Boundary analysis in sedimentation transport experiments: a procedure for obtaining sedimentation coefficient distributions using the time derivative of the concentration profile.

A procedure is described for computing sedimentation coefficient distributions from the time derivative of the sedimentation velocity concentration profile. Use of the time derivative, (delta c/delta t)r, instead of the radial derivative, (delta c/delta r)t, is desirable because it is independent of time-invariant contributions to the optical baseline. Slowly varying baseline changes also are significantly reduced. An apparent sedimentation coefficient distribution (i.e., uncorrected for the effects of diffusion), g*(s), can be calculated from (delta c/delta t)r as [formula: see text] where s is the sedimentation coefficient, omega is the angular velocity of the rotor, c0 is the initial concentration, r is the radius, rm is the radius of the meniscus, and t is time. An iterative procedure is presented for computing g*(s)t by taking into account the contribution to (delta c/delta t)r from the plateau region to give (delta c/delta t)corr. Values of g*(s)t obtained this way are identical to those of g*(s) calculated from the radial derivative to within the roundoff error of the computations. Use of (delta c/delta t)r, instead of (delta c/delta r)t, results in a significant increase (greater than 10-fold) in the signal-to-noise ratio of data obtained from both the uv photoelectric scanner and Rayleigh optical systems of the analytical ultracentrifuge. The use of (delta c/delta t)r to compute apparent sedimentation coefficient distributions for purposes of boundary analysis is exemplified with an antigen-antibody system.     

On the analysis of sedimentation velocity in the study of protein complexes

Sedimentation velocity analytical ultracentrifugation has experienced a significant transformation, precipitated by the possibility of efficiently fitting Lamm equation solutions to the experimental data. The precision of this approach depends on the ability to account for the imperfections of the experiment, both regarding the sample and the instrument. In the present work, we explore in more detail the relationship between the sedimentation process, its detection, and the model used in the mathematical data analysis. We focus on configurations that produce steep and fast-moving sedimentation boundaries, such as frequently encountered when studying large multi-protein complexes. First, as a computational tool facilitating the analysis of heterogeneous samples, we introduce the strategy of partial boundary modeling. It can simplify the modeling by restricting the direct boundary analysis to species with sedimentation coefficients in a predefined range. Next, we examine factors related to the experimental detection, including the magnitude of optical aberrations generated by out-of-focus solution columns at high protein concentrations, the relationship between the experimentally recorded signature of the meniscus and the meniscus parameter in the data analysis, and the consequences of the limited radial and temporal resolution of the absorbance optical scanning system. Surprisingly, we find that large errors can be caused by the finite scanning speed of the commercial absorbance optics, exceeding the statistical errors in the measured sedimentation coefficients by more than an order of magnitude. We describe how these effects can be computationally accounted for in SEDFIT and SEDPHAT.     

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.     

Malonyl CoA inhibition of carnitine palmityltransferase in rat heart mitochondria

The isolated beta 2-dimer of Escherichia coli tryptophan synthase exhibits reversible high-pressure deactivation and hybridization with an equilibrium transition at 690 and 870 bar for the apoenzyme and holoenzyme, respectively. To investigate the hypothetical dissociation mechanism ultracentrifugal analysis has been applied. In a conventional swing-out rotor (r(max) = 16 cm, fill-height 9 cm) a pressure gradient of 1 less than p less than 1840 bar is formed at maximum speed (40 000 rpm). Using a sucrose gradient to stabilize the particle distribution, pressure-dependent alterations of the state of association of oligomeric systems may be determined. In the present experiments ovalbumin (with a molecular mass close to the beta-monomer) has been used as a reference. The radial sedimentation velocity of the beta 2-dimer (in 5-20% sucrose, 10 degrees C) is found to decrease significantly at p approximately equal to 850 bar. From the slopes in an r-r(degrees) vs t plot the limiting values for the particle weight at the meniscus and the bottom of the tube are found to be the beta 2-dimer (M(r) = 85 800) and the beta-monomer (M(r) = 42 900), thus proving pressure-dependent dissociation. Since sucrose stabilizes the native quaternary structure, the beta 2 leads to 2 beta transition is shifted towards higher pressures compared to the dissociation in standard buffer. Conventional quench experiments in high-pressure cells in the presence of 13% (w/v) sucrose confirm the result of the sucrose gradient centrifugation with respect to the critical pressure where deactivation (and dissociation) occur.     

Diffusion of the Reaction Boundary of Rapidly Interacting Macromolecules in Sedimentation Velocity

Sedimentation velocity analytical ultracentrifugation combines relatively high hydrodynamic resolution of macromolecular species with the ability to study macromolecular interactions, which has great potential for studying dynamically assembled multiprotein complexes. Complicated sedimentation boundary shapes appear in multicomponent mixtures when the timescale of the chemical reaction is short relative to the timescale of sedimentation. Although the Lamm partial differential equation rigorously predicts the evolution of concentration profiles for given reaction schemes and parameter sets, this approach is often not directly applicable to data analysis due to experimental and sample imperfections, and/or due to unknown reaction pathways. Recently, we have introduced the effective particle theory, which explains quantitatively and in a simple physical picture the sedimentation boundary patterns arising in the sedimentation of rapidly interacting systems. However, it does not address the diffusional spread of the reaction boundary from the cosedimentation of interacting macromolecules, which also has been of long-standing interest in the theory of sedimentation velocity analytical ultracentrifugation. Here, effective particle theory is exploited to approximate the concentration gradients during the sedimentation process, and to predict the overall, gradient-average diffusion coefficient of the reaction boundary. The analysis of the heterogeneity of the sedimentation and diffusion coefficients across the reaction boundary shows that both are relatively uniform. These results support the application of diffusion-deconvoluting sedimentation coefficient distributions c(s) to the analysis of rapidly interacting systems, and provide a framework for the quantitative interpretation of the diffusional broadening and the apparent molar mass values of the effective sedimenting particle in dynamically associating systems.     

Dimer-tetramer association equilibria of human adult hemoglobin and its mutants as observed by analytical ultracentrifugation

Dimer-tetramer equilibrium of human adult hemoglobin in CO form (COHb A) and its mutants were measured by sedimentation velocity and sedimentation equilibrium. In sedimentation velocity, the association constants were estimated by measuring the concentration dependence of the weight average sedimentation coefficients at pH 6 and 7 and fitting the data to the theoretical binding isotherms with association constants as a parameter. Association constants of wild type Hb A and three mutant Hbs, Hb Hirose(βW37S), recombinant (r)Hb(βW37H) and rHb(αY42S), in which an amino acid was replaced at the α(1)β(2) interface, were measured in the presence and absence of inositol hexaphosphate (IHP). All the three mutations lowered the value of association constants, but the presence of IHP shifted the equilibrium toward tetramer. Although the association constant between dimer and tetramer of rHb(βW37H) and rHb(αY42S) were similar, sedimentation coefficient distribution function, c(s), analysis indicated that the association and dissociation rate constants of the former is higher than the latter.     

Seston sedimentation in a lowland river (River Spree,Germany): their spatial and temporal variations and controlling factors

Sedimentation of organic particles plays a decisive role in streams in relation to pelagic loss as well as retention of nutrients and other substances. The plate sediment trap allows for the direct measurement of these net fluxes. Biweekly measurements were undertaken in the eutrophic lowland River Spree (MQ 14 m³ s^–1) 10 km upstream of Berlin in 1999 and 2000. Trapping rates between 0.5 and 25 g DW m^–2 day^–1 were found near the bank. The variance of seston sedimentation is controlled by the seston concentration, the settling velocity of the particles and the flow velocity. The sinking velocity exhibits significant seasonal fluctuations with highest values in summer. It is shown that the critical flow velocity for sedimentation is another important parameter. This controls the distribution of sedimentation over the width of the river and thus the effective average sedimentation rate for the entire river segment. This average rate ranged between 0.9 and 6.6 g DW m^–2 day^–1.     

Simultaneous rapid estimation of sedimentation coefficient and molecular weight

Data obtained from the early portion of sedimentation velocity experiments may be analyzed to simultaneously estimate both s and s/D. The C versus r data obtained are analyzed using a nonlinear least squares algorithm and an approximate solution to the Lamm equation. This procedure was tested both with simulated noisy data and with experimental data obtained using ribonuclease, ovalbumin, and somatostatin.dodecylsulfate. The procedure assumes that both s and D are independent of concentration. The results suggest that optimal estimation of both s and s/D is obtained at values of (= 2D/(s.omega(2).r(a)(2))) in the range of 0.002 to 0.01 and values of tau(= 2 somega(2)t) less than 0.04. Appropriate selection of rotor speed allows the estimation of both s and s/D for nearly globular macromolecules in the range of 10(4) to 10(6) daltons with data obtained during the first 3000-5000 seconds of a sedimentation velocity experiment.     

A parameterized overspeeding method for the rapid attainment of low-speed sedimentation equilibrium

The approach of a solution of dilute, monodisperse, globular macromolecules to low-speed sedimentation equilibrium in an ultracentrifuge is simulated by numerical integration of the Lamm equation. Various combinations of overspeed time and angular velocity are used to assess the conditions needed to minimize the time it takes the solution to attain sedimentation equilibrium. The optimal overspeeding time and angular velocity are determined over a wide range of values of the molecular weight (relative molar mass) of the solute and the radial distance between the meniscus and base of the solution. The results may be expressed as simple functions which allow facile calculation of (a) the optimal overspeeding time and velocity, and (b) the time required to reach sedimentation equilibrium. The results are in reasonable agreement with previous analytical solutions which were based on several simplifying assumptions. The parameterized overspeeding procedure is shown to be robust over a wide range of conditions, and typically leads to a greater than 5-fold reduction in centrifugation time.     

Density contrast sedimentation velocity for the determination of protein partial-specific volumes

The partial-specific volume of proteins is an important thermodynamic parameter required for the interpretation of data in several biophysical disciplines. Building on recent advances in the use of density variation sedimentation velocity analytical ultracentrifugation for the determination of macromolecular partial-specific volumes, we have explored a direct global modeling approach describing the sedimentation boundaries in different solvents with a joint differential sedimentation coefficient distribution. This takes full advantage of the influence of different macromolecular buoyancy on both the spread and the velocity of the sedimentation boundary. It should lend itself well to the study of interacting macromolecules and/or heterogeneous samples in microgram quantities. Model applications to three protein samples studied in either H(2)O, or isotopically enriched H(2) (18)O mixtures, indicate that partial-specific volumes can be determined with a statistical precision of better than 0.5%, provided signal/noise ratios of 50-100 can be achieved in the measurement of the macromolecular sedimentation velocity profiles. The approach is implemented in the global modeling software SEDPHAT.     

Basic ideas and concepts for multiple comparison procedures

Around 1970, the author proposed a general theoretical approach to multiple decision problems (MDPs) of which multiple comparison problems (MCPs) are special cases. Suppose that a sample space Χ is given together with a set of probability distributions P = {P(θ), θ ∈ Ω} defined over Χ. Let a finite partition of the parameter space Ω = cupω(a), a ∈ A be given. Based on the observation X ∈ X, an MDP is to decide, which ω(a) the true parameter θ belongs to. An MD confidence procedure is a mapping ψ from X to the class of subsets of A, such that the probability that cupω(a), ω(a) C ψ(X) includes the true parameter θ is not smaller than 1-α(θ) . Here, 1-α(θ) is called the level of the confidence procedures and may vary depending on θ∈ω(a) . The MP confidence procedures are derived from the following proposition. When the ω(a) 's are mutually disjoint, there is a one-to-one correspondence between an MD confidence procedure ψ and a collection of (non-randomized) tests for the hypotheses H(a)?: θ∈ω(a) with level α(a) by rejecting the hypothesis H(a) if ω(a) ? ψ(X). In this paper we discuss in detail the problems of determining the signs or the orderings of normal means. The resulting confidence procedures from the LR tests are seen to be too complicated and difficult to understand. We therefore propose simplified, less powerful methods. We define an overlapping partition of Ω into simple sets, such that the original ω(a) 's can be expressed as an intersection of such simple sets. For each such set we define rejection regions corresponding to the levels α, α/2,...,α/k. Then we obtain the acceptance regions for H(a) :?θ∈ω(a) given as the intersection of all acceptance regions for the simple sets containing ω(a) at the level α/k, if there are k such simple sets. This method can be extended to obtain sequential confidence procedures.     

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.     

The use of metrizamide for stabilizing against convection in sedimentation equilibrium

In meniscus depletion sedimentation equilibrium experiments, the solute concentration gradient in much of the cell is insufficient to stabilize against thermal convection. Experiments with spectrin at 35 degrees C led to sufficient convection that, particularly at low concentrations of solute, no apparent redistribution occurred. At low angular velocity, the density gradient provided by moderate concentrations of NaCl or additional sucrose is inadequate to overcome the convective disturbances. We have found that the use of low concentrations of metrizamide can overcome these problems. The higher molecular weight and lower partial specific volume of metrizamide vis-a-vis sucrose allow a significant and stabilizing density gradient even at angular velocities below 8000 rpm.     

ANTIBODY-PRODUCING CELLS: ANALYSIS AND PURIFICATION BY VELOCITY SEDIMENTATION

Velocity sedimentation cell separation is a simple and reproducible method for obtaining highly enriched populations of viable antibody-producing cells. Using suspensions of spleen cells prepared from mice immunized with sheep erythrocytes, fractions containing up to 2% 19S-PFC and 25% 7S-PFC can be obtained. Granulocytes constitute almost all of the remaining cells in these fractions. The sedimentation profile of 7S-PFC is very broad in comparison with that of cell populations known to be homogeneous in size (e.g. mouse erythrocytes). Analysis of the profile of 7S-PFC at different times after immunization suggests that the heterogeneity arises largely from the doubling in cell volume as a cell moves from one mitosis to the next. Early in the immune response, when the majority of the PFC are proliferating, the variation in sedimentation velocities is consistent with such a two-fold variation in cell volume. Late in the response, when most PFC have stopped proliferating, the sedimentation profile is more homogeneous. This analysis suggests that the fractionation procedure is sensitive enough to separate PFC according to their position in the cell cycle. Sedimentation velocities were also measured for several other classes of cells found in spleen. Comparison of these values shows that sedimentation velocity is a useful parameter for characterizing different types of cells.     

Applications of analytical ultracentrifugation to protein size-and-shape distribution and structure-and-function analyses

The rebirth of modern analytical ultracentrifugation (AUC) began in 1990s. Since then many advanced AUC detectors have been developed that provide a vast range of versatile choices when characterizing the physical and chemical features of macromolecules. In addition, there have been remarkable advances in software that allow the analysis of AUC data using more sophisticated models, including quaternary structures, conformational changes, and biomolecular interactions. Here we report the application of AUC to protein size-and-shape distribution analysis and structure-and-function analysis in the presence of ligands or lipids. Using band-sedimentation velocity, quaternary structural changes and an enzyme's catalytic activity can be observed simultaneously. This provides direct insights into the correlation between quaternary structure and catalytic activity of the enzyme. On the other hand, also in this study, we have applied size-and-shape distribution analysis to a lipid-binding protein in either an aqueous or lipid environment. The sedimentation velocity data for the protein with or without lipid were evaluated using the c(s,f(r)) two-dimensional distribution model, which provides a precise and quantitative means of analyzing the protein's conformational changes.     

Conformational spectra--probing protein conformational changes

Stafford [Biophys. J. 17 (1996) MP452] has shown that it is possible, using the analytical ultracentrifuge in sedimentation velocity mode, to calculate the molecular weights of proteins with a precision of approximately 5%, by fitting Gaussian distributions to g(s*) profiles so long as partial specific volume and the radial position of the meniscus are known. This makes possible the analysis of systems containing several components by the fitting of multiple distributions to the total g(s*) profile. We have found the Stafford relationship to hold for a range of protein solutes, particularly good agreement being found when the g(s*) profiles are computed from Schlieren (dc/dr vs. r) data using the Bridgman equation [J. Am. Chem. Soc. 64 (1942) 2349] . On this basis, we have developed a new approach to the analysis of systems where two or more distinguishable conformations of a single species are present, either in the same sample cell or in different cells in the same rotor. In the former case, this allows us to analyse a given solution of pure protein (i.e. monodisperse with respect to M) to reveal the presence in that solution of two or more conformers under identical solvent conditions. In the latter case, we can detect with high sensitivity any conformational change occurring in the transition from one set of solvent conditions to another. Alternatively, in this case, we can analyse slightly different proteins (e.g. deletion mutants) for conformational changes under identical solvent conditions. Examples of these procedures using well-defined protein systems are given.     

An extrapolation method for reducing equilibration times in sedimentation equilibrium experiments.

We present a detailed investigation of the use of an extrapolation technique to decrease running times of sedimentation equilibrium experiments. If concentration profiles are available at time delta tau, 2delta tau, 3delta tau,...., cn(r) = c(r, n delta tau), then the Aitken transformation replaces the cn(r) + ĉn(r) = [cn + 1(r) cn - 1(r) - c2n(r)]/[cn + 1(r) + cn - 1(r) - 2cn(r)]. We show that the ĉn(r) converge to the equilibrium values c infinity (r) much more quickly than the cn(r). Savings in time are shown to range from a factor of approximately 2 for meniscus depletion experiments to factors of between 4 and 8 for lower speeds or smaller molecular weights. It is also shown that the technique is quite sensitive to noise, so that an accurate optical system is required to allow its optimal use.     

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.     

Size-distribution analysis of proteins by analytical ultracentrifugation: strategies and application to model systems.

Strategies for the deconvolution of diffusion in the determination of size-distributions from sedimentation velocity experiments were examined and developed. On the basis of four different model systems, we studied the differential apparent sedimentation coefficient distributions by the time-derivative method, g(s*), and by least-squares direct boundary modeling, ls-g*(s), the integral sedimentation coefficient distribution by the van Holde-Weischet method, G(s), and the previously introduced differential distribution of Lamm equation solutions, c(s). It is shown that the least-squares approach ls-g*(s) can be extrapolated to infinite time by considering area divisions analogous to boundary divisions in the van Holde-Weischet method, thus allowing the transformation of interference optical data into an integral sedimentation coefficient distribution G(s). However, despite the model-free approach of G(s), for the systems considered, the direct boundary modeling with a distribution of Lamm equation solutions c(s) exhibited the highest resolution and sensitivity. The c(s) approach requires an estimate for the size-dependent diffusion coefficients D(s), which is usually incorporated in the form of a weight-average frictional ratio of all species, or in the form of prior knowledge of the molar mass of the main species. We studied the influence of the weight-average frictional ratio on the quality of the fit, and found that it is well-determined by the data. As a direct boundary model, the calculated c(s) distribution can be combined with a nonlinear regression to optimize distribution parameters, such as the exact meniscus position, and the weight-average frictional ratio. Although c(s) is computationally the most complex, it has the potential for the highest resolution and sensitivity of the methods described.