Predicting protein folding pathways

A structured folding pathway, which is a time ordered sequence of folding events, plays an important role in the protein folding process and hence, in the conformational search. Pathway prediction, thus gives more insight into the folding process and is a valuable guiding tool to search the conformation space. In this paper, we propose a novel 'unfolding' approach to predict the folding pathway. We apply graph-based methods on a weighted secondary structure graph of a protein to predict the sequence of unfolding events. When viewed in reverse this yields the folding pathway. We demonstrate the success of our approach on several proteins whose pathway is partially known.     

Prediction of protein folding pathways.

Recent 1H nuclear magnetic resonance (n.m.r.) hydrogen exchange experiments on five different proteins have delineated the secondary structures formed in trapped, partially folded intermediates. The early forming structural elements are identifiable through a technique described in this work to predict folding pathways. The method assumes that the sequential selection of structural fragments such as alpha-helices and beta-strands involved in the folding process is founded upon the maximal burial of solvent accessible surface from both the formation of internal structure and substructure association. The substructural elements were defined objectively by major changes in main-chain direction. The predicted folding pathways are in complete correspondence with the n.m.r. results in that the formed structural fragments found in the folding intermediates are those predicted earliest in the pathways. The technique was also applied to proteins of known tertiary structure and with fold similar to one of the five proteins examined by 1H n.m.r. The pathways for these structures also showed general consistency with the n.m.r. observations, suggesting conservation of a secondary structural framework or molten globule about which folding nucleates and proceeds.     

An analysis of protein folding pathways

We have developed a model of the protein folding process based on three primary assumptions: that burying of hydrophobic area is the dominant contribution to the relative free energy of a conformation, that a record of the folding process is largely preserved in the final structure, and that the denatured state is a random coil. Detailed folding pathways are identified for 19 protein structures. The picture of the folding process that emerges from this analysis is one of nucleation by regions of 8-16 residues. Nucleation sites then lead to larger structures by two mechanisms: propagation and diffusion/collision. A Monte Carlo simulation is used to follow the folding pathway when propagation is the dominant mechanism. Because detailed pathways are derived for each protein, the models are susceptible to experimental verification.     

Equilibrium folding and unfolding pathways for a model protein

The folding–unfolding process of reduced bovine pancreatic trypsin inhibitor was investigated with an idealized model employing approximate free energies. The protein is regarded to consist of only C^α and C^β atoms. The backbone dihedral angles are the only conformational variables and are permitted to take discrete values at every 10°. Intraresidue energies consist of two terms: an empirical part taken from the observed frequency distributions of (?,ψ) and an additional favorable energy assigned to the native conformation of each residue. Interresidue interactions are simplified by assuming that there is an attractive energy operative only between residue pairs in close contact in the native structure. A total of 230,000 molecular conformations, with no atomic overlaps, ranging from the native state to the denatured state, are randomly generated by changing the sampling bias. Each conformation is classified according to its conformational energy, F; a conformational entropy, S(F) is estimated for each value of F from the number of samples. The dependence of S(F) on energy reveals that the folding–unfolding transition for this idealized model is an “all-or-none” type; this is attributable to the specific long-range interactions. Interresidue contact probabilities, averaged over samples representing various stages of folding, serve to characterize folding intermediates. Most probable equilibrium pathways for the folding–unfolding transition are constructed by connecting conformationally similar intermediates. The specific details obtained for bovine pancreatic trypsin inhibitor are as follows: (1) Folding begins with the appearance of nativelike medium-range contacts at a β-turn and at the α-helix. (2) These grow to include the native pair of interacting β-strands. This state includes intact regular secondary conformations, as well as the interstrand sheet contacts, and corresponds to an activated state with the highest free energy on the pathway. (3) Additional native long-range contacts are completely formed either toward the amino terminus or toward the carboxyl terminus. (4) In a final step, the missing contacts appear. Although these folding pathways for this model are not consistent with experimental reports, it does indicate multiple folding pathways. The method is general and can be applied to any set of calculated conformational energies and furthermore permits investigation of gross folding features.     

Constructing templates for protein structure prediction by simulation of protein folding pathways

How a one-dimensional protein sequence folds into a specific 3D structure remains a difficult challenge in structural biology. Many computational methods have been developed in an attempt to predict the tertiary structure of the protein; most of these employ approaches that are based on the accumulated knowledge of solved protein structures. Here we introduce a novel and fully automated approach for predicting the 3D structure of a protein that is based on the well accepted notion that protein folding is a hierarchical process. Our algorithm follows the hierarchical model by employing two stages: the first aims to find a match between the sequences of short independently-folding structural entities and parts of the target sequence and assigns the respective structures. The second assembles these local structural parts into a complete 3D structure, allowing for long-range interactions between them. We present the results of applying our method to a subset of the targets from CASP6 and CASP7. Our results indicate that for targets with a significant sequence similarity to known structures we are often able to provide predictions that are better than those achieved by two leading servers, and that the most significant improvements in comparison with these methods occur in regions of a gapped structural alignment between the native structure and the closest available structural template. We conclude that in addition to performing well for targets with known homologous structures, our method shows great promise for addressing the more general category of comparative modeling targets, which is our next goal.     

Hierarchical protein folding pathways: a computational study of protein fragments

We have previously presented a building block folding model. The model postulates that protein folding is a hierarchical top-down process. The basic unit from which a fold is constructed, referred to as a hydrophobic folding unit, is the outcome of combinatorial assembly of a set of "building blocks." Results obtained by the computational cutting procedure yield fragments that are in agreement with those obtained experimentally by limited proteolysis. Here we show that as expected, proteins from the same family give very similar building blocks. However, different proteins can also give building blocks that are similar in structure. In such cases the building blocks differ in sequence, stability, contacts with other building blocks, and in their 3D locations in the protein structure. This result, which we have repeatedly observed in many cases, leads us to conclude that while a building block is influenced by its environment, nevertheless, it can be viewed as a stand-alone unit. For small-sized building blocks existing in multiple conformations, interactions with sister building blocks in the protein will increase the population time of the native conformer. With this conclusion in hand, it is possible to develop an algorithm that predicts the building block assignment of a protein sequence whose structure is unknown. Toward this goal, we have created sequentially nonredundant databases of building block sequences. A protein sequence can be aligned against these, in order to be matched to a set of potential building blocks.     

Amino acid substitutions influencing intracellular protein folding pathways.

Though an increasing variety of chaperonins are emerging as important factors in directing polypeptide chain folding off the ribosome, the primary amino acid sequence remains the major determinant of final conformation. The ability to identify cytoplasmic folding intermediates in the formation of the tailspike endorhamnosidase of phage P22 has made it possible to isolate two classes of mutations influencing folding intermediates-temperature-sensitive folding mutations and global suppressors of tsf mutants. These and related amino acid substitutions in eukaryotic proteins are discussed in the context of inclusion body formation and problems in the recovery of correctly folded proteins.     

Parallel folding pathways in the SH3 domain protein

The transition-state ensemble (TSE) is the set of protein conformations with an equal probability to fold or unfold. Its characterization is crucial for an understanding of the folding process. We determined the TSE of the src-SH3 domain protein by using extensive molecular dynamics simulations of the Go model and computing the folding probability of a generated set of TSE candidate conformations. We found that the TSE possesses a well-defined hydrophobic core with variable enveloping structures resulting from the superposition of three parallel folding pathways. The most preferred pathway agrees with the experimentally determined TSE, while the two least preferred pathways differ significantly. The knowledge of the different pathways allows us to design the interactions between amino acids that guide the protein to fold through the least preferred pathway. This particular design is akin to a circular permutation of the protein. The finding motivates the hypothesis that the different experimentally observed TSEs in homologous proteins and circular permutants may represent potentially available pathways to the wild-type protein.     

Using motion planning to study protein folding pathways.

We present a framework for studying protein folding pathways and potential landscapes which is based on techniques recently developed in the robotics motion planning community. Our focus in this work is to study the protein folding mechanism assuming we know the native fold. That is, instead of performing fold prediction, we aim to study issues related to the folding process, such as the formation of secondary and tertiary structure, and the dependence of the folding pathway on the initial denatured conformation. Our work uses probabilistic roadmap (PRM) motion planning techniques which have proven successful for problems involving high-dimensional configuration spaces. A strength of these methods is their efficiency in rapidly covering the planning space without becoming trapped in local minima. We have applied our PRM technique to several small proteins (~60 residues) and validated the pathways computed by comparing the secondary structure formation order on our paths to known hydrogen exchange experimental results. An advantage of the PRM framework over other simulation methods is that it enables one to easily and efficiently compute folding pathways from any denatured starting state to the (known) native fold. This aspect makes our approach ideal for studying global properties of the protein's potential landscape, most of which are difficult to simulate and study with other methods. For example, in the proteins we study, the folding pathways starting from different denatured states sometimes share common portions when they are close to the native fold, and moreover, the formation order of the secondary structure appears largely independent of the starting denatured conformation. Another feature of our technique is that the distribution of the sampled conformations is correlated with the formation of secondary structure and, in particular, appears to differentiate situations in which secondary structure clearly forms first and those in which the tertiary structure is obtained more directly. Overall, our results applying PRM techniques are very encouraging and indicate the promise of our approach for studying proteins for which experimental results are not available.     

Malleability of protein folding pathways: a simple reason for complex behaviour

Although the structures of native proteins are generally unique, the pathways by which they form are often free to vary. Some proteins fold by a multitude of different pathways, whereas others seem restricted to only one choice. An explanation for this variation in folding behaviour has recently emerged from studies of transition state changes: the number of accessible pathways is linked to the number of nucleation motifs contained within the native topology. We refer to these nucleation motifs as 'foldons', as they approach the size of an independent cooperative unit. Thus, with respect to pathway malleability and the composition of the folding funnel, proteins can be seen as modular assemblies of competing foldons. For the split beta-alpha-beta fold, these foldons are two-strand-helix motifs coupled by spatial overlap.     

A unified mechanism for protein folding: predetermined pathways with optional errors

There is a fundamental conflict between two different views of how proteins fold. Kinetic experiments and theoretical calculations are often interpreted in terms of different population fractions folding through different intermediates in independent unrelated pathways (IUP model). However, detailed structural information indicates that all of the protein population folds through a sequence of intermediates predetermined by the foldon substructure of the target protein and a sequential stabilization principle. These contrary views can be resolved by a predetermined pathway--optional error (PPOE) hypothesis. The hypothesis is that any pathway intermediate can incorporate a chance misfolding error that blocks folding and must be reversed for productive folding to continue. Different fractions of the protein population will then block at different steps, populate different intermediates, and fold at different rates, giving the appearance of multiple unrelated pathways. A test of the hypothesis matches the two models against extensive kinetic folding results for hen lysozyme which have been widely cited in support of independent parallel pathways. The PPOE model succeeds with fewer fitting constants. The fitted PPOE reaction scheme leads to known folding behavior, whereas the IUP properties are contradicted by experiment. The appearance of a conflict with multipath theoretical models seems to be due to their different focus, namely on multitrack microscopic behavior versus cooperative macroscopic behavior. The integration of three well-documented principles in the PPOE model (cooperative foldons, sequential stabilization, optional errors) provides a unifying explanation for how proteins fold and why they fold in that way.     

Thermal folding and mechanical unfolding pathways of protein secondary structures

Mechanical stretching of secondary structures is studied through molecular dynamics simulations of a Go-like model. Force versus displacement curves are studied as a function of the stiffness and velocity of the pulling device. The succession of stretching events, as measured by the order in which contacts are ruptured, is compared to the sequencing of events during thermal folding and unfolding. Opposite cross-correlations are found for an alpha-helix and a beta-hairpin structure. In a tandem of two alpha-helices, the two constituent helices unravel nearly simultaneously. A simple condition for simultaneous versus sequential unraveling of repeat units is presented.     

Dynamics of unfolded polypeptide chains as model for the earliest steps in protein folding

The rate of formation of intramolecular interactions in unfolded proteins determines how fast conformational space can be explored during folding. Characterization of the dynamics of unfolded proteins is therefore essential for the understanding of the earliest steps in protein folding. We used triplet-triplet energy transfer to measure formation of intrachain contacts in different unfolded polypeptide chains. The time constants (1/k) for contact formation over short distances are almost independent of chain length, with a maximum value of about 5 ns for flexible glycine-rich chains and of 12 ns for stiffer chains. The rates of contact formation over longer distances decrease with increasing chain length, indicating different rate-limiting steps for motions over short and long chain segments. The effect of the amino acid sequence on local chain dynamics was probed by using a series of host-guest peptides. Formation of local contacts is only sixfold slower around the stiffest amino acid (proline) compared to the most flexible amino acid (glycine). Good solvents for polypeptide chains like EtOH, GdmCl and urea were found to slow intrachain diffusion and to decrease chain stiffness. These data allow us to determine the time constants for formation of the earliest intrachain contacts during protein folding.     

Prediction of protein folding pathways: Bovine pancreatic trypsin inhibitor

A previous theory for the prediction of protein folding pathways [1] is applied to bovine pancreatic trypsin inhibitor and the resulting sequence of folding events shown to corroborate well with available experimental data.     

The consistency principle in protein structure and pathways of folding

• 1.i) It is pointed out that various energy terms contributing to stabilize the native state of globular proteins are consistent in the first approximation with each other in the native state. This means that each energy term is individually minimized at the minimum point of the total energy. I proposed (1) to call this fact “the consistency principle in protein structure.”
• 2.ii) The fair success of various methods of prediction of the secondary structures in globular proteins from their amino acid sequence is often interpreted as indicating the dominance of the short-range interactions in determining the local structures of the polypeptide chains. Partly from such a point of view, the hierarchic condensation model has been popular for the process of protein folding. However the consistency principle indicates that the short-range interactions are just one type of intramolecular interaction which contributes to stabilization of the native structure together with other mutually consistent types of intramolecular interactions. Therefore the hierarchic condensation model is not necessarily a unique model of protein folding.
• 3.iii) Roles of a possible nonspecific globular state, stabilized by nonspecific long-range intramolecular interactions, in the folding process are discussed. It is expected that this nonspecific globular state is observed either as an equilibrium or a kinetic intermediate state between the unfolded and the folded native states. Observation as a kinetic intermediate state is expected to occur in experiments done under strongly refolding conditions. In this case the polypeptide chain in the unfolded state collapses into a nonspecific globule by the action of nonspecific long-range intramolecular interactions. Two possible mechanisms of the transition from the nonspecific globular state to the specific native folded state are discussed.
• 4.iv) In an experiment done under weakly refolding conditions, folding is expected to occur according to the embryo-nucleus model. This model is a refined version of the hierarchic condensation model. Refinement is done by taking into account the fact that the intermediate structures assumed in the hierarchic condensation model are unstable against both the native folded state and the unfolded state. A nucleus is an ordered structure of a certain size. Ordered structures of a size larger than a nucleus tend to fold further to become the native specific globule. Ordered structures of a size smaller than a nucleus tend to unfold. Embryos are intrinsically unstable ordered structures smaller than a nucleus. Folding occurs when embryos grow in size to become a nucleus. The intrinsic instability of embryos is the built-in mechanism to overcome the low resolving power of the short-range interactions in determining local conformations of the polypeptide chain.

     

Efficient traversal of beta-sheet protein folding pathways using ensemble models

Molecular dynamics (MD) simulations can now predict ms-timescale folding processes of small proteins; however, this presently requires hundreds of thousands of CPU hours and is primarily applicable to short peptides with few long-range interactions. Larger and slower-folding proteins, such as many with extended β-sheet structure, would require orders of magnitude more time and computing resources. Furthermore, when the objective is to determine only which folding events are necessary and limiting, atomistic detail MD simulations can prove unnecessary. Here, we introduce the program tFolder as an efficient method for modelling the folding process of large β-sheet proteins using sequence data alone. To do so, we extend existing ensemble β-sheet prediction techniques, which permitted only a fixed anti-parallel β-barrel shape, with a method that predicts arbitrary β-strand/β-strand orientations and strand-order permutations. By accounting for all partial and final structural states, we can then model the transition from random coil to native state as a Markov process, using a master equation to simulate population dynamics of folding over time. Thus, all putative folding pathways can be energetically scored, including which transitions present the greatest barriers. Since correct folding pathway prediction is likely determined by the accuracy of contact prediction, we demonstrate the accuracy of tFolder to be comparable with state-of-the-art methods designed specifically for the contact prediction problem alone. We validate our method for dynamics prediction by applying it to the folding pathway of the well-studied Protein G. With relatively very little computation time, tFolder is able to reveal critical features of the folding pathways which were only previously observed through time-consuming MD simulations and experimental studies. Such a result greatly expands the number of proteins whose folding pathways can be studied, while the algorithmic integration of ensemble prediction with Markovian dynamics can be applied to many other problems.     

Alternative pathways in diffusion–collision controlled protein folding

The diffusion–collision model of protein folding has been solved exactly for a three-microdomain protein subunit. Numerical analysis shows that the exact kinetics may be excellently approximated in all cases studied by a standard chemical kinetics approach with the forward rate constants calculated from the mean folding time formula found previously.