From everyday experience, the effect of sedimentation due to the influence of the Earth’s gravitational fi eld (G = g = 9.81 m s−2) versus the increased rate of sedimentation in a centrifugal fi eld (G > 9.81 m s−2) is apparent. To give a simple but illustrative example, crude sand particles added to a bucket of water travel slowly to the bottom of the bucket by gravitation, but sediment much faster when the bucket is swung around in a circle. Similarly, biological structures exhibit a drastic increase in sedimentation when they undergo acceleration in a centrifugal field. The relative centrifugal fi eld is usually expressed as a multiple of the acceleration due to gravity.
Below is a short description of equations used in practical centrifugation classes. When designing a centrifugation protocol, it is important to keep in mind that:
• the more dense a biological structure is, the faster it sediments in a centrifugal field
• the more massive a biological particle is, the faster it moves in a centrifugal field • the denser the biological buffer system is, the slower the particle will move in a centrifugal field
• the greater the frictional coefficient is, the slower a particle will move
• the greater the centrifugal force is, the faster the particle sediments
• the sedimentation rate of a given particle will be zero when the density of the particle and the surrounding medium are equal.
Biological particles moving through a viscous medium experience a frictional drag, whereby the frictional force acts in the opposite direction to sedimentation and equals the velocity of the particle multiplied by the frictional coefficient. The frictional coefficient depends on the size and shape of the biological particle. As the sample moves towards the bottom of a centrifuge tube in swing-out or fixed-angle rotors, its velocity will increase due to the increase in radial distance. At the same time, the particles also encounter a frictional drag that is proportional to their velocity. The frictional force of a particle moving through a viscous fluid is the product of its velocity and its frictional coefficient, and acts in the opposite direction to sedimentation.
When the conditions for the centrifugal separation of a biological particle are described, a detailed listing of rotor speed and radial dimensions of centrifugation has to be provided. Essentially, the rate of sedimentation, v, is dependent upon the applied centrifugal fi eld G (measured in cm s –2). G is determined by the radial distance, r, of the particle from the axis of rotation (in cm) and the square of the angular velocity, ω, of the rotor (in radians per second):
The average angular velocity of a rigid body that rotates around a fixed axis is defi ned as the ratio of the angular displacement in a given time interval. One radian, usually abbreviated as 1 rad, represents the angle subtended at the centre of a circle by an arc with length equal to the radius of the circle. Since 360° equals 2π radians (or rad), one revolution of the rotor can be expressed as 2π rad. Accordingly, the angular velocity of the rotor, given in rad s–1. Note that rad is treated as a scalar and is related to the rotor speed in revolutions per minute (rpm ≡ 1 min–1) by
and therefore the centrifugal fi eld can be expressed as:
where the variable rpm is the rotor speed (measured in revolutions per minute, i.e. the non- italicised ‘rpm' denotes the unit) and r is the radial distance from the centre of rotation. Note that 60 revolutions per minute is the same speed as one revolution per second, i.e. rpm = 60 min –1 = 1 s –1.
The centrifugal fi eld is generally expressed in multiples of the earth's gravitational fi eld, g (9.81 m s −2). The relative centrifugal fi eld RCF (or g - force) is the ratio of the centrifugal acceleration at a specified radius and the speed to the standard acceleration of gravity. The RCF can be calculated from the following equation:
RCF units are therefore dimensionless as they denote multiples of the gravitational constant g. Grouping numerical constants together leads to a more convenient form of Equation 12.4
with r given in cm. Although the relative centrifugal force can easily be calculated and can often be displayed on modern instruments, many centrifugation manuals contain a nomograph for the convenient conversion between relative centrifugal force and speed of the centrifuge at different radii of the centrifugation spindle to a point along the centrifuge tube. A nomograph consists of three columns representing the radial distance (in mm), the relative centrifugal fi eld and the rotor speed (in min –1). For the conversion between relative centrifugal force RCF and speed of the centrifuge spindle (rpm, in min –1) at different radii, a straight edge is aligned through known values in two columns, then the desired figure is read where the straight edge intersects the third column. See Figure 1 for an illustration of the usage of a nomograph.
Fig1. Nomograph for the determination of the relative centrifugal field for a given rotor speed and radius. The three columns represent the radial distance (in mm), the relative centrifugal fi eld and the rotor speed rpm (in min –1). For the conversion between relative centrifugal force and speed of the centrifuge spindle in revolutions per minute at different radii, draw a straight edge through known values in two columns. The desired figure can then be read where the straight edge intersects the third column. The example shown determines the relative centrifugal force for a 5 cm rotor operating at rpm = 40 000 min -1, yielding an applied centrifugal fi eld of about 90 000×g.
In a suspension of biological particles, the rate of sedimentation (ν) is dependent not only upon the applied centrifugal fi eld, but also on the nature of the particle, i.e. its density ρ p , its hydrodynamic radius Rhydro, and also the density (ρ m ) and viscosity (η) of the surrounding medium. Stokes’ law describes this relationship for the sedimentation of a rigid spherical particle:
where ν is the sedimentation rate of the sphere, Rhydro is the radius of particle, ρ p is the density of particle, ρm is the density of medium, g is the gravitational acceleration and η is the viscosity of the medium. Rhydro will depend on the shape of the particle. Following from Equation 12.6, the sedimentation time can be simply calculated as the ratio of distance sedimented and ν.
Accordingly, a mixture of biological particles exhibiting an approximately spherical shape can be separated in a centrifugal fi eld based on their density and/or their size. The time of sedimentation (in seconds) for a spherical particle is:
where t is the sedimentation time, η is the viscosity of medium, Rhydro is the hydrodynamic radius of the particle, rb is the radial distance from the centre of rotation to bottom of tube, rt is the radial distance from the centre of rotation to liquid meniscus, ρp is the density of the particle, ρ m is the density of the medium and ω is the angular velocity of the rotor.
The sedimentation rate or velocity of a biological particle can also be expressed as its sedimentation coefficient ( s ), whereby:
measured in the units of time (i.e. seconds).
Since the sedimentation rate per unit centrifugal fi eld can be determined at different temperatures and with various media of different densities and viscosities, experimental values of the sedimentation coefficient are often corrected or ‘normalised’ for comparison purposes to standard solvent conditions. These standard conditions are the density and viscosity of water at 20.0 °C and the symbol used for this normalised sedimentation coefficient is s20,w . Importantly, s20,w will depend on the size and shape or conformation of the macromolecule. The sedimentation coefficients (corrected or non-corrected) of biological macromolecules are relatively small, and are usually expressed as svedberg units denoted as S; one svedberg unit equals 10 −13 s.
