Differential interferometry in the analytical ultracentrifuge. II. General applications

Various ways of applying differential interferometry to ultracentrifugal analyses are examined and several analytical techniques are established. In transport and moving boundary methods, the sedimentation coefficient is more precisely determined in the differential interference system than in the schlieren optical system because fringe measurement accuracy is much higher in the former system. Compared to interference and absorption optics, the differential interferometer provides a more exact s value in the transport method since an accurate calculation procedure can be adopted. Moreover, the following advantages of differential interferometry are noted. Determination of the initial solute concentration, which must be done in the usual interference method, is unnecessary in this sedimentation equilibrium method. Regardless of the partial loss of solute from the observed system due to rapid precipitation or adsorption to the cell wall during centrifugation, the molecular weight of the rest of the solute can be determined exactly. The diffusion coefficient can be determined accurately by fringe displacement analysis at the hinge point during the transient state. Together with the molecular weight and diffusion coefficient, the partial specific volume and sedimentation coefficient of a solute can be obtained from the result of a single low-speed centrifugation when the sample solutions in H₂O and D₂O are compared.     

Molecular mass determination by sedimentation velocity experiments and direct fitting of the concentration profiles.

A new method for the direct molecular mass determination from sedimentation velocity experiments is presented. It is based on a nonlinear least squares fitting procedure of the concentration profiles and simultaneous estimation of the sedimentation and diffusion coefficients using approximate solutions of the Lamm equation. A computer program, LAMM, was written by using five different model functions derived by Fujita (1962, 1975) to describe the sedimentation of macromolecules during centrifugation. To compare the usefulness of these equations for the analysis of hydrodynamic results, the approach was tested on data sets of Claverie simulations as well as experimental curves of some proteins. A modification for one of the model functions is suggested, leading to more reliable sedimentation and diffusion coefficients estimated by the fitting procedure. The method seems useful for the rapid molecular mass determination of proteins larger than 10 kDa. One of the equations of the Archibald type is also suitable for compounds of low molecular mass, probably less than 10 kDa, because this model function requires neither the plateau region nor a meniscus free of solute.     

Automatic collection and processing of data from the ultracentrifuge using a programmable desk calculator.

Previous investigators [Trautman, R., Spragg, S. P., and Halsall, H. B. (1969) Anal. Biochem.28, 396–415] have published a detailed protocol for the analysis of sedimentation velocity measurements which is adaptable to data generated by an ultraviolet scanning system. The advent of programmable desk calculators capable of sampling the output of digital measuring devices has made it possible to develop inexpensive and highly convenient systems for collecting and processing scanner data. Basing our approach on the referenced protocol, we have developed algorithms for dealing with real data, that is, data characterized by a relatively high level of noise. The techniques are applieable to both sedimentation equilibrium and sedimentation velocity measurements using the seanning system and multicell rotors. With known concentration dependence, valid estimates of weight-average sedimentation coefficients, diffusion coefficients, and heterogeneity parameters have been obtained for both simulated and actual sedimenting anddiffusing macromolecular solutes. We find, however, that concentration dependence derived internally from a single sedimentation velocity measurement is unreliable.     

Some cellular diffusion problems based on Onsager's generalization of Fick's law

Some consequences of Onsager's, generalization of Fick's law are examined. It is found thatmetabolized solutes may flow continually against their concentration gradients.Inert solutes may exist in a higher or lower concentration inside of the cell than in the medium thus appearing to be accumulated or excluded. The magnitudes of such concentration differences are dependent upon the rates of metabolism of metabolized solutes. Alteration of these rates mayfurther increase the concentration disparity by causing inert solute to flow from a low to a high concentration. Due to the mutual dependence, demanded by Onsager's law, of the diffusion currents, the rates ofchemically independent reactions are mutually dependent. This so calledcoupling by diffusion implies that:The rate of metabolism of a given substrate is influenced by the rates of metabolism of metabolically unrelated substrates. Furthermore, the presence, in the medium, of aninert solute to which the membrane is permeable will influence the rates of concentration-dependent reactions in the cell. The spatial distribution of a catalyst in the diffusion field within the cell is examined. The general effect of including heat flow and thermal diffusion in the cellular diffusion problems is briefly pointed out.     

A method for the determination of diffusion coefficients for small molecules in aqueous solution

A method is described for determining the diffusion coefficients of small solutes in limited volumes (approximately equal to 4-9 ml) of fluid. Diffusion is measured in a three-chamber diffusion cell across a central unstirred compartment. Compartments are separated by nitrocellulose membranes. The instantaneous concentration gradient and the instantaneous flux of solute into the dilute end compartment are derived from changes in the concentration of solute in the two stirred end compartments through time. The diffusion coefficient is calculated from the slope of the least-squares regression line relating the magnitude of the instantaneous solute flux to that of the instantaneous concentration gradient. The apparatus is calibrated with a solute of known diffusivity (KCl). Diffusion coefficients thus determined in water at 25 degrees C for CaCl2 (7.54 X 10(-6) cm2.s-1), Na2-ATP (7.01 X 10(-6) cm2.s-1), 2-deoxyglucose (5.31 X 10(-6) cm2.s-1), and D-Na-lactate (5.62 X 10(-6) cm2.s-1) differed by an average of 3.7% from literature values. The method described results in accurate estimates of diffusion coefficients by a simple and relatively rapid procedure.     

Convection and Diffusion in Charged Hydrated Soft Tissues: A Mixture Theory Approach

The extracellular matrix of cartilage is a charged porous fibrous material. Transport phenomena in such a medium are very complex. In this study, solute diffusive flux and convective flux in porous fibrous media were investigated using a continuum mixture theory approach. The intrinsic diffusion coefficient of solute in the mixture was defined and its relation to drag coefficients was presented. The effect of mechanical loading on solute diffusion in cartilage under unconfined compression with a frictionless boundary condition was analyzed numerically using the model developed. Both strain-dependent hydraulic permeability and diffusivity were considered. Analyses and results show that (1) In porous media, the convective velocity for each solute phase is different. (2) The solute convection in tissue is governed by the relative convective velocity (i.e., relative to solid velocity). (3) Under the assumption that all the frictional interactions among solutes are negligible, the relative convective velocity for α-solute phase is equal to the relative solvent velocity multiplied by its convective coefficient (H ^α) which is also known as the hindrance factor in the literature. The relationship between the convective coefficient and the relative diffusivity of solute is presented. (4) Solute concentration profile within the cartilage sample depends on the phase of dynamic compression.     

Superhelix density heterogeneity in closed circular intracellular PM2 DNA

Covalently closed intracellular DNA obtained from Pseudomonas BAL 31 20 min after infection with PM2 phage has been shown to be heterogeneous in superhelix density. Analytical band sedimentation, in the presence of low concentrations of ethidium bromide, has been carried out on fractions centripetal and centrifugal to the mode of a single band of closed circular DNA in a preparative propidium iodide–CsCl buoyant density gradient. Different average sedimentation rates, as well as different band shapes, have been observed for upper and lower fractions centrifuged at a dye concentration near the minimum in s° versus ethidium bromide concentration titrations performed on DNA from proximate intermediate fractions. Similar differences, although not as pronounced, have been obtained at a dye concentration corresponding to a point in the steep region of the titrations. Differential band sedimentation experiments performed on the same fractions have confirmed these results. Differential band sedimentation experiments on similarly fractionated mature PM2 I DNA (closed circular form) have shown slight differences in the relative sedimentation rates of upper and lower fractions at dye concentrations corresponding to the steep regions in the titrations. The same experiments, when performed on nicked circular DNA obtained from heating both the mature and intracellular fractions, showed no evidence of differences in sedimentation coefficients. Such results may indicate slight heterogeneity in the superhelix density of viral PM2 I DNA; however, the degree of this heterogeneity would be somewhat less than that of the intracellular DNA. The possibility that superhelix density heterogeneity may arise from displacement loops, which have been found at low levels in intracellular PM2 DNA, has been subjected to experimental tests. Unless such structures are originally present and removed by the isolation procedure, this possibility may be rejected.     

Red Cell Membrane Permeability Deduced from Bulk Diffusion Coefficients

The permeability coefficients of dog red cell membrane to tritiated water and to a series of[¹⁴C]amides have been deduced from bulk diffusion measurements through a "tissue" composed of packed red cells. Red cells were packed by centrifugation inside polyethylene tubing. The red cell column was pulsed at one end with radiolabeled solute and diffusion was allowed to proceed for several hours. The distribution of radioactivity along the red cell column was measured by sequential slicing and counting, and the diffusion coefficient was determined by a simple plotting technique, assuming a one-dimensional diffusional model. In order to derive the red cell membrane permeability coefficient from the bulk diffusion coefficient, the red cells were assumed to be packed in a regular manner approximating closely spaced parallelopipeds. The local steady-state diffusional flux was idealized as a one-dimensional intracellular pathway in parallel with a one-dimensional extracellular pathway with solute exchange occurring within the series pathway and between the pathways. The diffusion coefficients in the intracellular and extracellular pathways were estimated from bulk diffusion measurements through concentrated hemoglobin solutions and plasma, respectively; while the volume of the extracellular pathway was determined using radiolabeled sucrose. The membrane permeability coefficients were in satisfactory agreement with the data of Sha'afi, R. I., C. M. Gary-Bobo, and A. K. Solomon (1971. J. Gen. Physiol. 58:238) obtained by a rapid-reaction technique. The method is simple and particularly well suited for rapidly permeating solutes.     

A method for measuring sedimentation and diffusion of macromolecules in capillary tubes by total intensity and quasi-elastic light-scattering techniques.

Light scattered from a macromolecular solution in a capillary tube is used to determine both the sedimentation and translational diffusion coefficients. The capillary tube is spun in a preparative centrifuge, removed, and placed in a light-scattering photometer equipped with a scanning mechanism. The intensity distribution of scattered light along the tube represents the concentration profile in the tube and provides the measure of boundary migration. The sedimentation coefficient is determined from this measure and the applied centrifugal field. The diffusion coefficient is obtained from a time-autocorrelation analysis of fluctuations in intensity of light scattered from any fixed point of the profile. These coefficients were obtained for two monodisperse systems, R17 bacteriophage and 28s ribosomal rat liver RNA. The molecular weights obtained from ratios of these coefficients are in good agreement with literature values. In the sedimentation analysis, deviations from linearity between boundary displacement and applied field were found to be less than 1%. This precision confirms that the boundary is stable for the capillary geometry even in the absence of a preformed density gradient. The sedimentation coefficients of identical samples were also measured with the Spinco Model E analytical ultracentrifuge; results of the two methods agree to within 4%. As a consequence of the capillary tube geometry and light-scattering detection, sedimentation coefficients can be obtained from sample volumes of less than 100 μl. This detection techniques is thus far demonstrated to be at least an order of magnitude more sensitive than Schlieren optics, thereby useful when uv absorption is not applicable. For diffusion measurements there are also several inherent advantages. The diffusion coefficient is obtained from the identical sample, and scanning provides the capability to measure D from various parts of the sedimentation profiles and thereby directly explore concentration dependence, homogeneity, and integrity of the sample. The capillary tube with a layer of silicone oil over the sample and centrifugation provides an effective method to cleanse the solution and trap all dust.     

Determination os sedimentation coefficient distributions for cartilage proteoglycans

Sedimentation coefficient distributions of widely polydisperse proteoglycan preparations were made using a previously developed transport sedimentation methodology. Boundary stability was improved by centrifuging samples in a preformed CsCl density gradient (0.016 g/cm⁴). The results were compared with the distributions obtained with an interferometric analytical centrifugation method. When these two techniques were applied to analyze A1 and A1–D1 proteoglycan preparations, results were in substantial agreement with respect to the mean sedimentation coefficients of the peaks, average S value, sedimentation coefficient distribution, skewness, proportion of monomer and aggregates, and linearity of the plot ln(s) versus C extrapolations to zero concentration. The lower solute concentration compatible with the transport (velocity gradient) method makes this technique particularly suitable for studying the details of proteoglycan distribution of molecular sizes, especially for aggregates.     

Precipitation of protein by ultracentrifuge with angle rotor

Precipitation of a protein by ultracentrifuge with an angle rotor was simulated by a model for sedimentation process. Assuming that the concentration of solute in an inclined ultracentrifugal tube is given by averaging the concentration in the imaginary horizontal tube, the governing equation describing the concentration in the rectangular-shaped tube with a uniform field of ultracentrifugal force for an inclined tube in an angle rotor was derived. The exact solution to this governing equation was obtained under the condition that the diffusion is absent or present. The dimensionless concentration which is reduced by the initial concentration can be expressed as the function of a dimensionless ultracentrifugal times ² t in case that the diffusion is absent, and as the function of dimensionless parameters andt ^*in case that the diffusion is present. From our first approximated model it is found that the precipitation of a protein by ultracentrifuge with an angle rotor proceeds more rapidly than that with a swing rotor whether diffusion is absent or present.List of Symbols c kg/m³ concentration of solute - c ⁰ kg/m³ initial concentration of solute - c ^A kg/m³ concentration of solute for angle rotor - c ^s kg/m³ concentration of solute for swing rotor - D cm²/s diffusion coefficient - d cm diameter of ultracentrifugal tube - k ₁ dimensionless constant - k ² dimensionless constant - r cm radial coordinate - r ₁ cm minimum radius of ultracentrifugal tube - r ₂ cm maximum radius of ultracentrifugal tube - r _m cm mean radius of ultracentrifugal tube - r _s cm radius from which sedimentation starts - s s sedimentation constant - t s time - z cm vertical coordinate - dimensionless parameter - ^m dimensionless parameter - deg inclination of ultracentrifugal tube - s^–1 angular velocity of rotation     

Boundary analysis of sedimentation-velocity experiments with monodisperse and paucidisperse solutes

A method is presented which allows one to calculate a distribution of sedimentation coefficients from the boundries of sedimentation-velocity experiments with mono- or pauci-disperse solutes. With two solutes differences in S value as small as 20% can be resolved. In absence of heterogeneity and concentration-dependent effects, the analysis also provides values for the diffusion coefficient within an accuracy of ?10 to +5%. Tests with both simulated data and ultracentrifugation experiments on short DNA fragments show the value and the limitations of the method.     

Sedimentation studies of giant DNA molecules present in unsheared whole-cell lysates of D. melanogaster cells.

We have used band sedimentation in shallow density gradients of CsCl in the preparative centrifuge to analyze the distribution of sedimentation coefficients present in tritium labelled DNA from D. melanogaster cells. The cells were lysed according to the method of Kavenoff and Zimm to preserve very high molecular weight DNA. Sedimentation measurements have been carried out as a function of speed of centrifugation. The resulting distribution functions have been interpreted with the aid of the Zimm-Schumaker equation for the speed dependence of the sedimentation coefficient of very high molecular weight DNA. Low-speed centrifugation (3000 rpm) indicates that DNA molecules from the lysate are evenly distributed over values of S_20,w from 0 to 514S. This distribution is very sensitive to changes in speed of centrifugation and is transformed into a bimodal distribution at 12,080 rpm. Analysis of this transformation allows us to postulate that perhaps 55% of the DNA in the lysate may have molecular weights in excess of 40 × 10⁹ g/mol. Some of these molecules may also possess a variety of configurations including partially replicated branched structures.     

Band centrifugation of macromolecules in self-generating density gradients. III. Conditions for convection-free band sedimentation

The conditions for convection-free hand sedimentation are analyzed in terms of the negative density gradients associated with the leading edge of a band and the positive density gradients generated during the experiment. The amount of material necessary to perform a band-centrifugation experiment depends on the diffusion coefficient of the macromolecules, which determines the rate at which the concentration at band maximum decreases, and on the extinction coefficient at the wavelength of observation. The maximum negative gradients in bands of macromolecules with diffusion and extinction coefficients typical of proteins, nucleic acids, and viruses are calculated. Positive density gradients generated by diffusion of small molecules between the thin lamella and the bulk solution are calculated for bulk solutions such us 1M NaCl or 95% D₂O. These “diffusion gradients” are generally adequate to stabilize bands of the above macromolecules. Positive density gradients generated by sedimentation of salts within the bulk solution may be significant in providing stability near the bottom of the cell. The effects of inadequate stabilizing gradients are discussed, and are found to cause forward spreading of the band.     

Osmoregulation in the Avena Coleoptile : CONTROL OF SOLUTE UPTAKE IN PEELED SECTIONS

Peeled Avena sativa coleoptile sections (i.e. sections from which the epidermis has been removed) have been used to study the control of solute uptake under conditions where the uptake is not limited by the cuticular barrier. In the presence of 2% sucrose, auxin enhances the rate at which the total osmotic solutes increase, but this appears to be a response to the increased growth rate, inasmuch as the auxin effect is eliminated when growth is inhibited osmotically. When sections are incubated in sucrose or in 20 millimolar NaCl, the osmotic concentration increases until a plateau is reached after 8 to 24 hours. Auxin has no effect on the initial rate of increase in osmotic concentration but causes the osmotic concentration to reach a plateau earlier and at a lower osmotic conentration value. This difference in steady-state osmotic concentration is, in part, a response to auxin itself, as it persists when auxin-induced growth is inhibited osmotically. The upper limit for osmotic concentration does not appear to be determined by the turgor pressure, inasmuch as a combination of sucrose and NaCl gave a higher plateau osmotic concentration than did either solute alone. We suggest that the rate of solute uptake is determined by the availability of absorbable solutes and by the surface area exposed to the solutes. Each absorbable solute reaches a maximum internal concentration independent of other absorbable solutes; the steady-state osmotic concentration is simply the sum of these individual internal concentrations.     

Molecular modeling and simulation of Mycobacterium tuberculosis cell wall permeability

The low permeability of the mycobacterial cell wall is thought to contribute to the intrinsic drug resistance of mycobacteria. In this study, the permeability of the Mycobacterium tuberculosis cell wall is studied by computer simulation. Thirteen known drugs with diverse chemical structures were modeled as solutes undergoing transport across a model for the M. tuberculosis cell wall. The properties of the solute-membrane complexes were investigated by means of molecular dynamics simulation, especially the diffusion coefficients of the solute molecules inside the cell wall. The molecular shape of the solute was found to be an important factor for permeation through the M. tuberculosis cell wall. Predominant lateral diffusion within, as opposed to transverse diffusion across, the membrane/cell wall system was observed for some solutes. The extent of lateral diffusion relative to transverse diffusion of a solute within a biological cell membrane may be an important finding with respect to absorption distribution, metabolism, elimination, and toxicity properties of drug candidates. Molecular similarity measures among the solutes were computed, and the results suggest that compounds having high molecular similarity will display similar transport behavior in a common membrane/cell wall environment. In addition, the diffusion coefficients of the solute molecules across the M. tuberculosis cell wall model were compared to those across the monolayers of dipalmitoylphosphatidylethanolamine and dimyristoylphosphatidylcholine, are two common phospholipids in bacterial and animal membranes. The differences among these three groups of diffusion coefficients were observed and analyzed.     

Hydrodynamic properties of aqueous solutions of galactomannans

Galactomannans were extracted from the seeds of seven different legumes to obtain samples which differed in average molecular weight, in polydispersity, and in average mannose-to-galactose ratio in the molecule. Measurements were made of the viscosities as a function of shear rate and of the sedimentation and diffusion coefficients of the aqueous solutions of all the galactomannans. Average molecular weights were calculated from the sedimentation and diffusion coefficients using the Svedberg equation. The Mark-Houwink relationship between the intrinsic viscosity and the average molecular weights was found to hold for the series of galactomannans despite differences in the mannose-to-galactose ratios of the solutes. Estimates were made of the galactomannan polydispersities from both the sedimentation-coeffient distributions and also from the diffusion-coefficient ratios, D_m/D_A. The two methods gave results in qualitative agreement. The hydrodynamic properties of the more homogeneous galactomannans were compared with the predictions of the Flory-Fox hydro-dynamic theory, with results very similar to those which were obtained for the water-soluble synthetic cellulose derivatives, hydroxyethylcellulose and ethydroxycellulose.     

Validation of theoretical framework explaining active solute uptake in dynamically loaded porous media

Solute transport in biological tissues is a fundamental process necessary for cell metabolism. In connective soft tissues, such as articular cartilage, cells are embedded within a dense extracellular matrix that hinders the transport of solutes. However, according to a recent theoretical study (Mauck et al., 2003, J. Biomech. Eng. 125, 602–614), the convective motion of a dynamically loaded porous solid matrix can also impart momentum to solutes, pumping them into the tissue and giving rise to concentrations which exceed those achived under passive diffusion alone. In this study, the theoretical predictions of this model are verified against experimental measurements. The mechanical and transport properties of an agarose–dextran model system were characterized from independent measurements and substituted into the theory to predict solute uptake or desorption under dynamic mechanical loading for various agarose concentrations and dextran molecular weights, as well as different boundary and initial conditions. In every tested case, agreement was observed between experiments and theoretical predictions as assessed by coefficients of determination ranging from R²=0.61 to 0.95. These results provide strong support for the hypothesis that dynamic loading of a deformable porous tissue can produce active transport of solutes via a pumping mechanisms mediated by momentum exchange between the solute and solid matrix.     

Theory of boundary spreading in velocity ultracentrifugation of polydisperse solutes

Under the assumptions that the rectangular approximation to the sector-shaped cell is valid and that both the sedimentation coefficient s and the diffusion coefficient D are independent of concentration, asymptotic solutions to the boundary spreading equation for velocity ultracentrifugation of polydisperse solutes have been derived for three cases: case A, D = constant for all s; case B, sD = constant; case C, \documentclass{article}\pagestyle{empty}\begin{document}$ \sqrt {sD} = {\rm constant} $\end{document}. Case A is the situation treated by all of the previous authors but supposed to be unrealistic for ordinary macromolecular solutes. Cases B and C may be associated with synthetic polymers under theta conditions and globular proteins in aqueous media, respectively. The solutions obtained have been used to explore the theoretical background of the empirical Gralén method for evaluating the distribution of s from sedimentation boundary curves, with special interest in the behavior of a plot for Sversus 1/t. Here S is the value of s for a fixed value of the apparent integral distribution of s and t is the time of centrifugation. It was found that when the distribution of s is Gaussian-like and fairly narrow, this plot becomes linear over a more extended range of t in the order case B > case C > case A.     

Conformation of LiDNA in solutions of LiCl

We have derived radii of gyratin, R_g, from the absolute intensity of the scattered light of mondisperse linear Col E₁ LiDNA in solution at various LiCl concentrations up to 5M. The second virial coefficients, A₂, decrease strongly with increasing LiCl concentration, and vanish between 3 and 5M LiCl. It was thus possible to calculate a limiting value at a high salt concentration of 28.5 nm for the persistence length, a₀, of LiDNA, without the necessity of applying excluded-volume corrections. The value obtained is in good agreement with the value previously obtained for NaDNA at high NaCl concentrations, and can be identified with the high salt limit of DNA flexibility, with long-range electrostatic interactions effectively screened. Sedimentation coefficients in the ultracentrifuge and apparent and translational diffusion coefficients (at finite and vanishing scattering vectors, respectively) from dynamic laser-light scattering have also been obtained up to 5M LiCl. From the sedimentation and apparent diffusion, D(90), (at 90° scattering angle only) above 5M, and up to 9M LiCl, it could be shown that the solutions are stable for reasonable periods of time, and the molecular parameters vary smoothly and moderately at high salt. Conformational transitions were not observed and precipitation occurs between 9 and 10M LiCl.