Allen Furst
Beckman Coulter, Inc.

The analytical ultracentrifuge permits observations of the behavior of macromolecules subjected to a centrifugal field. Such experiments permit determination of solution molecular weights, association constants, and studies of homogeneity, shape, and other molecular parameters. The instrument consists of a centrifuge and a rotor with a windowed sample compartment, so that the distribution of macromolecules along the radial dimension can be determined at any time by means of an appropriate optical system.

The main advantages of the analytical ultracentrifuge are 1) that it permits studies of molecules at varying concentrations in many buffers including widely varying salt and pH conditions, and 2) that the conclusions reached are based directly upon first principles, and do not require comparisons to standards, which must be assumed to behave in like manner to the molecules under investigation.

Two basic types of experiment can be performed with the analytical ultracentrifuge. In a sedimentation equilibrium experiment, a steady state condition is allowed to develop in which a sample's tendency to sediment in the centrifugal field is counterbalanced by its tendency to diffuse against the concentration gradient so established. This is the preferred method for accurate molecular weight determinations and for studies of homogeneity and molecular associations.

In a sedimentation velocity experiment, the speed with which a molecule moves toward the bottom (outermost boundary) of the cell is determined. This yields the sedimentation coefficient, s, which can be related to the molecular weight and to the frictional coefficient or shape of a particle. In an analogous method, the flotation velocity (i.e., towards the center of rotation) of materials having densities less than the solvent is measured. These methods are used for rapid estimates of the molecular weight and for determining particle size distributions in a sample.

Features of the Optima XL-A

Centrifuge and Rotor
The Optima XL-A ultracentrifuge is readily convertible between analytical and preparative configurations. The induction drive is equipped with a dynamic damping capability that reduces rotor precession to near zero. Rotor temperature is monitored by radiometry and regulated by thermoelectric modules. This system provides rotor temperature stability to within 0.13°C at equilibrium. The titanium rotor is designed to run double-sector sample cells, and includes a counterbalance with reference holes for radial calibration.

Absorption Optical System
The optical system is shown in Figure 1. A xenon flashlamp serves as the light source, providing a usable wavelength range of 190-800 nm. The lamp is fired as the sector of interest passes over the detector, with the timing regulated by monitoring the passage of a reference magnet in the bottom of the rotor. The maximum firing rate is 100 flashes per second, corresponding to once per ten revolutions at 60,000 rpm.

Figure 1. Schematic of the absorbance optical system of the Optima XL-A. Features of the optical system are described in the text.

A toroidally-curved diffraction grating selects single-wavelength light for projection onto the sample. The grating is rotated by a precision geartrain to select the wavelength of interest, and provides a nominal bandpass of 2 nm. A series of absorbing filters is also provided to block out light of other wavelengths, thus reducing stray light.

Since the intensity of light from the flash lamp varies somewhat from pulse to pulse, light from the diffraction grating is normalized by reflecting a small percentage onto a detector located at the virtual focal point of the monochromator system.

Monochromatic light passes through the sample cell, which is bounded by two quartz windows. This cell contains both a sample sector and a solvent sector so that the intensity of light transmitted through the sample can be expressed with reference to the solvent, as measured by a photomultiplier tube positioned beneath the rotor. A lens-slit assembly moves as a unit to provide radial scans of these sectors. Multiple readings can be acquired at each radial distance and averaged to reduce noise. Readings at several wavelengths can also be taken at each radial distance.

The absorption optical system of the XL-A is based upon the fact that many macromolecular solutes include chromophores that absorb incident radiation at particular wavelengths. For solutes obeying Beer's law, the absorption is linearly related to the molecular concentration. Thus, the radial distribution of the solute of interest, or C(r), is readily determined from a radial scan of optical density.

Absorption optics offer very high sensitivity. This allows the study of dilute solutions in which thermodynamic nonideality will be minimal. In typical salt concentrations (0.1 to 0.2 molar), nonideality can be ignored for all but the most asymmetric molecules.

A further advantage of absorption optics is the ability to discriminate between molecules with different chromophores. Thus, the radial distribution of a typical protein absorbing at 280 nm can be distinguished from that of a potential ligand if the latter has an absorption peak in the near UV or visible range. This permits characterization of the stoichiometry and association constant of the binding reaction.

Sedimentation Velocity

Mathematical Theory
Several excellent treatments of this topic are available in the literature (Svedberg and Pedersen, 1940; Schachman, 1959). What follows is a concise overview (see Figure 2). A particle of mass M in a centrifugal field generated by a spinning rotor is subjected to a centrifugal force

where omega is the angular velocity in radians per second and r is the distance in millimeters from the center of rotation. As this particle is moved through the solvent medium, it displaces solvent molecules and encounters an opposing force due to buoyancy.

where rho is the solvent density and v-bar_b the partial specific volume of the particle (the inverse of particle density). In addition, there will be frictional resistance to the motion of a particle through the solvent, with the frictional force equal to

where f is the frictional coefficient, and v the velocity of the particle. At constant rotor speeds, the velocity of the particle will be constant, and

Substituting and solving for the velocity,

The velocity is generally expressed in terms of the particle sedimentation coefficient

and expressed in Svedberg units (S; 1 S = 10^-13 seconds). This yields the basic equation applied to sedimentation velocity experiments, namely

Thus, the sedimentation velocity of a particle will increase with its mass, density, or with the rotational speed, and will decrease with increasing friction (related to particle asymmetry) or solution density.

Figure 2. Forces experienced by a particle in the centrifugal field. A particle, shown here in a sector-shaped cell, experiences three forces during centrifugation. These are the centrifugal force (F_cent), a force due to buoyancy (F_buoy), and frictional force (F_frict). The combined action of these three forces results in a constant velocity of the particle in the radial direction.

Experimental Considerations

For sedimentation velocity experiments it is essential that the cells holding the sample and solvent be sector-shaped, with walls aligned along the radii of the rotor. This prevents sedimenting particles from colliding with the walls (wall effects). The ability to resolve boundaries is proportional to omega²rL/theta where L is the column length, and theta is the width of the section (Svedberg and Pedersen, 1940). Thus, a long, narrow solution column is generally preferred for highest resolution. Typical solution columns for sedimentation velocity runs hold 0.45 mL.

Distribution of Concentrations
Sedimentation velocity experiments are normally of a type called boundary sedimentation. The experiment begins with the sample mixed uniformly throughout the cell, so that a plot of concentration vs radius is a horizontal line (C(r) = constant). As sedimentation proceeds, molecules are depleted from the top of the solution column. This results in the formation of a trailing boundary for the concentration distribution. In an alternative procedure, termed zonal sedimentation, a special sample cell is used to introduce the sample to the top of the solution column during the centrifuge run. This results in a discrete zone of sample molecules migrating through the solvent column.

Figure 3 shows typical data acquired during a boundary sedimentation experiment. This is simply a plot of the solute concentration as a function of radial distance, or C(r). Several key features of the data are pointed out in the figure. A pair of sharp peaks indicate the positions of the menisci. The sample compartment of the cell is normally filled with slightly less liquid than the solvent compartment, so that the inner meniscus is that of the solvent compartment and the outer meniscus is that of the sample compartment. An unusually large distance between these menisci, or one that shifts during a run, is an indication of leakage from the cell.

Figure 3. Features of boundary sedimentation data. The figure shows typical data from a boundary sedimentation experiment in the Optima XL-A. The data represent the absorbance of the fluid in the sample sector compared to the reference sector of the double-sector cell. Sharp peaks result from the refraction of light away from the photomultiplier by the menisci in each sector. The sedimentation coefficient is determined from the movement of the boundary region with time.

The remainder of the data consists of the boundary region in which the solute concentration increases rapidly to a reasonably constant value in the plateau region. Most of the information in a sedimentation velocity experiment is taken from analysis of the boundary. In a simple sedimentation involving one component, the boundary will be sharp, and the sedimentation coefficient can be derived from the motion of the boundary midpoint (see below). In more complex analyses involving two or more components, the boundary will also be divided into two or more rising segments. Assuming each component has the same extinction coefficient, the relative heights of the boundary segments return the relative concentration of each component. The radial motion of each segment can be analyzed independently to determine each sedimentation coefficient. At the extreme, a very broad boundary is indicative of a heterogeneous sample.

As an alternative representation, the data may be presented as the derivative of the concentration function, or dC/dr. In this representation, each boundary segment appears as a discrete peak, the sedimentation coefficient is obtained from the radial motion of these peaks. The relative concentration of each sample component is determined from the area under each peak.

One feature of the plateau region is worth noting. Particles at greater radii will move faster than those closer to the center of rotation, thus pulling away from the latter. In addition, as the experiment progresses, particles beginning near the outermost portion of the solution column will be pelleted against the outer wall of the sample cell, and will be replaced by particles from nearer the center of rotation. These latter particles enter a progressively increasing volume as they migrate outward through the sector-shaped cavity, and thus become more dilute. This phenomenon of radial dilution (Figure 4) accounts for the gradual decrease in optical density in the plateau (Trautman and Schumaker, 1954).

Figure 4. Radial dilution. These data represent sedimentation of a 2 S particle at 60,000 rpm during a run of approximately 3 h (10,000 S). As the particles sediment and the boundary moves to the right, the meniscus eventually becomes depleted (seen at left of figure). In addition, as the particles enter the increasing volume at the bottom of the sector-shaped cavity (right of figure), the concentration and the absorbance of the plateau region are seen to decrease.

Data Analysis

Follow the Midpoint
For all but the smallest monodisperse solutes, the sedimentation coefficient may be obtained by following the rate of motion of the boundary midpoint, r_b. This is most readily calculated from the slope of the equation

where the time, t, is plotted in seconds.

This approach will lead to incorrect results for solutes with s less than about 2 S. For such solutes, it is preferable to use the second moment method of Goldberg (1953). This requires integration of the concentration function C(t)rdr from the meniscus to the plateau.

Concentration Dependence
The sedimentation coefficient should be obtained over a range of solute concentrations and extrapolated to infinite dilution. Highly asymmetric molecules, or molecules forming associating systems, will show concentration dependence of s (Rowe, 1977).

Molecular volume and extension.
Highly asymmetric molecules tend to occupy a disproportionately large volume due to their rotational motion in solution. The net effect is to prevent solvent molecules from approaching them, increasing the apparent viscosity of the solvent and reducing the sedimentation rate of the asymmetric solute. Thus, the observed sedimentation coefficient for a highly asymmetric molecule can decrease precipitously with increasing concentration.

Associating systems.
Where macrosolute molecules tend to dimerize or form higher order associating systems, the sedimentation coefficient may increase with solute concentration. Such associating systems are best studied by methods of sedimentation equilibrium.

Johnston-Ogston effect for mixtures.
A similar effect is observed in the sedimentation of mixtures (Johnston and Ogston, 1946). At high concentrations, fast-moving macrosolute molecules must move through a layer of slow-moving macrosolutes as well as solvent. The slow-moving species increases the apparent viscosity of the solvent, again leading to a concentration-dependent decrease in the sedimentation coefficient. This effect is particularly evident for asymmetric molecules.

Speed Dependence

It is also good practice to check for speed dependence in the obtained values of s. Speed dependence is sometimes observed when sedimenting very large, asymmetric molecules, or highly polymerized, but dissociable molecules. Some of the causes are described here. Where speed dependence is observed, it can be avoided simply by working at the lowest practical rotational speeds.

Speed-dependent aggregation.
The observed sedimentation coefficient for some solutes may increase with increasing rotor speed. This phenomenon is believed to result from a wake left behind macrosolutes moving through the solution column, clearing buffer and salt molecules from the medium. This permits an increased velocity for trailing macrosolute molecules, resulting in the formation of macromolecular aggregates.

Speed-dependent distortion of large DNAs.
For highly asymmetric molecules, such as DNA, the sedimentation coefficient may appear to decrease with increasing speed. This is especially true for very large DNA fragments. This phenomenon is believed to be due to distortion of the molecule at high speeds caused by solvent friction.

Solvent Effects

Charge.
When charged macrosolutes, such as proteins or nucleic acids, are centrifuged through a polar solvent, they move more rapidly than the solvent counterions that normally envelope them. This results in a charge separation and potential difference that slows the macrosolute molecules, and results in a decrease in the observed sedimentation coefficient. This effect is generally avoided by using ionic strength in excess of 50 mM.

Viscosity and density.
Very dense or viscous solvents will reduce the observed sedimentation coefficient by increasing the forces of buoyancy and frictional drag experienced by the macrosolute. Sedimentation coefficients are therefore conventionally expressed in terms of a standard solvent, viz., water at 20°C. An observed sedimentation coefficient can be corrected to the standard, s_20,w value with the equation:

where rho_T is the solvent density and rho_20,w that of water at 20°C, eta_T/eta₂₀ is the relative viscosity of water at the temperature T with respect to 20°C, and eta/eta₀ is the relative viscosity of the solvent with respect to water.

Applications of Sedimentation Velocity

Analysis of Polydispersity
If a polydisperse solution is composed of particles large enough that there is no appreciable spreading of the boundary due to diffusion, then the boundary spreading that is observed will be solely attributable to the spread of s values within the sample. Under these circumstances, a distribution function g(s) can be defined such that g(s)ds will equal the weight fraction of macrosolute with sedimentation coefficients between s and s + ds (Signer and Gross, 1934; van Holde and Weischet, 1978). The function g(s) will depend upon dC/dr, r, t, and the angular velocity, omega.

Analysis by g(s) can be extended to cover a very broad range of particle sizes by means of a gravitational sweep experiment, in which the rotor speed is progressively increased during the run.

Molecular Weight Estimated from s and D
Estimates of the molecular weight of a macrosolute can be determined by sedimentation velocity experiments. More accurate values are obtained from sedimentation equilibrium runs. The advantage of the velocity procedure is that it permits simultaneous determination of the molecular weights for several components in a mixture, providing that the boundaries formed by these components can be well separated during the course of the experiment.

The molecular weight, M, can be determined from the following equation,

also known as the Svedberg equation. Here, R is the gas constant, T the temperature, v-bar_b the partial specific volume of the solute, D is its diffusion coefficient, and rho is the solvent density.

The diffusion coefficient may be obtained through separate analytical ultracentrifuge experiments in which the spreading of an artificial boundary is observed. Alternatively, both s and D may be estimated simultaneously by nonlinear regression (Holladay, 1979).

There are several ways to obtain the partial specific volume. It can be estimated from the sum of the partial specific volumes of the constituent amino acids (Cohn and Edsall, 1943). Accurate measurements of v-bar_b can be made by careful weighings of known solution volumes in a pycnometer. Finally, v-bar_brho can be measured in the analytical ultracentrifuge by determining the quantity M(1 - v-bar_brho) in solvents of different density, such as normal and deuterated water (Edelstein and Schachman, 1967). An estimated value of v-bar_brho is sometimes used, typically 0.735 mL/g. It should be noted, however, that a small error in v-bar_brho will lead to a considerably greater error in the determination of s or M.

Determination of Diffusion Coefficient

The diffusion coefficient, D, can be determined from the spreading of the boundary during a sedimentation velocity run. It is necessary that the sample consist of a single species or, if multiple species are present, that each boundary be completely resolved during the run. Details of this analysis are given by Baldwin (1957).

References

1. Baldwin, R. L. Boundary spreading in sedimentation velocity experiments. 5. Measurement of the diffusion coefficient of bovine albumin by Fujita's equation. Biochem. J. 54, 503-512 (1957)
2. Cohn, E. J. and Edsall, E. T. Proteins, Amino Acids, and Peptides as Ions and Dipolar Ions, pp. 374-377. New York, Reinhold Publ. Co., 1943.
3. Edelstein, S. J., Schachman, H. K. The simultaneous determination of partial specific volumes and molecular weights with microgram quantities. J. Biol. Chem. 242, 306-311 (1967)
4. Goldberg, R. J. Sedimentation in the Ultracentrifuge. J. Phys. Chem. 57, 194-202 (1953)
5. Holladay, L. A. Molecular weights from approach-to-sedimentation equilibrium data using nonlinear regression analysis. Biophys. Chem. 10, 183-185 (1979)
6. Johnston, J. P., Ogston, A. G. A boundary anomaly found in the ultracentrifugal sedimentation of mixtures. Trans. Faraday Soc. 42, 789-799 (1946)
7. Rowe, A. J. The concentration dependence of transport processes: a general description applicable to the sedimentation, translational diffusion, and viscosity coefficients of macromolecular solutes. Biopolymers 16, 2595-2611 (1977)
8. Schachman, H. K. Ultracentrifugation in Biochemistry. New York, Academic Press, 1959.
9. Signer, R., Gross, H. Ultrazentrifugale Polydispersitätsbestimmungen an hochpolymeren Stoffen. Helvet. Chim. Acta 17, 726-735 (1934)
10. Svedberg, T., Pedersen, K. O. The Ultracentrifuge. Oxford, Clarendon Press, 1940.
11. Trautman, R., Schumaker, V. Generalization of the radial dilution square law in ultracentrifugation. J. Chem. Phys. 22, 551-554 (1954)
12. van Holde, K. E., Weischet, W. Boundary analysis of sedimentation-velocity experiments with monodisperse and paucidisperse solutes. Biopolymers 17, 1387-1403 (1978)