Barrier crossing dynamics of a charged particle in the presence of a magnetic field: a new turnover phenomenon

In this paper we have investigated the effect of a magnetic field on the barrier crossing rate of a charged particle. At the low friction regime we have observed a new turnover phenomenon for the variation of rate as a function of field strength. Thus although the force due to the magnetic field is not dissipative in nature, it plays a role in the steady state barrier crossing rate similar to that of a dissipative force in the weak damping regime. For appreciable damping strength, the rate monotonically decreases with the increase of field strength. We have demonstrated an interesting resonance effect due to the variation of frequency of the harmonic oscillator associated with the y-component motion at low damping and magnetic field strength.     

Hydrophobic collapse in (in silico) protein folding

A model of hydrophobic collapse, which is treated as the driving force for protein folding, is presented. This model is the superposition of three models commonly used in protein structure prediction: (1) 'oil-drop' model introduced by Kauzmann, (2) a lattice model introduced to decrease the number of degrees of freedom for structural changes and (3) a model of the formation of hydrophobic core as a key feature in driving the folding of proteins. These three models together helped to develop the idea of a fuzzy-oil-drop as a model for an external force field of hydrophobic character mimicking the hydrophobicity-differentiated environment for hydrophobic collapse. All amino acids in the polypeptide interact pair-wise during the folding process (energy minimization procedure) and interact with the external hydrophobic force field defined by a three-dimensional Gaussian function. The value of the Gaussian function usually interpreted as a probability distribution is treated as a normalized hydrophobicity distribution, with its maximum in the center of the ellipsoid and decreasing proportionally with the distance versus the center. The fuzzy-oil-drop is elastic and changes its shape and size during the simulated folding procedure.     

Folding time distributions as an approach to protein folding kinetics

A 27-residue lattice heteropolymer subject to Monte Carlo dynamics on a simple cubic lattice is studied over a range of temperatures. Folding time distributions are used to obtain information concerning the details of folding kinetics. The results are compared with those from methods based on mean force surfaces expressed in terms of a reduced set of variables and on a disconnectivity graph for the same system. A detailed analysis of the folding trajectories is given, and the importance of dead-end traps in determining the folding time is demonstrated. We show that the calculated folding kinetics can be modeled by a system of kinetic equations, with the essential rate constants determined from the Monte Carlo simulations and the resulting folding time distributions. The kinetic equations make possible an analysis of the variation of the importance of different channels with temperature. In particular, we show that the presence of intermediates may be masked in the folding time distributions, with the mean folding time being independent of the height of the barrier between the intermediates and collapsed globule state of the system. This and other results demonstrate that care has to be used in interpreting experimental folding data in terms of the underlying kinetics. Correspondingly, simulations are shown to have to satisfy certain requirements to obtain proper sampling of the dead-end traps.     

The protein folding transition-state ensemble from a Gō-like model

Characterizing the structure of transition states (TS) is a first step towards understanding two-state protein folding mechanisms. However, a direct experimental characterization of these states is challenging and indirect information derived from protein engineering methodologies (?-value analysis) is often difficult to interpret. We present here a theoretical study on the nature of the transition state ensemble for three representative proteins covering the major structural classes using a mean-field C(α)-based Gō-model. We identify that transition state ensembles are dominated by local contacts, indicating that most non-local contacts form only upon crossing the macroscopic folding free energy barrier. We demonstrate that the mean ?-value corresponds to the fraction of stabilization energy gained at the barrier-top in two-state-like systems, and that it depends monotonically on the stability conditions. Furthermore, we show that there is a fundamental connection between small destabilization and large ?-values that in turn depends on the location of the mutated residue in the structure. These results that are in agreement with the recent empirical findings highlight the importance of local energetics in determining folding mechanisms.     

New force replica exchange method and protein folding pathways probed by force-clamp technique

We have developed a new extended replica exchange method to study thermodynamics of a system in the presence of external force. Our idea is based on the exchange between different force replicas to accelerate the equilibrium process. This new approach was applied to obtain the force-temperature phase diagram and other thermodynamical quantities of the three-domain ubiquitin. Using the C(alpha)-Go model and the Langevin dynamics, we have shown that the refolding pathways of single ubiquitin depend on which terminus is fixed. If the N end is fixed then the folding pathways are different compared to the case when both termini are free, but fixing the C terminal does not change them. Surprisingly, we have found that the anchoring terminal does not affect the pathways of individual secondary structures of three-domain ubiquitin, indicating the important role of the multidomain construction. Therefore, force-clamp experiments, in which one end of a protein is kept fixed, can probe the refolding pathways of a single free-end ubiquitin if one uses either the polyubiquitin or a single domain with the C terminus anchored. However, it is shown that anchoring one end does not affect refolding pathways of the titin domain I27, and the force-clamp spectroscopy is always capable to predict folding sequencing of this protein. We have obtained the reasonable estimate for unfolding barrier of ubiquitin, using the microscopic theory for the dependence of unfolding time on the external force. The linkage between residue Lys48 and the C terminal of ubiquitin is found to have the dramatic effect on the location of the transition state along the end-to-end distance reaction coordinate, but the multidomain construction leaves the transition state almost unchanged. We have found that the maximum force in the force-extension profile from constant velocity force pulling simulations depends on temperature nonlinearly. However, for some narrow temperature interval this dependence becomes linear, as have been observed in recent experiments.     

Two-state folding observed in individual protein molecules

The folding dynamics of small proteins are often described in terms of a simple two-state kinetic model. Within this notion, the behavior of individual molecules is expected to be stochastic, with a protein molecule residing in either the unfolded or the folded state for extended periods of time, with intermittent rapid jumps across the free energy barrier. However, a direct observation of this bistable behavior has not been made to date. Rather, previous reports of folding trajectories of individual proteins have shown an unexpected degree of complexity. This raises the question whether the simple kinetic properties derived from classical experiments on large ensembles of molecules are reflected in the folding paths taken by individual proteins. Here we report single-molecule folding/unfolding trajectories observed by fluorescence resonance energy transfer for a protein that meets all criteria of a two state-system. The trajectories, measured on molecules immobilized in lipid vesicles, demonstrate the anticipated bistable behavior, with steplike transitions between folded and unfolded conformations. They further allow us to put an upper bound on the barrier crossing time.     

Protein folding kinetics under force from molecular simulation

Despite a large number of studies on the mechanical unfolding of proteins, there are still relatively few successful attempts to refold proteins in the presence of a stretching force. We explore refolding kinetics under force using simulations of a coarse-grained model of ubiquitin. The effects of force on the folding kinetics can be fitted by a one-dimensional Kramers theory of diffusive barrier crossing, resulting in physically meaningful parameters for the height and location of the folding activation barrier. By comparing parameters obtained from pulling in different directions, we find that the unfolded state plays a dominant role in the refolding kinetics. Our findings explain why refolding becomes very slow at even moderate pulling forces and suggest how it could be practically observed in experiments at higher forces.     

Solvation dynamics of a protein in the pre molten globule state

The nature of solvent molecules around proteins in native and different non-native states is crucial for understanding the protein folding problem. We have characterized two compact denatured states of glutaminyl-tRNA synthetase (GlnRS) under equilibrium conditions in the presence of a naturally occurring osmolyte, l-glutamate. The solvation dynamics of the compact denatured states and the fully unfolded state has been studied using a covalently attached probe, acrylodan, near the active site. The solvation dynamics progressively becomes faster as the protein goes from the native to the molten globule to the pre molten globule to the fully unfolded state. Anisotropy decay measurements suggest that the pre-molten-globule intermediate is more flexible than the molten globule although the secondary structure is largely similar. Dynamic light scattering studies reveal that both the compact denatured states are aggregated under the measurement conditions. The implications of solvation dynamics in aggregated compact denatured states have been discussed.     

Thermodynamics of Go-type models for protein folding

Go-type potentials, based on the inter-residue contacts present in the native structure of a protein, are frequently used to predict dynamic and structural features of the folding pathways through computer simulations. However, the mathematical form used to define the model interactions includes several arbitrary choices, whose consequences are not usually analyzed. In this work, we use a simple off-lattice protein model and a parallel tempering Monte Carlo simulation technique to carry out such analysis, centered in the thermodynamic characteristics of the folding transition. We show how the definition of a native contact has a deep impact on the presence of simple or complex transitions, with or without thermodynamic intermediates. In addition, we have checked that the width of the attractive wells has a profound effect on the free-energy barrier between the folded and unfolded states, mainly through its influence on the entropy of the denatured state.     

Viscosity scaling for the glassy phase of protein folding

Although commendable progress has been made in the understanding of the physics of protein folding, a key unresolved issue is whether Kramers' diffusion model of chemical reactions is generally applicable to activated barrier crossing events during folding. To examine the solvent viscosity effect on the folding transition of native-like trapped intermediates, laser flash photolysis has been used to measure the microsecond folding kinetics of a natively folded state of CO-liganded ferrocytochrome c (M-state) in the 1-250 cP range of glycerol viscosity at pH 7.0, 20 degrees C. The single rate coefficient for the folding of the M-state to the native state of the protein (i.e., the M --> N folding process) decreases initially when the solvent viscosity is low (<10 cP), but saturates at higher viscosity, indicating that Kramers model is not general enough for scaling the viscosity dependence of post-transition folding involving glassy dynamics. Analysis based on the Grote-Hynes idea of time dependent friction in conjunction with defect diffusion dynamics can account for the observed non-Kramers scaling.     

Slow Unfolded-State Structuring in Acyl-CoA Binding Protein Folding Revealed by Simulation and Experiment

Protein folding is a fundamental process in biology, key to understanding many human diseases. Experimentally, proteins often appear to fold via simple two- or three-state mechanisms involving mainly native-state interactions, yet recent network models built from atomistic simulations of small proteins suggest the existence of many possible metastable states and folding pathways. We reconcile these two pictures in a combined experimental and simulation study of acyl-coenzyme A binding protein (ACBP), a two-state folder (folding time ～10 ms) exhibiting residual unfolded-state structure, and a putative early folding intermediate. Using single-molecule FRET in conjunction with side-chain mutagenesis, we first demonstrate that the denatured state of ACBP at near-zero denaturant is unusually compact and enriched in long-range structure that can be perturbed by discrete hydrophobic core mutations. We then employ ultrafast laminar-flow mixing experiments to study the folding kinetics of ACBP on the microsecond time scale. These studies, along with Trp-Cys quenching measurements of unfolded-state dynamics, suggest that unfolded-state structure forms on a surprisingly slow (～100 μs) time scale, and that sequence mutations strikingly perturb both time-resolved and equilibrium smFRET measurements in a similar way. A Markov state model (MSM) of the ACBP folding reaction, constructed from over 30 ms of molecular dynamics trajectory data, predicts a complex network of metastable stables, residual unfolded-state structure, and kinetics consistent with experiment but no well-defined intermediate preceding the main folding barrier. Taken together, these experimental and simulation results suggest that the previously characterized fast kinetic phase is not due to formation of a barrier-limited intermediate but rather to a more heterogeneous and slow acquisition of unfolded-state structure.     

Single molecule force spectroscopy using polyproteins

In recent years single molecule force spectroscopy has emerged as a powerful new tool to explore the mechanical stability and folding pathways of individual proteins. This technique is used to apply a stretching force between two points of a protein, unfolding the protein to an extended state. By measuring the unfolding and folding trajectories of individual proteins, insight can be gained into the physical mechanisms of protein folding. In this tutorial review we introduce the reader to single molecule force spectroscopy using the atomic force microscope (AFM), and explain the two main modes of operation of the AFM for force spectroscopy: force-extension and force-clamp. We introduce the approach of using polyproteins to obtain a clear mechanical fingerprint for monitoring the response of proteins to an applied mechanical force. In addition, we provide an informative and representative review of recent research on proteins using single molecule force spectroscopy. We focus on areas which have made a significant contribution to the single molecule protein folding field and highlight emerging areas of research which have wider implications for the general scientific community.     

Tunneling through a fluctuating barrier in the presence of a periodically driving field

The zero and finite temperature tunneling dynamics of a periodically driven particle moving in a bistable potential with a fluctuating barrier is studied. We have focused on the influence of barrier fluctuation and thermal modulation on the tunneling processes in the presence of a driving field. At zero temperature, for a fixed strength of the driving field, both the tunneling probability and rate passes through a well-defined minimum when plotted as a function of fluctuation frequency while it reveals a clear maximum as a function of driving frequency. However, at T > 0 the tunneling probability and rate show two maxima as a function of both fluctuation frequency and driving frequency. In both zero and finite temperature, the tunneling rate constant decreases with increasing fluctuation strength. So, the barrier fluctuation may enhance the stability of a periodically driven system. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2004     

Understanding the folding rates and folding nuclei of globular proteins

The first part of this paper contains an overview of protein structures, their spontaneous formation ("folding"), and the thermodynamic and kinetic aspects of this phenomenon, as revealed by in vitro experiments. It is stressed that universal features of folding are observed near the point of thermodynamic equilibrium between the native and denatured states of the protein. Here the "two-state" ("denatured state" <--> "native state") transition proceeds without accumulation of metastable intermediates, but includes only the unstable "transition state". This state, which is the most unstable in the folding pathway, and its structured core (a "nucleus") are distinguished by their essential influence on the folding/unfolding kinetics. In the second part of the paper, a theory of protein folding rates and related phenomena is presented. First, it is shown that the protein size determines the range of a protein's folding rates in the vicinity of the point of thermodynamic equilibrium between the native and denatured states of the protein. Then, we present methods for calculating folding and unfolding rates of globular proteins from their sizes, stabilities and either 3D structures or amino acid sequences. Finally, we show that the same theory outlines the location of the protein folding nucleus (i.e., the structured part of the transition state) in reasonable agreement with experimental data.     

De novo refolding and aggregation of insulin in a nonaqueous environment: an inside out protein remake

While thermodynamic penalties associated with protein-water interactions are the key driving force of folding, perturbed hydration of destabilized protein molecules may trigger aggregation, which in vivo often causes cellular and histological damage. Here we show, that the denatured state of an alpha-helical protein, insulin, converts to a non-native beta-sheet-rich structure upon de novo "refolding" in an anhydrous environment. The beta-pleated conformer precipitates from solutions of DMSO-denatured insulin upon dilution with chloroform. DMSO destroys hydrogen bond network of the native protein acting as a strong acceptor of main chain hydrogen bonds. Upon the addition of chloroform, which is a weak hydrogen bond donor per se, competitive hydrogen bonds between DMSO and chloroform are formed. This leads to the release of unfolded insulin molecules. In the absence of water, the imminent saturation of polypeptide's dandling hydrogen bonds does not produce the native and predominantly alpha-helical state but a beta-sheet-rich structure, which is morphologically and spectrally distinct from insulin amyloid fibrils. Unlike insulin fibrils, the beta-sheet conformer is metastable and refolds spontaneously to the native form in an aqueous environment. This implies that "folding" in the absence of water results in inefficient burial of hydrophobic side-chains, and thermodynamic frustration at the water-protein interface.     

Refolding of urea‐denatured α‐chymotrypsin by protein‐folding liquid chromatography

An approach for re‐folding denatured proteins during proteome research by protein folding liquid chromatography (PFLC) is presented. Standard protein, α‐chymotrypsin (α‐Chy), was selected as a model protein and hydrophobic interaction chromatography was performed as a typical PFLC; the three different α‐Chy states – urea‐denatured (U state), its folded intermediates (M state) and nature state (N state) – were studied during protein folding. Based on the test by matrix‐assisted laser desorption/ionization time of flight mass spectrometry and bioactivity, only one stable M state of the α‐Chy was identified and then it was prepared for further investigation. The specific bioactivity of the refolded α‐Chy was found to be higher than that of commercial α‐Chy as the urea concentration in the sample solution ranged from 1.0 to 3.0 m ; the highest specific bioactivity at urea concentration was 1.0 m , indicating the possibility for re‐folding some proteins that have partially or completely lost their bioactivity, as a dilute urea solution was employed for dissolving the sample. The experiment showed that the peak height of its M state increased with increasing urea concentration, and correspondingly decreased in the amount of the refolded α‐Chy. When the urea concentration reached 6.0 m , the unfolded α‐Chy could not be refolded at all. Copyright © 2012 John Wiley & Sons, Ltd.     

Denatured-state ensemble and the early-stage folding of the G29A mutant of the B-domain of protein A

The folding mechanism of the G29A mutant of the B-domain of protein A (BdpA) has been studied by all-atom molecular dynamics simulation using AMBER force field (ff03) and generalized Born continuum solvent model. Started from the extended chain conformation, a total of 16 simulations (400 ns each) at 300 K captured some early folding events of the G29A mutant of BdpA. In one of the 16 trajectories, the G29A mutant folded within 2.8 A (root mean square) of the wild-type NMR structure. We observed that the fast burial of hydrophobic residues was the driving force to bring the distant residues into close proximity. The initiation of the helix I and III occurred during the stage of hydrophobic collapse. The initiation and growth of the helix II was slow. Both the secondary structure formation and the development of the native tertiary contacts suggested a multistage folding process. Clustering analysis indicated that two helix species (helices I and III) could be intermediates. Further analysis revealed that the hydrophobic residues of partially folded helix II formed nativelike hydrophobic contacts with helices I and III that stabilized a nativelike state and delayed the completion of folding of the entire protein. The details of the early folding process were compared with other theoretical and experimental studies. It was found that a nativelike hydrophobic cluster was formed by residues including F(30), I(31), L(34), L(44), L(45), and A(48) that prevented further development of the native structures, and breaking the hydrophobic cluster like this one contributed to the rate-limiting step. This was in complete agreement with the recent kinetic measurements in which mutations of these residues to Gly and Ala substantially increased the folding rates by as much as 60 times. Apparently, destabilization of nonnative states dramatically enhanced the folding rates.     

Controlling activated processes of nonadiabatically, periodically driven dynamical systems: a multiple scale perturbation approach

We arrive at the escape rate from a metastable state for a system of Brownian particles driven periodically by a space dependent, rapidly oscillating external perturbation (with frequency ω) in one dimension (one of the most important class of nonequilibrium system). Though the problem may seem to be time-dependent, and is poised on the extreme opposite side of adiabaticity, there exists a multiple scale perturbation theory ("Kapitza window") by means of which the dynamics can be treated in terms of an effective time-independent potential that is derived as an expansion in orders of 1/ω to the order ω(-3). The resulting time-independent equation is then used to calculate the escape rate of physical systems from a metastable state induced by external monochromatic field in the moderate-to-large damping limit and to investigate the effect of ω on the resulting rate in conjunction with the thermal energy. With large value of ω, we find that the environment with moderate-to-large damping impedes the escape process of the particle while high amplitude of the periodic driving force allows the particle to cross the barrier with a large escape rate. A comparison of our theoretical expression with numerical simulation gives a satisfactory agreement.