DS-819
T e c h n i c a l
I n f o r m a t i o n
Analytical Ultracentrifugation
...............................................
Overview of Sedimentation Velocity
for the Optima™ XL-A Analytical Ultracentrifuge
Allen Furst
Beckman Instruments, Inc., Palo Alto, CA
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.
BECKMAN
Features of the Optima XL-A
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.
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.l3°C at equilibrium. The titanium rotor is
designed to run double-sector sample cells, and includes a counterbalance with reference holes for radial calibration.
Toroidal
Diffraction
Grating
Incident
Light
Detector
Reflector
Sample/Reference
Cell Assembly
Rotor
Imaging System for
Radial Scanning
Slit (2 nm)
Aperture
Xenon
Flash Lamp
Photomultiplier Tube
Figure 1. Schematic of the absorbance optical system
of the Optima XL-A. Features of the optical system
are described in the text.
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.
2
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
;
Fcent
Fbuoy
Fcent = Mω 2r
where ω 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.
Ffrict
Figure 2. Forces experienced by a particle in the
centrifugal field. A particle, shown here in a sectorshaped cell, experiences three forces during
centrifugation. These are the centrifugal force (Fcent),
a force due to buoyancy (Fbuoy), and frictional force
(Ffrict). The combined action of these three forces
results in a constant velocity of the particle in the
radial direction.
Fbuoy = Mω 2rvb ρ
where ρ is the solvent density and vb 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
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 ω 2rL/θ where L
is the column length, and θ 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.
Ffrict = fv
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
Fcent = Fbuoy + Ffrict.
Substituting and solving for the velocity,
v = Mω 2r (1 - vb ρ)/f.
The velocity is generally expressed in terms of the
particle sedimentation coefficient
s = v/ω 2r
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.
and expressed in Svedberg units (S; 1 S = 10-13 seconds). This yields the basic equation applied to sedimentation velocity experiments, namely
s = M(1 - vb ρ)/f.
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.
3
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 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.
Plateau Region
Boundary Region
Absorbance
Sample
Meniscus
0
Solvent
Meniscus
Radius
Concentration
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
1 000s
2000
s
300
0s
400
0s
500
0s
600
0
700 s
0s
80
00
s
90
0
10 0s
,00
0s
Initial
Concentration
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.
0
Radius
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.
4
Data Analysis
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.
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,
rb. This is most readily calculated from the slope of
the equation
ln(rb) = (ω 2s)t
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.
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, s20,w value with the
equation:
s20,w = sobs[(1 - vb ρ20,w)/(1 - vb ρT)](ηT/η20)(η/η0)
where ρT is the solvent density and ρ20,w that of water at 20°C, ηT/η20 is the relative viscosity of water
at the temperature T with respect to 20°C, and η/η0
is the relative viscosity of the solvent with respect to
water.
5
Applications of Sedimentation Velocity
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 vb can be made by careful weighings of known
solution volumes in a pycnometer. Finally, vb ρ can
be measured in the analytical ultracentrifuge by determining the quantity M(1 - vb ρ) in solvents of different density, such as normal and deuterated water
(Edelstein and Schachman, 1967). An estimated
value of vb ρ is sometimes used, typically
0.735 mL/g. It should be noted, however, that a
small error in vb ρ will lead to a considerably greater
error in the determination of s or M.
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, ω.
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,
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).
M = RTs/D(1 - vb ρ)
also known as the Svedberg equation. Here, R is the
gas constant, T the temperature, vb the partial specific volume of the solute, D is its diffusion coefficient, and ρ is the solvent density.
6
References
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)
Schachman, H. K. Ultracentrifugation in Biochemistry. New York, Academic Press, 1959.
Signer, R., Gross, H. Ultrazentrifugale
Polydispersitätsbestimmungen an hochpolymeren
Stoffen. Helvet. Chim. Acta 17, 726-735 (1934)
Svedberg, T., Pedersen, K. O. The Ultracentrifuge.
Oxford, Clarendon Press, 1940.
Trautman, R., Schumaker, V. Generalization of the
radial dilution square law in ultracentrifugation.
J. Chem. Phys. 22, 551-554 (1954)
van Holde, K. E., Weischet, W. Boundary analysis
of sedimentation-velocity experiments with
monodisperse and paucidisperse solutes. Biopolymers 17, 1387-1403 (1978)
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)
Cohn, E. J. and Edsall, E. T. Proteins, Amino Acids,
and Peptides as Ions and Dipolar Ions, pp. 374377. New York, Reinhold Publ. Co., 1943.
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)
Goldberg, R. J. Sedimentation in the Ultracentrifuge. J. Phys. Chem. 57,194-202 (1953)
Holladay, L. A. Molecular weights from approachto-sedimentation equilibrium data using nonlinear
regression analysis. Biophys. Chem. 10, 183-185
(1979)
Johnston, J. P., Ogston, A. G. A boundary anomaly
found in the ultracentrifugal sedimentation of
mixtures. Trans. Faraday Soc. 42, 789-799
(1946)
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