Lamm equation

The Lamm equation [O Lamm: (1929) "Die differentialgleichung der ultrazentrifugierung" Arkiv für matematik, astronomi och fysik" 21B No. 2, 1-4] describes the sedimentation and diffusion of a solute under ultracentrifugation in traditional sector-shaped cells. (Cells ofother shapes require much more complex equations.)

The Lamm equation can be written:cite book |title=Introduction to mathematical biology |author = SI Rubinow |page=pp. 235-244 |url=http://books.google.com/books?id=3j0gu63QWmQC&pg=PA250&dq=Lamm+equation&sig=mH9T2kyIE0PIqMGzh6WF-KGyApc#PPA235,M1 |isbn=0486425320 |year=2002 (1975) |publisher=Courier/Dover Publications] cite book |title=An Introduction to Mathematical Physiology and Biology |url=http://books.google.com/books?id=mqD6qSYGM-QC&pg=PA33&dq=Lamm+equation&lr=&sig=qiZ-6RAfQsRqDAzyuOJsiz4gOlE |page= pp. 33 ff |author= Jagannath Mazumdar |year=1999 |publisher=Cambridge University Press |location=Cambridge UK |isbn=0521646758 ]

:$frac\{partial\; c\}\{partial\; t\}\; =\; D\; left\; [\; left(\; frac\{partial^\{2\}\; c\}\{partial\; r^2\}\; ight)\; +\; frac\{1\}\{r\}\; left(\; frac\{partial\; c\}\{partial\; r\}\; ight)\; ight]\; -\; s\; omega^\{2\}\; left\; [\; r\; left(\; frac\{partial\; c\}\{partial\; r\}\; ight)\; +\; 2c\; ight]$

where "c" is the solute concentration, "t" and "r" are the time and radius, and the parameters "D", "s", and $omega$ represent the solute diffusion constant,sedimentation coefficient and the rotor angular velocity, respectively.The first and second terms on the right-hand side of the Lamm equation are proportional to "D" and $somega^\{2\}$, respectively, and describe the competing processes of diffusion and sedimentation. Whereas sedimentation seeks to concentrate the solute near the outer radius of the cell, diffusion seeks to equalize the solute concentration throughout the cell. The diffusion constant "D" can be estimated from the hydrodynamic radius and shape of the solute, whereas the buoyant mass $m\_\{b\}$ can be determined from the ratio of "s" and "D" :$frac\{s\}\{D\}\; =\; frac\{m\_\{b\{k\_\{B\}T\}$where $k\_\{B\}T$ is the thermal energy, i.e.,
Boltzmann's constant $k\_\{B\}$ multiplied by the temperature "T" in kelvin.

Solute molecules cannot pass through the inner and outer walls of the cell, resulting in the
boundary conditions on the Lammequation:$D\; left(\; frac\{partial\; c\}\{partial\; r\}\; ight)\; -\; s\; omega^\{2\}\; r\; c\; =\; 0$at the inner and outer radii, $r\_\{a\}$ and $r\_\{b\}$, respectively. By spinning samples at constant angular velocity $omega$ and observing the variation in the concentration $c(r,t)$, one may estimate theparameters "s" and "D" and, thence, the buoyant mass and shape of the solute.

Derivation of the Lamm equation

Faxén solution (no boundaries, no diffusion)

References and notes

ee also

*Ultracentrifuge
*Centrifugal force
* [http://www.nibib.nih.gov/Research/Intramural/lbps/pbr/auc/LammEqSolutions Solving the Lamm equation]
* [http://www.pubmedcentral.nih.gov/picrender.fcgi?artid=1299825&blobtype=pdf Peter Schuck: "Sedimentation analysis … using numerical solutions to the Lamm equation"]