# Sedimentation coefficient

The sedimentation coefficient is the quotient of the maximum sedimentation speed of a particle in a centrifuge and the strength of the centrifugal field. ${\ displaystyle s}$

The size of the sedimentation coefficient depends on the mass and shape of the particle as well as its interaction with the medium in which the particle sediments . It can therefore be used when using a medium with known properties to determine the nature of the particle, in particular its mass. Mainly in biology the masses of very small particles are determined by means of analytical ultracentrifugation , for example ribosomes , virions or protein molecules . In order to obtain a sufficient centrifugal field with such small particles, ultracentrifuges are usually used.

The dimension of the coefficient is time , its unit of measurement is Svedberg , abbreviated as  S , corresponding to 10 −13 s. The unit is named after the Swedish chemist Theodor Svedberg .

## Derivation

In the context of a centrifugation experiment with angular velocity  ω , several forces act on a particle of mass and density , which is located in a solvent with density in the radius position with respect to the axis of rotation . The sum of all relevant forces can be set up with particle acceleration as: ${\ displaystyle m}$ ${\ displaystyle \ rho _ {p}}$${\ displaystyle r}$${\ displaystyle \ rho _ {s}}$${\ displaystyle {\ vec {a}}}$

${\ displaystyle \ Sigma _ {i} {\ vec {F}} _ {i} = m {\ vec {a}} = {\ vec {F}} _ {Z} + {\ vec {F}} _ {A} + {\ vec {F}} _ {R}}$

The centrifugal force can be described with the radial basis vector in polar coordinates as: ${\ displaystyle {\ vec {e}} _ {r}}$

${\ displaystyle {\ vec {F}} _ {Z} = m \ omega ^ {2} r \ cdot {\ vec {e}} _ {r}}$

The static buoyancy counteracts the centrifugal force :

${\ displaystyle {\ vec {F}} _ {A} = - m \ omega ^ {2} r \ cdot {\ frac {\ rho _ {s}} {\ rho _ {p}}} \ cdot {\ vec {e}} _ {r}}$

as well as the friction force according to Stokes ' law , with particle velocity and the coefficient of friction , which depends on the shape, size and hydration of the particle and the viscosity of the medium. : ${\ displaystyle u}$${\ displaystyle f}$

${\ displaystyle {\ vec {F}} _ {R} = - fu \ cdot {\ vec {e}} _ {r}}$

Since all the forces considered only act in the radial direction, the basis vector is not considered in the following . ${\ displaystyle {\ vec {e}} _ {r}}$

Assuming that the forces involved are immediately in equilibrium, the result is ${\ displaystyle (\ Sigma _ {i} {\ vec {F}} _ {i} = {\ vec {0}})}$

${\ displaystyle m \ omega ^ {2} r (1 - {\ frac {\ rho _ {s}} {\ rho _ {p}}}) = f \ cdot u}$

If you divide the sedimentation speed by the strength of the centrifugal field, you get the sedimentation coefficient  , which only depends on the nature of the particle and the solvent: ${\ displaystyle \ omega ^ {2} r}$${\ displaystyle s}$

${\ displaystyle s = {\ frac {u} {\ omega ^ {2} r}} = {\ frac {m (1 - {\ frac {\ rho _ {s}} {\ rho _ {p}}} )} {f}}}$

The sign of and thus also of is determined by the sum of the centrifugal force and the counteracting buoyancy force and accordingly leads to flotation or sedimentation of the particle. ${\ displaystyle s}$${\ displaystyle u}$

The description of the sedimentation properties of a particle by the sedimentation coefficient offers the advantage of a radius and speed-independent consideration.

It can also be seen directly from the definition of the sedimentation coefficient that the sedimentation speed increases in the context of an experiment for sedimenting particles. This is due to the fact that the sum of centrifugal force and counteracting buoyancy force increases with higher radius positions. In the case of an instantaneous equilibrium of forces at every radius position, however, this is not a contradiction to the assumption . ${\ displaystyle (\ Sigma _ {i} {\ vec {F}} _ {i} = {\ vec {0}})}$

The dimension of results from ${\ displaystyle s}$

${\ displaystyle dim (s) = {\ frac {\ frac {\ mathrm {L {\ ddot {a}} nge}} {\ mathrm {time}}} {{\ frac {1} {\ mathrm {time} ^ {2}}} \ cdot \ mathrm {L {\ ddot {a}} nge}}} = {\ frac {\ mathrm {L {\ ddot {a}} nge} \ cdot \ mathrm {time} ^ { 2}} {\ mathrm {time} \ cdot \ mathrm {L {\ ddot {a}} nge}}} = \ mathrm {time}}$

In general, the sedimentation coefficient depends on the size, shape and density of the particles examined. For the special case of spherical particles, if the particle density , solvent density and viscosity are known, the following can be determined directly from the diameter : ${\ displaystyle \ rho _ {p}}$${\ displaystyle \ rho _ {s}}$ ${\ displaystyle \ eta}$${\ displaystyle s}$${\ displaystyle d}$

${\ displaystyle d = {\ sqrt {\ frac {18 \ eta s} {\ rho _ {p} - \ rho _ {s}}}}}$

If the particles are non-spherical, a corresponding equivalent diameter can be calculated.

When determining the mass of particles from their sedimentation coefficient, it must be taken into account that two particles of the same mass can have different sedimentation coefficients if they have different densities (influence on the buoyancy and thus on the effective mass) or shapes (influence on the friction coefficient). In addition, when two particles come together, their sedimentation coefficients cannot simply be added because, firstly, the contact surfaces do not contribute to friction and, secondly, the friction increases more slowly than the surface, see Stokes' law . For example, a complex ribosome from bacteria consisting of two ribosomal subunits of 30 S and 50 S has a sedimentation coefficient of only 70 S.

## determination

From the definition of sedimentation the particle path can be achieved by integration with the radius position of the particle at the time , and at the time , calculated. This means that by observing the movement of the sedimentation band, conclusions can be drawn about the sedimentation coefficient. This technique is used, for example, to determine the particle size by means of photo sedimentation in the earth's gravity field or by means of the disc centrifuge . Depending on the material system investigated, sedimentation coefficient and particle size distributions can be determined by using locally fixed detectors. ${\ displaystyle s = {\ frac {u} {\ omega ^ {2} r}}}$${\ displaystyle r (t) = r (0) \ cdot \ exp (s \ omega ^ {2} t)}$${\ displaystyle t}$${\ displaystyle r (t)}$${\ displaystyle 0}$${\ displaystyle r (0)}$

Other methods for determining the sedimentation coefficient are based on the spatial and temporal recording of the entire sedimentation process by means of analytical ultracentrifugation .

While local and temporal derivations of the sedimentation bands were used at the beginning of the development of the technology, later numerical solutions of the Lamm equation could be used for data evaluation. By modeling all the data available during the sedimentation process, it is possible to determine sedimentation coefficient distributions adjusted for diffusion broadening even for small particles. In addition, the diffusion information for the sedimentation coefficient can contain information that is complementary to the sedimentation coefficient, which, for example, enables the determination of the molar mass of proteins using the Svedberg equation .

## Individual evidence

1. a b Analytical Ultracentrifugation of Polymers and Nanoparticles (=  Springer Laboratory ). Springer-Verlag, New York 2006, ISBN 978-3-540-23432-6 , doi : 10.1007 / b137083 ( springer.com [accessed March 29, 2020]).
2. The Svedberg, J. Burton Nichols: DETERMINATION OF SIZE AND DISTRIBUTION OF SIZE OF PARTICLE BY CENTRIFUGAL METHODS . In: Journal of the American Chemical Society . tape 45 , no. December 12 , 1923, ISSN  0002-7863 , p. 2910-2917 , doi : 10.1021 / ja01665a016 .
3. ^ Wilbur B. Bridgman: Some Physical Chemical Characteristics of Glycogen . In: Journal of the American Chemical Society . tape 64 , no. October 10 , 1942, ISSN  0002-7863 , p. 2349-2356 , doi : 10.1021 / ja01262a037 .
4. Walter F. Stafford: Boundary analysis in sedimentation transport experiments: A procedure for obtaining sedimentation coefficient distributions using the time derivative of the concentration profile . In: Analytical Biochemistry . tape 203 , no. 2 , June 1992, pp. 295-301 , doi : 10.1016 / 0003-2697 (92) 90316-Y .
5. ^ Jean-Michel Claverie, Henri Dreux, René Cohen: Sedimentation of generalized systems of interacting particles. I. Solution of systems of complete lamb equations . In: Biopolymers . tape 14 , no. 8 , August 1975, ISSN  0006-3525 , pp. 1685-1700 , doi : 10.1002 / gdp . 1975.360140811 .
6. Borries Demeler, Hashim Saber: Determination of Molecular Parameters by Fitting Sedimentation Data to Finite-Element Solutions of the Lamm Equation . In: Biophysical Journal . tape 74 , no. 1 , January 1998, pp. 444–454 , doi : 10.1016 / S0006-3495 (98) 77802-6 , PMID 9449345 , PMC 1299397 (free full text).
7. Schuck, Peter: Sedimentation velocity analytical ultracentrifugation: discrete species and size distributions of macromolecules and particles . CRC Press, Boca Raton, FL, ISBN 978-1-4987-6895-5 ( [1] [accessed April 13, 2020]).