We assess the reliability of the recently developed approach denominated Dominant Reaction Pathways (DRP) by studying the folding of a 16-residue beta-hairpin, within a coarse-grained Go-type model. We show that the DRP predictions are in quantitative agreement with the results of Molecular Dynamics simulations, performed in the same model. On the other hand, in the DRP approach, the computational difficulties associated to the decoupling of time scales are rigorously bypassed. The analysis of the important transition pathways supports a picture of the beta-hairpin folding in which the reaction is initiated by the collapse of the hydrophobic cluster.
Within the frame of an effective, coarse-grained hydrophobic-polar protein model, we employ multicanonical Monte Carlo simulations to investigate free-energy landscapes and folding channels of exemplified heteropolymer sequences, which are permutations of each other. Despite the simplicity of the model, the knowledge of the free-energy landscape in dependence of a suitable system order parameter enables us to reveal complex folding characteristics known from real bioproteins and synthetic peptides, such as two-state folding, folding through weakly stable intermediates, and glassy metastability…
ZSPA-1 is an engineered protein that binds to its parent, the three-helix-bundle Z domain of staphylococcal protein A. Uncomplexed ZSPA-1 shows a reduced helix content and a melting behavior that is less cooperative, compared with the wild-type Z domain. Here we show that the difference in folding behavior between these two sequences can be partly understood in terms of an off-lattice model with 5-6 atoms per amino acid and a minimalistic potential, in which folding is driven by backbone hydrogen bonding and effective hydrophobic attraction.
In the past years, the folding kinetics of many small single-domain proteins has been characterized by mutational Phi-value analysis. In this article, a simple, essentially parameter-free model is introduced which derives folding routes from native structures by minimizing the entropic loop-closure cost during folding. The model predicts characteristic folding sequences of structural elements such as helices and beta-strand pairings. Based on few simple rules, the kinetic impact of these structural elements is estimated from the routes and compared to average experimental Phi-values for the he…
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, …
Understanding how monomeric proteins fold under in vitro conditions is crucial to describing their functions in the cellular context. Significant advances both in theory and experiments have resulted in a conceptual framework for describing the folding mechanisms of globular proteins. The experimental data and theoretical methods have revealed the multifaceted character of proteins. Proteins exhibit universal features that can be determined using only the number of amino acid residues (N) and polymer concepts. The sizes of proteins in the denatured and folded states, cooperativity of the foldi…
Phi-values are experimental measures of how the kinetics of protein folding is changed by single-site mutations. Phi-values measure energetic quantities, but are often interpreted in terms of the structures of the transition state ensemble. Here we describe a simple analytical model of the folding kinetics in terms of the formation of protein substructures. The model shows that Phi-values have both structural and energetic components. In addition, it provides a natural and general interpretation of "nonclassical" Phi-values (i.e., less than zero, or greater than one). The model reproduces the …
In this study, we explore nucleation and the transition state ensemble of the ribosomal protein S6 using a Monte Carlo Go model in conjunction with restraints from experiment. The results are analyzed in the context of extensive experimental and evolutionary data. The roles of individual residues in the folding nucleus are identified and the order of events in the S6 folding mechanism is explored in detail. Interpretation of our results agrees with, and extends the utility of, experiments that shift f-values by modulating denaturant concentration and presents strong evidence for the realism of…
Reaching a ground state of a spin system is analogous to a protein evolving into its native state. We study the ``folding'' times for various random Ising spin systems and determine characteristic temperatures that relate to the ``folding''. Under optimal kinetic conditions, the ``folding'' times scale with the system size as a power law with a non-universal exponent. This is similar to what happens in model proteins. On the other hand, the scaling behavior of the characteristic temperatures is different than in model proteins. Both in the spin systems and in proteins, the folding properties d…
The rates of protein folding with photon absorption or emission and the cross section of photon -protein inelastic scattering are calculated from the quantum folding theory by use of standard field-theoretical method. All these protein photo-folding processes are compared with common protein folding without interaction of photons (nonradiative folding). It is demonstrated that there exists a common factor (thermo-averaged overlap integral of vibration wave function, TAOI) for protein folding and protein photo-folding. Based on this finding it is predicted that: 1) the stimulated photo-folding …
The extent of coupling between the folding of a protein and its binding to a substrate varies from protein to protein. Some proteins have highly structured native states in solution, while others are natively disordered and only fold fully upon binding. In this Letter, we use Monte Carlo simulations to investigate how disordered polymer chains grafted around a binding site affect the folding and binding of three model proteins. The protein that approaches the substrate fully folded is more hindered during the binding process than those whose folding and binding are cooperative. The polymer cha…
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…
Scaling of folding properties of proteins is studied in a toy system -- the lattice Go model with various two- and three- dimensional geometries of the maximally compact native states. Characteristic folding times grow as power laws with the system size. The corresponding exponents are not universal. Scaling of the thermodynamic stability also indicates size-related deterioration of the folding properties.
We introduce a powerful iterative algorithm to compute protein folding pathways, with realistic all-atom force fields. Using the path integral formalism, we explicitly derive a modified Langevin equation which samples directly the ensemble of reactive pathways, exponentially reducing the cost of simulating thermally activated transitions. The algorithm also yields a rigorous stochastic estimate of the reaction coordinate. After illustrating this approach on a simple toy model, we successfully validate it against the results of ultra-long plain MD protein folding simulations for a fast folding …
We investigate the interplay between post-translational folding and escape of two small single-domain proteins at the ribosomal exit tunnel by using Langevin dynamics with coarse-grained models. It is shown that at temperatures lower or near the temperature of the fastest folding, folding proceeds concomitantly with the escape process, resulting in vectorial folding and enhancement of foldability of nascent proteins. The concomitance between the two processes, however, deteriorates as temperature increases. Our folding simulations as well as free energy calculation by using umbrella sampling s…
The assumption that similar structures have similar folding probabilities ($p_{fold}$) leads naturally to a procedure to evaluate $p_{fold}$ for every snapshot saved along an equilibrium folding-unfolding trajectory of a structured peptide or protein. The procedure utilizes a structurally homogeneous clustering and does not require any additional simulation. It can be used to detect multiple folding pathways as shown for a three-stranded antiparallel $β$-sheet peptide investigated by implicit solvent molecular dynamics simulations.
Two proteins, one belonging to the mainly alpha class and the other belonging to the alpha/beta class, are selected to test a kinetic mechanism for protein folding. Targeted molecular dynamics is applied to generate folding pathways for those two proteins, starting from two well defined initial conformations: a fully extended and a alpha-helical conformation. The results show that for both proteins the alpha-helical initial conformation provides overall lower energy pathways to the native state. For the alpha/beta protein, 30 % (40%) of the pathways from an initial alpha-helix (fully extended)…
We propose a general theory to describe the distribution of protein-folding transition paths. We show that transition paths follow a predictable sequence of high-free-energy transient states that are separated by free-energy barriers. Each transient state corresponds to the assembly of one or more discrete, cooperative units, which are determined directly from the native structure. We show that the transition state on a folding pathway is reached when a small number of critical contacts are formed between a specific set of substructures, after which folding proceeds downhill in free energy. Th…
Kinetics of folding of a protein held in a force-clamp are compared to an unconstrained folding. The comparison is made within a simple topology-based dynamical model of ubiquitin. We demonstrate that the experimentally observed variations in the end-to-end distance reflect microscopic events during folding. However, the folding scenarios in and out of the force-clamp are distinct.
As protein folding is a NP-complete problem, artificial intelligence tools like neural networks and genetic algorithms are used to attempt to predict the 3D shape of an amino acids sequence. Underlying these attempts, it is supposed that this folding process is predictable. However, to the best of our knowledge, this important assumption has been neither proven, nor studied. In this paper the topological dynamic of protein folding is evaluated. It is mathematically established that protein folding in 2D hydrophobic-hydrophilic (HP) square lattice model is chaotic as defined by Devaney. Consequ…
Novel numerical techniques, validated by an analysis of barnase and chymotrypsin inhibitor, are used to elucidate the paramount role played by the geometry of the protein backbone in steering the folding to the correct native state. It is found that, irrespective of the sequence, the native state of a protein has exceedingly large number of conformations with a given amount of structural overlap compared to other compact artificial backbones; moreover the conformational entropies of unrelated proteins of the same length are nearly equal at any given stage of folding. These results are suggesti…
Understanding protein folding has been one of the great challenges in biochemistry and molecular biophysics. Over the past 50 years, many thermodynamic and kinetic studies have been performed addressing the stability of globular proteins. In comparison, advances in the membrane protein folding field lag far behind. Although membrane proteins constitute about a third of the proteins encoded in known genomes, stability studies on membrane proteins have been impaired due to experimental limitations. Furthermore, no systematic experimental strategies are available for folding these biomolecules in…
We study the dynamics of protein folding via statistical energy-landscape theory. In particular, we concentrate on the local-connectivity case with the folding progress described by the fraction of native conformations. We obtain information for the first passage-time (FPT) distribution and its moments. The results show a dynamic transition temperature below which the FPT distribution develops a power-law tail, a signature of the intermittency phenomena of the folding dynamics. We also discuss the possible application of the results to single-molecule dynamics experiments.
Simple theoretical concepts and models have been helpful to understand the folding rates and routes of single-domain proteins. As reviewed in this article, a physical principle that appears to underly these models is loop closure.
We perform molecular dynamics simulations for a simple coarse-grained model of crambin placed inside of a softly repulsive sphere of radius R. The confinement makes folding at the optimal temperature slower and affects the folding scenarios, but both effects are not dramatic. The influence of crowding on folding are studied by placing several identical proteins within the sphere, denaturing them, and then by monitoring refolding. If the interactions between the proteins are dominated by the excluded volume effects, the net folding times are essentially like for a single protein. An introductio…
We introduce a method for calculating the extent to which chain non-crossing is important in the most efficient, optimal trajectories or pathways for a protein to fold. This involves recording all unphysical crossing events of a ghost chain, and calculating the minimal uncrossing cost that would have been required to avoid such events. A depth-first tree search algorithm is applied to find minimal transformations to fold $α$, $β$, $α/β$, and knotted proteins. In all cases, the extra uncrossing/non-crossing distance is a small fraction of the total distance travelled by a ghost chain. Different…
The folding of naturally occurring, single domain proteins is usually well-described as a simple, single exponential process lacking significant trapped states. Here we further explore the hypothesis that the smooth energy landscape this implies, and the rapid kinetics it engenders, arises due to the extraordinary thermodynamic cooperativity of protein folding. Studying Miyazawa-Jernigan lattice polymers we find that, even under conditions where the folding energy landscape is relatively optimized (designed sequences folding at their temperature of maximum folding rate), the folding of protein…
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…
A fundamental question in protein folding is whether the coil to globule collapse transition occurs during the initial stages of folding (burst-phase) or simultaneously with the protein folding transition. Single molecule fluorescence resonance energy transfer (FRET) and small angle X-ray scattering (SAXS) experiments disagree on whether Protein L collapse transition occurs during the burst-phase of folding. We study Protein L folding using a coarse-grained model and molecular dynamics simulations. The collapse transition in Protein L is found to be concomitant with the folding transition. In …
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.
A quantum theory on conformation-electron system is presented. Protein folding is regarded as the quantum transition between torsion states on polypeptide chain, and the folding rate is calculated by nonadiabatic operator method. The theory is used to study the temperature dependences of folding rate of 15 proteins and their non-Arrhenius behavior can all be deduced in a natural way. A general formula on the rate-temperature dependence has been deduced which is in good accordance with experimental data. These temperature dependences are further analyzed in terms of torsion potential parameters…
Exploring and understanding the protein-folding problem has been a long-standing challenge in molecular biology. Here, using molecular dynamics simulation, we reveal how parallel distributed adjacent planar peptide groups of unfolded proteins fold reproducibly following explicit physical folding codes in aqueous environments due to electrostatic attractions. Superfast folding of protein is found to be powered by the contribution of the formation of hydrogen bonds. Temperature-induced torsional waves propagating along unfolded proteins break the parallel distributed state of specific amino acid…
Nonnative residual interactions have attracted increasing attention in recent protein folding researches. Experimental and theoretical investigations had been set out to catch nonnative contacts that might dominate key events in protein folding. However, energetic frustrations caused by nonnative inter-residue interactions are not systematically characterized, due to the complicated folding mechanism. Recently, we studied the folding of a set of homologous all-α proteins and found that nonnative-contact-based energetic frustrations are closely related to protein native-contact networks. In thi…
We investigated the impact of hydrodynamic interactions (HI) on protein folding using a coarse-grained model. The extent of the impact of hydrodynamic interactions, whether it accelerates, retards, or has no effect on protein folding, has been controversial. Together with a theoretical framework of the energy landscape theory (ELT) for protein folding that describes the dynamics of the collective motion with a single reaction coordinate across a folding barrier, we compared the kinetic effects of HI on the folding rates of two protein models that use a chain of single beads with distinctive to…
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…
The protein folding is regarded as a quantum transition between torsion states on polypeptide chain. The deduction of the folding rate formula in our previous studies is reviewed. The rate formula is generalized to the case of frequency variation in folding. Then the following problems about the application of the rate theory are discussed: 1) The unified theory on the two-state and multi-state protein folding is given based on the concept of quantum transition. 2) The relationship of folding and unfolding rates vs denaturant concentration is studied. 3) The temperature dependence of folding r…
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…
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 …
We report 10 successfully folding events of trpzip2 by molecular dynamics simulation. It is found that the trizip2 can fold into its native state through different zipper pathways, depending on the ways of forming hydrophobic core. We also find a very fast non-zipper pathway. This indicates that there may be no inconsistencies in the current pictures of beta-hairpin folding mechanisms. These pathways occur with different probabilities. zip-out is the most probable one. This may explain the recent experiment that the turn formation is the rate-limiting step for beta-hairpin folding.
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.
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.
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.
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…
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.
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.…
Theory of multi-dimensional representation of free energy surface of protein folding is developed by adopting structural order parameters of multiple regions in protein as multiple coordinates. Various scenarios of folding are classified in terms of cooperativity within individual regions and interactions among multiple regions and thus obtained classification is used to analyze the folding process of several example proteins. Ribosomal protein S6, src-SH3 domain, CheY, barnase, and BBL domain are analyzed with the two-dimensional representation by using a structure-based Hamiltonian model. Ex…
Protein folding is a phenomenon that has been studied for about 50 years and still remains as an unsolved problem. The main feature of this process is that it occurs as an all or none process, so a protein, can jump directly between folded and unfolded states. It is proposed in this manuscript that the cooperative phenomena associated to protein folding has its origin in the synchronization oscillating phases of the instantaneously induced dipoles that gives rise to the van der Waals London dispersion interaction. When the oscillation of the induced dipoles is synchronized an enhanced interact…
As an example of topic where biology and physics meet, we present the issue of protein folding and stability, and the development of thermodynamics-based bioinformatics tools that predict the stability and thermal resistance of proteins and the change of these quantities upon amino acid substitutions. These methods are based on knowledge-driven statistical potentials, derived from experimental protein structures using the inverse Boltzmann law. We also describe an application of these predictors, which contributed to the understanding of the mechanisms of aggregation of a particular protein kn…
Here we study the effects of many-body interactions on rate and mechanism in protein folding, using the results of molecular dynamics simulations on numerous coarse-grained C-alpha-model single-domain proteins. After adding three-body interactions explicitly as a perturbation to a Go-like Hamiltonian with native pair-wise interactions only, we have found 1) a significantly increased correlation with experimental phi-values and folding rates, 2) a stronger correlation of folding rate with contact order, matching the experimental range in rates when the fraction of three-body energy in the nativ…
We have developed a simple, phenomenological, Monte-Carlo code that predicts the three-dimensional structure of globular proteins from the DNA sequences that define them. We have applied this code to two small proteins, the villin headpiece (1VII) and cole1 rop (1ROP). Our code folds both proteins to within 5 A rms of their native structures.
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…
The process of protein folding from an unfolded state to a biologically active, folded conformation is governed by many parameters e.g the sequence of amino acids, intermolecular interactions, the solvent, temperature and chaperon molecules. Our study, based on random matrix modeling of the interactions, shows however that the evolution of the statistical measures e.g Gibbs free energy, heat capacity, entropy is single parametric. The information can explain the selection of specific folding pathways from an infinite number of possible ways as well as other folding characteristics observed in …
A novel combination of discontinuous molecular dynamics and the Langevin equation, together with an intermediate-resolution model, are used to carry out long (several $μ$s) simulation and study folding transition and transport of proteins in slit nanopores. Both attractive ($U^+$) and repulsive ($U^-$) interaction potentials between the proteins and the pore walls are considered. Near the folding temperature $T_f$ and in the presence of $U^+$ the proteins undergo a repeating sequence of folding/partially-folding/ unfolding transitions, while $T_f$ decreases with decreasing pore sizes. The oppo…
De novo protein structure prediction from amino acid sequence is one of the most challenging problems in computational biology. As one of the extensively explored mathematical models for protein folding, Hydrophobic-Polar (HP) model enables thorough investigation of protein structure formation and evolution. Although HP model discretizes the conformational space and simplifies the folding energy function, it has been proven to be an NP-complete problem. In this paper, we propose a novel protein folding framework FoldingZero, self-folding a de novo protein 2D HP structure from scratch based on …
Different aspects of protein folding are illustrated by simplified polymer models. Stressing the diversity of side chains (residues) leads one to view folding as the freezing transition of an heteropolymer. Technically, the most common approach to diversity is randomness, which is usually implemented in two body interactions (charges, polar character,..). On the other hand, the (almost) universal character of the protein backbone suggests that folding may also be viewed as the crystallization transition of an homopolymeric chain, the main ingredients of which are the peptide bond and chirality…
We study the folding process in the shallowly knotted protein MJ0366 within two variants of a structure-based model. We observe that the resulting topological pathways are much richer than identified in previous studies. In addition to the single knot-loop events, we find novel, and dominant, two-loop mechanisms. We demonstrate that folding takes place in a range of temperatures and the conditions of most successful folding are at temperatures which are higher than those required for the fastest folding. We also demonstrate that nascent conditions are more favorable to knotting than off-riboso…
We propose a universal elastic energy for proteins, which depends only on the radius of gyration $R_{g}$ and the residue number $N$. It is constructed using physical arguments based on the hydrophobic effect and hydrogen bonding. Adjustable parameters are fitted to data from the computer simulation of the folding of a set of proteins using the CSAW (conditioned self-avoiding walk) model. The elastic energy gives rise to scaling relations of the form $R_{g}\sim N^ν$ in different regions. It shows three folding stages characterized by the progression with exponents $ν= 3/5, 3/7, 2/5$, which we i…
In this study we evaluate, at full atomic detail, the folding processes of two small helical proteins, the B domain of protein A and the Villin headpiece. Folding kinetics are studied by performing a large number of ab initio Monte Carlo folding simulations using a single transferable all-atom potential. Using these trajectories, we examine the relaxation behavior, secondary structure formation, and transition-state ensembles (TSEs) of the two proteins and compare our results with experimental data and previous computational studies. To obtain a detailed structural information on the folding d…
