Boundary centrifugation in isovolumetric and isokinetic cesium sulfate density gradients: application to cartilage proteoglycans and other macromolecules

A boundary sedimentation methodology is described that avoids plateau dilution and simplifies the calculation of centrifugal parameters. The technique is designed for the preparative ultracentrifuge and uses a newly developed sectorial cell. It is based on previous developments of the transport method and depends on isokinetic or isovolumetric Cs2SO4 density and viscosity gradients. These gradients are prepared with a single-chamber mixing device, and the only two parameters required for their calculations are presented in a tabulated form for general use with most available rotors and cell sizes. Conditions are specified (1) to assure that the density and shape of the sedimenting molecules remain invariant through the selected electrolytic gradient, (2) to monitor the gradient profiles, and (3) to verify attainment of isokinetic or isovolumetric sedimentations. A set of equations is presented to calculate the average and transport sedimentation coefficients and the differential sedimentation coefficient distribution for both the isokinetic and isovolumetric centrifugal regimes. The method was applied to slowly diffusing polydisperse proteoglycan monomers, to a paucidisperse DNA from bacteriophage PM2, and to a diffusible monodisperse system (purified bovine serum albumin). In all cases, the expected results were obtained.     

Diffusional boundary spreading in the ultracentrifuge

A common approximation for deriving solutions to the Lamm equation is to neglect diffusion. This paper presents a singular perturbation technique that allows one to estimate the band spreading due to nonzero diffusion coefficient. We illustrate the general mathematical technique by its application to sedimentation when pressure effects are important. Comparison of the approximate solution with accurate numerical solutions shows that the relative errors are of the order of 1% both for concentration and concentration gradient for parameters of chemical interest.     

Separation of hepatocytes from suspensions of mouse liver cells using programmed gradient sedimentation in gradients of ficoll in tissue sulture medium

Liver cells were obtained in suspension using a solution of lysozyme in Joklik's modification of minimum essential medium. Hepatocytes were separated in 74.2 ± 12.9% purity from other liver cells having different densities using isopycnic centrifugation, in 97.1 ± 1.9% purity from other liver cells having different diameters using velocity or rate-zonal centrifugation. A previously reported computer integration of the differential sedimentation equation was employed in determining the gradient design and the speed and duration of centrifugation which would permit purification of hepatocytes from other liver cells. More than 98% of the hepatocytes separated by velocity sedimentation excluded trypan blue. Velocity sedimentation is superior to isopycnic centrifugation for the separation of hepatocytes from liver cell suspensions because it gives more highly purified hepatocytes and because it requires lower centrifugal forces for shorter periods of time.     

A strategy for efficient characterization of macromolecular heteroassociations via measurement of sedimentation equilibrium.

A method is proposed for the selection of experimental conditions for sedimentation equilibrium experiments that will provide maximal information about the values of equilibrium association constants within a given scheme for heteroassociation of two solute components. A discriminator function is proposed that indicates the sensitivity of the experimentally observed gradient or gradients to alterations in the underlying association constants. The value of this function is plotted or tabulated as a function of the concentrations of the two components, over a broad range of solution compositions. It is suggested that experiments performed with loading compositions corresponding to large absolute values of the discriminator function will yield the most information with respect to determination of the underlying association constants. This method was tested by predicting optimal conditions for three different types of sedimentation equilibrium experiments: (i) measurement of total (natural) solute absorbance; (ii) measurement of individual component gradients via measurement of tracer absorbance; and (iii) global analysis of multiple experiments. Experimental data resulting from sedimentation equilibrium experiments carried out under the specified conditions were simulated by addition of realistic levels of random error to calculated equilibrium gradients. The simulated data were then analyzed exactly as real experimental data, i.e., without prior knowledge of the underlying association constants. It was found that the highest accuracy and precision in determination of heteroassociation constants are obtained by global analysis of multiple experiments performed using significantly different loading compositions, each of which is selected from 'sensitive' regions of the discriminator map.     

Membrane transport models with fast and slow reactions: General analytical solution for a single relaxation

Summary Membrane transport models are usually expressed on the basis of chemical kinetics. The states of a transporter are related by rate constants, and the time-dependent changes of these states are given by linear differential equations of first order. To calculate the time-dependent transport equation, it is necessary to solve a system of differential equations which does not have a general analytical solution if there are more than five states. Since transport measurements in a complex system rarely provide all the time constants because some of them are too rapid, it is more appropriate to obtain approximate analytical solutions, assuming that there are fast and slow reaction steps. The states of the fast steps are related by equilibrium constants, thus permitting the elimination of their differential equations and leaving only those for the slow steps. With a system having only two slow steps, a single differential equation is obtained and the state equations have a single relaxation. Initial conditions for the slow reactions are determined after the perturbation which redistribute the states related by fast reactions. Current and zero-trans uptake equations are calculated. Curve fitting programs can be used to implement the general procedure and obtain the model parameters.     

Determination of erythrocyte aggregation

One of the most common health criteria--erythrocyte sedimentation rate (ESR)--is considered in the paper. It is shown that the simple model presented, based on the generalized Stokes formula, the blood volume conservation law, and the Smoluchowski theory of particles coagulation, makes it possible, on the basis of experimentally recorded sedimentation curves, to identify quantitatively the values of the essential physical parameters of the coupled processes of erythrocyte aggregation and sedimentation. The analytical solution of Smoluchowski equation is used to evaluate the sedimentation and aggregation rate constants. The problem of determining the erythrocyte aggregation rate (EAR) is transformed to a minimization task in which only the experimental results for ESR are needed. Experimentally ESR is measured accurately enough by using an equipment set up just for the purpose. This method of identification could be used as a diagnostic test in hematological laboratories.     

A model for sedimentation in inhomogeneous media. I. Dynamic density gradients from sedimenting co-solutes

Macromolecular sedimentation in inhomogeneous media is of great practical importance. Dynamic density gradients have a long tradition in analytical ultracentrifugation, and are frequently used in preparative ultracentrifugation. In this paper, a new theoretical model for sedimentation in inhomogeneous media is presented, based on finite element solutions of the Lamm equation with spatial and temporal variation of the local solvent density and viscosity. It is applied to macromolecular sedimentation in the presence of a dynamic density gradient formed by the sedimentation of a co-solute at high concentration. It is implemented in the software SEDFIT for the analysis of experimental macromolecular concentration distributions. The model agrees well with the measured sedimentation profiles of a protein in a dynamic cesium chloride gradient, and may provide a measure for the effects of hydration or preferential solvation parameters. General features of protein sedimentation in dynamic density gradients are described.     

Bayesian inference for parameter estimation in lactoferrin-mediated iron transport across blood-brain barrier

BackgroundIn neurodegenerative diseases such as Alzheimer's and Parkinson's, excessive irons as well as lactoferrin (Lf), but not transferrin (Tf), have been found in and around the affected regions of the brain. These evidences suggest that lactoferrin plays a critical role during neurodegenerative diseases, although Lf-mediated iron transport across blood-brain barrier (BBB) is negligible compared to that of transferrin in normal condition. However, the kinetics of lactoferrins and lactoferrin-mediated iron transport are still unknown.MethodTo determine the kinetic rate constants of lactoferrin-mediated iron transport through BBB, a mass-action based ordinary differential equation model has been presented. A Bayesian framework is developed to estimate the kinetic rate parameters from posterior probability density functions. The iron transport across BBB is studied by considering both Lf- and Tf-mediated pathways for both normal and pathologic conditions.ResultsUsing the point estimates of kinetic parameters, our model can effectively reproduce the experimental data of iron transport through BBB endothelial cells. The robustness of the model and parameter estimation process are further verified by perturbation of kinetic parameters. Our results show that surge in high-affinity receptor density increases lactoferrin as well as iron in the brain.ConclusionsDue to the lack of a feedback loop such as iron regulatory proteins (IRPs) for lactoferrin, iron can transport to the brain continuously, which might increase brain iron to pathological levels and can contribute to neurodegeneration.General significanceThis study provides an improved understanding of presence of lactoferrin and iron in the brain during neurodegenerative diseases.     

Transient velocity sedimentation of cells at unit gravity.

A new kinetic method (TRANS-VELS) is described which allows for the first time the accurate determination of the sedimentation velocity of cells at unit gravity. This is accomplished by repetitive optical scanning of the cell distribution as a function of time and during transport through a shallow density gradient. Computer analysis of the statistical moments of the distribution is utilized for the measurement of the sedimentation velocity, its dispersion and the expected resolution. The latter two parameters being strongly time dependent have been estimated for the first time from kinetic data and bear important implications in the widely practiced preparative separation of cells by velocity sedimentation at unit gravity.     

Two enzyme active transport in vitro with ph induced asymmetrical functional structures: I. The model and its analytical treatment

A class of systems is characterized by the asymmetrical distribution of a sink and a source reaction, the asymmetry of the global chemical equation (energy liberation) and by an asymmetrical one-wave space profile. These systems belong to the family of primary chemical cells and can deplete and enrich the media they separate. A “ one way ” transport-reaction chain is needed for specific “ real ” active transport. A two enzyme model of this class is described in which the spatial asymmetry is due to a (diffusive) pH gradient; this distribution of “ potential ” enzyme activities is called the “ functional structure ”. Equal potential enzyme activities and absence of reactive back action on local pH are assumed in the “ square model ” version of the pump. Analytical expressions of the enzymatic diffusion reactions are derived for zero and first order kinetics, i.e. in function of substrate concentrations. Tables of equations are presented. The intrinsic properties of the pump are characterized by (dimensionless) transport reaction parameters, (membrane composition); the “ potential ” activity is controlled by the pH gradient; the “ effective ” pumping is also a function of the substrate concentrations on the boundaries.     

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.     

Analysis of heterologous interacting systems by sedimentation velocity: curve fitting algorithms for estimation of sedimentation coefficients, equilibrium and kinetic constants

Analytical ultracentrifugation (AUC) has played and will continue to play an important role in the investigation of protein-protein, protein-DNA and protein-ligand interactions. A major advantage of AUC over other methods is that it allows the analysis of systems free in solution in nearly any buffer without worry about spurious interactions with a supporting matrix. Large amounts of high-quality data can be acquired in relatively short times. Advances in software for the treatment of AUC data over the last decade have eliminated many of the tedious aspects of AUC data analysis, allowing relatively rapid analysis of complicated systems that were previously unapproachable. A software package called sedanal is described that can perform global fits to AUC sedimentation velocity data obtained for both interacting and non-interacting, macromolecular multi-species, multi-component systems, by combining data from multiple runs over a range of sample concentrations and component ratios. Interaction parameters include both forward and reverse rate constants, or equilibrium constants, for each reaction, as well as concentration dependence of both sedimentation and diffusion coefficients. sedanal fits to time-difference data to eliminate time-independent systematic errors inherent in AUC data. The sedanal software package is based on the use of finite-element numerical solutions of the Lamm equation.     

Structure refinement of the aldehyde oxidoreductase from Desulfovibrio gigas (MOP) at 1.28 Å

The sulfate-reducing bacterium aldehyde oxidoreductase from Desulfovibrio gigas (MOP) is a member of the xanthine oxidase family of enzymes. It has 907 residues on a single polypeptide chain, a molybdopterin cytosine dinucleotide (MCD) cofactor and two [2Fe-2S] iron-sulfur clusters. Synchrotron data to almost atomic resolution were collected for improved cryo-cooled crystals of this enzyme in the oxidized form. The cell constants of a=b=141.78 A and c=160.87 A are about 2% shorter than those of room temperature data, yielding 233,755 unique reflections in space group P6(1)22, at 1.28 A resolution. Throughout the entire refinement the full gradient least-squares method was used, leading to a final R factor of 14.5 and Rfree factor of 19.3 (4sigma cut-off) with "riding" H-atoms at their calculated positions. The model contains 8146 non-hydrogen atoms described by anisotropic displacement parameters with an observations/parameters ratio of 4.4. It includes alternate conformations for 17 amino acid residues. At 1.28 A resolution, three Cl- and two Mg2+ ions from the crystallization solution were clearly identified. With the exception of one Cl- which is buried and 8 A distant from the Mo atom, the other ions are close to the molecular surface and may contribute to crystal packing. The overall structure has not changed in comparison to the lower resolution model apart from local corrections that included some loop adjustments and alternate side-chain conformations. Based on the estimated errors of bond distances obtained by blocked least-squares matrix inversion, a more detailed analysis of the three redox centres was possible. For the MCD cofactor, the resulting geometric parameters confirmed its reduction state as a tetrahydropterin. At the Mo centre, estimated corrections calculated for the Fourier ripples artefact are very small when compared to the experimental associated errors, supporting the suggestion that the fifth ligand is a water molecule rather than a hydroxide. Concerning the two iron-sulfur centres, asymmetry in the Fe-S distances as well as differences in the pattern of NH.S hydrogen-bonding interactions was observed, which influences the electron distribution upon reduction and causes non-equivalence of the individual Fe atoms in each cluster.     

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.     

Size-distribution analysis of macromolecules by sedimentation velocity ultracentrifugation and lamm equation modeling

A new method for the size-distribution analysis of polymers by sedimentation velocity analytical ultracentrifugation is described. It exploits the ability of Lamm equation modeling to discriminate between the spreading of the sedimentation boundary arising from sample heterogeneity and from diffusion. Finite element solutions of the Lamm equation for a large number of discrete noninteracting species are combined with maximum entropy regularization to represent a continuous size-distribution. As in the program CONTIN, the parameter governing the regularization constraint is adjusted by variance analysis to a predefined confidence level. Estimates of the partial specific volume and the frictional ratio of the macromolecules are used to calculate the diffusion coefficients, resulting in relatively high-resolution sedimentation coefficient distributions c(s) or molar mass distributions c(M). It can be applied to interference optical data that exhibit systematic noise components, and it does not require solution or solvent plateaus to be established. More details on the size-distribution can be obtained than from van Holde-Weischet analysis. The sensitivity to the values of the regularization parameter and to the shape parameters is explored with the help of simulated sedimentation data of discrete and continuous model size distributions, and by applications to experimental data of continuous and discrete protein mixtures.     

Robust nonlinear parameter estimation in tracer kinetic analysis using infinity norm regularization and particle swarm optimization

In positron emission tomography (PET) studies, the voxel-wise calculation of individual rate constants describing the tracer kinetics is quite challenging because of the nonlinear relationship between the rate constants and PET data and the high noise level in voxel data. Based on preliminary simulations using a standard two-tissue compartment model, we can hypothesize that it is possible to reduce errors in the rate constant estimates when constraining the overestimation of the larger of two exponents in the model equation. We thus propose a novel approach based on infinity-norm regularization for limiting this exponent. Owing to the non-smooth cost function of this regularization scheme, which prevents the use of conventional Jacobian-based optimization methods, we examined a proximal gradient algorithm and the particle swarm optimization (PSO) through a simulation study. Because it exploits multiple initial values, the PSO method shows much better convergence than the proximal gradient algorithm, which is susceptible to the initial values. In the implementation of PSO, the use of a Gamma distribution to govern random movements was shown to improve the convergence rate and stability compared to a uniform distribution. Consequently, Gamma-based PSO with regularization was shown to outperform all other methods tested, including the conventional basis function method and Levenberg–Marquardt algorithm, in terms of its statistical properties.     

Efficient maximum likelihood estimation of kinetic rate constants from macroscopic currents

A new method is described that accurately estimates kinetic constants, conductance and number of ion channels from macroscopic currents. The method uses both the time course and the strength of correlations between different time points of macroscopic currents and utilizes the property of semiseparability of covariance matrix for computationally efficient estimation of current likelihood and its gradient. The number of calculation steps scales linearly with the number of channel states as opposed to the cubic dependence in a previously described method. Together with the likelihood gradient evaluation, which is almost independent of the number of model parameters, the new approach allows evaluation of kinetic models with very complex topologies. We demonstrate applicability of the method to analysis of synaptic currents by estimating accurately rate constants of a 7-state model used to simulate GABAergic macroscopic currents.     

Improved methods for calculating formation constants for nucleotide-cation complexes

Two data reduction methods that can be used to calculate formation constants for nucleotidecation complexes are described. Both methods are used to analyze data obtained by an anion-exchange resin method, and either method can improve the accuracy of the calculated formation constants by more than 50%. The key to this significant improvement in accuracy is the realization that the equation for the mathematical model describing such systems is always nonlinear in terms of the formation constants and, in the general case, is higher order in the cation concentration. There are two major reasons for the improved quality of the results associated with the new model. First, successive linear extrapolations are eliminated, and error propagation is reduced. Second, all of the data are used for the simultaneous calculation of all formation constants, and the uncertainty due to random experimental errors is minimized.     

Sedimentation of generalized systems of interacting particles. III. Concentration-dependent sedimentation and extension to other transport methods

The simulation method presented in the previous papers is related to the concentration-dependent sedimentation–diffusion. It can be shown that the efficiency of the program described previously is maintained. A simulation of a system exhibiting the Johnston–Ogston effect is presented. Through the similarity of their continuity equations with the Lamm equation, electrophoresis, molecular sieve, and chromatography are treated. A general simulation of transport for systems of many interacting components is thus presented, which is able to take into account kinetically controlled chemical reactions and nonideal phenomena.     

Isolation of preparative amounts of polyribosomes from normal rabbit and guinea pig spleen]

Preparative amounts of polyribosomes were isolated from normal rabbit and guinea pig spleen; up to 40 optical units of the polyribosome preparation could be obtained by centrifugation in a Spinco L-2B centrifuge with SW-27 rotor. The amount of polyribosomes isolated from spleens of immune animals was 2-3 higher than that isolated from normal animal spleens. Concentration of polyribosomal preparations by lyophylization and the storage of dried preparations do not alter the sedimentation properties of the polyribosomes. The distribution pattern of normal rabbit spleen polyribosomes in a linear sucrose gradient and the sedimentation constants of the polyribosome peaks are in good agreement with data reported by some other authors for plasmocytome polyribosomes. Using electrophoresis in agarose-polyacrylamide gel the radioactive proteins synthesized in the cell culture of normal rabbit spleen it was shown that in normal spleen the average amount of globulins makes up to 35% of total protein synthesis, as reported by some authors.