Peptide chain dynamics in light and heavy water: zooming in on internal friction

Frictional effects due to the chain itself, rather than the solvent, may have a significant effect on protein dynamics. Experimentally, such "internal friction" has been investigated by studying folding or binding kinetics at varying solvent viscosity; however, the molecular origin of these effects is hard to pinpoint. We consider the kinetics of disordered glycine-serine and α-helix forming alanine peptides and a coarse-grained protein folding model in explicit-solvent molecular dynamics simulations. By varying the solvent mass over more than two orders of magnitude, we alter only the solvent viscosity and not the folding free energy. Folding dynamics at the near-vanishing solvent viscosities accessible by this approach suggests that solvent and internal friction effects are intrinsically entangled. This finding is rationalized by calculation of the polymer end-to-end distance dynamics from a Rouse model that includes internal friction. An analysis of the friction profile along different reaction coordinates, extracted from the simulation data, demonstrates that internal as well as solvent friction varies substantially along the folding pathways and furthermore suggests a connection between friction and the formation of hydrogen bonds upon folding.     

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.     

Human telomere sequence DNA in water-free and high-viscosity solvents: g-quadruplex folding governed by kramers rate theory

Structures formed by human telomere sequence (HTS) DNA are of interest due to the implication of telomeres in the aging process and cancer. We present studies of HTS DNA folding in an anhydrous, high viscosity deep eutectic solvent (DES) comprised of choline choride and urea. In this solvent, the HTS DNA forms a G-quadruplex with the parallel-stranded ("propeller") fold, consistent with observations that reduced water activity favors the parallel fold, whereas alternative folds are favored at high water activity. Surprisingly, adoption of the parallel structure by HTS DNA in the DES, after thermal denaturation and quick cooling to room temperature, requires several months, as opposed to less than 2 min in an aqueous solution. This extended folding time in the DES is, in part, due to HTS DNA becoming kinetically trapped in a folded state that is apparently not accessed in lower viscosity solvents. A comparison of times required for the G-quadruplex to convert from its aqueous-preferred folded state to its parallel fold also reveals a dependence on solvent viscosity that is consistent with Kramers rate theory, which predicts that diffusion-controlled transitions will slow proportionally with solvent friction. These results provide an enhanced view of a G-quadruplex folding funnel and highlight the necessity to consider solvent viscosity in studies of G-quadruplex formation in vitro and in vivo. Additionally, the solvents and analyses presented here should prove valuable for understanding the folding of many other nucleic acids and potentially have applications in DNA-based nanotechnology where time-dependent structures are desired.     

Solvent viscosity dependence of the protein folding dynamics

Solvent viscosity has been frequently adopted as an adjustable parameter in various computational studies (e.g., protein folding simulations) with implicit solvent models. A common approach is to use low viscosities to expedite simulations. While using viscosities lower than that of aqueous is unphysical, such treatment is based on observations that the viscosity affects the kinetics (rates) in a well-defined manner as described by Kramers' theory. Here, we investigate the effect of viscosity on the detailed dynamics (mechanism) of protein folding. On the basis of a simple mathematical model, we first show that viscosity may indeed affect the dynamics in a complex way. By applying the model to the folding of a small protein, we demonstrate that the detailed dynamics is affected rather pronouncedly especially at unphysically low viscosities, cautioning against using such viscosities. In this regard, our model may also serve as a diagnostic tool for validating low-viscosity simulations. It is also suggested that the viscosity dependence can be further exploited to gain information about the protein folding mechanism.     

Molecular dynamics with the united-residue model of polypeptide chains. II. Langevin and Berendsen-bath dynamics and tests on model alpha-helical systems

The implementation of molecular dynamics (MD) with our physics-based protein united-residue (UNRES) force field, described in the accompanying paper, was extended to Langevin dynamics. The equations of motion are integrated by using a simplified stochastic velocity Verlet algorithm. To compare the results to those with all-atom simulations with implicit solvent in which no explicit stochastic and friction forces are present, we alternatively introduced the Berendsen thermostat. Test simulations on the Ala(10) polypeptide demonstrated that the average kinetic energy is stable with about a 5 fs time step. To determine the correspondence between the UNRES time step and the time step of all-atom molecular dynamics, all-atom simulations with the AMBER 99 force field and explicit solvent and also with implicit solvent taken into account within the framework of the generalized Born/surface area (GBSA) model were carried out on the unblocked Ala(10) polypeptide. We found that the UNRES time scale is 4 times longer than that of all-atom MD simulations because the degrees of freedom corresponding to the fastest motions in UNRES are averaged out. When the reduction of the computational cost for evaluation of the UNRES energy function is also taken into account, UNRES (with hydration included implicitly in the side chain-side chain interaction potential) offers about at least a 4000-fold speed up of computations relative to all-atom simulations with explicit solvent and at least a 65-fold speed up relative to all-atom simulations with implicit solvent. To carry out an initial full-blown test of the UNRES/MD approach, we ran Berendsen-bath and Langevin dynamics simulations of the 46-residue B-domain of staphylococcal protein A. We were able to determine the folding temperature at which all trajectories converged to nativelike structures with both approaches. For comparison, we carried out ab initio folding simulations of this protein at the AMBER 99/GBSA level. The average CPU time for folding protein A by UNRES molecular dynamics was 30 min with a single Alpha processor, compared to about 152 h for all-atom simulations with implicit solvent. It can be concluded that the UNRES/MD approach will enable us to carry out microsecond and, possibly, millisecond simulations of protein folding and, consequently, of the folding process of proteins in real time.     

Numerical simulation of the effect of solvent viscosity on the motions of a beta-peptide heptamer

This report examines the effect of a decrease in solvent viscosity on the simulated folding behaviour of a beta-peptide heptamer in methanol. Simulations of the molecular dynamics of the heptamer H-beta3-HVal-beta3-HAla-beta3-HLeu-(S,S)-beta3-HAla(alphaMe)-beta3-HVal-beta3-HAla-beta3-HLeu-OH in methanol, with an explicit representation of the methanol molecules, were performed for 80 ns at various solvent viscosities. The simulations indicate that at a solvent viscosity of one third of that of methanol, only the dynamic aspects of the folding process are altered, and that the rate of folding is increased. At a viscosity of one tenth of that of methanol, insufficient statistics are obtained within the 80 ns period. We suggest that 80 ns is an insufficient time to reach conformational equilibrium at very low viscosity because the dependence of the folding rate of a beta-peptide on solvent viscosity has two regimes; a result that was observed in another computational study for alpha-peptides.     

Engineering a beta-sheet protein toward the folding speed limit

Recent experimental studies have shown that alpha-helical proteins can approach the folding "speed limit", where folding switches from an activated to a downhill process in free energy. beta-sheet proteins are generally thought to fold more slowly than helix bundles. However, based on studies of hairpins, folding should still be able to approach the microsecond time scale. Here we demonstrate how the hPin1 WW domain, a triple-stranded beta-sheet protein with a sharp thermodynamic melting transition, can be engineered toward the folding "speed limit" without a significant loss in thermal denaturation cooperativity.     

Solvent viscosity dependence of the folding rate of a small protein: distributed computing study

By using distributed computing techniques and a supercluster of more than 20,000 processors we simulated folding of a 20-residue Trp Cage miniprotein in atomistic detail with implicit GB/SA solvent at a variety of solvent viscosities (gamma). This allowed us to analyze the dependence of folding rates on viscosity. In particular, we focused on the low-viscosity regime (values below the viscosity of water). In accordance with Kramers' theory, we observe approximately linear dependence of the folding rate on 1/gamma for values from 1-10(-1)x that of water viscosity. However, for the regime between 10(-4)-10(-1)x that of water viscosity we observe power-law dependence of the form k approximately gamma(-1/5). These results suggest that estimating folding rates from molecular simulations run at low viscosity under the assumption of linear dependence of rate on inverse viscosity may lead to erroneous results.     

Protein folding kinetics: barrier effects in chemical and thermal denaturation experiments

Recent experimental work on fast protein folding brings about an intriguing paradox. Microsecond-folding proteins are supposed to fold near or at the folding speed limit (downhill folding), but yet their folding behavior seems to comply with classical two-state analyses, which imply the crossing of high free energy barriers. However, close inspection of chemical and thermal denaturation kinetic experiments in fast-folding proteins reveals systematic deviations from two-state behavior. Using a simple one-dimensional free energy surface approach we find that such deviations are indeed diagnostic of marginal folding barriers. Furthermore, the quantitative analysis of available fast-kinetic data indicates that many microsecond-folding proteins fold downhill in native conditions. All of these proteins are then promising candidates for an atom-by-atom analysis of protein folding using nuclear magnetic resonance.1 We also find that the diffusion coefficient for protein folding is strongly temperature dependent, corresponding to an activation energy of approximately 1 kJ.mol-1 per protein residue. As a consequence, the folding speed limit at room temperature is about an order of magnitude slower than the approximately 1 micros estimates from high-temperature T-jump experiments. Our analysis is quantitatively consistent with the available thermodynamic and kinetic data on slow two-state folding proteins and provides a straightforward explanation for the apparent fast-folding paradox.     

Friction dependence of reaction rates in simple barrierless reactions

We have investigated the dynamics of simple chemical reactions which proceed without an activation barrier along the reaction coordinate. In the absence of the barrier the solvent friction is the only impediment to the reactive motion. By numerical simulation we show that in the low-friction limit the reaction rate increases as a fractional power of the friction coefficient. The power law dependence is sensitive to the initial conditions of the reactive coordinate. After exhibiting a maximum, the behaviour crosses over to that of inverse friction dependence in the high-friction limit. We have compared our results with earlier approximate analytical treatments and differences are pointed out.     

Intramolecular Diels-Alder reaction in ionic liquids: effect of ion-specific solvent friction

The present work aims at understanding the role of viscosity or solvent friction in ionic liquids for an intramolecular Diels-Alder (IMDA) reaction of (E)-1-phenyl-4-[2-(3-methyl-2-butenyloxy)benzylidene]-5-pyrazolone (1). The results have been analyzed on the basis of the current theoretical models, and their failure to account for the observed trends is discussed in terms of "effective" viscosity or microviscosity. The rates of the reaction decrease with the increasing viscosity of the ionic liquids. As evident from the anionic effect, the solute-solvent specific interactions play a role in governing the kinetics of the reaction. The lower viscosities of the bistrifluoromethanesulfonimide [NTf2](-) based ionic liquids as compared to those based on tetrafluoroborate [BF4](-) anion fail to result in a corresponding acceleration in the rates of the reaction. These contradictory results indicate that solvent microviscosity, rather than the bulk macroscopic viscosity, should be the criteria for selecting the ionic liquids as reaction media.     

Numerical Simulation of the Effect of Solvent Viscosity on the Motions of a β‐Peptide Heptamer

This report examines the effect of a decrease in solvent viscosity on the simulated folding behaviour of a β‐peptide heptamer in methanol. Simulations of the molecular dynamics of the heptamer H‐β³‐HVal‐β³‐HAla‐β³‐HLeu‐(S,S)‐β³‐HAla(αMe)‐β³‐HVal‐β³‐HAla‐β³‐HLeu‐OH in methanol, with an explicit representation of the methanol molecules, were performed for 80 ns at various solvent viscosities. The simulations indicate that at a solvent viscosity of one third of that of methanol, only the dynamic aspects of the folding process are altered, and that the rate of folding is increased. At a viscosity of one tenth of that of methanol, insufficient statistics are obtained within the 80 ns period. We suggest that 80 ns is an insufficient time to reach conformational equilibrium at very low viscosity because the dependence of the folding rate of a β‐peptide on solvent viscosity has two regimes; a result that was observed in another computational study for α‐peptides.     

Periodic force induced stabilization or destabilization of the denatured state of a protein

We have studied the effects of an external sinusoidal force in protein folding kinetics. The externally applied force field acts on the each amino acid residues of polypeptide chains. Our simulation results show that mean protein folding time first increases with driving frequency and then decreases passing through a maximum. With further increase of the driving frequency the mean folding time starts increasing as the noise-induced hoping event (from the denatured state to the native state) begins to experience many oscillations over the mean barrier crossing time period. Thus unlike one-dimensional barrier crossing problems, the external oscillating force field induces both stabilization or destabilization of the denatured state of a protein. We have also studied the parametric dependence of the folding dynamics on temperature, viscosity, non-Markovian character of bath in presence of the external field.     

Intramolecular excimer formation in polymers: Pyrene Labelled Polyvinylacetate

The rate constant for intramolecular excimer formation between pyrenyl side-groups, in a polyvinylacetate chain at a mean separation of 200 bonds, has been measured as a function of molecular weight, solvent viscosity and solvent thermodynamic power. Above M = 1 × 10⁵, the rate constant is 1.4 × 10⁷ sec^?1 in low viscosity bad solvents. This value is about twenty times that for the rate constant of the analogous reaction between the two terminal groups in a chain with a mean end-to-end separation of 200 bonds. Increases of the viscosity and of the thermodynamic power of the solvent depress the rate constant, in agreement with the behaviour expected for a diffusion controlled reaction.     

Watching solvent friction impede ultrafast barrier crossings: a direct test of Kramers theory

A systematic investigation of the solvent's dynamic influence on activated barrier crossings on an electronic ground state is performed using ultrafast two-dimensional infrared chemical exchange spectroscopy. These measurements facilitate a direct comparison with the widely adopted Kramers theory of condensed phase reaction kinetics, and for the first time avoid the significant complication of electronic excitation to probe directly in the time domain a ground electronic state reaction with a well-defined transition state. The picosecond timescale interconversion between two stable isomers of the metal carbonyl complex Co(2)(CO)(8) in a series of linear alkane solvents shows negligible energetic variation with solvent carbon chain length, providing an exclusive probe of the effects of solvent friction. Relative to the linear alkane series, cyclohexane does alter the potential energy surface by preferentially stabilizing one of the isomers. Despite this pronounced modification of the reaction barrier energetics, combination of experiment and computation enables the removal of the nondynamical barrier contribution to the rate constant, isolating the dynamical influence of solvent friction. The experimental data, supported with quantum and classical computations, show agreement with a simple Markovian Kramers theory for the isomerization rate constant's dependence on solvent viscosity.     

Ultrafast folding of a computationally designed Trp-cage mutant: Trp2-cage

Miniproteins provide useful model systems for understanding the principles of protein folding and design. These proteins also serve as useful test cases for theories of protein folding, and their small size and ultrafast folding kinetics put them in a regime of size and time scales that is now becoming accessible to molecular dynamics simulations. Previous estimates have suggested the "speed limit" for folding is on the order of 1 mus. Here a computationally designed mutant of the 20-residue Trp-cage miniprotein, Trp2-cage, is presented. The Trp2-cage has greater stability than the parent and folds on the ultrafast time scale of 1 mICROs at room temperature, as determined from infrared temperature-jump experiments.     

Concepts and misconcepts in the analysis of simple kinetics of protein folding

Unusually simple two-state kinetics characterizes the folding of a number of small proteins possessing a variety secondary structures. This limits dramatically the number of experimentally resolvable parameters that may characterize this process and also suggests the possibility to describe it based on simple theories borrowed from the field of ordinary chemical reactions. An attempt is made to critically evaluate the basic concepts, which are in the background of this approach. We demonstrate their limitations, which may cast doubt on the interpretation of experimental data. It is shown also that, in contrast to provisions of transition state theory, the simple kinetics of protein folding does not correlate with folded state stability or with the size of the folding unit. Moreover, the folding kinetics exhibits anomalous dependence on temperature and pressure and surprisingly strong dependence on solvent viscosity. The possible role in folding of fluctuations, relaxations and gradient dynamics is discussed. Being overlooked or underestimated, these mechanisms may determine the rate and specificity of the process.     

Critical importance of length-scale dependence in implicit modeling of hydrophobic interactions

The existence of length-scale dependence of hydrophobic solvation has important implications in the equilibrium of disordered, partially folded, and folded protein conformations. Neglecting this dependence, such as in popular solute surface-area based implicit solvent models with fixed surface tension coefficients, severely limits the ability to accurately model protein conformational equilibrium. We illustrate such fundamental limitations by examining the potentials of mean force of forming dimeric and trimeric nonpolar clusters and propose a new empirical model that effectively captures the context dependence of the local effective surface tension. Further optimization of the new model with other components of the implicit solvent force fields provides promise to significantly improve one's ability to simulate protein folding and conformational transitions. The existence of length-scale dependence of hydrophobic solvation has important implications in the equilibrium of disordered, partially folded, and folded protein conformations. Neglecting this dependence, such as in popular solute surface-area based implicit solvent models with fixed surface tension coefficients, severely limits the ability to accurately model protein conformational equilibrium. We illustrate such fundamental limitations by examining the potentials of mean force of forming dimeric and trimeric nonpolar clusters and propose a new empirical model that effectively captures the context dependence of the local effective surface tension. Further optimization of the new model with other components of the implicit solvent force fields provides promise to significantly improve one's ability to simulate protein folding and conformational transitions.     

Smaller and faster: the 20-residue Trp-cage protein folds in 4 micros

We have used laser temperature jump spectroscopy to measure the folding speed of the 20-residue Trp-cage, the smallest polypeptide known to exhibit truly cooperative folding behavior. The observed folding time (4 mus at room temperature) makes this not only the smallest foldable protein, but also the fastest, with a folding speed that exceeds contact-order predictions and approaches anticipated diffusional "speed limits" for protein folding.