Solution Of Lamm Equation Research Articles

A remark on solutions of Lame equations with elliptic potentials

A remark on solutions of Lame equations with elliptic potentials

Variable-Field Analytical Ultracentrifugation: I. Time-Optimized Sedimentation Equilibrium

Sedimentation equilibrium (SE) analytical ultracentrifugation (AUC) is a gold standard for the rigorous determination of macromolecular buoyant molar masses and the thermodynamic study of reversible interactions in solution. A significant experimental drawback is the long time required to attain SE, which is usually on the order of days. We have developed a method for time-optimized SE (toSE) with defined time-varying centrifugal fields that allow SE to be attained in a significantly (up to 10-fold) shorter time than is usually required. To achieve this, numerical Lamm equation solutions for sedimentation in time-varying fields are computed based on initial estimates of macromolecular transport properties. A parameterized rotor-speed schedule is optimized with the goal of achieving a minimal time to equilibrium while limiting transient sample preconcentration at the base of the solution column. The resulting rotor-speed schedule may include multiple over- and underspeeding phases, balancing the formation of gradients from strong sedimentation fluxes with periods of high diffusional transport. The computation is carried out in a new software program called TOSE, which also facilitates convenient experimental implementation. Further, we extend AUC data analysis to sedimentation processes in such time-varying centrifugal fields. Due to the initially high centrifugal fields in toSE and the resulting strong migration, it is possible to extract sedimentation coefficient distributions from the early data. This can provide better estimates of the size of macromolecular complexes and report on sample homogeneity early on, which may be used to further refine the prediction of the rotor-speed schedule. In this manner, the toSE experiment can be adapted in real time to the system under study, maximizing both the information content and the time efficiency of SE experiments.

Sedimentation in a Time-Varying Centrifugal Field for Rapid Attainment of Sedimentation Equilibrium

Sedimentation equilibrium (SE) analytical ultracentrifugation is a gold standard for the rigorous thermodynamic study of buoyant molecular weight and reversible interactions of macromolecules in solution. A significant drawback is the long experiment time, as it takes days to attain SE with standard solution columns. We have developed a new method for using a time-varying centrifugal field optimized such as to attain SE in significantly shorter time than usually required. Experimental data show that this permits long-column SE experiments to be carried out in times comparable to sedimentation velocity experiments, approximately fivefold shorter than standard SE. In contrast to the classical initial overspeeding method, which uses a single initial speed, we employ a freely varying rotor speed profile during an initial phase, for example, parameterized as a step-wise modulated exponential decay to the desired SE rotor speed. The rotor speed schedule is computationally optimized on the basis of numerical Lamm equation solutions for given macromolecular sedimentation parameter estimates, with the goal to provide a rapid attainment of equilibrium without the drawback of strong transient sample pre-concentration at the base of the solution column. The resulting rotor speed schedule frequently includes both over- and under-speeding sequences, and can be conveniently implemented on the Optima XLA/I analytical ultracentrifuge. We extended AUC data analysis models in SEDFIT to permit the analysis of concentration profiles in arbitrarily time-varying fields, to make it possible to exploit the migration in the initially high centrifugal field for estimates on macromolecular sedimentation parameters, which may be used in real-time to refine the prediction of the rotor speed schedule, so that the SE experiment can be optimized in both information content and time efficiency.

A Parametrically Constrained Optimization Method for Fitting Sedimentation Velocity Experiments

A method for fitting sedimentation velocity experiments using whole boundary Lamm equation solutions is presented. The method, termed parametrically constrained spectrum analysis (PCSA), provides an optimized approach for simultaneously modeling heterogeneity in size and anisotropy of macromolecular mixtures. The solutions produced by PCSA are particularly useful for modeling polymerizing systems, where a single-valued relationship exists between the molar mass of the growing polymer chain and its corresponding anisotropy. The PCSA uses functional constraints to identify this relationship, and unlike other multidimensional grid methods, assures that only a single molar mass can be associated with a given anisotropy measurement. A description of the PCSA algorithm is presented, as well as several experimental and simulated examples that illustrate its utility and capabilities. The performance advantages of the PCSA method in comparison to other methods are documented. The method has been added to the UltraScan-III software suite, which is available for free download from http://www.ultrascan.uthscsa.edu.

Axisymmetric vibrations of an infinite body with a thin elastic circular inclusion under conditions of smooth contact

We solve an axisymmetric problem of the interaction of harmonic waves with a thin elastic circular inclusion located in an elastic isotropic body (matrix). On both sides of the inclusion, between it and the body (matrix), conditions of smooth contact are realized. The method of solution is based on the representation of displacements in the matrix in terms of discontinuous solutions of Lame equations for harmonic vibrations. This enables us to reduce the problem to Fredholm integral equations of the second kind for functions related to jumps of normal stress and radial displacement on the inclusion.

Accounting for Solvent Signal Offsets in the Analysis of Interferometric Sedimentation Velocity Data

Sedimentation velocity (SV) analytical ultracentrifugation has re-emerged as an important tool in the characterization of biological macromolecules and nanoparticles. The computational analysis of the evolution of the macromolecular concentration profile allows the characterization of many hydrodynamic and thermodynamic properties of the macromolecules and their interactions. The Rayleigh interference optical system is often the detection method of choice, for its usually superior data quality and the wide applicability of refractive index sensitive detection. However, the interference optical system is also sensitive to the redistribution of co-solvent molecules, which are not of primary experimental interest. In principle, their contribution can be eliminated by an exact geometric and compositional match of the sample solution and the reference solution, achieving the complete optical subtraction of unwanted buffer signals. Unfortunately, in practice, this can often not be perfectly achieved for various reasons, leading to signal offsets arising from unmatched sedimentation of solvent components. If unrecognized, this can lead to significant misfit, accompanied by significant errors in the macromolecular sedimentation parameters. In the present work, we describe an approach of computationally accounting for signals from sedimenting buffer components through explicitly modeling their redistribution with Lamm equation solutions, implemented in the software SEDFIT. We demonstrate how this can restore the SV analysis to yield a high quality fit of the data and to provide correct macromolecular sedimentation parameters.

Interaction of plane nonstationary waves with a thin elastic inclusion under smooth contact conditions

We solve the problem of determining the stress state near a thin elastic inclusion in the form of a strip of finite width in an unbounded elastic body (matrix) with plane nonstationary waves propagating through it and with the forces exerted by the ambient medium taken into account. We assume that the matrix is in the plane strain state, and the smooth contact conditions are realized on both sides of the inclusion. The method for solving this problem consists in using the integral Laplace transform with respect to time and in representing the stress and displacement images in terms of the discontinuous solution of Lame equations in the case of plane strain. As a result, the initial problem is reduced to a system of singular integral equations for the transforms of the unknown stress and displacement jumps. To invert the Laplace transform, we use a numerical method based on replacing the Mellin integral by the Fourier series. As a result, we obtain approximate formulas for calculating the stress intensity factors (SIF) for the inclusion, which are used to study the SIF time-dependence and its influence on the values of the inclusion rigidity. We also studied the possibility of considering the inclusions of higher rigidity as absolutely rigid inclusions.

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.

Heparan Sulfate Regulates Fibrillin-1 N- and C-terminal Interactions

Fibrillin-1 N- and C-terminal heparin binding sites have been characterized. An unprocessed monomeric N-terminal fragment (PF1) induced a very high heparin binding response, indicating heparin-mediated multimerization. Using PF1 deletion and short fragments, a heparin binding site was localized within the domain encoded by exon 7 after the first hybrid domain. Rodent embryonic fibroblasts adhered to PF1 and deletion fragments, and, when cells were plated on fibrillin-1 or fibronectin Arg-Gly-Asp cell-binding fragments, cells showed heparin-dependent spreading and focal contact formation in response to soluble PF1. Within domains encoded by exons 59-62 near the fibrillin-1 C terminus are novel conformation-dependent high affinity heparin and tropoelastin binding sites. Heparin disrupted tropoelastin binding but did not disrupt N- and C-terminal fibrillin-1 interactions. Thus, fibrillin-1 N-terminal interactions with heparin/heparan sulfate directly influence cell behavior, whereas C-terminal interactions with heparin/heparan sulfate regulate elastin deposition. These data highlight how heparin/heparan sulfate controls fibrillin-1 interactions.

Macromolecular Size-and-Shape Distributions by Sedimentation Velocity Analytical Ultracentrifugation

Sedimentation velocity analytical ultracentrifugation is an important tool in the characterization of macromolecules and nanoparticles in solution. The sedimentation coefficient distribution c( s) of Lamm equation solutions is based on the approximation of a single, weight-average frictional coefficient of all particles, determined from the experimental data, which scales the diffusion coefficient to the sedimentation coefficient consistent with the traditional s ∼ M 2/3 power law. It provides a high hydrodynamic resolution, where diffusional broadening of the sedimentation boundaries is deconvoluted from the sedimentation coefficient distribution. The approximation of a single weight-average frictional ratio is favored by several experimental factors, and usually gives good results for chemically not too dissimilar macromolecules, such as mixtures of folded proteins. In this communication, we examine an extension to a two-dimensional distribution of sedimentation coefficient and frictional ratio, c( s, f r), which is representative of a more general set of size-and-shape distributions, including mass-Stokes radius distributions, c( M, R S), and sedimentation coefficient-molar mass distributions c( s, M). We show that this can be used to determine average molar masses of macromolecules and characterize macromolecular distributions, without the approximation of any scaling relationship between hydrodynamic and thermodynamic parameters.

Properties of the C-terminal Domain of Enzyme I of the Escherichia coli Phosphotransferase System

The bacterial phosphoenolpyruvate (PEP):glycose phosphotransferase system (PTS) mediates uptake/phosphorylation of sugars. The transport of all PTS sugars requires Enzyme I (EI) and a phosphocarrier histidine protein of the PTS (HPr). The PTS is stringently regulated, and a potential mechanism is the monomer/dimer transition of EI, because only the dimer accepts the phosphoryl group from PEP. EI monomer consists of two major domains, at the N and C termini (EI-N and EI-C, respectively). EI-N accepts the phosphoryl group from phospho-HPr but not PEP. However, it is phosphorylated by PEP(Mg(2+)) when complemented with EI-C. Here we report that the phosphotransfer rate increases approximately 25-fold when HPr is added to a mixture of EI-N, EI-C, and PEP(Mg(2+)). A model to explain this effect is offered. Sedimentation equilibrium results show that the association constant for dimerization of EI-C monomers is 260-fold greater than the K(a) for native EI. The ligands have no detectable effect on the secondary structure of the dimer (far UV CD) but have profound effects on the tertiary structure as determined by near UV CD spectroscopy, thermal denaturation, sedimentation equilibrium and velocity, and intrinsic fluorescence of the 2 Trp residues. The binding of PEP requires Mg(2+). For example, there is no effect of PEP on the T(m), an increase of 7 degrees C in the presence of Mg(2+), and approximately 14 degrees C when both are present. Interestingly, the dissociation constants for each of the ligands from EI-C are approximately the same as the kinetic (K(m)) constants for the ligands in the complete PTS sugar phosphorylation assays.

Structural and Functional Variations in Human Apolipoprotein E3 and E4

There are three major apolipoprotein E (apoE) isoforms. Although APOE-epsilon3 is considered a longevity gene, APOE-epsilon4 is a dual risk factor to atherosclerosis and Alzheimer disease. We have expressed full-length and N- and C-terminal truncated apoE3 and apoE4 tailored to eliminate helix and domain interactions to unveil structural and functional disturbances. The N-terminal truncated apoE4-(72-299) and C-terminal truncated apoE4-(1-231) showed more complicated or aggregated species than those of the corresponding apoE3 counterparts. This isoformic structural variation did not exist in the presence of dihexanoylphosphatidylcholine. The C-terminal truncated apoE-(1-191) and apoE-(1-231) proteins greatly lost lipid binding ability as illustrated by the dimyristoylphosphatidylcholine turbidity clearance. The low density lipoprotein (LDL) receptor binding ability, determined by a competition binding assay of 3H-LDL to the LDL receptor of HepG2 cells, showed that apoE4 proteins with N-terminal (apoE4-(72-299)), C-terminal (apoE4-(1-231)), or complete C-terminal truncation (apoE4-(1-191)) maintained greater receptor binding abilities than their apoE3 counterparts. The cholesterol-lowering abilities of apoE3-(72-299) and apoE3-(1-231) in apoE-deficient mice were decreased significantly. The structural preference of apoE4 to remain functional in solution may explain the enhanced opportunity of apoE4 isoform to display its pathophysiologic functions in atherosclerosis and Alzheimer disease.

Self-association and Chaperone Activity of Hsp27 Are Thermally Activated

The small heat shock protein 27 (Hsp27) is an oligomeric, molecular chaperone in vitro. This chaperone activity and other physiological roles attributed to Hsp27 have been reported to depend on the state of self-association. In the present work, we have used sedimentation velocity experiments to demonstrate that the self-association of Hsp27 is independent of pH and ionic strength but increases significantly as the temperature is increased from 10 to 40 degrees C. The largest oligomers formed at 10 degrees C are approximately 8-12 mer, whereas at 40 degrees C oligomers as large as 22-30 mer are observed. Similarly, the chaperone activity of Hsp27 as indicated by its ability to inhibit dithiothreitol-induced insulin aggregation also increases with increased temperature, with a particularly sharp increase in activity as temperature is increased from 34 to 43 degrees C. Similar studies of an Hsp27 triple variant that mimics the behavior of the phosphorylated protein establish that this protein has greatly diminished chaperone activity that responds minimally to increased temperature. We conclude that Hsp27 can exploit a large number of oligomerization states and that the range of oligomer size and the magnitude of chaperone activity increase significantly as temperature is increased over the range that is relevant to the physiological heat shock response.

Sedimentation Velocity Analysis of Heterogeneous Protein-Protein Interactions: Lamm Equation Modeling and Sedimentation Coefficient Distributions c( s)

We describe algorithms for solving the Lamm equations for the reaction-diffusion-sedimentation process in analytical ultracentrifugation, and examine the potential and limitations for fitting experimental data. The theoretical limiting case of a small, uniformly distributed ligand rapidly reacting with a larger protein in a “constant bath” of the ligand is recapitulated, which predicts the reaction boundary to sediment with a single sedimentation and diffusion coefficient. As a consequence, it is possible to express the sedimentation profiles of reacting systems as c( s) distribution of noninteracting Lamm equation solutions, deconvoluting the effects of diffusion. For rapid reactions, the results are quantitatively consistent with the “constant bath” approximation, showing c( s) peaks at concentration-dependent positions. For slower reactions, the deconvolution of diffusion is still partially successful, with c( s) resolving peaks that reflect the populations of sedimenting species. The transition between c( s) peaks describing reaction boundaries of moderately strong interactions ( K D ∼ 10 −6 M) or resolving sedimenting species was found to occur in a narrow range of dissociation rate constant between 10 −3 and 10 −4 s −1. The integration of the c( s) peaks can lead to isotherms of species populations or s-value of the reaction boundary, respectively, which can be used for the determination of the equilibrium binding constant.

Sedimentation Velocity Analysis of Heterogeneous Protein-Protein Interactions: Sedimentation Coefficient Distributions c( s) and Asymptotic Boundary Profiles from Gilbert-Jenkins Theory

Interacting proteins in rapid association equilibrium exhibit coupled migration under the influence of an external force. In sedimentation, two-component systems can exhibit bimodal boundaries, consisting of the undisturbed sedimentation of a fraction of the population of one component, and the coupled sedimentation of a mixture of both free and complex species in the reaction boundary. For the theoretical limit of diffusion-free sedimentation after infinite time, the shapes of the reaction boundaries and the sedimentation velocity gradients have been predicted by Gilbert and Jenkins. We compare these asymptotic gradients with sedimentation coefficient distributions, c( s), extracted from experimental sedimentation profiles by direct modeling with superpositions of Lamm equation solutions. The overall shapes are qualitatively consistent and the amplitudes and weight-average s-values of the different boundary components are quantitatively in good agreement. We propose that the concentration dependence of the area and weight-average s-value of the c( s) peaks can be modeled by isotherms based on Gilbert-Jenkins theory, providing a robust approach to exploit the bimodal structure of the reaction boundary for the analysis of experimental data. This can significantly improve the estimates for the determination of binding constants and hydrodynamic parameters of the complexes.

Reversible and Fast Association Equilibria of a Molecular Chaperone, gp57A, of Bacteriophage T4

The association of a molecular chaperone, gp57A, of bacteriophage T4, which facilitates formation of the long and short tail fibers, was investigated by analytical ultracentrifugation, differential scanning microcalorimetry, and stopped-flow circular dichroism (CD) to establish the association scheme of the protein. Gp57A is an oligomeric α-helix protein with 79 amino acids. Analysis of the sedimentation velocity data by direct boundary modeling with Lamm equation solutions together with a more detailed boundary analysis incorporating association schemes led us to conclude that at least three oligomeric species of gp57A are in reversible and fast association equilibria and that a 3mer-6mer-12mer model described the data best. On the other hand, differential scanning microcalorimetry revealed a highly reversible two-step transition of dissociation/denaturation, both of which accompanied decrease in CD at 222nm. The melting curve analysis revealed that it is consistent with a 6mer-3mer-1mer model. The refolding/association kinetics of gp57A measured by stopped-flow CD was consistent with the interpretation that the bimolecular reaction from trimer to hexamer was preceded by a fast α-helix formation in the dead-time. Trimer or hexamer is likely the functional oligomeric state of gp57A.

On the analysis of protein self-association by sedimentation velocity analytical ultracentrifugation

Analytical ultracentrifugation is one of the classical techniques for the study of protein interactions and protein self-association. Recent instrumental and computational developments have significantly enhanced this methodology. In this paper, new tools for the analysis of protein self-association by sedimentation velocity are developed, their statistical properties are examined, and considerations for optimal experimental design are discussed. A traditional strategy is the analysis of the isotherm of weight-average sedimentation coefficients s w as a function of protein concentration. From theoretical considerations, it is shown that integration of any differential sedimentation coefficient distribution c( s), ls- g *( s), or g( s *) can give a thermodynamically well-defined isotherm, as long as it provides a good model for the sedimentation profiles. To test this condition for the g( s *) distribution, a back-transform into the original data space is proposed. Deconvoluting diffusion in the sedimentation coefficient distribution c( s) can be advantageous to identify species that do not participate in the association. Because of the large number of scans that can be analyzed in the c( s) approach, its s w values are very precise and allow extension of the isotherm to very low concentrations. For all differential sedimentation coefficients, corrections are derived for the slowing of the sedimentation boundaries caused by radial dilution. As an alternative to the interpretation of the isotherm of the weight-average s value, direct global modeling of several sedimentation experiments with Lamm equation solutions was studied. For this purpose, a new software SEDPHAT is introduced, allowing the global analysis of several sedimentation velocity and equilibrium experiments. In this approach, information from the shape of the sedimentation profiles is exploited, which permits the identification of the association scheme and requires fewer experiments to precisely characterize the association. Further, under suitable conditions, fractions of incompetent material that are not part of the reversible equilibrium can be detected.

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.

Origin of giant dielectric response in ferroelectric thin film multilayers

A thermodynamic theory of dielectric response in ferroelectric thin film multilayers is developed. The solution of Lame equation for static dielectric susceptibility has shown that susceptibility diverges at the transition temperature of the thickness induced ferroelectric phase. This divergence is shown to be the origin of the giant dielectric response observed in some multilayers. The theory gives an excellent fit to the temperature dependence of the giant susceptibility observed recently in multilayers of PbTiO3-Pb1-xLaxTiO3 (x = 0.28).

Molecular parameters from sedimentation velocity experiments: Whole boundary fitting using approximate and numerical solutions of lamm equation

This chapter describes the molecular parameters from sedimentation velocity experiments. The chapter discusses two types of approaches that are recently implemented in software adapted for data obtained from the Beckman XL-A. In the first approach, the finite element method is used to solve the Lamm equation (and modifications of the Lamm equation) numerically, whereas in the second approach is analytical solutions of the Lamm equation, subjected to various approximations. Both approaches involve nonlinear least-squares, fitting of the moving boundary of sedimentation velocity, and approach-to-equilibrium experiments to solutions of the Lamm equation. Recent advances in analytical ultracentrifugation hardware brought about substantial improvements in the accuracy, precision, and range of data obtained from this instrumentation. These improvements are made possible by digital data acquisition, microprocessor-controlled experimental parameters, high-quality optical components, and by the introduction of a Raleigh interference system that permits the acquisition of data at lower protein concentration, with a higher signal-to-noise ratio and a higher data density than possible, with absorbance optics.