Using a beta-hairpin protein as a representative example of two-state folders, we studied how the exploration of native-like states affects the folding kinetics. It has been found that the first-passage time (FPT) distributions are essentially single-exponential not only for the times to overcome the free energy barrier that separates unfolded and native-like states but also for the times to find the native state among the native-like ones. If the protein explores native-like states for a time much longer than the time to overcome the free energy barrier, which was found to be characteristic of high temperatures, the resulting FPT distribution to reach the native state remains close to exponential but the mean FPT (MFPT) is determined not by the height of the free energy barrier but by the time to explore native-like states. The mean time to overcome the free energy barrier is found to be in reasonable agreement with the Kramers rate formula and generally far shorter than the MFPT to reach the native state. The time to find the native state among native-like ones increases with temperature, which explains the known U-shape dependence of the MFPTs on temperature.
We perform extensive Monte Carlo simulations of a lattice model and the Go potential to investigate the existence of folding pathways at the level of contact cluster formation for two native structures with markedly different geometries. Our analysis of folding pathways revealed a common underlying folding mechanism, based on nucleation phenomena, for both protein models. However, folding to the more complex geometry (i.e. that with more non-local contacts) is driven by a folding nucleus whose geometric traits more closely resemble those of the native fold. For this geometry folding is clearly…
Proteins are minimally frustrated polymers. However, for realistic protein models non-native interactions must be taken into account. In this paper we analyze the effect of non-native interactions on the folding rate and on the folding free energy barrier. We present an analytic theory to account for the modification on the free energy landscape upon introduction of non-native contacts, added as a perturbation to the strong native interactions driving folding. Our theory predicts a rate-enhancement regime at fixed temperature, under the introduction of weak, non-native interactions. We have th…
In protein folding the term plasticity refers to the number of alternative folding pathways encountered in response to free energy perturbations such as those induced by mutation. Here we explore the relation between folding plasticity and a gross, generic feature of the native geometry, namely, the relative number of local and non-local native contacts. The results from our study, which is based on Monte Carlo simulations of simple lattice proteins, show that folding to a structure that is rich in local contacts is considerably more plastic than folding to a native geometry characterized by h…
In this paper we investigate the role of native geometry on the kinetics of protein folding based on simple lattice models and Monte Carlo simulations. Results obtained within the scope of the Miyazawa-Jernigan indicate the existence of two dynamical folding regimes depending on the protein chain length. For chains larger than 80 amino acids the folding performance is sensitive to the native state's conformation. Smaller chains, with less than 80 amino acids, fold via two-state kinetics and exhibit a significant correlation between the contact order parameter and the logarithmic folding times.…
We use the Dominant Reaction Pathway (DRP) approach to study the dynamics of the folding of a beta-hairpin, within a model which accounts for both native and non-native interactions. We compare the most probable folding pathways calculated with the DRP method with those obtained directly from molecular dynamics (MD) simulations. We find that the two approaches give completely consistent results. We investigate the effects of the non-native hydrophobic interactions on the folding dynamics found them to be small.
Unavailability of experimental 3D structural data on protein folding dynamics and necessity for a new generation of structure prediction methods in this context
While all the information required for the folding of a protein is contained in its amino acid sequence, one has not yet learned how to extract this information to predict the three--dimensional, biologically active, native conformation of a protein whose sequence is known. Using insight obtained from simple model simulations of the folding of proteins, in particular of the fact that this phenomenon is essentially controlled by conserved (native) contacts among (few) strongly interacting ("hot"), as a rule hydrophobic, amino acids, which also stabilize local elementary structures (LES, hidden,…
Natural proteins fold to a unique, thermodynamically dominant state. Modeling of the folding process and prediction of the native fold of proteins are two major unsolved problems in biophysics. Here, we show successful all-atom ab initio folding of a representative diverse set of proteins, using a minimalist transferable energy model that consists of two-body atom-atom interactions, hydrogen-bonding, and a local sequence energy term that models sequence-specific chain stiffness. Starting from a random coil, the native-like structure was observed during replica exchange Monte Carlo (REMC) simul…
Cotranslational folding depends on the folding speed and stability of the nascent protein. It remains difficult, however, to predict which proteins cotranslationally fold. Here, we simulate evolution of model proteins to investigate how native structure influences evolution of cotranslational folding. We developed a model that connects protein folding during and after translation to cellular fitness. Model proteins evolved improved folding speed and stability, with proteins adopting one of two strategies for folding quickly. Low contact order proteins evolve to fold cotranslationally. Such pro…
Monte Carlo simulations show that long-range interactions play a major role in determining the folding rates of 48-mer three-dimensional lattice polymers modelled by the Go 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 ~50% towards zero. However, the dispersion of the simulated folding times depends strongly on the native geometry and Go polymers folding to one of the target structures exhibit folding times spanning…
We apply a simulational proxy of the phi-value analysis and perform extensive mutagenesis experiments to identify the nucleating residues in the folding reactions of two small lattice Go polymers with different native geometries. These results are compared with those obtained from an accurate analysis based on the reaction coordinate folding probability Pfold, and on structural clustering methods. For both protein models, the transition state ensemble is rather heterogeneous and splits-up into structurally different populations. For the more complex geometry the identified subpopulations are a…
A complex network approach to protein folding is proposed. The graph object is the network of shortcut edges present in a native-state protein (SCN0). Although SCN0s are found via an intuitive message passing algorithm (S. Milgram, Psychology Today, May 1967 pp. 61-67), they are meaningful enough that the logarithm form of their contact order (SCN0_lnCO) correlates significantly with protein kinetic rates, regardless of protein size. Further, the clustering coefficient of a SCN0 (CSCN0) can be used to combine protein segments iteratively within the Restricted Binary Collision model to form the…
We explore the consequences of very high dimensionality in the dynamical landscape of protein folding. Consideration of both typical range of stabilising interactions, and folding rates themselves, leads to a model of the energy hypersurface that is characterised by the structure of diffusive "hypergutters" as well as the familiar "funnels". Several general predictions result: (1) intermediate subspaces of configurations will always be visited; (2) specific but non-native interactions are important in stabilising these low-dimensional diffusive searches on the folding pathway; (3) sequential b…
We incorporate hydrodynamic interactions (HI) in a coarse-grained and structure-based model of proteins by employing the Rotne-Prager hydrodynamic tensor. We study several small proteins and demonstrate that HI facilitate folding. We also study HIV-1 protease and show that HI make the flap closing dynamics faster. The HI are found to affect time correlation functions in the vicinity of the native state even though they have no impact on same time characteristics of the structure fluctuations around the native state.
Models of protein energetics which neglect interactions between amino acids that are not adjacent in the native state, such as the Go model, encode or underlie many influential ideas on protein folding. Implicit in this simplification is a crucial assumption that has never been critically evaluated in a broad context: Detailed mechanisms of protein folding are not biased by non-native contacts, typically imagined as a consequence of sequence design and/or topology. Here we present, using computer simulations of a well-studied lattice heteropolymer model, the first systematic test of this oft-a…
Simplified Go models, where only native contacts interact favorably, have proven useful to characterize some aspects of the folding of small proteins. The success of these models is limited by the fact that all residues interact in the same way, so that the folding features of a protein are determined only by the geometry of its native conformation. We present an extended version of a C-alpha based Go model where different residues interact with different energies. The model is used to calculate the thermodynamics of three small proteins (Protein G, SrcSH3 and CI2) and the effect of mutations …
The folding rates of two-state proteins have been found to correlate with simple measures of native-state topology. The most prominent among these measures is the relative contact order (CO), which is the average CO or 'localness' of all contacts in the native protein structure, divided by the chain length. Here, we test whether such measures can be generalized to capture the effect of chain crosslinks on the folding rate. Crosslinks change the chain connectivity and therefore also the localness of some of the the native contacts. These changes in localness can be taken into account by the gra…
A theoretical framework is developed to study the dynamics of protein folding. The key insight is that the search for the native protein conformation is influenced by the rate r at which external parameters, such as temperature, chemical denaturant or pH, are adjusted to induce folding. A theory based on this insight predicts that (1) proteins with non-funneled energy landscapes can fold reliably to their native state, (2) reliable folding can occur as an equilibrium or out-of-equilibrium process, and (3) reliable folding only occurs when the rate r is below a limiting value, which can be calc…
For the vast majority of naturally occurring, small, single domain proteins folding is often described as a two-state process that lacks detectable intermediates. This observation has often been rationalized on the basis of a nucleation mechanism for protein folding whose basic premise is the idea that after completion of a specific set of contacts forming the so-called folding nucleus the native state is achieved promptly. Here we propose a methodology to identify folding nuclei in small lattice polymers and apply it to the study of protein molecules with chain length N=48. To investigate the…
Stochastic simulations of coarse-grained protein models are used to investigate the propensity to form knots in early stages of protein folding. The study is carried out comparatively for two homologous carbamoyltransferases, a natively-knotted N-acetylornithine carbamoyltransferase (AOTCase) and an unknotted ornithine carbamoyltransferase (OTCase). In addition, two different sets of pairwise amino acid interactions are considered: one promoting exclusively native interactions, and the other additionally including non-native quasi-chemical and electrostatic interactions. With the former model …
Understanding the biological function of knots in proteins and their folding process is an open and challenging question in biology. Recent studies classify the topology and geometry of knotted proteins by analysing the distribution of a protein's planar projections using topological objects called knotoids. We approach the analysis of proteins with the same topology by introducing a topologically inspired statistical metric between their knotoid distributions. We detect geometric differences between trefoil proteins by characterising their entanglement and we recover a clustering by sequence …
We review some of our recent results obtained within the scope of simple lattice models and Monte Carlo simulations that illustrate the role of native geometry in the folding kinetics of two state folders.
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 forty experiments shows that a particular topologic…
A theoretical model for the folding of proteins containing disulfide bonds is introduced. The model exploits the knowledge of the native state to favour the progressive establishment of native interactions. At variance with traditional approaches based on native topology, not all native bonds are treated in the same way; in particular, a suitable energy term is introduced to account for the special strength of disulfide bonds (irrespective of whether they are native or not) as well as their ability to undergo intra-molecular reshuffling. The model thus possesses the minimal ingredients necessa…
The folding of the alpha-helice domain hbSBD of the mammalian mitochondrial branched-chain alpha-ketoacid dehydrogenase (BCKD) complex is studied by the circular dichroism technique in absence of urea. Thermal denaturation is used to evaluate various thermodynamic parameters defining the equilibrium unfolding, which is well described by the two-state model with the folding temperature T_f = 317.8 K and the enthalpy change Delta H_g = 19.67 kcal/mol. The folding is also studied numerically using the off-lattice coarse-grained Go model and the Langevin dynamics. The obtained results, including t…
There are several simple criteria of folding to a native state in model proteins. One of them involves crossing of a threshold value of the RMSD distance away from the native state. Another checks whether all native contacts are established, i.e. whether the interacting amino acids come closer than some characteristic distance. We use Go-like models of proteins and show that such simple criteria may prompt one to declare folding even though fragments of the resulting conformations have a wrong sense of chirality. We propose that a better condition of folding should augment the simple criteria …
Native shortcut networks (SCN0) are sub-graphs of native protein residue networks (PRN0). In this paper, we propose the Network Dynamics (ND) model, which reconstructs a PRN0 by adding back its edges according to some recipe, while its nodes are fixed at native (PDB) locations. A PRN0 reconstruction will eventually reconstruct its SCN0, but only after exploring several non-native shortcut network (SCN) configurations. It is these other SCN configurations that are of interest to us, as they produce the statistics to evaluate the different recipes. The recipes vary from each other slightly to in…
By observing trends in the folding kinetics of experimental 2-state proteins at their transition midpoints, and by observing trends in the barrier heights of numerous simulations of coarse grained, C-alpha model, Go proteins, we show that folding rates correlate with the degree of heterogeneity in the formation of native contacts. Statistically significant correlations are observed between folding rates and measures of heterogeneity inherent in the native topology, as well as between rates and the variance in the distribution of either experimentally measured or simulated phi-values.
The fundamental law for protein folding is the Thermodynamic Principle: the amino acid sequence of a protein determines its native structure and the native structure has the minimum Gibbs free energy. If all chemical problems can be answered by quantum mechanics, there should be a quantum mechanics derivation of Gibbs free energy formula G(X) for every possible conformation X of the protein. We apply quantum statistics to derive such a formula. For simplicity, only monomeric self folding globular proteins are covered. We point out some immediate applications of the formula. We show that the fo…
Phi-values are experimental measures of the effects of mutations on the folding kinetics of a protein. A central question is which structural information Phi-values contain about the transition state of folding. Traditionally, a Phi-value is interpreted as the 'nativeness' of a mutated residue in the transition state. However, this interpretation is often problematic because it assumes a linear relation between the nativeness of the residue and its free-energy contribution. We present here a better structural interpretation of Phi-values for mutations within a given helix. Our interpretation i…
We study the folding thermodynamics of a beta-hairpin and two three-stranded beta-sheet peptides using a simplified sequence-based all-atom model, in which folding is driven mainly by backbone hydrogen bonding and effective hydrophobic attraction. The native populations obtained for these three sequences are in good agreement with experimental data. We also show that the apparent native population depends on which observable is studied; the hydrophobicity energy and the number of native hydrogen bonds give different results. The magnitude of this dependence matches well with the results obtain…
A generalized computational method for folding proteins with a fully transferable potential and geometrically realistic all-atom model is presented and tested on seven different helix bundle proteins. The protocol, which includes graph-theoretical analysis of the ensemble of resulting folded conformations, was systematically applied and consistently produced structure predictions of approximately 3 Angstroms without any knowledge of the native state. To measure and understand the significance of the results, extensive control simulations were conducted. Graph theoretic analysis provides a mean…
We report on atomistic simulation of the folding of a natively-knotted protein, MJ0366, based on a realistic force field. To the best of our knowledge this is the first reported effort where a realistic force field is used to investigate the folding pathways of a protein with complex native topology. By using the dominant-reaction pathway scheme we collected about 30 successful folding trajectories for the 82-amino acid long trefoil-knotted protein. Despite the dissimilarity of their initial unfolded configuration, these trajectories reach the natively-knotted state through a remarkably simila…
Proper folding of deeply knotted proteins has a very low success rate even in structure-based models which favor formation of the native contacts but have no topological bias. By employing a structure-based model, we demonstrate that cotranslational folding on a model ribosome may enhance the odds to form trefoil knots for protein YibK without any need to introduce any non-native contacts. The ribosome is represented by a repulsive wall that keeps elongating the protein. On-ribosome folding proceeds through a a slipknot conformation. We elucidate the mechanics and energetics of its formation. …
We present a simple model of protein folding dynamics that captures key qualitative elements recently seen in all-atom simulations. The goals of this theory are to serve as a simple formalism for gaining deeper insight into the physical properties seen in detailed simulations as well as to serve as a model to easily compare why these simulations suggest a different kinetic mechanism than previous simple models. Specifically, we find that non-native contacts play a key role in determining the mechanism, which can shift dramatically as the energetic strength of non-native interactions is changed…
In order to elucidate the role of the native state topology and the stability of subdomains in protein folding, we investigate free energy landscape of human lysozyme, which is composed of two subdomains, by Monte Carlo simulations. A realistic lattice model with Go-like interaction is used. We take the relative interaction strength (stability, in other word) of two subdomains as a variable parameter and study the folding process. A variety of folding process is observed and we obtained a phase diagram of folding in terms of temperature and the relative stability. Experimentally-observed diver…
The effects of cooperativity are studied within Go-Lennard-Jones models of proteins by making the contact interactions dependent on the proximity to the native conformation. The kinetic universality classes are found to remain the same as in the absence of cooperativity. For a fixed native geometry, small changes in the effective contact map may affect the folding times in a chance way and to the extent that is comparable to the shift in the folding times due to cooperativity. The contact order controlls folding scenarios: the average times necessary to bring pairs of amino acids into their ne…
De novo prediction of protein folding is an open scientific challenge. Many folding models and force fields have been developed, yet all face difficulties converging to native conformations. Hydrophobicity scales (HSs) play a crucial role in such simulations as they define the energetic interactions between protein residues, thus determining the energetically favorable conformation. While many HSs have been developed over the years using various methods, it is surprising that the scales show very weak consensus in their assignment of hydrophobicity indexes to the various residues. In this work…
RNA co-transcriptional folding has long been suspected to play an active role in helping proper native folding of ribozymes and structured regulatory motifs in mRNA untranslated regions. Yet, the underlying mechanisms and coding requirements for efficient co-transcriptional folding remain unclear. Traditional approaches have intrinsic limitations to dissect RNA folding paths, as they rely on sequence mutations or circular permutations that typically perturb both RNA folding paths and equilibrium structures. Here, we show that exploiting sequence symmetries instead of mutations can circumvent t…
The relevance of various residue positions for the stability and the folding characteristics of the prion protein are investigated by using molecular dynamics simulations of models exploiting the topology of the native state. Highly significant correlations are found between the most relevant sites in our analysis and the single point mutations known to be associated with the arousal of the genetic forms of prion disease (caused by the conformational change from the cellular to the scrapie isoform). Considerable insight into the conformational change is provided by comparing the folding proces…
We carry out a theoretical study of the vibrational and relaxation properties of naturally-occurring proteins with the purpose of characterizing both the folding and equilibrium thermodynamics. By means of a suitable model we provide a full characterization of the spectrum and eigenmodes of vibration at various temperatures by merely exploiting the knowledge of the protein native structure. It is shown that the rate at which perturbations decay at the folding transition correlates well with experimental folding rates. This validation is carried out on a list of about 30 two-state folders. Furt…
The folding of a protein towards its native state is a rather complicated process. However there are empirical evidences that the folding time correlates with the contact order, a simple measure of the spatial organisation of the native state of the protein. Contact order is related to the average length of the main chain loops formed by amino acids which are in contact. Here we argue that folding kinetics can be influenced also by the entanglement that loops may undergo within the overall three dimensional protein structure. In order to explore such possibility, we introduce a novel descripto…
Though the problem of sequence-reversed protein folding is largely unexplored, one might speculate that reversed native protein sequences should be significantly more foldable than purely random heteropolymer sequences. In this article, we investigate how the reverse-sequences of native proteins might fold by examining a series of small proteins of increasing structural complexity (α-helix, \b{eta}-hairpin, α-helix bundle, and α/\b{eta}-protein). Employing a tandem protein structure prediction algorithmic and molecular dynamics simulation approach, we find that the ability of reverse sequences…
We describe the results obtained from an improved model for protein folding. We find that a good agreement with the native structure of a 46 residue long, five-letter protein segment is obtained by carefully tuning the parameters of the self-avoiding energy. In particular we find an improved free-energy profile. We also compare the efficiency of the multidimensional replica exchange method with the widely used parallel tempering.
The high computational cost of carrying out molecular dynamics simulations of even small-size proteins is a major obstacle in the study, at atomic detail and in explicit solvent, of the physical mechanism which is at the basis of the folding of proteins. Making use of a biasing algorithm, based on the principle of the ratchet-and-pawl, we have been able to calculate eight folding trajectories (to an RMSD between 1.2A and 2.5A) of the B1 domain of protein G in explicit solvent without the need of high-performance computing. The simulations show that in the denatured state there is a complex net…
An increasing number of proteins are being discovered with a remarkable and somewhat surprising feature, a knot in their native structures. How the polypeptide chain is able to knot itself during the folding process to form these highly intricate protein topologies is not known. Here, we perform a computational study on the 160-amino acid homodimeric protein YibK which, like other proteins in the SpoU family of MTases, contains a deep trefoil knot in its C-terminal region. In this study, we use a coarse-grained C-alpha-chain representation and Langevin dynamics to study folding kinetics. We fi…
A simple statistical mechanical model proposed by Wako and Sait$\hat{\rm o}$ has explained the aspects of protein folding surprisingly well. This model was systematically applied to multiple proteins by Mu$\tilde{\rm n}$oz and Eaton and has since been referred to as the Wako-Sait$\hat{\rm o}$-Mu$\tilde{\rm n}$oz-Eaton (WSME) model. The success of the WSME model in explaining the folding of many proteins has verified the hypothesis that the folding is dominated by native interactions, which makes the energy landscape globally biased toward native conformation. Using the WSME and other related m…
We present a solvable model that predicts the folding kinetics of two-state proteins from their native structures. The model is based on conditional chain entropies. It assumes that folding processes are dominated by small-loop closure events that can be inferred from native structures. For CI2, the src SH3 domain, TNfn3, and protein L, the model reproduces two-state kinetics, and it predicts well the average Phi-values for secondary structures. The barrier to folding is the formation of predominantly local structures such as helices and hairpins, which are needed to bring nonlocal pairs of am…
We investigate the folding behavior of protein sequences by numerically studying all sequences with maximally compact lattice model through exhaustive enumeration. We get the prion-like behavior of protein folding. Individual proteins remaining stable in the isolated native state may change their conformations when they aggregate. We observe the folding properties as the interfacial interaction strength changes, and find that the strength must be strong enough before the propagation of the most stable structures happens.
The folding dynamics of small single-domain proteins is a current focus of simulations and experiments. Many of these proteins are 'two-state folders', i.e. proteins that fold rather directly from the denatured state to the native state, without populating metastable intermediate states. A central question is how to characterize the instable, partially folded conformations of two-state proteins, in particular the rate-limiting transition-state conformations between the denatured and the native state. These partially folded conformations are short-lived and cannot be observed directly in experi…
We consider six different secondary structures of proteins and construct two types of Go-type off-lattice models: with the steric constraints and without. The basic aminoacid-aminoacid potential is Lennard Jones for the native contacts and a soft repulsion for the non-native contacts. The interactions are chosen to make the target secondary structure be the native state of the system. We provide a thorough equilibrium and kinetic characterization of the sequences through the molecular dynamics simulations with the Langevin noise. Models with the steric constraints are found to be better folder…
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 MFP…
A central goal of protein-folding theory is to predict the stochastic dynamics of transition paths --- the rare trajectories that transit between the folded and unfolded ensembles --- using only thermodynamic information, such as a low-dimensional equilibrium free-energy landscape. However, commonly used one-dimensional landscapes typically fall short of this aim, because an empirical coordinate-dependent diffusion coefficient has to be fit to transition-path trajectory data in order to reproduce the transition-path dynamics. We show that an alternative, first-principles free-energy landscape …
Small single-domain proteins often exhibit only a single free-energy barrier, or transition state, between the denatured and the native state. The folding kinetics of these proteins is usually explored via mutational analysis. A central question is which structural information on the transition state can be derived from the mutational data. In this article, we model and structurally interpret mutational Phi-values for two small beta-sheet proteins, the PIN and the FBP WW domain. The native structure of these WW domains comprises two beta-hairpins that form a three-stranded beta-sheet. In our m…
Proteins with nontrivial topology, containing knots and slipknots, have the ability to fold to their native states without any additional external forces invoked. A mechanism is suggested for folding of these proteins, such as YibK and YbeA, which involves an intermediate configuration with a slipknot. It elucidates the role of topological barriers and backtracking during the folding event. It also illustrates that native contacts are sufficient to guarantee folding in around 1-2% of the simulations, and how slipknot intermediates are needed to reduce the topological bottlenecks. As expected, …
Predicting the three-dimensional (3D) functional structures of proteins remains an important computational milestone in molecular biology to be achieved. This feat is hinged on a clear understanding of the mechanism which proteins use to fold into their native structures. Since Levinthal's paradox, there has been a lot of progress in understanding this mechanism. Most of the earlier attempts were caught between assigning either hydrophobic interactions or hydrogen bonding as the dominant folding force. However, a consensus now seems to be emerging about hydrogen bonding being a stronger force.…
We propose a general method for predicting potentially good folders from a given number of amino acid sequences. Our approach is based on the calculation of the rate of convergence of each amino acid chain towards the native structure using only the very initial parts of the dynamical trajectories. It does not require any preliminary knowledge of the native state and can be applied to different kinds of models, including atomistic descriptions. We tested the method within both the lattice and off-lattice model frameworks and obtained several so far unknown good folders. The unbiased algorithm …
