Macromolecular size-and-shape distributions by sedimentation velocity analytical ultracentrifugation

Abstract

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 approximately 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.

FIGURE 1

Application of c(s,f_r) analysis to a model system of large macromolecules. (A) Simulated signal profiles for a mixture of three species with 50 kDa, 3.5 S, 100 kDa, 5 S, and 100 kDa, 6.5 S, respectively, loaded at equal weight concentrations in a solution column from 6.0 to 7.2 cm, and sedimenting at a rotor speed of 50,000 rpm at 20°C. Traces were calculated in time intervals of 300 s, for a period of 15,000 s, and 0.005 Gaussian noise was added. For clarity, only every fifth scan is shown. (B) c(s) analysis with maximum entropy regularization with P = 0.7 (black line). The transformation of c(s) to c(M) is shown in the inset. The residuals bitmap of this fit is shown as inset to panel A, scaled to ±0.02 black to white. Shown in red in panel B is the c(s,^*) distribution derived from the c(s,f_r) distribution by summation over all f_r-values. The colored squares are the weight-average f_r-values for each s-value from c(s,f_r) distribution, with the color density indicating the signal at each s-value. The blue dotted horizontal lines are the weight-average frictional ratios for a conventional, but segmented c(s) model with one segment for each peak. (C) The calculated c(s,f_r) distribution is shown as two-dimensional distribution with grid lines representing the s and f_r grid of the analysis. Below this c(s,f_r) surface is shown a contour plot of the distribution projected into the s-f_r plane, where the magnitude of c(s,f_r) is indicated by contour lines at constant c(s,f_r) in equidistant intervals of c.

FIGURE 2

Contour plots of the transformation of c(s,f_r) from Fig. 1 to a c(s,D) distribution (A), a c(s,R_S) distribution (B), and a c(s,M) distribution (C). The dotted lines indicate lines of constant f_r. The distributions are not normalized (see Methods section).

FIGURE 3

Application of c(s,f_r) analysis to a model system of small macromolecules. (A) Simulated signal profiles for a mixture of three species with 6 kDa, 1 S, 30 kDa, 2 S, and 30 kDa, 3 S, respectively, sedimenting at equal weight concentrations at a rotor speed of 50,000 rpm. Traces were calculated in time intervals of 600 s, for a period of 30,000 s, and 0.005 Gaussian noise was added. For clarity, only every fifth scan is shown. (B) c(s) analysis with maximum entropy regularization with P = 0.9, with the corresponding c(M) distribution shown in the inset. The residuals bitmap of this fit is shown as an inset to panel A, scaled to ±0.02 black to white. Shown in red is the one-dimensional c(s,^*) derived from the c(s,f_r) distribution by summation over all f_r-values for each s. (C and D) Calculated c(s,f_r) distribution, ranging from 0.2 to 4 S in 0.063 S steps, and from f_r = 1–2.2 in 0.1 steps. Shown are the transformations as c(s,R_S) and c(s,M), respectively. Also shown are lines of constant frictional ratio (dotted lines in C), and lines of constant s-value (dotted lines in D). The vertical red lines in panel D indicate the true molar mass values.

FIGURE 4

Analysis of a theoretical mixture of two species with buoyant molar mass of 25.6 kDa and 3 S (corresponding, for example, to 160 kDa extended polymer with ), and with buoyant molar mass 13.5 kDa and 4 S (corresponding, for example, to a folded protein of 50 kDa). Sedimentation was simulated under the same conditions as in Figs. 1 and 3. (A) The c(s) distribution is shown as a black line, with the residuals bitmap from the c(s) analysis in the inset (rmsd = 0.0074). The red line is the c(s,^*) trace from the c(s,f_r) analysis, which is shown in panel B as a transformation to a parameter space of buoyant molar mass and Stokes radius. The dotted lines indicate directions of constant f_r and constant s. The red crosses are the parameters underlying the simulation, the black crosses are values resulting from integration of an impostor c(M) transformation of c(s).

FIGURE 5

Analysis of experimental sedimentation velocity data from the study of the oligomeric state of a glycosylated NK receptor fragment. For experimental details, see Dam and Schuck (28). Panel A shows a representative subset of the raw data. Panel B shows c(s) analysis (black line) and transformation to c(M) (right inset). The left insets are residual bitmaps from (i) a single discrete species analysis, (ii) the c(s) analysis, and (iii) the c(s,f_r) analysis. The red line is the c(s,^*) trace of the c(s,f_r) distribution shown in panel C. Panel D shows the same analysis as in panel C, but without regularization.

FIGURE 6

Analysis of SV data from a bovine serum albumin sample (commercial sample used without further purification) sedimenting at 55,000 rpm, 22°C. Panel A shows c(s) distribution (black) and c(s,^*) distribution (red) derived from the c(s,f_r) fit shown in panel B as c(s₁M). Integration of the c(s,f_r) peak results in an average apparent molar mass of 61 kDa (assuming a value of 0.73 ml/g) for the monomer at ∼4.5 S, and 135 kDa for the dimer peak at ∼7 S, which amounts to ∼9% of the total sedimenting material. The dotted lines are lines of constant f_r. Panel C shows analysis of a mixture of a bovine serum albumin (taken from a different batch) with an IgG sample, sedimenting at 50,000 rpm, 20°C. Integration of the IgG peak leads to an average apparent molar mass of 154 kDa (assuming a value of 0.73 ml/g).

FIGURE 7

Application of c(s,f_r) to a reaction mixture with rapid kinetics on the timescale of sedimentation. Sedimentation velocity profiles were simulated for the interaction of a protein of 25 kDa, 2.5 S binding to a 40 kDa, 3.5 S species forming a 5 S complex with a equilibrium dissociation constant K_D = 3 μM, and a dissociation rate constant k_off = 0.01/s, studied at a rotor speed of 50,000 rpm. Interference optical detection was assumed, with conventional signal increments of 3.3 fringes/(mg/ml) and a noise level of 0.005 fringes. Concentrations were equimolar at 0.1-fold K_D (blue), 0.3-fold K_D (green), K_D (black), threefold K_D (red), and 10-fold K_D (magenta). c(s,f_r) distributions were calculated with s-values from 1 to 6 S and f_r-values from 0.8 to 2.0. Panel A shows c(s,^*) distributions (solid lines) and conventional c(s) distributions (dotted lines), both normalized to constant area. Panels B–F show c(s,f_r) distributions mapped into the c(s,M) plane, at the concentrations indicated.