# Proteomics/Protein Separations - Centrifugation/How the Centrifuge Works

## Forces of the Centrifuge

### Centripetal Force

A centrifuge works by spinning mixtures around a central axis (centrifugal force). As the sample spins the tendency of the inertia of the object is to move in a straight-line path. However, due to its confinement within the centrifuge, the path of the object must be bent into a circular one. The body of the centrifuge, or the body of the container within the centrifuge, provides a normal force that pushes the object toward the center of the circular path of travel. This inward force is refered to as a centripetal force, and its magnitude and direction are exactly what is needed to keep the object moving in a circular path around the axis of rotation of the centrifuge.

### Centrifugal Force

The strength of the outward force exerted by an object (due to its inertia trying to move it in a straight-line path) as it moves in a circle at a constant angular velocity depends on the angular velocity and the radius of rotation. This force is denoted F, the angular velocity is measured in radians and denoted as w, and the radius of rotation, r, is measured in centimeters.

```     $F=mw^{2}r$ ```

Usually, the value cited for the force applied to a suspension of particles during the centrifugation is a relative one, that is to say it is compared with the force that the earth's gravity would have on the same particles. This is referred to as relative centrifugal force (RCF). It was Galileo Galilei (1564-1642) who first systematically and scientifically investigated gravity as a natural phenomenon. The gravitational acceleration constant is customarily assigned the symbol "g" and for simplicity taken to be 9.80 m/sec sec. Based on this measure, relative centrifugal force is expressed as:

```  $RCF=mw^{2}r/9.80$ (RFC = F centrifugation / F gravity)
```

The common way of denoting the operating speed of a centrifuge is in “revolutions per minute” or rpm. The formula above may be converted so that this relationship expressed as such. If you are expressing the radius in cm then the formula for RCF is:

```    $RCF=11.17r(RPM/1000)^{2}$ ```

If you are expressing the radius in inches then the formula for RCF is:

```    $RCF=28.38r(RPM/1000)^{2}$ ```

The centrifugal force created by the spinning of the centrifuge is greater at the bottom of the tube (away from the central axis) and less at the top of the tube (closer to the central axis). This difference is almost twofold. Because denser items have a greater mass, this causes their centripetal force to be greater (F=ma), and the overall result is that they settle near the outside of the circular path. This places denser objects at the bottom of a test tube that has been run in a centrifuge. Less dense objects remain closer to the center of the path, or the top of a test tube that has been run in a centrifuge.

#### The Action of Centrifugal Force on Molecules

As samples spin in a centrifuge the particles in each sample are subjected to centrifugal force. However, this force is proportional to the mass of the particle. To express the centrifugal force applied to a particular molecule its molecular weight (M) is used in the formula:

```      Centrifugal force = $Mw^{2}r$ ```

Because weight takes into account the force of gravity, using molecular weight in the formula for centrifugal force removes the need to divide by the force of gravity as shown above. This formula shows how particles within a centrifuge are separated based on their molecular weight. In addition to this the size and shape of a particle also affects its migration in the gradient created by centrifugation. For example, plasmid DNA will travel farther down a gradient then chromosomal DNA. This is the result of the counteracting forces of buoyancy and friction counteracting the force of centrifugation.

### Buoyancy and Friction

While the centrifugal force acts to accelerate a particle away from the axis of rotation, the particle is also subjected to additional forces including the force of buoyancy and the force of friction.

• Buoyancy force refers to the interaction of the molecule within the solvent. It is calculated as the centrifugal force multiplied by the volume of solvent the molecule displaces (V, the “partial specific volume”) and by the density of the solvent itself (rho – r). Together this gives the formula:
```   Buoyant force = Mω2rVρ
```
• As the particles move through the solvent frictional force is also generated. The size and shape of a molecule are determinates in the measure of this force. These contribute to the rate of sedimentation which is expressed as the change in the axis of rotation over time (dr/dt). This force combines with the Buoyant force to counteract against the centrifugal force.
``` Frictional force = f(v) = f(dr/dt)
```

The final result of these forces acting together is that a particle will move through the solvent, away from the axis of rotation, until the centrifugal force is equivalent to the forced of buoyancy and friction. Using the formulas above the sedimentation coefficient (s) can be calculated from the molecular weight of a particle.

``` Sedimentation Coefficient (s) = M(1-Vr)D/RT
(R is equal to the gas constant
T is equal to absolute temperature)
```

The sedimentation coefficient of a molecule describes where it will settle in a gradient with the viscosity and density of water under centrifugation and is measured in seconds. For biological molecules these values range between 1 – 500 x$10^{-13}$ seconds. Instead of using $10^{-13}$ , this number is described as one Svedberg unit (S) after Theodore Svedberg; so that 12x$10^{-13}$ is expressed as 12S.

### Coriolis Force

In addition to centrifugal force, particles in suspension (and the body of suspending fluid itself) within a spinning rotor are subjected to Coriolis force. The Coriolis force which results from the inertia of the liquid and the suspended particles, is a small force directed at the right angles to both the axis of rotation clockwise. This force acts to deflect particles in a counterclockwise direction (and vice versa).Under nearly all experiment conditions, the coriolis force is very small in comparison with the centripetal force; however, its effects are magnified when the rotor's speed changes (e.g during acceleration and deceleration). While the centrifugal force acts to accelerate a particle away from the axis of rotation, the particle is also subjected to additional forces including frictional force, the force of buoyancy and gravitational force.

## Centrifuge – the Machine

The basic design of a centrifuge consists of a rotor which holds samples together rotates around a fixed axis driven by a motor (in modern centrifuges). More advanced centrifuges may also have lubrication and cooling systems. In addition to this, some centrifuges are capable of creating a vacuum environment around the rotor.

### Rotors

The most popular and widely used centrifuge rotors are the swinging-bucket and fixed-angle rotors. Two other types of rotors are the vertical rotor and the zonal rotor.

• Fixed-angle (or angle head) rotors are generally simpler in design than are swinging-bucket rotors. In this type of rotor, the centrifuge tubes are held at a specific and constant angle to the horizontal plane that is the tube does not reorient between the vertical and horizontal positions. This type of rotor works very well for simple pelleting centrifugation but has limited and variable success in rate-zonal sedimentation and isopycnic sedimentation respectively.
• Swinging-bucket rotors are able to pivot within the centrifuge. As speeds increase the angle of the rotor perpendicular to the axis of rotation also increases, positing it in a horizontal configuration. Conversely, as the centrifuge slows down the rotor returns to a vertical position. When the bucket swings out the pathlength is increased allowing for improved separation of individual particles, particularly in density gradient centrifugation. This type of rotor is inefficient when used for pelleting. However, it is good for use with both rate-zonal and isopycnic sedimentation.

Most swinging-bucket rotors are interchangeable so that different size test tubes can be used. Additionally, some have the ability to hold multiple test tubes in a single arm.

• Vertical rotors hold samples in a vertical position within the centrifuge. This type of rotor is not suitable for pelleting centrifugation but it does a good job when used for rate-zonal sedimentation, and an excellent job with isopycnic sedimentation.

### Centrifuge Tubes

Depending on the sample/ rotor size and speed of centrifugation different types of centrifuge tubes can be used. Proper selection of centrifuge tubes helps to ensure that leakage does not occur and none of the sample is lost, the chemical properties of the sample and the tube do not conflict, and that the sample can be recovered with little effort.

#### Rotor and Tube Materials

Early rotors such as the Svedberg rotors were made of steel and occasionally brass. The high density of these materials and the resulting high rotor weight produces an appreciable load on the centrifuge drive and significantly limits operating speeds. Most commercial rotors are now made of the partly or entirely of aluminum or titanium.

### Lubrication and Cooling Systems

The high speeds at which centrifuges operate generate a great deal of frictional force. Along with this friction comes heat. To prevent damage to the centrifuge and/or samples within the centrifuge, many present day centrifuges have lubrication and cooling systems to combat friction and the heat it produces.

Convection occurs whenever uniform suspensions of particles are sedimented in a conventional centrifuge. The term "convection" means the bulk movement of solute and/or solvent within the centrifuge tube. Unwanted convection can be caused by variations in temperature in different parts of the centrifuge. By controlling the temperature, lubrication and cooling systems help to guard against this.

### Superspeed vs. Ultraspeed Centrifuges

There are two basic types of preparative centrifuges; superspeed centrifuges and ultraspeed centrifuges or ultracentrifuges.

• Superspeed centrifuges generally operate at speeds up to about 20,000 rpm. These centrifuges usually do not require evacuation of the rotor chamber, and drive the rotor directly to through belts or gears. The picture below is of a Sorvall RC-5B refrigerated superspeed centrifuge.

• Ultracentrifuges can be operated at much greater speeds (up to 65,000 or 75,000 rpm). Due to these high speeds the rotor chamber must be evacuated of air to reduce friction and permit accurate rotor temperature control. In most ultracentrifuges the rotor is driven either by a motor and a set of gears or by an oil or air turbine system. the picture below is of a Beckman Coulter optima LE-80K ultracentrifuge.