are the slowest ones. Our result is in agreement with the experimental folding rates for 60 proteins. This suggests that structural and sequence properties are important determinants of protein folding rates.     

RNA and protein folding: common themes and variations

Visualizing the navigation of an ensemble of unfolded molecules through the bumpy energy landscape in search of the native state gives a pictorial view of biomolecular folding. This picture, when combined with concepts in polymer theory, provides a unified theory of RNA and protein folding. Just as for proteins, the major folding free energy barrier for RNA scales sublinearly with the number of nucleotides, which allows us to extract the elusive prefactor for RNA folding. Several folding scenarios can be anticipated by considering variations in the energy landscape that depend on sequence, native topology, and external conditions. RNA and protein folding mechanism can be described by the kinetic partitioning mechanism (KPM) according to which a fraction (Phi) of molecules reaches the native state directly, whereas the remaining fraction gets kinetically trapped in metastable conformations. For two-state folders Phi approximately 1. Molecular chaperones are recruited to assist protein folding whenever Phi is small. We show that the iterative annealing mechanism, introduced to describe chaperonin-mediated folding, can be generalized to understand protein-assisted RNA folding. The major differences between the folding of proteins and RNA arise in the early stages of folding. For RNA, folding can only begin after the polyelectrolyte problem is solved, whereas protein collapse requires burial of hydrophobic residues. Cross-fertilization of ideas between the two fields should lead to an understanding of how RNA and proteins solve their folding problems.     

Optimal relationship between average conformational entropy and average energy of interactions between residues for fast protein folding

Based on the known experimental data and using the theoretical modeling of protein folding, we demonstrate that there exists an optimal relationship between the average conformational entropy and the average energy of contacts per residue, that is an entropy capacity, for fast protein folding. Statistical analysis of conformational entropy and the number of contacts per residue for 5829 protein structures from four general structural classes (all-alpha, all-beta, +/-/beta, alpha+beta) demonstrates that each class of proteins has its own class-specific average number of contacts and average conformational entropy per residue. These class-specific features determine the folding rates: a proteins are the fastest folding proteins, then follow beta and alpha+beta proteins, and finally alpha/beta proteins are the slowest ones.