## Ultracentrifugation: The derivation and use of the Svedberg Equation

What happens to a particle (macromolecule) in a centrifugal field?

Consider a particle m in a centrifuge tube filled with a liquid.

The particle (m) is acted on by three forces:

FC: the centrifugal force

FB: the buoyant force

Ff: the frictional force between the particle and the liquid

We can derive an equation that describes the motion of this particle as follows:

You remember that the force on the particle is given by the mass times the acceleration:

where m is the mass of the particle and a is the acceleration. If the particle is moving at a constant velocity (v), then the acceleration is zero and the net force is given by:

Each of these forces can be described as follows:

Substituting these in the equation for the net force, we get:

Now the mass of the displaced solvent can be written in terms of the density of the solvent and the partial specific volume of the particle:

Substituting this in our developing equation, we get:

This equation can be rearranged to give:

Now we collect motion and distance terms on the left and particle and solution terms on the right:

If we multiply the top and bottom of the equation by Avagadro's number we get:

This is the Svedberg equation and is used to describe the motion of the particle in terms of molecular weight (a size term) and frictional coefficient (a shape term). The equation also relates the motion to the solvent density.

## Consequences of the Equation

Particles can be separated by size and shape criteria:

• particles of the same shape (e.g., both linear rods) but different sizes (M's) will separate
• the large particle (larger M) will move faster (have a large S)
• particles of the same size (M) but different shapes (e.g., linear versus globular) will separate
• the particle with the greater frictional coefficient (f ) will move slower (rod shaped moves slower than globular).
• This technique is called velocity gradient centrifugation (a gradient of sucrose is used to linearize the motion of the particles).

Particles can be separated by density:

• when the density in the solvent equals the density of the particle, the denominator of the equation equals zero and therefore velocity equals zero.
• the particle reaches its equilibrium density in the solvent
• this is called equilibrium density gradient centrifugation or isopycnic banding

The Svedberg coefficients are not additive. That is, 40S plus 60S does not equal 100S. This is the case for the ribosomal subunits, where the combination of a 40S small subunit and a 60S large subunit produces an 80S complete ribosome.