## Ultracentrifuge Studies of Solute Selfassociation

The use of analytical ultracentrifugation in protein chemistry for molecular mass determination is now usually bypassed in favour of its calculation from the amino acid sequence or its measurement by mass spectrometry. However, because such molecular mass values refer only to the covalently linked poly-peptides chain(s), they provide no information about the macromolecular state of the functional protein or enzyme. In the simplest current application of sedimentation equilibrium, molecular mass measurement is sometimes used to elucidate the nature of quaternary structure, which is an example of a self-association equilibrium displaced completely in favour of the polymeric state.

The aim of this section is to outline the basic principles of the use of analytical ultracentrifugation for the characterization of reversibly associating systems. Although more elaborate and intricate procedures are employed in practice, their description is beyond the scope of this introduction to the topic.

Characterization of Solute Self-association by Sedimentation Velocity

Because of the concentration-dependent variation in the average macromolecular state of a self-associating solute, the best approach in sedimentation velocity studies is to determine the dependence of the weight-average sedimentation coefficient, sA, upon total solute concentration, CA. Procedural details for the measurement of sA are as described above (eqn [1]), except that the asymmetric shape of the single reaction boundary that forms between solvent and the plateau region with concentration CA (Figure 1) necessitates the location of the boundary position, rp, as the square root of the second moment:

which allows an unequivocal estimate of the buoyant molecular mass to be obtained from the slope of ln cA(r) versus the square of radial distance.

to make allowance for the effects of migration in a sector-shaped cell rather than one with uniform cross-sectional area.

For a two-state self-association involving monomer (species 1) and dimer (species 2) the weight-average sedimentation coefficient is given by the expression:

where a1 = c1/cA is the weight fraction of monomer, and where the sedimentation coefficient of

Figure 3 Studies of solute self-association by sedimentation velocity: (A) Concentration dependence of the weight-average sedimentation coefficient simulated (eqn [10]) for a mono-mer-dimer system where K2 = 3.5Lg~\ s1=2.4S and s = 3.8 S. (B) Analysis of the data for the determination of K2 by means of eqn [13].

Figure 3 Studies of solute self-association by sedimentation velocity: (A) Concentration dependence of the weight-average sedimentation coefficient simulated (eqn [10]) for a mono-mer-dimer system where K2 = 3.5Lg~\ s1=2.4S and s = 3.8 S. (B) Analysis of the data for the determination of K2 by means of eqn [13].

monomer, s1, may be obtained by extrapolation of sA to zero solute concentration (Figure 3A). However, the evaluation of a1 from eqn [11] also depends upon knowledge of s2, the sedimentation coefficient of dimer, which is not readily obtained experimentally. An estimate is therefore usually made on the basis of the relationship s2 = s1(22/3), which follows from eqn [3] and assumed spherical geometry for monomeric and dimeric species. Knowledge of s2 then allows evaluation of the weight fraction of monomer from the following rearrangement of eqn [11]:

Characterization of Solute Self-association by Sedimentation Equilibrium

The above approach can be applied to the dependence of weight-average molecular mass MA upon total solute concentration cA by substituting molecular masses for the corresponding sedimentation coefficients in eqn [12]. Furthermore, there is no ambiguity about the magnitude of M2, which is twice the value of MA in the limit of zero solute concentration. Although this was indeed the original procedure used for characterizing solute self-association by sedimentation equilibrium, it has been superseded by a more accurate method involving direct analysis of the sedimentation equilibrium distribution(s).

Figure 4A presents simulated sedimentation equilibrium distributions at 15 000 and 25 000 rpm for a reversibly dimerizing protein with a buoyant molecular mass of 6.5 kDa for monomer (M1 + 25 kDa) and an equilibrium constant of 3.5 L g"1. Because the whereupon the dimerization constant, K2 = (1 — x1)ccA/ (a1c;A)2, may be determined.

Application of this approach to the data in Figure 3A, which have been simulated for a system with s1 = 2.4 S, s2 = 3.8 S and K2 = 3.5Lg~1, is illustrated in Figure 3B. From the logarithmic form of the expression for the dimerization constant (K2 = c2/c2), namely:

in present terminology, K2 may be obtained as the ordinate intercept of the linear dependence of log[(1 — a1)c;A] versus log(a1c;A); this has a mandatory slope of 2 for a monomer-dimer equilibrium.

Figure 4 Studies of solute self-association by sedimentation equilibrium. (A) Simulated concentration distributions reflecting reversible dimerization of a monomer (M1 = 25 kDa) governed by an association equilibrium constant of 3.5 L g — in sedimentation equilibrium experiments at 15000 (A) and 25000 (A) rev min-1. (B) Plot of data in readiness for global analysis according to eqn [15] with a fixed value of 0.4 g L_1 for the total solute concentration cA(rF) at reference radial position rF.

Figure 4 Studies of solute self-association by sedimentation equilibrium. (A) Simulated concentration distributions reflecting reversible dimerization of a monomer (M1 = 25 kDa) governed by an association equilibrium constant of 3.5 L g — in sedimentation equilibrium experiments at 15000 (A) and 25000 (A) rev min-1. (B) Plot of data in readiness for global analysis according to eqn [15] with a fixed value of 0.4 g L_1 for the total solute concentration cA(rF) at reference radial position rF.

distributions are at chemical equilibrium, the total solute concentration at each radial distance may be expressed in terms of monomer concentration by the relationship:

On the other hand, the condition of sedimentation equilibrium dictates that the distribution of monomer be described by eqn [7]. Combination of these two requirements leads to the expression:

The application of this approach is illustrated in Figure 4B, where global analysis of the two distributions from Figure 4A has been effected on the basis of a common CA(rF) value of 0.4 g L_1, thereby ensuring a common value of c1(rF). Magnitudes of ^1(r) at 20°C have then been calculated on the basis of the appropriate rF value, the buoyant molecular mass of monomer and the angular velocity. In as much as c1(rF) and K2 are both constants, their magnitudes are obtained by nonlinear curve-fitting of the combined [CA(r), ^1(r)] data sets to the above quadratic expression in ^1(r).

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