Sedimentation velocity centrifugation experiments measure the rate of movement of molecules through solution under centrifugal force and permit determining the sedimentation coefficient of a single macromolecule or the individual time-average distributions in a solution containing multiple sedimenting components. In addition to dynamic light scattering analyses, sedimentation velocity experiments are also useful for estimating the diffusion coefficients and frictional coefficients of the macromolecules and their size in terms of the radii of equivalent spherical molecules. From a single, short, sedimentation velocity run, it is possible to measure both the sedimentation and diffusion coefficients and to combine them to calculate Ms d, the sedimentation velocity-derived molecularweight. Finally, such information can be used with results from other physical measurements to evaluate the shape and hydration of a macromolecule in solution (1, 2).
1. Simple Graphical Analytical Methods
A sedimentation velocity experiment is illustrated in Figure 1. A sample of macromolecule is placed in one side of a centrifuge cell with only the buffer in the other side. It is centrifuged at a rate sufficient to sediment the macromolecule, depleting it at the meniscus and accumulating it at the bottom of the cell. At varying times, the sample cell is scanned for its absorbance at a wavelength where the macromolecule absorbs, giving the profile of the macromolecular concentration versus radial position. A series of profiles at different times represents the moving boundary of molecules sedimenting from the meniscus through the solution under the influence of the centrifugal field.
Figure 1. Sedimentation velocity experiment profiles at three different times (t = 1 to 3) in a centrifugation run scanned by absorption optics. The sample and reference solution menisci are indicated. The figure inset depicts the sample and reference channels of the centrifuge centerpiece.
The basic principle of sedimentation velocity is straightforward. The velocity of movement of a sedimenting boundary is defined by
or in its integrated form
where r^) is the position of the sedimenting boundary at time zero, r^ is the position of the sedimenting boundary at the time of measurement, w is the angular velocity of the rotor in radians/sec, and s is the sedimentation coefficient. The sedimentation coefficient can be determined from Eq. (2) by a plot of ln [rb^/r^] versus (t-^). There are typically three graphical/computermethods to choose the position of the sedimenting boundary (Figure 2). The first is simply to estimate or calculate the r^ value at the midpoint of the sedimenting boundary based on the(height) distance between the plateau region and the absorbance baseline of the sedimenting sample. The second is to use a first-derivative transformation of the data and to choose the r^ value at the peak position of the first derivative of each scan. The third and most accurate method for single-scan analysis using Eq. (2) is calculating the second-moment r^ boundary position (3).
Figure 2. Three different ways of analyzing sedimentation velocity data to identify the position of the sedimenting boundary. Raw data scans (upper panel) or smoothed data scans (middle panel) are evaluated at various times for the approximate values of rb(t) at (1) the transition midpoint (middle panel), (2) first-derivative maximum (lower panel), or (3) second-moment boundary position (lower panel) and plotted according to Eq. (2).
2. Computer-Based Analysis Methods
Until recently, graphical linear-fit analyses employing sedimentation velocity Eq. (2) have, been the most widely applied methods for analyzing sedimentation velocity data. Readily availability modern software for computerized graphical analysis has led, however, to the easy application of sophisticated hydrodynamic analytical methods to sedimentation velocity data to account for both the sedimentation and the diffusion of the macromolecule. In this way, it is possible to obtain values for both the sedimentation coefficient s and the translational diffusion coefficient D. For example, now two powerful analytical methods can be routinely applied to velocity data with only a modest investment of time and mathematical expertise.
2.1. Single-Scan Fitting to the Lamm Equation
One method now in common use is based on forms of Faxen’s two-component solution to the Lamm equation (4). The useful expressions are as follows:
where, F is the error function of the term enclosed in brackets,and s and D are, respectively, the sedimentation coefficient and the diffusion coefficient. Under suitable conditions (4), Eq. (3) can approximate the concentration profile of the sedimenting solute boundary closely (i.e., c versus r, as illustrated in Fig. 3). Recently, Behlke and Ristau (5) have described a very useful extension of this method, and Demeler et al. (6) demonstrated its value in modeling the sedimentation behavior of n-component samples. A short review of recent advances in this approach is provided by Laue (7).
Figure 3. Sedimentation velocity analysis of a 40-residue, b-amyloid peptide (Ref. 9 with permission). The absorbance profiles of the sedimenting peptide are given by the original curves after seven different times of centrifugation. In (a), the solid smooth lines indicate a single-component fit of the Lamm Eq. (3) to the data. The inset shows the values of s and D obtained from the scans at different times. In (b), the smooth solid lines represent a two-component fit (with two sedimenting species) of the Lamm Eq. (3) to the sample data. The inset represents the s and D values for the two different components at the various scan times. The sedimentation coefficients are in units of Svedbergs, and the diffusion coefficients are in units of Ficks.
2.2. Sedimentation Coefficient Distribution Analysis
A second modern approach, termed g*(s) analysis, was developed and has recently been reviewed by Stafford (8). Rather than analyzing a single scan, it uses a set of scans in a centrifugal run to compute the distribution of sedimentation coefficients of the various components of the sedimenting solute. This approach is extremely useful in analyzing otherwise poorly defined multicomponent samples. Because of this, it has found wide application in studying the size distributions of "real-world" samples of aggregated peptides (Fig. 4) and in defining the oligomeric state of recombinant proteins (9).
Figure 4. The apparent distribution of sedimentation coefficients for the data of Fig. 3 (Ref. 9 with permission). Whereas the data of Figure 3 show two sedimenting components of about 28S and 64S, the g*(s) analysis shows that the sample of b-peptide is better represented as a distribution of species ranging from about 10S to 70S. The amyloid b-peptide and its aggregated forms may play a role in familial Alzheimer’s disease.
3. The Sphere-Equivalent Radius
As indicated above, fitting to the Lamm equation yields the values of both s and D. One way of converting these sedimentation/diffusion coefficient values into more familiar terms is to employ the relationship between the diffusion coefficient and the frictional coefficient of the sedimenting particle (see Table 2 of Centrifugation). Although the details of the relationships between the frictional coefficient and molecule shape are beyond the scope of this discussion (see references 2, 10, 11 and Diffusion for more thorough treatments), a simple approximation is possible and is often very useful as a molecular "yardstick." The diffusion coefficient of a sphere is directly proportional to temperature and inversely proportional to the Stokes’ law frictional coefficient. Thus, knowing the viscosity of the solvent, it is possible to compute the radius of the equivalent sphere rsphere of the solute particle under study. This is typically done using the diffusion coefficient obtained from fitting to the Lamm equation. Alternatively, D can be estimated using the simple ratio of area/height of the first derivative of the sedimenting boundary (Fig. 1), as described by Fugita (4):
Equation (2) provides the sedimentation coefficient while Eq. (4) gives the value of D. Using the relationship between the frictional coefficient and the diffusion coefficient and combining all of the various constants, it can be shown that rsphere in nanometers is equal to kelvin/Din cm /sec) for a particle in a solvent with the viscosity of water. Thus a diffusion coefficient of 1 Fick, orrepresents an rsphere of 21.5 nm at 20°C in water. Using
this relationship with some typical proteins, it is possible to estimate their diameters from just their measured diffusion coefficients. For example, hen egg-white lysozyme has a diffusion coefficient of (10) and an rsphere of ~1.9nm at 20°C in water, and bovine serum albumin has a diffusion coefficient of(10) and an rsphere of ~3.1nm at 25°C in water. For comparison, the X-ray crystallographic structure of hen lysozyme has rough dimensions of 3.0×3.0×4.5nm.
4. Ms D, The Molecular Weight Derived from Sedimentation Velocity Data
The molecular weights of solutes are measured most accurately by sedimentation equilibrium centrifugation, but the sedimentation and diffusion coefficients obtained from sedimentation velocity experiments can be combined to give an approximate molecular weight (2, 10, 11) from the expression
where R is the gas constant, Tthe absolute temperature, r the density of the solvent, and v the partial specific volume of the macromolecule. It is most useful to calculate the M, d values during the sedimentation velocity phase of an analytical ultracentrifugation run profile. This is especially true for runs using "long-column" centerpieces (Fig. 1) and extended equilibrium run times (>48hours). However, it is also sometimes useful for data collected with multichannel equilibrium centerpieces, even though the time to equilibrium is shorter because of the reduced column height in the centerpieces. Because the value of M, d can be calculated quickly (within a few minutes), it is often useful in choosing the final speed for the sedimentation equilibrium phase of a centrifugation run after the high-speed, velocity phase of the run (see Analytical Ultracentrifugation).