Methods of local topology are introduced to the field of protein physics. This is achieved by explaining how the folding and unfolding processes of a globular protein alter the local topology of the protein's C-alpha backbone through conformational bifurcations. The mathematical formulation builds on the concept of Arnol'd's perestroikas, by extending it to piecewise linear chains using the discrete Frenet frame formalism. In the low-temperature folded phase, the backbone geometry generalizes the concept of a Peano curve, with its modular building blocks modeled by soliton solutions of a discretized nonlinear Schroedinger equation. The onset of thermal unfolding begins when perestroikas change the flattening and branch points that determine the centers of solitons. When temperature increases, the perestroikas cascade, which leads to a progressive disintegration of the modular structures. The folding and unfolding processes are quantitatively characterized by a correlation function that describes the evolution of perestroikas under temperature changes. The approach provides a comprehensive framework for understanding the Physics of protein folding and unfolding transitions, contributing to the broader field of protein structure and dynamics.
We introduce a novel generalization of the discrete nonlinear Schrödinger equation. It supports solitons that describe how proteins fold. As an example we scrutinize the villin headpiece HP35, an archetypal protein for testing both experimental and theoretical approaches to protein folding. Using explicit soliton profiles we construct its carbon backbone with an unprecedented accuracy.
We adopt the point of view that analysis of the stability of the protein folding process is central to understanding the underlying physics of folding. Stability of the folding process means that many perturbations do not disrupt the progress from the random coil to the native state. In this paper we explore the stability of folding using established methods from physics and mathematics. Our result is a preliminary theory of the physics of folding. We suggest some tests of these ideas using folding simulations. We begin by supposing that folding events are related in some way to mechanical wav…
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 principle of superposition to describe protein-folding nuclei and explain an evolutional conservation of a field pattern specifying native-state structure.
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…
We offer simple solutions to three kinematic problems that occur in the folding of proteins. We show how to construct suitably local elementary Monte Carlo moves, how to close a loop, and how to fold a loop without breaking the bond that closes it.
We study the protein folding problem on the base of the quantum approach we proposed recently by considering the model of protein chain with nine amino-acid residues. We introduced the concept of distance space and its projections on a $XY$-plane, and two characteristic quantities, one is called compactness of protein structure and another is called probability ratio involving shortest path. Our results not only confirmed the fast quantum folding time but also unveiled the existence of quantum intelligence hidden behind in choosing protein folding pathways.
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…
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…
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 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…
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…
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.
We present a novel statistical mechanics formalism for the theoretical description of the process of protein folding$\leftrightarrow$unfolding transition in water environment. The formalism is based on the construction of the partition function of a protein obeying two-stage-like folding kinetics. Using the statistical mechanics model of solvation of hydrophobic hydrocarbons we obtain the partition function of infinitely diluted solution of proteins in water environment. The calculated dependencies of the protein heat capacities upon temperature are compared with the corresponding results of e…
When described by a low-dimensional reaction coordinate, the rates of protein folding are determined by a subtle interplay between free-energy barriers and friction. While it is commonplace to extract free-energy profiles from molecular trajectories, a direct evaluation of friction is far more elusive, and one typically evaluates it indirectly via memoryless reaction rate theories. Here, using memory-kernel extraction methods founded on a generalised Langevin equation (GLE) formalism, we directly calculate the memory-dependent friction for eight fast-folding proteins, taken from a published se…
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…
Focusing on a small set of proteins that i) fold in a concerted, all-or-none fashion and ii) do not contain knots or slipknots, we show that the Gauss linking integral, the torsion and the number of sequence-distant contacts provide information regarding the folding rate. Our results suggest that the global topology/geometry of the proteins shifts from right-handed to left-handed with decreasing folding rate, and that this topological change is associated with an increase in the number of more sequence-distant contacts.
Protein folding produces characteristic and functional three-dimensional structures from unfolded polypeptides or disordered coils. The emergence of extraordinary complexity in the protein folding process poses astonishing challenges to theoretical modeling and computer simulations. The present work introduces molecular nonlinear dynamics (MND), or molecular chaotic dynamics, as a theoretical framework for describing and analyzing protein folding. We unveil the existence of intrinsically low dimensional manifolds (ILDMs) in the chaotic dynamics of folded proteins. Additionally, we reveal that …
We investigate the rate-length scaling law of protein folding, a key undetermined scaling law in the analytical theory of protein folding. We demonstrate that chain length is a dominant factor determining folding times, and that the unambiguous determination of the way chain length corre- lates with folding times could provide key mechanistic insight into the folding process. Four specific proposed laws (power law, exponential, and two stretched exponentials) are tested against one an- other, and it is found that the power law best explains the data. At the same time, the fit power law results…
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…
A coarse-grained variational model is used to investigate the polymer dynamics of barrier crossing for a diverse set of two-state folding proteins. The model gives reliable folding rate predictions provided excluded volume terms that induce minor structural cooperativity are included in the interaction potential. In general, the cooperative folding routes have sharper interfaces between folded and unfolded regions of the folding nucleus and higher free energy barriers. The calculated free energy barriers are strongly correlated with native topology as characterized by contact order. Increasing…
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…
Many aspects of the study of protein folding and dynamics have been affected by the recent advances in machine learning. Methods for the prediction of protein structures from their sequences are now heavily based on machine learning tools. The way simulations are performed to explore the energy landscape of protein systems is also changing as force-fields are started to be designed by means of machine learning methods. These methods are also used to extract the essential information from large simulation datasets and to enhance the sampling of rare events such as folding/unfolding transitions.…
It has been observed that the topology of the native state is an important determinant of protein folding kinetics and there is a significant correlation between folding rate and relative contact order (RCO) in two-state small single-domain proteins. However, as a pure topological property, RCO does not take into account residue interactions that also play an important role in folding kinetics. Using the inter-residue statistical contact potentials, we introduce weight into the residue network of contacts, and therefore define a weighted RCO. Using the weighted RCO, we can capture the folding …
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.
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…
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 …
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…
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 …
Folding protein dynamics has been an area of high interest for quite some time, especially given the increased focus on the field of Biophysics. Because folding dynamics occur on such short time scales, empirical techniques developed for more "static" protein events, such as X-ray crystallography, nuclear magnetic resonance, and green fluorescent protein (GFP) labelling, aren't as applicable. Instead, computational methods must often be used to simulate these short lived yet highly dynamic events. One such computational method that is proven to provide much valuable insight into protein foldin…
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 this paper, we introduce an approach to the protein folding problem from the point of view of statistical physics. Protein folding is a stochastic process by which a polypeptide folds into its characteristic and functional 3D structure from random coil. The process involves an intricate interplay between global geometry and local structure, and each protein seems to present special problems. We introduce CSAW (conditioned self-avoiding walk), a model of protein folding that combines the features of self-avoiding walk (SAW) and the Monte Carlo method. In this model, the unfolded protein chai…
The protein folding problem must ultimately be solved on all length scales from the atomic up through a hierarchy of complicated structures. By analyzing the stability of the folding process using physics and mathematics, this paper shows that features without length scales, i.e. topological features, are potentially of central importance. Topology is a natural mathematical tool for the study of shape and we avail ourselves of that tool to examine the relationship between the amino acid sequence and the shapes of protein molecules. We apply what we learn to conjectures about their biological e…
Protein folding is an indispensable process for the majority of proteins after their synthesis from ribosomes in the cell. Most in vitro protein folding studies have focused on single-domain proteins. Hence, it is important to understand the folding process of multi-domain proteins, especially when domains are discontinuous. We choose the Maltose binding protein (MBP) as a model system. In particular, we studied a mutant of MBP that folds slowly. Here, using two- and three-color single-molecule Foerster resonance energy transfer (smFRET) experiments, we study the refolding of both the domains …
A model to describe the mechanism of conformational dynamics in secondary protein based on matter interactions is proposed. The approach deploys the lagrangian method by imposing certain symmetry breaking. The protein backbone is initially assumed to be nonlinear and represented by the Sine-Gordon equation, while the nonlinear external bosonic sources is represented by $ϕ^4$ interaction. It is argued that the nonlinear source induces the folding pathway in a different way than the previous work with initially linear backbone. Also, the nonlinearity of protein backbone decreases the folding spe…
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…
In the traditional random-conformational-search model, various hypotheses with a series of meta-stable intermediate states were often proposed to resolve the Levinthal paradox. Here we introduce a quantum strategy to formulate protein folding as a quantum walk on a definite graph, which provides us a general framework without making hypotheses. Evaluating it by the mean of first passage time, we find that the folding time via our quantum approach is much shorter than the one obtained via classical random walks. This idea is expected to evoke more insights for future studies.
Protein folding and design are major biophysical problems, the solution of which would lead to important applications especially in medicine. Here a novel protein model capable of simultaneously provide quantitative protein design and folding is introduced. With computer simulations it is shown that, for a large set of real protein structures, the model produces designed sequences with similar physical properties to the corresponding natural occurring sequences. The designed sequences are not yet fully realistic and require further experimental testing. For an independent set of proteins, noto…
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. …
Neither of the two prevalent theories, namely thermodynamic stability and kinetic stability, provides a comprehensive understanding of protein folding. The thermodynamic theory is misleading because it assumes that free energy is the exclusive dominant mechanism of protein folding, and attributes the structural transition from one characteristic state to another to energy barriers. Conversely, the concept of kinetic stability overemphasizes dominant mechanisms that are related to kinetic factors. This article explores the stability condition of protein structures from the viewpoint of meso-sci…
We report the reproducible first-principles folding of the 40 amino acid, three-helix headpiece of the HIV accessory protein in a recently developed all-atom free-energy forcefield. Six of twenty simulations using an adapted basin-hopping method converged to better than 3 Åbackbone RMS deviation to the experimental structure. Using over 60,000 low-energy conformations of this protein, we constructed a decoy tree that completely characterizes its folding funnel.
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…
We compare numerically the effectiveness of three recently proposed algorithms, multicanonical simulations, simulations in a 1/k-ensemble, and simulated tempering for the protein folding problem. For this we perform simulations with high statistics for one of the simplest peptides, Met-enkephalin. While the performances of all three approaches is much better than traditional methods, we find that the differences among the three are only marginal.
A modified version of stochastic tunneling, a recently introduced global optimization technique, is introduced as a new generalized-ensemble technique and tested for a benchmark peptide, Met-enkephalin. It is demonstrated that the new technique allows to evaluate folding properties of this peptide
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 statistical properties of protein folding within the ϕ^4 model are investigated. The calculation is performed using statistical mechanics and path integral method. In particular, the evolution of heat capacity in term of temperature is given for various levels of the nonlinearity of source and the strength of interaction between protein backbone and nonlinear source. It is found that the nonlinear source contributes constructively to the specific heat especially at higher temperature when it is weakly interacting with the protein backbone. This indicates increasing energy absorption as the…
The protein folding problem is stated and a list of properties that do not depend upon specific molecules is compiled and analyzed. The relationship of this analysis to future simulations is emphasized. The choice of power and time as variables as opposed to energy and time is discussed. A wave motion model is reviewed and related to the action in classical mechanics. It is argued that the properties of the action support the idea that folding takes place in small steps. It is explained how catastrophe theory has been employed in wave motion models and how it can be used in examination of succ…
We propose a new deterministic methodology to predict RNA sequence and protein folding. Is stem enough for structure prediction? The main idea is to consider all possible stem formation in the given sequence. With the stem loop energy and the strength of stem, we explore how to deterministically utilize stem information for RNA sequence and protein folding structure prediction. We use graph notation, where all possible stems are represented as vertices, and co-existence as edges. This full Stem-graph presents all possible folding structure, and we pick sub-graph(s) which give the best matching…
We review theoretical approaches, experiments and numerical simulations that have been recently proposed to investigate the folding problem in single-domain proteins. From a theoretical point of view, we emphasize the energy landscape approach. As far as experiments are concerned, we focus on the recent development of single-molecule techniques. In particular, we compare the results obtained with two main techniques: single protein force measurements with optical tweezers and single-molecule fluorescence in studies on the same protein (RNase H). This allows us to point out some controversial i…
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…
The principles underlying protein folding remains one of Nature's puzzles with important practical consequences for Life. An approach that has gathered momentum since the late 1990's, looks at protein hetero-polymers and their folding process through the lens of complex network analysis. Consequently, there is now a body of empirical studies describing topological characteristics of protein macro-molecules through their contact networks and linking these topological characteristics to protein folding. The present paper is primarily a review of this rich area. But it delves deeper into certain …
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.
