When an object attached to a rope is whirled around, one can feel that the rope must be pulled inward towards the centre of the rotation in order to keep the object on the orbit. This force prevents the object from getting away and move with a constant speed along a straight tangential line. The inward force with which one has to pull the rope is called the centripetal force. One can also define the outward force, the centrifugal force, by which the object pulls the rope. This force is equal in magnitude to the centripetal force but has the opposite direction. The centrifugal force (F_{c}) is a virtual, socalled fictional force emerging due to the inertia of the object. Yet, because it leads to a simpler mathematical formalism, equations describing the processes when solutions are centrifuged use the F_{c} force.
According to the wellknown Newton equation:

(5.1) 
Upon centrifugation, acceleration equals the product of the radius of the orbit and the square of the angular velocity:

(5.2) 
The fictive centrifugal accelerating force in vacuum is therefore:

(5.3) 
The product of the radius and the square of the angular velocity equals the centrifugal accelerating potential. Traditionally, and perhaps somewhat misleadingly, the magnitude of this potential is compared to the Earth’s gravitational accelerating potential (g), and has been expressed in “g” units. The reason is quite simple. Earth’s gravitational potential, similarly to the accelerating potential provided by centrifugation, can also sediment particles dispersed in solution. This type of quantitation shows how many times centrifugation is more effective to sediment particles compared to the gravitational effect of Earth. In the fastest laboratory ultracentrifuges the applied accelerating potential can exceed 1 000 000 g.
When solutions are centrifuged, the particles are not in vacuum but in a solvent having a given density (mass/volume). Importantly, the centrifugal force acts not only on the particles, but on the solvent too. If the density of the particle equals the density of the solvent, the particle will not move relative to the solvent, and its velocity along the radius will be zero. If the density of the particle exceeds that of the solvent, the particle sediments (sinks), i.e. it moves outwards along the radius, while the displaced solvent molecules move inwards. In the opposite case when the density of the particle is lower than that of the solvent, the particle floats—it moves inwards while the displaced solvent molecules move outwards.
In order to provide a simple mathematical description of this phenomenon, the buoyancy factor has been defined as follows:

(5.4) 
The numerator of the fraction contains the density of the medium (solvent) while the denominator contains the density of the particle.
Introduction of the buoyancy factor leads to the following equation:

(5.5) 
This equation clearly shows that, upon centrifugation, the force acting on a given particle is a function of the mass of the particle, the relative density of the particle (compared to that of the medium), the angular velocity of the rotation and the distance of the particle from the centre of the rotation (i.e. from the spindle of the centrifuge).
The first two of these parameters, namely the mass and the density, are characteristic of the particle and differences in these parameters can allow for the physical separation of different types of particles. As we will see, there are two major types of centrifugationbased separation techniques. In one technique called differential centrifugation, the separation is based on both particle mass and density. In the case of the other, called equilibrium densitygradient centrifugation, the separation is based strictly on the density of the particles.
As soon as the particles are accelerated by the centrifugal force and start moving towards the spindle, a dragging force (F_{d}) called friction is exerted on them by the medium. This force, which has a direction opposite to the direction of the particle movement, is proportional to the velocity of the particle. At the typically very low speed of the sedimentation process, the F_{d} force is a linear function of the velocity. The ratio of the force and the velocity is defined as the frictional coefficient (f). The value of f is a function of the viscosity of the medium and of the size and shape of the particle as described below by Stokes’ law:

(5.6) 
In this equation “r” denotes the Stokes radius. If the particle is spherical, this equals the radius of the particle. If the particle is not spherical, “r” (a virtual value) denotes the radius of a spherical particle that has identical diffusion properties as the nonspherical particle in question and µ denotes viscosity of the medium. Note that the value of the frictional coefficient is proportional to the radius of the particle. The larger the particle, the higher dragging force is exerted on it by the medium.
In the course of centrifugation, the velocity of each particle is increasing due to the accelerating force F_{c}. However, as the velocity increases, the dragging force also increases. Therefore, the velocity of each particle can increase to a given value where the value of the dragging force F_{d} reaches the value of the accelerating F_{c} value. The magnitude of the two opposing forces becomes equal in a very short time:

(5.7) 
Once the magnitude of the two opposing forces becomes equal, the resultant force becomes zero. Therefore, the particle will move with a constant velocity characteristic to that particle at the given accelerating potential and medium. (A similar phenomenon is described in Chapter 7 on electrophoresis. There, the accelerating force is proportional to the charge instead of the mass of the particle, but the friction force and the phenomenon of two opposing forces leading to a characteristic particle velocity is analogous.) Substituting F_{c} into the previous equation leads to the following equation:

(5.8) 
If the above equation is rearranged by dividing particle velocity with the centrifugal acceleration potential, the resulting equation will lead to a useful parameter. This is the sedimentation coefficient (its unit of measure is one over seconds), which is usually expressed in Svedberg units. This coefficient describes the sedimentation propensity of the particle. It provides the characteristic sedimentation velocity of a particle triggered by a unit level of accelerating potential.

(5.9) 
The numerator of the equation contains all parameters that favour sedimentation. The higher the mass and relative density (compared to the medium) of the particle, the higher its sedimentation velocity will be when unit accelerating potential is applied. The mass of the particle of a given density, of course, is linearly proportional to its volume. In other words, the mass is a linear function of the cube of the particle radius. The denominator contains the parameter that negatively influences sedimentation speed. The larger the frictional coefficient, the lower velocity will be triggered by unit level acceleration potential. As we have seen, the frictional coefficient is a linear function of the particle radius. As the accelerating force is a linear function of the third power of the radius, while the dragging force is a linear function of the first power of the radius, the velocity of the particle will ultimately be proportional to the second power of the particle radius. If two particles have identical density, the larger particle will sediment faster and the ratio of the velocities will follow a square law with respect to the ratio of the particle radii. This relationship provides the basis for the socalled differential centrifugation methods.
The density of the various organelles differs on a smaller scale than their size. Therefore, while both size and density affect sedimentation velocity, their size difference dominates when organelles are separated by centrifugation.
In the procedure of differential centrifugation, cell constituents are separated from each other by their Svedberg value. Several consecutive centrifugation steps are applied in the order of increasing accelerating potential. Each individual centrifugation step relies on the different sedimentation speed of the different cell constituents at the given acceleration potential. At a properly chosen acceleration potential, almost 100 % of the largest component will sediment in the time span of the centrifugation. The sedimented organelles form a pellet at the bottom of the centrifuge tube. The potential should be set so that in the same period of time only a small portion of all smaller constituents latch on to the pellet (Figure 5.1).
The disrupted cell homogenate is centrifuged first at a relatively low accelerating potential of 500 g for 10 minutes. Under these conditions, only particles having the highest Svedberg value, intact cells and nuclei will form the pellet. All other cell constituents will sediment at a much lower rate and remain in the homogenate. The supernatant of the first centrifugation is transferred into an empty centrifugation tube and is subjected to another centrifugation step, now at a significantly higher accelerating potential of 10,000 g and for 20 minutes. These conditions favour sedimentation of mitochondria, lysosomes and peroxisomes having lower Svedberg values than nuclei. Many cell constituents still remain in the supernatant, which is again transferred into an empty tube. This tube is placed into an ultracentrifuge and, with an accelerating potential of 100 000 g in one hour, the socalled microsomal fraction sediments. This fraction contains mostly artificial vesicles with a diameter of 50150 nm that originate mostly from the endoplasmic reticulum and are generated by the cell disruption procedure. Other natural cell constituents of the same size range will also contribute to this fraction. After this third centrifugation step, the supernatant contains mostly macromolecules and supramolecular complexes such as ribosomes. By applying an accelerating potential as high as several hundred thousand g, ribosomes and large proteins can also be sedimented.
In the previous section we introduced the method of differential centrifugation. For simplicity, we stated that the constituents of the sample were separated in a medium of homogeneous density. This first approximation has didactical advantages as it makes the basic principle of differential centrifugation easier to comprehend. Nevertheless, it is sometimes advantageous to use a very shallow density gradient in the medium during differential centrifugation. This is done only to suppress convectional flows in the medium that could unsettle and mix layers of already separated cell constituents.
The essence of equilibrium densitygradient centrifugation is principally different. In this case, a rather steep density gradient is created in the medium—in such a manner that the density of the medium gradually increases towards the bottom of the centrifuge tube. This is achieved by using a very highdensity additive, for example caesium chloride (CsCl). The density gradient is created as follows. When the centrifuge tube is filled with the medium, a high concentration CsCl solution is added first. Subsequently, in the process of filling the tube, the concentration of CsCl is gradually decreased resulting in a CsCl gradient and, as a consequence, a density gradient in the tube. The sample is layered on the top of this special medium (Figure 5.2).
In the course of centrifugation, particles start to sediment moving towards the bottom of the centrifuge tube. By doing so, they travel through an increasing density medium. Each particle sediments to a section of the medium where its own density equals the density of the medium. At this section, the buoyancy factor becomes zero and, as a consequence, the accelerating force acting on the particle also becomes zero. The particle stops sedimenting. If it moved further towards the bottom of the tube, it would meet a higher density medium and a force opposing to its moving direction would be exerted on it, turning the particle back. If, by travelling backwards, it would meet a density lower than its own density, it would sediment again. As a consequence, this method separates particles exclusively based on their density. It is an equilibrium method in which, by the end of the separation, the system reaches a constant state. (In this aspect, this method shows an interesting analogy to the isoelectric focusing (IEF) method reviewed in Chapter 7. The two methods separate particles by entirely different characteristics (density versus isoelectric point), but in both cases, the separation leads to an equilibrium state. Both methods apply a gradient, but in the case of IEF a pH gradient is created.)
Note that the two centrifugation approaches introduced above separate particles by partially different characteristics. Consecutive combination of the two methods can lead to a more efficient separation than achieved by any of the methods alone. Therefore, to increase separation efficiency, fractions generated by differential centrifugation can be subjected to a subsequent densitygradient centrifugation step to further separate individual components (Figure 5.3).