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 kinetics of proteins having the same topology, but different sequence information. By constructing essential sub-networks based on the strength of the pairwise interactions, we are able to deduce the features of sequences redundant for folding events. We perform an analysis on 48 two-state proteins and the ultrafast-folding proteins, as well as mutants of the protein CI2, protein G, and protein L. Our results indicate that (i) both the weighted RCO of original residue network and that of essential sub-networks have significant correlations with the folding rate like RCO; (ii) the folding rate is critically dependent on the hydrophobic interactions for two-state folding; and (iii) the sub-networks distinguish the folding rate differences of the mutants and reveal the folding preferences of proteins.
Protein folding is a universal process, very fast and accurate, which works consistently (as it should be) in a wide range of physiological conditions. The present work is based on three premises, namely: ($i$) folding reaction is a process with two consecutive and independent stages, namely the search mechanism and the overall productive stabilization; ($ii$) the folding kinetics results from a mechanism as fast as can be; and ($iii$) at nanoscale dimensions, local thermal fluctuations may have important role on the folding kinetics. Here the first stage of folding process (search mechanism) …
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…
To function as gene regulatory elements in response to environmental signals, riboswitches must adopt specific secondary structures on appropriate time scales. We employ kinetic Monte Carlo simulation to model the time-dependent folding during transcription of TPP riboswitch expression platforms. According to our simulations, riboswitch transcriptional terminators, which must adopt a specific hairpin configuration by the time they have been transcribed, fold with higher efficiency than Shine-Dalgarno sequesterers, whose proper structure is required only at the time of ribosomal binding. Our fi…
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…
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.
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. …
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…
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 …
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.
In cellular environment, confinement and macromulecular crowding play an important role on thermal stability and folding kinetics of a protein. We have resorted to a generalized version of the Wako-Saito-Munoz-Eaton model for protein folding to study the behavior of six different protein structures confined between two walls. Changing the distance 2R between the walls, we found, in accordance with previous studies, two confinement regimes: starting from large R and decreasing R, confinement first enhances the stability of the folded state as long as this is compact and until a given value of R…
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 influence of the salts KCl, NaCl, and NaI at molar concentrations on the α-helical folding kinetics of the alanine-based oligopeptide Ace-AEAAAKEAAAKA-Nme is investigated by means of (explicit-water) molecular dynamics simulations and a diffusional analysis. The mean first passage times for folding and unfolding are found to be highly salt-specific. In particular, the folding times increase about one order of magnitude for the sodium salts. The drastic slowing down can be traced back to long-lived, compact configurations of the partially folded peptide, in which sodium ions are tightly bou…
Experimental observations suggest that proteins follow different pathways under different environmental conditions. We perform molecular dynamics simulations of a model of the SH3 domain over a broad range of temperatures, and identify distinct pathways in the folding transition. We determine the kinetic partition temperature --the temperature for which the SH3 domain undergoes a rapid folding transition with minimal kinetic barriers-- and observe that below this temperature the model protein may undergo a folding transition via multiple folding pathways. The folding kinetics is characterized …
Background: Guanine quadruplexes (GQs) play vital roles in many cellular processes and are of much interest as drug targets. In contrast to the availability of many structural studies, there is still limited knowledge on GQ folding. Scope of review: We review recent molecular dynamics (MD) simulation studies of the folding of GQs, with an emphasis paid to the human telomeric DNA GQ. We explain the basic principles and limitations of all types of MD methods used to study unfolding and folding in a way accessible to non-specialists. We discuss the potential role of G-hairpin, G-triplex and alter…
Non-coding RNA sequences play a great role in controlling a number of cellular functions, thus raising the need to understand their complex conformational dynamics in quantitative detail. In this perspective, we first show that single molecule pulling experiments when combined with with theory and simulations can be used to quantitatively explore the folding landscape of nucleic acid hairpins, and riboswitches with tertiary interactions. Applications to riboswitches, which are non-coding RNA elements that control gene expression by undergoing dynamical conformational changes in response to bin…
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…
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…
The mechanisms by which a protein's 3D structure can be determined based on its amino acid sequence have long been one of the key mysteries of biophysics. Often simplistic models, such as those derived from geometric constraints, capture bulk real-world 3D protein-protein properties well. One approach is using protein contact maps to better understand proteins' properties. Here, we investigate the emergent behaviour of contact maps for different geometrically constrained models and real-world protein systems. We derive an analytical approximation for the distribution of model amino acid distan…
The Metropolis Monte Carlo (MC) method is used to extract reaction kinetics from a given equilibrium distribution of states of a complex system. The approach is illustrated by the folding/unfolding reaction for two proteins - a model beta-hairpin and a-helical protein a3D. For beta-hairpin, the free energy surfaces (FESs) and free energy profiles (FEPs) are employed as the equilibrium distributions of states, playing a role of the potentials of mean force to determine the acceptance probabilities of new states in the MC simulations. Based on the FESs and PESs for a set of temperatures that wer…
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…
Cu,Zn superoxide dismutase (SOD1) has been implicated in the familial form of the neurodegenerative disease Amyotrophic Lateral Sclerosis (ALS). It has been suggested that mutant mediated SOD1 misfolding/aggregation is an integral part of the pathology of ALS. We study the folding thermodynamics and kinetics of SOD1 using a hybrid molecular dynamics approach. We reproduce the experimentally observed SOD1 folding thermodynamics and find that the residues which contribute the most to SOD1 thermal stability are also crucial for apparent two-state folding kinetics. Surprisingly, we find that these…
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. …
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…
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…
Intrinsically disordered proteins (IDPs) abound in cellular regulation. Their interactions are often transitory and highly sensitive to salt concentration and posttranslational modifications. However, little is known about the effect of macromolecular crowding on the kinetics and stability of the interactions of IDPs with their cellular targets. Here, we investigate the influence of crowding on the coupled folding and binding between two IDPs, using polyethylene glycol as a crowding agent across a broad size range. Single-molecule Förster resonance energy transfer allows us to quantify several…
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.
Single molecule force spectroscopy methods can be used to generate folding trajectories of biopolymers from arbitrary regions of the folding landscape. We illustrate the complexity of the folding kinetics and generic aspects of the collapse of RNA and proteins upon force quench, using simulations of an RNA hairpin and theory based on the de Gennes model for homopolymer collapse. The folding time, $τ_F$, depends asymmetrically on $δf_S = f_S - f_m$ and $δf_Q = f_m - f_Q$ where $f_S$ ($f_Q$) is the stretch (quench) force, and $f_m$ is the transition mid-force of the RNA hairpin. In accord with e…
Riboswitches, RNA elements found in the untranslated region, regulate gene expression by binding to target metaboloites with exquisite specificity. Binding of metabolites to the conserved aptamer domain allosterically alters the conformation in the downstream expression platform. The fate of gene expression is determined by the changes in the downstream RNA sequence. As the metabolite-dependent cotranscriptional folding and unfolding dynamics of riboswitches is the key determinant of gene expression, it is important to investigate both the thermodynamics and kinetics of riboswitches both in th…
The weighted ensemble (WE) simulation strategy provides unbiased sampling of non-equilibrium processes, such as molecular folding or binding, but the extraction of rate constants relies on characterizing steady state behavior. Unfortunately, WE simulations of sufficiently complex systems will not relax to steady state on observed simulation times. Here we show that a post-simulation clustering of molecular configurations into "microbins" using methods developed in the Markov State Model (MSM) community, can yield unbiased kinetics from WE data before steady-state convergence of the WE simulati…
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.
State-free reversible VAMPnets (SRVs) are a neural network-based framework capable of learning the leading eigenfunctions of the transfer operator of a dynamical system from trajectory data. In molecular dynamics simulations, these data-driven collective variables (CVs) capture the slowest modes of the dynamics and are useful for enhanced sampling and free energy estimation. In this work, we employ SRV coordinates as a feature set for Markov state model (MSM) construction. Compared to the current state of the art, MSMs constructed from SRV coordinates are more robust to the choice of input fea…
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…
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…
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 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 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 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…
Theoretical studies of protein folding on lattice models relie on the assumption that water close to amino-acids is always in thermal equilibrium all along the folding pathway. Within this framework, it has always been considered that out-of-equilibrium properties, such as folding time, could be evaluated equivalently from an averaging over a collection of trajectories of the protein with water described eitherexplicitly or through a mean-field approach. To critically assess this hypothesis, we built a two-dimensional lattice model of a protein in interaction with water molecules that can adop…
Computer experiments are performed to investigate why protein contact networks (networks induced by spatial contacts between amino acid residues of a protein) do not have shorter average shortest path lengths in spite of their importance to protein folding. We find that shorter average inter-nodal distances is no guarantee of finding a global optimum more easily. Results from the experiments also led to observations which parallel an existing view that neither short-range nor long-range interactions dominate the protein folding process. Nonetheless, runs where there was a slight delay in the u…
From the point of view of statistical mechanics, a full characterisation of a molecular system requires the experimental determination of its possible states, their populations and the respective interconversion rates. Well-established methods can incorporate in molecular dynamics simulations experimental information about states using structural restraints, and about populations using thermodynamic restraints. However, it is still unclear how to include experimental knowledge of interconversion rates. Here we introduce a method of imposing known rate constants as constraints in molecular dyna…
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)…
The concept of a protein diffusing in its free energy folding landscape has been fruitful for both theory and experiment. Yet the choice of the reaction coordinate (RC) introduces an undesirable degree of arbitrariness into the problem. We analyze extensive simulation data of an alpha-helix in explicit water solvent as it stochastically folds and unfolds. The free energy profiles for different RCs exhibit significant variation, some having an activation barrier, others not. We show that this variation has little effect on the predicted folding kinetics if the diffusivity profiles are properly …
