I'll try to shorten the post a little.
>Actually it
>would be useful if the equations were represented in text format to a
>post to relax-devel. This text could be included in the comments to
>the new code.
The equations are quite complex and would be incredibly tedious to write
them as text. I'll try when I'm working in the code, but I think leaving a
few references should be sufficient.
Surely it can't be as bad as equation 8.86 in the relax manual
(version 1.2.7)? Have a look at the comments in the
'maths_fns/jw_mf.py', especially the Hessian code, to see how I
presented some very long equations. The 'Values, gradients, and
Hessians' chapter of the manual also describes how I have broken the
model-free and relaxation equations into smaller components to speed
up calculations and simplify the maths.
>Another important question is whether the user inputs the CS tensor
>orientation (Euler angles, quaternions, etc). Is it necessary that
>the user inputs these values or are they dependent on the PDB
>coordinates? If we can remove this requirement away from the user and
>handle it in the code, it would be better. The less the user does,
>the better - this is an important philosophy behind relax. Otherwise
>we could again use the 'value.set()' function to set parameters such
>as 'CS_alpha', 'CS_beta', and 'CS_gamma'.
The simple answer is yes, it's necessary. The orientation and CSA
parameters are equivalent to knowing the appropriate bond length for a
dipolar interaction. What the relaxation equations need are the
orientations of the principle components with respect to the diffusion
frame. What I currently do is relatively simple, and I'll take a protein Ca
as an example: I find three nuclei in a plane, in this case Ca, N, and CO
would work, and generate a unit tensor for the nucleus of interest. I take
what is known about the tensors orientation, converted into euler angles,
and rotate the CSA tensor into it's molecular frame. I then assume that the
PDB is in the diffusion frame and calculate the euler angles from the
rotated unit tensor, and these are what enter into the relaxation equations.
My effort in keeping things simple for the user is to provide a table (or
tables) of CSA values and orientations. I can imagine a parameter such as
'CS_input' where one could choose DFT, SS NMR, Solution NMR, etc and the
program would read the necessary values from the designated table. These
values are just now being well characterized in RNA/DNA and I know they have
been studied in some depth for proteins. Long story short, there isn't a
final consensus on what CSA is right, every base and ribose nucleus is
different, and no study has really studied every nucleus fully (ie. in SS
NMR, angles are still derived from DFT calculations). This is the only way
I can think of to keep things relatively simple, given that the CSA is
rarely, if ever, simple.
This could be done by creating the 'csa.init()' function. The user
could simply select one of the tables you suggest and the program does
the rest. I.e. the only difference between 'standard' model-free
analysis and handling the more advanced CS tensor info is to use the
'csa.init()' function to selected the table rather than the
'value.set()' function to set the CSA value. That is unless I have
missed something?
>The most involved solution would be to create a new user function such
>as 'csa.init()' which mimics the 'diffusion_tensor.init()' function.
I can only imagine that some form of this will be necessary, but I can't say
for sure right now.
To select the table, 'csa.init()' would be best. In this case it
would not mimic the 'diffusion_tensor.init()' function at all as I'm
assuming the tensors will be hard coded into relax.
>3. Locating homonuclei.
>This could be either very easy or quite complex. If only the
>neighbouring bonded heteronuclei need to be taken into account then
>these could be easily extracted from the PDB.
This could be possible, but we would need to choose what's important for
every nucleus. This wouldn't necessarily be problematic for 13C/15N/X
nuclei but if someone were interested in 1H nuclei, this would be very
tricky.
>Otherwise if
>atoms from other residues need to be taken into account, an algorithm
>for identifying atoms within a certain radius may need to be coded.
I have written some C code to do this. It would likely need to be revamped
(perhaps considerably) for it to work in python.
>The final question is, what is then done with these spins?
Homonuclear R1 relaxation becomes very dominant in large molecules (the
spectral density term J(wi-wx) becomes a J(0) term). These homonuclei are
used to calcuate the contribution to auto-relaxation. Pulse sequences, if
written well, remove any and all cross-relaxation contributions. The
homonuclear R1 can be calculated using the heteronuclear equations if one
assumes that the nucleus is excited selectively, which is easily done for
many cases where the two homonuclei have significantly different isotropic
CS values (is this true for proteins?). Directly bonded nuclei of course
contribute the most but if there are any other nuclei that are in fixed
positions relative to the nucleus of interest and <~4 Ang away in a
relatively large molecule (tau(m) > 10-15 ns) these interactions may or may
not be negligible contributors to R1. This, of course, isn't necessary if
samples aren't uniformly 13C labeled (although most are) and another
parameter could be introduced such as 'DD_homonuc' which could be set as
'No', 'Hetero', or 'Homo' to declare how to calculate the contributions for
that particular spin. The XX contribution to R2 is rarely very significant.
Surely homonuclei aren't a problem? If you're looking at the nitrogen
with a single proton attached, decoupling in the experiment will
remove any spin diffusion problems. If you're looking at a carbon
with a single attached proton, can you not decouple it from the other
carbons and remove the cross-correlated relaxation mechanisms through
the pulse sequence?
>4. The optimisation of the new equations.
>A clear statement of the problem
>including which values are fixed, which are optimised, the equations
>used, etc, would be of help in determining the scope of the changes
required.
I would definitely refrain from optimizing anything more than what is
already done in relax. The number of variables is already increasing out of
control. Currently I assume (a very poor assumption in all likelihood) that
the CSA, XH, and XX relaxation mechanisms feel the same modelfree S2 and tau
parameters. Perhaps in the future more optimization could be addressed, but
if we worked on that now I would be submitting MY thesis before we have a
working version of it.
I understand, there is no part of the CS tensor which is optimised.
Essentially to the user the only difference is the selection of a more
complex chemical shift tensor. That should make the implementation
much easier.
>5. Misc.
>You said that the model-free parameters are the same for the CS and
>dipolar interactions, does this mean that you assume the same spectral
>density J(w) for both relaxation mechanisms? This would simplify
>implementation.
Yes and No. With diffusion more complex than the spherical case, the
asymmetry can not be pulled out of the spectral density equations. What I
did was write one function to calculate the spectral density for as complex
a system as I needed and then input the complexity whenever I call on it.
When looking at the extended model free equations, what this means is that
the c0, c1, and c2 terms include not only theta(or beta) but also eta,
alpha, and gamma. Gamma drops out when the diffusion is axially symmetric.
For dipolar relaxation, eta = 0, and so alpha and gamma both drop out as
well.
Sorry, I think I didn't state the question as I hoped to. What I
meant to ask is, is J_CSA(w) = J_DD(w)? This is really only the case
if the CS tensor is axially symmetric and the two interactions are
collinear, but is that an approximation you are making?
>Also relax handles all types of Brownian rotational
>diffusion including diffusion as a sphere, spheroid, and ellipsoid.
>The original reference has equations for an ellipsoid and you have
>derived the equations for a spheroid. Are there equations for the
>sphere (isotropic diffusion)?
For isotropic diffusion, no angular/asymmetry parameters are necessary. In
fact, for a spherical molecule, the current state of the equations is
perfect and no changes need to be made. Unfortunately, I don't know how
often a molecule is perfectly spherical.
Without the previous assumption, surely the non-collinearity of the DD
and CSA interactions together with certain anisotropic motions of the
XH bond would be of concern in spherical diffusion? As for a
macromolecule diffusing as a perfect sphere, despite that assumption
for many published model-free analyses, I can't see it as ever being
statistically significant.
Edward
Re: Improving expressions for the CSA interaction