Showing papers on "Contact order published in 2003"

The complete folding pathway of a protein from nanoseconds to microseconds

Abstract: Combining experimental and simulation data to describe all of the structures and the pathways involved in folding a protein is problematical. Transition states can be mapped experimentally by phi values, but the denatured state is very difficult to analyse under conditions that favour folding. Also computer simulation at atomic resolution is currently limited to about a microsecond or less. Ultrafast-folding proteins fold and unfold on timescales accessible by both approaches, so here we study the folding pathway of the three-helix bundle protein Engrailed homeodomain. Experimentally, the protein collapses in a microsecond to give an intermediate with much native alpha-helical secondary structure, which is the major component of the denatured state under conditions that favour folding. A mutant protein shows this state to be compact and contain dynamic, native-like helices with unstructured side chains. In the transition state between this and the native state, the structure of the helices is nearly fully formed and their docking is in progress, approximating to a classical diffusion-collision model. Molecular dynamics simulations give rate constants and structural details highly consistent with experiment, thereby completing the description of folding at atomic resolution.

Contact order revisited: Influence of protein size on the folding rate

Abstract: Guided by the recent success of empirical model predicting the folding rates of small two-state folding proteins from the relative contact order (CO) of their native structures, by a theoretical model of protein folding that predicts that logarithm of the folding rate decreases with the protein chain length L as L(2/3), and by the finding that the folding rates of multistate folding proteins strongly correlate with their sizes and have very bad correlation with CO, we reexamined the dependence of folding rate on CO and L in attempt to find a structural parameter that determines folding rates for the totality of proteins. We show that the Abs_CO = CO x L, is able to predict rather accurately folding rates for both two-state and multistate folding proteins, as well as short peptides, and that this Abs_CO scales with the protein chain length as L(0.70 +/- 0.07) for the totality of studied single-domain proteins and peptides.

How protein thermodynamics and folding mechanisms are altered by the chaperonin cage: Molecular simulations

Abstract: How the Escherichia coli GroEL/ES chaperonin assists folding of a substrate protein remains to be uncovered. Recently, it was suggested that confinement into the chaperonin cage itself can significantly accelerate folding of a substrate. Performing comprehensive molecular simulations of eight proteins confined into various sizes L of chaperonin-like cage, we explore how and to what extent protein thermodynamics and folding mechanisms are altered by the cage. We show that a substrate protein is remarkably stabilized by confinement; the estimated increase in denaturation temperature ΔTf is as large as ≈60°C. For a protein of size R0, the stabilization ΔTf scales as (R0/L)ν, where ν ≈ 3, which is consistent with a mean field theory of polymer. We also found significant free energy cost of confining a protein, which increases with R0/L, indicating that the confinement requires external work provided by the chaperonin system. In kinetic study, we show the folding is accelerated in a modestly well confined case, which is consistent with a recent experimental result on ribulose-1,5-bisphosphate carboxylase-oxygenase folding and simulation results of a β hairpin. Interestingly, the acceleration of folding is likely to be larger for a protein with more complex topology, as quantified by the contact order. We also show how ensemble of folding pathways are altered by the chaperonin-like cage calculating a variant of φ value used in the study of spontaneous folding.

Transition-path sampling of beta hairpin folding

Abstract: We examine the dynamical folding pathways of the C-terminal β-hairpin of protein G-B1 in explicit solvent at room temperature by means of a transition-path sampling algorithm. In agreement with previous free-energy calculations, the resulting path ensembles reveal a folding mechanism in which the hydrophobic residues collapse first followed by backbone hydrogen-bond formation, starting with the hydrogen bonds inside the hydrophobic core. In addition, the path ensembles contain information on the folding kinetics, including solvent motion. Using the recently developed transition interface sampling technique, we calculate the rate constant for unfolding of the protein fragment and find it to be in reasonable agreement with experiments. The results support the validation of using all-atom force fields to study protein folding.

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

Abstract: 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 ∼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.

Parallel protein-unfolding pathways revealed and mapped.

Abstract: Theoretical studies of protein folding suggest that multiple folding pathways should exist, but there is little experimental evidence to support this Here we demonstrate changes in the flux between different transition states on parallel folding pathways, resulting in unprecedented upward curvature in the denaturant-dependent unfolding kinetics of a β-sandwich protein As denaturant concentration increases, the highly compact transition state of one pathway becomes destabilized and the dominant flux of protein molecules shifts toward another pathway with a less structured transition state Furthermore, point mutations alter the relative accessibility of the pathways, allowing the structure of two transition states on separate, direct folding pathways to be mapped by systematic Φ-value analysis It has been suggested that pathways with diffuse rather than localized transition states are evolutionarily selected to prevent misfolding, and indeed we find that the transition state favored at high concentrations of denaturant is more polarized than the physiologically relevant one

Chain length is the main determinant of the folding rate for proteins with three-state folding kinetics.

Abstract: We demonstrate that chain length is the main determinant of the folding rate for proteins with the three-state folding kinetics. The logarithm of their folding rate in water (k(f)) strongly anticorrelates with their chain length L (the correlation coefficient being -0.80). At the same time, the chain length has no correlation with the folding rate for two-state folding proteins (the correlation coefficient is -0.07). Another significant difference of these two groups of proteins is a strong anticorrelation between the folding rate and Baker's "relative contact order" for the two-state folders and the complete absence of such correlation for the three-state folders.

The physics and bioinformatics of binding and folding-an energy landscape perspective.

Abstract: It has been recognized in the last few years that unstructured proteins play an important role in biological organisms, often participating in signal transduction, transcriptional regulation, and a variety of other regulatory activities. Various hypotheses have been put forward for the ubiquity of the unfolded state; rapid turnover, faster or more specific binding kinetics, multifunc- tionality may all possibly explain apparent ubiquitousness of unfolded proteins in eukaryotic cells. In this paper we extend the energy landscape theory of protein folding to construct an analytical model of how binding and folding are coupled thermodynamically when the energy landscape is partially rugged. To deduce the parameters that enter the theory, which is based on Generalized Random Energy Model, we have analyzed in a bioinformatic sense a large structural database of more than 500 protein complexes. We find that Miyazawa-Jernigan contact potential shows similar energy gaps for folding for both hydrophobic and hydrophilic proteins, but that for binding contacts hydrophobic interfaces turn out to be funneled while hydrophilic ones are antifunneled. This suggests evolution has found a mechanism for avoiding frustration between folding and binding by making use of indirect water-mediated interactions. By juxtaposing the monomeric protein folding free energy profile in the protein complex database with another database consisting of only well-folded monomers, we estimate that at least 15% of monomers in the former database are unfolded in the absence of partner protein interface interactions. When employing the parameters characteristic of these unfolded monomers to construct binding/folding phase diagrams, we find that these monomers would indeed fold if sufficiently stabilizing binding contacts, consistent with that fold, are formed. © 2003 Wiley Periodicals, Inc. Biopolymers 68: 333-349, 2003

Ab initio folding of helix bundle proteins using molecular dynamics simulations.

Abstract: We have demonstrated that ab initio fast folding simulations at 400 K using a GB implicit solvent model with an all-atom based force field can describe the spontaneous formation of nativelike structures for the 36-residue villin headpiece and the 46-residue fragment B of Staphylococcal protein A. An implicit solvent model combined with high-temperature MD makes it possible to perform direct folding simulations of small- to medium-sized proteins by reducing the computational requirements tremendously. In the early stage of folding of the villin headpiece and protein A, initial hydrophobic collapse and rapid formation of helices were found to play important roles. For protein A, the third helix forms first in the early stage of folding and exhibits higher stability. The free energy profiles calculated from the folding simulations suggested that both of the helix-bundle proteins show a two-state thermodynamic behavior and protein A exhibits rather broad native basins.

Fast protein folding kinetics

Abstract: Proteins are complex molecules, yet their folding kinetics is often fast (microseconds) and simple, involving only a single exponential function of time (called two-state kinetics). The main model for two-state kinetics has been transition-state theory, where an energy barrier defines a slow step to reach an improbable structure. But how can barriers explain fast processes, such as folding? We study a simple model with rigorous kinetics that explains the high speed instead as a result of the microscopic parallelization of folding trajectories. The single exponential results from a separation of timescales; the parallelization of routes is high at the start of folding and low thereafter. The ensemble of rate-limiting chain conformations is different from in transition-state theory; it is broad, overlaps with the denatured state, is not aligned along a single reaction coordinate, and involves well populated, rather than improbable, structures.

Effects of confinement and crowding on the thermodynamics and kinetics of folding of a minimalist β-barrel protein

Abstract: We report the results of a molecular dynamics (MD) study of the effects of confinement and molecular crowding on the thermodynamics and kinetics of folding of a 46 residue off-lattice minimalist β-barrel protein. Crowding was mimicked by restricting the protein to a sphere with a soft well repulsive potential. MD simulations were performed on the protein in an unconfined environment, as well as confined to spheres of two different radii, 5.88σ and 11.76σ. Here, σ is the bond length between two adjacent residues and the radius of gyration of the protein in its native state is 2.87σ. We find that for the larger sphere (11.76σ), the folding and collapse temperatures are virtually unchanged from their bulk values, but the average folding time is decreased by 35%. By contrast, the smaller sphere has a much more significant thermodynamic effect (the folding temperature is raised by 28%) but the average folding time is only decreased by 58%. Confinement is also seen to restrict the conformational space accessible to the protein in its denatured state. In addition, confinement appears to change the folding mechanism for this protein, as long-lived intermediates present in the bulk are both modified and have shorter lifetimes when the protein is confined.

First-passage time distribution and non-Markovian diffusion dynamics of protein folding

Abstract: We study the kinetics of protein folding via statistical energy landscape theory. We concentrate on the local-connectivity case, where the configurational changes can only occur among neighboring states, with the folding progress described in terms of an order parameter given by the fraction of native conformations. The non-Markovian diffusion dynamics is analyzed in detail and an expression for the mean first-passage time (MFPT) from non-native unfolded states to native folded state is obtained. It was found that the MFPT has a V-shaped dependence on the temperature. We also find that the MFPT is shortened as one increases the gap between the energy of the native and average non-native folded states relative to the fluctuations of the energy landscape. The second- and higher-order moments are studied to infer the first-passage time distribution. At high temperature, the distribution becomes close to a Poisson distribution, while at low temperatures the distribution becomes a Levy-type distribution with power-law tails, indicating a nonself-averaging intermittent behavior of folding dynamics. We note the likely relevance of this result to single-molecule dynamics experiments, where a power law (Levy) distribution of the relaxation time of the underlined protein energy landscape is observed.

Prediction of folding rates and transition‐state placement from native‐state geometry

Abstract: A variety of experimental and theoretical studies have established that the folding process of monomeric proteins is strongly influenced by the topology of the native state. In particular, folding times have been shown to correlate well with the contact order, a measure of contact locality. Our investigation focuses on identifying additional topologic properties that correlate with experimentally measurable quantities, such as folding rates and transition-state placement, for both two- and three-state folders. The validation against data from 40 experiments shows that a particular topological property that measures the interdependence of contacts, termed cliquishness or clustering coefficient, can account with statistically significant accuracy both for the transition state placement and especially for folding rates. The observed correlations can be further improved by optimally combining the distinct topological information captured by cliquishness and contact order.

Contact order dependent protein folding rates: Kinetic consequences of a cooperative interplay between favorable nonlocal interactions and local conformational preferences

Abstract: Physical mechanisms underlying the empirical correlation between relative contact order (CO) and folding rate among naturally occur- ring small single-domain proteins are investigated by evaluating postulated interaction schemes for a set of three-dimensional 27mer lattice protein mod- els with 97 different CO values. Many-body interac- tions are constructed such that contact energies become more favorable when short chain segments sequentially adjacent to the contacting residues adopt native-like conformations. At a given interac- tion strength, this scheme leads to folding rates that are logarithmically well correlated with CO (correla- tion coefficient r 0.914) and span more than 2.5 orders of magnitude, whereas folding rates of the corresponding Go models with additive contact en- ergies have much less logarithmic correlation with CO and span only approximately one order of magni- tude. The present protein chain models also exhibit calorimetric cooperativity and linear chevron plots similar to that observed experimentally for proteins with apparent simple two-state folding/unfolding kinetics. Thus, our findings suggest that CO-depen- dent folding rates of real proteins may arise partly from a significant positive coupling between nonlo- cal contact favorabilities and local conformational preferences. Proteins 2003;52:524 -533.

Simple two-state protein folding kinetics requires near-levinthal thermodynamic cooperativity.

Abstract: Simple two-state folding kinetics of many small single-domain proteins are characterized by chevron plots with linear folding and unfolding arms consistent with an apparent two-state description of equilibrium thermodynamics. This phenomenon is hereby recognized as a nontrivial heteropolymer property capable of providing fundamental insight into protein energetics. Many current protein chain models, including common lattice and continuum Gō models with explicit native biases, fail to reproduce this generic protein property. Here we show that simple two-state kinetics is obtainable from models with a cooperative interplay between core burial and local conformational propensities or an extra strongly favorable energy for the native structure. These predictions suggest that intramolecular recognition in real two-state proteins is more specific than that envisioned by common Gō-like constructs with pairwise additive energies. The many-body interactions in the present kinetically two-state models lead to high thermodynamic cooperativity as measured by their van't Hoff to calorimetric enthalpy ratios, implying that the native and denatured conformational populations are well separated in enthalpy by a high free-energy barrier. It has been observed experimentally that deviations from Arrhenius behavior are often more severe for folding than for unfolding. This asymmetry may be rationalized by one of the present modeling scenarios if the effective many-body cooperative interactions stabilizing the native structure against unfolding is less dependent on temperature than the interactions that drive the folding kinetics.

Specific non-native hydrophobic interactions in a hidden folding intermediate: implications for protein folding.

Abstract: Structures of intermediates and transition states in protein folding are usually characterized by amide hydrogen exchange and protein engineering methods and interpreted on the basis of the assumption that they have native-like conformations. We were able to stabilize and determine the high-resolution structure of a partially unfolded intermediate that exists after the rate-limiting step of a four-helix bundle protein, Rd-apocyt b(562), by multidimensional NMR methods. The intermediate has partial native-like secondary structure and backbone topology, consistent with our earlier native state hydrogen exchange results. However, non-native hydrophobic interactions exist throughout the structure. These and other results in the literature suggest that non-native hydrophobic interactions may occur generally in partially folded states. This can alter the interpretation of mutational protein engineering results in terms of native-like side chain interactions. In addition, since the intermediate exists after the rate-limiting step and Rd-apocyt b(562) folds very rapidly (k(f) approximately 10(4) s(-1)), these results suggest that non-native hydrophobic interactions, in the absence of topological misfolding, are repaired too rapidly to slow folding and cause the accumulation of folding intermediates. More generally, these results illustrate an approach for determining the high-resolution structure of folding intermediates.