The topomer-sampling model of protein folding

Synergetic Effects of Alanine and Glycine in Blob-Based Methods for Predicting Protein Folding Times

The polypeptide PGlyAlaGlu was prepared with 20 mol % glycine (Gly), 36 mol % d,l-alanine (Ala), and 44 mol % d,l-glutamic acid (Glu) and labeled with the dye 1-pyrenemethylamine to yield a series of Py-PGlyAlaGlu samples. The fluorescence decays of the Py-PGlyAlaGlu samples were analyzed according to the fluorescence blob model (FBM) to obtain the number N_blob^exp of amino acids (aa's) encompassed inside the subvolume V_blob of the polypeptide probed by an excited pyrene. An N_blob^exp value of 29 (±2) was retrieved for Py-PGlyAlaGlu, which was much larger than for any of the copolypeptide PGlyGlu or PAlaGlu prepared with either Gly and Glu or Ala and Glu, respectively. The continuous increase in N_blob^exp with decreasing side chain size (SCS) from 10 aa's for PGlu to 16 aa's for PAlaGlu and 22 aa's for PGlyGlu was used earlier to define the reach of an aa and determine the groups of aa's that could interact with each other along a polypeptide backbone according to their SCS. These groups of aa's, referred to as blobs, led to the implementation of blob-based models (BBM) to predict the folding time τ_F^theo,BBM of 145 proteins, which was found to match their experimental folding time τ_F^exp with a relatively high 0.71 correlation coefficient. Nevertheless, the much higher N_blob^exp value found for Py-PGlyAlaGlu compared to all other pyrene-labeled polypeptides studied to date indicates that the reach of aa's along a polypeptide sequence is affected not only by SCS but also by synergetic effects between different aa's. Following this new insight, a revised BBM was implemented to predict τ_F^theo,BBM for 195 proteins assuming the existence or absence of synergies to control the interactions between aa's along a polypeptide sequence. Similarly good correlation coefficients of 0.71 and 0.74 were obtained for a direct 1:1 comparison of τ_F^exp and τ_F^theo,BBM for the 195 proteins without and with synergies, respectively. This result suggests that synergetic effects between different aa's have little effect on τ_F^theo,BBM predicted from BBM underlying the robustness of this methodology.

Protein folding problem: enigma, paradox, solution

The ability of protein chains to spontaneously form their three-dimensional structures is a long-standing mystery in molecular biology. The most conceptual aspect of this mystery is how the protein chain can find its native, "working" spatial structure (which, for not too big protein chains, corresponds to the global free energy minimum) in a biologically reasonable time, without exhaustive enumeration of all possible conformations, which would take billions of years. This is the so-called "Levinthal's paradox." In this review, we discuss the key ideas and discoveries leading to the current understanding of protein folding kinetics, including folding landscapes and funnels, free energy barriers at the folding/unfolding pathways, and the solution of Levinthal's paradox. A special role here is played by the "all-or-none" phase transition occurring at protein folding and unfolding and by the point of thermodynamic (and kinetic) equilibrium between the "native" and the "unfolded" phases of the protein chain (where the theory obtains the simplest form). The modern theory provides an understanding of key features of protein folding and, in good agreement with experiments, it (i) outlines the chain length-dependent range of protein folding times, (ii) predicts the observed maximal size of "foldable" proteins and domains. Besides, it predicts the maximal size of proteins and domains that fold under solely thermodynamic (rather than kinetic) control. Complementarily, a theoretical analysis of the number of possible protein folding patterns, performed at the level of formation and assembly of secondary structures, correctly outlines the upper limit of protein folding times.

De novo protein folding on computers. Benefits and challenges

There has been recent success in prediction of the three-dimensional folded native structures of proteins, most famously by the AlphaFold Algorithm running on Google's/Alphabet's DeepMind computer. However, this largely involves machine learning of protein structures and is not a de novo protein structure prediction method for predicting three-dimensional structures from amino acid residue sequences. A de novo approach would be based almost entirely on general principles of energy and entropy that govern protein folding energetics, and importantly do so without the use of the amino acid sequences and structural features of other proteins. Most consider that problem as still unsolved even though it has occupied leading scientists for decades. Many consider that it remains one of the major outstanding issues in modern science. There is crucial continuing help from experimental findings on protein unfolding and refolding in the laboratory, but only to a limited extent because many researchers consider that the speed by which real proteins folds themselves, often from milliseconds to minutes, is itself still not fully understood. This is unfortunate, because a practical solution to the problem would probably have a major effect on personalized medicine, the pharmaceutical industry, biotechnology, and nanotechnology, including for example "smaller" tasks such as better modeling of flexible "unfolded" regions of the SARS-COV-2 spike glycoprotein when interacting with its cell receptor, antibodies, and therapeutic agents. Some important ideas from earlier studies are given before moving on to lessons from periodic and aperiodic crystals, and a possible role for quantum phenomena. The conclusion is that better computation of entropy should be the priority, though that is presented guardedly.

Solution of Levinthal's Paradox and a Physical Theory of Protein Folding Times

“How do proteins fold?” Researchers have been studying different aspects of this question for more than 50 years. The most conceptual aspect of the problem is how protein can find the global free energy minimum in a biologically reasonable time, without exhaustive enumeration of all possible conformations, the so-called “Levinthal’s paradox.” Less conceptual but still critical are aspects about factors defining folding times of particular proteins and about perspectives of machine learning for their prediction. We will discuss in this review the key ideas and discoveries leading to the current understanding of folding kinetics, including the solution of Levinthal’s paradox, as well as the current state of the art in the prediction of protein folding times.

There and back again: Two views on the protein folding puzzle

The ability of protein chains to spontaneously form their spatial structures is a long-standing puzzle in molecular biology. Experimentally measured folding times of single-domain globular proteins range from microseconds to hours: the difference (10-11 orders of magnitude) is the same as that between the life span of a mosquito and the age of the universe. This review describes physical theories of rates of overcoming the free-energy barrier separating the natively folded (N) and unfolded (U) states of protein chains in both directions: "U-to-N" and "N-to-U". In the theory of protein folding rates a special role is played by the point of thermodynamic (and kinetic) equilibrium between the native and unfolded state of the chain; here, the theory obtains the simplest form. Paradoxically, a theoretical estimate of the folding time is easier to get from consideration of protein unfolding (the "N-to-U" transition) rather than folding, because it is easier to outline a good unfolding pathway of any structure than a good folding pathway that leads to the stable fold, which is yet unknown to the folding protein chain. And since the rates of direct and reverse reactions are equal at the equilibrium point (as follows from the physical "detailed balance" principle), the estimated folding time can be derived from the estimated unfolding time. Theoretical analysis of the "N-to-U" transition outlines the range of protein folding rates in a good agreement with experiment. Theoretical analysis of folding (the "U-to-N" transition), performed at the level of formation and assembly of protein secondary structures, outlines the upper limit of protein folding times (i.e., of the time of search for the most stable fold). Both theories come to essentially the same results; this is not a surprise, because they describe overcoming one and the same free-energy barrier, although the way to the top of this barrier from the side of the unfolded state is very different from the way from the side of the native state; and both theories agree with experiment. In addition, they predict the maximal size of protein domains that fold under solely thermodynamic (rather than kinetic) control and explain the observed maximal size of the "foldable" protein domains.

Computational and theoretical methods for protein folding

A computational approach is essential whenever the complexity of the process under study is such that direct theoretical or experimental approaches are not viable. This is the case for protein folding, for which a significant amount of data are being collected. This paper reports on the essential role of in silico methods and the unprecedented interplay of computational and theoretical approaches, which is a defining point of the interdisciplinary investigations of the protein folding process. Besides giving an overview of the available computational methods and tools, we argue that computation plays not merely an ancillary role but has a more constructive function in that computational work may precede theory and experiments. More precisely, computation can provide the primary conceptual clues to inspire subsequent theoretical and experimental work even in a case where no preexisting evidence or theoretical frameworks are available. This is cogently manifested in the application of machine learning methods to come to grips with the folding dynamics. These close relationships suggested complementing the review of computational methods within the appropriate theoretical context to provide a self-contained outlook of the basic concepts that have converged into a unified description of folding and have grown in a synergic relationship with their computational counterpart. Finally, the advantages and limitations of current computational methodologies are discussed to show how the smart analysis of large amounts of data and the development of more effective algorithms can improve our understanding of protein folding.

The principle of stationary action in biophysics: stability in protein folding

We conceptualize protein folding as motion in a large dimensional dihedral angle space. We use Lagrangian mechanics and introduce an unspecified Lagrangian to study the motion. The fact that we have reliable folding leads us to conjecture the totality of paths forms caustics that can be recognized by the vanishing of the second variation of the action. There are two types of folding processes: stable against modest perturbations and unstable. We also conjecture that natural selection has picked out stable folds. More importantly, the presence of caustics leads naturally to the application of ideas from catastrophe theory and allows us to consider the question of stability for the folding process from that perspective. Powerful stability theorems from mathematics are then applicable to impose more order on the totality of motions. This leads to an immediate explanation for both the insensitivity of folding to solution perturbations and the fact that folding occurs using very little free energy. The theory of folding, based on the above conjectures, can also be used to explain the behavior of energy landscapes, the speed of folding similar to transition state theory, and the fact that random proteins do not fold.

Why do protein folding rates correlate with metrics of native topology?

For almost 15 years, the experimental correlation between protein folding rates and the contact order parameter has been under scrutiny. Here, we use a simple simulation model combined with a native-centric interaction potential to investigate the physical roots of this empirical observation. We simulate a large set of circular permutants, thus eliminating dependencies of the folding rate on other protein properties (e.g. stability). We show that the rate-contact order correlation is a consequence of the fact that, in high contact order structures, the contact order of the transition state ensemble closely mirrors the contact order of the native state. This happens because, in these structures, the native topology is represented in the transition state through the formation of a network of tertiary interactions that are distinctively long-ranged.

A versatile method for systematic conformational searches: application to CheY

A novel molecular structure prediction method, the Z Method, is described. It provides a versatile platform for the development and use of systematic, grid-based conformational search protocols, in which statistical information (i.e., rotamers) can also be included. The Z Method generates trial structures by applying many changes of the same type to a single starting structure, thereby sampling the conformation space in an unbiased way. The method, implemented in the CHARMM program as the Z Module, is applied here to an illustrative model problem in which rigid, systematic searches are performed in a 36-dimensional conformational space that describes the relative positions of the 10 secondary structural elements of the protein CheY. A polar hydrogen representation with an implicit solvation term (EEF1) is used to evaluate successively larger fragments of the protein generated in a hierarchical build-up procedure. After a final refinement stage, and a total computational time of about two-and-a-half CPU days on AMD Opteron processors, the prediction is within 1.56 Å of the native structure. The errors in the predicted backbone dihedral angles are found to approximately cancel. Monte Carlo and simulated annealing trials on the same or smaller versions of the problem, using the same atomic model and energy terms, are shown to result in less accurate predictions. Although the problem solved here is a limited one, the findings illustrate the utility of systematic searches with atom-based models for macromolecular structure prediction and the importance of unbiased sampling in structure prediction methods.

What have we learned from the studies of two-state folders, and what are the unanswered questions about two-state protein folding?

Small proteins with globular structures often fold by simple all-or-none mechanisms, both in an equilibrium and a kinetic sense, despite the very large number of partly folded conformations available. This type of 'two-state' folding will be discussed in terms of experimental tests, underlying molecular mechanisms, and limits to two-state behavior. Factors that appear to be important for two-state folding include topology (sequence distance of contacts in the native structure), molecular cooperativity and local energy distribution. Because their local stability distributions and cooperativities can be dissected and analyzed separately from topological features, recent studies of the folding of symmetric proteins will be discussed as a means to better understand the origins of two-state folding.

The complex folding pathways of protein A suggest a multiple-funnelled energy landscape

Folding proteins into their native states requires the formation of both secondary and tertiary structures. Many questions remain, however, as to whether these form into a precise order, and various pictures have been proposed that place the emphasis on the first or the second level of structure in describing folding. One of the favorite test models for studying this question is the B domain of protein A, which has been characterized by numerous experiments and simulations. Using the activation-relaxation technique coupled with a generic energy model (optimized potential for efficient peptide structure prediction), we generate more than 50 folding trajectories for this 60-residue protein. While the folding pathways to the native state are fully consistent with the funnel-like description of the free energy landscape, we find a wide range of mechanisms in which secondary and tertiary structures form in various orders. Our nonbiased simulations also reveal the presence of a significant number of non-native beta and alpha conformations both on and off pathway, including the visit, for a non-negligible fraction of trajectories, of fully ordered structures resembling the native state of nonhomologous proteins.

Intra-molecular cross-linking of acidic residues for protein structure studies

Intra-molecular cross-linking has been suggested as a method of obtaining distance constraints that would help to develop structural models of proteins. Recent work published on intra-molecular cross-linking for protein structural studies has employed commercially available primary amine (lysine, the amino terminus) selective reagents. Previous work using these cross-linkers has shown that for several proteins of known structure, the number of cross-links that can be obtained experimentally may be small compared to what would be expected from the known structure, due to the relative reactivity, distribution and solvent accessibility of the lysines in the protein sequence. To overcome these limitations, we have investigated the use of cross-linking reagents that can react with other reactive side chains in proteins. We used 1-ethyl-3-(3-dimethylaminopropyl) carbodiimide hydrochloride (EDC) to activate the carboxylic acid containing residues, aspartic acid (D), glutamic acid (E) and the carboxy terminus (O), for cross-linking reactions. Once activated, the DEO side chains can react to form "zero-length" cross-links with nearby primary amine containing residues, lysines (K) and the amino terminus (X), via the formation of a new amide bond. We also show that the EDC-activated DEO side chains can be cross-linked to each other using dihydrazides, two hydrazide moieties connected by an alkyl cross-linker arm of variable length. Using these reagents, we have found three new "zero-length" cross-links in ubiquitin consistent with its known structure (M1-E16, M1-E18 and K63-E64). Using the dihydrazide cross-linkers, we have identified two new cross-links (D21-D32 and E24-D32) unambiguously. Using a library of dihydrazide cross-linkers with varying arm length, we have shown that there is a minimum arm length required for the DEO-DEO cross-links of 5.8 A. These results show that additional structural information can be obtained by exploiting new cross-linker chemistry, increasing the probability that the protein target will yield sufficient distance constraints to develop a structural model.

The protein folding problem

The "protein folding problem" consists of three closely related puzzles: (a) What is the folding code? (b) What is the folding mechanism? (c) Can we predict the native structure of a protein from its amino acid sequence? Once regarded as a grand challenge, protein folding has seen great progress in recent years. Now, foldable proteins and nonbiological polymers are being designed routinely and moving toward successful applications. The structures of small proteins are now often well predicted by computer methods. And, there is now a testable explanation for how a protein can fold so quickly: A protein solves its large global optimization problem as a series of smaller local optimization problems, growing and assembling the native structure from peptide fragments, local structures first.

Loop-closure principles in protein folding

Simple theoretical concepts and models have been helpful to understand the folding rates and routes of single-domain proteins. As reviewed in this article, a physical principle that appears to underly these models is loop closure.

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.

Exploring the relationship between funneled energy landscapes and two-state protein folding

It should take an astronomical time span for unfolded protein chains to find their native state based on an unguided conformational random search. The experimental observation that folding is fast can be rationalized by assuming that protein energy landscapes are sloped towards the native state minimum, such that rapid folding can proceed from virtually any point in conformational space. Folding transitions often exhibit two-state behavior, involving extensively disordered and highly structured conformers as the only two observable kinetic species. This study employs a simple Brownian dynamics model of "protein particles" moving in a spherically symmetrical potential. As expected, the presence of an overall slope towards the native state minimum is an effective means to speed up folding. However, the two-state nature of the transition is eradicated if a significant energetic bias extends too far into the non-native conformational space. The breakdown of two-state cooperativity under these conditions is caused by a continuous conformational drift of the unfolded proteins. Ideal two-state behavior can only be maintained on surfaces exhibiting large regions that are energetically flat, a result that is supported by other recent data in the literature (Kaya and Chan, Proteins: Struct Funct Genet 2003;52:510-523). Rapid two-state folding requires energy landscapes exhibiting the following features: (i) A large region in conformational space that is energetically flat, thus allowing for a significant degree of random sampling, such that unfolded proteins can retain a random coil structure; (ii) a trapping area that is strongly sloped towards the native state minimum.

Quantifying Polypeptide Conformational Space: Sensitivity to Conformation and Ensemble Definition

Quantifying the density of conformations over phase space (the conformational distribution) is needed to model important macromolecular processes such as protein folding. In this work, we quantify the conformational distribution for a simple polypeptide (N-mer polyalanine) using the cumulative distribution function (CDF), which gives the probability that two randomly selected conformations are separated by less than a "conformational" distance and whose inverse gives conformation counts as a function of conformational radius. An important finding is that the conformation counts obtained by the CDF inverse depend critically on the assignment of a conformation's distance span and the ensemble (e.g., unfolded state model): varying ensemble and conformation definition (1 --> 2 A) varies the CDF-based conformation counts for Ala(50) from 10(11) to 10(69). In particular, relatively short molecular dynamics (MD) relaxation of Ala(50)'s random-walk ensemble reduces the number of conformers from 10(55) to 10(14) (using a 1 A root-mean-square-deviation radius conformation definition) pointing to potential disconnections in comparing the results from simplified models of unfolded proteins with those from all-atom MD simulations. Explicit waters are found to roughen the landscape considerably. Under some common conformation definitions, the results herein provide (i) an upper limit to the number of accessible conformations that compose unfolded states of proteins, (ii) the optimal clustering radius/conformation radius for counting conformations for a given energy and solvent model, (iii) a means of comparing various studies, and (iv) an assessment of the applicability of random search in protein folding.

Study of the Villin headpiece folding dynamics by combining coarse-grained Monte Carlo evolution and all-atom molecular dynamics

The folding mechanism of the Villin headpiece (HP36) is studied by means of a novel approach which entails an initial coarse-grained Monte Carlo (MC) scheme followed by all-atom molecular dynamics (MD) simulations in explicit solvent. The MC evolution occurs in a simplified free-energy landscape and allows an efficient selection of marginally-compact structures which are taken as viable initial conformations for the MD. The coarse-grained MC structural representation is connected to the one with atomic resolution through a "fine-graining" reconstruction algorithm. This two-stage strategy is used to select and follow the dynamics of seven different unrelated conformations of HP36. In a notable case the MD trajectory rapidly evolves towards the folded state, yielding a typical root-mean-square deviation (RMSD) of the core region of only 2.4 A from the closest NMR model (the typical RMSD over the whole structure being 4.0 A). The analysis of the various MC-MD trajectories provides valuable insight into the details of the folding and mis-folding mechanisms and particularly about the delicate influence of local and nonlocal interactions in steering the folding process.

Are structural biases at protein termini a signature of vectorial folding?

Experimental investigations of the biosynthesis of a number of proteins have pointed out that part of the native structure may be acquired already during translation. We carried out a comprehensive statistical analysis of some average structural properties of proteins that have been put forward as possible signatures of this progressive buildup process. Contrary to a widespread belief, we found that there is no major propensity of the amino acids to form contacts with residues that are closer to the N-terminus. Moreover, we found that the C-terminus is significantly more compact and locally organized than the N-terminus. This bias, though, is unlikely to be related to vectorial effects, since it correlates with subtle differences in the primary sequence. These findings indicate that even if proteins acquire their structure vectorially, no signature of this seems to be detectable in their average structural properties.

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

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

How well can simulation predict protein folding kinetics and thermodynamics?

Simulation of protein folding has come a long way in five years. Notably, new quantitative comparisons with experiments for small, rapidly folding proteins have become possible. As the only way to validate simulation methodology, this achievement marks a significant advance. Here, we detail these recent achievements and ask whether simulations have indeed rendered quantitative predictions in several areas, including protein folding kinetics, thermodynamics, and physics-based methods for structure prediction. We conclude by looking to the future of such comparisons between simulations and experiments.

The Gō model revisited: Native structure and the geometric coupling between local and long-range contacts

Monte Carlo simulations show that long-range interactions play a major role in determining the folding rates of 48-mer three-dimensional lattice polymers modeled by the Gō potential. For three target structures with different native geometries we found a sharp increase in the folding time when the relative contribution of the long-range interactions to the native state's energy is decreased from approximately 50% towards zero. However, the dispersion of the simulated folding times is strongly dependent on native geometry and Gō polymers folding to one of the target structures exhibits folding times spanning three orders of magnitude. We have also found that, depending on the target geometry, a strong geometric coupling may exist between local and long-range contacts, which means that, when this coupling exists, the formation of long-range contacts is forced by the previous formation of local contacts. The absence of a strong geometric coupling results in a kinetics that is more sensitive to the interaction energy parameters; in this case, the formation of local contacts is not capable of promoting the establishment of long-range ones when the latter are strongly penalized energetically and this results in longer folding times.

The heme iron coordination of unfolded ferric and ferrous cytochrome c in neutral and acidic urea solutions. Spectroscopic and electrochemical studies

The heme iron coordination of unfolded ferric and ferrous cytochrome c in the presence of 7-9 M urea at different pH values has been probed by several spectroscopic techniques including magnetic and natural circular dichroism (CD), electrochemistry, UV-visible (UV-vis) absorption and resonance Raman (RR). In 7-9 M urea at neutral pH, ferric cytochrome c is found to be predominantly a low spin bis-His-ligated heme center. In acidic 9 M urea solutions the UV-vis and near-infrared (NIR) magnetic circular dichroism (MCD) measurements have for the first time revealed the formation of a high spin His/H(2)O complex. The pK(a) for the neutral to acidic conversion is 5.2. In 9 M urea, ferrous cytochrome c is shown to retain its native ligation structure at pH 7. Formation of a five-coordinate high spin complex in equilibrium with the native form of ferrous cytochrome c takes place below the pK(a) 4.8. The formal redox potential of the His/H(2)O complex of cytochrome c in 9 M urea at pH 3 was estimated to be -0.13 V, ca. 100 mV more positive than E degrees ' estimated for the bis-His complex of cytochrome c in urea solution at pH 7.

Protein folding and the organization of the protein topology universe

The mechanism by which proteins fold to their native states has been the focus of intense research in recent years. The rate-limiting event in the folding reaction is the formation of a conformation in a set known as the transition-state ensemble. The structural features present within such ensembles have now been analysed for a series of proteins using data from a combination of biochemical and biophysical experiments together with computer-simulation methods. These studies show that the topology of the transition state is determined by a set of interactions involving a small number of key residues and, in addition, that the topology of the transition state is closer to that of the native state than to that of any other fold in the protein universe. Here, we review the evidence for these conclusions and suggest a molecular mechanism that rationalizes these findings by presenting a view of protein folds that is based on the topological features of the polypeptide backbone, rather than the conventional view that depends on the arrangement of different types of secondary-structure elements. By linking the folding process to the organization of the protein structure universe, we propose an explanation for the overwhelming importance of topology in the transition states for protein folding.

Using protein folding rates to test protein folding theories

The fastest simple, kinetically two-state protein folds a million times more rapidly than the slowest. Here we review many recent theories of protein folding kinetics in terms of their ability to qualitatively rationalize, if not quantitatively predict, this fundamental experimental observation.

Transition states for protein folding have native topologies despite high structural variability

We present a structural analysis of the folding transition states of three SH3 domains. Our results reveal that the secondary structure is not yet fully formed at this stage of folding and that the solvent is only partially excluded from the interior of the protein. Comparison of the members of the transition state ensemble with a database of native folds shows that, despite substantial local variability, the transition state structures can all be classified as having the topology characteristic of an SH3 domain. Our results suggest a mechanism for folding in which the formation of a network of interactions among a subset of hydrophobic residues ensures that the native topology is generated. Such a mechanism enables high fidelity in folding while minimizing the need to establish a large number of specific interactions in the conformational search.

Critical nucleation size in the folding of small apparently two-state proteins

For apparently two-state proteins, we found that the size (number of folded residues) of a transition state is mostly encoded by the topology, defined by total contact distance (TCD) of the native state, and correlates with its folding rate. This is demonstrated by using a simple procedure to reduce the native structures of the 41 two-state proteins with native TCD as a constraint, and is further supported by analyzing the results of eight proteins from protein engineering studies. These results support the hypothesis that the major rate-limiting process in the folding of small apparently two-state proteins is the search for a critical number of residues with the topology close to that of the native state.

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.

Interplay among tertiary contacts, secondary structure formation and side-chain packing in the protein folding mechanism: all-atom representation study of protein L

Experimental and theoretical results suggest that, since proteins are energetically minimally frustrated, the native fold, or topology, plays a primary role in determining the structure of the transition state ensemble and on-pathway intermediate states in protein folding. Although the central role of native state topology in determining the folding mechanism is thought to be a quite general result-at least for small two-state folding proteins-there are remarkable exceptions. Recent experimental findings have shown that topology alone cannot always determine the folding mechanism, and demonstrated that the balance between topology and energetics is very delicate. This balance seems to be particularly critical in proteins with a highly symmetrical native structure, such as proteins L and G, which have similar native structure topology but fold by different mechanisms. Simplified, C(alpha)-atom only protein models have shown not be sufficient to differentiate these mechanisms. An all-atom Gō model provides a valuable intermediate model between structurally simplified protein representations and all-atom protein simulations with explicit/implicit solvent descriptions. We present here a detailed study of an all-atom Gō-like representation of protein L, in close comparison with the experimental results and with the results obtained from a simple C(alpha)-atom representation of the same protein. We also perform simulations for protein G, where we obtain a folding mechanism in which the protein symmetry is broken exactly in the opposite way to protein L as has been observed experimentally. A detailed analysis for protein L also shows that the role of specific residues is correctly and quantitatively reproduced by the all-atom Gō model over almost the entire protein.

Cooperativity, smooth energy landscapes and the origins of topology-dependent protein folding rates

The relative folding rates of simple, single-domain proteins, proteins whose folding energy landscapes are smooth, are highly dispersed and strongly correlated with native-state topology. In contrast, the relative folding rates of small, Gō-potential lattice polymers, which also exhibit smooth energy landscapes, are poorly dispersed and insignificantly correlated with native-state topology. Here, we investigate this discrepancy in light of a recent, quantitative theory of two-state folding kinetics, the topomer search model. This model stipulates that the topology-dependence of two-state folding rates is a direct consequence of the extraordinarily cooperative equilibrium folding of simple proteins. We demonstrate that traditional Gō polymers lack the extreme cooperativity that characterizes the folding of naturally occurring, two-state proteins and confirm that the folding rates of a diverse set of Gō 27-mers are poorly dispersed and effectively uncorrelated with native state topology. Upon modestly increasing the cooperativity of the Gō-potential, however, significantly increased dispersion and strongly topology-dependent kinetics are observed. These results support previous arguments that the cooperative folding of simple, single-domain proteins gives rise to their topology-dependent folding rates. We speculate that this cooperativity, and thus, indirectly, the topology-rate relationship, may have arisen in order to generate the smooth energetic landscapes upon which rapid folding can occur.

The topomer search model: A simple, quantitative theory of two-state protein folding kinetics

Most small, single-domain proteins fold with the uncomplicated, single-exponential kinetics expected for diffusion on a smooth energy landscape. Despite this energetic smoothness, the folding rates of these two-state proteins span a remarkable million-fold range. Here, we review the evidence in favor of a simple, mechanistic description, the topomer search model, which quantitatively accounts for the broad scope of observed two-state folding rates. The model, which stipulates that the search for those unfolded conformations with a grossly correct topology is the rate-limiting step in folding, fits observed rates with a correlation coefficient of approximately 0.9 using just two free parameters. The fitted values of these parameters, the pre-exponential attempt frequency and a measure of the difficulty of ordering an unfolded chain, are consistent with previously reported experimental constraints. These results suggest that the topomer search process may dominate the relative barrier heights of two-state protein-folding reactions.

Cytochrome c' folding triggered by electron transfer: fast and slow formation of four-helix bundles

Reduced (Fe(II)) Rhodopseudomonas palustris cytochrome c' (Cyt c') is more stable toward unfolding ([GuHCl](1/2) = 2.9(1) M) than the oxidized (Fe(III)) protein ([GuHCl](1/2) = 1.9(1) M). The difference in folding free energies (Delta Delta G(f) degrees = 70 meV) is less than half of the difference in reduction potentials of the folded protein (100 mV vs. NHE) and a free heme in aqueous solution ( approximately -150 mV). The spectroscopic features of unfolded Fe(II)-Cyt c' indicate a low-spin heme that is axially coordinated to methionine sulfur (Met-15 or Met-25). Time-resolved absorption measurements after CO photodissociation from unfolded Fe(II)(CO)-Cyt c' confirm that methionine can bind to the ferroheme on the microsecond time scale [k(obs) = 5(2) x 10(4) s(-1)]. Protein folding was initiated by photoreduction (two-photon laser excitation of NADH) of unfolded Fe(III)-Cyt c' ([GuHCl] = 2.02--2.54 M). Folding kinetics monitored by heme absorption span a wide time range and are highly heterogeneous; there are fast-folding ( approximately 10(3) s(-1)), intermediate-folding (10(2)-10(1) s(-1)), and slow-folding (10(-1) s(-1)) populations, with the last two likely containing methionine-ligated (Met-15 or Met-25) ferrohemes. Kinetics after photoreduction of unfolded Fe(III)-Cyt c' in the presence of CO are attributable to CO binding [1.4(6) x 10(3) s(-1)] and Fe(II)(CO)-Cyt c' folding [2.8(9) s(-1)] processes; stopped-flow triggered folding of Fe(III)-Cyt c' (which does not contain a protein-derived sixth ligand) is adequately described by a single kinetics phase with an estimated folding time constant of approximately 4 ms [Delta G(f) degrees = -33(3) kJ mol(-1)] at zero denaturant.

Learning effective amino acid interactions through iterative stochastic techniques

The prediction of the three-dimensional structures of the native states of proteins from the sequences of their amino acids is one of the most important challenges in molecular biology. An essential task for solving this problem within coarse-grained models is the deduction of effective interaction potentials between the amino acids. Over the years, several techniques have been developed to extract potentials that are able to discriminate satisfactorily between the native and nonnative folds of a preassigned protein sequence. In general, when these potentials are used in actual dynamical folding simulations, they lead to a drift of the native structure outside the quasinative basin. In this article, we present and validate an approach to overcome this difficulty. By exploiting several numerical and analytical tools, we set up a rigorous iterative scheme to extract potentials satisfying a prerequisite of any viable potential: the stabilization of proteins within their native basin (less than 3-4 A RMSD). The scheme is flexible and is demonstrated to be applicable to a variety of parameterizations of the energy function, and it provides in each case the optimal potentials.

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.

Disulfide bonds and protein folding

The applications of disulfide-bond chemistry to studies of protein folding, structure, and stability are reviewed and illustrated with bovine pancreatic ribonuclease A (RNase A). After surveying the general properties and advantages of disulfide-bond studies, we illustrate the mechanism of reductive unfolding with RNase A, and discuss its application to probing structural fluctuations in folded proteins. The oxidative folding of RNase A is then described, focusing on the role of structure formation in the regeneration of the native disulfide bonds. The development of structure and conformational order in the disulfide intermediates during oxidative folding is characterized. Partially folded disulfide species are not observed, indicating that disulfide-coupled folding is highly cooperative. Contrary to the predictions of "rugged funnel" models of protein folding, misfolded disulfide species are also not observed despite the potentially stabilizing effect of many nonnative disulfide bonds. The mechanism of regenerating the native disulfide bonds suggests an analogous scenario for conformational folding. Finally, engineered covalent cross-links may be used to assay for the association of protein segments in the folding transition state, as illustrated with RNase A.

First principles prediction of protein folding rates

Experimental studies have demonstrated that many small, single-domain proteins fold via simple two-state kinetics. We present a first principles approach for predicting these experimentally determined folding rates. Our approach is based on a nucleation-condensation folding mechanism, where the rate-limiting step is a random, diffusive search for the native tertiary topology. To estimate the rates of folding for various proteins via this mechanism, we first determine the probability of randomly sampling a conformation with the native fold topology. Next, we convert these probabilities into folding rates by estimating the rate that a protein samples different topologies during diffusive folding. This topology-sampling rate is calculated using the Einstein diffusion equation in conjunction with an experimentally determined intra-protein diffusion constant. We have applied our prediction method to the 21 topologically distinct small proteins for which two-state rate data is available. For the 18 beta-sheet and mixed alpha-beta native proteins, we predict folding rates within an average factor of 4, even though the experimental rates vary by a factor of approximately 4 x 10(4). Interestingly, the experimental folding rates for the three four-helix bundle proteins are significantly underestimated by this approach, suggesting that proteins with significant helical content may fold by a faster, alternative mechanism. This method can be applied to any protein for which the structure is known and hence can be used to predict the folding rates of many proteins prior to experiment.