I think you lost me a bit there. I don't understand 'fully
anisotropic'? Does that mean rhombic (all three eigenvalues are
different), hence the tensor is ellipsoidal?
Sorry, that's my bad. I'm catching a cold and having trouble reading and thinking clearly :-(. I meant to say fully asymmetric, ie eta (your R, I believe) is ~ 1.
But because you say dzz
~= dyy, then the tensor must be spheroidal (the two eigenvalues dzz
and dyy are equal), hence the tensor is anisotropic but not rhombic.
Most of the references to 'fully anisotropic' that I have encountered
are synonymous with 'rhombic'. Also, how is 'eta' defined? I have
seen at least four different ways of expressing the shape of the CSA
tensor which doesn't include the three eigenvalues, be that d11 >= d22
>= d33, |dzz| >= |dyy| >= |dxx|, etc. Symbols are symbols, as long as
you define them you can use anything.
Here's my little tutorial on CSA nomenclature. It's one of the most annoying things I have found in NMR and I've made it a significant part of my thesis to boot! So, I'm going to start from scratch and assume almost no previous understanding (it's a large message board, so why not?). First, CSA means two things: either 'chemical SHIFT anisotropy' or 'chemical SHIELDING anisotropy'. The differences come from, I believe, the discipline of NMR you are most familiar with: solution state people always display chemical shifts whereas solid state people tend towards the shielding side. These two terms are synonymous, and yet opposite each other. Shift tensors are always described as delta(ii) and shielding tensors are sigma(ii) where:
delta(ii) = sigma(reference) - sigma(ii)
The three principle values are then organized as del(11) > del(22) > del(33) and sig(11) < sig(22) < sig(33), ie del/sig(11) has the largest chemical shift value (most downfield) while the del/sig(33) value is the most shielded value (most upfield). To make things even more ridiculous, some people reverse the sigma organization (sig(11) > sig(22) > sig(33)) where del11 is now equivalent to sig33, etc! So, beware! As I said before, I prefer the delta organization as I am in solution state and there isn't the added sigma confusion.
Now, the CSA tensor that defines the position of a peak in your NMR spectrum has both the isotropic and anisotropic component:
delta(ij) = delta(iso) + delta(ani,ij)
In relaxation and RCSA studies, the isotropic part is completely irrelevant and therefore we are only interested in the anisotropic part which is traceless (SUM[del(ani,ii) from 1to3] = 0) and assumed to be symmetric (del(ani,ij) = delta(ani,ji)), although this isn't necessarily true, but for molecular symmetry reasons almost always is true. From here on I'm just refering to the anisotropic part of the tensor and delta(11) is always positive and delta(33) is always negative.
For a number of mathematical reasons, the CSA tensor is often reorganized once again in terms of x,y,z such |deltz(zz)| > |delta(yy)| > |delta(xx)|, similar to diffusion tensor and alignment tensor definitions. The asymmetry of the tensor is defined as:
eta = (delta(xx) - delta(yy))/delta(zz)
and varies from 0 (fully symmetric) to 1 (fully asymmetric). This is the parameter that is ubiquitously assymed to be zero in relaxation in order to keep the CSA term simple. In the situation where eta ~= 1, del(11) ~= del(33) or del(zz) ~= del(yy) and del(22) = del(xx) ~= 0. Now the confusing part (if this hasn't been confusing enough!): there are many values proposed in the literature for the principle values of various nuclei and in some cases |del(11)| > |del(33)| and in others |del(33)| > |del(11)| ! Given the the CSA is a tensor, the orientation of all 3 components are orthogonal, so this seemingly harmless difference represents a 90 degree shift in the dominant principle component value in terms of x,y, and z. So, the question is, how does this relate to relaxation studies?
In much of the literature (which assumes eta = 0), the ANISOTROPY of the tensor is given as:
DELTAdelta = delta(zz) - (delta(xx)+delta(yy))/2
or, with a little trivial math:
DELTAdelta = (3/2) delta(zz)
What you find in most relaxation equations for R1 and R2 is that the CSA constant is:
(2/5) (omega(i) * DELTAdelta)^2 / 3 or equivalently (3/10) (omega(i) * delta(zz))^2
(For those of you wondering where 2/5 comes from, I simply prefer to pull it out of the extended model free spectral density function.)
Let's keep things simple for now (ie, isotropic diffusiong). If you want to include asymmetry, one should really use the following as the CSA constant:
(3/10) omega(i)^2 delta(zz)^2 (1+eta^2/3)
where delta(zz) * sqrt((1+(eta^2)/3)) is the asymmetric anisotropy, or CSAa. The CSAa is equivalently:
sqrt((3/2) (del(xx)^2 + del(yy)^2 + del(zz)^2))
So, for eta = 0, DELTAdelta = CSAa, but when eta != 0, CSAa > DELTAdelta. Now, how many biomolecules diffuse isotropically? Can I see a show of hands? Ok, I don't see a lot of hands, although ubiquitin is sitting in the back waving his/her arms wildly :). Here's the kicker, when a molecule tumbles anisotropically the asymmetry term of the CSA can NOT be pulled out of the spectral density function! The CSA constant for these need to be:
(3/10) omega(i)^2 delta(zz)^2
and the extended model free coefficients contain the asymmetry as well as euler angles relating the CSA tensor with the diffusion tensor (but let's not go there for right now).
So, where am I going with all this? When is Alex going to stop rambling on and on about CSAs?! If you can remember back, I said that the literature is unclear about some CSAs and that the delta(zz) term for some nuclei switches between delta(11) and delta(33). Well, there you go, in all relaxation delta(zz) is the principle component that matters, and if your delta(zz) = delta(11), you'll be oriented in the least shielded direction, whereas if your delta(zz) = delta(33), you'll be oriented 90 degrees away in the most shielded direction. It's possible that this isn't too detrimental of a problem, given that the asymmetry is ~1, but I'm still looking into some simulations of this problem.
That's about everything I can think of as important for right now. Let me know if you have any questions. :)
Another thing that confuses me is that the dxx eigenvalue is
approximately zero ppm? Is that possible? Also, doesn't traceless
refer to a tensor in which the sum of the eigenvalues is zero? That
would only occur if the isotropic chemical shift is zero ppm. Sorry
about my ignorance, but what does 'traceless anisotropic part' and
'anti-symmetric part' of the CSA tensor mean? Does this involve the
diagonalisation of the matrix and then the subtraction of the
isotropic shift times the identity matrix from the diagonalised CSA
matrix?
I was trying to be a little scary, for some reason. The CSA tensor CAN, have an anti-symmetric part (delta(ij) != delta(ji)) which leads to terms in the correlation functions, f_lm(tau), and their Fourier transforms, g_lm(omega), whith l = 1. !! I'm not very good at describing, let alone understanding, this but hopefully you know what I mean. It's an interesting bit of trivia, but for now let's pretend I never said anything ;)
One last question, if the CSA tensor for RNA/DNA base 13C and 15N
spins is not rhombic, does this mean that the angle between the unique
axis of the CSA tensor and the XH dipolar interaction vector is ~90
degrees?
More often than not, yes, but there's really two principle components to the tensor that are important. I don't quite understand the rhombic term, but I'm assuming it means assymetry (?)
As you can probably see, I am a bit lost. This is probably because of
my ignorance of the variety of symbols used to express the same thing
and the terminology used in each instance.
That's completely understandable! The whole topic is filled with duplicate and triplicate nomenclature! If I had my way, I would chose yet another way to describe the CSA tensor that I didn't even discuss that keeps everything in terms of the delta(11) , (22), and (33) that I prefer. But that would take a lot of derivations from scratch on my part, so I'm leaving that alone ... for now ;) One day, I hope to confuse the literature even more!!! Mwa ha ha ha!! I think that's a prerequisite for anyone studying CSAs. I hope you've enjoyed this mini tutorial of my knowledge on CSA tensors.
Cheers,
Alex
Edward
On 10/6/06, Alexandar Hansen <
viochemist@xxxxxxxxx> wrote:
> In most of the base nuclei in RNA/DNA, 13C and 15N, the CSA tensor is almost
> fully anisotropic (eta >0.9), so dzz ~= dyy and dxx ~= 0. Of course, this is
> the traceless anisotropic part (let's pray there's no anti-symmetric part!!!
> :) ) and |dzz| >= |dyy| >= |dxx|. I prefer the d11 >= d22 >= d33 version
> where you always know that d33 is the most shielded and d11 has the largest
> CS, but that's just me. dzz is more often than not the direction
> perpindicular to the base plane, although it varies with your choice of CSA
> tensor source (DFT, SS NMR, Solution NMR, etc.).
>
> Alex
>
>
> On 10/5/06, Edward d'Auvergne <edward.dauvergne@xxxxxxxxx> wrote:
> > That's a good idea, the eigenvector of the CSA tensor perpendicular to
> > the base rings and parallel to the long axis of the diffusion tensor
> > should contain a bit of that missing information about Da. Is the
> > amplitude of the perpendicular CSA component (say sigma_z)
> > significantly different from the other two eigenvalues (sigma_x and
> > sigma_y)? relax is currently incapable of using that information
> > though. Maybe that will soon change ;)
> >
> > Edward
> >
> >
> >
> > On 10/6/06, Alexandar Hansen < viochemist@xxxxxxxxx> wrote:
> > > I had meant to say more on this but had to run to a meeting.
> > >
> > > In addition to ribose residues being helpful, the 13C CSA tensors of the
> > > base are highly asymetric and anisotropic. One of the components of the
> CSA
> > > tensor is perpindicular to the plane of the base (I think, perhaps, 13CO
> has
> > > a similar situation?) so that the CSA part of the relaxation will be
> > > sensitive to both orientations and should help to adequately span the 3D
> > > environment of anisotropic diffusion tensors. We have shown this to be
> true
> > > when measuring residual CSAs (RCSAs) as complementary to RDCs (Me, JMR
> > > (2006) 179, p323)
> > >
> > > Alright, off to another meeting!
> > >
> > >
> > > Alex
> > >
> > >
> > >
> > > On 10/4/06, Alexandar Hansen <viochemist@xxxxxxxxx > wrote:
> > > > You have it right. Measuring ribose, or simply anything that's not
> also
> > > perpindicular to the base, should adequately sample more of the 3D
> space.
> > > We find this to be the case frequently when analyzing RDCs measured in
> RNA.
> > > Of particular interest would be the C1'-H1's. Having just a handful of
> > > those would like be highly beneficial.
> > > >
> > > > Alex
> > > >
> > > >
> > > >
> > > >
> > > > On 10/4/06, Edward d'Auvergne < edward.dauvergne@xxxxxxxxx> wrote:
> > > > > Hi,
> > > > >
> > > > > In relaxation data analysis, you can only view the components of the
> > > > > Brownian rotational diffusion tensor that the XH bond vectors
> sample.
> > > > > So if your macromolecule diffuses as a prolate spheroid but the XH
> > > > > bond vectors are close to perpendicular to the unique axis of the
> > > > > tensor, the only component of the diffusion tensor that the
> relaxation
> > > > > data contains information about is the eigenvalue Dper (the
> > > > > perpendicular component of the tensor). The result is that the
> > > > > diffusion will appear to be spherical where Diso has the value of
> > > > > Dper! In relax the parameters tm (which is essentially Diso) and Da
> > > > > are optimised. For this case, Da (and hence Dratio) would be
> > > > > undefined - it can have any geometrically possible value while
> having
> > > > > zero effect on the results.
> > > > >
> > > > > Have you tried starting with the calculated Da value (or Dratio if
> you
> > > > > wish)? This is not possible using the 'full_analysis.py' script,
> but
> > > > > the other sample scripts can be modified to do this. As these
> > > > > parameters will be statistically undefined, the final optimised
> values
> > > > > should be pretty close to the input values. This assumes tm (or
> Diso)
> > > > > is set to be close to the Dper value as the curvature of the space
> may
> > > > > cause optimisation to shift Da. The parameter Dr would also be
> > > > > undefined and this would fully explain the Dr value of 1 reported in
> > > > > bug #7297 ( https://gna.org/bugs/?7297 ).
> > > > >
> > > > > The problem of the undefined Da and Dr, and hence the molecule
> > > > > appearing to diffuse as a sphere, could be resolved by having a few
> > > > > vectors which deviate from the perpendicular. However this is only
> > > > > important if you are actually interested in characterising the
> > > > > Brownian rotational diffusion. In any case, attempting to optimise
> > > > > these values using relaxation data of perpendicular XH's will only
> > > > > result in statistically insignificant values - it's not
> statistically
> > > > > possible to pull out these parameters. It is almost guaranteed that
> > > > > AIC model selection will select spherical diffusion. Would the
> ribose
> > > > > CH's together with the base XH's adequately sample three-dimensional
> > > > > space?
> > > > >
> > > > > I hope this info helps,
> > > > >
> > > > > Edward
> > > > >
> > > > >
> > > > >
> > > > > On 10/5/06, Alexandar Hansen < viochemist@xxxxxxxxx
> wrote:
> > > > > > Hello all,
> > > > > >
> > > > > > In studying RNA you run into a number of limiting factors of your
> data
> > > set.
> > > > > > a) NH data is available only on half of the residues (G's and
> U's), b)
> > > these
> > > > > > G's and U's must be in a helix, or the NH becomes exchanged with
> > > solvent,
> > > > > > and c) the NH vectors on the bases in a helix don't sample space
> > > randomly
> > > > > > and are oriented ~perpindicular to the diffusion axis (RNA is
> almost
> > > always
> > > > > > prolate shaped). This last scenario, for you protein folks, would
> be
> > > > > > similar to the situation where you had a single alpha helix and
> only
> > > NH
> > > > > > data, ie. sample only directions paralell to the helix axis.
> > > > > >
> > > > > > With this in mind, one can easily imagine that any relaxation
> analysis
> > > would
> > > > > > be happy to fit them to a lower diffusion model, such as
> spherical,
> > > than
> > > > > > what is in reality highly anisotropic. What I'd like to know how
> to
> > > do is
> > > > > > impose additional limits on the minimization step such that, for
> > > instance,
> > > > > > the Dratio could be fixed between some values. With the data I've
> > > been
> > > > > > analyzing, relax happily fits my NH data to the spherical case
> and,
> > > for the
> > > > > > prolate model, fits the Dratio to 1 ->
1.1. From hydrodynamic
> > > simulation,
> > > > > > we know, however, that the Dratio should be between 4-5. Are
> there
> > > any
> > > > > > thoughts on how to do this? On one level, it appears to be
> forcing
> > > the data
> > > > > > into a particular model. But if you can know something about the
> > > diffusion
> > > > > > parameters or anything else a priori from a different source than
> NMR,
> > > > > > shouldn't that be allowed to factor into the analysis?
> > > > > >
> > > > > > Thanks,
> > > > > > Alex Hansen
> > > > > >
> > > > > >
> > > > > > _______________________________________________
> > > > > > relax ( http://nmr-relax.com
)
> > > > > >
> > > > > > This is the relax-users mailing list
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> > > > > > visit the list information page at
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> > > >
> > > >
> > >
> > >
> >
>
Re: analysis of limited data sets