Unification of the folding mechanisms of non-two-state and two-state proteins

We have collected the kinetic folding data for non-two-state and two-state globular proteins reported in the literature, and investigated the relationships between the folding kinetics and the native three-dimensional structure of these proteins. The rate constants of formation of both the intermediate and the native state of non-two-state folders were found to be significantly correlated with protein chain length and native backbone topology, which is represented by the absolute contact order and sequence-distant native pairs. The folding rate of two-state folders, which is known to be correlated with the native backbone topology, apparently does not correlate significantly with protein chain length. On the basis of a comparison of the folding rates of the non-two-state and two-state folders, it was found that they are similarly dependent on the parameters that reflect the native backbone topology. This suggests that the mechanisms behind non-two-state and two-state folding are essentially identical. The present results lead us to propose a unified mechanism of protein folding, in which folding occurs in a hierarchical manner, reflecting the hierarchy of the native three-dimensional structure, as embodied in the case of non-two-state folding with an accumulation of the intermediate. Apparently, two-state folding is merely a simplified version of hierarchical folding caused either by an alteration in the rate-limiting step of folding or by destabilization of the intermediate.     

Contact order revisited: influence of protein size on the folding rate

Guided by the recent success of empirical model predicting the folding rates of small two-state folding proteins from the relative contact order (CO) of their native structures, by a theoretical model of protein folding that predicts that logarithm of the folding rate decreases with the protein chain length L as L(2/3), and by the finding that the folding rates of multistate folding proteins strongly correlate with their sizes and have very bad correlation with CO, we reexamined the dependence of folding rate on CO and L in attempt to find a structural parameter that determines folding rates for the totality of proteins. We show that the Abs_CO = CO x L, is able to predict rather accurately folding rates for both two-state and multistate folding proteins, as well as short peptides, and that this Abs_CO scales with the protein chain length as L(0.70 +/- 0.07) for the totality of studied single-domain proteins and peptides.     

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.     

Chain length is the main determinant of the folding rate for proteins with three-state folding kinetics

We demonstrate that chain length is the main determinant of the folding rate for proteins with the three-state folding kinetics. The logarithm of their folding rate in water (k(f)) strongly anticorrelates with their chain length L (the correlation coefficient being -0.80). At the same time, the chain length has no correlation with the folding rate for two-state folding proteins (the correlation coefficient is -0.07). Another significant difference of these two groups of proteins is a strong anticorrelation between the folding rate and Baker's "relative contact order" for the two-state folders and the complete absence of such correlation for the three-state folders.     

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.     

A novel topology for representing protein folds

Various topologies for representing 3D protein structures have been advanced for purposes ranging from prediction of folding rates to ab initio structure prediction. Examples include relative contact order, Delaunay tessellations, and backbone torsion angle distributions. Here, we introduce a new topology based on a novel means for operationalizing 3D proximities with respect to the underlying chain. The measure involves first interpreting a rank‐based representation of the nearest neighbors of each residue as a permutation, then determining how perturbed this permutation is relative to an unfolded chain. We show that the resultant topology provides improved association with folding and unfolding rates determined for a set of two‐state proteins under standardized conditions. Furthermore, unlike existing topologies, the proposed geometry exhibits fine scale structure with respect to sequence position along the chain, potentially providing insights into folding initiation and/or nucleation sites.     

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.     

Experimental characterization of the folding kinetics of the notch ankyrin domain

Proteins constructed from linear arrays of tandem repeats provide a simplified architecture for understanding protein folding. Here, we examine the folding kinetics of the ankyrin repeat domain from the Drosophila Notch receptor, which consists of six folded ankyrin modules and a seventh partly disordered N-terminal ankyrin repeat sequence. Both the refolding and unfolding kinetics are best described as a sum of two exponential phases. The slow, minor refolding phase is limited by prolyl isomerization in the denatured state (D). The minor unfolding phase, which appears as a lag during fluorescence-detected unfolding, is consistent with an on-pathway intermediate (I). This intermediate, although not directly detected during refolding, is shown to be populated by interrupted refolding experiments. When plotted against urea, the rate constants for the major unfolding and refolding phases define a single non-linear v-shaped chevron, as does the minor unfolding phase. These two chevrons, along with unfolding amplitudes, are well-fitted by a sequential three-state model, which yields rate constants for the individual steps in folding and unfolding. Based on these fitted parameters, the D to I step is rate-limiting, and closely matches the major observed refolding phase at low denaturant concentrations. I appears to be midway between N and D in folding free energy and denaturant sensitivity, but has Trp fluorescence properties close to N. Although the Notch ankyrin domain has a simple architecture, folding is slow, with the limiting refolding rate constant as much as seven orders of magnitude smaller than expected from topological predictions.     

The energy landscape of modular repeat proteins: topology determines folding mechanism in the ankyrin family

Proteins consisting of repeating amino acid motifs are abundant in all kingdoms of life, especially in higher eukaryotes. Repeat-containing proteins self-organize into elongated non-globular structures. Do the same general underlying principles that dictate the folding of globular domains apply also to these extended topologies? Using a simplified structure-based model capturing a perfectly funneled energy landscape, we surveyed the predicted mechanism of folding for ankyrin repeat containing proteins. The ankyrin family is one of the most extensively studied classes of non-globular folds. The model based only on native contacts reproduces most of the experimental observations on the folding of these proteins, including a folding mechanism that is reminiscent of a nucleation propagation growth. The confluence of simulation and experimental results suggests that the folding of non-globular proteins is accurately described by a funneled energy landscape, in which topology plays a determinant role in the folding mechanism.     

Kinetic barriers and the role of topology in protein and RNA folding

This review compares the folding behavior of proteins and RNAs. Topics covered include the role of topology in the determination of folding rates, major folding events including collapse, properties of denatured states, pathway heterogeneity, and the influence of the mode of initiation on the folding pathway.     

Rapid exploration of the folding topology of helical membrane proteins using paramagnetic perturbation

An understanding of the folding states of α-helical membrane proteins in detergent systems is important for functional and structural studies of these proteins. Here, we present a rapid and simple method for identification of the folding topology and assembly of transmembrane helices using paramagnetic perturbation in nuclear magnetic resonance spectroscopy. By monitoring the perturbation of signals from glycine residues located at specific sites, the folding topology and the assembly of transmembrane helices of membrane proteins were easily identified without time-consuming backbone assignment. This method is validated with Mistic (membrane-integrating sequence for translation of integral membrane protein constructs) of known structure as a reference protein. The folding topologies of two bacterial histidine kinase membrane proteins (SCO3062 and YbdK) were investigated by this method in dodecyl phosphocholine (DPC) micelles. Combing with analytical ultracentrifugation, we identified that the transmembrane domain of YbdK is present as a parallel dimer in DPC micelle. In contrast, the interaction of transmembrane domain of SCO3062 is not maintained in DPC micelle due to disruption of native structure of the periplasmic domain by DPC micelle.     

Surprisingly high correlation between early and late stages in non-two-state protein folding

We collected quantitative kinetic data on early and late stages of folding in non-two-state proteins from the literature, and studied the relationship between the kinetics of the two stages. There was a surprisingly high correlation between the rate constants of these stages. The correlation coefficient of the logarithmic rate constants was as high as 0.97, which could not be caused by chance. We also studied relationships of the logarithmic rate constants of the two stages with native three-dimensional structures represented by the residue-residue contact map. There were again surprisingly high correlations between the logarithmic rate constants and the number of non-local contact clusters obtained from the contact maps. Because the number of non-local contact clusters represents overall arrangement of substructures in a native protein, the results strongly suggested the importance of the arrangement of the substructures for the kinetics of both early and late stages of protein folding.     

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.     

Exploring zipping and assembly as a protein folding principle

It has been proposed that proteins fold by a process called "Zipping and Assembly" (Z&A). Zipping refers to the growth of local substructures within the chain, and assembly refers to the coming together of already-formed pieces. Our interest here is in whether Z&A is a general method that can fold most of sequence space, to global minima, efficiently. Using the HP model, we can address this question by enumerating full conformation and sequence spaces. We find that Z&A reaches the global energy minimum native states, even though it searches only a very small fraction of conformational space, for most sequences in the full sequence space. We find that Z&A, a mechanism-based search, is more efficient in our tests than the replica exchange search method. Folding efficiency is increased for chains having: (a) small loop-closure steps, consistent with observations by Plaxco et al. 1998;277;985-994 that folding rates correlate with contact order, (b) neither too few nor too many nucleation sites per chain, and (c) assembly steps that do not occur too early in the folding process. We find that the efficiency increases with chain length, although our range of chain lengths is limited. We believe these insights may be useful for developing faster protein conformational search algorithms.     

Local secondary structure content predicts folding rates for simple,two-state proteins

Many single-domain proteins exhibit two-state folding kinetics, with folding rates that span more than six orders of magnitude. A quantity of much recent interest for such proteins is their contact order, the average separation in sequence between contacting residue pairs. Numerous studies have reached the surprising conclusion that contact order is well-correlated with the logarithm of the folding rate for these small, well-characterized molecules. Here, we investigate the physico-chemical basis for this finding by asking whether contact order is actually a composite number that measures the fraction of local secondary structure in the protein; viz. turns, helices, and hairpins. To pursue this question, we calculated the secondary structure content for 24 two-state proteins and obtained coefficients that predict their folding rates. The predicted rates correlate strongly with experimentally determined rates, comparable to the correlation with contact order. Further, these predicted folding rates are correlated strongly with contact order. Our results suggest that the folding rate of two-state proteins is a function of their local secondary structure content, consistent with the hierarchic model of protein folding. Accordingly, it should be possible to utilize secondary structure prediction methods to predict folding rates from sequence alone.     

Cooperativity and the origins of rapid, single-exponential kinetics in protein folding

The folding of naturally occurring, single-domain proteins is usually well described as a simple, single-exponential process lacking significant trapped states. Here we further explore the hypothesis that the smooth energy landscape this implies, and the rapid kinetics it engenders, arises due to the extraordinary thermodynamic cooperativity of protein folding. Studying Miyazawa-Jernigan lattice polymers, we find that, even under conditions where the folding energy landscape is relatively optimized (designed sequences folding at their temperature of maximum folding rate), the folding of protein-like heteropolymers is accelerated when their thermodynamic cooperativity is enhanced by enhancing the nonadditivity of their energy potentials. At lower temperatures, where kinetic traps presumably play a more significant role in defining folding rates, we observe still greater cooperativity-induced acceleration. Consistent with these observations, we find that the folding kinetics of our computational models more closely approximates single-exponential behavior as their cooperativity approaches optimal levels. These observations suggest that the rapid folding of naturally occurring proteins is, in part, a consequence of their remarkably cooperative folding.     

Cooperativity in two-state protein folding kinetics

We present a solvable model that predicts the folding kinetics of two-state proteins from their native structures. The model is based on conditional chain entropies. It assumes that folding processes are dominated by small-loop closure events that can be inferred from native structures. For CI2, the src SH3 domain, TNfn3, and protein L, the model reproduces two-state kinetics, and it predicts well the average Phi-values for secondary structures. The barrier to folding is the formation of predominantly local structures such as helices and hairpins, which are needed to bring nonlocal pairs of amino acids into contact.     

Secondary structure determines protein topology

Using a test set of 13 small, compact proteins, we demonstrate that a remarkably simple protocol can capture native topology from secondary structure information alone, in the absence of long-range interactions. It has been a long-standing open question whether such information is sufficient to determine a protein's fold. Indeed, even the far simpler problem of reconstructing the three-dimensional structure of a protein from its exact backbone torsion angles has remained a difficult challenge owing to the small, but cumulative, deviations from ideality in backbone planarity, which, if ignored, cause large errors in structure. As a familiar example, a small change in an elbow angle causes a large displacement at the end of your arm; the longer the arm, the larger the displacement. Here, correct secondary structure assignments (alpha-helix, beta-strand, beta-turn, polyproline II, coil) were used to constrain polypeptide backbone chains devoid of side chains, and the most stable folded conformations were determined, using Monte Carlo simulation. Just three terms were used to assess stability: molecular compaction, steric exclusion, and hydrogen bonding. For nine of the 13 proteins, this protocol restricts the main chain to a surprisingly small number of energetically favorable topologies, with the native one prominent among them.     

The "first in-last out" hypothesis on protein folding revisited

We calculated profiles for mean residue depth, contact order, and number of contacts in the native structure of a series of proteins for which folding has been studied extensively, the chymotrypsin inhibitor 2, the SH3 module from the src tyrosine kinase, the small ribonuclease barnase, the bacterial immunity protein Im7, and apomyoglobin. We compared these profiles with experimental data from equilibrium or pulse labeling hydrogen-deuterium exchange obtained from NMR and phi values obtained from the protein engineering approach. We find a good qualitative agreement between the hierarchy of formation of topological elements during the folding process and the ranking of secondary structure elements in terms of residue depth. Residues that are most deeply buried in the core of the native protein usually belong to stretches of secondary structure elements that are formed early in the folding pathway. Residue depth can thus provide a useful and simple tool for the design of folding experiments.