Solution properties of γ‐crystallins: Hydration of fish and mammal γ‐crystallins

Lens γ crystallins are found at the highest protein concentration of any tissue, ranging from 300 mg/mL in some mammals to over 1000 mg/mL in fish. Such high concentrations are necessary for the refraction of light, but impose extreme requirements for protein stability and solubility. γ‐crystallins, small stable monomeric proteins, are particularly associated with the lowest hydration regions of the lens. Here, we examine the solvation of selected γ‐crystallins from mammals (human γD and mouse γS) and fish (zebrafish γM2b and γM7). The thermodynamic water binding coefficient B₁ could be probed by sucrose expulsion, and the hydrodynamic hydration shell of tightly bound water was probed by translational diffusion and structure‐based hydrodynamic boundary element modeling. While the amount of tightly bound water of human γD was consistent with that of average proteins, the water binding of mouse γS was found to be relatively low. γM2b and γM7 crystallins were found to exhibit extremely low degrees hydration, consistent with their role in the fish lens. γM crystallins have a very high methionine content, in some species up to 15%. Structure‐based modeling of hydration in γM7 crystallin suggests low hydration is associated with the large number of surface methionine residues, likely in adaptation to the extremely high concentration and low hydration environment in fish lenses. Overall, the degree of hydration appears to balance stability and tissue density requirements required to produce and maintain the optical properties of the lens in different vertebrate species.     

Frictional models for stochastic simulations of proteins

As a first step toward a systematic parametrization of friction constants of atoms in proteins, a model in which frictional resistance is placed explicitly on each atom accessible to solvent is used to calculate overall translational and rotational diffusion constants. It is found that these quantities are relatively insensitive to the precise value of the atomic friction constant, as long as the effective hydrodynamic radius of the surface atoms is approximately 1 Å. However, if only protein atoms are included in the calculation, no reasonable range atomic of radii can reproduce the experimental translational diffusion constant to better than 20% for lysozyme and 5% for ribonuclease. When a hydration shell of approximately 70% coverage for lysozyme and 20% for ribonuclease is included, there is quantitative agreement with experimental results. The sensitivity of peptide diffusion to levels of hydration is also investigated; it is found that for glycine, two bound waters are required to provide agreement with experiment. These findings imply that the effects of solvent damping will be underestimated in stochastic simulations of proteins and peptides unless bound waters are taken into account.     

Translational diffusion and intrinsic viscosity of globular proteins. Theoretical predictions using hydrated hydrodynamic models. Application to BPTI.

In this paper, we present a way to make hydrodynamic models of globular proteins, including the hydration shell associated with them in aqueous solutions. Theoretical calculations using these models are made in order to determine the hydrodynamic properties of these proteins, employing rigorous and approximate methods of calculation. These will be applied to the bovine pancreatic trypsin inhibitor, BPTI. Several hydrodynamic models are constructed: the A-model for the unhydrated protein BPTI and a set of H-models for hydrated protein with different hydration degrees. Theoretical results for the translational diffusion coefficient Dt and the intrinsic viscosity [eta] are obtained from different models. From the analysis of the A-model and hydrodynamic properties, there is not a clear assignation of an ellipsoidal shape to this protein molecule. An amount of approximately 0.5 g H2O/g protein could be assigned to the BPTI.     

A unified picture of protein hydration: prediction of hydrodynamic properties from known structures.

Hydration is essential for the structural and functional integrity of globular proteins. How much hydration water is required for that integrity? A number of techniques such as X-ray diffraction, nuclear magnetic resonance (NMR) spectroscopy, calorimetry, infrared spectroscopy, and molecular dynamics (MD) simulations indicate that the hydration level is 0.3-0.5 g of water per gram of protein for medium sized proteins. Hydrodynamic properties, when accounted for by modeling proteins as ellipsoids, appear to give a wide range of hydration levels. In this paper we describe an alternative numerical technique for hydrodynamic calculations that takes account of the detailed protein structures. This is made possible by relating hydrodynamic properties (translational and rotational diffusion constants and intrinsic viscosity) to electrostatic properties (capacitance and polarizability). We show that the use of detailed protein structures in predicting hydrodynamic properties leads to hydration levels in agreement with other techniques. A unified picture of protein hydration emerges. There are preferred hydration sites around a protein surface. These sites are occupied nearly all the time, but by different water molecules at different times. Thus, though a given water molecule may have a very short residence time (approximately 100-500 ps from NMR spectroscopy and MD simulations) in a particular site, the site appears fully occupied in experiments in which time-averaged properties are measured.     

Calculation of translational friction and intrinsic viscosity. II. Application to globular proteins.

The translational friction coefficients and intrinsic viscosities of four proteins (ribonuclease A, lysozyme, myoglobin, and chymotrypsinogen A) are calculated using atomic-level structural details. Inclusion of a 0.9-A-thick hydration shell allows calculated results for both hydrodynamic properties of each protein to reproduce experimental data. The use of detailed protein structures is made possible by relating translational friction and intrinsic viscosity to capacitance and polarizability, which can be calculated easily. The 0.9-A hydration shell corresponds to a hydration level of 0.3-0.4 g water/g protein. Hydration levels within this narrow range are also found by a number of other techniques such as nuclear magnetic resonance spectroscopy, infrared spectroscopy, calorimetry, and computer simulation. The use of detailed protein structures in predicting hydrodynamic properties thus allows hydrodynamic measurement to join the other techniques in leading to a unified picture of protein hydration. In contrast, earlier interpretations of hydrodynamic data based on modeling proteins as ellipsoids gave hydration levels that varied widely from protein to protein and thus challenged the existence of a unified picture of protein hydration.     

Calculation of hydrodynamic properties of small nucleic acids from their atomic structure

Hydrodynamic properties (translational diffusion, sedimentation coefficients and correlation times) of short B-DNA oligonucleotides are calculated from the atomic-level structure using a bead modeling procedure in which each non-hydrogen atom is represented by a bead. Using available experimental data of hydrodynamic properties for several oligonucleotides, the best fit for the hydrodynamic radius of the atoms is found to be ~2.8 Å. Using this value, the predictions for the properties corresponding to translational motion and end-over-end rotation are accurate to within a few percent error. Analysis of NMR correlation times requires accounting for the internal flexibility of the double helix, and allows an estimation of ~0.85 for the Lipari–Szabo generalized order parameter. Also, the degree of hydration can be determined from hydrodynamics, with a result of ~0.3 g (water)/g (DNA). These numerical results are quite similar to those found for globular proteins. If the hydrodynamic model for the short DNA is simply a cylindrical rod, the predictions for overall translation and rotation are slightly worse, but the NMR correlation times and the degree of hydration, which depend more on the cross-sectional structure, are more severely affected.     

Modeling the hydration of proteins: prediction of structural and hydrodynamic parameters from X-ray diffraction and scattering data

The implications of protein-water interactions are of importance for understanding the solution behavior of proteins and for analyzing the fine structure of proteins in aqueous solution. Starting from the atomic coordinates, by bead modeling the scattering and hydrodynamic properties of proteins can be predicted reliably (Debye modeling, program HYDRO). By advanced modeling techniques the hydration can be taken into account appropriately: by some kind of rescaling procedures, by modeling a water shell, by iterative comparisons to experimental scattering curves (ab initio modeling) or by special hydration algorithms. In the latter case, the surface topography of proteins is visualized in terms of dot surface points, and the normal vectors to these points are used to construct starting points for placing water molecules in definite positions on the protein envelope. Bead modeling may then be used for shaping the individual atomic or amino acid residues and also for individual water molecules. Among the tuning parameters, the choice of the scaling factor for amino acid hydration and of the molecular volume of bound water turned out to be crucial. The number and position of bound water molecules created by our hydration modeling program HYDCRYST were compared with those derived from X-ray crystallography, and the capability to predict hydration, structural and hydrodynamic parameters (hydrated volume, radius of gyration, translational diffusion and sedimentation coefficients) was compared with the findings generated by the water-shell approach CRYSOL. If the atomic coordinates are unknown, ab initio modeling approaches based on experimental scattering curves can provide model structures for hydrodynamic predictions.     

Recent advances in macromolecular hydrodynamic modeling

The modern implementation of the boundary element method [23] has ushered unprecedented accuracy and precision for the solution of the Stokes equations of hydrodynamics with stick boundary conditions. This article begins by reviewing computations with the program BEST of smooth surface objects such as ellipsoids, the dumbbell, and cylinders that demonstrate that the numerical solution of the integral equation formulation of hydrodynamics yields very high precision and accuracy. When BEST is used for macromolecular computations, the limiting factor becomes the definition of the molecular hydrodynamic surface and the implied effective solvation of the molecular surface. Studies on 49 different proteins, ranging in molecular weight from 9 to over 400kDa, have shown that a model using a 1.1? thick hydration layer describes all protein transport properties very well for the overwhelming majority of them. In addition, this data implies that the crystal structure is an excellent representation of the average solution structure for most of them. In order to investigate the origin of a handful of significant discrepancies in some multimeric proteins (about -20% observed in the intrinsic viscosity), the technique of Molecular Dynamics simulation (MD) has been incorporated into the research program. A preliminary study of dimeric α-chymotrypsin using approximate implicit water MD is presented. In addition I describe the successful validation of modern protein force fields, ff03 and ff99SB, for the accurate computation of solution structure in explicit water simulation by comparison of trajectory ensemble average computed transport properties with experimental measurements. This work includes small proteins such as lysozyme, ribonuclease and ubiquitin using trajectories around 10ns duration. We have also studied a 150kDa flexible monoclonal IgG antibody, Trastuzumab, with multiple independent trajectories encompassing over 320ns of simulation. The close agreement within experimental error of the computed and measured properties allows us to conclude that MD does produce structures typical of those in solution, and that flexible molecules can be properly described using the method of ensemble averaging over a trajectory. We review similar work on the study of a transfer RNA molecule and DNA oligomers that demonstrate that within 3% a simple uniform hydration model 1.1? thick provides agreement with experiment for these nucleic acids. In the case of linear oligomers, the precision can be improved close to 1% by a non-uniform hydration model that hydrates mainly in the DNA grooves, in agreement with high resolution X-ray diffraction. We conclude with a vista on planned improvements for the BEST program to decrease its memory requirements and increase its speed without sacrificing accuracy.     

Hydration from hydrodynamics. General considerations and applications of bead modelling to globular proteins.

The effect of hydration on hydrodynamic properties of globular proteins can be expressed in terms of two quantities: the delta (g/g) parameter and the thickness of the hydration layer. The two paradigms on hydration that originate these alternative measures are described and compared. For the numerical calculation of hydrodynamic properties, from which estimates of hydration can be made, we employ the bead modelling with atomic resolution implemented in programs HYDROPRO and HYDRONMR. As typical, average values, we find 0.3 g/g and a thickness of only approximately 1.2 A. However, noticeable differences in this parameter are found from one protein to another. We have made a numerical analysis, which leaves apart marginal influences of modelling imperfections by simulating properties of a spherical protein. This analysis confirms that the errors that one can attribute to the experimental quantities suffice to explain the observed fluctuations in the hydration parameters. However, for the main purpose of predicting protein solution properties, the above mentioned typical values may be safely used. Particularly for atomic bead modelling, a hydrodynamic radius of approximately 3.2 A yields predictions in very good agreement with experiments.     

Lateral Diffusion of Membrane Proteins: Consequences of Hydrophobic Mismatch and Lipid Composition

Biological membranes are composed of a large number lipid species differing in hydrophobic length, degree of saturation, and charge and size of the headgroup. We now present data on the effect of hydrocarbon chain length of the lipids and headgroup composition on the lateral mobility of the proteins in model membranes. The trimeric glutamate transporter (GltT) and the monomeric lactose transporter (LacY) were reconstituted in giant unilamellar vesicles composed of unsaturated phosphocholine lipids of varying acyl chain length (14-22 carbon atoms) and various ratios of DOPE/DOPG/DOPC lipids. The lateral mobility of the proteins and of a fluorescent lipid analog was determined as a function of the hydrophobic thickness of the bilayer (h) and lipid composition, using fluorescence correlation spectroscopy. The diffusion coefficient of LacY decreased with increasing thickness of the bilayer, in accordance with the continuum hydrodynamic model of Saffman-Delbrück. For GltT, the mobility had its maximum at diC18:1 PC, which is close to the hydrophobic thickness of the bilayer in vivo. The lateral mobility decreased linearly with the concentration of DOPE but was not affected by the fraction of anionic lipids from DOPG. The addition of DOPG and DOPE did not affect the activity of GltT. We conclude that the hydrophobic thickness of the bilayer is a major determinant of molecule diffusion in membranes, but protein-specific properties may lead to deviations from the Saffman-Delbrück model.     

The dynamics of protein hydration water: a quantitative comparison of molecular dynamics simulations and neutron-scattering experiments

We present results from an extensive molecular dynamics simulation study of water hydrating the protein Ribonuclease A, at a series of temperatures in cluster, crystal, and powder environments. The dynamics of protein hydration water appear to be very similar in crystal and powder environments at moderate to high hydration levels. Thus, we contend that experiments performed on powder samples are appropriate for discussing hydration water dynamics in native protein environments. Our analysis reveals that simulations performed on cluster models consisting of proteins surrounded by a finite water shell with free boundaries are not appropriate for the study of the solvent dynamics. Detailed comparison to available x-ray diffraction and inelastic neutron-scattering data shows that current generation force fields are capable of accurately reproducing the structural and dynamical observables. On the time scale of tens of picoseconds, at room temperature and high hydration, significant water translational diffusion and rotational motion occur. At low hydration, the water molecules are translationally confined but display appreciable rotational motion. Below the protein dynamical transition temperature, both translational and rotational motions of the water molecules are essentially arrested. Taken together, these results suggest that water translational motion is necessary for the structural relaxation that permits anharmonic and diffusive motions in proteins. Furthermore, it appears that the exchange of protein-water hydrogen bonds by water rotational/librational motion is not sufficient to permit protein structural relaxation. Rather, the complete exchange of protein-bound water molecules by translational displacement seems to be required.     

Facilitated diffusion: the elasticity of oxygen supply.

It is well established that the presence of oxygen-carrying proteins such as haemoglobin can facilitate the diffusion of oxygen through a solution. In this paper, it is shown that some properties of a facilitated flow are substantially different from those of unfacilitated flux, including especially the stability of the tension at which the oxygen arrives at the end of the diffusion path. The concept of the “output resistance” of the supply is introduced, and a facilitated pathway is shown to have a lowered resistance. A role for the storage capacity of the bound oxygen reservoir is also developed; it is shown that delivery oxygen tensions are stabilized against transient changes in oxygen demand. In treating the equations of facilitated diffusion, a simplified approach is used to take account of boundary layers in the solution where deviations from oxygen-protein equilibrium are significant. A measure of the thickness and importance of such boundary layers is calculated.     

Diffusion coefficients of segmentally flexible macromolecules with two spherical subunits

The translational and rotational diffusion coefficients have been calculated for a simple, segmentally flexible model: the hinged dumbbell (HD). In the HD, two spherical subunits are attached to an universal joint by means of frictionless connectors. In addition to the case in which hydrodynamic interactions are neglected (NI), we have also considered two more cases, including hydrodynamic interaction by means of the Kirkwood-Riseman approximate treatment (KR) and using accurate procedure based in the series expansions for the two-sphere diffusion tensor (SE). Expressions for the friction coefficients of the HD are given for the three cases, and the diffusion coefficients are evaluted inverting the 9 × 9 resistance matrix, for two HDs with different dimensions. The KR treatment, which includes a contribution from the finite volume of the subunits, is shown to be an excellent approximation to the more rigorous procedure. In the NI case for rotation, the various coefficients present different deviations with respect to the SE results. A rough estimate of the errors of the NI relaxation times indicates that they may be smaller than 15% for a HD with identical beads. However, the influence of hydrodynamic interaction should be more important for the rotational diffusivity of a small sphere attached to a larger one. The error of the NI result for the translational diffusion coefficient is of about 25% for the two HDs.     

Human erythrocyte Ca2+-Mg2+-ATPase: mechanism of stimulation by Ca2+.

We have critically evaluated hydrodynamic data from 21 proteins whose molecular dimensions are known from X-ray crystallography. We present two useful equations relating the molecular weights and sedimentation coefficients of globular proteins. The hydrodynamic data combined with data for small molecules from the literature indicate that failure of the Stokes equation occurs only for molecular weights <850. Calculated hydration values for the 21 proteins have a mean value and standard deviation of 0.53 ± 0.26 g H₂O/g protein. Furthermore, statistical arguments indicate that only 5.3% of the variance is due to experimental error. The mean value and especially the dispersion of values are in sharp contrast to the values 0.36 ± 0.04 obtained by others from nmr measurements on frozen protein solutions. Hydration values calculated from nmr measurements are closely correlated with the number of charged and polar amino acid residues. In contrast to this result, our analysis of the amino acid compositions of the four proteins with the lowest hydration and the four monomeric proteins with the highest shows that the range of values we observe cannot be accounted for on the basis of amino acid composition. In fact there appears to be a weak correlation between the number of apolar residues and hydrodynamic hydration. We therefore conclude that the dispersion must result from variations in fine details of the surface structures of individual proteins. We propose a model of hemispherical clathrate cages which if correct, would account for the differences in the data obtained by these two methods.     

Transport properties and hydrodynamic centers of rigid macromolecules with arbitrary shapes

A general formalism, which includes translation–rotation coupling, is proposed for calculating translational and rotational transport properties, as well as intrinsic viscosities, of rigid macromolecules with an arbitrary shape. This formalism is based on Brenner's theory of translational–rotational dynamics and on methods for the calculation of hydrodynamic properties that have been already presented, and can be regarded as a generalization of the one proposed by Nakajima and Wada. The calculated transport properties depend on the origin as predicted by Brenner's theory, but in a disagreement with him, the center of resistance and the center of diffusion do not coincide. As one can define several hydrodynamic centers, which in practice turn out to be located at different points, the influence of the choice of the center on the calculated transport properties is discussed. An analysis of the translation–rotation coupling effects in translational diffusion reveals that they arise exclusively from hydrodynamic interactions and are rather small in some cases of interest. Finally, we present a study of the rotational diffusion of rigid bent rods with a fixed length-to-diameter ratio. The diffusion coefficients obtained can be useful to estimate changes with respect to a straight rod.     

Characterization of Phospholipid Mixed Micelles by Translational Diffusion

The concentration dependence of the translational self diffusion rate, D (s), has been measured for a range of micelle and mixed micelle systems. Use of bipolar gradient pulse pairs in the longitudinal eddy current delay experiment minimizes NOE attenuation and is found critical for optimizing sensitivity of the translational diffusion measurement of macromolecules and aggregates. For low volume fractions Phi (Phi\\ le 15% v/v) of the micelles, experimental measurement of the concentration dependence, combined with use of the D (s)= D (o)(1-3.2lambdaPhi) relationship, yields the hydrodynamic volume. For proteins, the hydrodynamic volume, derived from D (s) at infinitely dilute concentration, is found to be about 2.6 times the unhydrated molecular volume. Using the data collected for hen egg white lysozyme as a reference, diffusion data for dihexanoyl phosphatidylcholine (DHPC) micelles indicate approximately 27 molecules per micelle, and a critical micelle concentration of 14 mM. Differences in translational diffusion rates for detergent and long chain phospholipids in mixed micelles are attributed to rapid exchange between free and micelle-bound detergent. This difference permits determination of the free detergent concentration, which, for a high detergent to long chain phospholipid molar ratio, is found to depend strongly on this ratio. The hydrodynamic volume of DHPC/POPC bicelles, loaded with an M2 channel peptide homolog, derived from translational diffusion, predicts a rotational correlation time that slightly exceeds the value obtained from peptide (15)N relaxation data.     

Photon correlation spectroscopy of human IgG

The translational diffusion coefficient D _20,w ⁰ , of monomeric human immunoglobulin G (IgG) has been studied by photon-correlation spectroscopy as a function of pH and protein concentration. At pH 7.6, we find D _20,w ⁰ =3.89×10^–7±0.02 cm²/sec, in good agreement with the value determined by classic mehods. This value corresponds to an effective hydrodynamic radius R, of 55.1±0.3 Å. As pH is increased to 8.9; with the same ionic strength, the molecule appears to expand slightly (3.5% increase in hydrodynamic radius). The concentration dependence of the IgG diffusion constant is interpreted in terms of solution electrostatic effects and shows that long-range repulsive interactions are negligible in the buffer used. The diffusion coefficient for dimeric IgG has also been determined to be D_20,w=2.81×10^–7±0.04 cm²/sec at 1.6 mg/ml, which corresponds to a hydrodynamic radius of 75 Å. For light-scattering studies of protein molecules in the dimension range of 5–10 nm (M_r=10⁵–10⁷) we find monomeric horse spleen ferritin well suited as a reference standard. Ferritin is a spherical molecule with a hydrodynamic radius R of 6.9±0.1 nm and is stable for years in our standard Tris-HCl-NaCl buffer even at room temperature.     

Protein-protein association in polymer solutions: from dilute to semidilute to concentrated

In a typical cell, proteins function in the crowded cytoplasmic environment where 30% of the space is occupied by macromolecules of varying size and nature. This environment may be simulated in vitro using synthetic polymers. Here, we followed the association and diffusion rates of TEM1-beta-lactamase (TEM) and the beta-lactamase inhibitor protein (BLIP) in the presence of crowding agents of varying molecular mass, from monomers (ethylene glycol, glycerol, or sucrose) to polymeric agents such as different polyethylene glycols (PEGs, 0.2-8 kDa) and Ficoll. An inverse linear relation was found between translational diffusion of the proteins and viscosity in all solutions tested, in accordance with the Stokes-Einstein (SE) relation. Conversely, no simple relation was found between either rotational diffusion rates or association rates (k(on)) and viscosity. To assess the translational diffusion-independent steps along the association pathway, we introduced a new factor, alpha, which corrects the relative change in k(on) by the relative change in solution viscosity, thus measuring the deviations of the association rates from SE behavior. We found that these deviations were related to the three regimes of polymer solutions: dilute, semidilute, and concentrated. In the dilute regime PEGs interfere with TEM-BLIP association by introducing a repulsive force due to solvophobic preferential hydration, which results in slower association than predicted by the SE relation. Crossing over from the dilute to the semidilute regime results in positive deviations from SE behavior, i.e., relatively faster association rates. These can be attributed to the depletion interaction, which results in an effective attraction between the two proteins, winning over the repulsive force. In the concentrated regime, PEGs again dramatically slow down the association between TEM and BLIP, an effect that does not depend on the physical dimensions of PEGs, but rather on their mass concentration. This is probably a manifestation of the monomer-like repulsive depletion effect known to occur in concentrated polymer solutions. As a transition from moderate to high crowding agent concentration can occur in the cellular milieu, this behavior may modulate protein association in vivo, thereby modulating biological function.     

Size-dependent diffusion of membrane inclusions

Experimentally determined diffusion constants are often used to elucidate the size and oligomeric state of membrane proteins and domains. This approach critically relies on the knowledge of the size-dependence of diffusion. We have used mesoscopic simulations to thoroughly quantify the size-dependent diffusion properties of membrane inclusions. For small radii R, we find that the lateral diffusion coefficient D is well described by the Saffman-Delbrück relation, which predicts a logarithmic decrease of D with R. However, beyond a critical radius Rc approximately hetam/(2etac) (h, bilayer thickness; etam/c, viscosity of the membrane/surrounding solvent) we observe significant deviations and the emergence of an asymptotic scaling D approximately 1/R2. The latter originates from the asymptotic hydrodynamics and the inclusion's internal degrees of freedom that become particularly relevant on short timescales. In contrast to the lateral diffusion, the size dependence of the rotational diffusion constant Dr follows the predicted hydrodynamic scaling Dr approximately 1/R2 over the entire range of sizes studied here.