Radius of gyration as an indicator of protein structure compactness

Identification and study of the main principles underlying the kinetics and thermodynamics of protein folding generate a new insight into the factors that control this process. Statistical analysis of the radius of gyration for 3769 protein domains of four major classes (α, β, α/β, and α + β) showed that each class has a characteristic radius of gyration that determines the protein structure compactness. For instance, α proteins have the highest radius of gyration throughout the protein size range considered, suggesting a less tight packing as compared with β-and (α + β)-proteins. The lowest radius of gyration and, accordingly, the tightest packing are characteristic of α/β-proteins. The protein radius of gyration normalized by the radius of gyration of a ball with the same volume is independent of the protein size, in contrast to compactness and the number of contacts per residue.     

Radius of gyration is indicator of compactness of protein structure

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. Statistical analysis of radii of gyration for 3769 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 radius of gyration, which determines compactness of protein structures: alpha-proteins have the largest radius of gyration. This indicates that they are less tightly packed than beta- and alpha + beta-proteins. Finally, alpha/beta-proteins are the most tightly packed proteins with the least radius of gyration. It should be underlined that radius of gyration normalized on the radius of gyration of ball with the same volume, is independent of the length in comparison with such parameters as compactness and number of contacts per residue.     

Protein folding rates estimated from contact predictions

Folding rates of small single-domain proteins that fold through simple two-state kinetics can be estimated from details of the three-dimensional protein structure. Previously, predictions of secondary structure had been exploited to predict folding rates from sequence. Here, we estimate two-state folding rates from predictions of internal residue-residue contacts in proteins of unknown structure. Our estimate is based on the correlation between the folding rate and the number of predicted long-range contacts normalized by the square of the protein length. It is well known that long-range order derived from known structures correlates with folding rates. The surprise was that estimates based on very noisy contact predictions were almost as accurate as the estimates based on known contacts. On average, our estimates were similar to those previously published from secondary structure predictions. The combination of these methods that exploit different sources of information improved performance. It appeared that the combined method reliably distinguished fast from slow two-state folders.     

Protein unfolded states populated at high and ambient pressure are similarly compact

The relationship between the dimensions of pressure-unfolded states of proteins compared with those at ambient pressure is controversial; resolving this issue is related directly to the mechanisms of pressure denaturation. Moreover, a significant pressure dependence of the compactness of unfolded states would complicate the interpretation of folding parameters from pressure perturbation and make comparison to those obtained using alternative perturbation approaches difficult. Here, we determined the compactness of the pressure-unfolded state of a small, cooperatively folding model protein, CTL9-I98A, as a function of temperature. This protein undergoes both thermal unfolding and cold denaturation, and the temperature dependence of the compactness at atmospheric pressure is known. High-pressure small angle x-ray scattering studies, yielding the radius of gyration and high-pressure diffusion ordered spectroscopy NMR experiments, yielding the hydrodynamic radius were carried out as a function of temperature at 250 MPa, a pressure at which the protein is unfolded. The radius of gyration values obtained at any given temperature at 250 MPa were similar to those reported previously at ambient pressure, and the trends with temperature are similar as well, although the pressure-unfolded state appears to undergo more pronounced expansion at high temperature than the unfolded state at atmospheric pressure. At 250 MPa, the compaction of the unfolded chain was maximal between 25 and 30°C, and the chain expanded upon both cooling and heating. These results reveal that the pressure-unfolded state of this protein is very similar to that observed at ambient pressure, demonstrating that pressure perturbation represents a powerful approach for observing the unfolded states of proteins under otherwise near-native conditions.     

Compactness Determines the Success of Cube and Octahedron Self-Assembly

Nature utilizes self-assembly to fabricate structures on length scales ranging from the atomic to the macro scale. Self-assembly has emerged as a paradigm in engineering that enables the highly parallel fabrication of complex, and often three-dimensional, structures from basic building blocks. Although there have been several demonstrations of this self-assembly fabrication process, rules that govern a priori design, yield and defect tolerance remain unknown. In this paper, we have designed the first model experimental system for systematically analyzing the influence of geometry on the self-assembly of 200 and 500 µm cubes and octahedra from tethered, multi-component, two-dimensional (2D) nets. We examined the self-assembly of all eleven 2D nets that can fold into cubes and octahedra, and we observed striking correlations between the compactness of the nets and the success of the assembly. Two measures of compactness were used for the nets: the number of vertex or topological connections and the radius of gyration. The success of the self-assembly process was determined by measuring the yield and classifying the defects. Our observation of increased self-assembly success with decreased radius of gyration and increased topological connectivity resembles theoretical models that describe the role of compactness in protein folding. Because of the differences in size and scale between our system and the protein folding system, we postulate that this hypothesis may be more universal to self-assembling systems in general. Apart from being intellectually intriguing, the findings could enable the assembly of more complicated polyhedral structures (e.g. dodecahedra) by allowing a priori selection of a net that might self-assemble with high yields.     

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.     

BCL::Fold - De Novo Prediction of Complex and Large Protein Topologies by Assembly of Secondary Structure Elements

Computational de novo protein structure prediction is limited to small proteins of simple topology. The present work explores an approach to extend beyond the current limitations through assembling protein topologies from idealized α-helices and β-strands. The algorithm performs a Monte Carlo Metropolis simulated annealing folding simulation. It optimizes a knowledge-based potential that analyzes radius of gyration, β-strand pairing, secondary structure element (SSE) packing, amino acid pair distance, amino acid environment, contact order, secondary structure prediction agreement and loop closure. Discontinuation of the protein chain favors sampling of non-local contacts and thereby creation of complex protein topologies. The folding simulation is accelerated through exclusion of flexible loop regions further reducing the size of the conformational search space. The algorithm is benchmarked on 66 proteins with lengths between 83 and 293 amino acids. For 61 out of these proteins, the best SSE-only models obtained have an RMSD100 below 8.0 Å and recover more than 20% of the native contacts. The algorithm assembles protein topologies with up to 215 residues and a relative contact order of 0.46. The method is tailored to be used in conjunction with low-resolution or sparse experimental data sets which often provide restraints for regions of defined secondary structure.     

Microsecond hydrophobic collapse in the folding of Escherichia coli dihydrofolate reductase, an alpha/beta-type protein

Using small-angle X-ray scattering combined with a continuous-flow mixing device, we monitored the microsecond compaction dynamics in the folding of Escherichia coli dihydrofolate reductase, an alpha/beta-type protein. A significant collapse of the radius of gyration from 30 A to 23.2 A occurs within 300 micros after the initiation of refolding by a urea dilution jump. The subsequent folding after the major chain collapse occurs on a considerably longer time-scale. The protein folding trajectories constructed by comparing the development of the compactness and the secondary structure suggest that the specific hydrophobic collapse model rather than the framework model better explains the experimental observations. The folding trajectory of this alpha/beta-type protein is located between those of alpha-helical and beta-sheet proteins, suggesting that native structure determines the folding landscape.     

Folding rate prediction using n-order contact distance for proteins with two- and three-state folding kinetics

It is a challenging task to understand the relationship between sequences and folding rates of proteins. Previous studies are found that one of contact order (CO), long-range order (LRO), total contact distance (TCD), chain topology parameter (CTP), and effective length (Leff) has a significant correlation with folding rate of proteins. In this paper, we introduce a new parameter called n-order contact distance (nOCD) and use it to predict folding rate of proteins with two- and three-state folding kinetics. A good linear correlation between the folding rate logarithm lnkf and nOCD with n=1.2, alpha=0.6 is found for two-state folders (correlation coefficient is -0.809, P-value<0.0001) and n=2.8, alpha=1.5 for three-state folders (correlation coefficient is -0.816, P-value<0.0001). However, this correlation is completely absent for three-state folders with n=1.2, alpha=0.6 (correlation coefficient is 0.0943, P-value=0.661) and for two-state folders with n=2.8, alpha=1.5 (correlation coefficient is -0.235, P-value=0.2116). We also find that the average number of contacts per residue Pm in the interval of m for two-state folders is smaller than that for three-state folders. The probability distribution P(gamma) of residue having gamma pairs of contacts fits a Gaussian distribution for both two- and three-state folders. We observe that the correlations between square radius of gyration S2 and number of residues for two- and three-state folders are both good, and the correlation coefficient is 0.908 and 0.901, and the slope of the fitting line is 1.202 and 0.795, respectively. Maybe three-state folders are more compact than two-state folders. Comparisons with nTCD and nCTP are also made, and it is found that nOCD is the best one in folding rate prediction.     

Protein folding pathways and kinetics: molecular dynamics simulations of beta-strand motifs

The folding pathways and the kinetic properties for three different types of off-lattice four-strand antiparallel beta-strand protein models interacting via a hybrid Go-type potential have been investigated using discontinuous molecular dynamics simulations. The kinetic study of protein folding was conducted by temperature quenching from a denatured or random coil state to a native state. The progress parameters used in the kinetic study include the squared radius of gyration R(2)(g), the fraction of native contacts within the protein as a whole Q, and between specific strands Q(ab). In the time series of folding, the denatured proteins undergo a conformational change toward the native state. The model proteins exhibit a variety of kinetic folding pathways that include a fast-track folding pathway without passing through an intermediate and multiple pathways with trapping into more than one intermediate. The kinetic folding behavior of the beta-strand proteins strongly depends on the native-state geometry of the model proteins and the size of the bias gap g, an artificial measure of a model protein's preference for its native state.     

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。     

蛋白质折叠速率预测研究进展

蛋白质折叠速率预测是当今生物物理学最具挑战性的课题之一。近年来,该领域的研究取得了很大的进展,提出了许多经验参数,例如:接触序、长程序、总接触距离、链拓扑参数、二级结构含量、有效长度、螺旋参数、n-阶接触距离等。这些参数都和蛋白质的折叠速率有很好的相关性,基于这些参数的各种预测方法所得到的预测结果也与实验数据较好地吻合。     

About the structural role of disulfide bridges in serum albumins: evidence from protein simulated unfolding

The role of the 17 disulfide (S-S) bridges in preserving the native conformation of human serum albumin (HSA) is investigated by performing classical molecular dynamics (MD) simulations on protein structures with intact and, respectively, reduced S-S bridges. The thermal unfolding simulations predict a clear destabilization of the protein secondary structure upon reduction of the S-S bridges as well as a significant distortion of the tertiary structure that is revealed by the changes in the protein native contacts fraction. The effect of the S-S bridges reduction on the protein compactness was tested by calculating Gibbs free energy profiles with respect to the protein gyration radius. The theoretical results obtained using the OPLS-AA and the AMBER ff03 force fields are in agreement with the available experimental data. Beyond the validation of the simulation method, the results here reported provide new insights into the mechanism of the protein reductive/oxidative unfolding/folding processes. It is predicted that in the native conformation of the protein, the thiol (-SH) groups belonging to the same reduced S-S bridge are located in potential wells that maintain them in contact. The -SH pairs can be dispatched by specific conformational transitions of the peptide chain located in the neighborhood of the cysteine residues.     

Protein unfolding rates correlate as strongly as folding rates with native structure

Although the folding rates of proteins have been studied extensively, both experimentally and theoretically, and many native state topological parameters have been proposed to correlate with or predict these rates, unfolding rates have received much less attention. Moreover, unfolding rates have generally been thought either to not relate to native topology in the same manner as folding rates, perhaps depending on different topological parameters, or to be more difficult to predict. Using a dataset of 108 proteins including two-state and multistate folders, we find that both unfolding and folding rates correlate strongly, and comparably well, with well-established measures of native topology, the absolute contact order and the long range order, with correlation coefficient values of 0.75 or higher. In addition, compared to folding rates, the absolute values of unfolding rates vary more strongly with native topology, have a larger range of values, and correlate better with thermodynamic stability. Similar trends are observed for subsets of different protein structural classes. Taken together, these results suggest that choosing a scaffold for protein engineering may require a compromise between a simple topology that will fold sufficiently quickly but also unfold quickly, and a complex topology that will unfold slowly and hence have kinetic stability, but fold slowly. These observations, together with the established role of kinetic stability in determining resistance to thermal and chemical denaturation as well as proteases, have important implications for understanding fundamental aspects of protein unfolding and folding and for protein engineering and design.     

Protein aggregation determinants from a simplified model: cooperative folders resist aggregation

Two-chain aggregation simulations using minimalist models of proteins G, L, and mutants were used to investigate the fundamentals of protein aggregation. Mutations were selected to break up repeats of hydrophobic beads in the sequence while maintaining native topology and folding ability. Data are collected under conditions in which all chain types have similar folded populations and after equilibrating the separated chains to minimize competition between folding and aggregation. Folding cooperativity stands out as the best single-chain determinant under these conditions and for these simple models. It can be experimentally measured by the width of the unfolding transition during thermal denaturation and loosely related to population of intermediate-like states during folding. Additional measures of cooperativity and other properties such as radius of gyration fluctuations and patterning of hydrophobic residues are also examined. Initial contact system states with transition-state characteristics can be identified and are more expanded than average initial contact states. Two-chain minimalist model aggregates are considerably less structured than their native states and have minimal domain-swapping features.     

Contact order dependent protein folding rates: kinetic consequences of a cooperative interplay between favorable nonlocal interactions and local conformational preferences

Physical mechanisms underlying the empirical correlation between relative contact order (CO) and folding rate among naturally occurring small single-domain proteins are investigated by evaluating postulated interaction schemes for a set of three-dimensional 27mer lattice protein models with 97 different CO values. Many-body interactions are constructed such that contact energies become more favorable when short chain segments sequentially adjacent to the contacting residues adopt native-like conformations. At a given interaction strength, this scheme leads to folding rates that are logarithmically well correlated with CO (correlation coefficient r = 0.914) and span more than 2.5 orders of magnitude, whereas folding rates of the corresponding Gō models with additive contact energies have much less logarithmic correlation with CO and span only approximately one order of magnitude. The present protein chain models also exhibit calorimetric cooperativity and linear chevron plots similar to that observed experimentally for proteins with apparent simple two-state folding/unfolding kinetics. Thus, our findings suggest that CO-dependent folding rates of real proteins may arise partly from a significant positive coupling between nonlocal contact favorabilities and local conformational preferences.     

Compactness determines protein folding type

We have demonstrated here that protein compactness, which we define as the ratio of the accessible surface area of a protein to that of the ideal sphere of the same volume, is one of the factors determining the mechanism of protein folding. Proteins with multi-state kinetics, on average, are more compact (compactness is 1.49 +/- 0.02 for proteins within the size range of 101-151 amino acid residues) than proteins with two-state kinetics (compactness is 1.59 +/- 0.03 for proteins within the same size range of 101-151 amino acid residues). We have shown that compactness for homologous proteins can explain both the difference in folding rates and the difference in folding mechanisms.     

Dependence of Protein Folding Stability and Dynamics on the Density and Composition of Macromolecular Crowders

We investigate the effect of macromolecular crowding on protein folding, using purely repulsive crowding particles and a self-organizing polymer model of protein folding. We find that the variation in folding stability with crowder size for typical α-, β-, and α/β-proteins is well described by an adaptation of the scaled particle theory. The native state, the transition state, and the unfolded protein are treated as effective hard spheres, with the folded and transition state radii independent of the size and concentration of the crowders. Remarkably, we find that, as the effective unfolded state radius is very weakly dependent on the crowder concentration, it can also be approximated by a single size. The same model predicts the effect of crowding on the folding barrier and therefore refolding rates with no adjustable parameters. A simple extension of the scaled-particle theory model, assuming additivity, can also describe the behavior of mixtures of crowding particles.