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 barriers will commonly be found, even in "two-state"proteins; (4) very early times will show charactreristic departures from single-exponential kinetics; (5) contributions of non-native interactions to phi-values are calculable, and may be significant. The example of a three-helix bundle is treated in more detail as an illustration. The model also shows that high-dimensional structures provide conceptual relations between the "folding funnel", "diffusion-collision", "nucleation-condensation" and "topomer search" models of protein folding. It suggests that kinetic strategies for fast folding may be encoded rather generally in non-native, rather than native interactions. The predictions are related to very recent findings in experiment and simulation.
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 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…
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 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 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 …
Many protein systems fold in a two-state manner. Random models, however, rarely display two-state kinetics and thus such behavior should not be accepted as a default. To date, many theories for the prevalence of two-state kinetics have been presented, but none sufficiently explain the breadth of experimental observations. A model, making a minimum of assumptions, is introduced that suggests two-state behavior is likely for any system with an overwhelmingly populated native state. We show two-state folding is emergent and strengthened by increasing the occupancy population of the native state. …
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 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…
We assume that the protein folding process follows two autonomous steps: the conformational search for the native, mainly ruled by the hydrophobic effect; and, the final adjustment stage, which eventually gives stability to the native. Our main tool of investigation is a 3D lattice model provided with a ten-letter alphabet, the stereochemical model. This model was conceived for Monte Carlo (MC) simulations when one keeps in mind the kinetic behavior of protein-like chains in solution. In order to characterize the folding characteristic time (τ) by two distinct sampling methods, first we presen…
We propose a principle of superposition to describe protein-folding nuclei and explain an evolutional conservation of a field pattern specifying native-state structure.
The statistical mechanics of heteropolymer structure formation is studied in the context of RNA secondary structures. A designed RNA sequence biased energetically towards a particular native structure (a hairpin) is used to study the transition between the native and molten phase of the RNA as a function of temperature. The transition is driven by a competition between the energy gained from the polymer's overlap with the native structure and the entropic gain of forming random contacts. A simplified Go-like model is proposed and solved exactly. The predicted critical behavior is verified via …
A major challenge in molecular simulations is to describe denaturant-dependent folding of proteins order to make direct comparisons with {\it in vitro} experiments. We use the molecular transfer model, which is currently the only method that accomplishes this goal albeit phenomenologically, to quantitatively describe urea-dependent folding of PDZ domain, which plays a significant role in molecular recognition and signaling. Experiments show that urea-dependent unfolding rates of the PDZ2 domain exhibit a downward curvature at high urea concentrations, which has been interpreted by invoking the…
Motivated by claims about the nature of the observed timescales in protein systems said to fold downhill, we have studied the finite, linear master equation which is a model of the downhill process. By solving for the system eigenvalues, we prove the often stated claim that in situations where there is no free energy barrier, a transition between single and multi-exponential kinetics occurs at sufficient bias (towards the native state). Consequences for protein folding, especially the downhill folding scenario, are briefly discussed.
We develop a coarse-grained protein model with a simplified amino acid interaction potential. We perform discrete molecular dynamics folding simulations of a small 20 residue protein - Trp-cage - from a fully extended conformation. We demonstrate the ability of the Trp-cage model to consistently reach conformations within 2angstrom backbone root-mean-square distance (RMSD) from the corresponding NMR structures. The minimum RMSD of Trp-cage conformations in the simulation can be smaller than 1.00angstrom. Our findings suggest that, at least for the case of Trp-cage, a detailed all-atom protein …
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…
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)…
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…
The dynamics of two 12-monomer heteropolymers on the square lattice is studied exactly within the master equation approach. The time evolution of the occupancy of the native state is determined. At low temperatures, the median folding time follows the Arrhenius law and is governed by the longest relaxation time. For good folders, significant kinetic traps appear in the folding funnel whereas for bad folders, the traps also occur in non-native energy valleys.
Using the Helmholtz decomposition of the vector field of folding fluxes in a two-dimensional space of collective variables, a potential of the driving force for protein folding is introduced. The potential has two components. One component is responsible for the source and sink of the folding flows, which represent, respectively, the unfolded states and the native state of the protein, and the other, which accounts for the flow vorticity inherently generated at the periphery of the flow field, is responsible for the canalization of the flow between the source and sink. The theoretical consider…
The master equation for 12-monomer lattice heteropolymers is solved numerically and the time evolution of the occupancy of the native state is determined. At low temperatures, the median folding time follows the Arrhenius law and is governed by the longest relaxation time. For good folders, significant kinetic traps appear in the folding funnel whereas for bad folders, the traps also occur in non-native energy valleys.
Extensive Monte Carlo folding simulations for four proteins of various structural classes are carried out, using a single atomistic potential. In all cases, collapse occurs at a very early stage, and proteins fold into their native-like conformations at appropriate temperatures. The results demonstrate that the folding mechanism is controlled not only by thermodynamic factors but also by kinetic factors: The way a protein folds into its native structure, is also determined by the convergence point of early folding trajectories, which cannot be obtained by the free energy surface.
Molecular chaperones are ATP-consuming biological machines, which facilitate the folding of proteins and RNA molecules that are kinetically trapped in misfolded states for long times. Unassisted folding occurs by the kinetic partitioning mechanism according to which folding to the native state, with low probability as well as misfolding to one of the many metastable states, with high probability, occur rapidly on similar time scales. GroEL is an all-purpose stochastic machine that assists misfolded substrate proteins (SPs) to fold. The RNA chaperones (CYT-19) help the folding of ribozymes that…
Acylphosphatase (AcP) is a small protein with 98 amino acid residues that catalyzes the hydrolysis of carboxyl-phosphate bonds. AcP is a typical two-state protein with slow folding rate due to its relatively large contact order in the native structure. The mechanical properties and unfolding behavior of AcP has been studied by atomic force microscope. But the folding and unfolding dynamics at low forces has not been reported. Here using stable magnetic tweezers, we measured the force-dependent folding rates within a force range from 1 pN to 3 pN, and unfolding rates from 15 pN to 40 pN. The ob…
Energy landscape theory describes how a full-length protein can attain its native fold after sampling only a tiny fraction of all possible structures. Although protein folding is now understood to be concomitant with synthesis on the ribosome there have been few attempts to modify energy landscape theory by accounting for cotranslational folding. This paper introduces a model for cotranslational folding that leads to a natural definition of a nested energy landscape. By applying concepts drawn from submanifold differential geometry the dynamics of protein folding on the ribosome can be explore…
CSAW (conditioned self-avoiding walk) is a model of protein folding that combines SAW (self-avoiding walk) with Monte-Carlo. It simulates the Brownian motion of a chain molecule in the presence of interactions, both among chain residues, and with the environment. In a first model that includes the hydrophobic effect and hydrogen bonding, a chain of 30 residues folds into a native state with stable secondary and tertiary structures. The process starts with a rapid collapse into an intermediate "molten globule", which slowly decays into the native state afer a relatively long quiescent period. T…
Evolutionally conserved quantity that specifies folding nuclei is pursued by a case study for a small protein (PDB code: 1ten). First it is demonstrated that the sequences of amino acids at folding nuclei are not conserved. Then 3D (3-dimensional) information of the structure is considered and it is found that a 3D hydrophobicity profile is essential to specify the folding nuclei and evolutionally conserved. This profile is maintained by the interaction, including entropic effect, among amino acids realized in the native state structure. Experimentally observed phi-value is correlated to this …
As a consequence of the rugged landscape of RNA molecules their folding is described by the kinetic partitioning mechanism according to which only a small fraction ($ϕ_F$) reaches the folded state while the remaining fraction of molecules is kinetically trapped in misfolded intermediates. The transition from the misfolded states to the native state can far exceed biologically relevant time. Thus, RNA folding in vivo is often aided by protein cofactors, called RNA chaperones, that can rescue RNAs from a multitude of misfolded structures. We consider two models, based on chemical kinetics and ch…
The thermodynamic properties for three different types of off-lattice four-strand beta-sheet protein models interacting via a hybrid Go-type potential have been investigated. Discontinuous molecular dynamic simulations have been performed for different sizes of the bias gap g, an artificial measure of a model protein's preference for its native state. The thermodynamic transition temperatures are obtained by calculating the squared radius of gyration, the root-mean-squared pair separation fluctuation, the specific heat, the internal energy of the system, and the Lindemann disorder parameter. I…
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.
A reduced protein model with five to six atoms per amino acid and five amino acid types is developed and tested on a three-helix-bundle protein, a 46-amino acid fragment from staphylococcal protein A. The model does not rely on the widely used Go approximation where non-native interactions are ignored. We find that the collapse transition is considerably more abrupt for the protein A sequence than for random sequences with the same composition. The chain collapse is found to be at least as fast as helix formation. Energy minimization restricted to the thermodynamically favored topology gives a…
Protein folding processes are generally described statistically with the help of multidimensional free energy landscape, typically reduced to a 1-D free energy profile along good reaction co-ordinate. There are many physical parameters which are responsible for protein molecule to hop between the native and unfolded states. The transition path region across the barrier is the region corresponding to the minimum fluctuation. The extent to which this transition region can extend beyond the obvious 1/2 kBT region has been a question of interest for a long time. We propose a new method to locate t…
An all-atom model of proteins is used to show that the same sequence of amino acids can have many alternative structures, that are very distant from, and that can be as stable as, the corresponding native structure. Such alternative structures are not easily rationalized as belonging to the native basin and indicate instead that the free energy landscape of proteins is multi-funnel-shaped and that Anfinsen's thermodynamic hypothesis alone cannot explain protein folding. An alternative two-step process for folding is proposed and its consistency with the experimental evidence available is discu…
A small model polypeptide represented in atomic detail is folded using Monte Carlo dynamics. The polypeptide is designed to have a native conformation similar to the central part of the helix-turn-helix protein ROP. Starting from a beta-strand conformation or two different loop conformations of the protein glutamine synthetase, six trajectories are generated using the so-called window move in dihedral angle space. This move changes conformations locally and leads to realistic, quasi-continuously evolving trajectories. Four of the six trajectories end in stable native-like conformations. Their …
We study folding dynamics of protein-like sequences on square lattice using physical move set that exhausts all possible conformational changes. By analytically solving the master equation, we follow the time-dependent probabilities of occupancy of all 802,075 conformations of 16-mers over 7-orders of time span. We find that (i) folding rates of these protein-like sequences of same length can differ by 4-orders of magnitude, (ii) folding rates of sequences of the same conformation can differ by a factor of 190, and (iii) parameters of the native structures, designability, and thermodynamic pro…
The prediction of protein secondary and tertiary structures from the primary amino acid sequence is both an incredibly important and incredibly difficult problem. Accurate prediction of a protein's native structure can provide critical insights about its function, ultimately leading to breakthoughs in drug design and disease diagnosis. The field has a rich history, from the earliest folding experiments in the 1960's to the use of state-of-the-art algorithms today; this article reviews protein folding's history with an emphasis on how modern methods are tackling the protein folding problem. Ass…
We propose a model for protein folding in vivo based on a Brownian-ratchet mechanism in the multidimensional energy landscape space. The device is able to produce directed transport taking advantage of the assumed intrinsic asymmetric properties of the proteins and employing the consumption of energy provided by an external source. Through such a directed transport phenomenon, the polypeptide finds the native state starting from any initial state in the energy landscape with great efficacy and robustness, even in the presence of different type of obstacles. This model solves Levinthal's parado…
Loops are abundant in native RNA structures and proliferate close to the unfolding transition. By including a statistical weight ~ l^{-c} for loops of length l in the recursion relation for the partition function, we show that the calculated heat capacity depends sensitively on the presence and value of the exponent c, even of short t-RNA. For homo-RNA we analytically calculate the critical temperature and critical exponents which exhibit a non-universal dependence on c.
Understanding the protein folding process is an outstanding issue in biophysics; recent developments in molecular dynamics simulation have provided insights into this phenomenon. However, the large freedom of atomic motion hinders the understanding of this process. In this study, we applied persistent homology, an emerging methods to analyze topological features in a dataset, to reveal protein folding dynamics. We developed a new method to characterize protein structure based on persistent homology and applied this method to molecular dynamics simulations of chignolin. Using principle componen…
The processes by which protein sidechains reach equilibrium during a folding reaction are investigated using both lattice and all-atom simulations. We find that rates of sidechain relaxation exhibit a distribution over the protein structure, with the fastest relaxing sidechains being involved in kinetically important positions. Traversal of the major folding transition state corresponds to the freezing of a small number of residues, while the rest of the chain proceeds towards equilibrium via backbone fluctuations around the native fold. The post-nucleation processes by which sidechains relax …
We report a single-copy tempering method for simulating large complex systems. In a generalized ensemble, the method uses runtime estimate of the thermal average energy computed from a novel integral identity to guide a continuous temperature-space random walk. We first validated the method in a two-dimensional Ising model and a Lennard-Jones liquid system. It was then applied to folding of three small proteins, trpzip2, trp-cage, and villin headpiece in explicit solvent. Within 0.5~1 microsecond, all three systems were folded into atomic accuracy: the alpha carbon root mean square deviations …
