Protein folding mechanisms and the multidimensional folding funnel

An important idea that emerges from the energy landscape theory of protein folding is that subtle global features of the protein landscape can profoundly affect the apparent mechanism of folding. The relationship between various characteristic temperatures in the phase diagrams and landmarks in the folding funnel at fixed temperatures can be used to classify different folding behaviors. The one-dimensional picture of a folding funnel classifies folding kinetics into four basic scenarios, depending on the relative location of the thermodynamic barrier and the glass transition as a function of a single-order parameter. However, the folding mechanism may not always be quantitatively described by a single-order parameter. Several other order parameters, such as degree of secondary structure formation, collapse and topological order, are needed to establish the connection between minimalist models and proteins in the laboratory. In this article we describe a simple multidimensional funnel based on two-order parameters that measure the degree of collapse and topological order. The appearance of several different “mechanisms” is illustrated by analyzing lattice models with different potentials and sequences with different degrees of design. In most cases, the two-dimensional analysis leads to a classification of mechanisms totally in keeping with the one-dimensional scheme, but a topologically distinct scenario of fast folding with traps also emerges. The nature of traps depends on the relative location of the glass transition surface and the thermodynamic barrier in the multidimensional funnel. Proteins 32:136–158, 1998. © 1998 Wiley-Liss, Inc.     

Toward minimalist models of larger proteins: a ubiquitin-like protein

Our recently developed off-lattice bead model capable of simulating protein structures with mixed alpha/beta content has been extended to model the folding of a ubiquitin-like protein and provides a means for examining the more complex kinetics involved in the folding of larger proteins. Using trajectories generated from constant-temperature Langevin dynamics simulations and sampling with the multiple multi-histogram method over five-order parameters, we are able to characterize the free energy landscape for folding and find evidence for folding through compact intermediates. Our model reproduces the observation that the C-terminus loop structure in ubiquitin is the last to fold in the folding process and most likely plays a spectator role in the folding kinetics. The possibility of a productive metastable intermediate along the folding pathway consisting of collapsed states with no secondary structure, and of intermediates or transition structures involving secondary structural elements occurring early in the sequence, is also supported by our model. The kinetics of folding remain multi-exponential below the folding temperature, with glass-like kinetics appearing at T/T(f) approximately 0.86. This new physicochemical model, designed to be predictive, helps validate the value of modeling protein folding at this level of detail for genomic-scale studies, and motivates further studies of other protein topologies and the impact of more complex energy functions, such as the addition of solvation forces.     

Diverse pathways of oxidative folding of disulfide proteins: underlying causes and folding models

The pathway of oxidative folding of disulfide proteins exhibits a high degree of diversity, which is manifested mainly by distinct structural heterogeneity and diverse rearrangement pathways of folding intermediates. During the past two decades, the scope of this diversity has widened through studies of more than 30 disulfide-rich proteins by various laboratories. A more comprehensive landscape of the mechanism of protein oxidative folding has emerged. This review will cover three themes. (1) Elaboration of the scope of diversity of disulfide folding pathways, including the two opposite extreme models, represented by bovine pancreatic trypsin inhibitor (BPTI) and hirudin. (2) Demonstration of experimental evidence accounting for the underlying mechanism of the folding diversity. (3) Discussion of the convergence between the extreme models of oxidative folding and models of conventional conformational folding (framework model, hydrophobic collapse model).     

Why do proteins divide into domains? Insights from lattice model simulations

It is known that larger globular proteins are built from domains, relatively independent structural units. A domain size seems to be limited, and a single domain consists of from few tens to a couple of hundred amino acids. Based on Monte Carlo simulations of a reduced protein model restricted to the face centered simple cubic lattice, with a minimal set of short-range and long-range interactions, we have shown that some model sequences upon the folding transition spontaneously divide into separate domains. The observed domain sizes closely correspond to the sizes of real protein domains. Short chains with a proper sequence pattern of the hydrophobic and polar residues undergo a two-state folding transition to the structurally ordered globular state, while similar longer sequences follow a multistate transition. Homopolymeric (uniformly hydrophobic) chains and random heteropolymers undergo a continuous collapse transition into a single globule, and the globular state is much less ordered. Thus, the factors responsible for the multidomain structure of proteins are sufficiently long polypeptide chain and characteristic, protein-like, sequence patterns. These findings provide some hints for the analysis of real sequences aimed at prediction of the domain structure of large proteins.     

Aromatic residues engineered into the beta-turn nucleation site of ubiquitin lead to a complex folding landscape, non-native side-chain interactions, and kinetic traps

The fast folding of small proteins is likely to be the product of evolutionary pressures that balance the search for native-like contacts in the transition state with the minimum number of stable non-native interactions that could lead to partially folded states prone to aggregation and amyloid formation. We have investigated the effects of non-native interactions on the folding landscape of yeast ubiquitin by introducing aromatic substitutions into the beta-turn region of the N-terminal beta-hairpin, using both the native G-bulged type I turn sequence (TXTGK) as well as an engineered 2:2 XNGK type I' turn sequence. The N-terminal beta-hairpin is a recognized folding nucleation site in ubiquitin. The folding kinetics for wt-Ub (TLTGK) and the type I' turn mutant (TNGK) reveal only a weakly populated intermediate, however, substitution with X = Phe or Trp in either context results in a high propensity to form a stable compact intermediate where the initial U-->I collapse is visible as a distinct kinetic phase. The introduction of Trp into either of the two host turn sequences results in either complex multiphase kinetics with the possibility of parallel folding pathways, or formation of a highly compact I-state stabilized by non-native interactions that must unfold before refolding. Sequence substitutions with aromatic residues within a localized beta-turn capable of forming non-native hydrophobic contacts in both the native state and partially folded states has the undesirable consequence that folding is frustrated by the formation of stable compact intermediates that evolutionary pressures at the sequence level may have largely eliminated.     

The experimental survey of protein-folding energy landscapes

We review what has been learned about the protein-folding problem from experimental kinetic studies. These studies reveal patterns of both great richness and surprising simplicity. The patterns can be interpreted in terms of proteins possessing an energy landscape which is largely, but not completely, funnel-like. Issues such as speed limitations of folding, the robustness of folding, the origin of barriers and cooperativity and the ensemble nature of transition states, intermediate and traps are assessed using the results from several experimental groups highlighting energy-landscape ideas as an interpretive framework.     

RNA tertiary interactions mediate native collapse of a bacterial group I ribozyme

Large RNAs collapse into compact intermediates in the presence of counterions before folding to the native state. We previously found that collapse of a bacterial group I ribozyme correlates with the formation of helices within the ribozyme core, but occurs at Mg2+ concentrations too low to support stable tertiary structure and catalytic activity. Here, using small-angle X-ray scattering, we show that Mg2+-induced collapse is a cooperative folding transition that can be fit by a two-state model. The Mg2+ dependence of collapse is similar to the Mg2+ dependence of helix assembly measured by partial ribonuclease T1 digestion and of an unfolding transition measured by UV hypochromicity. The correspondence between multiple probes of RNA structure further supports a two-state model. A mutation that disrupts tertiary contacts between the L9 tetraloop and its helical receptor destabilized the compact state by 0.8 kcal/mol, while mutations in the central triplex were less destabilizing. These results show that native tertiary interactions stabilize the compact folding intermediates under conditions in which the RNA backbone remains accessible to solvent.     

Computational protein design and large-scale assessment by I-TASSER structure assembly simulations

Protein design aims at designing new protein molecules of desired structure and functionality. One of the major obstacles to large-scale protein design are the extensive time and manpower requirements for experimental validation of designed sequences. Recent advances in protein structure prediction have provided potentials for an automated assessment of the designed sequences via folding simulations. We present a new protocol for protein design and validation. The sequence space is initially searched by Monte Carlo sampling guided by a public atomic potential, with candidate sequences selected by the clustering of sequence decoys. The designed sequences are then assessed by I-TASSER folding simulations, which generate full-length atomic structural models by the iterative assembly of threading fragments. The protocol is tested on 52 nonhomologous single-domain proteins, with an average sequence identity of 24% between the designed sequences and the native sequences. Despite this low sequence identity, three-dimensional models predicted for the first designed sequence have an RMSD of < 2 Å to the target structure in 62% of cases. This percentage increases to 77% if we consider the three-dimensional models from the top 10 designed sequences. Such a striking consistency between the target structure and the structural prediction from nonhomologous sequences, despite the fact that the design and folding algorithms adopt completely different force fields, indicates that the design algorithm captures the features essential to the global fold of the target. On average, the designed sequences have a free energy that is 0.39 kcal/(mol residue) lower than in the native sequences, potentially affording a greater stability to synthesized target folds.     

Energy barriers, cooperativity, and hidden intermediates in the folding of small proteins

Current theoretical views of the folding process of small proteins (< approximately 100 amino acids) postulate that the landscape of potential mean force (PMF) for the formation of the native state has a funnel shape and that the free energy barrier to folding arises from the chain configurational entropy only. However, recent theoretical studies on the formation of hydrophobic clusters with explicit water suggest that a barrier should exist on the PMF of folding, consistent with the fact that protein folding generally involves a large positive activation enthalpy at room temperature. In addition, high-resolution structural studies of the hidden partially unfolded intermediates have revealed the existence of non-native interactions, suggesting that the correction of the non-native interactions during folding should also lead to barriers on PMF. To explore the effect of a PMF barrier on the folding behavior of proteins, we modified Zwanzig's model for protein folding with an uphill landscape of PMF for the formation of transition states. We found that the modified model for short peptide segments can satisfy the thermodynamic and kinetic criteria for an apparently two-state folding. Since the Levinthal paradox can be solved by a stepwise folding of short peptide segments, a landscape of PMF with a locally uphill search for the transition state and cooperative stabilization of folding intermediates/native state is able to explain the available experimental results for small proteins. We speculate that the existence of cooperative hidden folding intermediates in small proteins could be the consequence of the highly specific structures of the native state, which are selected by evolution to perform specific functions and fold in a biologically meaningful time scale.     

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.     

Fold prediction by a hierarchy of sequence, threading, and modeling methods.

Several fold recognition algorithms are compared to each other in terms of prediction accuracy and significance. It is shown that on standard benchmarks, hybrid methods, which combine scoring based on sequence-sequence and sequence-structure matching, surpass both sequence and threading methods in the number of accurate predictions. However, the sequence similarity contributes most to the prediction accuracy. This strongly argues that most examples of apparently nonhomologous proteins with similar folds are actually related by evolution. While disappointing from the perspective of the fundamental understanding of protein folding, this adds a new significance to fold recognition methods as a possible first step in function prediction. Despite hybrid methods being more accurate at fold prediction than either the sequence or threading methods, each of the methods is correct in some cases where others have failed. This partly reflects a different perspective on sequence/structure relationship embedded in various methods. To combine predictions from different methods, estimates of significance of predictions are made for all methods. With the help of such estimates, it is possible to develop a "jury" method, which has accuracy higher than any of the single methods. Finally, building full three-dimensional models for all top predictions helps to eliminate possible false positives where alignments, which are optimal in the one-dimensional sequences, lead to unsolvable sterical conflicts for the full three-dimensional models.     

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

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

An empirical energy function for threading protein sequence through the folding motif

In this paper we present a new residue contact potantial derived by statistical analysis of protein crystal structures. This gives mean hydrophobic and pairwise contact energies as a function of residue type and distance interval. To test the accuracy of this potential we generate model structures by “threading” different sequences through backbone folding motifs found in the structural data base. We find that conformational energies calculated by summing contact potentials show perfect specificity in matching the correct sequences with each globular folding motif in a 161-protcin data set. They also identify correct models with the core folding motifs of heme-rythrin and immunoglobulin McPC603 V₁-do- main, among millions of alternatives possible when we align subsequences with α-helices and β-strands, and allow for variation in the lengths of intervening loops. We suggest that contact potentials reflect important constraints on nonbonded interaction in native proteins, and that “threading” may be useful for structure prediction by recognition of folding motif. © 1993 Wiley-Liss, Inc.     

Polymer principles of protein calorimetric two-state cooperativity

The experimental calorimetric two-state criterion requires the van't Hoff enthalpy DeltaH(vH) around the folding/unfolding transition midpoint to be equal or very close to the calorimetric enthalpy DeltaH(cal) of the entire transition. We use an analytical model with experimental parameters from chymotrypsin inhibitor 2 to elucidate the relationship among several different van't Hoff enthalpies used in calorimetric analyses. Under reasonable assumptions, the implications of these DeltaH(vH)'s being approximately equal to DeltaH(cal) are equivalent: Enthalpic variations among denatured conformations in real proteins are much narrower than some previous lattice-model estimates, suggesting that the energy landscape theory "folding to glass transition temperature ratio" T(f) /T(g) may exceed 6.0 for real calorimetrically two-state proteins. Several popular three-dimensional lattice protein models, with different numbers of residue types in their alphabets, are found to fall short of the high experimental standard for being calorimetrically two-state. Some models postulate a multiple-conformation native state with substantial pre-denaturational energetic fluctuations well below the unfolding transition temperature, or predict a significant post-denaturational continuous conformational expansion of the denatured ensemble at temperatures well above the transition point, or both. These scenarios either disagree with experiments on protein size and dynamics, or are inconsistent with conventional interpretation of calorimetric data. However, when empirical linear baseline subtractions are employed, the resulting DeltaH(vH)/DeltaH(cal)'s for some models can be increased to values closer to unity, and baseline subtractions are found to correspond roughly to an operational definition of native-state conformational diversity. These results necessitate a re-assessment of theoretical models and experimental interpretations.     

Alternate pathways for folding in the flavodoxin fold family revealed by a nucleation-growth model

A recent study of experimental results for flavodoxin-like folds suggests that proteins from this family may exhibit a similar, signature pattern of folding intermediates. We study the folding landscapes of three proteins from the flavodoxin family (CheY, apoflavodoxin, and cutinase) using a simple nucleation and growth model that accurately describes both experimental and simulation results for the transition state structure, and the structure of on-pathway and misfolded intermediates for CheY. Although the landscape features of these proteins agree in basic ways with the results of the study, the simulations exhibit a range of folding behaviours consistent with two alternate folding routes corresponding to nucleation and growth from either side of the central beta-strand.     

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.     

Matching simulation and experiment: a new simplified model for simulating protein folding.

Simulations of simplified protein folding models have provided much insight into solving the protein folding problem. We propose here a new off-lattice bead model, capable of simulating several different fold classes of small proteins. We present the sequence for an alpha/beta protein resembling the IgG-binding proteins L and G. The thermodynamics of the folding process for this model are characterized using the multiple multihistogram method combined with constant-temperature Langevin simulations. The folding is shown to be highly cooperative, with chain collapse nearly accompanying folding. Two parallel folding pathways are shown to exist on the folding free energy landscape. One pathway contains an intermediate--similar to experiments on protein G, and one pathway contains no intermediates-similar to experiments on protein L. The folding kinetics are characterized by tabulating mean-first passage times, and we show that the onset of glasslike kinetics occurs at much lower temperatures than the folding temperature. This model is expected to be useful in many future contexts: investigating questions of the role of local versus nonlocal interactions in various fold classes, addressing the effect of sequence mutations affecting secondary structure propensities, and providing a computationally feasible model for studying the role of solvation forces in protein folding.     

An obligate intermediate along the slow folding pathway of a group II intron ribozyme

Most RNA molecules collapse rapidly and reach the native state through a pathway that contains numerous traps and unproductive intermediates. The D135 group II intron ribozyme is unusual in that it can fold slowly and directly to the native state, despite its large size and structural complexity. Here we use hydroxyl radical footprinting and native gel analysis to monitor the timescale of tertiary structure collapse and to detect the presence of obligate intermediates along the folding pathway of D135. We find that structural collapse and native folding of Domain 1 precede assembly of the entire ribozyme, indicating that D1 contains an on-pathway intermediate to folding of the D135 ribozyme. Subsequent docking of Domains 3 and 5, for which D1 provides a preorganized scaffold, appears to be very fast and independent of one another. In contrast to other RNAs, the D135 ribozyme undergoes slow tertiary collapse to a compacted state, with a rate constant that is also limited by the formation D1. These findings provide a new paradigm for RNA folding and they underscore the diversity of RNA biophysical behaviors.     

Protein folding in the landscape perspective: Chevron plots and non-arrhenius kinetics

We use two simple models and the energy landscape perspective to study protein folding kinetics. A major challenge has been to use the landscape perspective to interpret experimental data, which requires ensemble averaging over the microscopic trajectories usually observed in such models. Here, because of the simplicity of the model, this can be achieved. The kinetics of protein folding falls into two classes: multiple-exponential and two-state (single-exponential) kinetics. Experiments show that two-state relaxation times have “chevron plot” dependences on denaturant and non-Arrhenius dependences on temperature. We find that HP and HP+ models can account for these behaviors. The HP model often gives bumpy landscapes with many kinetic traps and multiple-exponental behavior, whereas the HP+ model gives more smooth funnels and two-state behavior. Multiple-exponential kinetics often involves fast collapse into kinetic traps and slower barrier climbing out of the traps. Two-state kinetics often involves entropic barriers where conformational searching limits the folding speed. Transition states and activation barriers need not define a single conformation; they can involve a broad ensemble of the conformations searched on the way to the native state. We find that unfolding is not always a direct reversal of the folding process. Proteins 30:2–33, 1998. © 1998 Wiley-Liss, Inc.     

Two-state expansion and collapse of a polypeptide

The initial phase of folding for many proteins is presumed to be the collapse of the polypeptide chain from expanded to compact, but still denatured, conformations. Theory and simulations suggest that this collapse may be a two-state transition, characterized by barrier-crossing kinetics, while the collapse of homopolymers and random heteropolymers is continuous and multi-phasic. A new rapid-mixing flow technique has been used to resolve the late stages of polypeptide collapse, at time scales >/=45 microseconds. We have used a laser temperature-jump with fluorescence spectroscopy to resolve the complete time-course of the collapse of denatured cytochrome c with nanosecond time resolution. We find the process to be exponential in time and thermally activated, with an apparent activation energy approximately 9 k(B)T (after correction for solvent viscosity). These results indicate that polypeptide collapse is kinetically a two-state transition. Because of the observed free energy barrier, the time scale of polypeptide collapse is dramatically slower than is predicted by Langevin models for homopolymer collapse.