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Centrifugation is a mechanical process which involves the use of the
centrifugal force In Newtonian mechanics, the centrifugal force is an inertial force (also called a "fictitious" or "pseudo" force) that appears to act on all objects when viewed in a rotating frame of reference. It is directed away from an axis which is paralle ...
to separate particles from a solution according to their size, shape, density, medium viscosity and rotor speed. The denser components of the mixture migrate away from the axis of the
centrifuge A centrifuge is a device that uses centrifugal force to separate various components of a fluid. This is achieved by spinning the fluid at high speed within a container, thereby separating fluids of different densities (e.g. cream from milk) or ...
, while the less dense components of the mixture migrate towards the axis. Chemists and biologists may increase the effective gravitational force of the test tube so that the precipitate (pellet) will travel quickly and fully to the bottom of the tube. The remaining liquid that lies above the precipitate is called a supernatant or supernate. There is a correlation between the size and
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
of a particle and the rate that the particle separates from a heterogeneous mixture, when the only force applied is that of gravity. The larger the size and the larger the density of the particles, the faster they separate from the mixture. By applying a larger effective gravitational force to the mixture, like a centrifuge does, the separation of the particles is accelerated. This is ideal in industrial and lab settings because particles that would naturally separate over a long period of time can be separated in much less time. The rate of centrifugation is specified by the
angular velocity In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an object ...
usually expressed as
revolutions per minute Revolutions per minute (abbreviated rpm, RPM, rev/min, r/min, or with the notation min−1) is a unit of rotational speed or rotational frequency for rotating machines. Standards ISO 80000-3:2019 defines a unit of rotation as the dimensio ...
(RPM), or acceleration expressed as ''g''. The conversion factor between RPM and ''g'' depends on the
radius In classical geometry, a radius (plural, : radii) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', ...
of the centrifuge rotor. The particles'
settling Settling is the process by which particulates move towards the bottom of a liquid and form a sediment. Particles that experience a force, either due to gravity or due to centrifugal motion will tend to move in a uniform manner in the direction ...
velocity in centrifugation is a function of their size and shape, centrifugal acceleration, the volume fraction of solids present, the
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
difference between the particle and the liquid, and the
viscosity The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the int ...
. The most common application is the separation of solid from highly concentrated suspensions, which is used in the treatment of sewage sludges for dewatering where less consistent sediment is produced. The centrifugation method has a wide variety of industrial and laboratorial applications; not only is this process used to separate two miscible substances, but also to analyze the
hydrodynamic In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) ...
properties of macromolecules. It is one of the most important and commonly used research methods in
biochemistry Biochemistry or biological chemistry is the study of chemical processes within and relating to living organisms. A sub-discipline of both chemistry and biology, biochemistry may be divided into three fields: structural biology, enzymology and ...
, cell and
molecular biology Molecular biology is the branch of biology that seeks to understand the molecular basis of biological activity in and between cells, including biomolecular synthesis, modification, mechanisms, and interactions. The study of chemical and phys ...
. In the chemical and food industries, special centrifuges can process a continuous stream of particle turning into separated liquid like plasma. Centrifugation is also the most common method used for uranium enrichment, relying on the slight mass difference between atoms of U-238 and U-235 in uranium hexafluoride gas.


Mathematical formula

In a liquid suspension, many particles or cells will gradually fall to the bottom of the container due to
gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
; however, the amount of time taken for such separations is not feasible. Other particles, which are very small, can not be isolated at all in solution until they are exposed to a high
centrifugal force In Newtonian mechanics, the centrifugal force is an inertial force (also called a "fictitious" or "pseudo" force) that appears to act on all objects when viewed in a rotating frame of reference. It is directed away from an axis which is paralle ...
. As the suspension is rotated at a certain speed or
revolutions per minute Revolutions per minute (abbreviated rpm, RPM, rev/min, r/min, or with the notation min−1) is a unit of rotational speed or rotational frequency for rotating machines. Standards ISO 80000-3:2019 defines a unit of rotation as the dimensio ...
(RPM), the centrifugal force allows the particles to travel radially away from the rotation axis. The general formula for calculating the revolutions per minute (RPM) of a centrifuge is: RPM= \sqrt, where ''g'' represents the respective force of the centrifuge and ''r'' the radius from the center of the rotor to a point in the sample. However, depending on the centrifuge model used, the respective angle of the rotor and the radius may vary, thus the formula gets modified. For example, the Sorvall #SS-34 rotor has a maximum radius of 10.8 cm, so the formula becomes RPM= 299\sqrt, which can further simplify to RPM = 91\sqrt. When compared to gravity, the particle force is called the 'Relative Centrifugal Force' (RCF). It is the perpendicular force exerted on the contents of the rotor as a result of the rotation, always relative to the gravity of the earth, which measures the strength of rotors of different types and sizes. For instance, the RCF of 1000 x g means that the centrifugal force is 1000 times stronger than the earthly gravitational force. RCF is dependent on the speed of rotation in rpm and the distance of the particles from the center of rotation. The most common formula used for calculating RCF is: RCF = 1.118 \times 10^ \times r \times (rpm)^2, where 1.118 \times 10^ is a constant; ''r'' is the radius, expressed in centimetres, between the axis of rotation and the center; and ''rpm'' is the speed in revolutions per minute. Historically, many separations have been carried out at the speed of 3000 rpm; a rough guide to the ‘g’ force exerted at this speed is to multiply the centrifugation radius by a factor of 10, so a radius of 160 mm gives approximately 1600 x g. This is a rather arbitrary approach, since the RCF applied is linearly dependent on the radius, so a 10% larger radius means that a 10% higher RCF is applied at the same speed. Roughly, the above formula can be simplified to RCF = 10 \times r, with an error of only 0.62%.


Centrifugation in biological research


Microcentrifuges

Microcentrifuges are specially designed table-top models with light, small-volume rotors capable of very fast acceleration up to approximately 17,000 rpm. They are lightweight devices which are primarily used for short-time centrifugation of samples up to around 0.2–2.0 mL. However, due to their small scale, they are readily transportable and, if necessary, can be operated in a cold room. They can be refrigerated or not. The microcentrifuge is normally used in research laboratories where small samples of biological molecules,
cells Cell most often refers to: * Cell (biology), the functional basic unit of life Cell may also refer to: Locations * Monastic cell, a small room, hut, or cave in which a religious recluse lives, alternatively the small precursor of a monastery w ...
, or nuclei are required to be subjected to high RCF for relatively short time intervals. Microcentrifuges designed for high-speed operation can reach up to 35,000 rpm, giving RCF up to 30000×g, and are called high-speed microcentrifuges.


Low-speed centrifuges

Low-speed centrifuges are used to harvest chemical precipitates, intact cells (animal, plant and some microorganisms), nuclei, chloroplasts, large mitochondria and the larger plasma-membrane fragments. Density gradients for purifying cells are also run in these centrifuges. Swinging-bucket rotors tend to be used very widely because of the huge flexibility of sample size through the use of adaptors. These machines have maximum rotor speeds of less than 10 000 rpm and vary from small, bench-top to large, floor-standing centrifuges.


High-speed centrifuges

High-speed centrifuges are typically used to harvest microorganisms,
virus A virus is a submicroscopic infectious agent that replicates only inside the living cells of an organism. Viruses infect all life forms, from animals and plants to microorganisms, including bacteria and archaea. Since Dmitri Ivanovsk ...
es,
mitochondria A mitochondrion (; ) is an organelle found in the cells of most Eukaryotes, such as animals, plants and fungi. Mitochondria have a double membrane structure and use aerobic respiration to generate adenosine triphosphate (ATP), which is used ...
, lysosomes, peroxisomes and intact tubular Golgi membranes. The majority of the simple pelleting tasks are carried out in fixed angle rotors. Some density-gradient work for purifying cells and organelles can be carried out in swinging-bucket rotors, or in the case of Percoll gradients in fixed-angle rotors. High-speed or superspeed centrifuges can handle larger sample volumes, from a few tens of millilitres to several litres. Additionally, larger centrifuges can also reach higher angular velocities (around 30,000 rpm). The rotors may come with different adapters to hold various sizes of test tubes, bottles, or microtiter plates.


Ultracentrifugations

Ultracentrifugation makes use of high centrifugal force for studying properties of biological particles at exceptionally high speeds. Current ultracentrifuges can spin to as much as 150,000 rpm (equivalent to 1,000,000 x g). They are used to harvest all membrane vesicles derived from the plasma membrane,
endoplasmic reticulum The endoplasmic reticulum (ER) is, in essence, the transportation system of the eukaryotic cell, and has many other important functions such as protein folding. It is a type of organelle made up of two subunits – rough endoplasmic reticulum ...
(ER) and Golgi membrane,
endosomes Endosomes are a collection of intracellular sorting organelles in eukaryotic cells. They are parts of endocytic membrane transport pathway originating from the trans Golgi network. Molecules or ligands internalized from the plasma membrane can ...
, ribosomes, ribosomal subunits,
plasmids A plasmid is a small, extrachromosomal DNA molecule within a cell that is physically separated from chromosomal DNA and can replicate independently. They are most commonly found as small circular, double-stranded DNA molecules in bacteria; how ...
, DNA, RNA and proteins in fixed-angle rotors. Compared to microcentrifuges or high-speed centrifuges, ultracentrifuges can isolate much smaller particles and, additionally, while microcentrifuges and supercentrifuges separate particles in batches (limited volumes of samples must be handled manually in test tubes or bottles), ultracentrifuges can separate molecules in batch or continuous flow systems. Ultracentrifugation is employed for separation of macromolecules/ligand binding kinetic studies, separation of various lipoprotein fractions from plasma and deprotonisation of physiological fluids for amino acid analysis. They are the most commonly used centrifuge for the density-gradient purification of all particles except cells, and, while swinging buckets have been traditionally used for this purpose, fixed-angle rotors and vertical rotors are also used, particularly for self-generated gradients and can improve the efficiency of separation greatly. There are two kinds of ultracentrifuges: the analytical and the preparative.


Analytical ultracentrifugation

Analytical ultracentrifugation (AUC) can be used for determination of the properties of macromolecules such as shape, mass, composition, and conformation. It is a commonly used biomolecular analysis technique used to evaluate sample purity, to characterize the assembly and disassembly mechanisms of
biomolecular complex A biomolecule or biological molecule is a loosely used term for molecules present in organisms that are essential to one or more typically biological processes, such as cell division, morphogenesis, or development. Biomolecules include large ...
es, to determine subunit stoichiometries, to identify and characterize macromolecular conformational changes, and to calculate equilibrium constants and thermodynamic parameters for self-associating and hetero-associating systems. Analytical ultracentrifuges incorporate a scanning visible/
ultraviolet light Ultraviolet (UV) is a form of electromagnetic radiation with wavelength from 10 nm (with a corresponding frequency around 30  PHz) to 400 nm (750  THz), shorter than that of visible light, but longer than X-rays. UV radiatio ...
-based optical detection system for real-time monitoring of the sample’s progress during a spin. Samples are centrifuged with a high-density solution such as
sucrose Sucrose, a disaccharide, is a sugar composed of glucose and fructose subunits. It is produced naturally in plants and is the main constituent of white sugar. It has the molecular formula . For human consumption, sucrose is extracted and refine ...
,
caesium chloride Caesium chloride or cesium chloride is the inorganic compound with the formula Cs Cl. This colorless salt is an important source of caesium ions in a variety of niche applications. Its crystal structure forms a major structural type where each ca ...
, or
iodixanol Iodixanol, sold under the brand name Visipaque, is an iodine-containing non-ionic radiocontrast agent. It is available as a generic medication. Medical uses The radiocontrast agent is given intravenously for computed tomography (CT) imaging o ...
. The high-density solution may be at a uniform concentration throughout the test tube ("cushion") or a varying concentration ("
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
"). Molecular properties can be modeled through sedimentation velocity analysis or sedimentation equilibrium analysis. During the run, the particle or molecules will migrate through the test tube at different speeds depending on their physical properties and the properties of the solution, and eventually form a pellet at the bottom of the tube, or bands at various heights.


Preparative ultracentrifugation

Preparative ultracentrifuges are often used for separating particles according to their densities, isolating and/or harvesting denser particles for collection in the pellet, and clarifying suspensions containing particles. Sometimes researchers also use preparative ultracentrifuges if they need the flexibility to change the type of rotor in the instrument. Preparative ultracentrifuges can be equipped with a wide range of different rotor types, which can spin samples of different numbers, at different angles, and at different speeds.


Fractionation process

In biological research, cell fractionation typically includes the isolation of cellular components while retaining the individual roles of each component. Generally, the cell sample is stored in a suspension which is: *Buffered—neutral pH, preventing damage to the structure of proteins including enzymes (which could affect ionic bonds) *Isotonic (of equal water potential)—this prevents water gain or loss by the organelles *Cool—reducing the overall activity of enzyme released later in the procedure Centrifugation is the first step in most fractionations. Through low-speed centrifugation, cell debris may be removed, leaving a supernatant preserving the contents of the cell. Repeated centrifugation at progressively higher speeds will fractionate homogenates of cells into their components. In general, the smaller the subcellular component, the greater is the centrifugal force required to sediment it. The soluble fraction of any
lysate Lysis ( ) is the breaking down of the membrane of a cell, often by viral, enzymic, or osmotic (that is, "lytic" ) mechanisms that compromise its integrity. A fluid containing the contents of lysed cells is called a ''lysate''. In molecular bi ...
can then be further separated into its constituents using a variety of methods.


Differential centrifugation

Differential centrifugation is the simplest method of fractionation by centrifugation, commonly used to separate organelles and membranes found in cells. Organelles generally differ from each other in density and in size, making the use of differential centrifugation, and centrifugation in general, possible. The organelles can then be identified by testing for indicators that are unique to the specific organelles. The most widely used application of this technique is to produce crude subcellular fractions from a tissue homogenate such as that from rat liver. Particles of different densities or sizes in a suspension are sedimented at different rates, with the larger and denser particles sedimenting faster. These sedimentation rates can be increased by using centrifugal force. A suspension of cells is subjected to a series of increasing centrifugal force cycles to produce a series of pellets comprising cells with a declining sedimentation rate. Homogenate includes nuclei, mitochondria, lysosomes, peroxisomes, plasma membrane sheets and a broad range of vesicles derived from a number of intracellular membrane compartments and also from the plasma membrane, typically in a buffered medium.


Density gradient centrifugation

Density gradient centrifugation In biochemistry and cell biology, differential centrifugation (also known as differential velocity centrifugation) is a common procedure used to separate organelles and other sub-cellular particles based on their sedimentation rate. Although o ...
is known to be one of the most efficient methods for separating suspended particles, and is used both as a separation technique and as a method for measuring the density of particles or molecules in a mixture. It is used to separate particles on the basis of size, shape, and density by using a medium of graded densities. During a relatively short or slow centrifugation, the particles are separated by size, with larger particles sedimenting farther than smaller ones. Over a long or fast centrifugation, particles travel to locations in the gradient where the density of the medium is the same as that of the particle density; (ρp – ρm) → 0. Therefore, a small, dense particle initially sediments less readily than a large, low density particle. The large particles reach their equilibrium density position early, while the small particles slowly migrate across the large particle zone and ultimately take up an equilibrium position deeper into the gradient. A tube, after being centrifuged by this method, has particles in order of density based on height. The object or particle of interest will reside in the position within the tube corresponding to its density. Nevertheless, some non-ideal sedimentations are still possible when using this method. The first potential issue is the unwanted aggregation of particles, but this can occur in any centrifugation. The second possibility occurs when droplets of solution that contain particles sediment. This is more likely to occur when working with a solution that has a layer of suspension floating on a dense liquid, which in fact have little to no density gradient.


Other applications

A centrifuge can be used to isolate small quantities of solids retained in suspension from liquids, such as in the separation of
chalk Chalk is a soft, white, porous, sedimentary carbonate rock. It is a form of limestone composed of the mineral calcite and originally formed deep under the sea by the compression of microscopic plankton that had settled to the sea floor. C ...
powder from water. In biological research, it can be used in the purification of mammalian cells, fractionation of subcellular organelles, fractionation of membrane vesicles, fractionation of macromolecules and macromolecular complexes, etc. Centrifugation is used in many different ways in the
food industry The food industry is a complex, global network of diverse businesses that supplies most of the food consumed by the world's population. The food industry today has become highly diversified, with manufacturing ranging from small, traditional, ...
. For example, in the dairy industry, it is typically used in the clarification and skimming of
milk Milk is a white liquid food produced by the mammary glands of mammals. It is the primary source of nutrition for young mammals (including breastfed human infants) before they are able to digest solid food. Immune factors and immune-modulat ...
, extraction of cream, production and recovery of
casein Casein ( , from Latin ''caseus'' "cheese") is a family of related phosphoproteins ( αS1, aS2, β, κ) that are commonly found in mammalian milk, comprising about 80% of the proteins in cow's milk and between 20% and 60% of the proteins in hum ...
,
cheese Cheese is a dairy product produced in wide ranges of flavors, textures, and forms by coagulation of the milk protein casein. It comprises proteins and fat from milk, usually the milk of cows, buffalo, goats, or sheep. During productio ...
production, removing bacterial contaminants, etc. This processing technique is also used in the production of beverages, juices, coffee, tea, beer,
wine Wine is an alcoholic drink typically made from Fermentation in winemaking, fermented grapes. Yeast in winemaking, Yeast consumes the sugar in the grapes and converts it to ethanol and carbon dioxide, releasing heat in the process. Different ...
, soy milk, oil and fat processing/recovery,
cocoa butter Cocoa butter, also called theobroma oil, is a pale-yellow, edible fat extracted from the cocoa bean. It is used to make chocolate, as well as some ointments, toiletries, and pharmaceuticals. Cocoa butter has a cocoa flavor and aroma. Its melt ...
, sugar production, etc. It is also used in the clarification and stabilization of wine. In forensic and research laboratories, it can be used in the separation of
urine Urine is a liquid by-product of metabolism in humans and in many other animals. Urine flows from the kidneys through the ureters to the urinary bladder. Urination results in urine being excreted from the body through the urethra. Cellul ...
and
blood Blood is a body fluid in the circulatory system of humans and other vertebrates that delivers necessary substances such as nutrients and oxygen to the cells, and transports metabolic waste products away from those same cells. Blood in the cir ...
components. It also aids in separation of proteins using purification techniques such as
salting out Salting out (also known as salt-induced precipitation, salt fractionation, anti-solvent crystallization, precipitation crystallization, or drowning out) is a purification technique that utilizes the reduced solubility of certain molecules in a s ...
, e.g. ammonium sulfate precipitation. Centrifugation is also an important technique in waste treatment, being one of most common processes used for sludge dewatering. This process also plays a role in
cyclonic separation Cyclonic separation is a method of removing particulates from an air, gas or liquid stream, without the use of filters, through vortex separation. When removing particulate matter from liquid, a hydrocyclone is used; while from gas, a gas cyclone ...
, where particles are separated from an air-flow without the use of filters. In a cyclone collector, air moves in a helical path. Particles with high inertia are separated by the centrifugal force whilst smaller particles continue with the air-flow. Centrifuges have also been used to a small degree to isolate lighter-than-water compounds, such as oil. In such situations, the aqueous discharge is obtained at the opposite outlet from which solids with a specific gravity greater than one are the target substances for separation.


History

By 1923
Theodor Svedberg Theodor Svedberg (30 August 1884 – 25 February 1971) was a Swedish chemist and Nobel laureate for his research on colloids and proteins using the ultracentrifuge. Svedberg was active at Uppsala University from the mid 1900s to late 1940s. ...
and his student H. Rinde had successfully analyzed large-grained sols in terms of their gravitational sedimentation.Van Holde, K. E. (1998). Analytical ultracentrifugation from 1924 to the present: A remarkable history. Chemtracts – Biochemistry and Molecular Biology. 11:933-943 Sols consist of a substance evenly distributed in another substance, also known as a
colloid A colloid is a mixture in which one substance consisting of microscopically dispersed insoluble particles is suspended throughout another substance. Some definitions specify that the particles must be dispersed in a liquid, while others extend ...
.Svedberg, T. (1927). The Ultracentrifuge Nobel Lecture However, smaller grained sols, such as those containing gold, could not be analyzed. To investigate this problem Svedberg developed an analytical centrifuge, equipped with a photographic absorption system, which would exert a much greater centrifugal effect. In addition, he developed the theory necessary to measure molecular weight. During this time, Svedberg's attention shifted from gold to proteins. By 1900, it had been generally accepted that proteins were composed of amino acids; however, whether proteins were colloids or
macromolecule A macromolecule is a very large molecule important to biophysical processes, such as a protein or nucleic acid. It is composed of thousands of covalently bonded atoms. Many macromolecules are polymers of smaller molecules called monomers. The ...
s was still under debate.Tanford, C., and Reynolds, J. 2001. Nature’s robots: A history of proteins. Oxford University Press. pp. 303-305 One protein being investigated at the time was
hemoglobin Hemoglobin (haemoglobin BrE) (from the Greek word αἷμα, ''haîma'' 'blood' + Latin ''globus'' 'ball, sphere' + ''-in'') (), abbreviated Hb or Hgb, is the iron-containing oxygen-transport metalloprotein present in red blood cells (erythroc ...
. It was determined to have 712 carbon, 1,130 hydrogen, 243 oxygen, two sulfur atoms, and at least one iron atom. This gave hemoglobin a resulting weight of approximately 16,000 dalton (Da) but it was uncertain whether this value was a multiple of one or four (dependent upon the number of iron atoms present).Simoni, D. S., Hill, R. L., and Vaughan, M. (2002). The structure and function of hemoglobin: Gilbery Smithson Adair and the Adair equations. The Journal of Biological Chemistry. 277(31): e1-e2 Through a series of experiments utilizing the
sedimentation equilibrium Sedimentation equilibrium in a suspension of different particles, such as molecules, exists when the rate of transport of each material in any one direction due to sedimentation equals the rate of transport in the opposite direction due to diffusi ...
technique, two important observations were made: hemoglobin has a molecular weight of 68,000 Da, suggesting that there are four iron atoms present rather than one, and that ,no matter where the hemoglobin was isolated from, it had exactly the same molecular weight. How something of such a large molecular mass could be consistently found, regardless of where it was sampled from in the body, was unprecedented and favored the idea that proteins are macromolecules rather than colloids. In order to investigate this phenomenon, a centrifuge with even higher speeds was needed, and thus the ultracentrifuge was created to apply the theory of sedimentation-diffusion. The same molecular mass was determined, and the presence of a spreading boundary suggested that it was a single compact particle. Further application of centrifugation showed that under different conditions the large homogeneous particles could be broken down into discrete subunits. The development of centrifugation was a great advance in experimental protein science. Linderstorm-Lang, in 1937, discovered that density gradient tubes could be used for density measurements. He discovered this when working with potato yellow-dwarf virus. This method was also used in Meselson and Stahl's famous experiment in which they proved that DNA replication is semi-conservative by using different isotopes of nitrogen. They used density gradient centrifugation to determine which isotope or isotopes of nitrogen were present in the DNA after cycles of replication.


See also

*
Centrifuge A centrifuge is a device that uses centrifugal force to separate various components of a fluid. This is achieved by spinning the fluid at high speed within a container, thereby separating fluids of different densities (e.g. cream from milk) or ...


References


Sources

*Harrison, Roger G., Todd, Paul, Rudge, Scott R., Petrides D.P. ''Bioseparations Science and Engineering''. Oxford University Press, 2003. *Dishon, M., Weiss, G.H., Yphantis, D.A. ''Numerical Solutions of the Lamm Equation. I. Numerical Procedure''. Biopolymers, Vol. 4, 1966. pp. 449–455. *Cao, W., Demeler B. ''Modeling Analytical Ultracentrifugation Experiments with an Adaptive Space-Time Finite Element Solution for Multicomponent Reacting Systems''. Biophysical Journal, Vol. 95, 2008. pp. 54–65. *Howlett, G.J., Minton, A.P., Rivas, G. ''Analytical Ultracentrifugation for the Study of Protein Association and Assembly''. Current Opinion in Chemical Biology, Vol. 10, 2006. pp. 430–436. *Dam, J., Velikovsky, C.A., Mariuzza R.A., et al. ''Sedimentation Velocity Analysis of Heterogeneous Protein-Protein Interactions: Lamm Equation Modeling and Sedimentation Coefficient Distributions c(s)''. Biophysical Journal, Vol. 89, 2005. pp. 619–634. *Berkowitz, S.A., Philo, J.S. ''Monitoring the Homogeneity of Adenovirus Preparations (a Gene Therapy Delivery System) Using Analytical Ultracentrifugation''. Analytical Biochemistry, Vol. 362, 2007. pp. 16–37. {{Separation processes