A new approximate whole boundary solution of the Lamm differential equation for the analysis of sedimentation velocity experiments

Sedimentation velocity is one of the best-suited physical methods for determining the size and shape of macromolecular substances or their complexes in the range from 1 to several thousand kDa. The moving boundary in sedimentation velocity runs can be described by the Lamm differential equation. Fitting of suitable model functions or solutions of the Lamm equation to the moving boundary is used to obtain directly sedimentation and diffusion coefficients, thus allowing quick determination of size, shape and other parameters of macromolecules. Here we present a new approximate whole boundary solution of the Lamm equation that simultaneously allows the specification of sedimentation and diffusion coefficients with deviations smaller than 1% from the expected values.     

An improved approximate solution of the Lamm equation for the simultaneous estimation of sedimentation and diffusion coefficients from sedimentation velocity experiments

Sedimentation and diffusion coefficients are important parameters to describe size and shape of macromolecules in solution. The data can be obtained from sedimentation velocity experiments by a nonlinear fitting procedure using approximate solutions for the Lamm equation. Here, we present a modification of such a model function that was originally proposed by Fujita [H. Fujita, Mathematical Theory of Sedimentation Analysis, Wiley, New York, 1962]. The extended model function is well suitable to study low molecular mass compounds. The improvement of this solution given here is based on using an adjustable value for the explicit integration variable, z, the reduced radius. This modification leads to more accurate sedimentation and diffusion coefficients compared to using a constant value of 0.5 as used by Fujita. The advantage of our modification was demonstrated by the analysis of noise-free curves calculated using the finite element method, as well as experimental curves obtained for the peptides angiotensin I and II. The relatively low sedimentation and diffusion coefficients found for both substances indicate that the peptides exist as extended chains of about 3.65 nm (angiotensin I) or 3.04 nm length (angiotensin II) in solution. The lack of higher-order structure of the peptides that was derived also from CD spectra might facilitate receptor binding, and could be one reason for the fast proteolytic digestion of the free peptides.     

A new simple method to determine the diffusion coefficient from Active Enzyme Centrifugation experiments

A simple and accurate procedure to determine the diffusion coefficient from Active Enzyme Centrifugation experiments is presented. Using computer-simulated concentration distributions we demonstrate that the procedure, derived by Vinograd for the conventional band sedimentation, is suitable to exploit Active Enzyme Centrifugation experiments when the Cohen's Difference Curves Method is used. This new empirical method avoids all the difficulties of the rigorous method previously proposed by Cohen et al., without any loss of accuracy. Optimal conditions are described which allow the determination of the enzyme diffusion coefficient with a 5% uncertainty. Such an easy determination of the sedimentation and diffusion coefficients by the AEC technique can provide a good and rapid estimation of the active enzyme molecular weight, either with a low amount of material or in very impure preparations.     

The effect of spermidine on endonuclease inhibition by agarose contaminants

A simple method for the determination of molecular weight and effective size of proteins is proposed. The procedure consists in comparison of sedimentation coefficients of reversed micelles of aerosol OT in octane in the presence and in the absence of solubilized protein.     

A method for directly fitting the time derivative of sedimentation velocity data and an alternative algorithm for calculating sedimentation coefficient distribution functions

The time-derivative method for deriving the sedimentation coefficient distribution, g(s*), from sedimentation velocity data that was developed by Walter Stafford has many advantages and is now widely used. By fitting Gaussian functions to the g(s*) distribution both sedimentation and diffusion coefficients (and therefore molecular masses) for individual species can be obtained. However, some of the approximations used in these procedures limit the accuracy of the results. An alternative approach is proposed in which the dc/dt data are fitted rather than g(s*). This new approach gives improved accuracy, extends the range to sedimentation coefficients below 1 S, and enhances resolution of multiple species. For both approaches the peaks from individual species are broadened when the data cover too wide a time span, and this effect is explored and quantified. An alternative algorithm for calculating ?(s*) from the dc/dt curves is presented and discussed. Rather than first averaging the dc/dt data for individual scan pairs and then calculating ?(s*) from that average, the ?(s*) distributions are calculated for every scan pair and then subsequently averaged. This alternative procedure yields smaller error bars for g(s*) and somewhat greater accuracy for fitted hydrodynamic properties when the time span becomes large.     

Development of macroporous poly(ethylene glycol) hydrogel arrays within microfluidic channels

The mass transport of solutes through hydrogels is an important design consideration in materials used for tissue engineering, drug delivery, and protein arrays used to quantify protein concentration and activity. We investigated the use of poly(ethylene glycol) (PEG) as a porogen to enhance diffusion of macromolecules into the interior of polyacrylamide and PEG hydrogel posts photopatterned within microfluidic channels. The diffusion of GST-GFP and dextran-FITC into hydrogels was monitored and effective diffusion coefficients were determined by fitting to the Fickian diffusion equations. PEG-diacrylate (M(r) 700) with porogen formed a macroporous structure and permitted significant penetration of 250 kDa dextran. Proteins copolymerized in these macroporous hydrogels retained activity and were more accessible to antibody binding than proteins copolymerized in nonporous gels. These results suggest that hydrogel macroporosity can be tuned to regulate macromolecular transport in applications such as tissue engineering and protein arrays.     

Determination of molecular parameters by fitting sedimentation data to finite-element solutions of the Lamm equation.

A method for fitting experimental sedimentation velocity data to finite-element solutions of various models based on the Lamm equation is presented. The method provides initial parameter estimates and guides the user in choosing an appropriate model for the analysis by preprocessing the data with the G(s) method by van Holde and Weischet. For a mixture of multiple solutes in a sample, the method returns the concentrations, the sedimentation (s) and diffusion coefficients (D), and thus the molecular weights (MW) for all solutes, provided the partial specific volumes (v) are known. For nonideal samples displaying concentration-dependent solution behavior, concentration dependency parameters for s(sigma) and D(delta) can be determined. The finite-element solution of the Lamm equation used for this study provides a numerical solution to the differential equation, and does not require empirically adjusted correction terms or any assumptions such as infinitely long cells. Consequently, experimental data from samples that neither clear the meniscus nor exhibit clearly defined plateau absorbances, as well as data from approach-to-equilibrium experiments, can be analyzed with this method with enhanced accuracy when compared to other available methods. The nonlinear least-squares fitting process was accomplished by the use of an adapted version of the "Doesn't Use Derivatives" nonlinear least-squares fitting routine. The effectiveness of the approach is illustrated with experimental data obtained from protein and DNA samples. Where applicable, results are compared to methods utilizing analytical solutions of approximated Lamm equations.     

RNA dimerization monitored by fluorescence correlation spectroscopy

Fluorescence correlation spectroscopy (FCS) provides a versatile tool to investigate molecular interaction under native conditions, approximating infinite dilution. One precondition for its application is a sufficient difference between the molecular weights of the fluorescence-labelled unbound and bound ligand. In previous studies, an 8-fold difference in molecular weights or correspondingly a 1.6-fold difference in diffusion coefficients was required to accurately distinguish between two diffusion species by FCS. In the presented work, the hybridization of two complementary equally sized RNA single strands was investigated at an excellent signal-to-noise ratio enabled by the highly photostable fluorophore Atto647N. The fractions of ssRNA and dsRNA were quantified by applying multicomponent model analysis of single autocorrelation functions and globally fitting several autocorrelation functions. By introducing a priori knowledge into the fitting procedure, 1.3- to 1.4-fold differences in diffusion coefficients of single- and double-stranded RNA of 26, 41, and 54 nucleotides could be accurately resolved. Global fits of autocorrelation functions of all titration steps enabled a highly accurate quantification of diffusion species fractions and mobilities. At a high signal-to-noise ratio, the median of individually fitted autocorrelation functions allowed a robust representation of heterogeneous data. These findings point out the possibility of studying molecular interaction of equally sized molecules based on their diffusional behavior, which significantly broadens the application spectrum of FCS.     

Structure of initiation factor eIF-3 from rat liver. Hydrodynamic and electron microscopic investigations

On the basis of hydrodynamic, electron microscopic and biochemical investigations a new model of the structure of initiation factor eIF-3 is proposed. From sedimentation and diffusion coefficients of 16.35 S and 2.13 X 10(-7) cm2/s, respectively, as well as from sedimentation equilibrium measurements, a molecular mass of about 650 kDa was determined for isolated eIF-3. This is in agreement with molecular mass estimations by sodium dodecyl sulphate gel electrophoresis. A partial specific volume of 0.723 cm3/g was determined by means of the amino acid composition and the specific volume increments of the amino acids. From this value and from the molecular mass, a volume of 780 nm3 was calculated for eIF-3. In electron micrographs of isolated eIF-3, images with triangular profiles and side lengths of 14 nm, 16 nm, and 17 nm have been observed. Taking into account the calculated volume and considering the triangular image as one face of the particle, it is suggested that eIF-3 has the shape of a flat triangular prism with a height of about 7 nm and the above-mentioned side-lengths. This model is in agreement with results of electron microscopic investigations of eIF-3 in native small ribosomal subunits [Lutsch, G., Benndorf, R., Westermann, P., Bommer, U.-A. & Bielka, H. (1986) Eur. J. Cell Biol. 40/2, in press]. The high frictional ratio of about 1.7 also supports eIF-3 to be rather of a flat than of a globular shape.     

Small angle x-ray scattering of the hemoglobin from Biomphalaria glabrata

The hemoglobin from Biomphalaria glabrata is an extracellular respiratory protein of high molecular mass composed by subunits of 360 kDa, each one containing two 180 kDa chains linked by disulfide bridges. In this work, small angle x-ray scattering (SAXS) measurements were performed with the hemoglobin at pH 5.0 and 7.5. Radii of gyration of 98.6 +/- 0.5 and 101.8 +/- 0.2 A and maximum diameters of 300 +/- 10 and 305 +/- 10 A, respectively, were obtained from Guinier plot extrapolation and analytical curve fitting. The pair distance distribution functions p(r) corresponded to globular particles with a somewhat anisotropic shape for both preparations. Computer analysis of the low angle part of the scattering curve led to the determination of the low resolution envelope of the protein, revealing a P(222) symmetry. Shape reconstruction from ab initio calculations using the complete scattering curve furnished a compact prolate three-dimensional (3D) bead model for the protein. Hydrodynamic parameters were obtained from experiments and theoretical calculations using the 3D model. The results of the structural and biochemical studies reported herein indicate that the multisubunit structure of this hemoglobin is compatible with a tetrameric arrangement.     

Determination of sedimentation coefficients for small peptides.

Direct fitting of sedimentation velocity data with numerical solutions of the Lamm equations has been exploited to obtain sedimentation coefficients for single solutes under conditions where solvent and solution plateaus are either not available or are transient. The calculated evolution was initialized with the first experimental scan and nonlinear regression was employed to obtain best-fit values for the sedimentation and diffusion coefficients. General properties of the Lamm equations as data analysis tools were examined. This method was applied to study a set of small peptides containing amphipathic heptad repeats with the general structure Ac-YS-(AKEAAKE)nGAR-NH2, n = 2, 3, or 4. Sedimentation velocity analysis indicated single sedimenting species with sedimentation coefficients (s(20,w) values) of 0.37, 0.45, and 0.52 S, respectively, in good agreement with sedimentation coefficients predicted by hydrodynamic theory. The described approach can be applied to synthetic boundary and conventional loading experiments, and can be extended to analyze sedimentation data for both large and small macromolecules in order to define shape, heterogeneity, and state of association.     

Sizes and mass distributions of clathrin-coated vesicles from bovine brain

Clathrin-coated vesicles obtained from bovine brain have been studied by ultracentrifugation and dynamic light scattering techniques to provide information on their sedimentation and mass distributions and their average diffusion coefficients. "Uncoated" vesicles, obtained by removing the protein coat from coated vesicles, have been similarly characterized. For typical preparations, maximal values of approximately 210 and 95 S are observed for the sedimentation coefficients of coated and uncoated vesicles, respectively. Corresponding values for the average molecular weights, determined from values of average sedimentation and diffusion coefficients, are 49 X 10(6) and 13 X 10(6); values obtained by equilibrium sedimentation are 37.2 X 10(6) and 10.6 X 10(6). In order to obtain these results, some minor modifications of sedimentation and light-scattering techniques have been devised which may have application to other studies of size distributions of large particles.     

Purification and Characterization of High Molecular Mass and Low Molecular Mass Cystatin from Goat Brain

Cystatin are thiol proteinase inhibitors ubiquitously present in mammalian body and serve various important physiological functions. In the present study two cystatins were isolated from goat brain using alkaline treatment, ammonium sulphate fractionation, gel filtration and ion exchange chromatography. The high molecular mass cystatin of 70.8 kDa was named as HM-GBC (high molecular mass goat brain cystatin) and the low molecular mass cystatin of 12.72 kDa was named as LM-GBC (low molecular mass goat brain cystatin). The molecular mass determined by SDS-PAGE was found to be 70.8 and 12.88 kDa for HM-GBC and LM-GBC, respectively, however with gel filtration the masses were found to be 70.8 and 12.58 kDa. Both the cystatins were found to be stable in broad range of pH and temperature. HM-GBC was found to have 2% carbohydrate content while LM-GBC lacks any carbohydrate content. Both cystatins were found to be devoid of any sulphydryl content. Stoke's radii of 36 and 16 A, and diffusion coefficient of 6.189 x 10(-15) and 1.392 x 10(-14) cm(2)/s were calculated for HM-GBC and LM-GBC. K (i) values with papain were found to be 1.875 x 10(-8) and 3.125 x 10(-8) M for HM-GBC and LM-GBC, respectively. K (+1), K (-1) and half-life calculated along with K (i) values obtained showed that HM-GBC inhibited papain more specifically as compared to LM-GBC. The IC(50) values obtained for HM-GBC and LM-GBC also showed that HM-GBC binds more effectively to papain than LM-GBC. Ultraviolet and fluorescence spectra indicated that upon formation of papain-HM-GBC/LM-GBC complex there is significant conformational change after interaction in one or both the proteins of the complex.     

Ultracentrifugal studies in capillary cells. I. Determination of sedimentation coefficients

A technique has been developed which is based on transport method equations and allows determination of sedimentation coefficients using less than 5 ng of a biological active principle. Glass capillaries of 0.30 mm inside diameter and 3.0 mm length are used as sedimentation cells. About 200 nl of pure or impure solutions with concentration as low as 0.05 mg/ml are ultracentrifuged in a swinging-bucket rotor with a conventional preparative ultracentrifuge. The capillary microcells are easily sectioned after centrifugation through a plane previously marked on the glass surface. The technique was successfully extended to the use of larger glass capillaries, up to 1.25 mm inside diameter, and plastic centrifuge tubes of 5.0 mm diameter and 0.40 ml capacity. The method has been experimentally verified with proteins and protein-polysaccharides of known sedimentation constants.     

Annular arrangement and collaborative actions of four domains of protein-disulfide isomerase: a small angle X-ray scattering study in solution

We presented for the first time a small angle x-ray scattering study of intact protein-disulfide isomerase (PDI) in solution. The restored model revealed that PDI is a short and roughly elliptical cylinder with a molecular mass of 69 kDa and dimensions of 105 x 65 x 40 A, and the four thioredoxin-fold domains in the order a-b-b'-a' are arranged in an annular fashion. Atomic force microscope imaging also supported the finding that PDI appears as an approximately flat elliptical cylinder. A PDI species with apparent molecular mass of 116 kDa measured by using size-exclusion chromatography, previously assumed to be a dimer, was determined to exist mainly as a monomer by using analytical ultracentrifugation. The C-terminal fragment 441-491 contributed to the anomalous molecular mass determination of PDI by size-exclusion chromatography. The annular model of PDI accounted for the cooperative properties of the four domains in both the isomerase and chaperone functions of PDI.     

Mobility shift detection of phosphorylation on large proteins using a Phos‐tag SDS‐PAGE gel strengthened with agarose

We describe a novel technique of phosphate‐affinity SDS‐PAGE using Phos‐tag to analyze large phosphoproteins with molecular masses of more than 200 kDa. The protein phosphoisotypes were clearly separated as up‐shifted migration bands in a 3% w/v polyacrylamide gel containing 20 μM Phos‐tag and 0.5% w/v agarose. In subsequent immunoblotting, the procedure permitted the determination of the phosphoisotypes of high‐molecular‐mass proteins, such as mTOR (289 kDa), ATM kinase (350 kDa), and 53BP1 (213 kDa).     

Determination of the mass of viruses by quantitative electron microscopy.

The photometric method of quantitative determination of dry mass by electron microscopy has been applied to the study of various types of viruses: animal, plant, insect, and bacterial. The method is applicable to all viruses having a mass of 1 x 10-18g or greater. The molecular weight of viruses can be calculated from the mass value by multiplying it by Avogadro's number. In comparison to other methods of determining the molecular weight of viruses, sedimentation and diffusion, sedimentation equilibrium, light scattering, and electron microscopy counting, the method of quantitative electron microscopy is competitive. In some ways quantitative electron microscopy is superior to other methods for the determination of molecular weight: There is no limitation to the size of the virus, no experimental time involved and no concentration and purity of virus preparations required, and finally it is independent of the geometry of the virion. In one important aspect it is unique when compared to other methods; namely, it affords one the capacity to analyse individual virus particles.     

The use of sedimentation coefficients to distinguish between models for protein oligomers.

The sedimentation coefficients of proteins are dependent on their sizes, shapes and densities and on the density and viscosity of the solvent. However, when the sedimentation coefficients of an oligomeric protein and its protomer are measured under the same experimental conditions, the ratio of the two coefficients depends only on the protomer shape and the mode of aggregation. This property, which we shall call the sedimentation ratio, therefore provides a way of distinguishing between models for oligomeric proteins. To allow examination of the behaviour of the sedimentation ratio, sedimentation coefficients are calculated for a comprehensive range of protomer shapes and modes of aggregation in hexameric systems using equations derived by Kirkwood. As illustrations of the method the resulting sedimentation ratios are compared with experimental values for insulin and arthroped hemocyanin, which eliminates many of the possible structures for these proteins. When experimental estimates of degree of hydration and molecular dimensions are also considered, all but a group of virtually identical structures are eliminated for the insulin hexamer and a single most likely structure remains for arthropod hemocyanin. The insulin structure is in good agreement with that determined by X-ray crystallography while the hemocyanin hexameric structure is a hexagonal prism formed by the cyclic aggregation of prolate ellipsoids of axial ratio about 2.5 : 1.     

Simultaneous determination of the individual concentration gradients of two solute species in a centrifuged mixture: application to analytical ultracentrifugation

A procedure for determining the absolute activity of 14C-labeled and 3H-labeled solutes in a mixture from the measured counts per minute in two scintillation energy windows is described. It is shown that the method described here provides a substantially more accurate determination of 3H activity in the presence of a larger 14C activity, and a more accurate determination of 14C activity in the presence of a larger 3H activity, than does the standard dual label analysis implemented in a Beckman LS 3801 scintillation counter. The new dual label procedure is combined with the automated fractionation procedure of Attri and Minton [(1986) Anal. Biochem. 152, 319-328] to permit the gradients of each of two differently radiolabeled solute species in a mixture to be individually determined following centrifugation. It is shown that the sedimentation coefficients of each of two differently labeled noninteracting proteins in a mixture may be readily determined in a sedimentation velocity experiment, and that the molecular weights of each of two such proteins in a mixture may be readily determined in a sedimentation equilibrium experiment.     

Accurate quantification of diffusion and binding kinetics of non-integral membrane proteins by FRAP

Non-integral membrane proteins frequently act as transduction hubs in vital signaling pathways initiated at the plasma membrane (PM). Their biological activity depends on dynamic interactions with the PM, which are governed by their lateral and cytoplasmic diffusion and membrane binding/unbinding kinetics. Accurate quantification of the multiple kinetic parameters characterizing their membrane interaction dynamics has been challenging. Despite a fair number of approximate fitting functions for analyzing fluorescence recovery after photobleaching (FRAP) data, no approach was able to cope with the full diffusion-exchange problem. Here, we present an exact solution and matlab fitting programs for FRAP with a stationary Gaussian laser beam, allowing simultaneous determination of the membrane (un)binding rates and the diffusion coefficients. To reduce the number of fitting parameters, the cytoplasmic diffusion coefficient is determined separately. Notably, our equations include the dependence of the exchange kinetics on the distribution of the measured protein between the PM and the cytoplasm, enabling the derivation of both k(on) and k(off) without prior assumptions. After validating the fitting function by computer simulations, we confirm the applicability of our approach to live-cell data by monitoring the dynamics of GFP-N-Ras mutants under conditions with different contributions of lateral diffusion and exchange to the FRAP kinetics.