Outlining folding nuclei in globular proteins

Our theoretical approach for prediction of folding/unfolding nuclei in three-dimensional protein structures is based on a search for free energy saddle points on networks of protein unfolding pathways. Under some approximations, this search is performed rapidly by dynamic programming and results in prediction of Phi values, which can be compared with those found experimentally. In this study, we optimize some details of the model (specifically, hydrogen atoms are taken into account in addition to heavy atoms), and compare the theoretically obtained and experimental Phi values (which characterize involvement of residues in folding nuclei) for all 17 proteins, where Phi values are now known for many residues. We show that the model provides good Phi value predictions for proteins whose structures have been determined by X-ray analysis (the average correlation coefficient is 0.65), with a more limited success for proteins whose structures have been determined by NMR techniques only (the average correlation coefficient is 0.34), and that the transition state free energies computed from the same model are in a good anticorrelation with logarithms of experimentally measured folding rates at mid-transition (the correlation coefficient is -0.73).     

Size and folding in globular proteins

We have modeled protein folding by packing a unified length of regular structural elements (alpha-helices and beta-sheets) into a 'cube'. In a globular protein with m alpha-helices and n beta-strands, this unified length is expressed in units of heptapeptides in alpha-helices, and in units of tripeptides in beta-strands. Calculations using published data show that a 4-helix bundle (m = 4, n = 0) has at least 2 x 2 x 2 helical heptapeptides; the 16-strand beta-barrel of porin (m = 0, n = 16) is at most 4 x 4 x 4 tripeptides in beta-strands. Compact, recurring protein modules with mixed helices and beta-strands are the ones that actually acquire a geometrically quasi-spherical, or cubic, shape.     

Theory for the folding and stability of globular proteins

Using lattice statistical mechanics, we develop theory to account for the folding of a heteropolymer molecule such as a protein to the globular and soluble state. Folding is assumed to be driven by the association of solvophobic monomers to avoid solvent and opposed by the chain configurational entropy. Theory predicts a phase transition as a function of temperature or solvent character. Molecules that are too short or too long or that have too few solvophobic residues are predicted not to fold. Globular molecules should have a largely solvophobic core, but there is an entropic tendency for some residues to be "out of place", particularly in small molecules. For long chains, molecules comprised of globular domains are predicted to be thermodynamically more stable than spherical molecules. The number of accessible conformations in the globular state is calculated to be an exceedingly small fraction of the number available to the random coil. Previous estimates of this number, which have motivated kinetic theories of folding, err by many tens of orders of magnitude.     

Denaturants can accelerate folding rates in a class of globular proteins.

We present a lattice Monte Carlo study to examine the effect of denaturants on the folding rates of simplified models of proteins. The two-dimensional model is made from a three-letter code mimicking the presence of hydrophobic, hydrophilic, and cysteine residues. We show that the rate of folding is maximum when the effective hydrophobic interaction epsilon H is approximately equal to the free energy gain epsilon S upon forming disulfide bonds. In the range 1 < or = epsilon H/ epsilon S < or = 3, multiple paths that connect several intermediates to the native state lead to fast folding. It is shown that at a fixed temperature and epsilon S the folding rate increases as epsilon H decreases. An approximate model is used to show that epsilon H should decrease as a function of the concentration of denaturants such as urea or guanidine hydrochloride. Our simulation results, in conjunction with this model, are used to show that increasing the concentration of denaturants can lead to an increase in folding rates. This occurs because denaturants can destabilize the intermediates without significantly altering the energy of the native conformation. Our findings are compared with experiments on the effects of denaturants on the refolding of bovine pancreatic trypsin inhibitor and ribonuclease T1. We also argue that the phenomenon of denaturant-enhanced folding of proteins should be general.     

Secondary structure prediction and folding of globular protein: refolding of ferredoxin

The physicochemical mechanism of protein folding has been elucidated by the island model, describing a growth type of folding. The folding pathway is closely related with nucleation on the polypeptide chain and thus the formation of small local structures or secondary structures at the earliest stage of folding is essential to all following steps. The island model is applicable to any protein, but a high precision of secondary structure prediction is indispensable to folding simulation. The secondary structures formed at the earliest stage of folding are supposed to be of standard form, but they are usually deformed during the folding process, especially at the last stage, although the degree of deformation is different for each protein. Ferredoxin is an example of a protein having this property. According to X-ray investigation (1FDX), ferredoxin is not supposed to have secondary structures. However, if we assumed that in ferredoxin all the residues are in a coil state, we could not attain the correct structure similar to the native one. Further, we found that some parts of the chain are not flexible, suggesting the presence of secondary structures, in agreement with the recent PDB data (1DUR). Assuming standard secondary structures (-helices and -strands) at the nonflexible parts at the early stage of folding, and deforming these at the final stage, a structure similar to the native one was obtained. Another peculiarity of ferredoxin is the absence of disulfide bonds, in spite of its having eight cysteines. The reason cysteines do not form disulfide bonds became clear by applying the lampshade criterion, but more importantly, the two groups of cysteines are ready to make iron complexes, respectively, at a rather later stage of folding. The reason for poor prediction accuracy of secondary structure with conventional methods is discussed.     

Contact potential that recognizes the correct folding of globular proteins.

We have devised a continuous function of interresidue contacts in globular proteins such that the X-ray crystal structure has a lower function value than that of thousands of protein-like alternative conformations. Although we fit the adjustable parameters of the potential using only 10,000 alternative structures for a selected training set of 37 proteins, a grand total of 530,000 constraints was satisfied, derived from 73 proteins and their numerous alternative conformations. In every case where the native conformation is adequately globular and compact, according to objective criteria we have developed, the potential function always favors the native over all alternatives by a substantial margin. This is true even for an additional three proteins never used in any way in the fitting procedure. Conformations differing only slightly from the native, such as those coming from crystal structures of the same protein complexed with different ligands or from crystal structures of point mutants, have function values very similar to the native's and always less than those of alternatives derived from substantially different crystal structures. This holds for all 95 structures that are homologous to one or another of various proteins we used. Realizing that this potential should be useful for modeling the conformation of new protein sequences from the body of protein crystal structures, we suggest a test for deciding whether a nearly correct approximation to the native conformation has been found.     

Influence of medium- and long-range interactions in different folding types of globular proteins

Recognition of protein fold from amino acid sequence is a challenging task. The structure and stability of proteins from different fold are mainly dictated by inter-residue interactions. In our earlier work, we have successfully used the medium- and long-range contacts for predicting the protein folding rates, discriminating globular and membrane proteins and for distinguishing protein structural classes. In this work, we analyze the role of inter-residue interactions in commonly occurring folds of globular proteins in order to understand their folding mechanisms. In the medium-range contacts, the globin fold and four-helical bundle proteins have more contacts than that of DNA-RNA fold although they all belong to all-alpha class. In long-range contacts, only the ribonuclease fold prefers 4-10 range and the other folding types prefer the range 21-30 in alpha/beta class proteins. Further, the preferred residues and residue pairs influenced by these different folds are discussed. The information about the preference of medium- and long-range contacts exhibited by the 20 amino acid residues can be effectively used to predict the folding type of each protein.     

Loop folds in proteins and evolutionary conservation of folding nuclei

We show that loops of close contacts involving hydrophobic residues are important in protein folding. Contrary to Berezovsky Berezovsky and Trifonov (J Biomol Struct Dyn 20, 5-6, 2002) the loops important in protein folding usually are much larger in size than 23-31 residues, being instead comparable to the size of the protein for single domain proteins. Additionally what is important are not single loop contacts, but a highly interconnected network of such loop contacts, which provides extra stability to a protein fold and which leads to their conservation in evolution.     

On the environment of ionizable groups in globular proteins

The environment of ionizable groups in 36 proteins is characterized in terms of solvent-accessibility, salt-bridge formation and hydrogen-bonding. Possible implications of our results as to the protonation state of buried ionizable groups are considered and patterns useful for model building studies on proteins are derived. The most interesting finding is that there are on average two completely buried ionizable groups per protein of which at least 20% do not form saltbridges. However, all buried ionizable groups form hydrogen bonds with neutral polar groups.     

On the thermal unfolding character of globular proteins

A theoretical model is presented to study the stepwise thermal unfolding of globular proteins using the stabilizing/destabilizing characters of amino acid residues in protein crystals. A multiple regression relation connecting the melting temperature and the amounts of stabilizing and destabilizing groups of residues in a protein, when used for the thermal behavior of peptide segments, provides reliable results on the stepwise unfolding nature of the protein. In ribonuclease A, the shell residues 16–22 are predicted to unfold earlier in the temperature range 30–45°C; the -sheet structures undergo thermal denaturation as a single cooperative unit and there is evidence indicating the segment 106–118 as a nucleation site. In ribonuclease S, the S-peptide unfolds earlier than S-protein. The predicted average and the range of melting temperatures, and the folding pathways of a set of globular proteins, agree very well with the experimental results. The results obtained in the present study indicate that (i) most of the nucleation parts possess high relative thermal stability, (ii) the unfolded state retains some residual structure, and (iii) some segments undergo gradual and overlapping thermal denaturation.     

Analysis and prediction of inter-strand packing distances between beta-sheets of globular proteins

Any two beta-strands belonging to two different beta-sheets in a protein structure are considered to pack interactively if each beta-strand has at least one residue that undergoes a loss of one tenth or more of its solvent contact surface area upon packing. A data set of protein 3-D structures (determined at 2.5 A resolution or better), corresponding to 428 protein chains, contains 1986 non-identical pairs of beta-strands involved in interactive packing. The inter-axial distance between these is significantly correlated to the weighted sum of the volumes of the interacting residues at the packing interface. This correlation can be used to predict the changes in the inter-sheet distances in equivalent beta-sheets in homologous proteins and, therefore, is of value in comparative modelling of proteins.     

Noninteracting local-structure model of folding and unfolding transition in globular proteins. II. Application to two-dimensional lattice proteins

The noninteracting local-structure model of the folding and unfolding transition in globular proteins, the formulation of which was given in the preceding paper, is applied to the analysis of the two-dimensional lattice model of proteins. The lattice model of proteins is a theoretical tool designed to study the statistical-mechanical aspect of the folding and unfolding transition. Its dynamics have been studied by a method of Monte Carlo simulation. The noninteracting local-structure model reproduces the equilibrium properties of the lattice model obtained previously by computer simulation remarkably well, when the specificity of the long-range interactions is strong. This observation indicates that the basic assumption of the noninteracting local-structure model is equivalent to the assumption of strong specificity of intramolecular interactions. It is argued that by assuming this strong specificity, we can emphasize the correct main paths of folding and unfolding transition. The way local structures grow and/or merge along the most probable path of folding in the lattice model is discussed by the noninteracting local-structure model.     

Topology of globular proteins

This paper inquires whether it is reasonable to expect the native structure of proteins to be “knotted”. To this end, some topological properties of polypeptides containing disulfide bridges are discussed using notions from mathematical knot theory and graph theory. The probability of occurrence of knots in random cyclic polymers is calculated as a function of chain length by elementary Monte Carlo methods. The implications of this for protein renaturation and for determining the tertiary structure of proteins are discussed.     

In silico chaperonin-like cycle helps folding of proteins for structure prediction

Currently, one of the most serious problems in protein-folding simulations for de novo structure prediction is conformational sampling of medium-to-large proteins. In vivo, folding of these proteins is mediated by molecular chaperones. Inspired by the functions of chaperonins, we designed a simple chaperonin-like simulation protocol within the framework of the standard fragment assembly method: in our protocol, the strength of the hydrophobic interaction is periodically modulated to help the protein escape from misfolded structures. We tested this protocol for 38 proteins and found that, using a certain defined criterion of success, our method could successfully predict the native structures of 14 targets, whereas only those of 10 targets were successfully predicted using the standard protocol. In particular, for non-α-helical proteins, our method yielded significantly better predictions than the standard approach. This chaperonin-inspired protocol that enhanced de novo structure prediction using folding simulations may, in turn, provide new insights into the working principles underlying the chaperonin system.     

Different circular permutations produced different folding nuclei in proteins: a computational study

There have been many studies about the effect of circular permutation on the transition state/folding nucleus of proteins, with sometimes conflicting conclusions from different proteins and permutations. To clarify this important issue, we have studied two circular permutations of a lattice protein model with side-chains. Both permuted sequences have essentially the same native state as the original (wild-type) sequence. Circular permutant 1 cuts at the folding nucleus of the wild-type sequence. As a result, the permutant has a drastically different nucleus and folds more slowly than wild-type. In contrast, circular permutant 2 involves an incision at a site unstructured in the wild-type transition state, and the wild-type nucleus is largely retained in the permutant. In addition, permutant 2 displays both two-state and multi-state folding, with a native-like intermediate state occasionally populated. Neither the wild-type nor permutant 1 has a similar intermediate, and both fold in an apparently two-state manner. Surprisingly, permutant 2 folds at a rate identical with that of the wild-type. The intermediate in permutant 2 is stabilised by native and non-native interactions, and cannot be classified simply as on or off-pathway. So we advise caution in attributing experimental data to on or off-pathway intermediates. Finally, our work illuminates the results on alpha-spectrin SH3, chymotrypsin inhibitor 2 and beta-lactoglobulin, and supports a key assumption in the experimental efforts to locate potential nucleation sites of real proteins via circular permutations.     

Disulphide bridges in globular proteins

Disulphide bridges in proteins of known sequence, connectivity and structure were studied to search for common features. Their distribution, topology, conformation and conservation were analysed in detail. Several general patterns emerge which to some extent dictate disulphide bridge formation. For example, there is a strong preference for shorter connections, with half-cystines separated by less than 24 residues in 49% of all disulphides. Right- and left-handed disulphides occur equally; the left-handed structures adopt one predominant conformation (symmetric χ₁ = ?60 °, χ₂ = ?80 °, χ₃ = t-90 °). Cystines are generally very well conserved, in contrast to cysteines, with a free —SH group, which mutate rapidly. If a disulphide is not conserved, both cystines are mutated. The role of disulphide bridges in globular proteins is discussed.