> > > 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.
I thought that tc term (relax uses the tm notation) would be
problematic! Doh. Is there an alternative equation (probably an
intermediate in it's derivation) in which the (w.tc)^2 term is
replaced by J(w). I'm assuming that the intermediate equations depend
on J(w) and assumptions are made to replace it with (w.tc)^2. If you
did use J(w) instead, the equation would be much more accurate and it
would be a lot easier to implement the equations into relax (as the
J(w) terms will already be calculated by the time you reach the R1,
R2, and NOE relaxation equations). Then again, the tm value will be
accessible at that time as well. Therefore the only benefit of using
J(w) over (w.tc)^2 would be that no assumptions about the global
tumbling and internal motions are made. Actually, now that I think a
little more about it, you would need the J(w) component parallel to
the CC vector for each CC interaction. That would significantly
complicate the situation over the isotropic assumption of (w.tc)^2.
Forget I just wrote that paragraph!
Ok ... paragraph forgotten :) One thing though. Correct me if I'm wrong, but I think of tau( c ) and tau( m ) interchangeably. In fact tau( e ), tau( f ), and tau( s ) are pretty much the same thing to me too. Just to be clear, the _expression_ I gave is a huge approximation and drops about 4 J(w) terms, keeping only the dominant one for D_cc and D_ch relaxation (J(0) and J(w_c) respectively). It also assumes (for J(w_c)) that (
w.tc)^2 >> 1, which is pretty true. The _expression_ is for a rigid interaction.
Therefore it's obvious that the separation of the R1 rates is not
possible prior to analysis because it depends on the Brownian
rotational diffusion. It would therefore be useful to have an option
so that the user can select between uniform labelling and natural
abundance when inputting the 13C R1 values.
I agree. Also, for uniformly labeled, it would be useful to choose if the spin was detected selectively or not. If selective, one can use heteronuclear dipolar equations to calculate the relaxation rate. If nonselective (ie you hit both Ca and CO, or C5 and C6) then you need to use the homonuclear expressions for dipolar relaxation.
a) R1 like spins:
0.1 d^2 [ 3 J(w_i) + 12 J(2 w_i) ]
b) R1 unlike spins (should be familiar):
0.1 d^2 [ J (w_i - w_s) + 3 J (w_i) + 6 J(w_i + w_s) ]
c) R1 of like spins, treated as unlike (selective excitation):
0.1 d^2 [ J ( 0 ) + 3 J (w_i) + 6 J(2 w_i) ]
d = (mu0 * gam_i * gam_s * h)/(8 pi^2 * r^3)
J = extended model free
These I determined from using the Spiess reference I keep talking about :) [NMR Basic Principles and Progress (1978) 15, 55-214]
> > > > > >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.
Got it!!! I think I fully understand all the assumptions now. Oh, it
would be best to keep ideas for future work secret as these posts are
archived and will be Google searchable in a month or two - minimising
the amount of info the competition has access to would be best
(assuming you have competition). I would guess that the resultant S2
would be closer to the true order parameter as you are measuring
spectral density functions in all three dimensions (x, y, z of the CS
tensor) and approximating them by the single function J(w). Would
this mean that S2 ~= S2_DD * S2_CSA_parallel * S2_CSA_perpendicuar? I
haven't really put much thought into correlation functions and
spectral densities of rotational motions in all three dimensions and
how they relate to the true order parameter. Model-free analysis of
the backbone N or C_alpha in proteins is essentially the motion of a
vector (due to the collinearity of the relaxation mechanisms) rather
than the motion of a tensor (which is probably a better description
for non-collinear relaxation mechanisms). Interesting!
I would imagine that S2_true = S2_x * S2_y * S2_z ~= S2_csa, if that makes any sense. I haven't given much thought to any relationships like that.
Now that I think that I fully understand the problem, I should be able
to help with the implementation of the analysis techniques. Oh, I
have one more question. How does the measured 13C R1 relate to the
rho(HC)/rho(CC) ratio (or ratios for each proximal 13C)? Do you need
to know rho(CC)?
So, the measured R1 rate ( rho(C,auto) ) = rho(csa) + rho(CC dip) + rho(CH dip), assuming that all cross and cross-correlated relaxation is eliminated with the pulse sequence. Similarly, R2(C,auto) = R2(csa) + R2(CC dip) + R2(CH dip). However R2(CC dip) is never more than 5% of the R2(CH dip) rate (in my calculations anyways) and more often than not much less. My reasoning is that R2 already contains a J(0) term so there's no significant gain with homonuclear relaxation as there is in R1, which has no J(0) term to start with, only in the homonuclear case.
Alex
Re: Improving expressions for the CSA interaction