Theory of Ultracentrifugation: Svedberg Equation
A centrifuge is a device for separating particles from a solution according to their size, shape, density, viscosity of the medium and rotor speed. In biology, the particles are usually cells, sub cellular organelles, viruses, large molecules such as proteins and nucleic acids. To simplify mathematical terminology we will refer to all biological material as spherical particles. There are many ways to classify centrifugation.
Basic Theory of Sedimentation
There are many type of preparative centrifugation such as rate zonal, differential, isopycnic centrifugation.
Molecules separate according to their size, shape, density, viscosity, and centrifugal force. The simplest case is a spherical molecule. If the liquid has the density of do and the molecule has a density of d and if d>do then the protein will sediment. In gravitational field, the motor force (P) equals the acceleration of gravity (g) multiplied by the difference between the mass of the molecule and the mass of a corresponding volume of medium.
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
The Svedberg Equation
The single most important advance in the use of centrifugal force to separate biologically important substances. was the coupling of mechanics, optics and mathematics by T. Svedberg and J.W. Williams in the 1920's. They initiated the mathematics and advanced the instrumentation to a point where it was possible to prove that proteins were large molecules that could be weighed in a centrifuge. In honor of that work, the value for a molecule's (or organelle's) sedimentation velocity in a centrifugal field is known as its Svedberg constant or S value for short.
The instrumentation has progressed quite far since the early work of Svedberg and Williams. Today, any technique employing the quantitative application of centrifugal force is known as ultracentrifugation. The design of the basic instruments, the rotors and the optical systems for measurement are too extensive to cover in this volume. For our purposes, we will concentrate on two types of rotor, and a few selected parameters to be measured.
Calculation of S
vr = partial specific volume of the molecule
N = Avogadro's number
f = frictional coefficient
s = sedimentation coefficient (units: 1 Svedberg = 10-13 sec)
The above equation depends on the size of the molecule (M), however the shape of the molecule plays an important role in its behavior under centrifugal force so it is appropriate to take this (f) into account.
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.
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.
Examples of S values
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)
The centrifugation technique which makes use of this property is called:
Moving boundary/Zone Centrifugation
In moving boundary (or differential centrifugation), the entire tube is filled with sample and centrifuged. Through centrifugation, one obtains a separation of two particles but any particle in the mixture may end up in the supernatant or in the pellet or it may be distributed in both fractions, depending upon it size, shape, density, and conditions of centrifugation. The pellet is a mixture of all of the sedimented components, and it is contaminated with whatever unsedimented particles were in the bottom of the tube initially. The only component which is purified is the slowest sedimenting one, but its yield is often very low. The two fractions are recovered by decanting the supernatant solution from the pellet. The supernatant can be recentrifuged at higher speed to obtain further purification, with the formation of a new pellet and supernatant.
Rate Zonal Centrifugation
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).
In rate zonal centrifugation, the sample is applied in a thin zone at the top of the centrifuge tube on a density gradient. Under centrifugal force, the particles will begin sedimenting through the gradient in separate zones according to their size shape and density. The run must be terminated before any of the separated particles reach the bottom of the tube.
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.
In isopycnic technique, the density gradient column encompasses the whole range of densities of the sample particles. The sample is uniformly mixed with the gradient material. Each particle will sediment only to the position in the centrifuge tube at which the gradient density is equal to its own density, and there it will remain. The isopycnic technique, therefore, separate particles into zone solely on the basis of their density differences, independent of time. In many density gradient experiments, particles of both the rate zonal and the isopycnic principles may enter into the final separations. For example, the gradient may be of such a density range that one component sediments to its density in the tube and remains there, while another component sediments to the bottom of the tube. The self generating gradient technique often requires long hours of centrifugation.
Isopynically banding DNA, for example, takes 36 to 48 hours in a self-generating cesium chloride gradient. It is important to note that the run time cannot be shortened by increasing the rotor speed; this only results in changing the position of the zones in the tube since the gradient material will redistribute farther down the tube under greater centrifugal force.
An equilibrium density gradient centrifugation experiment
using mouse DNA. Most, random sequence, DNA has a similar buoyant density and forms part of the main band. However, one short sequence is repeated many thousands of times in a tandem array. All the fragments of DNA derived from this repeat have the same base composition and hence the same buoyant density. They band together at a lighter density than the main band DNA.
Analytical centrifugation involves measuring the physical properties of the sedimenting particles such as sedimentation coefficient or molecular weight. Optimal methods are used in analytical ultracentrifugation. Molecules are observed by optical system during centrifugation, to allow observation of macromolecules in solution as they move in gravitational field. The samples are centrifuged in cells (tubes with quartz windows. See figure 1) having windows that lie paralleled to the plan of rotation of the rotor head. As the rotor turns, the images of the cell (proteins) are projected by an optical system on to film or a computer. The concentration of the solution at various points in the cell is determined by absorption of a light of the appropriate wavelength (Beer's law is followed). This can be accomplished either by measuring the degree of blackening of a photographic film or by the pen deflection of the recorder of the scanning system and fed into a computer.
Fortunately, through a simple relationship due to Einstein we can solve for f and hence calculate M based on the opposed forces of sedimentation (s) and diffusion (D).
To determine molecular weight Eqn. becomes:
Sedimentation coefficient s is measured in the ultracentrifuge and the diffusion coefficient, D, is measured separately.
k = Boltzmann constant
T = absolute temperature
D = diffusion coefficient
R = gas constant ( x k) [ Units; 8.314 erg × deg-1 × mol-1 ]
Rotors, Tubes and Relative Centrifugal Force
Sedimenting particles have only short
distance to travel before pelleting.
Shorter run time.
The most widely used rotor type.
Figure F.1. Cross section of Sorvall SS-34 rotor ( a fixed angle rotor ).
Rotors for a centrifuge are either fixed angles , swinging buckets , continuous flow, or zonal, depending upon whether the sample is held at a given angle to the rotation plane, is allowed to swing out on a pivot and into the plane of rotation, designed with inlet and outlet ports for separation of large volumes, or a combination of these. Figure F.1 demonstrates the characteristics of each of these.
Fixed angles generally work faster; substances precipitate faster in a given rotational environment, or they have an increased relative centrifugal force for a given rotor speed and radius. They also have few (or no) moving parts on the rotor itself and thus have virtually no major mechanical failures, other than potential metal stress, which all rotors undergo. These rotors are the work-horse elements of a cell laboratory, and the most common is a rotor holding 8 centrifuge tubes at an angle of 34 ° C from the vertical (such as the Sorvall SS-34 rotor or the Beckman JA-20). Figure F.1 presents a cross- sectional diagram of the Sorvall SS-34.
Swinging bucket rotors
(also known as horizontal rotors) have the advantage that there is usually a clean meniscus of minimum area. In a fixed angle rotor, the materials are forced against the side of the centrifuge tube, and then slide down the wall of the tube. This action is the primary reason for their apparent faster separation, but also leads to abrasion of the particles along the wall of the centrifuge tube. For a swinging bucket, the materials must travel down the entire length of the centrifuge tube and always through the media within the tube. Since the media is usually a viscous substance, the swinging bucket appears to have a lower relative centrifugal force, that is it takes longer to precipitate anything contained within. If, however, the point of centrifugation is to separate molecules or organelles on the basis of their movements through a viscous field, then the swinging bucket is the rotor of choice. Moreover, if there is a danger or scraping off an outer shell of a particle (such as the outer membrane of a chloroplast), then the swinging bucket is the rotor of choice. Most common clinical centrifuges have swinging buckets. Since the buckets are easy to interchange, this type of rotor is extremely versatile. Its major drawback is the number of moving parts which are prone to failure with extended use.
Nearly all cell biology laboratories will have several examples of fixed angle and horizontal rotors. While the sample volumes of these rotors can be significant, they are limiting. To overcome this limitation, a continuous flow centrifuge can be used.
Limnologists often employ such a device to separate plankton from gallons of lake water. Cell biologists employ zonal rotors for the large scale separation of particles on density gradients. Zonal rotors can contain up to 2 liters of solution and can work with tissue samples measured in ounces (or even pounds). The rotors are brought up to about 3000 RPM while empty, and the density media and tissues are added through specialized ports. This type of rotor has a distinct preparative advantage over the gradient capacity of more typical rotors.
In using either a fixed angle or swinging bucket rotor, it is necessary to contain the sample in some type of holder. Continuous and zonal rotors are designed to be used without external tubes.
For biological work, the tubes are divided into functional groups, made of regular glass, Corex glass, nitrocellulose, or polyallomer. Regular glass centrifuge tubes can be used at speeds below 3,000 RPM, that is in a standard clinical centrifuge. Above this speed, the xg forces will shatter the glass.
A special high speed glass with the tradename of Corex (Corning Glass Works) has been developed to handle speeds up to 15-18,000 RPM. These tubes can be used in most routine cell organelle preparations, if, and only if, the proper adapters are also used within the centrifuge rotors. These tubes are relatively expensive (about $3.50 each) and should never be used for any purpose other than the centrifuge. Any tubes with scratches or chips should be disposed of immediately. These high- speed glass tubes will shatter above 18,000 RPM.
For work in the higher speed ranges, centrifuge tubes are made of plastic or nitrocellulose. Preparative centrifuge tubes are made of polypropylene (sometimes polyethylene) and can withstand speeds up to 20,000 RPM. These tubes should be carefully examined for stress fractures before use. A tube with a fracture will hold fluids before centrifugation, but the cracks will open under centrifugal force.
Nitrocellulose are inexpensive and used for most ultracentrifugation. They are meant to be used only once and then discarded. Repeated use increases the chance of tube collapse due to internal molecular stress within the tube walls. There is no way to pre-determine this, so it is best to always use a new tube for ultracentrifugation. Nitrocellulose also becomes less flexible with age, and the purchase date for all tubes should be noted. Tubes older than 1 year should be discarded. A centrifuge tube is inexpensive when compared to the loss of time and materials for a typical ultracentrifuge run.
Polyallomer tubes are re-usable, more expensive, and slippery. Molecules will slide down the walls of these tubes more easily, and thus are the tubes of choice for precipitation centrifugations. They are also more chemically inert.
RELATIVE CENTRIFUGAL FORCE
Figure F.2. Nomogram for R.C.F.
Modern day ultracentrifuges can generate forces in excess of 300,000 times that of gravity, forces sufficient to overcome the very cohesion of most molecules (including the metal of the rotor). The force is usually given as some value times that of gravity.
The centrifugal force is dependent upon the radius of the rotation of the rotor, the speed at which it rotates, and the design of the rotor itself (fixed angle, vs swinging bucket). Rotor speed and design can be held constant, but the radius will vary from the top of a centrifuge tube to the bottom. If a measurement for the radius is taken as the mid-point, or as an average radius, and all forces are mathematically related to gravity, then one obtains a relative centrifugal force, labeled as xg. Centrifugation procedures are given as xg measures, since RPM and other parameters will vary with the particular instrument and rotor used. Relative Centrifugal Force is a constant that is independent of the apparatus used.
Figure F.2 presents a Nomogram for calculation of R.C.F. for a given radius and RPM. A simple formula for calculating this value is:
RCF = 1.12r (RPM/1000)
where r = radius in millimeters
RPM = revolutions per minute
The difficulty with using the formula is establishing the value for r. Typically, there are three r values given (by the manufacturer) for a rotor: the maximum, minimum and average r. These correspond to the distances from the center of rotation to the bottom, top and middle of the sample tube.
The Cornell Accident:
Description of the Cornell Accident -- On December 16, 1998, milk samples were running in a Beckman.L2-65B ultracentrifuge using a large aluminum rotor. The rotor had been used for this procedure many times before. Approximately one hour into the operation, the rotor failed due to excessive mechanical stress caused by the "G" forces of the high rotation speed. The subsequent explosion completely destroyed the centrifuge. The safety shielding in the unit did not contain all the metal fragments. The half-inch thick sliding steel door on top of the unit buckled allowing fragments, including the steel rotor top, to escape. Fragments ruined a nearby refrigerator and an ultra-cold freezer in addition to making holes in the walls and ceiling. The unit itself was propelled sideways and damaged cabinets and shelving that contained over a hundred containers of chemicals. Sliding cabinet doors prevented the containers from falling to the floor and breaking. A shock wave from the accident shattered all four windows in the room. The shock wave also destroyed the control system for an incubator and shook an interior wall causing shelving on the wall to collapse. Fortunately the room was not occupied at the time and there were no personal injuries.
There are a number of safety precautions that must be adhered to when using any centrifuge and rotor.
All rotors are subject to stress and with time will undergo metal fatigue. This is a given, and consequently, a detailed history of the rotor use should be kept. This is usually not done with clinical centrifuges, but is an absolute for an ultracentrifuge rotor.
EVERY USE OF THE ULTRA SPEED CENTRIFUGE ROTOR MUST BE RECORDED IN THE CENTRIFUGE LOG. ABSOLUTELY NO EXCEPTIONS!
After a period of use, each rotor will in turn be derated , that is its maximum RPM will be lowered. The Beckman rotors may contain optical speed control rings at their base - be sure they are present, and clean before use. These devices will strobiscopically monitor the maximum speed that a rotor can be used at. They are replaced as the rotor is derated.
By far the most common cause of rotor failure is corrosion stress. Salts, highly alkaline detergents and of course corrosive acids and alkali's will cause decomposition of the coatings on aluminum rotors, which in turn will concentrate stress and eventually result in cracks and total rotor failure. Titanium rotors are more corrosion resistant, but more expensive. Ultracentrifuge rotors are expensive (in excess of $6,000 each on average) and can be potentially hazardous. At the forces generated in an ultracentrifuge, a rotor failure is the equivalent of a small bomb.
THEREFORE THE FOLLOWING RULES MUST BE OBSERVED.
1.Before running a centrifuge, check the classification decal on the centrifuge to ensure that the rotor is safe to use in the centrifuge at hand.
2.Never use an alkali detergent on a rotor (most are highly alkaline - be sure to check before use).
3.Always clean and completely dry the rotor after every use. Any spilled materials, especially salts and corrosive solvents must be removed immediately with running water. Fixed angle rotors are stored upside down, to drain after thorough cleaning and rinsing. Swinging buckets have only the buckets cleaned and dried, and stored inverted and with the caps removed. NEVER immerse the rotor portion of a swinging bucket rotor. Inevitably the linkage pins will rust, as it is virtually impossible to remove all fluids from them.
4.Be especially careful not to scratch the surface of a rotor or bucket. Use plastic brushes only. Normal wire brushes will scratch the anodized surface of aluminum rotors which will increase the likelihood of corrosion. The anodized layer is extremely thin and is the main defense against corrosion of an aluminum rotor.
5.Always use the proper centrifuge tube. Glass tubes are used in clinical centrifuges only. High Speed Corex tubes can be used up to 15,000 RPM (in SS34 rotor) IF there are no scratches or imperfections in the glass, and if the proper rubber or plastic adapter is employed. All ultracentrifugation use employs nitrocellulose or polyallomer tubes. Nitrocellulose ages and will collapse in a strong centrifugal field if old.
6.Always fill the centrifuge tubes to the proper level. (usually full to within 1/2 inch of the top). The tubes are thin walled and will collapse if improperly filled.
7.Always balance the rotor properly. Use a precision scale for most work. Always balance the tube with a medium that is identical to that being centrifuged, i.e. do not balance an alcohol solution with water, or a dense sucrose solution with water only -- the distribution of the densities will be incorrect. For swinging buckets, be sure the buckets are weighed with their caps in place, that the seals are intact and that the caps are secure. Be careful in the placement of tubes within a rotor to ensure proper balance - check the manufacturer’s guides for complex rotors that hold multiple tubes.
8.Ensure that the rotor is properly seated within the centrifuge. For swinging buckets, ensure that they are hanging properly. - Double or Triple check! For preparative rotors, be sure the rotor cover is in place and properly screwed down, where appropriate. NEVER use a rotor without its lid, when one is supplied - in some cases the screw actually holds the rotor to the motor shaft; in others, it seals the contents of the rotor from explosively evaporating into the vacuum within the centrifuge chamber.
9.Check that the centrifuge chamber is clean, defrosted and that all membranes or measuring devices are intact and functional (Beckman speed and temperature controls) and that the lid is securely closed.
10.Adjust acceleration rates, deceleration rates, temperature and RPM controls as appropriate. Set brake on or off as appropriate and check vacuum level where appropriate.
11.Set the timer and speed controls and start the centrifuge. Do not attempt to open the centrifuge until the rotor has come to a complete stop.
12.Before opening the centrifuge, record the appropriate information in the centrifuge log.
Note: If properly balanced and used, the rotor should accelerate smoothly and
with a constant change in the pitch of the motor sound. Any vibrations, or
unusual sounds should alert the operator to IMMEDIATELY bring the run to an end by the safest means possible (turn off power, turn down time, etc.) . NEVER leave the centrifuge until you are certain that it has reached its operating
speed and is functioning properly. All rotors go through a minor vibration phase
when they first start. There will be a minor flutter when the rotor reaches its harmonic
vibration point - do not confuse this with a serious vibration caused by imbalance. A further vibration may occur whenever the referigeration unit cuts in and out.
If in doubt, halt the centrifuge and get assistance.