Hydrodynamic multibead modeling: problems, pitfalls, and solutions. 1. Ellipsoid models

The shape of macromolecules can be approximated by filling models, if both hydrodynamic and scattering properties should be predicted. Modeling of complex biological macromolecules, such as oligomeric proteins, or of molecule details calls for usage of many beads to preserve the original features. However, the calculation of precise values for structural and hydrodynamic parameters has to consider many problems and pitfalls. Among these, the huge number of beads required for modeling details and the choice of appropriate volume corrections for the calculation of intrinsic viscosities are pestering problems to date. As a first step to tackle these problems, various tests with multibead models (ellipsoids of different axial ratios) were performed. The agreement of the predicted molecular properties with those derived from whole-body approaches can be used as evaluation criteria. Modification of previously available versions of García de la Torre’s program HYDRO allows hydrodynamic modeling of macromolecules composed of a maximum of about 11,000 beads. Moreover, application of our recently suggested “reduced volume correction” enables a fast and efficient anticipation of intrinsic viscosities. Correct parameter predictions were obtained for all models analyzed. The data obtained were compared to the results of calculations based on HYDRO programs available to the public. The calculations revealed some unexpected results and allowed founded conclusions of general importance for precise calculations on multibead models (e.g., the requirement of calculations in the double-precision mode).     

Intrinsic viscosity of bead models for macromolecules and bioparticles

A new method based on the fractal dimension dependence of the hydrodynamic radius is proposed for calculation of the intrinsic viscosity of bead models. The method describes properly the viscosity increment except for elongated structures such as linear aggregates and ellipsoids. It is expected to be useful for very compact structures, for which the volume correction does not improve the results calculated by the modified Oseen tensor. The results obtained for the viscosity increment lie between the volume corrected ones and those determined by the cubic substitution procedure. They are close to the values recalculated from the falling velocities of the models analyzed.     

Hydrodynamic multibead modeling: problems, pitfalls, and solutions. 2. Proteins

The shape of macromolecules can be approximated by filling models, if both hydrodynamic and scattering properties should be predicted. Modeling of complex biological macromolecules, such as oligomeric proteins, or of molecule details calls for usage of many beads to preserve the original features. However, the calculation of precise values for structural and hydrodynamic parameters has to consider many problems and pitfalls. Among these, the huge number of beads required for modeling details and the choice of appropriate volume corrections for the calculation of intrinsic viscosities are pestering problems to date. As a first step to tackle these problems, various tests with multibead models (ellipsoids of different axial ratios) were performed. The agreement of the predicted molecular properties with those derived from whole-body approaches can be used as evaluation criteria. Modification of previously available versions of García de la Torre’s program HYDRO allows hydrodynamic modeling of macromolecules composed of a maximum of about 11,000 beads. Moreover, application of our recently suggested “reduced volume correction” enables a fast and efficient anticipation of intrinsic viscosities. Correct parameter predictions were obtained for all models analyzed. The data obtained were compared to the results of calculations based on HYDRO programs available to the public. The calculations revealed some unexpected results and allowed founded conclusions of general importance for precise calculations on multibead models (e.g., the requirement of calculations in the double-precision mode).     

Hydrodynamic multibead modeling: problems, pitfalls and solutions. 3. Comparison of new approaches for improved predictions of translational properties

Modeling simple and complex biopolymers in solution requires the shapes of these molecules to be approximated by bead modeling procedures, primarily for the prediction of hydrodynamic and scattering quantities. Though several bead modeling strategies (strict, shell and filling models) and a variety of computer programs (preferably the HYDRO suite by the García de la Torre group) are available, several subtle questions remain to be answered, in particular concerning the appropriate volume correction for intrinsic viscosity computations. In this context, various versions of the HYDRO programs and different types of volume corrections, as well as the novel, alternative program ZENO of the Mansfield group, were applied to a plethora of thoroughly designed multibead models of spherical, ellipsoidal, cylindrical and prismatic shapes. A critical comparison of the results obtained reveals a variety of new aspects, useful for many future applications. Among these, application of our recently suggested “reduced volume correction” (RVC) together with specially adapted HYDRO versions and use of ZENO turned out to be highly effective, in particular when aiming at filling model strategies and using high bead numbers, a domain not fully supported by the recent HYDRO++ versions. By our approaches, the values of translational properties (diffusion coefficients, D, and intrinsic viscosities, [η]) of all multibead models applied were anticipated correctly.     

The Limitations of an Exclusively Colloidal View of Protein Solution Hydrodynamics and Rheology

Proteins are complex macromolecules with dynamic conformations. They are charged like colloids, but unlike colloids, charge is heterogeneously distributed on their surfaces. Here we overturn entrenched doctrine that uncritically treats bovine serum albumin (BSA) as a colloidal hard sphere by elucidating the complex pH and surface hydration-dependence of solution viscosity. We measure the infinite shear viscosity of buffered BSA solutions in a parameter space chosen to tune competing long-range repulsions and short-range attractions (2 mg/mL ≤ [BSA] ≤ 500 mg/mL and 3.0 ≤ pH ≤ 7.4). We account for surface hydration through partial specific volume to define volume fraction and determine that the pH-dependent BSA intrinsic viscosity never equals the classical hard sphere result (2.5). We attempt to fit our data to the colloidal rheology models of Russel, Saville, and Schowalter (RSS) and Krieger-Dougherty (KD), which are each routinely and successfully applied to uniformly charged suspensions and to hard-sphere suspensions, respectively. We discover that the RSS model accurately describes our data at pH 3.0, 4.0, and 5.0, but fails at pH 6.0 and 7.4, due to steeply rising solution viscosity at high concentration. When we implement the KD model with the maximum packing volume fraction as the sole floating parameter while holding the intrinsic viscosity constant, we conclude that the model only succeeds at pH 6.0 and 7.4. These findings lead us to define a minimal framework for models of crowded protein solution viscosity wherein critical protein-specific attributes (namely, conformation, surface hydration, and surface charge distribution) are addressed.     

Modeling Complex Biological Macromolecules: Reduction of Multibead Models

The shape of simple and complex biological macromolecules can be approximated by bead modeling procedures. Such approaches are required, for example, for the analysis of the scattering and hydrodynamic behavior of the models under analysis and the prediction of their molecular properties. Using the atomic coordinates of proteins for modeling inevitably leads to models composed of a multitude of beads. In particular, for hydrodynamic modeling, a drastic reduction of the bead number may become unavoidable to enable computation. A systematic investigation of different approaches and computation modes shows that the ‘running mean’, ‘cubic grid,’ and ‘hexagonal grid’ approaches are successful, provided that the extent of reduction does not exceed a factor of 100 and the grid approaches use beads of unequal size and the beads are located at the centers of gravity. Further precautions to be taken include usage of appropriate interaction tensors for overlapping beads of unequal size and appropriate volume corrections when calculating intrinsic viscosities. The applied procedures were tested with the small protein lysozyme in a case study and were then applied to the huge capsid of the phage fr and its trimeric building block. The appearance of the models and the agreement of molecular properties and distance distribution functions of unreduced and reduced models can be used as evaluation criteria.     

Calculation of hydrodynamic properties of macromolecular bead models with overlapping spheres

For the calculation of hydrodynamic properties of rigid macromolecules using bead modelling, models with overlapping beads of different sizes are used in some applications. The hydrodynamic interaction tensor between unequal overlapping beads is unknown, and an oversimplified treatment with the Oseen tensor may introduce important errors. Here we discuss some aspects of the overlapping problem, and explore an ad hoc form of the interaction tensor, proposed by Zipper and Durchschlag. We carry out a systematic numerical study of the hydrodynamic properties of a two-spheres model, showing how the Zipper-Durchschlag correction removes efficiently the numerical instabilities, and predicts the correct limits. Received: 15 February 1999 / Revised version: 29 April 1999 / Accepted: 11 May 1999     

Influence of the Volume Fraction,Size and Surface Coating of Hard Spheres on the Microstructure and Rheological Properties of Model Mozzarella Cheese

This study investigated the influence of model filler particles (glass beads) on the microstructure and rheological properties of Mozzarella cheese. Model Mozzarella cheese composites with increasing volume fractions of glass beads of various sizes and surface properties were processed in a Rapid Visco Analyser (RVA). Confocal laser scanning microscope images showed that all the hard spheres were dispersed in the protein phase, rather than in the fat phase. Dynamic oscillatory rheology revealed that the volume fraction of the glass beads had a major influence on the complex modulus (G*) of the cheese composites, whereas the size and the coating of the glass beads had no influence. However, the zero shear viscosity (η ₀), measured using the creep-compliance test, was affected by both the size and the volume fraction of the glass beads. This indicated that there were some interactions between the glass beads and the cheese matrix. Filler–matrix interactions played a major role in the fracture properties of the cheese composites. The fracture stress (σ _f ) was highly dependent on the coating and the size of the glass beads. Simple equations for filled gels from the literature fitted well with the experimental results and could be successfully applied for future predictions. According to this study, the transfer of knowledge from filled polymer composites to model cheese appears relevant. This can provide a good basis for designing new dairy product structures.     

Hydrodynamic properties of rigid particles: comparison of different modeling and computational procedures.

The hydrodynamic properties of rigid particles are calculated from models composed of spherical elements (beads) using theories developed by Kirkwood, Bloomfield, and their coworkers. Bead models have usually been built in such a way that the beads fill the volume occupied by the particles. Sometimes the beads are few and of varying sizes (bead models in the strict sense), and other times there are many small beads (filling models). Because hydrodynamic friction takes place at the molecular surface, another possibility is to use shell models, as originally proposed by Bloomfield. In this work, we have developed procedures to build models of the various kinds, and we describe the theory and methods for calculating their hydrodynamic properties, including approximate methods that may be needed to treat models with a very large number of elements. By combining the various possibilities of model building and hydrodynamic calculation, several strategies can be designed. We have made a quantitative comparison of the performance of the various strategies by applying them to some test cases, for which the properties are known a priori. We provide guidelines and computational tools for bead modeling.     

Determination of diffusion coefficients and diffusion characteristics for chlorferon and diethylthiophosphate in Ca-alginate gel beads

Diffusion characteristics of chlorferon and diethylthiophosphate (DETP) in Ca-alginate gel beads were studied to assist in designing and operating bioreactor systems. Diffusion coefficients for chlorferon and DETP in Ca-alginate gel beads determined at conditions suitable for biodegradation studies were 2.70 x 10(-11) m(2)/s and 4.28 x 10(-11) m(2)/s, respectively. Diffusivities of chlorferon and DETP were influenced by several factors, including viscosity of the bulk solution, agitation speed, and the concentrations of diffusing substrate and immobilized cells. Diffusion coefficients increased with increasing agitation speed, probably due to poor mixing at low speed and some attrition of beads at high speeds. Diffusion coefficients also increased with decreasing substrate concentration. Increased cell concentration in the gel beads caused lower diffusivity. Theoretical models to predict diffusivities as a function of cell weight fraction overestimated the effective diffusivities for both chlorferon and DETP, but linear relations between effective diffusivity and cell weight fraction were derived from experimental data. Calcium-alginate gel beads with radii of 1.65-1.70 mm used in this study were not subject to diffusional limitations: external mass transfer resistances were negligible based on Biot number calculations and effectiveness factors indicated that internal mass transfer resistance was negligible. Therefore, the degradation rates of chlorferon and DETP inside Ca-alginate gel beads were reaction-limited.     

The Limitations of an Exclusively Colloidal View of Protein Solution Hydrodynamics and Rheology

Proteins are complex macromolecules with dynamic conformations. They are charged like colloids, but unlike colloids, charge is heterogeneously distributed on their surfaces. Here we overturn entrenched doctrine that uncritically treats bovine serum albumin (BSA) as a colloidal hard sphere by elucidating the complex pH and surface hydration-dependence of solution viscosity. We measure the infinite shear viscosity of buffered BSA solutions in a parameter space chosen to tune competing long-range repulsions and short-range attractions (2 mg/mL ≤ [BSA] ≤ 500 mg/mL and 3.0 ≤ pH ≤ 7.4). We account for surface hydration through partial specific volume to define volume fraction and determine that the pH-dependent BSA intrinsic viscosity never equals the classical hard sphere result (2.5). We attempt to fit our data to the colloidal rheology models of Russel, Saville, and Schowalter (RSS) and Krieger-Dougherty (KD), which are each routinely and successfully applied to uniformly charged suspensions and to hard-sphere suspensions, respectively. We discover that the RSS model accurately describes our data at pH 3.0, 4.0, and 5.0, but fails at pH 6.0 and 7.4, due to steeply rising solution viscosity at high concentration. When we implement the KD model with the maximum packing volume fraction as the sole floating parameter while holding the intrinsic viscosity constant, we conclude that the model only succeeds at pH 6.0 and 7.4. These findings lead us to define a minimal framework for models of crowded protein solution viscosity wherein critical protein-specific attributes (namely, conformation, surface hydration, and surface charge distribution) are addressed.     

Significance of the surface of immobilized Saccharomyces cerevisiae cells in ethanolic fermentation

S. cerevisiae cells immobilized in alginate beads show in many cases an increase of mean single cell volume during long-time fermentations (successive batch cycles). The biomass loading capacity of the gel beads is characterized by a maximum volume but not by a maximum number of cells occupying the gel volume. In our system this loading capacity, i.e. the maximum volume fraction of cells per volume of beads, amounted to about 0.54. As a more important result it must be stated that the specific product formation rate in the case of fermentations negligibly influenced by diffusion hindrance is related to the total surface of the viable cells but not to their total number, total volume or total dry weight.     

Laser-Induced Heating in Optical Traps

In an optical tweezers experiment intense laser light is tightly focused to intensities of MW/cm² in order to apply forces to submicron particles or to measure mechanical properties of macromolecules. It is important to quantify potentially harmful or misleading heating effects due to the high light intensities in biophysical experiments. We present a model that incorporates the geometry of the experiment in a physically correct manner, including heat generation by light absorption in the neighborhood of the focus, balanced by outward heat flow, and heat sinking by the glass surfaces of the sample chamber. This is in contrast to the earlier simple models assuming heat generation in the trapped particle only. We find that in the most common experimental circumstances, using micron-sized polystyrene or silica beads, absorption of the laser light in the solvent around the trapped particle, not in the particle itself, is the most important contribution to heating. To validate our model we measured the spectrum of the Brownian motion of trapped beads in water and in glycerol as a function of the trapping laser intensity. Heating both increases the thermal motion of the bead and decreases the viscosity of the medium. We measured that the temperature in the focus increased by 34.2 ± 0.1 K/W with 1064-nm laser light for 2200-nm-diameter polystyrene beads in glycerol, 43.8 ± 2.2 K/W for 840-nm polystyrene beads in glycerol, 41.1 ± 0.7 K/W for 502-nm polystyrene beads in glycerol, and 7.7 ± 1.2 K/W for 500-nm silica beads and 8.1 ± 2.1 K/W for 444-nm silica beads in water. Furthermore, we observed that in glycerol the heating effect increased when the bead was trapped further away from the cover glass/glycerol interface as predicted by the model. We show that even though the heating effect in water is rather small it can have non-negligible effects on trap calibration in typical biophysical experimental circumstances and should be taken into consideration when laser powers of more than 100 mW are used.     

A comparison of five methods for obtaining the intrinsic viscosity of bovine serum albumin

Intrinsic viscosity [η] is a characteristic of proteins and other molecules related directly to their ability to disturb flow and indirectly to their size and shape. It is usually determined by extrapolating reduced viscosity to zero concentration. Four other methods for deriving [η] have been utilized by previous investigators. Studies of the intrinsic viscosity of bovine serum albumin had been carried out two years apart as a test of viscometry technique; the data obtained were used to compare the five methods. Four of the five produced [η] values ranging from 3.92 to 4.21 ml/g. Agreement was good between the two studies. The five methods were compared to each other using linearity of regression, statistical error of determination, effect of varying solvent time, and result obtained in different concentration ranges. By these four criteria, use of the regression of specific fluidity (1 ? 1/η_rel) with concentration was found superior to other methods. Its only deficiency was a requirement that solution density be corrected for at each concentration studied rather than applying a single correction for density after using kinematic viscosity data. All methods for deriving intrinsic viscosity are based on one of three equations; flow is expressed either in terms of reduced viscosity (Huggins), inherent viscosity (Kraemer), or specific fluidity. Of these three equations, specific fluidity is the most closely related both to theoretical analyses and to experimental studies of rigid spheres. There is abundant evidence in past reports that in contrast to rigid spheres, flexible polymers do not produce a linear rise in specific fluidity as their concentration increases, strongly suggesting that their molecular conformation is changing with concentration. A linear relation between fluidity and concentration has been observed for almost all proteins and protein mixtures studied. When this linear relation is present it indicates both that molecular conformation during flow is independent of concentration in the range studied and that the specific fluidity method for deriving intrinsic viscosity is the most appropriate.     

Do organic solvents affect the catalytic properties of lipase? Intrinsic kinetic parameters of lipases in ester hydrolysis and formation in various organic solvents

When it is assumed that organic solvents do not interfere with the binding process nor with the catalytic mechanism, the contribution of substrate-solvent interactions to enzyme kinetics can be accounted for by just replacing substrate concentrations in the equations by thermodynamic activities. It appears from the transformation that only the affinity parameters (K(m), K(sp)) are affected by this. Thus, in theory, the values of these corrected, intrinsic parameters (K(m) (int), k(sp) (int)) and the maximal rate (V(1)) should be equal for all media. This was tested for hydrolysis, transesterification, and esterification reactions catalyzed by pig pancreas lipase and Pseudomonas cepacia lipase in various organic solvents. Correction was carried out via experimentally determined activity coefficients for the substrates in these solvents or, if not feasible, from values in data bases. However, although the kinetic performances of each enzyme in the solvents became much more similar after correction, differences still remained. Analysis of the enzyme suspensions revealed massive particles, which explains the low activity of enzymes in organic solvents. However, no correlation was found between estimates of the amount of catalytically available enzyme (present at the surface of suspended particles or immobilized on beads) and the maximal rates observed. Moreover, the solvents had similar effects on the intrinsic parameters of suspended and immobilized enzyme. The possible causes for the effects of the solvents on the catalytic performance of the enzymes, remaining after correction for solvent-substrate interactions and the amount of participating enzyme, are discussed with respect to the premises on which the correction method is based. (c) 1995 John Wiley & Sons, Inc.     

Sedimentation and intrinsic viscosity behavior of PM2 bacteriophage DNA in alkaline solution

The sedimentation coefficient and intrinsic viscosity of nicked and closed circular PM2 bacteriophage DNA have been measured as a function of pH in the alkaline region. A gradual increase in the sidimentation coefficient, and a corresponding decrease in the intrinsic viscosity, are observed for the superhelical (closed) circle in the pH region from 10.5 to about 10.9. This has been tentatively interpreted in terms of the known dependence of sedimentation coefficient upon the number of superhelical turns. At slightly higher pH values, the curve passes through the minimum (sedimentation coefficient) and maximum (intrinsic viscosity) expected when the superhelical turns present at neutral pH are unwound by partial alkaline denaturation. Sedimentation studies of the relaxed (nicked) circular species have revealed the existence of DNA forms in the pH region from 11.27 to 11.37 which sediment considerably faster than the closed circle in the same pH region. These have been identified as partially denatured nicked circles, in which varying fractions of the duplex structure have undergone alkaline denaturation, but strand separation has not yet occurred. Varying fractions of a slower species, either undenatured or completely denatured nicked circles, are also observed in some of these experiments. A corresponding result is observed in the intrinsic viscosity vs. pH curve. When nicked circular PM2 DNA is exposed to various alkaline pH's, rapidly neutralized, and sedimented at neutral pH, the expected sharp transition from native to denatured (strand-separated) molecules is seen. However, a very narrow pH range is noted in which native and denatured forms coexist in a single experiment. The above experiments carried out upon the closed form also reveal a narrow pH range in which the bulk of the transition from native closed circles to the collapsed cyclic coil takes place, in acccord with an earlier study on a different DNA. This transition is shown never to be completely effected, however, as there is a fraction (7–8%)of the closed circles which renature to the native form, regardless of the alkaline pH employed. This same phenomenon was not observed in the case of artificially closed λb₂b₅c DNA circles. Possible explanations for some of the above results are discussed.     

Enhanced Tethered-Particle Motion Analysis Reveals Viscous Effects

Tethered-particle motion experiments do not require expensive or technically complex hardware, and increasing numbers of researchers are adopting this methodology to investigate the topological effects of agents that act on the tethering polymer or the characteristics of the polymer itself. These investigations depend on accurate measurement and interpretation of changes in the effective length of the tethering polymer (often DNA). However, the bead size, tether length, and buffer affect the confined diffusion of the bead in this experimental system. To evaluate the effects of these factors, improved measurements to calibrate the two-dimensional range of motion (excursion) versus DNA length were carried out. Microspheres of 160 or 240 nm in radius were tethered by DNA molecules ranging from 225 to 3477 basepairs in length in aqueous buffers containing 100 mM potassium glutamate and 8 mM MgCl₂ or 10 mM Tris-HCl and 200 mM KCl, with or without 0.5% Tween added to the buffer, and the motion was recorded. Different buffers altered the excursion of beads on identical DNA tethers. Buffer with only 10 mM NaCl and >5 mM magnesium greatly reduced excursion. Glycerol added to increase viscosity slowed confined diffusion of the tethered beads but did not change excursion. The confined-diffusion coefficients for all tethered beads were smaller than those expected for freely diffusing beads and decreased for shorter tethers. Tethered-particle motion is a sensitive framework for diffusion experiments in which small beads on long leashes most closely resemble freely diffusing, untethered beads.     

The use of microcarrier beads in the production of endothelium-derived relaxing factor by freshly harvested endothelial cells.

This study is concerned with the use of freshly harvested bovine endothelial cells attached to microcarrier beads in the production of the endothelium-derived relaxing factor (EDRF). The results are compared to production of EDRF by endothelial cells grown in tissue cultures. We found that freshly harvested cells attach themselves to microcarrier beads within minutes. This results in large surface/area volume ratio and permits superfusion of cells suspension on a filter (pore size of 25-30 microns), resulting in cell free filtrate. When superfusing an endothelium-deprived pulmonary artery strip, the effluent causes relaxation; the response depends on the number of superfused endothelial cells. The number of viable freshly harvested cells attached to microcarrier beads in 5 ml Krebs-Henseleit solution is small (30%), as compared to almost 100% for cultured cells. Despite this difference, percent relaxation induced for the same number of viable cells is identical for both groups. Scanning electromicrographs confirm anchorage of endothelial cells to microcarrier beads. While cultured cells cover the entire surface and are individually attached, freshly harvested cells are anchored as cell aggregates leaving some of the surface free. Attachment of freshly harvested endothelial cells to microcarrier beads offers an alternative for the study of the role of endothelial cells in the production of vasoactive substances.     

Estimation of the relative stiffness of the molecular chain in polyelectrolytes from measurements of viscosity at different ionic strengths

A method was developed that allowed comparison of the stiffness of the chain in different polyelectrolyte from measurements of the intrinsic viscosity at different concentrations of added monovalent (sodium) salt. The response to salt was quantitatively expressed as the slope of straight lines relating the intrinsic viscosity to the reciprocal of the square-root of the ionic strength. This slope increased considerably with increasing molecular weight of the polyelectrolyte, and could serve to characterize the response to salt of different substances only when comparison was made at a constant molecular weight. An empirical parameter, B, which is the slope corresponding to an intrinsic viscosity of 1.0 at an ionic strength of 0.1 M could be correlated to the unperturbed dimensions of the molecules. A method of extrapolation, enabling the determination of B from measurements of viscosity on only one sample of unknown molecular weight, was evaluated. The empirically found correlation between Band some well established parameters of stiffness did not contrast predictions from the “fuzzy-sphere model” of Fixman, provided that reasonable assumptions regarding ion-binding and the interaction between polymer and solvent were made.