Re: [sr #3154] Implementation of Baldwin (2014) B14 model - 2-site exact solution model for all time scales -- May 01, 2014 - 10:08

Title: 110214 exchange13

The Carver and Richards code in relax is fast enough, though Troels
might have an interest in squeezing out a little more speed.  Though
it would require different value checking to avoid NaNs, divide by
zeros, trig function overflows (which don't occur for math.sqrt), etc.
 Any changes would have to be heavily documented in the manual, wiki,
etc.

The prove that the two definitions are equivalent is relatively straightforward. The trig-to-exp command in the brilliant (and free) symbolic program sage might prove helpful in doing that.

For the tau_c, tau_cp, tau_cpmg, etc. definitions, comparing the relax
code to your script will ensure that the correct definition is used.
That's how I've made sure that all other dispersion models are
correctly handled - simple comparison to other software.  I'm only
aware of two definitions though:

tau_cpmg = 1 / (4 nu_cpmg)
tau_cpmg = 1 / (2 nu_cpmg)

tau_cpmg = 1/(nu_cpmg).

Carver and Richard's switch definitions half way through the paper somewhere.

What is the third?  Internally in relax, the 1 / (4 nu_cpmg)
definition is used.  But the user inputs nu_cpmg.  This avoids the
user making this mistake as nu_cpmg only has one definition.

I've seen people use nu_cpmg defined as the pulse frequency. It's just an error that students make when things aren't clear. I've often seen brave student from a lab that has never tried CPMG before do this. Without someone to tell them that this is wrong, it's not obvious to them that they've made a mistake. I agree with you that this is solved with good documentation.

You guys are free to use my code (I don't mind the gnu license) or of course
implement from scratch as needed.

Cheers!  For a valid copyright licensing agreement, you'll need text
something along the lines of:

"I agree to licence my contributions to the code in the file
http://gna.org/support/download.php?file_id=20615 attached to
http://gna.org/support/?3154 under the GNU General Public Licence,
version three or higher."

Feel free to copy and paste.

No problem:

"I agree to licence my contributions to the code in the file
http://gna.org/support/download.php?file_id=20615 attached to
http://gna.org/support/?3154 under the GNU General Public Licence,
version three or higher."

I'd like to note again though that anyone using this formula to fit data,
though exact in the case of 2site exchange/inphase magnetisation, evaluated
at double floating point precision should not be doing so! Neglecting scalar
coupling/off resonance/spin flips/the details of the specific pulse sequence
used will lead to avoidable foobars. I do see value in this, as described in
the paper, as being a feeder for initial conditions for a more fancy
implemenation. But beyond that, 'tis a bad idea. Ideally this should appear
in big red letters in your (very nice!) gui when used.

In relax, we allow the user to do anything though, via the
auto-analysis (hence the GUI), we direct the user along the best path.
 The default is to use the CR72 and 'NS CPMG 2-site expanded' models
(Carver & Richards, and Martin Tollinger and Nikolai Skrynnikov's
Maple expansion numeric model).  We use the CR72 model result as the
starting point for optimisation of the numeric models, allowing a huge
speed up in the analysis.  The user can also choose to not use the
CR72 model results for the final model selection - for determining
dispersion curve significance.

Here's the supp from my CPMG formula paper (attached). Look at the last figure. Maybe relevant. Nikolai's algorithm blows up sometimes when you evaluate to double float point precision (as you will when you have a version in python or matlab). The advantage of Nicolai's formula, or mine is that they won't fail when Pb starts to creep above a per cent or so.

Using the simple formula as a seed for the more complex on is a good idea. The most recent versions of CATIA have something analogous.

As for the scalar coupling and spin flips, I am unaware of any
dispersion software that handles this.  

CATIA. Also cpmg_fig I believe. In fact, I think we may have had this discussion before?

https://plus.google.com/s/cpmg%20glove

If you consider experimental validation a reasonable justification for inclusion of the effects then you might find this interesting:

Spin flips are also quite relevant to NH/N (and in fact any spin system). The supplementary section of Flemming and Pramodh go into it here for NH/N
http://www.pnas.org/content/105/33/11766.full.pdf

And this:
JACS (2010) 132: 10992-5
Figure 2:
r2 0.97, rmsd  8.0 ppb (no spin flips)
r2 0.99, rmsd  5.7 ppb (spin flips).

The improvements are larger than experimental uncertainties.

When we design these experiments and test them, we need to think about the details. This is in part because Lewis beats us if we don't. You can imagine that it comes as a surprise then when we see people neglecting this. In short, the parameters you extract from fitting data will suffer if the details are not there. In the case of spin flips, the bigger the protein, the bigger the effect. In your code, you have the opportunity to do things properly. This leaves the details behind the scenes, so the naive user doesn't have to think about them.

And only Flemming's CATIA
handles the CPMG off-resonance effects.  This is all explained in the
relax manual.  I have asked the authors of most dispersion software
about this too, just to be sure.  I don't know how much of an effect
these have though.  But one day they may be implemented in relax as
well, and then the user can perform the comparison themselves and see
if all these claims hold.

Myint, W. & Ishima, R. Chemical exchange effects during refocusing pulses in constant-time CPMG relaxation dispersion experiments. J Biomol Nmr 45, 207-216, (2009).

also:

Bain, A. D., Kumar Anand, C. & Nie, Z. Exact solution of the CPMG pulse sequence with phase variation down the echo train: application to R(2) measurements. J Magn Reson 209, 183- 194, (2011).


Or just simulate the off-resonance effects yourself to see what happens. For NH you see the effects clearly for glycines and side chains, if the carrier is in the centre around 120ppm. The problem generally gets worse the higher field you go though this of course depends when you bought your probe. You seem to imply that these effects are almost mythical. I assure you that they come out happily from the Bloch equations.

Out of curiosity, from a philosophical perspective, I wonder if you'd agree with this statement:

"the expected deviations due to approximations in a model should be lower than the experimental errors on each datapoint."

Anyway, the warnings about analytic versers numeric are described in
the manual.  But your new model which adds a new factor to the CR72
model, just as Dimitry Korzhnev's cpmg_fit software does for his
multiple quantum extension of the equation (from his 2004 and 2005
papers), sounds like it removes the major differences between the
analytic and numeric results anyway.  In any case, I have never seen
an analytic result which is more than 10% different in parameter
values (kex specifically) compared to the numeric results.  I am
constantly on the lookout for a real or synthetic data set to add to
relax to prove this wrong though.

I think there's a misunderstanding in what I mean by numerical modelling. For the 2x2 matrix (basis: I+, ground and excited) from the Bloch McConnell equation, you can manipulate this to get an exact solution. Nikolai's approach does this, though his algorithm can trip up sometimes for relatively exotic parameters when you use doubles (see attached). My formula also does this. I agree with you entirely that in this instance, numerically solving the equations via many matrix exponentials is irrelevant as you'll get an identical result to using a formula.

My central thesis here though is that to get an accurate picture of the experiment you need more physics. This means a bigger basis. To have off resonance effects, you need a minimum 6x6 (Ix,Iy,Iz, ground and excited). To include scalar coupling and off resonance, you need a 12x12 (Ix,Iy,Iz,IxHz,IyHz,IzHz, ground and excited). Including R1 means another element and so on. The methyl group, for example, means you need 24.

So when we use the term numerical treatment, we generally mean a calculation in a larger basis, as is necessary to take into account the appropriate spin physics. There aren't formulas to deal with these situations. In the 6x6 case for example, you need 6 Eigenvalues, which is going to make life very rough for someone brave enough to attempt a close form solution. The Palmer and Trott trick used in 2002 for R1rho is a smart way of ducking the problem of having 6 Eigenvalues, but for CPMG unfortunately you need several Eigenvalues, not just the smallest.

The 2x2 matrix that Nikolai and Martin, Carver Richard's and I analyse does not include scalar coupling, as magnetisation is held in-phase (in addition to neglecting all the other stuff detailed above). So it is a good representation for describing the continuous wave in-phase experiments introduced here (neglecting relaxation effects and off resonance):

Vallurupalli, P.; Hansen, D. F.; Stollar, E. J.; Meirovitch, E.; Kay, L. E.
Proc. Natl. Acad. Sci. U.S.A. 2007, 104, 18473–18477.

and here:

Baldwin, A. J.; Hansen, D. F.; Vallurupalli, P.; Kay, L. E. J. Am. Chem.
Soc. 2009, 131, 11939–48.

But I think are the only two where this formula is directly applicable. Only if you have explicit decoupling during the CPMG period do you satisfy this condition. So this is not the case for all other experiments and certainly not true for those used most commonly.

Anyhow. Best of luck with the software. I would recommend that you consider implementing these effects and have a look at some of the references. The physics are fairly complex, but the implementations are relatively straightforward and amount to taking many matrix exponentials. If you do this, I think you'd end up with a solution that really is world-leading.

As it stands though, in your position, I would worry that on occasion, users will end up getting slightly wrong parameters out from your code by neglecting these effects. If a user trusts this code then, in turn, they might lead themselves to dodgy biological conclusions. For the time being, I'll stick to forcing my students to code things up themselves.

All best wishes,

Andy.

Attachment: 240514 exchange14_supp.pdf
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