Long time dynamics of Met-enkephalin: comparison of explicit and implicit solvent models

Met-enkephalin is one of the smallest opiate peptides. Yet, its dynamical structure and receptor docking mechanism are still not well understood. The conformational dynamics of this neuron peptide in liquid water are studied here by using all-atom molecular dynamics (MD) and implicit water Langevin dynamics (LD) simulations with AMBER potential functions and the three-site transferable intermolecular potential (TIP3P) model for water. To achieve the same simulation length in physical time, the full MD simulations require 200 times as much CPU time as the implicit water LD simulations. The solvent hydrophobicity and dielectric behavior are treated in the implicit solvent LD simulations by using a macroscopic solvation potential, a single dielectric constant, and atomic friction coefficients computed using the accessible surface area method with the TIP3P model water viscosity as determined here from MD simulations for pure TIP3P water. Both the local and the global dynamics obtained from the implicit solvent LD simulations agree very well with those from the explicit solvent MD simulations. The simulations provide insights into the conformational restrictions that are associated with the bioactivity of the opiate peptide dermorphin for the delta-receptor.     

Solvent Friction Changes the Folding Pathway of the Tryptophan Zipper TZ2

Because the rate of a diffusional process such as protein folding is controlled by friction encountered along the reaction pathway, the speed of folding is readily tunable through adjustment of solvent viscosity. The precise relationship between solvent viscosity and the rate of diffusion is complex and even conformation-dependent, however, because both solvent friction and protein internal friction contribute to the total reaction friction. The heterogeneity of the reaction friction along the folding pathway may have subtle consequences. For proteins that fold on a multidimensional free-energy surface, an increase in solvent friction may drive a qualitative change in folding trajectory. Our time-resolved experiments on the rapidly and heterogeneously folding β-hairpin TZ2 show a shift in the folding pathway as viscosity increases, even though the energetics of folding is unaltered. We also observe a nonlinear or saturating behavior of the folding relaxation time with rising solvent viscosity, potentially an experimental signature of the shifting pathway for unfolding. Our results show that manipulations of solvent viscosity in folding experiments and simulations may have subtle and unexpected consequences on the folding dynamics being studied.     

15N relaxation study of the cold shock protein CspB at various solvent viscosities

For a detailed NMR study of the dynamics of the cold shock protein CspB from Bacillus subtilis, we determined ¹⁵N transverse and longitudinal relaxation rates and heteronuclear nuclear Overhauser effects at different solvent viscosities. Up to a relative viscosity of 2, which is equivalent to 27% ethylene glycol (EG), the overall correlation time follows the linear Stokes-Einstein equation. At a relative viscosity of 6 (70% EG) the correlation time deviates from linearity by 30%, indicating that CspB tumbles at a higher rate as expected from the solvent viscosity probably due to a preferential binding of water molecules at the protein surface. The corresponding hydrodynamic radii, determined by NMR diffusion experiments, show no variation with viscosity. The amplitudes of intramolecular motions on a sub-nanosecond time scale revealed by an extended Lipari–Szabo analysis were mainly independent of the solvent viscosity. The lower limit of the NMR `observation window' for the internal correlation time shifts above 0.5 ns at 70% EG, which is directly reflected in the experimentally derived internal correlation times. Chemical exchange contributions to the transverse relaxation rates derived from the Lipari-Szabo approach coincide with the experimentally determined values from the transverse ¹H-¹⁵N dipolar/¹⁵N chemical shift anisotropy relaxation interference. These contributions originate from fast protein folding reactions on a millisecond timescale, which get retarded at increased solvent viscosities.     

Viscosity scaling and protein dynamics

The rates of molecular motions in the interior of some proteins were found to scale with an inverse power of the external solvent viscosity. The data were explained by a flexible protein structure whose dynamics is partially controlled by the solvent. Reaction dynamics in the presence of structural fluctuations with finite lifetimes lead to a dynamic friction coefficient defined by a generalized Langevin equation and a fluctuation-dissipation theorem. A model for the dynamic friction is derived assuming that the fluctuation spectrum at the reaction site involves two components: solvent-independent diffusion of local structural defects in the protein matrix and global fluctuations coupled to the solvent. The theory is applied to the viscosity dependence of molecular oxygen-binding rates in sperm whale myoglobin.     

Effects of viscous solvent on DNA polymer in a fiber

The viscous forces acting on a DNA macromolecule in a fiber are calculated. The DNA polymer is modeled as an infinite rod of elliptical cross section with a grooved surface. The viscous solvent is hydrodynamic water. Appropriate boundary conditions for determining the viscous forces on the acoustic vibrational modes are discussed. The viscous forces acting on each mode are then calculated as functions of both frequency and amount of water in the fiber. The mass loading of the DNA due to water in the grooves is shown to reduce the longitudinal acoustic velocity, which agrees with recent experimental results. The longitudinal modes are determined to be underdamped and correspondingly sharp over a range of frequencies and humidities appropriate to experimental conditions. The torsional and transverse acoustic modes are still strongly overdamped.     

Viscosity dependence of protein dynamics

The influence of solvent viscosity on protein dynamics was investigated with molecular dynamics simulations of factor Xa in two solvents differing only in viscosity, by a factor of 10. We obtained this viscosity change by changing the masses of the solvent atoms by a factor of 100. Equilibrium properties of the protein, that is, the average structure, its fluctuations, and the secondary structure, show no significant dependence on the solvent viscosity. The dynamic properties of the protein, that is, the atom-positional correlation times and torsional angle transitions, however, depend on the solvent viscosity. The protein appears to be much more mobile in the solvent of lower viscosity. It feels the influence of the solvent not only on the surface but even in its core. With increasing solvent viscosity, the positional relaxation times of atoms in the protein core increase as much as those of atoms on the protein surface, and the relative increase in the core is even larger than on the surface.     

A hierarchy of functionally important relaxations within myoglobin based on solvent effects, mutations and kinetic model

Geminate CO rebinding in myoglobin is studied for two viscous solvents, trehalose and sol-gel (bathed in 100% glycerol) at several temperatures. Mutations in key distal hemepocket residues are used to eliminate or enhance specific relaxation modes. The time-resolved data are analyzed with a modified Agmon-Hopfield model which is capable of providing excellent fits in cases where a single relaxation mode is dominant. Using this approach, we determine the relaxation rate constants of specific functionally important modes, obtaining also their Arrhenius activation energies. We find a hierarchy of distal pocket modes controlling the rebinding kinetics. The "heme access mode" (HAM) is responsible for the major slow-down in rebinding. It is a solvent-coupled cooperative mode which restricts ligand return from the xenon cavities. Bulky side-chains, like those His64 and Trp29 (in the L29W mutant), operate like overdamped pendulums which move over and block the binding site. They may be either unslaved (His64) or moderately slaved (Trp29) to the solvent. Small side-chain relaxations, most notably of leucines, are revealed in some mutants (V68L, V68A). They are conjectured to facilitate inter-cavity ligand motion. When all relaxations are arrested (H64L in trehalose), we observe pure inhomogeneous kinetics with no temperature dependence, suggesting that proximal relaxation is not a factor on the investigated timescale.     

Viscoelastic relaxation of bilayer lipid membranes. Frequency-dependent tension and membrane viscosity.

Photon correlation spectroscopy has been used to study capillary waves on black lipid membranes of glycerol monooleate at temperatures above the lipid transition. For the first time the tension and viscosity of solvent-free bilayers have been observed to display a frequency dependence. The variations of both parameters can be accounted for by a Maxwell viscoelastic fluid model having a relaxation time of 37 microseconds. The equilibrium (omega = 0) tension is compatible with literature values. The present results do not suffice to precisely define the specific molecular processes involved, but relaxation times similar to the present are associated with certain phenomena in phospholipid vesicles. Bilayers containing hydrocarbon solvent do not show such relaxation, presumably due to their weaker intermolecular interactions.     

Modeling the correlation functions of conformational motions in proteins

A first principles calculation of the correlation function for conformational motion (CM) in proteins is carried out within the framework of a microscopic model of a protein as a heterogeneous system. The fragments of the protein are assumed to be identical hard spheres undergoing the CM within their conformational potentials about some mean equilibrium positions assigned by the tertiary structure. The memory friction function (MFF) for the generalized Langevin equation describing the CM of the particle is obtained on the basis of the direct calculation which is feasible for the present model of the protein due to the existence of a natural large parameter, viz. the ratio of the minimal distance between the mean equilibrium positions of the particles (approximately 7A) to the amplitude of their CM (<1A). A relationship between the MFF and the correlation functions of the CM of the particles is derived which makes their calculation to be a self-consistent mathematical problem. The general analysis of the MFF is exemplified by a simple model case in which the mean equilibrium positions of the particles form a regular lattice so that the correlation functions for all particles are the same. In this case the MFF is shown to be an infinite series of the powers of the auto-correlation function whose coefficients are independent on temperature. The latter is a result of the abstraction of the interaction potential by that of hard spheres which actually corresponds to the high temperature limit. On the examples of cubic and triangular lattices the coefficients are shown to be non-negative values which increase with the increase of the packing density of the particles and quickly tend to zero with the increase of their index. Thus the MFF can be approximated by a polynomial of the correlation function and the resulting mathematical equation is analogous to the one from the dynamic theory of liquids. The correlation function of the CM is obtained by numerical solution of the equation. At realistic packing densities for proteins it exhibits transparent non-exponential decay and includes two relaxation processes: the first one on the intermediate timescale (tens of picoseconds) and the second on the long timescale (its characteristic time is about tens of nanoseconds at small values of the friction coefficient and increases by orders of the magnitude with the increase of the latter). Thus the present approach provides the microscopic basis for previous phenomenological models of cooperative dynamics in proteins.     

Water-coupled low-frequency modes of myoglobin and lysozyme observed by inelastic neutron scattering.

Conformational changes of proteins often involve the relative motion of rigid structural domains. Normal mode analysis and molecular dynamics simulations of small globular proteins predict delocalized vibrations with frequencies below 20 cm(-1), which may be overdamped in solution due to solvent friction. In search of these modes, we have studied deuterium-exchanged myoglobin and lysozyme using inelastic neutron scattering in the low-frequency range at full and low hydration to modify the degree of damping. At room temperature, the hydrated samples exhibit a more pronounced quasielastic spectrum due to diffusive motions than the dehydrated samples. The analysis of the corresponding lineshapes suggests that water modifies mainly the amplitude, but not the characteristic time of fast protein motions. At low temperatures, in contrast, the dehydrated samples exhibit larger motional amplitudes than the hydrated ones. The excess scattering, culminating at 16 cm(-1), is suggested to reflect water-coupled librations of polar side chains that are depressed in the hydrated system by strong intermolecular hydrogen bonding. Both myoglobin and lysozyme exhibit ultra-low-frequency modes below 10 cm(-1) in the dry state, possibly related to the breathing modes predicted by harmonic analysis.     

Modeling the Correlation Functions of Conformational Motions in Proteins

Abstract

A first principles calculation of the correlation function for conformational motion (CM) in proteins is carried out within the framework of a microscopic model of a protein as a heterogeneous system. The fragments of the protein are assumed to be identical hard spheres undergoing the CM within their conformational potentials about some mean equilibrium positions assigned by the tertiary structure. The memory friction function (MFF) for the generalized Langevin equation describing the CM of the particle is obtained on the basis of the direct calculation which is feasible for the present model of the protein due to the existence of a natural large parameter, viz. the ratio of the minimal distance between the mean equilibrium positions of the particles (～7A) to the amplitude of their CM (<1A). A relationship between the MFF and the correlation functions of the CM of the particles is derived which makes their calculation to be a self-consistent mathematical problem. The general analysis of the MFF is exemplified by a simple model case in which the mean equilibrium positions of the particles form a regular lattice so that the correlation functions for all particles are the same. In this case the MFF is shown to be an infinite series of the powers of the auto-correlation function whose coefficients are independent on temperature. The latter is a result of the abstraction of the interaction potential by that of hard spheres which actually corresponds to the high temperature limit. On the examples of cubic and triangular lattices the coefficients are shown to be non-negative values which increase with the increase of the packing density of the particles and quickly tend to zero with the increase of their index. Thus the MFF can be approximated by a polynomial of the correlation function and the resulting mathematical equation is analogous to the one from the dynamic theory of liquids. The correlation function of the CM is obtained by numerical solution of the equation. At realistic packing densities for proteins it exhibits transparent non-exponential decay and includes two relaxation processes: the first one on the intermediate timescale (tens of picoseconds) and the second on the long timescale (its characteristic time is about tens of nanoseconds at small values of the friction coefficient and increases by orders of the magnitude with the increase of the latter). Thus the present approach provides the microscopic basis for previous phenomenologi- cal models of cooperative dynamics in proteins.     

Harmonic analysis of DNA dynamics in a viscous medium

The harmonic dynamics of normal modes of double-stranded DNA in a viscous fluid are investigated. The model DNA consists of two backbone-supported DNA strands coiling around a common helix axis with base stacking, sugar puckering, interstrand hydrogen bonding, and intrastrand sugar-base interactions assigned values based on published data. Assuming that the DNA bases are shielded from direct bombardment by the solvent, analytical solutions are obtained. The dissipation and fluctuation of the normal modes of the bases moving along the spirals display the effect of the medium indirectly through interactions with the backbone. The dynamics of the backbone are found to be overdamped with the characteristic damping times extending to the picosecond region for disturbance in position and to the sub-picosecond region for disturbance in velocity. In addition to the dynamic mode of a rigid rod, the motions of the bases are coupled to the motions of the backbone with comparable amplitudes for disturbance in position. For disturbance in velocity, however, the bases are effectively at rest, not being able to follow the motions of the backbone. The angular frequencies of the underdamped vibrational modes, identified as the ringing modes of the bases with the backbone effectively at rest, are insensitive to the viscosity and lie in the low frequency region of the Raman spectrum. These findings indicate that the backbone of DNA plays a significant role in modulating the dynamics of double-stranded DNA in an overdamping environment. This modulation of the dynamics of the motions of the bases in DNA by environmental impediments to molecular motion is briefly discussed in connection with protein- and drug- DNA interactions as well as gene regulation.     

Internal friction controls the speed of protein folding from a compact configuration

Several studies have found millisecond protein folding reactions to be controlled by the viscosity of the solvent: Reducing the viscosity allows folding to accelerate. In the limit of very low solvent viscosity, however, one expects a different behavior. Internal interactions, occurring within the solvent-excluded interior of a compact molecule, should impose a solvent-independent upper limit to folding speed once the bulk diffusional motions become sufficiently rapid. Why has this not been observed? We have studied the effect of solvent viscosity on the folding of cytochrome c from a highly compact, late-stage intermediate configuration. Although the folding rate accelerates as the viscosity declines, it tends toward a finite limiting value approximately 10(5) s(-1) as the viscosity tends toward zero. This limiting rate is independent of the cosolutes used to adjust solvent friction. Therefore, interactions within the interior of a compact denatured polypeptide can limit the folding rate, but the limiting time scale is very fast. It is only observable when the solvent-controlled stages of folding are exceedingly rapid or else absent. Interestingly, we find a very strong temperature dependence in these "internal friction"-controlled dynamics, indicating a large energy scale for the interactions that govern reconfiguration within compact, near-native states of a protein.     

Conformational motion correlations in the formation of polypeptide secondary structure in a viscous medium

The Langevin dynamics method and statistical correlation analysis were used to study the α-helical structure folding dynamics of the (Ala)₅₀, (AlaGly)₂₅, and (AlaGly)₇₅ polypeptides depending on the viscosity of the medium. Friction forces that arise when the effective viscosity of the medium is similar to the viscosity of water were found to result in strong correlations between the backbone torsion angles. The polypeptides under study folded mainly to produce α-helical structures. A structure of two contacting α-helices that were approximately equal in length and had a loop between them was observed for a longer chain of 150 residues. A method to visualize the correlation matrix of the dihedral angles of a polypeptide chain was developed for analyzing the effects of the dynamic correlation of conformational degrees of freedom. The analysis of the dynamics of the correlation matrix showed that rotations involving angles of the same type (φ–φ and ψ–ψ) occur predominantly in one direction. Rotations invoving different angles (φ–ψ) occur predominantly in opposite directions, so that the total macromolecule does not rotate. A significant reduction in the effective viscosity of the medium disrupts the correlation and makes the rotations stochastic, thus distorting the formation of the regular (helical) structure. The effects of correlated conformational motions are consequences of viscous friction forces. This conclusion agrees with our previous results that outlined the principle of the minimum rate of energy dissipation and the equipartition of energy dissipation rate between conformational degrees of freedom.     

Intramolecular and overall motion of proline: the influence of viscosity on carbon-13 spin-lattice relaxation times.

Nuclear magnetic resonance of ¹³C is used to probe the overall and internal motions of proline. Spin-lattice relaxation times (T₁) are reported for proline monomer dissolved in water/glycerol mixtures. Rates of overall molecular motion and internal motion depend on solvent composition but to different degrees. The effective correlation times (τ_eff) of the various proton-bearing carbon atoms in proline vary linearly as a function of solvent composition (%v/v) rather than of solution viscosity. The effective correlation time for molecular motion (τ_eff) is separated into contributions from overall molecular motion (τ_mol) and internal motion (τ_int). The γ-carbon of proline shows the smallest dependence of τ_int on solvent composition. The data indicate a high degree of intramolecular motion for the γ-carbon of proline. Inclusion of anisotropic molecular reorientation in the data analysis was found not to affect the above conclusions. The observed values of τ_eff indicate that the rotational diffusion model of molecular reorientations should apply to proline. The values of τ_eff calculated for proline using the Stokes-Einstein relation are larger than those observed; the discrepancy is discussed in terms of solvent-solute interactions.     

Solvent damping of internal processes in myoglobin studied by specific heat spectroscopy and flash photolysis.

We address the question of dynamic coupling between protein and solvent by comparing the enthalpy relaxation of the solvent (75% v/v glycerol-water) to internal ligand binding in myoglobin. When the solvent relaxation is slow compared to intramolecular events we observe decoupling of protein motions from the solvent. In the opposite limit there is a significant contribution of the solvent to internal friction. The solvent enhances the apparent activation energy of transitions in myoglobin. This result is discussed in terms of a generalized Kramer's law involving a dynamic friction coefficient.     

The influence of glycerol on hydrogen isotope exchange in lysozyme

Hydrogen isotope exchange rates for lysozyme in glycerol cosolvent mixtures [D. G. Knox and A. Rosenberg (1980) Biopolymers 19 , 1049–1068] have been analyzed as functions of solvent viscosity and glycerol activity in an attempt to determine which solvent properties influence protein internal dynamics. The effect of glycerol on the fast- and slow-exchanging protons is different. Slow-exchanging protons [H(t) < 20] are slowed by ever-increasing amounts as H(t) decreases. However, comparison with data for the effect of glycerol on the thermal unfolding of lysozyme [K. Gekko (1982) J. Biochem. 19 , 1197–1204] indicates that the large decrease in exchange rates for the slow protons is not consistent with a local unfolding mechanism of exchange. These effects are also too large to be easily rationalized in terms of solvent viscosity. Instead, we suggest that the large effect of glycerol on exchange of the slow protons is due to a “compression” of the protein, as a result of thermodynamically unfavorable interactions of glycerol with the protein surface. This reduces the protein void volume, which in turn decreases the probability of conformational transitions required for exchange of the slowest protons. Present data do not allow a distinction to be made between thermodynamic (glycerol activity) and dynamic (solvent viscosity) influences on exchange rates for the fast-exchanging protons, although the effect of glycerol on these protons is also probably too large to be consistent with a local unfolding mechanism. In this case, glycerol decreases the rate of catalyst diffusion within the protein matrix, either by decreasing the probabilities or amplitudes of “gating” reactions that allow passage of the catalyst from the solvent to the exchange site, or by increasing the relaxation times for these conformational rearrangements.     

Modeling truncated hemoglobin vibrational dynamics

We present a study on the near equilibrium dynamics of two small proteins in the family of truncated hemoglobins, developed under the framework of a Gaussian network approach. Effective beta carbon atoms are taken into account besides Calphas for all residues but glycines in the coarse-graining procedure, without leading to an increase in the degrees of freedom (beta Gaussian Model). Normalized covariance matrix and deformation along slowest modes with collective character are analyzed, pointing out anticorrelations between functionally relevant sites for the proteins under study. In particular, we underline the functional motions of an extended tunnel-cavity system running inside the protein matrix, which provide a pathway for small ligands binding with the iron in the heme group. We give a rough estimate of the order of magnitude of the relaxation times of the slowest two overdamped modes and compare results with previous studies on globins.     

Langevin dynamics of peptides: the frictional dependence of isomerization rates of N-acetylalanyl-N'-methylamide.

The rate constant for the transition between the equatorial and axial conformations of N-acetylalanyl-N'-methylamide has been determined from Langevin dynamics (LD) simulations with no explicit solvent. The isomerization rate is maximum at collision frequency gamma = 2 ps-1, shows diffusive character for gamma greater than or equal to 10 ps-1, but does not approach zero even at gamma = 0.01 ps-1. This behavior differs from that found for a one-dimensional bistable potential and indicates that both collisional energy transfer with solvent and vibrational energy transfer between internal modes are important in the dynamics of barrier crossing for this system. It is suggested that conformational searches of peptides be carried out using LD with a collision frequency that maximizes the isomerization rate (i.e., gamma approximately 2 ps-1). This method is expected to be more efficient than either molecular dynamics in vacuo (which corresponds to LD with gamma = 0) or molecular dynamics in solvent (where dynamics is largely diffusive).