> 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.
So, you would propose taking the rate input by the user and (if the indicate in some way that it's uniformly labeled) just subtract any CC rates from it? I'm not sure that's a good idea for a couple of reasons. 1) we would need to know a priori what it's relaxation properties are (diffusion model, model-free model, correlation time, etc. that _expression_ I wrote is just an approximation of the relation between CH and CC dipolar relaxation), and 2) for larger molecules, the CC becomes the primary means of relaxation and so we would be ignoring the majority of the relaxation. Granted, except for some sort of alkyne, the CC bond will be NOT parallel to the CH bond and also lead into the complication you mention below.
> > > >5. Misc.
> 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?
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).
Spectral density mapping is an interesting approach to interpreting relaxation, but I've never been to keen on the idea.
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.
I don't think you're completely lost at all. What you say is absolutely right and is at the crux of the assumption I make. There is some, but not a lot, of literature about the model free parameters derived from NH dipolar relaxation vs other relaxation. I think it has been shown that in ubiquitin, S2_csa = S2_dd, but it's not necessarily the case as you already mentioned. Fundamentally, I like to think of a fully asymmetric CSA tensor as having two main components orthogonal to one another. If one were able to measure CSA relaxation alone (no CH, no CC, no quadrupole, whatever :) ) the S2 and tau values report the on the motion of both components without specific information on either orientation. The orientation of those two components almost guarantees that one will be significantly different from the CH vector. In RNA bases, one component (the most shielded component) is oriented orthogonal to the plane, while the other is usually between 5-30 degrees from the CH orientation. The hypothetical situation you represent, in fact, isn't necessarily that impossible. Nucleic acid bases undergo internal wobbling motions that, I would assume, "pivot" about the hydrogen bonds or perhaps the glycosidic bond. In the case of pyrimidines, this hydrogen bond pivoting would be close to your hypothetical situation for the C6-H6 position (and the N-H)!
As you can see, and I'm glad you understand the problem, this is the reason I said originally that the assumption that the order paramaters being the same is likely not a good one, but a necessary one to begin with. I think I understand what you were saying about this also being a problem for isotropic diffusion and I think I agree with you that it should contribute differently ... somehow. :) In addition, with CC relaxation contributing, it should also be sensitive to a different set of internal motions. I can only think of four ways to ultimately address this issue: 1) measure more rates which contain the rate(s) of interest (C1q, C2q C1qC2q, CqNq, CqHq, Hq, cross-correlated, cross relaxation) and expand model free to have S2_dd, S2_csa, S2_cc, etc. or 2) (and I hope I don't give away too much of some future ideas) develop models similar to the gaussian axial fluctuation (GAF) model used in proteins and fit rates using that instead of/in addition to ModelFree, 3) measure relaxation with and without uniform isotopic labelling and expand model free as in 1), and 4) measure relaxation at tons of field strengths and expand model free as in 1). All of these are, of course, incredibly complex and outside of the scope of what we currently would like to accomplish. And as you already said, who knows if any of these additional variations or components would be statistically significant, or even physically comprehensible.
In the short term, simply knowing this, we can expect that model free using the assumption that the mechansims sense the same motions and have the same model free parameters should decrease S2's and affect tau's as well, although I don't know if that would go up or down. I imagine that would depend on the motion.
Now I think we're on the same page. Please let me know what you think about all of this and if you can see any other problems/scenarios of interest.
Alex
Re: Improving expressions for the CSA interaction