Glycerol-slaved 1H-1H NMR cross-relaxation in quasi-native lysozyme

1H-1H nuclear cross-relaxation experiments have been carried out with lysozyme in variable glycerol viscosity to study intramolecular motion, self-diffusion, and isotropic rigid-body rotational tumbling at 298 K, pH 3.8. Dynamics of intramolecular 1H-1H cross-relaxation rates, the increase in internuclear spatial distances, and lateral and rotational diffusion coefficients all show fractional viscosity dependence with a power law exponent κ in the 0.17-0.83 range. The diffusion coefficient of glycerol Ds with the bulk viscosity itself is non-Stokesian, having a fractional viscosity dependence on the medium viscosity (Ds ∼ η-κ, κ ≈ 0.71). The concurrence and close similarity of the fractional viscosity dependence of glycerol diffusion on the one hand, and diffusion and intramolecular cross-relaxation rates of the protein on the other lead to infer that relaxation of glycerol slaves protein relaxations. Glycerol-transformed native lysozyme to a quasi-native state does not affect the conclusion that both global and internal fluctuations are slaved to glycerol relaxation.

Effects of Hydrodynamic Backflow on the Transmission Coefficient of a Barrier-Crossing Brownian Particle

The slow power law decay of the velocity autocorrelation function of a particle moving stochastically in a condensed-phase fluid is widely attributed to the momentum that fluid molecules displaced by the particle transfer back to it during the course of its motion. The forces created by this backflow effect are known as Basset forces, and they have been found in recent analytical work and numerical simulations to be implicated in a number of interesting dynamical phenomena, including boosted particle mobility in tilted washboard potentials. Motivated by these findings, the present paper is an investigation of the role of backflow in thermally activated barrier crossing, the governing process in essentially all condensed-phase chemical reactions. More specifically, it is an exact analytical calculation, carried out within the framework of the reactive-flux formalism, of the transmission coefficient κ(t) of a Brownian particle that crosses an inverted parabola under the influence of a colored noise process originating in the Basset force and a Markovian time-local friction. The calculation establishes that κ(t) is significantly enhanced over its backflow-free limit.

Particle dynamics in viscoelastic media: Effects of non-thermal white noise on barrier crossing rates

The growing interest in the dynamics of self-driven particle motion has brought increased attention to the effects of non-thermal noise on condensed phase diffusion. Thanks to data recently collected by Ferrer et al. on activated dynamics in the presence of memory [Phys. Rev. Lett. 126, 108001 (2021)], some of these effects can now be characterized quantitatively. In the present paper, the data collected by Ferrer et al. are used to calculate the extent to which non-thermal white noise alters the time taken by single micron-sized silica particles in a viscoelastic medium to cross the barrier separating the two wells of an optically created bistable potential. The calculation-based on a generalized version of Kramers's flux-over-population approach-indicates that the added noise causes the barrier crossing rate (compared to the noise-free case) to first increase as a function of the noise strength and then to plateau to a constant value. The precise degree of rate enhancement may depend on how the data from the experiments conducted by Ferrer et al. are used in the flux-over-population approach. As claimed by Ferrer et al., this approach predicts barrier crossing times for the original silica-fluid system that agree almost perfectly with their experimental counterparts. However, this near-perfect agreement between theory and experiment is only achieved if the theoretical crossing times are obtained from the most probable values of a crossing time distribution constructed from the distributions of various parameters in Kramers's rate expression. If the mean values of these parameters are used in the expression instead, as would be commonly done, the theoretical crossing times are found to be as much as 1.5 times higher than the experimental values. However, these times turn out to be consistent with an alternative model of viscoelastic barrier crossing based on a mean first passage time formalism, which also uses mean parameter values in its rate expression. The rate enhancements predicted for barrier crossing under non-thermal noise are based on these mean parameter values and are open to experimental verification.

Conformational Transitions of Amyloid-β: A Langevin and Generalized Langevin Dynamics Simulation Study

The dynamics of conformational transitions of the disordered protein, amyloid-β, is studied via Langevin and generalized Langevin dynamics simulations. The transmission coefficient for the unfold-misfold transition of amyloid-β is calculated from multiple independent trajectories that originate at the transition state with different initial velocities and are directly correlated to Kramers and Grote-Hynes theories. For lower values of the frictional coefficient, a well-defined rate constant is obtained, whereas, for higher values, the transmission coefficient decays with time, indicating a breakdown of the Kramers and Grote-Hynes theories and the emergence of a dynamic disorder, which demonstrates the presence of multiple local minima in the misfolding potential energy surface. The calculated free energy profile describes a two-state transition of amyloid-β in the energy landscape. The transition path time distribution computed from these simulations is compared with the related experimental and theoretical results for the unfold-misfold transition of amyloid-β. The high free energy barrier for this transition confirms the misfolding of amyloid-β. These findings offer an insight into the dynamics of the unfold-misfold transition of this protein.

Temperature Dependence of Internal Friction of Peptides

Internal friction is a valuable concept to describe the kinetics of proteins. As is well known, internal friction can be modulated by solvent features (such as viscosity). How can internal friction be affected by environmental temperature? The answer to this question is not evident. In the present work, we approach this problem with simulations on two model peptides. The thermodynamics and relaxation kinetics are characterized through long molecular dynamics simulations, with the viscosity modulated by varying the mass of solvent molecules. Based on the extrapolation to zero viscosity together with scaling of the relaxation time scales, we discover that internal friction is almost invariant at various temperatures. Controlled simulations further support the idea that internal friction is independent of environmental temperature. Comparisons between the two model peptides help us to understand the diverse phenomena in experiments.

Effect of elastic colored noise in the hopping dynamics of single molecules in stretching experiments

The influence of colored noise induced by elastic fluctuations in single-molecule stretching experiments is theoretically and numerically studied. Unlike in the thermal white noise case currently considered in the literature, elastically induced hopping dynamics between folded and unfolded states is manifested through critical oscillations showing smaller end-to-end distance fluctuations (δx∼1.25nm) within the free energy wells corresponding to both states. Our results are derived by analyzing the elastic coupling between the Handle-Molecule-Handle system and the laser optical tweezers (LOT) array. It is shown that an Ornstein-Uhlenbeck process related to this elastic coupling may trigger the hopping transitions via a colored noise with an intensity proportional to the elastic constant of the LOT array. Evolution equations of the variables of the system were derived by using the irreversible thermodynamics of small systems recently proposed. Theoretical expressions for the corresponding stationary probability densities are provided and the viability of inferring the shape of the free energy from direct measurements is discussed.

Viscosity Dependence of Some Protein and Enzyme Reaction Rates: Seventy-Five Years after Kramers

Kramers rate theory is a milestone in chemical reaction research, but concerns regarding the basic understanding of condensed phase reaction rates of large molecules in viscous milieu persist. Experimental studies of Kramers theory rely on scaling reaction rates with inverse solvent viscosity, which is often equated with the bulk friction coefficient based on simple hydrodynamic relations. Apart from the difficulty of abstraction of the prefactor details from experimental data, it is not clear why the linearity of rate versus inverse viscosity, k ∝ η(-1), deviates widely for many reactions studied. In most cases, the deviation simulates a power law k ∝ η(-n), where the exponent n assumes fractional values. In rate-viscosity studies presented here, results for two reactions, unfolding of cytochrome c and cysteine protease activity of human ribosomal protein S4, show an exceedingly overdamped rate over a wide viscosity range, registering n values up to 2.4. Although the origin of this extraordinary reaction friction is not known at present, the results indicate that the viscosity exponent need not be bound by the 0-1 limit as generally suggested. For the third reaction studied here, thermal dissociation of CO from nativelike cytochrome c, the rate-viscosity behavior can be explained using Grote-Hynes theory of time-dependent friction in conjunction with correlated motions intrinsic to the protein. Analysis of the glycerol viscosity-dependent rate for the CO dissociation reaction in the presence of urea as the second variable shows that the protein stabilizing effect of subdenaturing amounts of urea is not affected by the bulk viscosity. It appears that a myriad of factors as diverse as parameter uncertainty due to the difficulty of knowing the exact reaction friction and both mode and consequences of protein-solvent interaction work in a complex manner to convey as though Kramers rate equation is not absolute.

Effect of interference between two colored noises on the stationary states of a Brownian particle

In this paper we present properties of an external colored cross-correlated noise-driven Brownian system which is coupled to a thermal bath. Multiplicative cross-correlated noises can stabilize the transition state. Thus by monitoring the interference between the noises one can understand the mechanism of a chemical reaction. At the same time, we have investigated how the interference affects the barrier-crossing dynamics. In its presence breakdown of the Arrhenius result occurs. The breakdown becomes prominent if the multiplicative noises become additive in nature. We have also investigated how the power law behavior of the rate constant as a function of damping strength is affected by the properties of external colored noises. Furthermore, we have observed that multiplicative colored cross-correlated noises can induce the resonant activation phenomenon.

Shannon entropic temperature and its lower and upper bounds for non-Markovian stochastic dynamics

In this article we have studied Shannon entropic nonequilibrium temperature (NET) extensively for a system which is coupled to a thermal bath that may be Markovian or non-Markovian in nature. Using the phase-space distribution function, i.e., the solution of the generalized Fokker Planck equation, we have calculated the entropy production, NET, and their bounds. Other thermodynamic properties like internal energy of the system, heat, and work, etc. are also measured to study their relations with NET. The present study reveals that the heat flux is proportional to the difference between the temperature of the thermal bath and the nonequilibrium temperature of the system. It also reveals that heat capacity at nonequilibrium state is independent of both NET and time. Furthermore, we have demonstrated the time variations of the above-mentioned and related quantities to differentiate between the equilibration processes for the coupling of the system with the Markovian and the non-Markovian thermal baths, respectively. It implies that in contrast to the Markovian case, a certain time is required to develop maximum interaction between the system and the non-Markovian thermal bath (NMTB). It also implies that longer relaxation time is needed for a NMTB compared to a Markovian one. Quasidynamical behavior of the NMTB introduces an oscillation in the variation of properties with time. Finally, we have demonstrated how the nonequilibrium state is affected by the memory time of the thermal bath.

Two-point approximation to the Kramers problem with coloured noise

We present a method, founded on previous renewal approaches as the classical Wilemski-Fixman approximation, to describe the escape dynamics from a potential well of a particle subject to non-Markovian fluctuations. In particular, we show how to provide an approximated expression for the distribution of escape times if the system is governed by a generalized Langevin equation (GLE). While we show that the method could apply to any friction kernel in the GLE, we focus here on the case of power-law kernels, for which extensive literature has appeared in the last years. The method presented (termed as two-point approximation) is able to fit the distribution of escape times adequately for low potential barriers, even if conditions are far from Markovian. In addition, it confirms that non-exponential decays arise when a power-law friction kernel is considered (in agreement with related works published recently), which questions the existence of a characteristic reaction rate in such situations.

Colored noise in the fluctuations of an extended DNA molecule detected by optical trapping

We studied fluctuations of an optically trapped bead connected to a single DNA molecule anchored between the bead and a cover glass or between two optically trapped beads. Power spectral densities of the bead position for different extensions of the molecule were compared with the power spectral density of the position fluctuations of the same bead without the molecule attached. Experiments showed that the fluctuations of the DNA molecule extended up to 80% by a force of 3 pN include the colored noise contribution with spectral dependence 1/f (α) with α ~ 0.75.

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.