Protein folding through kinetic discrimination

Proteins fold on a micros-ms time scale. However, the number of possible conformations of the polypeptide backbone is so large that random sampling would not allow the protein to fold within the lifetime of the universe, the Levinthal paradox. We show here that a protein chain can fold efficiently with high fidelity if on average native contacts survive longer than non-native ones, that is, if the dissociation rate constant for breakage of a contact is lower for native than for non-native interactions. An important consequence of this finding is that no pathway needs to be specified for a protein to fold. Instead, kinetic discrimination among formed contacts is a sufficient criterion for folding to proceed to the native state. Successful protein folding requires that productive contacts survive long enough to obtain a certain level of probability that other native contacts form before the first interacting unit dissociates. If native contacts survive longer than non-native ones, this prevents misfolding and provides the folding process with directionality toward the native state. If on average all contacts survive equally long, the protein chain is deemed to fold through random search through all possible conformations (i.e., the Levinthal paradox). A modest degree of cooperativity among the native contacts, that is, decreased dissociation rate next to neighboring contacts, shifts the required ratio of dissociation rates into a realistic regime and makes folding a stochastic process with a nucleation step. No kinetic discrimination needs to be invoked in regards to the association process, which is modeled as dependent on the diffusion rate of chain segments.     

Folding dynamics of Trp-cage in the presence of chemical interference and macromolecular crowding. I

Proteins fold and function in the crowded environment of the cell's interior. In the recent years it has been well established that the so-called "macromolecular crowding" effect enhances the folding stability of proteins by destabilizing their unfolded states for selected proteins. On the other hand, chemical and thermal denaturation is often used in experiments as a tool to destabilize a protein by populating the unfolded states when probing its folding landscape and thermodynamic properties. However, little is known about the complicated effects of these synergistic perturbations acting on the kinetic properties of proteins, particularly when large structural fluctuations, such as protein folding, have been involved. In this study, we have first investigated the folding mechanism of Trp-cage dependent on urea concentration by coarse-grained molecular simulations where the impact of urea is implemented into an energy function of the side chain and/or backbone interactions derived from the all-atomistic molecular dynamics simulations with urea through a Boltzmann inversion method. In urea solution, the folding rates of a model miniprotein Trp-cage decrease and the folded state slightly swells due to a lack of contact formation between side chains at the terminal regions. In addition, the equilibrium m-values of Trp-cage from the computer simulations are in agreement with experimental measurements. We have further investigated the combined effects of urea denaturation and macromolecular crowding on Trp-cage's folding mechanism where crowding agents are modeled as hard-spheres. The enhancement of folding rates of Trp-cage is most pronounced by macromolecular crowding effect when the extended conformations of Trp-cast dominate at high urea concentration. Our study makes quantitatively testable predictions on protein folding dynamics in a complex environment involving both chemical denaturation and macromolecular crowding effects.     

Foldable subunits of helix protein

Detection of foldable subunits in proteins is an important approach to understand their evolutions and find building motifs for de novo protein design. Using united-residue model, we simulated the folding of a six-helix protein with a length of 120 amino acids (C-terminal domain of Ku86). The folding behaviors, structural topology and sequence repetition of this protein all suggest that it may have a two-fold quasi-repetition or symmetry in its sequence and structure. Therefore, we simulated the folding of its two halves (1–60 and 61–120 amino acids) and find that they can fold into native conformations independently. It is also found that their folding behaviors are very similar to other three-helix bundles. This suggests that this protein may be divided into two foldable halves.     

Understanding the growth rates of polymer cocrystallization in the binary mixtures of different chain lengths

Polymer materials often contain a polydispersity of molecular lengths. We studied the linear growth rates of polymer lamellar crystals in the binary mixtures of different chain lengths by means of dynamic Monte Carlo simulations. Both chain lengths were chosen large enough to perform chain folding upon crystal growth but not very large to avoid the effect of chain entanglement in the bulk phase. We found that the crystal growth rates exhibit a linear dependence upon the compositions of mixtures. This linear relation implies that the overall crystal growth rates are integrated by the separate contributions of variable-length single polymers, supporting the model of intramolecular crystal nucleation. In each event of crystal growth of single polymers, long chains yield more crystallinity than short chains. This high efficiency explains higher crystal growth rates of long chains than that of short chains, and the explanation is quite different from the traditional view on the basis of their different melting points. In addition, with a partial release of sliding diffusion for crystal thickening, a new dependence of crystal growth rates occurs near the dilute end of long-chain compositions at high temperatures, which can be attributed to the preference of integer-number chain folding at the crystal growth front. The preferred fold lengths may vary with chain lengths and thus influence the crystal growth rates.     

Single‐Molecule Detection Reveals Knot Sliding in TrmD Denaturation

An increasing number of proteins are found to contain a knot in their polypeptide chain. Although some studies have looked into the folding mechanism of knotted proteins, why and how these complex topologies form are still far from being fully answered. Moreover, no experimental information about how the knot moves during the protein‐folding process is available. Herein, by combining single‐molecule fluorescence resonance energy transfer (smFRET) experiments with molecular dynamics (MD) simulations, we performed a detailed study to characterize the knot in the denatured state of TrmD, a knotted tRNA (guanosine‐1) methyltransferase from Escherichia coli, as a model system. We found that the knot still existed in the unfolded state of TrmD, consistent with the results for two other knotted proteins, YibK and YbeA. More interestingly, both smFRET experiments and MD simulations revealed that the knot slid towards the C‐terminal during the unfolding process, which could be explained by the relatively strong interactions between the β‐sheet core at the N terminal of the native knot region. The size of the knot in the unfolded state is not larger than that in the native state. In addition, the knot slid in a “downhill” mode with simultaneous chain collapse in the denatured state.     

Statistical mechanics of helix bundles using a dynamic programming approach

Despite much study, biomolecule folding cooperativity is not well understood. There are quantitative models for helix-coil transitions and for coil-to-globule transitions, but no accurate models yet treat both chain collapse and secondary structure formation together. We develop here a dynamic programming approach to statistical mechanical partition functions of foldamer chain molecules. We call it the ascending levels model. We apply it to helix-coil and helix-bundle folding and cooperativity. For 14- to 50-mer Baldwin peptides, the model gives good predictions for the heat capacity and helicity versus temperature and urea. The model also gives good fits for the denaturation of Oas's three-helix bundle B domain of protein A (F13W*) and synthetic protein alpha3C by temperature and guanidine. The model predicts the conformational distributions. It shows that these proteins fold with transitions that are two-state, although the transitions in the Baldwin helices are nearly higher order. The model shows that the recently developed three-helix bundle polypeptoids of Lee et al. fold anti-cooperatively, with a predicted value of DeltaHvH/DeltaHcal = 0.72. The model also predicts that two-helix bundles are unstable in proteins but stable in peptoids. Our dynamic programming approach provides a general way to explore cooperativity in complex foldable polymers.     

Folding kinetics of a naturally occurring helical peptide: implication of the folding speed limit of helical proteins

The folding mechanism and dynamics of a helical protein may strongly depend on how quickly its constituent alpha-helices can fold independently. Thus, our understanding of the protein folding problem may be greatly enhanced by a systematic survey of the folding rates of individual alpha-helical segments derived from their parent proteins. As a first step, we have studied the relaxation kinetics of the central helix (L9:41-74) of the ribosomal protein L9 from the bacterium Bacillus stearothermophilus , in response to a temperature-jump ( T-jump) using infrared spectroscopy. L9:41-74 has been shown to exhibit unusually high helicity in aqueous solution due to a series of side chain-side chain interactions, most of which are electrostatic in nature, while still remaining monomeric over a wide concentration range. Thus, this peptide represents an excellent model system not only for examining how the folding rate of naturally occurring helices differs from that of the widely studied alanine-based peptides, but also for estimating the folding speed limit of (small) helical proteins. Our results show that the T-jump induced relaxation rate of L9:41-74 is significantly slower than that of alanine-based peptides. For example, at 11 degrees C its relaxation time constant is about 2 micros, roughly seven times slower than that of SPE(5), an alanine-rich peptide of similar chain length. In addition, our results show that the folding rate of a truncated version of L9:41-74 is even slower. Taken together, these results suggest that individual alpha-helical segments in proteins may fold on a time scale that is significantly slower than the folding time of alanine-based peptides. Furthermore, we argue that the relaxation rate of L9:41-74 measured between 8 and 45 degrees C provides a realistic estimate of the ultimate folding rate of (small) helical proteins over this temperature range.     

Variationally determined free energy profiles for structural models of proteins: characteristic temperatures for folding and trapping

Characterizing the phase diagram for proteins is important both for laboratory studies and for the development of structure prediction algorithms. Using a variational scheme, we calculated the generic features of the protein thermostability over a large range of temperatures for a set of more than 50 different proteins using a model based on native structure alone. Focusing on a specific system, protein G, we further examined, using a more realistic model that includes the nonnative interaction, the thermostability of both the native state and a collection of trap structures. By surveying the native structures for many proteins and by paying closer attention to the various trap structures of protein G, we obtained an overall understanding of the folding dynamics far from the conditions usually focused on; namely, those near the folding temperature alone. Two characteristic temperatures (shown to scale with folding temperature in general) signal drastic changes in the folding mechanism. The variational calculations suggest that most proteins would, indeed, fold in a barrierless manner below a critical temperature analogous to a spinodal in crystallization. For fixed interaction strengths, this temperature, however, seems to be generally very low, approximately 50% of the equilibrium folding temperature. Likewise, native proteins, in general, would unfold in a completely barrierless way at a temperature 25% above folding temperature according to these variational calculations. We also studied the distribution of free energy profiles for escape from a set of trap structures generated by simulations.     

Trigger factor slows co-translational folding through kinetic trapping while sterically protecting the nascent chain from aberrant cytosolic interactions

The E. coli chaperone trigger factor (TF) interacts directly with nascent polypeptide chains as they emerge from the ribosome exit tunnel. Small protein domains can fold under the cradle created by TF, but the co-translational folding of larger proteins is slowed down by its presence. Because of the great experimental challenges in achieving high spatial and time resolution, it is not yet known whether or not TF alters the folding properties of small proteins and if the reduced rate of folding of larger proteins is the result of kinetic or thermodynamic effects. We show, by molecular simulations employing a coarse-grained model of a series of ribosome nascent-chain complexes, that TF does not alter significantly the co-translational folding process of a small protein G domain but delays that of a large β-galactosidase domain as a result of kinetic trapping of its unfolded ensemble. We demonstrate that this trapping occurs through a combination of three distinct mechanisms: a decrease in the rate of structural rearrangements within the nascent chain, an increase in the effective exit tunnel length due to folding outside the cradle, and entanglement of the nascent chain with TF. We present evidence that this TF-induced trapping represents a trade-off between promoting co-translational folding and sterically shielding the nascent chain from aberrant cytosolic interactions that could lead to its aggregation or degradation.     

Hierarchy of folding and unfolding events of protein G, CI2, and ACBP from explicit-solvent simulations

The study of the mechanism which is at the basis of the phenomenon of protein folding requires the knowledge of multiple folding trajectories under biological conditions. Using a biasing molecular-dynamics algorithm based on the physics of the ratchet-and-pawl system, we carry out all-atom, explicit solvent simulations of the sequence of folding events which proteins G, CI2, and ACBP undergo in evolving from the denatured to the folded state. Starting from highly disordered conformations, the algorithm allows the proteins to reach, at the price of a modest computational effort, nativelike conformations, within a root mean square deviation (RMSD) of approximately 1 A?. A scheme is developed to extract, from the myriad of events, information concerning the sequence of native contact formation and of their eventual correlation. Such an analysis indicates that all the studied proteins fold hierarchically, through pathways which, although not deterministic, are well-defined with respect to the order of contact formation. The algorithm also allows one to study unfolding, a process which looks, to a large extent, like the reverse of the major folding pathway. This is also true in situations in which many pathways contribute to the folding process, like in the case of protein G.     

Evidence of an intermediate and parallel pathways in protein unfolding from single-molecule fluorescence

Determining how proteins fold into their native structures is a subject of great importance, since ultimately it will allow protein structure and function to be predicted from primary sequence data. In addition, there is now a clear link between protein unfolding and misfolding events and many disease states. However, since proteins fold over rugged, multidimensional energy landscapes, this is a challenging experimental and theoretical problem. Single-molecule fluorescence methods developed over the past decade have the potential to follow the unfolding/folding of individual molecules. Mapping out the landscape without ensemble averaging will enable the identification of intermediate states which may not be significantly populated, in addition to the presence of multiple pathways. To date, there have been only a limited number of single-molecule folding/unfolding studies under nonequilibrium conditions and no intermediates have been observed. Here, for the first time, we present a single-molecule study of the unfolding of a large autofluorescent protein, Citrine, a variant of green fluorescent protein. Single-molecule fluorescence techniques are used to directly detect an intermediate on the unfolding/folding pathway and the existence of parallel unfolding pathways. This work, and the novel methods used, shows that single-molecule fluorescence can now provide new, hitherto experimentally inaccessible, insights into the folding/unfolding of proteins.     

Temperature effects on the nucleation mechanism of protein folding and on the barrierless thermal denaturation of a native protein

Recently, the authors proposed a kinetic model for the nucleation mechanism of protein folding where a protein was treated as a heteropolymer with all the bonds and bond angles equal and constant. As a crucial idea of the model, an overall potential around a cluster of native residues in which a protein residue performs a chaotic motion is considered to be a combination of three potentials: effective pairwise, average dihedral, and confining. The overall potential as a function of the distance from the cluster center has a double well shape which allows one to determine the rates with which the cluster emits and absorbs residues by using a first passage time analysis. One can then develop a kinetic theory for the nucleation mechanism of protein folding and evaluate the protein folding time. In the present paper we evaluate the optimal temperature at which the protein folding time is the shortest. A method is also proposed to determine the temperature dependence of the folding time without carrying out the time consuming calculations for a series of temperatures. Using Taylor series expansions in the formalism of the first passage time analysis, one can calculate the temperature dependence of the cluster emission and absorption rates in the vicinity of some temperature T(0) if they are known at T(0). Thus one can evaluate the protein folding time t(f) at any other temperature T in the vicinity of T(0) at which the folding time t(f) is known. We also present a model for the thermal denaturation of a protein occurring via the decay of the native structure of the protein. Due to a sufficiently large temperature increase or decrease, the rate with which a cluster of native residues within a protein emits residues becomes larger than the absorption rate in the whole range of cluster sizes up to the size of the whole protein. This leads to the unfolding of the protein in a barrierless way, i.e., as spinodal decomposition. Knowing the cluster emission and absorption rates as functions of temperature and cluster size, one can find the threshold temperatures of cold and hot barrierless denaturation as well as the corresponding unfolding times. Both proposed methods are illustrated by numerical calculations for two model proteins, one consisting of 124 amino acids, the other consisting of 2500 residues. The first one roughly mimicks a bovine pancreatic ribonuclease while the second one is a representative of the largest proteins which are extremely difficult to study by straightforward Monte Carlo or molecular dynamics simulations.     

Frustration and hydrophobicity interplay in protein folding and protein evolution

A lattice model is used to study mutations and compacting effects on protein folding rates and folding temperature. In the context of protein evolution, we address the question regarding the best scenario for a polypeptide chain to fold: either a fast nonspecific collapse followed by a slow rearrangement to form the native structure or a specific collapse from the unfolded state with the simultaneous formation of the native state. This question is investigated for optimized sequences, whose native state has no frustrated contacts between monomers, and also for mutated sequences, whose native state has some degree of frustration. It is found that the best scenario for folding may depend on the amount of frustration of the native structure. The implication of this result on protein evolution is discussed.     

How does the first water shell fold proteins so fast?

First shells of hydration and bulk solvent play a crucial role in the folding of proteins. Here, the role of water in the dynamics of proteins has been investigated using a theoretical protein-solvent model and a statistical physics approach. We formulate a hydration model where the hydrogen bonds between water molecules pertaining to the first shell of the protein conformation may be either mainly formed or broken. At thermal equilibrium, hydrogen bonds are formed at low temperature and are broken at high temperature. To explore the solvent effect, we follow the folding of a large sampling of protein chains, using a master-equation evolution. The dynamics shows a clear mechanism. Above the glass-transition temperature, a large ratio of chains fold very rapidly into the native structure irrespective of the temperature, following pathways of high transition rates through structures surrounded by the solvent with broken hydrogen bonds. Although these states have an infinitesimal probability, they act as strong dynamical attractors and fast folding proceeds along these routes rather than pathways with small transition rates between configurations of much higher equilibrium probabilities. At a given low temperature, a broad jump in the folding times is observed. Below this glass temperature, the pathways where hydrogen bonds are mainly formed become those of highest rates although with conformational changes of huge relaxation times. The present results reveal that folding obeys a double-funnel mechanism.     

Protein folding kinetics: barrier effects in chemical and thermal denaturation experiments

Recent experimental work on fast protein folding brings about an intriguing paradox. Microsecond-folding proteins are supposed to fold near or at the folding speed limit (downhill folding), but yet their folding behavior seems to comply with classical two-state analyses, which imply the crossing of high free energy barriers. However, close inspection of chemical and thermal denaturation kinetic experiments in fast-folding proteins reveals systematic deviations from two-state behavior. Using a simple one-dimensional free energy surface approach we find that such deviations are indeed diagnostic of marginal folding barriers. Furthermore, the quantitative analysis of available fast-kinetic data indicates that many microsecond-folding proteins fold downhill in native conditions. All of these proteins are then promising candidates for an atom-by-atom analysis of protein folding using nuclear magnetic resonance.1 We also find that the diffusion coefficient for protein folding is strongly temperature dependent, corresponding to an activation energy of approximately 1 kJ.mol-1 per protein residue. As a consequence, the folding speed limit at room temperature is about an order of magnitude slower than the approximately 1 micros estimates from high-temperature T-jump experiments. Our analysis is quantitatively consistent with the available thermodynamic and kinetic data on slow two-state folding proteins and provides a straightforward explanation for the apparent fast-folding paradox.     

Packing of protein structures in clusters with magic numbers

Recently we have proposed a model for folding proteins into packed’ clusters’. We have constructed a local homology measure for protein fold classes by projecting consecutively secondary structures onto a lattice. Taking into account hydrophobic forces we have found a mechanism for formation of clusters containing magic numbers of secondary structures and multipla of these clusters. A scheme for the relation between the sequence information and the native fold is given. We have performed a statistical analysis of available protein structures and found agreement with the predicted preferred abundances. In this paper we demonstrate that the results are robust to variations in the coordination number of the model.     

Pathways to folding, nucleation events, and native geometry

We perform extensive Monte Carlo simulations of a lattice model and the Gō potential [N. Gō and H. Taketomi, Proc. Natl. Acad. Sci. U.S.A. 75, 559563 (1978)] to investigate the existence of folding pathways at the level of contact cluster formation for two native structures with markedly different geometries. Our analysis of folding pathways revealed a common underlying folding mechanism, based on nucleation phenomena, for both protein models. However, folding to the more complex geometry (i.e., that with more nonlocal contacts) is driven by a folding nucleus whose geometric traits more closely resemble those of the native fold. For this geometry folding is clearly a more cooperative process.     

Expanding the realm of ultrafast protein folding: gpW, a midsize natural single-domain with alpha+beta topology that folds downhill

All ultrafast folding proteins known to date are either very small in size (less than 45 residues), have an alpha-helix bundle topology, or have been artificially engineered. In fact, many of them share two or even all three features. Here we show that gpW, a natural 62-residue alpha+beta protein expected to fold slowly in a two-state fashion, folds in microseconds (i.e., from tau = 33 micros at 310 K to tau = 1.7 micros at 355 K). Thermodynamic analyses of gpW reveal probe dependent thermal denaturation, complex coupling between two denaturing agents, and differential scanning calorimetry (DSC) thermogram characteristic of folding over a negligible thermodynamic folding barrier. The free energy surface analysis of gpW folding kinetics also produces a marginal folding barrier of about thermal energy ( RT) at the denaturation midpoint. From these results we conclude that gpW folds in the downhill regime and is close to the global downhill limit. This protein seems to be poised toward downhill folding by a loosely packed hydrophobic core with low aromatic content, large stabilizing contributions from local interactions, and abundance of positive charges on the native surface. These special features, together with a complex functional role in bacteriophage lambda assembly, suggest that gpW has been engineered to fold downhill by natural selection.