In a post to the relax-devel mailing list
(https://mail.gna.org/public/relax-devel/2006-10/msg00136.html,
Message-id: <7f080ed10610252045r3dcbba38q58aa53142af96e5a@xxxxxxxxxxxxxx>)
I discussed an idea about representing the Brownian rotational
diffusion tensor superimposed onto the 3D molecular structure. For a
better understanding of the representation, I'll create a
mini-tutorial on diffusion tensors to explain the different notations,
representations, etc. This text should be useful for future
reference.
The best way to think of the tensor is as a geometric object.
Isotropic diffusion is diffusion as a perfectly round sphere. Prolate
axial symmetry is where one axis has been stretched - i.e. the shape
of a rugby/AFL/gridiron ball. Oblate axial symmetry is where one axis
has been squashed - i.e. the shape of a discus. These two geometric
objects are known as spheroids. Finally fully anisotropic diffusion
is the shape of an ellipsoid, you can think of it as a partially flat
rugby/AFL/gridiron ball that you have sat on. These geometric objects
will form the representation of the tensor superimposed onto the 3D
structure.
For directly comparing diffusion tensors it is the eigenvalues and
eigenvectors, the geometric description, which is important. The
three eigenvectors of the ellipsoid are the three axes of the tensor
(two are undefined in the spheroids and all three are undefined in the
sphere). The ellipsoidal eigenvalues Dx, Dy, and Dz measure the rate
of rotational diffusion of a vector parallel to the corresponding axis
(or eigenvector). In the spheroids, Dx = Dy = Dper and Dz = Dpar. In
the sphere, Dx = Dy = Dz = Diso.
The eigenvalues tell you the geometric length of the axes of the
ellipsoid and if two are equal you have a spheroid, and if all three
are equal you have sphere. There are a few other ways of describing
the geometric objects though. The method I prefer is the isotropic
factor Diso, the anisotropic factor Da, and the rhombic or asymmetric
factor Dr. You can physically picture these as follows. Diso is the
starting spherical structure and is equal to 1/(6tm). Da is the
amount you then stretch (Da > 0) or squash (Da < 0) this sphere. Dr
is then the amount you stretch along the y dimension. This approach
simplifies the parameterisation of the model. All three values are
non-zero in the ellipsoid. In the spheroids, Dr = 0. In the sphere
Da = 0 and Dr = 0. Hence parameters drop out of the model as it
simplifies! Dratio is another way of expressing Da, but I don't like
it much as it isn't a good way to describe the ellipsoid.
To see all the relationships between the geometric parameters, have a
look at the documentation for the 'diffusion_tensor.init()' user
function in relax, either in the current 1.2.7 version by typing
'help(diffusion_tensor.init)' or in the current relax manual at
http://www.nmr-relax.com/docs.html or
http://download.gna.org/relax/manual/relax.pdf.
Edward
Brownian rotational diffusion mini-tutorial.