## A1 Sedimentation Velocity

At the start of an ultracentrifugation experiment, the macromolecules are distributed homogeneously in the test solution. When the rotor begins to spin, the molecules begin to be pushed toward the bottom of the sector-shaped cell (Figure 10.1). This creates a moving boundary that travels

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Figure 10.1 Diagram of a sector shaped cell (Adapted from Van Holde, Physical Biochemistry, © 1971, 99. Reprinted by permission of Prentice-Hall, Inc., Englewood Cliffs, N.J.)

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Figure 10.1 Diagram of a sector shaped cell (Adapted from Van Holde, Physical Biochemistry, © 1971, 99. Reprinted by permission of Prentice-Hall, Inc., Englewood Cliffs, N.J.)

toward the bottom of the cell. Above this boundary, there are no macromolecules, which all become concentrated below the boundary.

The rate of movement of the boundary depends upon the sedimentation coefficient s of the molecule. In the absence of diffusion, s is expressed as [1]

Ma f where Mw is molecular weight, v is the partial specific volume of the molecule, p is the density of the solution, NA is Avagadro's number, and / is the frictional coefficient. The velocity dr/dt of the moving boundary, where r is the distance from the center of rotation, is directly proportional to s:

dt where r is the distance of the center of the boundary from the center of rotation and &> is the angular velocity. Integrating this equation gives r(t) = r(t0) exp[co2s(i — f0)]- (10-3)

The use of eq. (10.3) requires an accurate determination of r(t) at various times at a fixed angular velocity. The linearized version of eq. (10.3) is:

Thus, data in the form of ln[r(i)/r(/0)] vs. (t - t0) could be subjected to linear regression to give u>2s as the slope. However, as discussed in Section 2.C.1, the error distributions in the independent variable will change accompanying linearization. If linear regression is to be used, it must be properly weighted. Again, the use of nonlinear regression directly onto eq. (10.3) avoids these problems.

The diffusion of macromolecules during sedimentation causes the moving boundary to spread with time [1, 2]. This spreading can be estimated by analysis of the shape of the boundary. Nonlinear regression can be employed for the estimation of spreading factors and r(t) values during the sedimentation velocity experiment.

The model is the integral of the normal curve of error or Gaussian peak shape and can be computed from an integrated, series representation of the error function. [3, 4], The spreading factor is the width at half-height of the integrated Gaussian peak, and r(t) is the position of integrated peak. If s and D are constant throughout the boundary, then a graph of the square of the spreading factor (W2) in cm2 vs. time in seconds gives the apparent diffusion coefficient of the macromolecule in cm2 s 1 as (slope)/2 of the linear regression line. The complete model for sedimentation velocity experiments assuming that absorbance is used as a detection method is given in Table 10.1.

Sedimentation velocity data for a solution of freshly purified bovine 02-microglobulin shows excellent agreement with the model in Table 10.1 (Figure 10.2). Diffusion and sedimentation coefficients are obtained by analysis as described previously. Mw is obtained from s using eq. (10.1), after computing values of v and / from appropriate models [5-7]. These parameter values are generally corrected to 20°C in water. Analysis of a full set of r(t) vs. (f - i0) data for bovine ^-microglobulin2 gave s2o,w = 1-86,

Table 10.1 Models for Analysis of Sedimentation Velocity Data for a Single Macromolecule

Assumptions: One macromolecule, no association or dissociation; detection by absorbance or refractive index Regression equations:

Position of boundary at t:

series approximation to integral of Gaussian peak [2]

A = h[l - (0.3084284rj + 0.0849713r;2 + 0.6627698t;3) exp(-*2)] Velocity of boundary: r(t) = r(t0) exp[w2j(i - r0)] Regression parameters: Data:

Position of boundary: h r0 W A vs. r

Special considerations: Better accuracy for A may be obtained by using more terms in the series [5]; D is obtained from linear plot of W2 vs. t2 as slope/2

Figure 10.2 A sedimentation velocity pattern of /^-microglobulin in 0.08 M NaCl, 0.02 M sodium phosphate, pH 5.0 at 25°C at 52,000 rpm for 96 min. in a 1.2 cm cell. Initial concentration <0.3 mg mL-1. The points are experimental data, and the solid line is the best fit onto A vs. r model in Table 10.1. (Reprinted from [5] with permission.)

DISTANCE FROM CENTER OF ROTATION, cm

Figure 10.2 A sedimentation velocity pattern of /^-microglobulin in 0.08 M NaCl, 0.02 M sodium phosphate, pH 5.0 at 25°C at 52,000 rpm for 96 min. in a 1.2 cm cell. Initial concentration <0.3 mg mL-1. The points are experimental data, and the solid line is the best fit onto A vs. r model in Table 10.1. (Reprinted from [5] with permission.)

D201W = 1-37 X 10"6 cm2 s "'. Knowledge of these parameters allowed an estimation of M = 11,900.

Slow aggregation of proteins under conditions of storage is quite common and can lead to the presence of multiple boundaries in the sedimentation velocity experiment. These data can be modeled by summing the model for A vs. r in Table 10.1. This model is summarized in Table 10.2. An application of this model to bovine /^-microglobulin [5] showed that this

Table 10.2 Model for Analysis of Sedimentation Velocity Data for Samples Containing Two Macromolecules

Assumptions: Two macromolecules, or association; detection by absorbance or refractive index Regression equations: Position of boundary at t:

Series approximation to sum of two integrals of Gaussian peaks [2]

A = ht[ 1 - (0.30842847J1 + 0.0849713^ + 0.66276987^) exp(-A^)] + h2[ 1 - (0.3084284rj2 + 0.0849713r^ + 0.6627698tj|) exp(~ Xi)} Regression parameters: Data h\ h2 r0,i ro,i A vs. r

Special considerations: D is obtained from plot of W2 vs. t

Figure 10.3 A sedimentation velocity pattern of /^-microglobulin in 0.08 M NaCl, 0.02 M sodium phosphate after incubation for seven days at pH 5.0 at 25°C; 60,000 rpm for 40 min. in a 1.2 cm cell. The points are experimental data and the solid line is the best fit onto A vs. r model in Table 10.3. (Reprinted from [5] with permission.)

DISTANCE FROM CENTER OF ROTATION, cm

Figure 10.3 A sedimentation velocity pattern of /^-microglobulin in 0.08 M NaCl, 0.02 M sodium phosphate after incubation for seven days at pH 5.0 at 25°C; 60,000 rpm for 40 min. in a 1.2 cm cell. The points are experimental data and the solid line is the best fit onto A vs. r model in Table 10.3. (Reprinted from [5] with permission.)

protein dimerized when stored at 25°C for a week (Figure 10.3). The two sigmoid-shaped components are clearly seen, and the results of analysis of the r(t) vs. (t - t{]) for each of them is given in Table 10.3. These data were used to show that the radius of the aggregate defined by the Stokes-Einstein equation (eq. (6.15)) was twice that of the monomer.

The models in Tables 10.1 and 10.2 are appropriate for ultracentrifuge experiments in which either absorbance or refractive index is used as the method of detecting the moving boundary. The absorbance is directly related to the concentration of the macromolecule. Another major detection method uses Schlieren optics, which measures the concentration gradient dC/dr. When this method is used, the models for the shape of the boundary

Table 10,3 Parameters for Irreversible Aggregation of Bovine ß2-Microglobulin from Sedimentation Velocity Experiments [6]

 Species S2CI,K> f'tf Stokes radius,' A Monomer 1.21 1.56 17 Aggregate 5.10 1.48 34

a Normalized frictional coefficient. b Computed as in [6].

a Normalized frictional coefficient. b Computed as in [6].

are the derivatives of those presented in Tables 10.1 and 10.2. The model for the velocity of the boundary is the same irrespective of the detection method.