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

Orientationally Averaged Version of the Rotne-Prager-Yamakawa Tensor Provides a Fast but Still Accurate Treatment of Hydrodynamic Interactions in Brownian Dynamics Simulations of Biological Macromolecules

The Brownian dynamics (BD) simulation technique is widely used to model the diffusive and conformational dynamics of complex systems comprising biological macromolecules. For the diffusive properties of macromolecules to be described correctly by BD simulations, it is necessary to include hydrodynamic interactions (HIs). When modeled at the Rotne-Prager-Yamakawa (RPY) level of theory, for example, the translational and rotational diffusion coefficients of isolated macromolecules can be accurately reproduced; when HIs are neglected, however, diffusion coefficients can be underestimated by an order of magnitude or more. The principal drawback to the inclusion of HIs in BD simulations is their computational expense, and several previous studies have sought to accelerate their modeling by developing fast approximations for the calculation of the correlated random displacements. Here, we explore the use of an alternative way to accelerate the calculation of HIs, i.e., by replacing the full RPY tensor with an orientationally averaged (OA) version which retains the distance dependence of the HIs but averages out their orientational dependence. We seek here to determine whether such an approximation can be justified in application to the modeling of typical proteins and RNAs. We show that the use of an OA-RPY tensor allows translational diffusion of macromolecules to be modeled with very high accuracy at the cost of rotational diffusion being underestimated by ∼25%. We show that this finding is independent of the type of macromolecule simulated and the level of structural resolution employed in the models. We also show, however, that these results are critically dependent on the inclusion of a non-zero term that describes the divergence of the diffusion tensor: when this term is omitted from simulations that use the OA-RPY model, unfolded macromolecules undergo rapid collapse. Our results indicate that the orientationally averaged RPY tensor is likely to be a useful, fast, approximate way of including HIs in BD simulations of intermediate-scale systems.

Effects of Homogeneous and Heterogeneous Crowding on Translational Diffusion of Rigid Bovine Serum Albumin and Disordered Alfa-Casein

Intracellular environment includes proteins, sugars, and nucleic acids interacting in restricted media. In the cytoplasm, the excluded volume effect takes up to 40% of the volume available for occupation by macromolecules. In this work, we tested several approaches modeling crowded solutions for protein diffusion. We experimentally showed how the protein diffusion deviates from conventional Brownian motion in artificial conditions modeling the alteration of medium viscosity and rigid spatial obstacles. The studied tracer proteins were globular bovine serum albumin and intrinsically disordered α-casein. Using the pulsed field gradient NMR, we investigated the translational diffusion of protein probes of different structures in homogeneous (glycerol) and heterogeneous (PEG 300/PEG 6000/PEG 40,000) solutions as a function of crowder concentration. Our results showed fundamentally different effects of homogeneous and heterogeneous crowded environments on protein self-diffusion. In addition, the applied "tracer on lattice" model showed that smaller crowding obstacles (PEG 300 and PEG 6000) create a dense net of restrictions noticeably hindering diffusing protein probes, whereas the large-sized PEG 40,000 creates a "less restricted" environment for the diffusive motion of protein molecules.

An Orientationally Averaged Version of the Rotne-Prager-Yamakawa Tensor Provides A Fast But Still Accurate Treatment Of Hydrodynamic Interactions In Brownian Dynamics Simulations Of Biological Macromolecules

The Brownian dynamics (BD) simulation technique is widely used to model the diffusive and conformational dynamics of complex systems comprising biological macromolecules. For the diffusive properties of macromolecules to be described correctly by BD simulations, it is necessary to include hydrodynamic interactions (HI). When modeled at the Rotne-Prager-Yamakawa (RPY) level of theory, for example, the translational and rotational diffusion coefficients of isolated macromolecules can be accurately reproduced; when HIs are neglected, however, diffusion coefficients can be underestimated by an order of magnitude or more. The principal drawback to the inclusion of HIs in BD simulations is their computational expense, and several previous studies have sought to accelerate their modeling by developing fast approximations for the calculation of the correlated random displacements. Here we explore the use of an alternative way to accelerate calculation of HIs, i.e., by replacing the full RPY tensor with an orientationally averaged (OA) version which retains the distance dependence of the HIs but averages out their orientational dependence. We seek here to determine whether such an approximation can be justified in application to the modeling of typical proteins and RNAs. We show that the use of an OA RPY tensor allows translational diffusion of macromolecules to be modeled with very high accuracy at the cost of rotational diffusion being underestimated by ∼25%. We show that this finding is independent of the type of macromolecule simulated and the level of structural resolution employed in the models. We also show, however, that these results are critically dependent on the inclusion of a non-zero term that describes the divergence of the diffusion tensor: when this term is omitted from simulations that use the OA RPY model, unfolded macromolecules undergo rapid collapse. Our results indicate that the orientationally averaged RPY tensor is likely to be a useful, fast approximate way of including HIs in BD simulations of intermediate-scale systems.

Effect of Protein–Protein Interactions on Translational Diffusion of Spheroidal Proteins

One of the commonly accepted approaches to estimate protein–protein interactions (PPI) in aqueous solutions is the analysis of their translational diffusion. The present review article observes a phenomenological approach to analyze PPI effects via concentration dependencies of self- and collective translational diffusion coefficient for several spheroidal proteins derived from the pulsed field gradient NMR (PFG NMR) and dynamic light scattering (DLS), respectively. These proteins are rigid globular α-chymotrypsin (ChTr) and human serum albumin (HSA), and partly disordered α-casein (α-CN) and β-lactoglobulin (β-Lg). The PPI analysis enabled us to reveal the dominance of intermolecular repulsion at low ionic strength of solution (0.003–0.01 M) for all studied proteins. The increase in the ionic strength to 0.1–1.0 M leads to the screening of protein charges, resulting in the decrease of the protein electrostatic potential. The increase of the van der Waals potential for ChTr and α-CN characterizes their propensity towards unstable weak attractive interactions. The decrease of van der Waals interactions for β-Lg is probably associated with the formation of stable oligomers by this protein. The PPI, estimated with the help of interaction potential and idealized spherical molecular geometry, are in good agreement with experimental data.

Temperature Dependence of Protein Solution Viscosity and Protein-Protein Interactions: Insights into the Origins of High-Viscosity Protein Solutions

Protein solution viscosity (η) as a function of temperature was measured at a series of protein concentrations under a range of formulation conditions for two monoclonal antibodies (MAbs) and a globular protein (aCgn). Based on theoretical arguments, a strong temperature dependence for protein-protein interactions (PPI) indicates highly anisotropic, short-ranged attractions that could lead to higher solution viscosities. The semi-empirical Ross-Minton model was used to determine the apparent intrinsic viscosity, shape, and "crowding" factors for each protein as a function of temperature and formulation conditions. The apparent intrinsic viscosity was independent of temperature for aCgn, while a slight decrease with increasing temperature was observed for the MAbs. The temperature dependence of solution viscosity was analyzed using the Andrade-Eyring equation to determine the effective activation energy of viscous flow (E_a,η). While E_a,η values were different for each protein, they were independent of formulation conditions for a given protein. PPI were quantified via the osmotic second virial coefficient (B₂₂) and the protein diffusion interaction parameter (k_D) as a function of temperature under the same formulation conditions as the viscosity measurements. Net interactions ranged from strongly attractive to repulsive by changing formulation pH and ionic strength for each protein. Overall, larger activation energies for PPI corresponded to larger activation energies for η, and those were predictive of the highest η values at higher protein concentrations.

Evolutionary drivers of protein shape

Diffusional motion within the crowded environment of the cell is known to be crucial to cellular function as it drives the interactions of proteins. However, the relationships between protein diffusion, shape and interaction, and the evolutionary selection mechanisms that arise as a consequence, have not been investigated. Here, we study the dynamics of triaxial ellipsoids of equivalent steric volume to proteins at different aspect ratios and volume fractions using a combination of Brownian molecular dynamics and geometric packing. In general, proteins are found to have a shape, approximately Golden in aspect ratio, that give rise to the highest critical volume fraction resisting gelation, corresponding to the fastest long-time self-diffusion in the cell. The ellipsoidal shape also directs random collisions between proteins away from sites that would promote aggregation and loss of function to more rapidly evolving nonsticky regions on the surface, and further provides a greater tolerance to mutation.

Mechanism-Based Mathematical Model for Gating of Ionotropic Glutamate Receptors

We present a mathematical model for ionotropic glutamate receptors (iGluR's) that is built on mechanistic understanding and yields a number of thermodynamic and kinetic properties of channel gating. iGluR's are ligand-gated ion channels responsible for the vast majority of fast excitatory neurotransmission in the central nervous system. The effects of agonist-induced closure of the ligand-binding domain (LBD) are transmitted to the transmembrane channel (TMC) via interdomain linkers. Our model demonstrates that, relative to full agonists, partial agonists may reduce either the degree of LBD closure or the curvature of the LBD free energy basin, leading to less stabilization of the channel open state and hence lower channel open probability. A rigorous relation is derived between the channel closed-to-open free energy difference and the tension within the linker. Finally, by treating LBD closure and TMC opening as diffusive motions, we obtain gating trajectories that resemble stochastic current traces from single-channel recordings and calculate the rate constants for transitions between the channel open and closed states. Our model can be implemented by molecular dynamics simulations to realistically depict iGluR gating and may guide functional experiments in gaining deeper insight into this essential family of channel proteins.

A Multi-Scale Method for Dynamics Simulation in Continuum Solvent Models I: Finite-Difference Algorithm for Navier-Stokes Equation

A multi-scale framework is proposed for more realistic molecular dynamics simulations in continuum solvent models by coupling a molecular mechanics treatment of solute with a fluid mechanics treatment of solvent. This article reports our initial efforts to formulate the physical concepts necessary for coupling the two mechanics and develop a 3D numerical algorithm to simulate the solvent fluid via the Navier-Stokes equation. The numerical algorithm was validated with multiple test cases. The validation shows that the algorithm is effective and stable, with observed accuracy consistent with our design.

Rationalizing 5000-fold differences in receptor-binding rate constants of four cytokines

The four cytokines erythropoietin (EPO), interleukin-4 (IL4), human growth hormone (hGH), and prolactin (PRL) all form four-helix bundles and bind to type I cytokine receptors. However, their receptor-binding rate constants span a 5000-fold range. Here, we quantitatively rationalize these vast differences in rate constants by our transient-complex theory for protein-protein association. In the transient complex, the two proteins have near-native separation and relative orientation, but have yet to form the short-range specific interactions of the native complex. The theory predicts the association rate constant as k(a)=k(a0)exp(-ΔG(el)(∗)/k(B)T) where k(a0) is the basal rate constant for reaching the transient complex by random diffusion, and the Boltzmann factor captures the rate enhancement due to electrostatic attraction. We found that the vast differences in receptor-binding rate constants of the four cytokines arise mostly from the differences in charge complementarity among the four cytokine-receptor complexes. The basal rate constants (k(a0)) of EPO, IL4, hGH, and PRL were similar (5.2 × 10(5) M(-1)s(-1), 2.4 × 10(5) M(-1)s(-1), 1.7 × 10(5) M(-1)s(-1), and 1.7 × 10(5) M(-1)s(-1), respectively). However, the average electrostatic free energies (ΔG(e1)(∗)) were very different (-4.2 kcal/mol, -2.4 kcal/mol, -0.1 kcal/mol, and -0.5 kcal/mol, respectively, at ionic strength=160 mM). The receptor-binding rate constants predicted without adjusting any parameters, 6.2 × 10(8) M(-1)s(-1), 1.3 × 10(7) M(-1)s(-1), 2.0 × 10(5) M(-1)s(-1), and 7.6 × 10(4) M(-1)s(-1), respectively, for EPO, IL4, hGH, and PRL agree well with experimental results. We uncover that these diverse rate constants are anticorrelated with the circulation concentrations of the cytokines, with the resulting cytokine-receptor binding rates very close to the limits set by the half-lives of the receptors, suggesting that these binding rates are functionally relevant and perhaps evolutionarily tuned. Our calculations also reproduced well-observed effects of mutations and ionic strength on the rate constants and produced a set of mutations on the complex of hGH with its receptor that putatively enhances the rate constant by nearly 100-fold through increasing charge complementarity. To quantify charge complementarity, we propose a simple index based on the charge distribution within the binding interface, which shows good correlation with ΔG(e1)(∗). Together these results suggest that protein charges can be manipulated to tune k(a) and control biological function.

Prediction of hydrodynamic and other solution properties of rigid proteins from atomic- and residue-level models

Here we extend the ability to predict hydrodynamic coefficients and other solution properties of rigid macromolecular structures from atomic-level structures, implemented in the computer program HYDROPRO, to models with lower, residue-level resolution. Whereas in the former case there is one bead per nonhydrogen atom, the latter contains one bead per amino acid (or nucleotide) residue, thus allowing calculations when atomic resolution is not available or coarse-grained models are preferred. We parameterized the effective hydrodynamic radius of the elements in the atomic- and residue-level models using a very large set of experimental data for translational and rotational coefficients (intrinsic viscosity and radius of gyration) for >50 proteins. We also extended the calculations to very large proteins and macromolecular complexes, such as the whole 70S ribosome. We show that with proper parameterization, the two levels of resolution yield similar and rather good agreement with experimental data. The new version of HYDROPRO, in addition to considering various computational and modeling schemes, is far more efficient computationally and can be handled with the use of a graphical interface.

A comparison of the activation energy of viscous flow for hen egg-white lysozyme obtained on the basis of different models of viscosity for glass-forming liquids

The paper presents the results of viscosity determinations on aqueous solutions of hen egg-white lysozyme at a wide range of concentrations and at temperatures ranging from 5°C to 55°C. On the basis of these measurements and different models of viscosity for glass-forming liquids, the activation energy of viscous flow for solutions and the studied protein, at different temperatures, was calculated. The analysis of the results obtained shows that the activation energy monotonically decreases with increasing temperature both for solutions and the studied protein. The numerical values of the activation energy for lysozyme, calculated on the basis of discussed models, are very similar in the range of temperatures from 5°C to 35°C.

Insight into the integrase-DNA recognition mechanism. A specific DNA-binding mode revealed by an enzymatically labeled integrase

Integration catalyzed by integrase (IN) is a key process in the retrovirus life cycle. Many biochemical or structural human immunodeficiency virus, type 1 (HIV-1) IN studies have been severely impeded by its propensity to aggregate. We characterized a retroviral IN (primate foamy virus (PFV-1)) that displays a solubility profile different from that of HIV-1 IN. Using various techniques, including fluorescence correlation spectroscopy, time-resolved fluorescence anisotropy, and size exclusion chromatography, we identified a monomer-dimer equilibrium for the protein alone, with a half-transition concentration of 20-30 mum. We performed specific enzymatic labeling of PFV-1 IN and measured the fluorescence resonance energy transfer between carboxytetramethylrhodamine-labeled IN and fluorescein-labeled DNA substrates. FRET and fluorescence anisotropy highlight the preferential binding of PFV-1 IN to the 3'-end processing site. Sequence-specific DNA binding was not observed with HIV-1 IN, suggesting that the intrinsic ability of retroviral INs to bind preferentially to the processing site is highly underestimated in the presence of aggregates. IN is in a dimeric state for 3'-processing on short DNA substrates, whereas IN polymerization, mediated by nonspecific contacts at internal DNA positions, occurs on longer DNAs. Additionally, aggregation, mediated by nonspecific IN-IN interactions, occurs preferentially with short DNAs at high IN/DNA ratios. The presence of either higher order complex is detrimental for specific activity. Ionic strength favors catalytically competent over higher order complexes by selectively disrupting nonspecific IN-IN interactions. This counteracting effect was not observed with polymerization. The synergic effect on the selection of specific/competent complexes, obtained by using short DNA substrates under high salt conditions, may have important implications for further structural studies in IN.DNA complexes.

Electrostatic rate enhancement and transient complex of protein-protein association

The association of two proteins is bounded by the rate at which they, via diffusion, find each other while in appropriate relative orientations. Orientational constraints restrict this rate to approximately 10(5)-10(6) M(-1) s(-1). Proteins with higher association rates generally have complementary electrostatic surfaces; proteins with lower association rates generally are slowed down by conformational changes upon complex formation. Previous studies (Zhou, Biophys J 1997;73:2441-2445) have shown that electrostatic enhancement of the diffusion-limited association rate can be accurately modeled by $k_{\bf D}$ = $k_{D}0\ {exp} ( - \langle U_{el} \rangle;{\star}/k_{B} T),$ where k(D) and k(D0) are the rates in the presence and absence of electrostatic interactions, respectively, U(el) is the average electrostatic interaction energy in a "transient-complex" ensemble, and k(B)T is the thermal energy. The transient-complex ensemble separates the bound state from the unbound state. Predictions of the transient-complex theory on four protein complexes were found to agree well with the experiment when the electrostatic interaction energy was calculated with the linearized Poisson-Boltzmann (PB) equation (Alsallaq and Zhou, Structure 2007;15:215-224). Here we show that the agreement is further improved when the nonlinear PB equation is used. These predictions are obtained with the dielectric boundary defined as the protein van der Waals surface. When the dielectric boundary is instead specified as the molecular surface, electrostatic interactions in the transient complex become repulsive and are thus predicted to retard association. Together these results demonstrate that the transient-complex theory is predictive of electrostatic rate enhancement and can help parameterize PB calculations.

Prediction of protein-protein association rates from a transition-state theory

We recently developed a theory for the rates of protein-protein association. The theory is based on the concept of a transition state, which separates the bound state, with numerous short-range interactions but restricted translational and rotational freedom, and the unbound state, with, at most, a small number of interactions but expanded configurational freedom. When not accompanied by large-scale conformational changes, protein-protein association becomes diffusion limited. The association rate is then predicted as k(a)=k(a)(0)exp(-DeltaG(el)(double dagger)/k(B)T), where DeltaG(el)(double dagger) is the electrostatic interaction free energy in the transition state, k(a)(0) is the rate in the absence of electrostatic interactions, and k(B)T is thermal energy. Here, this transition-state theory is used to predict the association rates of four protein complexes. The predictions for the wild-type complexes and 23 mutants are found to agree closely with experimental data over wide ranges of ionic strength.

The effect of the layer structure on the activity of immobilized enzymes in ultrathin films

The molecular engineering capability of the layer-by-layer (LbL) method for fabricating thin films has been exploited in order to immobilize glucose oxidase (GOD) in films with alternating layers of chitosan. Chitosan was proven a good scaffolding material, as GOD molecules preserved their catalytic activity towards glucose oxidation. Using electrochemical measurements, we showed that chitosan/GOD LbL films can be used to detect glucose with a limit of detection of 0.2 mmol l-1 and an activity of 40.5 microA mmol-1 L microg-1, which is highly sensitive when compared to other sensors in previous reports in the literature. The highest sensitivity of the LbL film was achieved when only the top layer contained GOD, thus indicating that GOD in inner layers did not contribute to glucose oxidation, probably because it hampers analyte diffusion and electron transport through the deposited layers. This may be explained by the dense packing of GOD molecules in the LbL films with chitosan, as inferred from estimates of the amount of GOD adsorbed per layer using a quartz crystal microbalance.

Precise boundary element computation of protein transport properties: Diffusion tensors, specific volume, and hydration

A precise boundary element method for the computation of hydrodynamic properties has been applied to the study of a large suite of 41 soluble proteins ranging from 6.5 to 377 kDa in molecular mass. A hydrodynamic model consisting of a rigid protein excluded volume, obtained from crystallographic coordinates, surrounded by a uniform hydration thickness has been found to yield properties in excellent agreement with experiment. The hydration thickness was determined to be delta = 1.1 +/- 0.1 A. Using this value, standard deviations from experimental measurements are: 2% for the specific volume; 2% for the translational diffusion coefficient, and 6% for the rotational diffusion coefficient. These deviations are comparable to experimental errors in these properties. The precision of the boundary element method allows the unified description of all of these properties with a single hydration parameter, thus far not achieved with other methods. An approximate method for computing transport properties with a statistical precision of 1% or better (compared to 0.1-0.2% for the full computation) is also presented. We have also estimated the total amount of hydration water with a typical -9% deviation from experiment in the case of monomeric proteins. Both the water of hydration and the more precise translational diffusion data hint that some multimeric proteins may not have the same solution structure as that in the crystal because the deviations are systematic and larger than in the monomeric case. On the other hand, the data for monomeric proteins conclusively show that there is no difference in the protein structure going from the crystal into solution.

How thick is the layer of thermal volume surrounding the protein?

Investigation on the volume properties of protein hydration layers is reported. Presented results are based on combination of Monte Carlo modeling and available experimental data. Six globular proteins with known data are chosen for analysis. Analyzing the model and the experimental results we found that water molecules bound to proteins by hydrogen bond are preferentially located at the places with local depressions on the protein surface. Consequently, the hydration level is not strictly proportional to the area of charged and polar surfaces, but also depends on the shape of the molecular surface. The thickness of the thermal volume layer as calculated in the framework of the scaled particle theory is 0.6-0.65 A for chosen proteins. The obtained value is significantly lower than that presented for proteins in earlier papers (where proportionality between the hydration level and the area of charged and polar surfaces was assumed), but is close to the value published for small solute molecules. Discussion including the influence of protein size and the thermal motion of the surface is presented.

Characterization of the nonregular regions of proteins by a contortion index

Nonstructured regions in proteins that provide the link between two regular structured regions play a significant role in maintaining the scaffold of the protein. Not only do they act as connectors between two regular secondary structural elements of proteins but they also provide the necessary turn or reversal in the polypeptide chain. This incorporates flexibility in the structure. Thus an understanding of the structural aspects of the nonregular regions is necessary to have a better insight into these features. We can assume the nonregular region to be a contorted polypeptide segment tethered by regular secondary structured regions at both ends. To describe the undulating nature of the nonregular regions, we introduce a parameter called the "contortion index." This index describes how tortuously the region is organized. Our analysis shows that the contortion index is related to other physicochemical parameters and can be used to characterize the nonregular regions of proteins.

SOMO (SOlution MOdeler) differences between X-Ray- and NMR-derived bead models suggest a role for side chain flexibility in protein hydrodynamics

Reduced numbers of frictional/scattering centers are essential for tractable hydrodynamic and small-angle scattering data modeling. We present a method for generating medium-resolution models from the atomic coordinates of proteins, basically by using two nonoverlapping spheres of differing radii per residue. The computed rigid-body hydrodynamic parameters of BPTI, RNase A, and lysozyme models were compared with a large database of critically assessed experimental values. Overall, very good results were obtained, but significant discrepancies between X-ray- and NMR-derived models were found. Interestingly, they could be accounted for by properly considering the extent to which highly mobile surface side chains differently affect translational/rotational properties. Models of larger structures, such as fibrinogen fragment D and citrate synthase, also produced consistent results. Foremost among this method's potential applications is the overall conformation and dynamics of modular/multidomain proteins and of supramolecular complexes. The possibility of merging data from high- and low-resolution structures greatly expands its scope.

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.

Comparative investigations of biopolymer hydration by physicochemical and modeling techniques

The comparative investigation of biopolymer hydration by physicochemical techniques, particularly by small-angle X-ray scattering, has shown that the values obtained differ over a wide range, depending on the nature of the polymer and the environmental conditions. In the case of simple proteins, a large number of available data allow the derivation of a realistic average value for the hydration (0.35 g of water per gram of protein). As long as the average properties of proteins are considered, the use of such a default value is sufficient. Modeling approaches may be used advantageously, in order to differentiate between different assumptions and hydration contributions, and to correctly predict hydrodynamic properties of biopolymers on the basis of their three-dimensional structure. Problems of major concern are the positioning and the properties of the water molecules on the biopolymer surface. In this context, different approaches for calculating the molecular volume and surface of biopolymers have been applied, in addition to the development of appropriate hydration algorithms.

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.

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.

Calculation of hydrodynamic properties of globular proteins from their atomic-level structure

The solution properties, including hydrodynamic quantities and the radius of gyration, of globular proteins are calculated from their detailed, atomic-level structure, using bead-modeling methodologies described in our previous article (, Biophys. J. 76:3044-3057). We review how this goal has been pursued by other authors in the past. Our procedure starts from a list of atomic coordinates, from which we build a primary hydrodynamic model by replacing nonhydrogen atoms with spherical elements of some fixed radius. The resulting particle, consisting of overlapping spheres, is in turn represented by a shell model treated as described in our previous work. We have applied this procedure to a set of 13 proteins. For each protein, the atomic element radius is adjusted, to fit all of the hydrodynamic properties, taking values close to 3 A, with deviations that fall within the error of experimental data. Some differences are found in the atomic element radius found for each protein, which can be explained in terms of protein hydration. A computational shortcut makes the procedure feasible, even in personal computers. All of the model-building and calculations are carried out with a HYDROPRO public-domain computer program.

Solution structure of biopolymers: a new method of constructing a bead model

We propose a new, automated method of converting crystallographic data into a bead model used for the calculations of hydrodynamic properties of rigid macromolecules. Two types of molecules are considered: nucleic acids and small proteins. A bead model of short DNA fragments has been constructed in which each nucleotide is represented by two identical, partially overlapping spheres: one for the base and one for the sugar and phosphate group. The optimum radius sigma = 5.0 A was chosen on the basis of a comparison of the calculated translational diffusion coefficients (D(T)) and the rotational relaxation times (tau(R)) with the corresponding experimental data for B-DNA fragments of 8, 12, and 20 basepairs. This value was assumed for the calculation D(T) and tau(R) of tRNA(Phe). Better agreement with the experimental data was achieved for slightly larger sigma = 5.7 A. A similar procedure was applied to small proteins. Bead models were constructed such that each amino acid was represented by a single sphere or a pair of identical, partially overlapping spheres, depending on the amino acid's size. Experimental data of D(T) of small proteins were used to establish the optimum value of sigma = 4.5 A for amino acids. The lack of experimental data on tau(R) for proteins restricted the tests to the translational diffusion properties.

Novel size-independent modeling of the dilute solution conformation of the immunoglobulin IgG Fab' domain using SOLPRO and ELLIPS

The proliferation of hydrodynamic modeling strategies to represent the shape of quasirigid macromolecules in solution has been hampered by ambiguities caused by size. Universal shape parameters, independent of size, developed originally for ellipsoid modeling, are now available for modeling using the bead-shell approximation via the algorithm SOLPRO. This paper validates such a "size-independent" bead-shell approach by comparison with the exact hydrodynamics of 1) an ellipsoid of revolution and 2) a general triaxial ellipsoid (semiaxial ratios a/b, b/c) based on a fit using the routine ELLIPSE (. J. Mol. Graph. 1:30-38) to the chimeric (human/mouse) IgG Fab' B72.3; a similar fit is obtained for other Fabs. Size-independent application of the bead-shell approximation yields errors of only approximately 1% in frictional ratio based shape functions and approximately 3% in the radius of gyration. With the viscosity increment, errors have been reduced to approximately 3%, representing a significant improvement on earlier procedures. Combination of the Perrin frictional ratio function with the experimentally measured sedimentation coefficient for the same Fab' from B72.3 yields an estimate for the molecular hydration of the Fab' fragment of approximately (0.43 +/- 0.07) g/g. This value is compared to values obtained in a similar way for deoxyhemoglobin (0.44) and ribonuclease (0.27). The application of SOLPRO to the shape analysis of more complex macromolecules is indicated, and we encourage such size-independent strategies. The utility of modern sedimentation data analysis software such as SVEDBERG, DCDT, LAMM, and MSTAR is also clearly demonstrated.

COVOL: an interactive program for evaluating second virial coefficients from the triaxial shape or dimensions of rigid macromolecules

An interactive program is described for calculating the second virial coefficient contribution to the thermodynamic nonideality of solutions of rigid macromolecules based on their triaxial dimensions. The FORTRAN-77 program, available in precompiled form for the PC, is based on theory for the covolume of triaxial ellipsoid particles [Rallison, J. M., and S.E Harding. (1985). J. Colloid Interface Sci. 103:284-289]. This covolume has the potential to provide a magnitude for the second virial coefficient of macromolecules bearing no net charge. Allowance for a charge-charge contribution is made via an expression based on Debye-Hückel theory and uniform distribution of the net charge over the surface of a sphere with dimensions governed by the Stokes radius of the macromolecule. Ovalbumin, ribonuclease A, and hemoglobin are used as model systems to illustrate application of the COVOL routine.

Bead modeling using HYDRO and SOLPRO of the conformation of multisubunit proteins: sunflower and rape-seed 11S globulins

Oil seed globulins from sunflower and rape seed are multi-subunit, oligomeric proteins whose native 11S form is a hexamer. In this work we try to determine the spatial structure in which the six subunits of 11S globulin are arranged. Experimental values of solution properties, including radius of gyration, sedimentation and diffusion coefficients and intrinsic viscosity, are compared with theoretical predictions for hexamers of various geometries. Bead model calculations of solution properties are carried out using the HYDRO and SOLPRO computer programs. A most compact shape, the regular octahedron, is the hexameric structure that fits best the experimental values.

An empirical relationship between rotational correlation time and solvent accessible surface area

Structure-dynamics interrelationships are important in understanding protein function. We have explored the empirical relationship between rotational correlation times (tau(c) and the solvent accessible surface areas (SASA) of 75 proteins with known structures. The theoretical correlation between SASA and tau(c) through the equation SASA = K(r)tau(c) ((2/3)) is also considered. SASA was determined from the structure, tau(c) (calc) was determined from diffusion tensor calculations, and tau(c) (expt) was determined from NMR backbone(13) C or (15)N relaxation rate measurements. The theoretical and experimental values of tau(c) correlate with SASA with regression analyses values of K(r) as 1696 and 1896 m(2)s(-(2/3)), respectively, and with corresponding correlation coefficients of 0.92 and 0.70.

Electrostatic enhancement of diffusion-controlled protein-protein association: comparison of theory and experiment on barnase and barstar

The electrostatic enhancement of the association rate of barnase and barstar is calculated using a transition-state theory like expression and atomic-detail modeling of the protein molecules. This expression predicts that the rate enhancement is simply the average Boltzmann factor in the region of configurational space where association occurs instantaneously in the diffusion-controlled limit. Based on experimental evidence, this "transition state" is defined by configurations in which, relative to the stereospecifically bound complex, the two proteins are shifted apart by approximately 8 A (so a layer of water can be accommodated in the interface) and the two binding surfaces are rotated away by 0 degrees to 3 degrees. The values of the average Boltzmann factor, calculated by solving the Poisson-Boltzmann equation, for the wild-type complex and 16 complexes with single mutations are found to correlate well with experimental results for the electrostatic rate enhancement. The predicted rate enhancement is found to be somewhat insensitive to the precise definition of the transition state, due to the long-range nature of electrostatic interactions. The experimental ionic strength dependence of the rate enhancement is also reasonably reproduced.

Concentration and temperature dependence of viscosity in lysozyme aqueous solutions

The paper presents the results of viscosity determinations on aqueous solutions of hen egg-white lysozyme at a wide range of concentrations and at temperatures ranging from 5 degrees C to 55 degrees C. It has been proved that, at each fixed concentration, the viscosity-temperature dependence may be quantitatively described by the modified Arrhenius formula. On the basis of the generalized Arrhenius formula, the parameters of the Mooney approximation were calculated. It has been concluded that lysozyme molecules in aqueous solution behave as hard quasi-spherical particles. By applying an asymptotic form of the generalized Arrhenius formula, such rheological quantities as the intrinsic viscosity and Huggins coefficient were calculated.

Modeling of protein conformational fluctuations in pKa predictions

A method is presented to account for conformational fluctuations of a protein in predicting the pK(a) values of its titrating groups. Conformations of the protein are generated by conventional molecular dynamics or Monte Carlo simulations, in which the protonations of the titrating groups are fixed. For each protein conformation, the electrostatic free energies required to add a proton charge to a titrating group while other groups are either unprotonated or protonated are calculated within a dielectric continuum model. These are used to determine the mean protonations of the titrating groups in the conformation at a series of pH values. The mean protonations are then used to determine the relative weight of the particular conformation with the titrating groups having all possible protonations. A conformationally averaged mean protonation for each titrating group is finally obtained by the weighted sum of the group's mean protonations in all the conformations. This method is applied to yeast iso-1-ferricytochrome c. The predicted pK(a) values are in general agreement with experimental results.

A pulsed field gradient NMR study of the aggregation and hydration of parvalbumin

Pulsed field gradient NMR is a convenient alternative to traditional methods for measuring diffusion of biological macromolecules. In the present study, pulsed field gradient NMR was used to study the effects of calcium binding and hydration on carp parvalbumin. Carp parvalbumin is known to undergo large changes in tertiary structure with calcium loading. The diffusion coefficient is a sensitive guide to changes in molecular shape and in the present study the large changes in tertiary structure were clearly reflected in the measured diffusion coefficient upon calcium loading. The (monomeric) calcium-loaded form had a diffusion coefficient of 1.4 x 10(-10) m(2) s(-1) at 298 K, which conforms with the structure being a nearly spherical prolate ellipsoid from X-ray studies. The calcium-free form had a significantly lower diffusion coefficient of 1.1 x 10(-10) m(2) s(-1). The simplest explanation consistent with the change in diffusion coefficient is that the parvalbumin molecules form dimers upon the removal of Ca(2+) at the protein concentration studied (1 mM).