## A 2 Sedimentation Equilibrium

When an ultracentrifuge experiment reaches equilibrium, there is no net transport of matter in the sector cell. Hence, the concentration boundary does not move. In the Yphantis method [6] of high-speed sedimentation equilibrium, the concentration of macromolecules is negligible at the meniscus in the cell and can be used as a point of reference. This is also called the meniscus depletion method. Analysis of sedimentation equilibrium data using this method will be discussed in this section.

The sedimentation equilibrium experiment for a single ideal solute can be described by fc — = f awrdr (10.5)

jc0 c jra where the symbols are defined as follows: <rw = Mwa)2( 1 - vp)/RT,

Mw - weight average molecular weight of the macromolecule, v = partial specific volume of the macromolecule, p = density of the solvent,

R = ideal gas constant,

T = temperature in Kelvins, c0 = concentration in mg mL ' at the meniscus, c = concentration at a distance r from the center of rotation, ra = distance of the meniscus from the center of rotation, r = distance from the center of rotation, a) = angular velocity of the rotor in radians s"1.

Absorbance vs. distance data for sedimentation equilibrium can be analyzed by the integral of eq. (10.5). This model is summarized in Table 10.4.

Table 10.4 Model for Analysis of Sedimentation Equilibrium Data Obtained by the Yphantis Meniscus Depletion Method [6] for a Single Macromolecule

Assumptions: One macromolecule, no association or dissociation; detection by absorbance,

A, which is proportional to c Regression equations:

c = co exp[(<r„ / 2)(r2 - r|)] Regression parameters: Data:

Special considerations: M„ is obtained from = Mw<o2(l — vp)IRT; other symbols are defined in the text

We shall see that sedimentation equilibrium experiments can also be used to test for macromolecular aggregation. First, we show the results of an experiment on a monomeric macromolecule, holoRiboflavin-binding protein (Figure 10.4). This monomeric protein was found to have a Mw of 32,500 ± 1000 Daltons and to be free from aggregation or higher Mw impurities.

Proteins often associate reversibly to form dimers, trimers, and higher aggregates. Some typical association schemes follow:

A limiting model arises from the assumption that mainly monomer and n-mers exist in the solution:

The equilibrium constant for this association is given by

This monomer/ra-mer model is summarized in Table 10.5.

The model in Table 10.5 is illustrated for sedimentation equilibrium of bovine ^-microglobulin (Figure 10.5). This protein forms a tetramer (n = 4) reversibly when the monomer concentration is above 0.4 mg mL

ui O

o to CD

DISTANCE FROM CENTER OF ROTATION, cm

Figure 10.4 Sedimentation equilibrium data for hoioRiboflavin binding protein, pH 7.0 in 0.1 M NaCl, 0.03 M sodium phosphate at 25°C; 8,000 rpm in a 1.2 cm cell. The points are experimental data, and the solid line is the best fit of the model in Table 10.4. (Reprinted from [7] with permission.)

Figure 10.4 Sedimentation equilibrium data for hoioRiboflavin binding protein, pH 7.0 in 0.1 M NaCl, 0.03 M sodium phosphate at 25°C; 8,000 rpm in a 1.2 cm cell. The points are experimental data, and the solid line is the best fit of the model in Table 10.4. (Reprinted from [7] with permission.)

Table 10.5 Model for Analysis of Sedimentation Equilibrium Data Obtained by the Yphantis Meniscus Depletion Method [6] for Monomer/n-Mer Equilibrium

Assumptions: Detection by absorbance A; nM ^ M„ Regression equations:

c = c0exp[(crJ2)(r2 - r2)] + nKc$ exp[(naJ2)(r2 - r2)] Regression parameters: Data:

Special considerations: M„ is obtained from an = Mwoj 2(1 - vp)/RT; other symbols are defined in the text. If the model in Table 10.4 gives a poor fit, do a series of regression with integral n values beginning with n = 2. Choose the model with the smallest standard deviation of regression as the best fit

The models in Table 10.4 and 10.5 are appropriate for ultracentrifuge experiments in which absorbance is used as the method of detecting the boundary. When Schlieren optics are used, the models for the shape of the boundary are the derivatives of those presented in Table 10.4 and 10.5. Alternatively, if the Schlieren data are integrated, the models in Table 10.4 and 10.5 would be applicable.

The model in Table 10.4 assumes that the sedimentation experiments were done on ideal systems. For nonideal systems, virial coefficients must be considered. These and other aspects of analysis of data from sedimentation equilibrium experiments have been discussed in detail [4],

7.00

7.04

7.08

7.12

7.16

Figure 10.5 Sedimentation equilibrium data for /^-microglobulin in 0.08 M NaCl, 0.02 M sodium phosphate, pH 5.0 at 25°C at 40,000 rpm in a 0.6 cm cell. Initial concentration >0.4 mg mL~!. The points are experimental data, and the solid line is the best fit (n = 4) onto model in Table 10.5. (Reprinted from [5] with permission.)

7.00

7.04

7.08

7.12

7.16

DISTANCE FROM CENTER OF ROTATION, cm

Figure 10.5 Sedimentation equilibrium data for /^-microglobulin in 0.08 M NaCl, 0.02 M sodium phosphate, pH 5.0 at 25°C at 40,000 rpm in a 0.6 cm cell. Initial concentration >0.4 mg mL~!. The points are experimental data, and the solid line is the best fit (n = 4) onto model in Table 10.5. (Reprinted from [5] with permission.)

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