Rotational diffusion

Rotational diffusion

Rotational diffusion is a process by which the equilibrium statistical distribution of the overall orientation of particles or molecules is maintained or restored. Rotational diffusion is the counterpart of translational diffusion, which maintains or restores the equilibrium statistical distribution of particles' position in space.

The random re-orientation of molecules (or larger systems) is an important process for many biophysical probes. Due to the equipartition theorem, larger molecules re-orient more slowly than do smaller objects and, hence, measurements of the re-orientation times can give insight into the overall mass and distribution of mass within an object. Quantitatively, the mean square of the angular velocity about each of an object's principal axes is inversely proportional to its moment of inertia about that axis. Therefore, there should be three independent relaxation time constants for re-orientation, corresponding to each of the three principal axes; however, two or even three of these time constants may be the same if the object is symmetrical in its principal axes. For example, spheroidal particles have two time constants for rotational diffusion; these time constants may be determined from the Perrin friction factors, in analogy with the Einstein relation of translational diffusion.

These time constants may be determined experimentally from fluorescence anisotropy, flow birefringence, dielectric spectroscopy, the linewidths of liquid-state NMR peaks and other related biophysical methods. However, it is very difficult to discern the three different time constants; usually, only one is possible. Two time constants can sometimes be measured when there is a great difference between them, e.g., for very long, thin ellipsoids such as certain viruses.

Rotational version of Fick's law

A rotational version of Fick's law of diffusion can be defined. Let each rotating molecule be associated with a vector n of unit length n·n=1; for example, n might represent the orientation of an electric or magnetic dipole moment. Let "f(θ, φ, t)" represent the probability density distribution for the orientation of n at time "t". Here, θ and φ represent the spherical angles, with θ being the polar angle between n and the "z"-axis and φ being the azimuthal angle of n in the "x-y" plane. The rotational version of Fick's law states

:$frac\left\{1\right\}\left\{D_\left\{mathrm\left\{rot\right\} frac\left\{partial f\right\}\left\{partial t\right\} = abla^\left\{2\right\} f = frac\left\{1\right\}\left\{sin heta\right\} frac\left\{partial\right\}\left\{partial heta\right\}left\left( sin heta frac\left\{partial f\right\}\left\{partial heta\right\} ight\right) + frac\left\{1\right\}\left\{sin^\left\{2\right\} heta\right\} frac\left\{partial^\left\{2\right\} f\right\}\left\{partial phi^\left\{2$

This partial differential equation (PDE) may be solved by expanding "f(θ, φ, t)" in spherical harmonics for which the mathematical identity holds

:$frac\left\{1\right\}\left\{sin heta\right\} frac\left\{partial\right\}\left\{partial heta\right\}left\left( sin heta frac\left\{partial Y^\left\{m\right\}_\left\{l\left\{partial heta\right\} ight\right) + frac\left\{1\right\}\left\{sin^\left\{2\right\} heta\right\} frac\left\{partial^\left\{2\right\} Y^\left\{m\right\}_\left\{l\left\{partial phi^\left\{2 = -l\left(l+1\right) Y^\left\{m\right\}_\left\{l\right\}$

Thus, the solution of the PDE may be written

:$f\left( heta, phi, t\right) = sum_\left\{l=0\right\}^\left\{infty\right\} sum_\left\{m=-l\right\}^\left\{l\right\} C_\left\{lm\right\} Y^\left\{m\right\}_\left\{l\right\}\left( heta, phi\right) e^\left\{-t/ au_\left\{l$

where "Clm" are constants fitted to the initial distribution and the time constants equal

:$au_\left\{l\right\} = frac\left\{1\right\}\left\{D_\left\{mathrm\left\{rotl\left(l+1\right)\right\}$

ee also

* Perrin friction factors

References

*

Wikimedia Foundation. 2010.

Look at other dictionaries:

• Diffusion (disambiguation) — Diffusion is a time dependent random process causing a spread in space. Diffusion may also refer to: In physical sciences Molecular diffusion, spontaneous dispersion of mass (distinct from migration, caused by an external force) Conduction of… …   Wikipedia

• Diffusion — This article is about the generic concept of the time dependent random process. For other uses, see Diffusion (disambiguation). Diffusion describes the spread of particles through random motion from regions of higher concentration to regions of… …   Wikipedia

• Fluorescence correlation spectroscopy — (FCS) is a common technique used by physicists, chemists, and biologists to experimentally characterize the dynamics of fluorescent species (e.g. single fluorescent dye molecules in nanostructured materials, autofluorescent proteins in living… …   Wikipedia

• Flow birefringence — In biochemistry, flow birefringence is a hydrodynamic technique for measuring the rotational diffusion constants (or, equivalently, the rotational drag coefficients). The birefringence of a solution sandwiched between two concentric cylinders is… …   Wikipedia

• Quaternary structure — In biochemistry, quaternary structure is the arrangement of multiple folded protein molecules in a multi subunit complex.Description and examplesMany proteins are actually assemblies of more than one polypeptide chain, which in the context of the …   Wikipedia

• Protein quaternary structure — In biochemistry, quaternary structure is the arrangement of multiple folded protein or coiling protein molecules in a multi subunit complex. Contents 1 Description and examples 2 Nomenclature of quaternary structures 3 Determination of qua …   Wikipedia

• Circular dichroism — (CD) refers to the differential absorption of left and right circularly polarized light.[1][2] This phenomenon was discovered by Jean Baptiste Biot, Augustin Fresnel, and Aimé Cotton in the first half of the 19th century.[3] It is exhibited in… …   Wikipedia

• Список научных публикаций Альберта Эйнштейна — Альберт Эйнштейн (1879 1955) был известным специалистом по теоретической физике, который наиболее известен как разработчик общей и специальной теорий относительности. Он также внёс большой вклад в развитие статистической механики, особенно… …   Википедия

• Fluorescence anisotropy — In chemistry, fluorescence anisotropy assays the rotational diffusion of a molecule from the decorrelation of polarization in fluorescence, i.e., between the exciting and emitted (fluorescent) photons. This decorrelation can measure the tumbling… …   Wikipedia

• Chemotaxis — is the phenomenon in which somatic cells, bacteria, and other single cell or multicellular organisms direct their movements according to certain chemicals in their environment. This is important for bacteria to find food (for example, glucose) by …   Wikipedia