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.     

Effects of confinement in chaperonin assisted protein folding: rate enhancement by decreasing the roughness of the folding energy landscape

Chaperonins, such as the GroE complex of the bacteria Escherichia coli, assist the folding of proteins under non-permissive folding conditions by providing a cavity in which the newly translated or translocated protein can be encapsulated. Whether the chaperonin cage plays a passive role in protecting the protein from aggregation, or an active role in accelerating folding rates, remains a matter of debate. Here, we investigate the role of confinement in chaperonin mediated folding through molecular dynamics simulations. We designed a substrate protein with an alpha/beta sandwich fold, a common structural motif found in GroE substrate proteins and confined it to a spherical hydrophilic cage which mimicked the interior of the GroEL/ES cavity. The thermodynamics and kinetics of folding were studied over a wide range of temperature and cage radii. Confinement was seen to significantly raise the collapse temperature, T(c), as a result of the associated entropy loss of the unfolded state. The folding temperature, T(f), on the other hand, remained unaffected by encapsulation, a consequence of the folding mechanism of this protein that involves an initial collapse to a compact misfolded state prior to rearranging to the native state. Folding rates were observed to be either accelerated or retarded compared to bulk folding rates, depending on the temperature of the simulation. Rate enhancements due to confinement were observed only at temperatures above the temperature T(m), which corresponds to the temperature at which the protein folds fastest. For this protein, T(m) lies above the folding temperature, T(f), implying that encapsulation alone will not lead to a rate enhancement under conditions where the native state is stable (T相似文献   

基于序列疏水值震荡的折叠速率预测

蛋白质折叠速率的正确预测对理解蛋白质的折叠机理非常重要。本文从伪氨基酸组成的方法出发,提出利用序列疏水值震荡的方法来提取蛋白质氨基酸的序列顺序信息,建立线性回归模型进行折叠速率预测。该方法不需要蛋白质的任何二级结构、三级结构信息或结构类信息,可直接从序列对蛋白质折叠速率进行预测。对含有62个蛋白质的数据集,经过Jack．knife交互检验验证,相关系数达到0．804,表示折叠速率预测值与实验值有很好的相关性,说明了氨基酸序列信息对蛋白质折叠速率影响重要。同其他方法相比,本文的方法具有计算简单,输入参数少等特点。     

Desolvation Barrier Effects Are a Likely Contributor to the Remarkable Diversity in the Folding Rates of Small Proteins

The variation in folding rate among single-domain natural proteins is tremendous, but common models with explicit representations of the protein chain are either demonstrably insufficient or unclear as to their capability for rationalizing the experimental diversity in folding rates. In view of the critical role of water exclusion in cooperative folding, we apply native-centric, coarse-grained chain modeling with elementary desolvation barriers to investigate solvation effects on folding rates. For a set of 13 proteins, folding rates simulated with desolvation barriers cover ∼ 4.6 orders of magnitude, spanning a range essentially identical to that observed experimentally. In contrast, folding rates simulated without desolvation barriers cover only ∼ 2.2 orders of magnitude. Following a Hammond-like trend, the folding transition-state ensemble (TSE) of a protein model with desolvation barriers generally has a higher average number of native contacts and is structurally more specific, that is, less diffused, than the TSE of the corresponding model without desolvation barriers. Folding is generally significantly slower in models with desolvation barriers because of their higher overall macroscopic folding barriers as well as slower conformational diffusion speeds in the TSE that are ≈ 1/50 times those in models without desolvation barriers. Nonetheless, the average root-mean-square deviation between the TSE and the native conformation is often similar in the two modeling approaches, a finding suggestive of a more robust structural requirement for the folding rate-limiting step. The increased folding rate diversity in models with desolvation barriers originates from the tendency of these microscopic barriers to cause more heightening of the overall macroscopic folding free-energy barriers for proteins with more nonlocal native contacts than those with fewer such contacts. Thus, the enhancement of folding cooperativity by solvation effects is seen as positively correlated with a protein's native topological complexity.     

Deconvoluting Protein (Un)folding Structural Ensembles Using X-Ray Scattering,Nuclear Magnetic Resonance Spectroscopy and Molecular Dynamics Simulation

The folding and unfolding of protein domains is an apparently cooperative process, but transient intermediates have been detected in some cases. Such (un)folding intermediates are challenging to investigate structurally as they are typically not long-lived and their role in the (un)folding reaction has often been questioned. One of the most well studied (un)folding pathways is that of Drosophila melanogaster Engrailed homeodomain (EnHD): this 61-residue protein forms a three helix bundle in the native state and folds via a helical intermediate. Here we used molecular dynamics simulations to derive sample conformations of EnHD in the native, intermediate, and unfolded states and selected the relevant structural clusters by comparing to small/wide angle X-ray scattering data at four different temperatures. The results are corroborated using residual dipolar couplings determined by NMR spectroscopy. Our results agree well with the previously proposed (un)folding pathway. However, they also suggest that the fully unfolded state is present at a low fraction throughout the investigated temperature interval, and that the (un)folding intermediate is highly populated at the thermal midpoint in line with the view that this intermediate can be regarded to be the denatured state under physiological conditions. Further, the combination of ensemble structural techniques with MD allows for determination of structures and populations of multiple interconverting structures in solution.     

The dynamical contact order: Protein folding rate parameters based on quantum conformational transitions

Protein folding is regarded as a quantum transition between the torsion states of a polypeptide chain. According to the quantum theory of conformational dynamics, we propose the dynamical contact order (DCO) defined as a characteristic of the contact described by the moment of inertia and the torsion potential energy of the polypeptide chain between contact residues. Consequently, the protein folding rate can be quantitatively studied from the point of view of dynamics. By comparing theoretical calculations and experimental data on the folding rate of 80 proteins, we successfully validate the view that protein folding is a quantum conformational transition. We conclude that (i) a correlation between the protein folding rate and the contact inertial moment exists; (ii) multi-state protein folding can be regarded as a quantum conformational transition similar to that of two-state proteins but with an intermediate delay. We have estimated the order of magnitude of the time delay; (iii) folding can be classified into two types, exergonic and endergonic. Most of the two-state proteins with higher folding rate are exergonic and most of the multi-state proteins with low folding rate are endergonic. The folding speed limit is determined by exergonic folding.     

Elastic properties of proteins: insight on the folding process and evolutionary selection of native structures

We carry out a theoretical study of the vibrational and relaxation properties of naturally occurring proteins with the purpose of characterizing both the folding and equilibrium thermodynamics. By means of a suitable model, we provide a full characterization of the spectrum and eigenmodes of vibration at various temperatures by merely exploiting the knowledge of the protein native structure. It is shown that the rate at which perturbations decay at the folding transition correlates well with experimental folding rates. This validation is carried out on a list of about 30 two-state folders. Furthermore, the qualitative analysis of residues mean square displacements (shown to reproduce crystallographic data accurately) provides a reliable and statistically accurate method to identify crucial folding sites/contacts. This novel strategy is validated against clinical data for human immunodeficiency virus type 1 (HIV-1) protease. Finally, we compare the spectra and eigenmodes of vibration of natural proteins against randomly generated compact structures and regular random graphs. The comparison reveals a distinctive enhanced flexibility of natural structures accompanied by slow relaxation times at the folding temperature. The fact that these properties are connected intimately to the presence and assembly of secondary motifs hints at the special criteria adopted by evolution in the selection of viable folds.     

Multiple folding mechanisms of protein ubiquitin

Based on the C(alpha) Go-type model, the folding kinetics and mechanisms of protein ubiquitin with mixed alpha/beta topology are studied by molecular dynamics simulations. The relaxation kinetics shows that there are three phases, namely the major phase, the intermediate phase and the slowest minor phase. The existence of these three phases are relevant to the phenomenon found in experiments. According to our simulations, the folding at high temperatures around the folding transition temperature T(f) is of a two-state process, and the folding nucleus is consisted of contacts between the front end of alpha-helix and the turn(4). The folding at low temperature (approximately T = 0.8) is also studied, where an A-state like structure is found lying on the major folding pathway. The appearance of this structure is related to the stability of the first part (residue 1-51) of protein ubiquitin. As the temperature decreases, the formation of secondary structures, tertiary structures and collapse of the protein are found to be decoupled gradually and the folding mechanism changes from the nucleation-condensation to the diffusion-collision. This feature indicates a unifying common folding mechanism for proteins. The intermediate phase is also studied and is found to represent a folding process via a long-lived intermediate state which is stabilized by strong interactions between the beta(1) and the beta(5) strand. These strong interactions are important for the function of protein ubiquitin as a molecular chaperone. Thus the intermediate phase is assumed as a byproduct of the requirement of protein function. In addition, the validity of the current Go-model is also investigated, and a lower limited temperature for protein ubiquitin T(limit) = 0.8 is proposed. At temperatures higher than this value, the kinetic traps due to glass dynamics cannot be significantly populated and the intermediate states can be reliably identified although there is slight chevron rollover in the folding rates. At temperature lower than T(limit), however, the traps due to glass dynamics become dominant and may be mistaken for real intermediate states. This limitation of valid temperature range prevents us to reveal the burst phase intermediate in the major folding phase since it might only be stabilized at temperatures lower than T(limit), according to experiments. Our works show that caution must be taken when studying low-temperature intermediate states by using the C(alpha) Go-models.     

Submillisecond folding of the peripheral subunit-binding domain.

Folding and unfolding rates have been measured for the peripheral subunit-binding domain, a small three-helix protein. The protein folds very fast, with rates too rapid to be measured using traditional stopped-flow techniques. Folding and unfolding rates were measured as a function of temperature using dynamic NMR lineshape analysis. At the lowest temperature at which there is sufficient broadening to measure rates, 41 degrees C, the folding rate is 16,050 s(-1). Thus, the halftime required for folding is 43 micros. At the same temperature, the unfolding rate is 2800 s(-1). Identical rates were measured using resolved resonances from Val16 in the loop and Val21 at the end of the 310-helix. Folding rates have been correlated with protein topology, and this correlation is consistent with the rapid folding of the peripheral subunit-binding domain. The results presented here show that the peripheral subunit-binding domain is the third fastest folding protein for which rates have been estimated. The folding rate is the fastest that has been directly measured and provides further support for the importance of chain topology as a major determinant of folding rates.     

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.     

Nucleation‐based prediction of the protein folding rate and its correlation with the folding nucleus size

The problem of protein self‐organization is in the focus of current molecular biology studies. Although the general principles are understood, many details remain unclear. Specifically, protein folding rates are of interest because they dictate the rate of protein aggregation which underlies many human diseases. Here we offer predictions of protein folding rates and their correlation with folding nucleus sizes. We calculated free energies of the transition state and sizes of folding nuclei for 84 proteins and peptides whose other parameters were measured at the point of thermodynamic equilibrium between their unfolded and native states. We used the dynamic programming method where each residue was considered to be either as folded as in its native state or completely disordered. The calculated and measured folding rates showed a good correlation at the temperature mid‐transition point (the correlation coefficient was 0.75). Also, we pioneered in demonstrating a moderate (‐0.57) correlation coefficient between the calculated sizes of folding nuclei and the folding rates. Predictions made by different methods were compared. The established good correlation between the estimated free energy barrier and the experimentally found folding rate of each studied protein/peptide indicates that our model gives reliable results for the considered data set. Proteins 2012; © 2012 Wiley Periodicals, Inc.     

Equilibrium collapse and the kinetic 'foldability' of proteins.

An important element of protein folding theory has been the identification of equilibrium parameters that might uniquely distinguish rapidly folding polypeptide sequences from those that fold slowly. One such parameter, termed sigma, is a dimensionless, equilibrium measure of the coincidence of chain compaction and folding that is predicted to be an important determinant of relative folding kinetics. To test this prediction and improve our understanding of the putative relationship between nonspecific compaction of the unfolded state and protein folding kinetics, we have used small-angle X-ray scattering and circular dichroism spectroscopy to measure the sigma of five well-characterized proteins. Consistent with theoretical predictions, we find that near-perfect coincidence of the unfolded state contraction and folding (sigma approximately 0) is associated with the rapid kinetics of these naturally occurring proteins. We do not, however, observe any significant correlation between sigma and either the relative folding rates of these proteins or the presence or absence of well-populated kinetic intermediates. Thus, while sigma approximately 0 may be a necessary condition to ensure rapid folding, differences in sigma do not account for the wide range of rates and mechanisms with which naturally occurring proteins fold.     

Machine Learning: How Much Does It Tell about Protein Folding Rates?

The prediction of protein folding rates is a necessary step towards understanding the principles of protein folding. Due to the increasing amount of experimental data, numerous protein folding models and predictors of protein folding rates have been developed in the last decade. The problem has also attracted the attention of scientists from computational fields, which led to the publication of several machine learning-based models to predict the rate of protein folding. Some of them claim to predict the logarithm of protein folding rate with an accuracy greater than 90%. However, there are reasons to believe that such claims are exaggerated due to large fluctuations and overfitting of the estimates. When we confronted three selected published models with new data, we found a much lower predictive power than reported in the original publications. Overly optimistic predictive powers appear from violations of the basic principles of machine-learning. We highlight common misconceptions in the studies claiming excessive predictive power and propose to use learning curves as a safeguard against those mistakes. As an example, we show that the current amount of experimental data is insufficient to build a linear predictor of logarithms of folding rates based on protein amino acid composition.     

An approach to quality management in structural biology: biophysical selection of proteins for successful crystallization

Aggregation, incorrect folding and low stability are common obstacles for protein structure determination, and are often discovered at a very late state of protein production. In many cases, however, the reasons for failure to obtain diffracting crystals remain entirely unknown. We report on the contribution of systematic biophysical characterization to the success in structural determination of human proteins of unknown fold. Routine analysis using dynamic light scattering (DLS), differential scanning calorimetry (DSC) and Fourier-transform infrared spectroscopy (FTIR) was employed to evaluate fold and stability of 263 purified protein samples (98 different human proteins). We found that FTIR-monitored temperature scanning may be used to detect incorrect folding and discovered a positive correlation between unfolding enthalpy measured with DSC and the size of small, globular proteins that may be used to estimate the quality of protein preparations. Furthermore, our work establishes that the risk of aggregation during concentration of proteins may be reduced through DLS monitoring. In summary, our study demonstrates that biophysical characterization provides an ideal tool to facilitate quality management for structural biology and many other areas of biological research.     

蛋白质折叠速率与其氨基酸序列极性的关系

以大肠杆菌60个蛋白酶以及几种常见病毒（SARS病毒、艾滋病病毒、丙型肝炎病毒及乙型肝炎病毒）各蛋白质序列中的所有α-螺旋和β-折叠片段为研究对象,计算了各片段的折叠速率和平均极性,分别在各物种的α-螺旋和β-折叠两类二级结构片段中分析了二者的相关性。得到结论：不论是大肠杆菌中的蛋白酶还是病毒蛋白,其中的两类氨基酸片段的平均极性与折叠速率都是极显著相关的：对于所有的α片段,二者呈线性正相关,而对于所有的β片段,二者成线性负相关。结果证实了在蛋白质折叠中,氨基酸的极性起着重要的作用。     

Reduced alphabet for protein folding prediction

What are the key building blocks that would have been needed to construct complex protein folds? This is an important issue for understanding protein folding mechanism and guiding de novo protein design. Twenty naturally occurring amino acids and eight secondary structures consist of a 28‐letter alphabet to determine folding kinetics and mechanism. Here we predict folding kinetic rates of proteins from many reduced alphabets. We find that a reduced alphabet of 10 letters achieves good correlation with folding rates, close to the one achieved by full 28‐letter alphabet. Many other reduced alphabets are not significantly correlated to folding rates. The finding suggests that not all amino acids and secondary structures are equally important for protein folding. The foldable sequence of a protein could be designed using at least 10 folding units, which can either promote or inhibit protein folding. Reducing alphabet cardinality without losing key folding kinetic information opens the door to potentially faster machine learning and data mining applications in protein structure prediction, sequence alignment and protein design. Proteins 2015; 83:631–639. © 2015 Wiley Periodicals, Inc.     

Probing the folding and unfolding dynamics of secondary and tertiary structures in a three-helix bundle protein

Fast relaxation kinetics studies of the B-domain of staphylococcal protein A were performed to characterize the folding and unfolding of this small three-helix bundle protein. The relaxation kinetics were initiated using a laser-induced temperature jump and probed using time-resolved infrared spectroscopy. The kinetics monitored within the amide I' absorbance of the polypeptide backbone exhibit two distinct kinetics phases with nanosecond and microsecond relaxation times. The fast kinetics relaxation time is close to the diffusion limits placed on protein folding reactions. The fast kinetics phase is dominated by the relaxation of the solvated helix (nu = 1632 cm(-1)), which reports on the fast relaxation of the individual helices. The slow kinetics phase follows the cooperative relaxation of the native helical bundle core that is monitored by both solvated (nu = 1632 cm(-1)) and buried helical IR bands (nu = 1652 cm(-1)). The folding rates of the slow kinetics phase calculated over an extended temperature range indicate that the core formation of this protein follows a pathway that is energetically downhill. The unfolding rates are much more strongly temperature-dependent indicating an activated process with a large energy barrier. These results provide significant insight into the primary process of protein folding and suggest that fast formation of helices can drive the folding of helical proteins.     

Ultrafast light harvesting dynamics in the cryptophyte phycocyanin 645.

Steady-state and femtosecond time-resolved optical methods have been used to study spectroscopic features and energy transfer dynamics in the soluble antenna protein phycocyanin 645 (PC645), isolated from a unicellular cryptophyte Chroomonas CCMP270. Absorption, emission and polarization measurements as well as one-colour pump-probe traces are reported in combination with complementary quantum chemical calculations of electronic transitions of the bilins. Estimation of bilin spectral positions and energy transfer rates aids in the development of a model for light harvesting by PC645. At higher photon energies light is absorbed by the centrally located dimer (DBV, beta50/beta61) and the excitation is subsequently funneled through a complex interference of pathways to four peripheral pigments (MBV alpha19, PCB beta158). Those chromophores transfer the excitation energy to the red-most bilins (PCB beta82). We suggest that the final resonance energy transfer step occurs between the PCB 82 bilins on a timescale estimated to be approximately 15 ps. Such a rapid final energy transfer step cannot be rationalized by calculations that combine experimental parameters and quantum chemical calculations, which predict the energy transfer time to be 40 ps.     

Characterization of the folding degree of proteins

MOTIVATION: The characterization of the folding degree of chains is central to the elucidation of structure--function relationships in proteins. Here we present a new index for characterizing the folding degree of a (protein) chain. This index shows a range of features that are desirable for the study of the relation between structure and function in proteins. RESULTS: A novel index characterizing the folding degree of (protein) chains is developed based on the spectral moments of a matrix representing the dihedral angles (phi, omega and epsilon) of the protein main chain. The proposed index is normalized to the chain size, is not correlated to the gyration radius of the backbone chain and is able to distinguish between structures for which the sum of the main-chain dihedral angles is identical. The index is well correlated to the percentages of helix and strand in proteins, shows a linear dependence with temperature changes, and is able to differentiate among protein families. AVAILABILITY: On request from the author.     

Double mutant MBP refolds at same rate in free solution as inside the GroEL/GroES chaperonin chamber when aggregation in free solution is prevented

Under "permissive" conditions at 25°C, the chaperonin substrate protein DM-MBP refolds 5-10 times more rapidly in the GroEL/GroES folding chamber than in free solution. This has been suggested to indicate that the chaperonin accelerates polypeptide folding by entropic effects of close confinement. Here, using native-purified DM-MBP, we show that the different rates of refolding are due to reversible aggregation of DM-MBP while folding free in solution, slowing its kinetics of renaturation: the protein exhibited concentration-dependent refolding in solution, with aggregation directly observed by dynamic light scattering. When refolded in chloride-free buffer, however, dynamic light scattering was eliminated, refolding became concentration-independent, and the rate of refolding became the same as that in GroEL/GroES. The GroEL/GroES chamber thus appears to function passively toward DM-MBP.