Numerical solutions of the Lamm equation. III. Velocity centrifugation

We have generated solutions to the Lamm equation to examine the effects of concentration dependence on velocity experiments. Two forms of c dependence are considered: s/s₀ = 1 – kc and s/s₀ = (1 + kc)^?1. Features of these solutions are discussed. The magnitude of the errors resulting from the usual procedure of measuring the rate of movement of schlieren maxima or of the position at which the concentration is one half the plateau value have been examined. These errors are usually negligible after sufficient centrifugation time. The errors in using the half-plateau concentration are less than those using the movement of the peak. We have also examined a technique due to Fujita for determining D from boundary spreading when s/s₀ = (1+kc)^?1. This method is satisfactory when s/s₀ is actually of this form, or under certain limitations when s/s₀ = (1 + kc)^?1. Creeth has shown that under certain conditions the concentration gradient, curve remains virtually unchanged in shape after separating from the meniscus. When this occurs it is possible to estimate s/D from the data. The condition for such a steady state is that kc₀ be sufficiently large. Numerical confirmation of this method is presented in the final section.     

Numerical solutions of the Lamm equation. II. Equilibrium sedimentation

The Lamm equation has been solved numerically for conditions corresponding to equilibrium runs for a nonlinear concentration dependence of the form s/s₀ = (1 + kc)^?1. It is shown that the approach to equilibrium is very close to being exponential (in time) as in the case k = 0. We also compare results for the nonlinear case given above with results obtained for linear c-dependence of the form s/s₀ = 1 – kc. For relatively high speeds the time required to attain equilibrium may be greatly underestimated by use of the latter approximation. Finally, we present analytic approximations for the concentration distribution at equilibrium and as a function of time.     

Numerical solutions of the Lamm equation. IV. Rotor slowing experiments

This paper presents the results of a numerical solution of the Lamm equation for roton slowing specified by ω² = ω₀² exp {– λτ}, for parameters relevant to equilibrium experiments. It. is shown that in the two-component system it is theoretically possible to deduce s/D from measurements of the difference of concentration across the. cell with rotor slowing, provided that the time at which the difference reaches a maximum is known.     

Sedimentation analysis of noninteracting and self-associating solutes using numerical solutions to the Lamm equation.

The potential of using the Lamm equation in the analysis of hydrodynamic shape and gross conformation of proteins and reversibly formed protein complexes from analytical ultracentrifugation data was investigated. An efficient numerical solution of the Lamm equation for noninteracting and rapidly self-associating proteins by using combined finite-element and moving grid techniques is described. It has been implemented for noninteracting solutes and monomer-dimer and monomer-trimer equilibria. To predict its utility, the error surface of a nonlinear regression of simulated sedimentation profiles was explored. Error contour maps were calculated for conventional independent and global analyses of experiments with noninteracting solutes and with monomer-dimer systems at different solution column heights, loading concentrations, and centrifugal fields. It was found that the rotor speed is the major determinant for the shape of the error surface, and that global analysis of different experiments can allow substantially improved characterization of the solutes. We suggest that the global analysis of the approach to equilibrium in a short-column sedimentation equilibrium experiment followed by a high-speed short-column sedimentation velocity experiment can result in sedimentation and diffusion coefficients of very high statistical accuracy. In addition, in the case of a protein in rapid monomer-dimer equilibrium, this configuration was found to reveal the most precise estimate of the association constant.     

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.     

An approximate solution to the Lamm equation

An approximate solution to the Lamm equation subject to the initial and boundary conditions for conventional sedimentation velocity experiments is derived and compared with the approximate solution of Fujita and MacCosham. Calculations with this solution demonstrate that the half-height method of estimating sedimentation coefficients yields correct values for epsilon < 0.02.     

Sedimentation of generalized systems of interacting particles. I. Solution of systems of complete Lamm equations.

A very general approach to the chemical equilibria between many interacting molecules during sedimentation (boundary, band, or active enzyme) taking into account boundary conditions, cell geometry, equilibrium constants, diffusion, enzyme kinetics, etc., is presented. Through a Fortran program, the method has been applied to two very simple but typical cases. With only minor adjustments, the method presented here for sedimentation studies can be extended to all sorts of problems in which “pools” of various species are interacting with each other.     

Modeling analytical ultracentrifugation experiments with an adaptive space-time finite element solution of the Lamm equation

Analytical ultracentrifugation experiments can be accurately modeled with the Lamm equation to obtain sedimentation and diffusion coefficients of the solute. Existing finite element methods for such models can cause artifactual oscillations in the solution close to the endpoints of the concentration gradient, or fail altogether, especially for cases where somega(2)/D is large. Such failures can currently only be overcome by an increase in the density of the grid points throughout the solution at the expense of increased computational costs. In this article, we present a robust, highly accurate and computationally efficient solution of the Lamm equation based on an adaptive space-time finite element method (ASTFEM). Compared to the widely used finite element method by Claverie and the moving hat method by Schuck, our ASTFEM method is not only more accurate but also free from the oscillation around the cell bottom for any somega(2)/D without any increase in computational effort. This method is especially superior for cases where large molecules are sedimented at faster rotor speeds, during which sedimentation resolution is highest. We describe the derivation and grid generation for the ASTFEM method, and present a quantitative comparison between this method and the existing solutions.     

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

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

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

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

Relaxation and electrophoretic effects in polyelectrolyte solutions. I. Salt-free solutions

Relations between relaxation and electrophoretic effects in salt-free polyelectrolyte solutions are studied in terms of nonequilibrium thermodynamics and of binary friction coefficients. The significance and the method of determining these coefficients are explained. Consideration of experimental results and of long- and short-range interactions suggests that the concept of “ion condensation” is needed to provide a consistent desciption of transport processes for linear flexible polyelectrolytes. Empirical relations previously used are shown to have only limited validity.     

Asymptotic solutions to the matrix Riccati equation

This paper describes a new computational method for calculating the asymptotic solution to the matrix Riccati equation. This method is fast, efficient and gives all possible solutions to the matrix quadratic form. Matrix sign functions are used to find the asymptotic solutions.     

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) approximately 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.     

Singular solutions of the diffusion equation of population genetics

The forward diffusion equation for gene frequency dynamics is solved subject to the condition that the total probability is conserved at all times. This can lead to solutions developing singular spikes (Dirac delta functions) at the gene frequencies 0 and 1. When such spikes appear in solutions they signal gene loss or gene fixation, with the "weight" associated with the spikes corresponding to the probability of loss or fixation. The forward diffusion equation is thus solved for all gene frequencies, namely the absorbing frequencies of 0 and 1 along with the continuous range of gene frequencies on the interval (0,1) that excludes the frequencies of 0 and 1. Previously, the probabilities of the absorbing frequencies of 0 and 1 were found by appeal to the backward diffusion equation, while those in the continuous range (0,1) were found from the forward diffusion equation. Our unified approach does not require two separate equations for a complete dynamical treatment of all gene frequencies within a diffusion approximation framework. For cases involving mutation, migration and selection, it is shown that a property of the deterministic part of gene frequency dynamics determines when fixation and loss can occur. It is also shown how solution of the forward equation, at long times, leads to the standard result for the fixation probability.