> Surely it can't be as bad as equation 8.86 in the relax manual
> (version 1.2.7)?
My equations for the axially symmetric diffusion aren't too bad, but for
the
ellipsoidal diffusion and asymmetric CSA, it's not good. I'll look into
though and post what I can (hopefully without typos :) )
Maybe it will be better to embed the equations into the comments in the
code?
> 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?
Yes, absolutely the C-C cross and cross-correlated relaxation can be
supressed through appropriate means in the pulse sequence and we can just
assume that the user is taking care of those appropriately. However, C-C
auto relaxation can't be suppressed and this should be significant for a
number of nuclei in uniformly labeled samples. The relation of the CC
auto-relaxtion to CH dipolar relaxation is given in Yamazaki et al (1994)
JACS, 116, 8266-8278 (equ 1).
I've always wanted to get around to reading that one. As I've
currently only been working with NH data, coding, model-free and
Brownian rotational diffusion theory, model selection statistical
theory, etc, I haven't found a reason to read that reference yet.
Maybe now is a good time ;)
rho(HC)/rho(CC) =
3*(gamma_H^2/gamma_C^2)*(R_CC/R_HC)^6*(omega_c * tau_c)^ -2
I don't know the specifics for proteins, but in pyrimidine bases (uracil and
cytodine) and in the ribose rings, the C-C bond length are 134pm and 153pm
respectively. At 600 Mhz and as small as a 5 ns correlation time, the CC
contributes ~13% to the R1 rate of the base relaxation and ~6% for the
ribose. The CC rate is equal to the CH rate at a tau( c ) of ~14 ns for the
base and ~20 ns for the ribose.
Can we handle this situation prior to model-free analysis? I'll read
the paper, as what I'm asking is quite likely to be answered there,
but is it possible to tease out the auto-relaxation rate of the CH
carbon? Hence we input the pure R1 rate of the carbon (free of the CC
rate) into the model-free analysis and leave the current relaxation
equations as they are? We could therefore completely ignore the CC
rate in the analysis. That would significantly simplify the task of
coding.
> > >5. Misc.
>
> 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?
Then no, i'm not making that assumption. I calculate a different spectral
density for the two/three mechanisms ... is that a flaw in the way I'm
analyzing things? If so, I'm not sure I understand. It does say in Spiess
that "The [c_i] contain the coupling parameters delta_lambda and, therefore,
are not spectral densities. ... It should be noted, however, that even in
the general case, for the l = 2 terms, the coupling constant delta_lambda
enters as a common factor .. and, therefore, reduced g_2,lambda = g_2,lambda
/ delta_lambda^2 are independent of the strength of the coupling and only
depend on the assymetry parameter eta_lambda, a number between 0 and 1." I
guess, in this light, what I use are reduced spectral densities (?) or is
that yet something else entirely? What affect does this have on what we
want to do?
> > 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.
I'm still not sure I understand. In model free analysis, aren't the
internal motions assumed isotropic? To my knowledge, no one has attempted
to address anisotropic internal motions, whether the interaction is
symmetric (dipole) or not (csa). However, I am certain that, assuming
isotropic tumbling and model free as it currently is used, the
non-collinearity is not necessary for the understanding of auto-relaxation.
If cross-correlated relaxation were to be added to relax, that would be a
different story altogether ...
Reduced spectral density mapping (Lefevre, J. F., Dayie, K. T., Peng,
J. W., and Wagner, G. (1996) Biochem, 35(8), 2674-2686), is where the
assumption J(wX-wH) = J(wH) = J(wX+xH), i.e. the three higher
frequency spectral densities, are equal. Hence you can map the R1,
R2, and NOE directly into the three spectral densities J(0), J(wX),
and J(~wH).
As for comprehending the problem, it's me who is the one who doesn't
fully understand! I'll try to explain what my perception currently
is. Firstly the model-free spectral density function is (the original
Lipari-Szabo formula)
_k_
2 \ / S2 (1 - S2)(te + ti)te \
J(w) = - > ci . ti | ------------ + ------------------------- |,
5 /__ \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 /
i=-k
where ci are the weights and ti are the global correlation times of
the diffusion tensor. I hope that that equation doesn't get too
mangled! This is the formula used in relax. It is a generic equation
that handles all types of Brownian rotational diffusion of the
macromolecule - spherical (k = {0}), spheroidal (k = {-1, 0, 1}), and
ellipsoidal (k = {-2, -1, 0, 1, 2}). This is a numerically stabilised
form of the equations used to prevent round-off errors.
I'll create a hypothetical, yet impossible, example in which the
model-free parameters would be different. It should illustrate my
confusion ;) Say the DD interaction is perpendicular to the major
axis of the CSA interaction. Now construct a restricted motion which
stochastically rotates about the DD vector. For the spectral density
function experienced by the DD interaction (J_DD(w)), the S2 value
would be 1 and te would be undefined (== statistically zero). Yet
the S2 value for the spectral density function parallel to the major
axis of the CS tensor (J_CSA(w)) would not be one and the te value may
not be statistically zero.
In model-free analysis, there are no assumptions about the internal
motions. The actual model of the physical motion is not taken into
account, but you can map the Lipari-Szabo order parameter (and
effective correlation time) back onto a certain motion to define that
motion. Hence isotropy in the internal motion is not important, you
only see the component of the motion parallel to the DD and CSA
interactions.
Is there an assumption in the theory that as J_DD(w) and J_CSA(w) are
often close to parallel that you can assume J_DD(w) ~= J_CSA(w) and
hence S2_DD ~= S2_CSA and te_DD ~= te_CSA? As you can see, I'm
completely lost.
Edward
Re: Improving expressions for the CSA interaction