Microscopic viscosity and rotational diffusion of proteins in a macromolecular environment

The Stokes-Einstein-Debye equation is currently used to obtain information on protein size or on local viscosity from the measurement of the rotational correlation time. However, the implicit assumptions of a continuous and homogeneous solvent do not hold either in vivo, because of the high density of macromolecules, or in vitro, where viscosity is adjusted by adding viscous cosolvents of various size. To quantify the consequence of nonhomogeneity, we have measured the rotational Brownian motion of three globular proteins with molecular mass from 66 to 4000 kD in presence of 1.5 to 2000 kD dextrans as viscous cosolvents. Our results indicate that the linear viscosity dependence of the Stokes-Einstein relation must be replaced by a power law to describe the rotational Brownian motion of proteins in a macromolecular environment. The exponent of the power law expresses the fact that the protein experiences only a fraction of the hydrodynamic interactions of macromolecular cosolvents. An explicit expression of the exponent in terms of protein size and cosolvent's mass is obtained, permitting definition of a microscopic viscosity. Experimental data suggest that a similar effective microviscosity should be introduced in Kramers' equation describing protein reaction rates.     

Measuring how two proteins affect each other's net charge in a crowded environment

Theory predicts that the net charge (Z) of a protein can be altered by the net charge of a neighboring protein as the two approach one another below the Debye length. This type of charge regulation suggests that a protein''s charge and perhaps function might be affected by neighboring proteins without direct binding. Charge regulation during protein crowding has never been directly measured due to analytical challenges. Here, we show that lysine specific protein crosslinkers (NHS ester‐Staudinger pairs) can be used to mimic crowding by linking two non‐interacting proteins at a maximal distance of ~7.9 Å. The net charge of the regioisomeric dimers and preceding monomers can then be determined with lysine‐acyl “protein charge ladders” and capillary electrophoresis. As a proof of concept, we covalently linked myoglobin (Z _monomer = −0.43 ± 0.01) and α‐lactalbumin (Z _monomer = −4.63 ± 0.05). Amide hydrogen/deuterium exchange and circular dichroism spectroscopy demonstrated that crosslinking did not significantly alter the structure of either protein or result in direct binding (thus mimicking crowding). Ultimately, capillary electrophoretic analysis of the dimeric charge ladder detected a change in charge of ΔZ = −0.04 ± 0.09 upon crowding by this pair (Z _dimer = −5.10 ± 0.07). These small values of ΔZ are not necessarily general to protein crowding (qualitatively or quantitatively) but will vary per protein size, charge, and solvent conditions.     

Large-scale rotational motions of proteins detected by electron paramagnetic resonance and fluorescence.

Direct spectroscopic measurements of rotational motions of proteins and large protein segments are crucial to understanding the molecular dynamics of protein function. Fluorescent probes and spin labels attached to proteins have proved to be powerful tools in the study of large-scale protein motions. Fluorescence depolarization and conventional electron paramagnetic resonance (EPR) are applicable to the study of rotational motions in the nanosecond-to-microsecond time range, and have been used to demonstrate segmental flexibility in an antibody and in myosin. Very slow rotational motions, occurring in the microsecond-to-millisecond time range, are particularly important in supramolecular assemblies, where protein motions are restricted by association with other molecules. Saturation transfer spectroscopy (ST-EPR), a recently developed electron paramagnetic resonance (EPR) technique that permits the detection of rotational correlation times as long as 1 ms, has been used to detect large-scale rotational motions of spin-labeled proteins in muscle filaments and in membranes, providing valuable insights into energy transduction mechanisms in these assemblies.     

Time-dependent phosphorescence anisotropy measurements of the slow rotational motions of proteins in viscous solution

A method of monitoring slow rotational motions of proteins from the decay of the intrinsic phosphorescence is described. The phosphorescence is excited with a 10-μsec pulse of vertically polarized light from an air gap lamp, and the anisotropy was computed as a function of time from the simultaneously detected vertically and horizontally polarized components of the emission. The approach is illustrated with time-dependent measurements of the anisotropy of the tryptophan phosphorescence of Staphylococcus aureus nuclease, bovine carbonic anhydrase, and liver alcohol dehydrogenase in glycerol-phosphate buffer between ?90 and ?70°C. The temperature- and molecular-weight dependence of the exponential decays in the anisotropy indicate that overall rotation of the proteins is at the origin of the depolarization. The potential of the approach as a probe of the slow rotational motions of proteins in membranes and other macromolecular complexes is stressed.     

Coupling between overall rotational diffusion and domain motions in proteins and its effect on dielectric spectra

In this work, we formulate a closed‐form solution of the model of a semirigid molecule for the case of fluctuating and reorienting molecular electric dipole moment. We illustrate with numeric calculations the impact of protein domain motions on dielectric spectra using the example of the 128 kDa protein dimer of Enzyme I. We demonstrate that the most drastic effect occurs for situations when the characteristic time of protein domain dynamics is comparable to the time of overall molecular rotational diffusion. We suggest that protein domain motions could be a possible explanation for the high‐frequency contribution that accompanies the major relaxation dispersion peak in the dielectric spectra of protein aqueous solutions. We propose that the presented computational methodology could be used for the simultaneous analysis of dielectric spectroscopy and nuclear magnetic resonance data. Proteins 2015; 83:1571–1581. © 2015 Wiley Periodicals, Inc.     

Domain motions in proteins

Although proteins require and possess well defined spatial structures, ever more cases are emerging where parts of proteins have moved relative to each other. These parts can be as small as single side chains or as large as domains of 50–150 residues. Analysis of these motions yields valuable data on the functions of the respective proteins.     

Local motions in a benchmark of allosteric proteins

Allosteric proteins have been studied extensively in the last 40 years, but so far, no systematic analysis of conformational changes between allosteric structures has been carried out. Here, we compile a set of 51 pairs of known inactive and active allosteric protein structures from the Protein Data Bank. We calculate local conformational differences between the two structures of each protein using simple metrics, such as backbone and side-chain Cartesian displacement, and torsion angle change and rearrangement in residue-residue contacts. Thresholds for each metric arise from distributions of motions in two control sets of pairs of protein structures in the same biochemical state. Statistical analysis of motions in allosteric proteins quantifies the magnitude of allosteric effects and reveals simple structural principles about allostery. For example, allosteric proteins exhibit substantial conformational changes comprising about 20% of the residues. In addition, motions in allosteric proteins show strong bias toward weakly constrained regions such as loops and the protein surface. Correlation functions show that motions communicate through protein structures over distances averaging 10-20 residues in sequence space and 10-20 A in Cartesian space. Comparison of motions in the allosteric set and a set of 21 nonallosteric ligand-binding proteins shows that nonallosteric proteins also exhibit bias of motion toward weakly constrained regions and local correlation of motion. However, allosteric proteins exhibit twice as much percent motion on average as nonallosteric proteins with ligand-induced motion. These observations may guide efforts to design flexibility and allostery into proteins.     

Fluorescence recovery spectroscopy as a probe of slow rotational motions.

Pump-and-probe techniques can be used to follow the slow rotational motions of fluorescent labels bound to macromolecules in solution. A strong pulse of polarized light initially anisotropically depletes the ground-state population. A continuous low-intensity beam of variable polarization then probes the anisotropic ground-state distribution. Using an additional emission polarizer, the generated fluorescence can be recorded as it rises towards its prepump value. A general theory of fluorescence recovery spectroscopy (FRS) is presented that allows for irreversible depletion processes like photobleaching as well as slowly reversible processes like triplet formation. In either case, rotational motions modulate recovery through cosine-squared laws for dipolar absorption and emission processes. Certain pump, probe, and emission polarization directions eliminate the directional dependence of either dipole and simplify the resulting expressions. Two anisotropy functions can then be constructed to independently monitor the rotations of either dipole. These functions are identical in form to the anisotropy used in fluorescence depolarization measurements and all rotational models developed there apply here with minor modifications. Several setups are discussed that achieve the necessary polarization alignments. These include right-angle detection equipment that is commonly available in laboratories using fluorescence methods.     

Fluorescence anisotropy as a probe to study tracer proteins in crowded solutions

Fluorescence anisotropy has been widely used to study the dynamics and interactions of biomolecules in diluted solutions. Comparable studies on single tracer macromolecules at the cellular level are now feasible because of the recent development of non-invasive fluorescence markers, like the growing family of the green fluorescence proteins (GFPs), and the advances in time-resolved fluorescence microscopy instrumentation. The interpretation of fluorescence polarization data in terms of dynamics and biological function of the macromolecular complexes in these physiological environments requires a deep understanding of the tracer rotational diffusion in such complex media. In this work we have studied the rotational diffusion of a tracer protein, apomyoglobin labeled with 1-anilino-8-naphthalene sulfonate, in crowded solutions of an unrelated protein, ribonuclease A. We have evaluated the deviation of the different tracer rotational motions from the Stokes-Einstein-Debye diffusion behavior, and its relation to the properties of the transient molecular cavities where the tracer is rotating in the fluorescence lifetime window. Finally, we have analyzed the application of fluorescence polarization methods to determine the apparent equilibrium constants of homo and hetero-associations of macromolecules in crowded conditions.     

Charge transport in DNA model with vibrational and rotational coupling motions

The dynamics of the Peyrard-Bishop model for vibrational motion of DNA dynamics, which has been extended by taking into account the rotational motion for the nucleotides (Silva et al., J. Biol. Phys. 34, 511–519, 2018) is studied. We report on the presence of the modulational instability (MI) of a plane wave for charge migration in DNA and the generation of soliton-like excitations in DNA nucleotides. We show that the original differential-difference equation for the DNA dynamics can be reduced in the continuum approximation to a set of three coupled nonlinear equations. The linear stability analysis of continuous wave solutions of the coupled systems is performed and the growth rate of instability is found numerically. Numerical simulations show the validity of the analytical approach with the generation of wave packets provided that the wave numbers fall in the instability domain.     

Translational and rotational joint power terms in a six degree-of-freedom model of the normal ankle complex

We hypothesized that defining joint power (JP) merely on the basis of joint rotations ignores important translational power terms, and may not adequately represent the energy flow profile for a given muscle group. A novel six degree-of-freedom (6 DOF) model of the ankle complex was implemented, accounting for previously ignored joint translations as well as traditional rotations. Foot and shank kinematic and kinetic data were collected over a stride cycle on five male and five female adults, walking five trials each at 0.69 statures s⁻¹. During intra-subject analyses, ensemble averages were calculated (n=5) for JP associated with each DOF, and for related velocity and force/moment data. Translational joint velocities typically peaked below 10% of the mean walking velocity. The largest peak in JP occurred for the rotational DOF associated with dorsi/planter flexion (360 W). The next largest peak in JP was for the vertical translational DOF, and was nearly 10% of the predominant peak. Positive work during push-off was significantly less p≤0.05) for the 6 DOF model (27.9 J) than for either 1 or 3 DOF rotational models (30.3 and 29.9 J, respectively). Negative work during early stance was significantly less for the 6 DOF model (−10.3 J) than for either the 1 or 3 DOF models (−13.1 and −12.6 J, respectively). Inter-subject analyses (n=50) were conducted for JP data only, with similar results. We conclude that translational JP terms are of practical importance in mechanical energy studies, and may be of particular concern when evaluating energy storing prostheses, when summing total power at several joints, and when studying pathologies that disturb joint geometry.     

Contribution of translational and rotational motions to molecular association in aqueous solution

Much uncertainty and controversy exist regarding the estimation of the enthalpy, entropy, and free energy of overall translational and rotational motions of solute molecules in aqueous solutions, quantities that are crucial to the understanding of molecular association/recognition processes and structure-based drug design. A critique of the literature on this topic is given that leads to a classification of the various views. The major stumbling block to experimentally determining the translational/rotational enthalpy and entropy is the elimination of vibrational perturbations from the measured effects. A solution to this problem, based on a combination of energy equi-partition and enthalpy-entropy compensation, is proposed and subjected to verification. This method is then applied to analyze experimental data on the dissociation/unfolding of dimeric proteins. For one translational/rotational unit at 1 M standard state in aqueous solution, the results for enthalpy (H degrees (tr)), entropy (S degrees (tr)), and free energy (G degrees (tr)) are H (degrees) (tr) = 4.5 +/- 1.5RT, S (degrees) (tr) = 5 +/- 4R, and G (degrees) (tr) = 0 +/- 5RT. Therefore, the overall translational and rotational motions make negligible contribution to binding affinity (free energy) in aqueous solutions at 1 M standard state.     

Membrane proteins in reverse micelles: myelin basic protein in a membrane-mimetic environment

The solubility, reactivity, and conformational dynamics of myelin basic protein (MBP) from bovine brain were studied in reverse micelles of sodium bis(2-ethylhexyl) sulfosuccinate (AOT)-isooctane and water. Such a membrane-mimetic system resembles the aqueous spaces of native myelin sheath in terms of physicochemical properties as reflected in the high affinity of MBP for interfacial bound water. This is marked by the unusual profile of the solubility curve of the protein in reverse micelles, which shows optimal solubility at a much lower molar ratio of water to surfactant ([ H2O]/[AOT] = w0) than that reported for other water-soluble proteins. The role of counterions and/or charged polar head groups in the solubilization process is revealed by comparison of the solubility of MBP in nonionic surfactant micellar solutions. Whereas MBP is unfolded in aqueous solutions, insertion into reverse micelles generates a more folded structure, characterized by the presence of 20% alpha-helix. This conformation is unaffected by variations in the water content of the system (in the 2.0-22.4 w0 range). The reactivity of epsilon-amino groups of lysine residues with aqueous solutions of o-phthalaldehyde demonstrates that segments of the peptide chain are accessible to water. Similar results were obtained with the sequence involved in heme binding. In contrast, the sole tryptophan residue, Trp-117, is shielded from the aqueous solvent, as indicated by lack of reaction with N-bromosuccinimide. The invariance of the wavelength maximum emission in the fluorescence spectra as a function of w0 is consistent with this result.(ABSTRACT TRUNCATED AT 250 WORDS)     

Radially softening diffusive motions in a globular protein

Molecular dynamics simulation, quasielastic neutron scattering and analytical theory are combined to characterize diffusive motions in a hydrated protein, C-phycocyanin. The simulation-derived scattering function is in approximate agreement with experiment and is decomposed to determine the essential contributions. It is found that the geometry of the atomic motions can be modeled as diffusion in spheres with a distribution of radii. The time dependence of the dynamics follows stretched exponential behavior, reflecting a distribution of relaxation times. The average side chain and backbone dynamics are quantified and compared. The dynamical parameters are shown to present a smooth variation with distance from the core of the protein. Moving outward from the center of the protein there is a progressive increase of the mean sphere size, accompanied by a narrowing and shifting to shorter times of the relaxation time distribution. This smooth, "radially softening" dynamics may have important consequences for protein function. It also raises the possibility that the dynamical or "glass" transition with temperature observed experimentally in proteins might be depth dependent, involving, as the temperature decreases, progressive freezing out of the anharmonic dynamics with increasing distance from the center of the protein.     

The microsecond rotational motions of eosin-labelled fatty acids in multilamellar vesicles

The rotational properties of two eosin-labelled fatty acids of different alkyl chain length have been studied in large multilamellar dimyristoylphosphatidylcholine vesicles. The location of the probes at the surface region were ascertained by quenching experiments using a hydrophilic divalent cation solubilized in the aqueous phase (Cu2+) and a hydrophobic aromatic aniline (N,N-dimethylaniline) associated with the lipid. Phosphorescence anisotropy measurements reveal that above the phospholipid phase transition the polarization of eosin luminescence decays monoexponentially in the micro-to-millisecond time range, while below the phase transition a biexponential decay is observed. A model is proposed which attributes the time constants to two separate motions, discrete jumps or 'flipping' of the eosin moiety within restricted boundaries and long-axis rotation. The value of the time-independent term changes with probe position and temperature and reflects orientational constraints imposed by lipid-chromophore interactions. The implications of these results for the study of protein rotations in membranes are discussed.     

Protein self-association in crowded protein solutions: a time-resolved fluorescence polarization study

The self-association equilibrium of a tracer protein, apomyoglobin (apoMb), in highly concentrated crowded solutions of ribonuclease A (RNase A) and human serum albumin (HSA), has been studied as a model system of protein interactions that occur in crowded macromolecular environments. The rotational diffusion of the tracer protein labeled with two different fluorescent dyes, 8-anilinonaphthalene-1-sulfonate and fluorescein isothiocyanate, was successfully recorded as a function of the two crowder concentrations in the 50-200 mg/mL range, using picosecond-resolved fluorescence anisotropy methods. It was found that apoMb molecules self-associate at high RNase A concentration to yield a flexible dimer. The apparent dimerization constant, which increases with RNase A concentration, could also be estimated from the fractional contribution of monomeric and dimeric species to the total fluorescence anisotropy of the samples. In contrast, an equivalent mass concentration of HSA does not result in tracer dimerization. This different effect of RNase A and HSA is much larger than that predicted from simple models based only on the free volume available to apoMb, indicating that additional, nonspecific interactions between tracer and crowder should come into play. The time-resolved fluorescence polarization methods described here are expected to be of general applicability to the detection and quantification of crowding effects in a variety of macromolecules of biological relevance.     

Combined effect of restricted rotational diffusion plus jumps on nuclear magnetic resonance and fluorescence probes of aromatic ring motions in proteins.

A simple model is presented for the motion of phenylalanine and tyrosine rings in proteins. The model consists of restricted rotational diffusion of the rings about the side chain C alpha--C beta (chi 1) and C beta--C gamma (chi 2) axes combined with 180 degree ring flips. The model is used to evaluate order parameters for nuclear magnetic resonance relaxation and fluorescence depolarization probes of aromatic ring motions in proteins. The dependence of the order parameters on orientation in the ring plane is examined and it is demonstrated that in the presence of ring flips, additional ring librations can have a large effect on the probe order parameters.