> > 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.
Re: Improving expressions for the CSA interaction