Monte Carlo Simulations of Proteins in Cages: Influence of Confinement on the Stability of Intermediate States

We study the folding of small proteins inside confining potentials. Proteins are described using an effective potential model that contains the Ramachandran angles as degrees of freedom and does not need any a priori information about the native state. Hydrogen bonds, dipole-dipole-, and hydrophobic interactions are taken explicitly into account. An interesting feature displayed by this potential is the presence of metastable intermediates between the unfolded and native states. We consider different types of confining potentials to describe proteins folding inside cages with repulsive or attractive walls. Using the Wang-Landau algorithm, we determine the density of states and analyze in detail the thermodynamical properties of the confined proteins for different sizes of the cages. We show that confinement dramatically reduces the phase space available to the protein and that the presence of intermediate states can be controlled by varying the properties of the confining potential. Cages with strongly attractive walls destabilize the intermediate states and lead to a two-state folding into a configuration that is less stable than the native structure. However, cages with slightly attractive walls enhance the stability of native structure and induce a folding process, which occurs through intermediate configurations.     

Rapid knot detection and application to protein structure prediction

MOTIVATION: Knots in polypeptide chains have been found in very few proteins, and consequently should be generally avoided in protein structure prediction methods. Most effective structure prediction methods do not model the protein folding process itself, but rather seek only to correctly obtain the final native state. Consequently, the mechanisms that prevent knots from occurring in native proteins are not relevant to the modeling process, and as a result, knots can occur with significantly higher frequency in protein models. Here we describe Knotfind, a simple algorithm for knot detection that is fast enough for structure prediction, where tens or hundreds of thousands of conformations may be sampled during the course of a prediction. We have used this algorithm to characterize knots in large populations of model structures generated for targets in CASP 5 and CASP 6 using the Rosetta homology-based modeling method. RESULTS: Analysis of CASP5 models suggested several possible avenues for introduction of knots into these models, and these insights were applied to structure prediction in CASP 6, resulting in a significant decrease in the proportion of knotted models generated. Additionally, using the knot detection algorithm on structures in the Protein Data Bank, a previously unreported deep trefoil knot was found in acetylornithine transcarbamylase. AVAILABILITY: The Knotfind algorithm is available in the Rosetta structure prediction program at http://www.rosettacommons.org.     

pH-induced equilibrium unfolding of apomyoglobin: Substitutions at conserved Trp14 and Met131 and non-conserved Val17 positions

A number of residues in globins family are well conserved but are not directly involved in the primary oxygen-carrying function of these proteins. A possible role for these conserved, non-functional residues has been suggested in promoting a rapid and correct folding process to the native tertiary structure. To test this hypothesis, we have studied pH-induced equilibrium unfolding of mutant apomyoglobins with substitutions of the conserved residues Trp14 and Met131, which are not involved in the function of myoglobin, by various amino acids. This allowed estimating their impact on the stability of various conformational states of the proteins and selecting conditions for a folding kinetics study. The results obtained from circular dichroism, tryptophan fluorescence, and differential scanning microcalorimetry for these mutant proteins were compared with those for the wild type protein and for a mutant with the non-conserved Val17 substituted by Ala. In the native folded state, all of the mutant apoproteins have a compact globular structure, but are destabilized in comparison to the wild type protein. The pH-induced denaturation of the mutant proteins occurs through the formation of a molten globule-like intermediate similar to that of the wild type protein. Thermodynamic parameters for all of the proteins were calculated using the three state model. Stability of equilibrium intermediates at pH ~4.0 was shown to be slightly affected by the mutations. Thus, all of the above substitutions influence the stability of the native state of these proteins. The cooperativity of conformational transitions and the exposed to solvent protein surface were also changed, but not for the substitution at Val17.     

Translocation boost protein-folding efficiency of double-barreled chaperonins

Incorrect folding of proteins in living cells may lead to malfunctioning of the cell machinery. To prevent such cellular disasters from happening, all cells contain molecular chaperones that assist nonnative proteins in folding into the correct native structure. One of the most studied chaperone complexes is the GroEL-GroES complex. The GroEL part has a "double-barrel" structure, which consists of two cylindrical chambers joined at the bottom in a symmetrical fashion. The hydrophobic rim of one of the GroEL chambers captures nonnative proteins. The GroES part acts as a lid that temporarily closes the filled chamber during the folding process. Several capture-folding-release cycles are required before the nonnative protein reaches its native state. Here we report molecular simulations that suggest that translocation of the nonnative protein through the equatorial plane of the complex boosts the efficiency of the chaperonin action. If the target protein is correctly folded after translocation, it is released. However, if it is still nonnative, it is likely to remain trapped in the second chamber, which then closes to start a reverse translocation process. This shuttling back and forth continues until the protein is correctly folded. Our model provides a natural explanation for the prevalence of double-barreled chaperonins. Moreover, we argue that internal folding is both more efficient and safer than a scenario where partially refolded proteins escape from the complex before being recaptured.     

Molecular basis of cooperativity in protein folding IV. Core: A general cooperative folding model

The cooperative nature of the protein folding process is independent of the characteristic fold and the specific secondary structure attributes of a globular protein. A general folding/unfolding model should, therefore, be based upon structural features that transcend the peculiarities of α-helices, β-sheets, and other structural motifs found in proteins. The studies presented in this paper suggest that a single structural characteristic common to all globular proteins is essential for cooperative folding. The formation of a partly folded state from the native state results in the exposure to solvent of two distinct regions: (1) the portions of the protein that are unfolded; and (2) the “complementary surfaces,” located in the regions of the protein that remain folded. The cooperative character of the folding/unfolding transition is determined largely by the energetics of exposing complementary surface regions to the solvent. By definition, complementary regions are present only in partly folded states; they are absent from the native and unfolded states. An unfavorable free energy lowers the probability of partly folded states and increases the cooperativity of the transition. In this paper we present a mathematical formulation of this behavior and develop a general cooperative folding/unfolding model, termed the “complementary region” (CORE) model. This model successfully reproduces the main properties of folding/unfolding transitions without limiting the number of partly folded states accessible to the protein, thereby permitting a systematic examination of the structural and solvent conditions under which intermediates become populated. It is shown that the CORE model predicts two-state folding/unfolding behavior, even though the two-state character is not assumed in the model. © 1993 Wiley-Liss, Inc.     

Dynamic Monte Carlo simulations of globular protein folding/unfolding pathways. II. Alpha-helical motifs

Dynamic Monte Carlo simulations of the folding pathways of alpha-helical protein motifs have been undertaken in the context of a diamond lattice model of globular proteins. The first question addressed in the nature of the assembly process of an alpha-helical hairpin. While the hairpin could, in principle, be formed via the diffusion-collision-adhesion of isolated performed helices, this is not the dominant mechanism of assembly found in the simulations. Rather, the helices that form native hairpins are constructed on-site, with folding initiating at or near the turn in almost all cases. Next, the folding/unfolding pathways of four-helix bundles having tight bends and one and two long loops in the native state are explored. Once again, an on-site construction mechanism of folding obtains, with a hairpin forming first, followed by the formation of a three-helix bundle, and finally the fourth helix of the native bundle assembles. Unfolding is essentially the reverse of folding. A simplified analytic theory is developed that reproduces the equilibrium folding transitions obtained from the simulations remarkably well and, for the dominant folding pathway, correctly identifies the intermediates seen in the simulations. The analytic theory provides the free energy along the reaction co-ordinate and identifies the transition state for all three motifs as being quite close to the native state, with three of the four helices assembled, and approximately one turn of the fourth helix in place. The transition state is separated from the native conformation by a free-energy barrier of mainly energetic origin and from the denatured state by a barrier of mainly entropic origin. The general features of the folding pathway seen in all variants of the model four-helix bundles are similar to those observed in the folding of beta-barrel, Greek key proteins; this suggests that many of the qualitative aspects of folding are invariant to the particular native state topology and secondary structure.     

All-atom ab initio folding of a diverse set of proteins

Natural proteins fold to a unique, thermodynamically dominant state. Modeling of the folding process and prediction of the native fold of proteins are two major unsolved problems in biophysics. Here, we show successful all-atom ab initio folding of a representative diverse set of proteins by using a minimalist transferable-energy model that consists of two-body atom-atom interactions, hydrogen bonding, and a local sequence-energy term that models sequence-specific chain stiffness. Starting from a random coil, the native-like structure was observed during replica exchange Monte Carlo (REMC) simulation for most proteins regardless of their structural classes; the lowest energy structure was close to native-in the range of 2-6 A root-mean-square deviation (rmsd). Our results demonstrate that the successful folding of a protein chain to its native state is governed by only a few crucial energetic terms.     

Kinetics, thermodynamics and evolution of non-native interactions in a protein folding nucleus

A lattice model with side chains was used to investigate protein folding with computer simulations. In this model, we rigorously demonstrate the existence of a specific folding nucleus. This nucleus contains specific interactions not present in the native state that, when weakened, slow folding but do not change protein stability. Such a decoupling of folding kinetics from thermodynamics has been observed experimentally for real proteins. From our results, we conclude that specific non-native interactions in the transition state would give rise to straight phi-values that are negative or larger than unity. Furthermore, we demonstrate that residue Ile 34 in src SH3, which has been shown to be kinetically, but not thermodynamically, important, is universally conserved in proteins with the SH3 fold. This is a clear example of evolution optimizing the folding rate of a protein independent of its stability and function.     

Protein misfolding: optional barriers, misfolded intermediates, and pathway heterogeneity

To investigate the character and role of misfolded intermediates in protein folding, a recombinant cytochrome c without the normally blocking histidine to heme misligation was studied. Folding remains heterogeneous as in the wild-type protein. Half of the population folds relatively rapidly to the native state in a two-state manner. The other half collapses (fluorescence quenching) and forms a full complement of helix (CD) with the same rate and denaturant dependence as the fast folding fraction but then is blocked and reaches the native structure (695nm absorbance) much more slowly. The factors that transiently block folding are not intrinsic to the folding process but depend on ambient conditions, including protein aggregation (f(concentration)), N terminus to heme misligation (f(pH)), and proline mis-isomerization (f(U state equilibration time)). The misfolded intermediate populated by the slowly folding fraction was characterized by hydrogen exchange pulse labeling. It is very advanced with all of the native-like elements fairly stably formed but not the final Met80-S to heme iron ligation, similar to a previously studied molten globule form induced by low pH. To complete final native state acquisition, some small back unfolding is required (error repair) but the misfolded intermediate does not revisit the U state before proceeding to N. These properties show that the intermediate is a normal on-pathway form that contains, in addition, adventitious misfolding errors that transiently block its forward progress. Related observations for other proteins (partially misfolded intermediates, pathway heterogeneity) might be similarly explained in terms of the optional insertion of error-dependent barriers into a classical folding pathway.     

Sequential Steps in the Assembly of the Multimeric Outer Membrane Secretin PulD

Investigations into protein folding are largely dominated by studies on monomeric proteins. However, the transmembrane domain of an important group of membrane proteins is only formed upon multimerization. Here, we use in vitro translation-coupled folding and insertion into artificial liposomes to investigate kinetic steps in the assembly of one such protein, the outer membrane secretin PulD of the bacterial type II secretion system. Analysis of the folding kinetics, measured by the acquisition of distinct determinants of the native state, provides unprecedented evidence for a sequential multistep process initiated by membrane-driven oligomerization. The effects of varying the lipid composition of the liposomes indicate that PulD first forms a “prepore” structure that attains the native state via a conformational switch.     

Knotted fusion proteins reveal unexpected possibilities in protein folding

Proteins that contain a distinct knot in their native structure are impressive examples of biological self-organization. Although this topological complexity does not appear to cause a folding problem, the mechanisms by which such knotted proteins form are unknown. We found that the fusion of an additional protein domain to either the amino terminus, the carboxy terminus, or to both termini of two small knotted proteins did not affect their ability to knot. The multidomain constructs remained able to fold to structures previously thought unfeasible, some representing the deepest protein knots known. By examining the folding kinetics of these fusion proteins, we found evidence to suggest that knotting is not rate limiting during folding, but instead occurs in a denatured-like state. These studies offer experimental insights into when knot formation occurs in natural proteins and demonstrate that early folding events can lead to diverse and sometimes unexpected protein topologies.     

Topological and energetic factors: what determines the structural details of the transition state ensemble and "en-route" intermediates for protein folding? An investigation for small globular proteins

Recent experimental results suggest that the native fold, or topology, plays a primary role in determining the structure of the transition state ensemble, at least for small, fast-folding proteins. To investigate the extent of the topological control of the folding process, we studied the folding of simplified models of five small globular proteins constructed using a Go-like potential to retain the information about the native structures but drastically reduce the energetic frustration and energetic heterogeneity among residue-residue native interactions. By comparing the structure of the transition state ensemble (experimentally determined by Phi-values) and of the intermediates with those obtained using our models, we show that these energetically unfrustrated models can reproduce the global experimentally known features of the transition state ensembles and "en-route" intermediates, at least for the analyzed proteins. This result clearly indicates that, as long as the protein sequence is sufficiently minimally frustrated, topology plays a central role in determining the folding mechanism.     

Failures of inverse folding and threading with gapped alignment

To calculate the tertiary structure of a protein from its amino acid sequence, the thermodynamic approach requires a potential function of sequence and conformation that has its global minimum at the native conformation for many different proteins. Here we study the behavior of such functions for the simplest model system that still has some of the features of the protein folding problem, namely two-dimensional square lattice chain configurations involving two residue types. First we show that even the given contact potential, which by definition is used to identify the folding sequences and their unique native conformations, cannot always correctly select which sequences will fold to a given structure. Second, we demonstrate that the given contact potential is not always able to favor the native alignment of a native sequence on its own native conformation over other gapped alignments of different folding sequences onto that same conformation. Because of these shortcomings, even in this simple model system in which all conformations and all native sequences are known and determined directly by the given potential, we must reexamine our expectations for empirical potentials used for inverse folding and gapped alignment on more realistic representations of proteins. © 1996 Wiley-Liss, Inc.     

A critical assessment of the topomer search model of protein folding using a continuum explicit-chain model with extensive conformational sampling

Recently, a series of closely related theoretical constructs termed the "topomer search model" (TSM) has been proposed for the folding mechanism of small, single-domain proteins. A basic assumption of the proposed scenarios is that the rate-limiting step in folding is an essentially unbiased, diffusive search for a conformational state called the native topomer defined by an overall native-like topological pattern. Successes in correlating TSM-predicted folding rates with that of real proteins have been interpreted as experimental support for the model. To better delineate the physics entailed, key TSM concepts are examined here using extensive Langevin dynamics simulations of continuum C(alpha) chain models. The theoretical native topomers of four experimentally well-studied two-state proteins are characterized. Consistent with the TSM perspective, we found that the sizes of the native topomers increase with experimental folding rate. However, a careful determination of the corresponding probabilities that the native topomers are populated during a random search fails to reproduce the previously predicted folding rates. Instead, our results indicate that an unbiased TSM search for the native topomer amounts to a Levinthal-like process that would take an impossibly long average time to complete. Furthermore, intraprotein contacts in all four native topomers considered exhibit no apparent correlation with the experimental phi-values determined from the folding kinetics of these proteins. Thus, the present findings suggest that certain basic, generic yet essential energetic features in protein folding are not accounted for by TSM scenarios to date.     

Molecular dynamics simulations of folding processes of a beta-hairpin in an implicit solvent

Computer simulations of beta-hairpin folding are relatively difficult, especially those based on the explicit water model. This greatly limits the complete analysis and understanding of their folding mechanisms. In this paper, we use the generalized Born/solvent accessible implicit solvent model to simulate the folding processes of a nine-residue beta-hairpin. We find that the beta-hairpin can fold into its native structure very easily, even using the traditional molecular dynamics method. This allows us to extract 21 complete folding events and investigate the folding process sufficiently. Our results show that there exist four most stable states on the free energy landscape of the short peptide, one native state and three intermediates. We find that two of the non-native stable states have almost the same potential energy as the native state but with lower entropy. This suggests that the native state can be stabilized entropically. Furthermore, we find that the folding processes of this peptide have common features: to fold into its native state, the peptide undergoes a continuous collapsing-extending-recollapsing process to adjust the positions of the side chains in order to form the native middle inter-strand hydrogen bonds. The formations of these bonds are the key step of the folding process. Once these bonds are formed, the peptide can fold into the native state quickly.     

Shortening a loop can increase protein native state entropy

Protein loops are essential structural elements that influence not only function but also protein stability and folding rates. It was recently reported that shortening a loop in the AcP protein may increase its native state conformational entropy. This effect on the entropy of the folded state can be much larger than the lower entropic penalty of ordering a shorter loop upon folding, and can therefore result in a more pronounced stabilization than predicted by polymer model for loop closure entropy. In this study, which aims at generalizing the effect of loop length shortening on native state dynamics, we use all‐atom molecular dynamics simulations to study how gradual shortening a very long or solvent‐exposed loop region in four different proteins can affect their stability. For two proteins, AcP and Ubc7, we show an increase in native state entropy in addition to the known effect of the loop length on the unfolded state entropy. However, for two permutants of SH3 domain, shortening a loop results only with the expected change in the entropy of the unfolded state, which nicely reproduces the observed experimental stabilization. Here, we show that an increase in the native state entropy following loop shortening is not unique to the AcP protein, yet nor is it a general rule that applies to all proteins following the truncation of any loop. This modification of the loop length on the folded state and on the unfolded state may result with a greater effect on protein stability. Proteins 2015; 83:2137–2146. © 2015 Wiley Periodicals, Inc.     

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.     

Implications of protein folding. Additivity schemes for volumes and compressibilities

Partial specific volume and compressibility properties of the extended state of proteins are estimated from additivity schemes using revised amino acid and peptide data. These calculated properties are compared with the experimental data of the native state in order to assess the contribution from folding. Results of this treatment show that, in the case of partial specific volumes, there is close agreement between the two data sets for a number of proteins. The implication is that subtle compensatory contributions in volume occur during the folding process. In the case of compressibilities, however, a substantial difference is observed which is believed to arise because of the hydrophobic interior created in the native protein as a result of the folding process. Using suitable measures of protein hydrophobicities and estimates of the fraction of buried apolar residues, a "micellar model of protein compressibility" is proposed and tested for several proteins. Results obtained from this model show good agreement with the experimental data for the native state of a number of proteins.     

Denatured state is critical in determining the properties of model proteins designed on different folds

The thermodynamics of proteins designed on three common folds (SH3, chymotrypsin inhibitor 2 [CI2], and protein G) is studied with a simplified C(alpha) model and compared with the thermodynamics of proteins designed on random-generated folds. The model allows to design sequences to fold within a dRMSD ranging from 1.2 to 4.2 A from the crystallographic native conformation and to study properties that are hard to be measured experimentally. It is found that the denatured state of all of them is not random but is, to different extents, partially structured. The degree of structure is more abundant for SH3 and protein G, giving rise to a weaker stability but a more efficient folding kinetics than CI2 and, even more, than the random-generated folds. Consequently, the features of the unfolded state seem to be as important in the determination of the thermodynamic properties of these proteins as the features of the native state.