Comparison between long-range interactions and contact order in determining the folding rate of two-state proteins: application of long-range order to folding rate prediction.

The contact order is believed to be an important factor for understanding protein folding mechanisms. In our earlier work, we have shown that the long-range interactions play a vital role in protein folding. In this work, we analyzed the contribution of long-range contacts to determine the folding rate of two-state proteins. We found that the residues that are close in space and are separated by at least ten to 15 residues in sequence are important determinants of folding rates, suggesting the presence of a folding nucleus at an interval of approximately 25 residues. A novel parameter "long-range order" has been proposed to predict protein folding rates. This parameter shows as good a relationship with the folding rate of two-state proteins as contact order. Further, we examined the minimum limit of residue separation to determine the long-range contacts for different structural classes. We observed an excellent correlation between long-range order and folding rate for all classes of globular proteins. We suggest that in mixed-class proteins, a larger number of residues can serve as folding nuclei compared to all-alpha and all-beta proteins. A simple statistical method has been developed to predict the folding rates of two-state proteins using the long-range order that produces an agreement with experimental results that is better or comparable to other methods in the literature.     

A simple measure of native-state topology and chain connectivity predicts the folding rates of two-state proteins with and without crosslinks

The folding rates of two-state proteins have been found to correlate with simple measures of native-state topology. The most prominent among these measures is the relative contact order (CO), which is the average CO, or localness, of all contacts in the native protein structure, divided by the chain length. Here, we test whether such measures can be generalized to capture the effect of chain crosslinks on the folding rate. Crosslinks change the chain connectivity and therefore also the localness of some of the native contacts. These changes in localness can be taken into account by the graph-theoretical concept of effective contact order (ECO). The relative ECO, however, the natural extension of the relative CO for proteins with crosslinks, overestimates the changes in the folding rates caused by crosslinks. We suggest here a novel measure of native-state topology, the relative logCO, and its natural extension, the relative logECO. The relative logCO is the average value for the logarithm of the CO of all contacts, divided by the logarithm of the chain length. The relative log(E)CO reproduces the folding rates of a set of 26 two-state proteins without crosslinks with essentially the same high correlation coefficient as the relative CO. In addition, it also captures the folding rates of eight two-state proteins with crosslinks.     

Non-native interactions,effective contact order,and protein folding: a mutational investigation with the energetically frustrated hydrophobic model

By Monte Carlo simulations, we explored the effect of single mutations on the thermodynamics and kinetics of the folding of a two-dimensional, energetically frustrated, hydrophobic protein model. Phi-Value analysis, corroborated by simulations beginning from given sets of judiciously chosen initial contacts, suggests that the transition state of the model consists of a limited region of the native structure, that is, a folding nucleus. It seems that the most important contacts in the transition state (large and positive Phi) are not the ones with the highest contact order, because in this case the entropic cost of their formation would be too high, but exactly the ones that decrease the entropic cost of difficult contacts, reducing their effective contact order. Mutations of internal monomers involved in high-order contacts were actually the ones resulting in the fastest kinetics (and Phi < 0), indicating they tend to make low order, non-native contacts of low entropic cost that stabilize the unfolded state with respect to the transition state. Folding acceleration by other non-native interactions was also observed and a simple general mechanism is proposed according to which non-native contacts can act indirectly over the folding nucleus, "chelating" out potentially harmful contacts. The polymer graph of our model, which facilitates the visualization of effective contact orders, successfully suggests the relative kinetic importance of different contacts and is reasonably consistent with analogous graphs for the well characterized family of SH3 domains.     

Non-native interactions play an effective role in protein folding dynamics

Systematic Monte Carlo simulations of simple lattice models show that the final stage of protein folding is an ordered process where native contacts get locked (i.e., the residues come into contact and remain in contact for the duration of the folding process) in a well‐defined order. The detailed study of the folding dynamics of protein‐like sequences designed as to exhibit different contact energy distributions, as well as different degrees of sequence optimization (i.e., participation of non‐native interactions in the folding process), reveals significant differences in the corresponding locking scenarios—the collection of native contacts and their average locking times, which are largely ascribable to the dynamics of non‐native contacts. Furthermore, strong evidence for a positive role played by non‐native contacts at an early folding stage was also found. Interestingly, for topologically simple target structures, a positive interplay between native and non‐native contacts is observed also toward the end of the folding process, suggesting that non‐native contacts may indeed affect the overall folding process. For target models exhibiting clear two‐state kinetics, the relation between the nucleation mechanism of folding and the locking scenario is investigated. Our results suggest that the stabilization of the folding transition state can be achieved through the establishment of a very small network of native contacts that are the first to lock during the folding process.     

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

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

Cysteine-cysteine contact preference leads to target-focusing in protein folding

We perform a statistical analysis of amino-acid contacts to investigate possible preferences of amino-acid interactions. We include in the analysis only tertiary contacts, because they are less constrained--compared to secondary contacts--by proteins' backbone rigidity. Using proteins from the protein data bank, our analysis reveals an unusually high frequency of cysteine pairings relative to that expected from random. To elucidate the possible effects of cysteine interactions in folding, we perform molecular simulations on three cysteine-rich proteins. In particular, we investigate the difference in folding dynamics between a Gō-like model (where attraction only occurs between amino acids forming a native contact) and a variant model (where attraction between any two cysteines is introduced to mimic the formation/dissociation of native/nonnative disulfide bonds). We find that when attraction among cysteines is nonspecific and comparable to a solvent-averaged interaction, they produce a target-focusing effect that expedites folding of cysteine-rich proteins as a result of a reduction of conformational search space. In addition, the target-focusing effect also helps reduce glassiness by lowering activation energy barriers and kinetic frustration in the system. The concept of target-focusing also provides a qualitative understanding of a correlation between the rates of protein folding and parameters such as contact order and total contact distance.     

Sequence determinants of protein folding rates: positive correlation between contact energy and contact range indicates selection for fast folding

In comparison with intense investigation of the structural determinants of protein folding rates, the sequence features favoring fast folding have received little attention. Here, we investigate this subject using simple models of protein folding and a statistical analysis of the Protein Data Bank (PDB). The mean-field model by Plotkin and coworkers predicts that the folding rate is accelerated by stronger-than-average interactions at short distance along the sequence. We confirmed this prediction using the Finkelstein model of protein folding, which accounts for realistic features of polymer entropy. We then tested this prediction on the PDB. We found that native interactions are strongest at contact range l = 8. However, since short range contacts tend to be exposed and they are frequently formed in misfolded structures, selection for folding stability tends to make them less attractive, that is, stability and kinetics may have contrasting requirements. Using a recently proposed model, we predicted the relationship between contact range and contact energy based on buriedness and contact frequency. Deviations from this prediction induce a positive correlation between contact range and contact energy, that is, short range contacts are stronger than expected, for 2/3 of the proteins. This correlation increases with the absolute contact order (ACO), as expected if proteins that tend to fold slowly due to large ACO are subject to stronger selection for sequence features favoring fast folding. Our results suggest that the selective pressure for fast folding is detectable only for one third of the proteins in the PDB, in particular those with large contact order.     

On the relation between native geometry and conformational plasticity

In protein folding the term plasticity refers to the number of alternative folding pathways encountered in response to free energy perturbations such as those induced by mutation. Here we explore the relation between folding plasticity and a gross, generic feature of the native geometry, namely, the relative number of local and non-local native contacts. The results from our study, which is based on Monte Carlo simulations of simple lattice proteins, show that folding to a structure that is rich in local contacts is considerably more plastic than folding to a native geometry characterized by having a very large number of long-range contacts (i.e., contacts between amino acids that are separated by more than 12 units of backbone distance). The smaller folding plasticity of native geometries is probably a direct consequence of their higher folding cooperativity that renders the folding reaction more robust against single- and multiple-point mutations.     

Equilibrium folding-unfolding pathways of model proteins: effect of myoglobin-heme contacts

The equilibrium of protein folding–unfolding has been investigated with a simple two-state, native and random-coil, model; we have termed this the globule–coil model. Energies are calculated by favoring native long-range contact pairs. Most-probable native domains are obtained at all stages of the transition; plausible folding pathways are constructed by connecting these domains by assuming simple growth. Even though native heme–protein contacts represent less than 6% of the total number of native contact pairs, their inclusion appears to change the folding pathway of apomyoglobin from the growth and merging of two native domains to the growth of a single domain. This indicates that pathways derived with this method may be critically sensitive to the details of the contact map and physical constraints during the folding process.     

Predicting protein folding rates from geometric contact and amino acid sequence

Protein folding speeds are known to vary over more than eight orders of magnitude. Plaxco, Simons, and Baker (see References) first showed a correlation of folding speed with the topology of the native protein. That and subsequent studies showed, if the native structure of a protein is known, its folding speed can be predicted reasonably well through a correlation with the "localness" of the contacts in the protein. In the present work, we develop a related measure, the geometric contact number, N (alpha), which is the number of nonlocal contacts that are well-packed, by a Voronoi criterion. We find, first, that in 80 proteins, the largest such database of proteins yet studied, N (alpha) is a consistently excellent predictor of folding speeds of both two-state fast folders and more complex multistate folders. Second, we show that folding rates can also be predicted from amino acid sequences directly, without the need to know the native topology or other structural properties.     

Hierarchy of structure loss in MD simulations of src SH3 domain unfolding.

To complement experimental studies of the src SH3 domain folding, we studied 30 independent, high-temperature, molecular dynamics simulations of src SH3 domain unfolding. These trajectories were observed to differ widely from each other. Thus, rather than analyzing individual trajectories, we sought to identify the recurrent features of the high-temperature unfolding process. The conformations from all simulations were combined and then divided into groups based on the number of native contacts. Average occupancies of each side-chain hydrophobic contact and hydrogen bond in the protein were then determined. In the symmetric funnel limit, the occupancies of all contacts should decrease in concert with the loss in total number of native contacts. If there is a lack of symmetry or hierarchy to the unfolding process, the occupancies of some contacts should decrease more slowly, and others more rapidly. Despite the heterogeneity of the individual trajectories, the ensemble averaging revealed an order to the unfolding process: contacts between the N and C-terminal strands are the first to disappear, whereas contacts within the distal beta-hairpin and a hydrogen-bonding network involving the distal loop beta-turn and the diverging turn persist well after the majority of the native contacts are lost. This hierarchy of events resembles but is somewhat less pronounced than that observed in our experimental studies of the folding of src SH3 domain.     

Kinetics and Reaction Coordinates of the Reassembly of Protein Fragments Via Forward Flux Sampling

We studied the mechanism of the reassembly and folding process of two fragments of a split lattice protein by using forward flux sampling (FFS). Our results confirmed previous thermodynamics and kinetics analyses that suggested that the disruption of the critical core (of an unsplit protein that folds by a nucleation mechanism) plays a key role in the reassembly mechanism of the split system. For several split systems derived from a parent 48-mer model, we estimated the reaction coordinates in terms of collective variables by using the FFS least-square estimation method and found that the reassembly transition is best described by a combination of the total number of native contacts, the number of interchain native contacts, and the total conformational energy of the split system. We also analyzed the transition path ensemble obtained from FFS simulations using the estimated reaction coordinates as order parameters to identify the microscopic features that differentiate the reassembly of the different split systems studied. We found that in the fastest folding split system, a balanced distribution of the original-core amino acids (of the unsplit system) between protein fragments propitiates interchain interactions at early stages of the folding process. Only this system exhibits a different reassembly mechanism from that of the unsplit protein, involving the formation of a different folding nucleus. In the slowest folding system, the concentration of the folding nucleus in one fragment causes its early prefolding, whereas the second fragment tends to remain as a detached random coil. We also show that the reassembly rate can be either increased or decreased by tuning interchain cooperativeness via the introduction of a single point mutation that either strengthens or weakens one of the native interchain contacts (prevalent in the transition state ensemble).     

Reduced contact order and RNA folding rates

We investigated the relationship between RNA structure and folding rates accounting for hierarchical structural formation. Folding rates of two-state folding proteins correlate well with relative contact order, a quantitative measure of the number and sequence distance between tertiary contacts. These proteins do not form stable structures prior to the rate-limiting step. In contrast, most secondary structures are stably formed prior to the rate-limiting step in RNA folding. Accordingly, we introduce "reduced contact order", a metric that reflects only the number of residues available to participate in the conformational search after the formation of secondary structure. Plotting the folding rates and the reduced contact order from ten different RNAs suggests that RNA folding can be divided into two classes. To examine this division, folding rates of circularly permutated isomers are compared for two RNAs, one from each class. Folding rates vary by tenfold for circularly permuted Bacillus subtilis RNase P RNA isomers, whereas folding rates vary by only 1.2-fold for circularly permuted catalytic domains. This difference is likely related to the dissimilar natures of their rate-limiting steps.     

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.     

Identifying folding nucleus based on residue contact networks of proteins

In the native structure of a protein, all the residues are tightly parked together in a specific order following its folding and every residue contacts with some spatially neighbor residues. A residue contact network can be constructed by defining the residues as nodes and the native contacts as edges. During the folding of small single-domain proteins, there is a set of contacts (or bonds), defined as the folding nucleus (FN), which is formed around the transition state, i.e., a rate-limiting barrier located at about the middle between the unfolded states and the native state on the free energy landscape. Such a FN plays an essential role in the folding dynamics and the residues, which form the related contacts called as folding nucleus residues (FNRs). In this work, the FNRs in proteins are identified by using quantities which characterize the topology of residue contact networks of proteins. By comparing the specificities of residues with the network quantities K(R), L(R), and D(R), up to 90% FNRs of six typical proteins found experimentally are identified. It is found that the FNRs behave the full-closeness centrals rather than degree or closeness centers in the residue contact network, implying that they are important to the folding cooperativity of proteins. Our study shows that the FNRs can be identified solely from the native structures of proteins based on the analysis of residue contact network without any knowledge of the transition state ensemble.     

The Limited Role of Nonnative Contacts in the Folding Pathways of a Lattice Protein

Models of protein energetics that neglect interactions between amino acids that are not adjacent in the native state, such as the Gō model, encode or underlie many influential ideas on protein folding. Implicit in this simplification is a crucial assumption that has never been critically evaluated in a broad context: Detailed mechanisms of protein folding are not biased by nonnative contacts, typically argued to be a consequence of sequence design and/or topology. Here we present, using computer simulations of a well-studied lattice heteropolymer model, the first systematic test of this oft-assumed correspondence over the statistically significant range of hundreds of thousands of amino acid sequences that fold to the same native structure. Contrary to previous conjectures, we find a multiplicity of folding mechanisms, suggesting that Gō-like models cannot be justified by considerations of topology alone. Instead, we find that the crucial factor in discriminating among topological pathways is the heterogeneity of native contact energies: The order in which native contacts accumulate is profoundly insensitive to omission of nonnative interactions, provided that native contact heterogeneity is retained. This robustness holds over a surprisingly wide range of folding rates for our designed sequences. Mirroring predictions based on the principle of minimum frustration, fast-folding sequences match their Gō-like counterparts in both topological mechanism and transit times. Less optimized sequences dwell much longer in the unfolded state and/or off-pathway intermediates than do Gō-like models. For dynamics that bridge unfolded and unfolded states, however, even slow folders exhibit topological mechanisms and transit times nearly identical with those of their Gō-like counterparts. Our results do not imply a direct correspondence between folding trajectories of Gō-like models and those of real proteins, but they do help to clarify key topological and energetic assumptions that are commonly used to justify such caricatures.     

Transition state contact orders correlate with protein folding rates

We have used molecular dynamics simulations restrained by experimental phi values derived from protein engineering experiments to determine the structures of the transition state ensembles of ten proteins that fold with two-state kinetics. For each of these proteins we then calculated the average contact order in the transition state ensemble and compared it with the corresponding experimental folding rate. The resulting correlation coefficient is similar to that computed for the contact orders of the native structures, supporting the use of native state contact orders for predicting folding rates. The native contacts in the transition state also correlate with those of the native state but are found to be about 30% lower. These results show that, despite the high levels of heterogeneity in the transition state ensemble, the large majority of contributing structures have native-like topologies and that the native state contact order captures this phenomenon.     

Diffusion-collision model for the folding kinetics of myoglobin

The diffusion-collision model has been used to analyze the folding kinetics of myoglobin. The microdomains, which are the basic units that coalesce during the folding, are identified with the helices and the stabilizing contacts between helices are determined from the native structure. Both association and dissociation reactions are included and a range of stabilization parameters is investigated to determine the variation in overall rate and the relative contributions made by different intermediates during the folding process. In a comparison of folding to the native state and to the midpoint of the folding transition (i.e., 50% native protein at the completion of the reaction) significant differences in the contributing intermediates are found.     

Topological determinants of protein unfolding rates

For proteins that fold by two-state kinetics, the folding and unfolding processes are believed to be closely related to their native structures. In particular, folding and unfolding rates are influenced by the native structures of proteins. Thus, we focus on finding important topological quantities from a protein structure that determine its unfolding rate. After constructing graphs from protein native structures, we investigate the relationships between unfolding rates and various topological quantities of the graphs. First, we find that the correlation between the unfolding rate and the contact order is not as prominent as in the case of the folding rate and the contact order. Next, we investigate the correlation between the unfolding rate and the clustering coefficient of the graph of a protein native structure, and observe no correlation between them. Finally, we find that a newly introduced quantity, the impact of edge removal per residue, has a good overall correlation with protein unfolding rates. The impact of edge removal is defined as the ratio of the change of the average path length to the edge removal probability. From these facts, we conclude that the protein unfolding process is closely related to the protein native structure.