Protein folding: simple models for a complex process

Twenty-eight years after its original publication, the diffusion-collision model has successfully been applied to describe the folding kinetics of two proteins with the same native structure but different sequences (Islam et al., this issue of Structure). The calculations show the relative importance of the primary and tertiary structure on the sequence of events and folding. For both proteins, the model suggests parallel folding pathways, a finding which has wide implications for the interpretations of experiments.     

Brownian dynamics simulation of protein folding: a study of the diffusion-collision model

The dynamic aspects of protein folding are described by a series of diffusion-collision steps involving structural units (microdomains) of various sizes that combine to form the protein in its native state. A method is introduced for obtaining the rate constants for the basic diffusion-collision step by use of Brownian dynamics. The method is applied to an investigation of the folding dynamics of two α-helices connected by a flexible (random-coil) polypeptide chain. The results of this full three-dimensional treatment are compared with simplified model calculations for the diffusion-collision step. Of particular interest are the nature of the collision dynamics and the role of the intervening peptide chain.     

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.     

Combining experiment and simulation in protein folding: closing the gap for small model systems

All-atom molecular dynamics (MD) simulations on increasingly powerful computers have been combined with experiments to characterize protein folding in detail over wider time ranges. The folding of small ultrafast folding proteins is being simulated on micros timescales, leading to improved structural predictions and folding rates. To what extent is 'closing the gap' between simulation and experiment for such systems providing insights into general mechanisms of protein folding?     

Rattling the cage: computational models of chaperonin-mediated protein folding

Chaperonins are known to maintain the stability of the proteome by facilitating the productive folding of numerous misfolded or aggregation-prone proteins and are thus essential for cell viability. Despite their established importance, the mechanism by which chaperonins facilitate protein folding remains unknown. Computer simulation techniques are now being employed to complement experimental ones in order to shed light on this mystery. Here we review previous computational models of chaperonin-mediated protein folding in the context of the two main hypotheses for chaperonin function: iterative annealing and landscape modulation. We then discuss new results pointing to the importance of solvent (a previously neglected factor) in chaperonin activity. We conclude with our views on the future role of simulation in studying chaperonin activity as well as protein folding in other biologically relevant confined contexts.     

Structure of stable protein folding intermediates by equilibrium phi-analysis: the apoflavodoxin thermal intermediate

Protein intermediates in equilibrium with native states may play important roles in protein dynamics but, in cases, can initiate harmful aggregation events. Investigating equilibrium protein intermediates is thus important for understanding protein behaviour (useful or pernicious) but it is hampered by difficulties in gathering structural information. We show here that the phi-analysis techniques developed to investigate transition states of protein folding can be extended to determine low-resolution three-dimensional structures of protein equilibrium intermediates. The analysis proposed is based solely on equilibrium data and is illustrated by determination of the structure of the apoflavodoxin thermal unfolding intermediate. In this conformation, a large part of the protein remains close to natively folded, but a 40 residue region is clearly unfolded. This structure is fully consistent with the NMR data gathered on an apoflavodoxin mutant designed specifically to stabilise the intermediate. The structure shows that the folded region of the intermediate is much larger than the proton slow-exchange core at 25 degrees C. It also reveals that the unfolded region is made of elements whose packing surface is more polar than average. In addition, it constitutes a useful guide to rationally stabilise the native state relative to the intermediate state, a far from trivial task.     

Effect of the thermostat in the molecular dynamics simulation on the folding of the model protein chignolin

Molecular dynamics simulations of the model protein chignolin with explicit solvent were carried out, in order to analyze the influence of the Berendsen thermostat on the evolution and folding of the peptide. The dependence of the peptide behavior on temperature was tested with the commonly employed thermostat scheme consisting of one thermostat for the protein and another for the solvent. The thermostat coupling time of the protein was increased to infinity, when the protein is not in direct contact with the thermal bath, a situation known as minimally invasive thermostat. In agreement with other works, it was observed that only in the last situation the instantaneous temperature of the model protein obeys a canonical distribution. As for the folding studies, it was shown that, in the applications of the commonly utilized thermostat schemes, the systems are trapped in local minima regions from which it has difficulty escaping. With the minimally invasive thermostat the time that the protein needs to fold was reduced by two to three times. These results show that the obstacles to the evolution of the extended peptide to the folded structure can be overcome when the temperature of the peptide is not directly controlled.     

Redesigning the folding energetics of a model three-helix bundle protein by site-directed mutagenesis

Because of their limited size and complexity, de novo designed proteins are particularly useful for the detailed investigation of folding thermodynamics and mechanisms. Here, we describe how subtle changes in the hydrophobic core of a model three-helix bundle protein (GM-0) alter its folding energetics. To explore the folding tolerance of GM-0 toward amino acid sequence variability, two mutant proteins (GM-1 and GM-2) were generated. In the mutants, cavities were created in the hydrophobic core of the protein by either singly (GM-1; L35A variant) or doubly (GM-2; L35A/I39A variant) replacing large hydrophobic side chains by smaller Ala residues. The folding of GM-0 is characterized by two partially folded intermediate states exhibiting characteristics of molten globules, as evidenced by pressure-unfolding and pressure-assisted cold denaturation experiments. In contrast, the folding energetics of both mutants, GM-1 and GM-2, exhibit only one folding intermediate. Our results support the view that simple but biologically important folding motifs such as the three-helix bundle can reveal complex folding plasticity, and they point to the role of hydrophobic packing as a determinant of the overall stability and folding thermodynamic of the helix bundle.     

Roles of beta-turns in protein folding: from peptide models to protein engineering

Reverse turns are a major class of protein secondary structure; they represent sites of chain reversal and thus sites where the globular character of a protein is created. It has been speculated for many years that turns may nucleate the formation of structure in protein folding, as their propensity to occur will favor the approximation of their flanking regions and their general tendency to be hydrophilic will favor their disposition at the solvent-accessible surface. Reverse turns are local features, and it is therefore not surprising that their structural properties have been extensively studied using peptide models. In this article, we review research on peptide models of turns to test the hypothesis that the propensities of turns to form in short peptides will relate to the roles of corresponding sequences in protein folding. Turns with significant stability as isolated entities should actively promote the folding of a protein, and by contrast, turn sequences that merely allow the chain to adopt conformations required for chain reversal are predicted to be passive in the folding mechanism. We discuss results of protein engineering studies of the roles of turn residues in folding mechanisms. Factors that correlate with the importance of turns in folding indeed include their intrinsic stability, as well as their topological context and their participation in hydrophobic networks within the protein's structure.     

Molecular dynamics simulation of protein folding by essential dynamics sampling: folding landscape of horse heart cytochrome c

A new method for simulating the folding process of a protein is reported. The method is based on the essential dynamics sampling technique. In essential dynamics sampling, a usual molecular dynamics simulation is performed, but only those steps, not increasing the distance from a target structure, are accepted. The distance is calculated in a configurational subspace defined by a set of generalized coordinates obtained by an essential dynamics analysis of an equilibrated trajectory. The method was applied to the folding process of horse heart cytochrome c, a protein with approximately 3000 degrees of freedom. Starting from structures, with a root-mean-square deviation of approximately 20 A from the crystal structure, the correct folding was obtained, by utilizing only 106 generalized degrees of freedom, chosen among those accounting for the backbone carbon atoms motions, hence not containing any information on the side chains. The folding pathways found are in agreement with experimental data on the same molecule.     

Visualization of the nature of protein folding by a study of a distance constraint approach in two-dimensional models

A distance constraint approach is applied to two-dimensional models of proteins in order to visualize the nature of protein folding and to examine the relative roles of different ranges of interaction. Three different native structures (I, II, and III) are considered; they have two different kinds of residues, viz., hydrophobic and hydrophilic, and different sequences of these residues. We examine how the distance constraint approach functions in the prediction of protein folding when we know the sequence of the residues, the (fixed) bond lengths, the mean distances between residues i and i + 2, and i and i + 3, and the mean distances for hydrophobic–hydrophobic, hydrophobic–hydrophilic, and hydrophilic–hydrophilic contacts between residues i and i + j, where j ≥ 4. This approach involves optimization of an object function with respect to 98 variables and is not free of the multiple-minimum problem. The optimization is always terminated if the chain is entangled and/or the segments (residues) are packed too compactly to move. In order to escape from such situations and to take the excluded-volume effect into account, a Monte Carlo method is used after the optimization is trapped in local minima. Success in the prediction of folding is found to depend on the starting conformations and on the native conformations. Fair success is obtained in predicting the helix-like structure in protein I and the overall structure of protein III, but not the β-like structures of proteins I and II. Insofar as the prediction of the structure of protein III is reasonable, it appears that some sequences of residues produce greater constraints on their conformations than others, if one considers only the hydrophobic and hydrophilic nature of the residues. These results imply that, in the folding of real proteins in three dimensions, the competition for hydrophobic (and hydrophilic) residues for inside (outside) positions in the molecule probably constitutes a necessary but not a sufficient condition to form and stabilize the native structure. The failure to predict the structure of protein II, and part of that of protein I, suggests that there are two types of long-range interactions. One (which we considered here) is nonspecific (i.e., is defined only in terms of contacts between residues of the same or different polarity) and acts at any stage of protein folding; the other (which we did not consider here) is a specific interaction between residues in pairs and contributes only when the residues in the specific pair take on the native conformation. Presumably, incorporation of such specific long-range interactions, together with the nonspecific ones, is necessary for successful protein folding, using the distance constraint approach.     

Fast tree search for a triangular lattice model of protein folding

Using a triangular lattice model to study the designability of protein folding, we overcame the parity problem of previous cubic lattice model and enumerated all the sequences and compact structures on a simple two-dimensional triangular lattice model of size 4 5 6 5 4. We used two types of amino acids, hydrophobic and polar, to make up the sequences, and achieved 223W212 different sequences excluding the reverse symmetry sequences. The total string number of distinct compact structures was 219,093, excluding reflection symmetry in the self-avoiding path of length 24 triangular lattice model. Based on this model, we applied a fast search algorithm by constructing a cluster tree. The algorithm decreased the computation by computing the objective energy of non-leaf nodes. The parallel experiments proved that the fast tree search algorithm yielded an exponential speed-up in the model of size 4 5 6 5 4. Designability analysis was performed to understand the search result.     

A framework for describing topological frustration in models of protein folding

In a natively folded protein of moderate or larger size, the protein backbone may weave through itself in complex ways, raising questions about what sequence of events might have to occur in order for the protein to reach its native configuration from the unfolded state. A mathematical framework is presented here for describing the notion of a topological folding barrier, which occurs when a protein chain must pass through a hole or opening, formed by other regions of the protein structure. Different folding pathways encounter different numbers of such barriers and therefore different degrees of frustration. A dynamic programming algorithm finds the optimal theoretical folding path and minimal degree of frustration for a protein based on its natively folded configuration. Calculations over a database of protein structures provide insights into questions such as whether the path of minimal frustration might tend to favor folding from one or from many sites of folding nucleation, or whether proteins favor folding around the N terminus, thereby providing support for the hypothesis that proteins fold co-translationally. The computational methods are applied to a multi-disulfide bonded protein, with computational findings that are consistent with the experimentally observed folding pathway. Attention is drawn to certain complex protein folds for which the computational method suggests there may be a preferred site of nucleation or where folding is likely to proceed through a relatively well-defined pathway or intermediate. The computational analyses lead to testable models for protein folding.     

A heteropolymer model study for the mechanism of protein folding

A heteropolymer model of randomly self-interacting chains in two dimensions is studied with numerical simulations in order to elucidate the folding mechanism of protein. We find that the model occasionally shows folding propensity depending on the sequence of random numbers given to the chain. We study the thermodynamic and kinematic roles in the folding mechanism by grouping the local energy minima found in the simulations into clusters according to the similarity of their conformations. It is suggested that the local minima to which some heteropolymers show a folding tendency are always the lowest energy states of the energy spectrum within a cluster, though which cluster is selected depends on the sequence. For the eight random sequences we study, we find that the energy gap between the ground state and excited states is little correlated with folding or nonfolding. We rather find that folding propensities are correlated with the global structure of the average energy surface, implying a dominant kinetic role in the folding mechanism, although thermal factors cannot be ignored as the mechanism of choosing the ground state within a cluster of states connected by small deformations. We suggest that a hierarchical cluster structure plays an important role in selecting a unique folded state out of the huge number of local minima of heteropolymers. © 1997 John Wiley & Sons, Inc.     

On the accuracy of metadynamics and its variations in a protein folding process

Metadynamics and its variations are powerful tools for exploring the free energy landscape of physical, chemical and biophysical systems. However, previous tests of their accuracy were based on either low-dimensional systems or complicated systems without exact answers. Therefore their accuracy, particularly when used for high-dimensional biophysical systems, has not been rigorously tested. In this work we performed a series of simulations with metadynamics and its variations for a typical biophysical process – the folding of protein chymotrypsin inhibitor 2 (CI2) based on a coarse-grained structure-based model. The results revealed the power as well as limitations of the algorithms and provided some guidelines for using metadynamics and its variations in high-dimensional biological systems.     

Kinetics of a finite one-dimensional spin system as a model for protein folding

Simple spin models are used to analyze the kinetic nature of lowest energy state formation of the spin systems as models of protein folding kinetics. The models employed in the present paper are based on the spin systems as models of biopolymers previously proposed by the author for the analysis of the equilibrium nature of transitions [T. Kikuchi, Biophys. Chem. 65 (1997) 109]. In particular, the effect of frustrations on the kinetics is investigated with the Monte Carlo simulations in this study. The results show that the kinetics of the present systems are characterized by the ratio of foldables (pathways on the energy landscape that to lead to the lowest energy state) and the temperature dependence of the mean first passage time of foldables. We also discuss the free energy profile of the present models and the relation of the present results to the kinetics of actual proteins.     

Equilibrium folding and stability of myotrophin: a model ankyrin repeat protein

Proteins containing stretches of repeating amino acid sequences are prevalent throughout nature, yet little is known about the general folding and assembly mechanisms of these systems. Here we propose myotrophin as a model system to study the folding of ankyrin repeat proteins. Myotrophin is folded over a large pH range and is soluble at high concentrations. Thermal and urea denaturation studies show that the protein displays cooperative two-state folding properties despite its modular nature. Taken together with previous studies on other ankyrin repeat proteins, our data suggest that the two-state folding pathway may be characteristic of ankyrin repeat proteins and other integrated alpha-helical repeat proteins in general.     

Gatekeepers in the ribosomal protein s6: thermodynamics, kinetics, and folding pathways revealed by a minimalist protein model

We investigate the effect of structural gatekeepers on the folding of the ribosomal protein S6. Folding thermodynamics and early refolding kinetics are studied for this system utilizing computer simulations of a minimalist protein model. When gatekeepers are eliminated, the thermodynamic signature of a folding intermediate emerges, and a marked decrease in folding efficiency is observed. We explain the prerequisites that determine the "strength" of a given gatekeeper. The investigated gatekeepers are found to have distinct functions, and to guide the folding and time-dependent packing of non-overlapping secondary structure elements in the protein. Gatekeepers avoid kinetic traps during folding by favoring the formation of "productive topologies" on the way to the native state. The trends in folding rates in the presence/absence of gatekeepers observed for our minimalist model of S6 are in very good agreement with experimental data on this protein.