## Sedimentation Velocity and Sedimentation Equilibrium

A complex in rapid association equilibrium with reac-tant species has no separate existence in the absence of the reactants. The characterization of such interactions requires methods that can accommodate the equilibrium coexistence of several species. This requirement is readily accommodated by either of the commonly used techniques in analytical ultracen-trifugation - sedimentation velocity and sedimentation equilibrium. In the former, the centrifuge is operated at a sufficiently high angular velocity (speed) for the centrifugal force on a solute to dominate its migration. In sedimentation equilibrium the same instrument is operated at a much lower angular velocity to effect a balance between the radially outward flux of solute and the back-diffusional flow in response to the concentration gradient being generated. Before considering their application to the characterization of interacting systems, it is appropriate to describe the two techniques in relation to the information derived for a single noninteracting solute.

### Measurement of a Sedimentation Coefficient

In sedimentation velocity, a solution is placed in a sector-shaped cell which allows unimpeded migration of solute molecules in a radially outward direction in response to the applied centrifugal field. At the commencement of the experiment, the solute concentration is uniform throughout the cell, but the application of a strong centrifugal field (typically 200 000-300 000 g operating at 50 000-60 000 rpm) leads to a progressive removal of solute from the inner region of the cell (Figure 1).

Migration of the resultant solute boundary is recorded optically and the sedimentation coefficient of the solute, sA , is then determined from the rate of migration. Specifically:

where rp denotes the radial position (cm) of the solute boundary after centrifugation at angular velocity w (rads"1) for time t (s). Traditionally, the sedimentation coefficient has thus been determined from the slope (w2sA) of the time dependence of ln rp. Despite being the rate of migration per unit field, sA has the dimensions of time, but is usually reported in Svedberg units S (1 S = 10"13 s).

At the limit of zero solute concentration, the sedimentation coefficient, sA, is related to molecular

Radial distance (r)

Figure 1 Schematic representation of migration in a sedimentation velocity experiment, showing the generation of a boundary between solvent and the solution subjected to centri-fugation: rp denotes the mean boundary position at time t3, and ra the air-liquid meniscus (boundary position at zero time).

Radial distance (r)

Figure 1 Schematic representation of migration in a sedimentation velocity experiment, showing the generation of a boundary between solvent and the solution subjected to centri-fugation: rp denotes the mean boundary position at time t3, and ra the air-liquid meniscus (boundary position at zero time).

parameters by the expression:

where fA denotes the shape-dependent translational frictional coefficient of the solute with molecular mass Ma and partial specific volume vA: ps is the solvent density and N is Avogadro's number. Elimination of the frictional coefficient on the assumption of spherical geometry for the solute (fA = 6n^saA) leads to the relationship:

where is the solvent density and aA is the Stokes radius of the solute (radius of the equivalent hy-drodynamic sphere). The sedimentation coefficient is thus a function of the size (shape) as well as the buoyant molecular mass, MA(1 — vAps), of the solute.

### Experimental Aspects of Sedimentation Equilibrium

Unequivocal determination of the buoyant molecular mass by analytical ultracentrifugation is clearly conditional upon replacement of the frictional coefficient in eqn [2]. On the grounds that the translational diffusion coefficient, DA, is related to fA by the expression:

where R and T refer to the universal gas constant and the absolute temperature (K), respectively, the influence of fA upon the separate magnitudes of the sedimentation and diffusion coefficients disappears from their ratio. Thus:

In sedimentation equilibrium, the solute distribution is governed by this ratio of soA/DoA. The parameter to emerge from the analysis of such distributions is the buoyant molecular mass of the solute, MA(1 — vAps). Its conversion to a molecular mass requires assignment of a magnitude to (1 — vAps), which may be determined experimentally by density measurements and the relationship (1 — vAps) + (p — ps)/cA, where p is the density of a solute solution with weight concentration cA.

Sedimentation equilibrium experiments are conducted in a double-sector cell. One sector contains the solution of macromolecular solute and the other the appropriate solvent, which for charged solutes is the buffer with which the solution of macro-ion is in dialysis equilibrium. As in sedimentation velocity, the solute concentration is initially uniform throughout the column of solution (Figure 2). Application of the centrifugal field then results in depletion of solute in the vicinity of the air-liquid meniscus (ra) and its accumulation at the cell base (rb). However, at the relatively low speeds of such experiments (say, 10 000 rpm for a 50 kDa protein), these tendencies are countered by back-diffusion in response to the concentration gradient being formed. The net result is a progression towards

Radial distance (r)

Figure 2 Schematic representation of the approach to a time-independent distribution in a sedimentation equilibrium experiment: the initially uniform concentration gradually changes into an exponential distribution described by eqn [7] for a single solute.

Radial distance (r)

Figure 2 Schematic representation of the approach to a time-independent distribution in a sedimentation equilibrium experiment: the initially uniform concentration gradually changes into an exponential distribution described by eqn [7] for a single solute.

a time-independent exponential increase in solute concentration across the cell (Figure 2).

Because the time taken to attain sedimentation equilibrium varies inversely with the square of the column length, columns longer than 3 mm are rarely used. Whereas 3-4 weeks of centrifugation would be required to attain time independence of the solute distribution in a completely filled cell (rb — ra + 1 cm), effective sedimentation equilibrium can be reached within 16-36 h by decreasing the column length to 2-3 mm.

In the original treatise on analytical ultracentrifu-gation published by Svedberg and Pedersen in 1940, sedimentation equilibrium was considered in terms of the balance between the processes of sedimentation and diffusion. However, later consideration led to the realization that results from such experiments were amenable to rigorous thermodynamic analysis by expressing the diffusional flow in terms of the negative gradient of solute chemical potential. Consequently, even though the experimental record is in terms of solute concentration as a function of radial distance, the distribution of a single solute at sedimentation equilibrium is defined in terms of its ther-modynamic activity:

which relates the thermodynamic activity of solute at any given radius, zA(r), to that at a chosen reference radial position, rF. Although procedures have now been devised which make allowance for effects of thermodynamic nonideality, the situation is simplified for the present purpose by restricting consideration to ideal systems in which the two thermodyn-amic activities may be replaced by concentrations cA(r) and cA(rF): eqn [6] thus becomes:

From the logarithmic form of eqn [7] namely: lnCA(r) = lnca(M # MA(1 - ^A^r2 - rF)/(2RT)

it is evident that:

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