Sedimentation of generalized systems of interacting particles. II. Active enzyme centrifugation--theory and extensions of its validity range.

The first detailed application of a recently published very general approach to chemical equilibria during sedimentation is presented. As a consequence of the very extensive theoretical treatment, made possible by this approach, the active enzyme analytical centrifugation method can now be used under a far wider set of conditions than before, including the study of many interacting active molecule systems. It has also been shown that this method is as precise as the more conventional ones.     

Broad-zone active-enzyme chromatography: Keto-acid dehydrogenases as associating systems

We have extended the method of active-enzyme chromatography to include the use of broad zones of enzyme. This allows examination of interacting systems in a way formally analogous to sedimentation velocity so that simulation of the observed activity profiles is possible. The method has been applied using pyridine nucleotide-linked active enzyme assays. At the concentrations presently accessible by this technique, hexokinase and glucose-6-phosphate dehydrogenase, both associating systems, show single symmetrical boundaries, as does isolated diaphorase, while pyruvate and α-ketoglutarate dehydrogenases show more complex patterns, with the position of the reaction boundary for diaphorase activity being dependent on enzyme concentration.     

Boundary spreading in sedimentation velocity experiments on partially liganded aspartate transcarbamoylase. A ligand-mediated isomerization

Transport theory for rapidly reversible interacting systems was used to analyze boundary spreading in sedimentation velocity experiments on partially liganded aspartate transcarbamoylase. In the presence of sub-stoichiometric amounts of a bisubstrate analog, N-(phosphonacetyl)-L-aspartate, which is bound with high affinity to the enzyme (Kd approximately 100 nM), broad boundaries were observed consistent with the presence of two conformational forms. The theoretical treatment showed that under these conditions, the interconversion between the compact (11.7 S) and swollen (11.3 S) forms of the enzyme appears uncoupled, due to the formation of a gradient of free ligand that is caused by the re-equilibration resulting from the differential sedimentation of the two enzyme forms. Sedimentation velocity patterns for such systems are interpretable in terms of two independent species. When, however, the enzyme is in the presence of a sub-saturating amount of the weakly bound ligand, succinate (Kd approximately 1 mM), the re-equilibration caused by the differential sedimentation does not perturb the large background of free ligand and form a gradient. Instead, the two different forms of the enzyme are in dynamic equilibrium, resulting in a boundary having average sedimentation and diffusion coefficients. The observed boundary spreading experiments with different ligands are satisfactorily interpreted in terms of a ligand-mediated isomerization of aspartate transcarbamoylase from a compact to a swollen conformation.     

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.     

Macro and micro rate zonal analytical centrifugation of polydisperse and slowly diffusing sedimenting systems in isovolumetric density gradients. Application to cartilage proteoglycans

A method to study the polydispersity of zonally sedimenting and slowly diffusing macromolecules or particles in isokinetic or isovolumetric density gradients is presented. First, a brief theory is given for predicting the zonal profile after a "triangular" (or "inverse") zone is centrifuged. This type of zone is essential to preserve hydrodynamic stability of the very slowly diffusing polydisperse solutes. It is proven, both by semitheoretical considerations and by computer calculations, that the resulting concentration profile of macrosolute is almost identical with that obtainable with a rectangular zone coextensive with the triangular one and carrying the same total mass. Next, practical procedures are described for the convectionless layering of very small triangular zones (50 microL or less). The linearity and stability of the zones are experimentally tested and verified. Finally, the method is applied to cartilage proteoglycan preparations that included either the monomeric molecules only or both the monomeric and the aggregated ones. The zonal results are compared with those obtained by using conventional boundary sedimentation. The two sets of results are seen to coincide fairly well, thus proving that the present technique can add to preparative zonal centrifugation the analytical precision of boundary sedimentation. A multimodal polydisperse system is suggested to describe the aggregated proteoglycan macromolecules.     

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.     

Sedimentation velocity analysis of highly heterogeneous systems

This article discusses several improvements to the van Holde-Weischet (vHW) method [Biopolymers 17 (1978) 1387] that address its capability to deal with sedimentation coefficient distributions spanning a large range of s values. The method presented here allows the inclusion of scans early and late in the experiment that ordinarily would need to be excluded from the analysis due to ultracentrifuge cell end effects. Scans late in the experiment are compromised by the loss of a defined plateau region and by back-diffusion from the bottom of the cell. Early scans involve partial boundaries that have not fully cleared the meniscus. In addition, a major refinement of the algorithm for determining the boundary fractions is introduced, taking into account different degrees of radial dilution for different species in the system. The method retains its desirable model-independent properties (the analysis of sedimentation data does not require prior knowledge of a user-imposed model or range of sedimentation coefficients) and reports diffusion-corrected s value distributions, which can be presented either in a histogram format or the traditional integral distribution format. Data analyzed with the traditional vHW method are compared with those of the improved method to demonstrate the benefit from the added information in the analysis.     

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.     

Transient-time analysis of substrate-channelling in interacting enzyme systems.

The kinetics of dynamically interacting enzyme systems is examined, in the light of increasing evidence attesting to the widespread occurrence of this mode of organization in vivo. The transient time, a key phenomenological parameter for the coupled reaction, is expressed as a function of the lifetime of the intermediate substrate. The relationships between the transient time and the pseudo-first-order rate constants for the coupled reaction by the complexed and uncomplexed enzyme species are indicative of the mechanism of intermediate transfer ('channelling'). In a dynamically interacting enzyme system these kinetic parameters are composite functions of those for the processes catalysed by the complex and by the separated enzymes. The mathematical paradigm can be extended to a linear sequence of N coupled reactions catalysed by dynamically (pair-wise) interacting enzymes.     

Theory and practice of the analytical centrifugation of an active substrate-enzyme complex

The active enzyme centrifugal-ion (AEC) method presented here permits the hydrodynamic study of active enzyme–substrate(s) complexes in solutions containing micro-gram amounts of the studied enzyme, even if the enzyme preparation is very impure. The AEC method can he used only when the specific enzymatie reaction can be measured directly in a spectrophotometer. The general equations relevant to the method and their solutions are presented in detail. Their use requires some numerical calculations. A practical summary of the AEC method is given, and the precision of the measured values of the sedimentation and diffusion coefficients is discussed.     

Analytical ultracentrifugation as a tool for studying protein interactions

The last two decades have led to significant progress in the field of analytical ultracentrifugation driven by instrumental, theoretical, and computational methods. This review will highlight key developments in sedimentation equilibrium (SE) and sedimentation velocity (SV) analysis. For SE, this includes the analysis of tracer sedimentation equilibrium at high concentrations with strong thermodynamic non-ideality, and for ideally interacting systems, the development of strategies for the analysis of heterogeneous interactions towards global multi-signal and multi-speed SE analysis with implicit mass conservation. For SV, this includes the development and applications of numerical solutions of the Lamm equation, noise decomposition techniques enabling direct boundary fitting, diffusion deconvoluted sedimentation coefficient distributions, and multi-signal sedimentation coefficient distributions. Recently, effective particle theory has uncovered simple physical rules for the co-migration of rapidly exchanging systems of interacting components in SV. This has opened new possibilities for the robust interpretation of the boundary patterns of heterogeneous interacting systems. Together, these SE and SV techniques have led to new approaches to study macromolecular interactions across the entire spectrum of affinities, including both attractive and repulsive interactions, in both dilute and highly concentrated solutions, which can be applied to single-component solutions of self-associating proteins as well as the study of multi-protein complex formation in multi-component solutions.     

Analysis of the ligand-promoted global conformational change in aspartate transcarbamoylase. Evidence for a two-state transition from boundary spreading in sedimentation velocity experiments

A global conformational change in the regulatory enzyme aspartate transcarbamoylase of Escherichia coli was demonstrated 20 years ago by the 3.5% decrease in the sedimentation coefficient of the enzyme upon its interaction with carbamoyl phosphate and saturating amounts of the aspartate analog succinate. This "swelling" of aspartate transcarbamoylase attributable to the T----R allosteric transition was observed also in subsequent studies when the enzyme was completely saturated with the bisubstrate analog N-(phosphonacetyl)-L-aspartate. In neither of these studies was a direct attempt made by an analysis of boundary spreading (expressed as an apparent diffusion coefficient) on partially liganded enzyme to determine whether the solution contained only T and R-state molecules, as expected for a concerted transition, or a mixture of more than two distinct conformational states. The sensitivity of boundary spreading measurements was tested with a known mixture of fully liganded wild-type enzyme (R-state) and an inactive T-state mutant that did not bind either succinate or the bisubstrate ligand. This experiment yielded broad boundaries with an apparent diffusion coefficient about 10% greater than that of T-state enzyme, due to the differential sedimentation of the two independent species. Identical boundary spreading was obtained theoretically by simulating an equimolar mixture of T and R-state aspartate transcarbamoylase. These results proved that the boundary spreading measurement was sensitive to the presence of heterogeneity. Analogous experiments with only wild-type enzyme in the presence of sub-stoichiometric amounts of the tightly bound bisubstrate ligand sufficient to promote a 1.8% decrease in sedimentation coefficient also exhibited broader boundaries, corresponding to a 10% increase in the apparent diffusion coefficient relative to the unliganded enzyme. In contrast, such broad boundaries were not observed in experiments when the weakly bound succinate was present in quantities sufficient to cause the same 1.8% decrease in sedimentation coefficient. The differences in boundary spreading observed with the two active-site ligands were accounted for by the affinities of the respective ligands for the enzyme and the transport theory of a ligand-promoted isomerization of the protein. In the presence of sub-stoichiometric levels of the tight-binding bisubstrate ligand, the dynamic equilibrium between the T and the R-state is essentially uncoupled and the species sediment at slightly different rates to give broad boundaries.(ABSTRACT TRUNCATED AT 250 WORDS)     

A comparison of weight average and direct boundary fitting of sedimentation velocity data for indefinite polymerizing systems

Analysis of sedimentation velocity data for indefinite self-associating systems is often achieved by fitting of weight average sedimentation coefficients (s(20,w)) However, this method discriminates poorly between alternative models of association and is biased by the presence of inactive monomers and irreversible aggregates. Therefore, a more robust method for extracting the binding constants for indefinite self-associating systems has been developed. This approach utilizes a set of fitting routines (SedAnal) that perform global non-linear least squares fits of up to 10 sedimentation velocity experiments, corresponding to different loading concentrations, by a combination of finite element simulations and a fitting algorithm that uses a simplex convergence routine to search parameter space. Indefinite self-association is analyzed with the software program isodesfitter, which incorporates user provided functions for sedimentation coefficients as a function of the degree of polymerization for spherical, linear and helical polymer models. The computer program hydro was used to generate the sedimentation coefficient values for the linear and helical polymer assembly mechanisms. Since this curve fitting method directly fits the shape of the sedimenting boundary, it is in principle very sensitive to alternative models and the presence of species not participating in the reaction. This approach is compared with traditional fitting of weight average data and applied to the initial stages of Mg(2+)-induced tubulin self-associating into small curved polymers, and vinblastine-induced tubulin spiral formation. The appropriate use and limitations of the methods are discussed.     

Determination of the sedimentation coefficient distribution by least-squares boundary modeling

A new method is presented for the calculation of apparent sedimentation coefficient distributions g*(s) for the size-distribution analysis of polymers in sedimentation velocity experiments. Direct linear least-squares boundary modeling by a superposition of sedimentation profiles of ideal nondiffusing particles is employed. It can be combined with algebraic noise decomposition techniques for the application to interference optical ultracentrifuge data at low loading concentrations with significant systematic noise components. Because of the use of direct boundary modeling, residuals are available for assessment of the quality of the fits and the consistency of the g*(s) distribution with the experimental data. The method can be combined with regularization techniques based on F statistics, such as used in the program CONTIN, or alternatively, the increment of s values can be adjusted empirically. The method is simple, has advantageous statistical properties, and reveals precise sedimentation coefficients. The new least-squares ls-g*(s) exhibits a very high robustness and resolution if data acquired over a large time interval are analyzed. This can result in a high resolution for large particles, and for samples with a high degree of heterogeneity. Because the method does not require a high frequency of scans, it can also be easily used in experiments with the absorbance optical scanning system. Published 2000 John Wiley & Sons, Inc.     

Equilibrium electropotentials in electrolyte systems

The determinationof electric potentials in finite regions of symmetrical electrolyte in one-dimensional equilibrium situations requires the solution of the one-dimensional Poisson-Boltzmann equation in which the dependent variable is linearly related to the electric potential and contains unknown parameters. These require evaluation as part of the solution to a given boundary value problem. The general solution of the equation is presented. This involves elliptic functions and integrals and is sectionally isomorphic with respect to an integration parameter. The application to problems posed in terms of both initial values and two-point boundary values is discussed. The solution is used to determine the potential and concentration distributions between two flat-faced charged particles immersed in an electrolyte liquid and having interacting double layers.     

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.     

Sedimentation Patterns of Rapidly Reversible Protein Interactions

The transport behavior of macromolecular mixtures with rapidly reversible complex formation is of great interest in the study of protein interactions by many different methods. Complicated transport patterns arise even for simple bimolecular reactions, when all species exhibit different migration velocities. Although partial differential equations are available to describe the spatial and temporal evolution of the interacting system given particular initial conditions, a general overview of the phase behavior of the systems in parameter space has not yet been reported. In the case of sedimentation of two-component mixtures, this study presents simple analytical solutions that solve the underlying equations in the diffusion-free limit previously subject to Gilbert-Jenkins theory. The new expressions describe, with high precision, the average sedimentation coefficients and composition of each boundary, which allow the examination of features of the whole parameter space at once, and may be used for experimental design and robust analysis of experimental boundary patterns to derive the stoichiometry and affinity of the complex. This study finds previously unrecognized features, including a phase transition between boundary patterns. The model reveals that the time-average velocities of all components in the reaction mixture must match—a condition that suggests an intuitive physical picture of an effective particle of the coupled cosedimentation of an interacting system. Adding to the existing numerical solutions of the relevant partial differential equations, the effective particle model provides physical insights into the relationships of the parameters that govern sedimentation patterns.     

High concentration active enzyme centrifugation: analysis of active polymeric forms at up to 10 000-fold higher concentrations than with conventional methods.

This paper describes the theoretical basis, experimental technique, and experimental evaluation of a new method of analysis called "high concentration active enzyme centrifugation". It extends by up to four orders of magnitude the upper concentration limits at which the technique of "active enzyme centrifugation" can be used for analysis of enzyme structure. This new theory is largely based on certain properties of Gaussian curves which we have described in previous publications [Wei, G.J., & Deal, W.C., Jr. (1976) Anal. Biochem. 75, 113-121; Anal. Biochem. (1978) 87, 433-446]. One of the most important aspects of this development is that it extends the concentration range upward so that experiments can be performed on enzymes in the active polymeric forms corresponding to their in vivo states. Furthermore, this expansion includes the range in which most enzymes go through all their association-dissociation transitions from one polymeric form to another. Hence, the method can be used to define the various concentration-dependent transitions and also to ascertain which of the various polymeric forms of an enzyme are active, under various conditions. This method also retains the many favorable characteristics inherent in the active enzyme centrifugation technique. In studies with lactate dehydrogenase, the results from this method of band sedimentation were identical within experimental error (about 1.5%) with results from conventional boundary sedimentation velocity studies.     

Regulation of enzyme activity in adsorptive enzyme systems. III. Interaction of pig muscle lactate dehydrogenase with F-actin

The binding of pig skeletal muscle lactate dehydrogenase by F-actin has been studied using the sedimentation method in 10 mM Tris-acetate buffer, pH 6.0 at 20 degrees C. Adsorption capacity of F-actin is equal to (1 +/- 0.1) . 10(-5) moles of lactate dehydrogenase per 1 g of actin. NADH decreases the affinity of F-actin with respect to lactate dehydrogenase. The binding of lactate dehydrogenase by F-actin in diminishing the rate of enzymatic reduction of alpha-ketoglutarate. The microscopic dissociation constant for the complex of the enzyme with F-actin which is estimated from the dependence of the enzymatic reaction rate of F-actin concentration at saturating NADH concentrations is equal (3.0 +2- 0.5) . 10(-7) M. It has been shown that the bound enzyme is characterized by the greater value of Km and the lower value of Vmax in comparison to the free enzyme.