Dear Troels,
The existence of typos is exceedingly probable! The editors have added more
errors in the transcrpition, so it'll be curious to see how accurate the
final draft will be. Your comments come just in time for alterations.
So I'll pay a pint of beer for every identified error. I didn't use that
particular supplementary section to directly calculate anything, so this is
the place in the manu where errors are most likely to occur. I do note the
irony that this is probably one of few bits in the article that people are
likely to read.
1) Yep - that's a typo. alpha- should definitely be KEG-KGE (score 1 pint).
2) Yep! zs should be zetas (score 1 pint).
3) This one is more interesting. Carver and Richard's didn't define a
deltaR2. To my knowledge, I think Nikolai was the first to do that in his
2002 paper on reconstructing invisible states. So here, I think you'll find
that my definition is the correct one. There is a typo in the Carver
Richard's equation. I've just done a brief lit review, and as far as I can
see, this mistake has propagated into every transcription of the equation
since. I should note that in the manu in this section. Likely this has gone
un-noticed as differences in R2 are difficult to measure by CPMG, instead
R2A=R2B being assumed. Though thanks to Flemming, it is now possible to
break this assumption:
J Am Chem Soc. 2009 Sep 9;131(35):12745-54. doi: 10.1021/ja903897e.
Also, in the Ishima and Torchia paper looking at the effects of relaxation
differences:
Journal of biomolecular NMR. 2006 34(4):209-19
I understand their derivations started from Nikolai's formula, which is
exact, so the error wouldn't come to light there either, necessarily.
To prove the point try this: numerically evaluate R2,effs over a big grid of
parameters, and compare say Nikolai's formula to your implementation of
Carver Richard's with a +deltaR2 and a -deltaR2 in the alpha- parameter.
Which agrees better? You'll see that dR2=R2E-R2G being positive wins.
The attached should also help. It's some test code I used for the paper that
has been briefly sanitised. In it, the numerical result, Carver Richards,
Meiboom, Nikolai's and my formula are directly compared for speed (roughly)
and precision.
The code added for Nikolai's formula at higher precision was contributed by
Nikolai after the two of us discussed some very odd results that can come
from his algorithm. Turns out evaluating his algorithm at double precision
(roughly 9 decimal places, default in python and matlab) is insuffficient.
About 23 decimal places is necessary for exactness. This is what is
discussed in the supplementary info in the recent paper. In short, on my
laptop:
formula / relative speed / biggest single point error in s-1 over grid
Mine 1 7.2E-10
Meiboom 0.3 9E7
Carver Richards 0.53 8
Nikolai (double precision 9dp) 3.1 204
Nikolai (long double 18dp) 420 3E-2
Nikolai (23 dp) 420 2E-7
Numerical 118 0
The error in Carver Richard's with the alpha- sign made consistent with the
Carver Richard's paper gives an error of 4989 s-1 in the same test. Ie, it's
a bad idea.
Note that the precise calculations in python at arbitrary precision are slow
not because of the algorithm, but because of the mpmath module, which deals
with C code via strings. Accurate but slow. This code has a small amount of
additional overhead that shouldn't strictly be included in the time. I've no
doubt also that algorithms here could be sped up further by giving them a
bit of love. Those prefaces aside, I'm surprised that your benchmarking last
Friday showed Nikolai's to be so much slower than mine? In my hands
(attached) at double precision, Nikolai's formula is only 3x slower than
mine using numpy. Though this is not adequate precision for mistake free
calculations.
Perhaps more physically compelling, to get my equation to reduce to Carver
Richard's, alpha- has to be defined as I show it. Note that the origin of
this parameter is its role in the Eigenvalue of the 2x2 matrix. This is the
one advantage of a derivation - you can see where the parameters come from
and get a feel for what their meaning is.
So this typo, and the fact that Carver and Richards inadvertently change
their definition of tau_cpmg half way through the paper, are both errors in
the original manuscript. Though given that proof reading and type-setting in
1972 would have been very very painful, that there are this few errors is
still actually quite remarkable. I notice in your CR72 code, you are
currently using the incorrect definition, so I would suggest you change
that.
So overall, I think you're still ahead, so I'm calling that 2 pints to
Troels.
I would actually be very grateful if you can bring up any other
inconsistencies in the paper! Also from an editorial perspective, please
feel free to note if there are parts of the paper that were particularly
unclear when reading. If you let me know, there's still time to improve it.
Thusfar, I think you are the third person to read this paper (the other two
being the reviewers who were forced to read it), and you're precisely its
target demographic!
Best,
Andy.
On 03/05/2014 13:54, Troels Emtekær Linnet wrote:
Dear Andrew Baldwin.
I am in the process of implementing your code in relax.
I am getting myself into a terrible mess by trying to compare
equations in papers, code and conversions.
But I hope you maybe have a little time to clear something up?
I am looking at the current version of your paper:
dx.doi.org/10.1016/j.jmr.2014.02.023
which still is still "early version of the manuscript. The manuscript
will undergo copyediting, typesetting, and review of the resulting
proof before it is published in its final form."
1) Equations are different ?
In Appendix 1 – recipe for exact calculation of R2,eff
h1 = 2 dw (dR2 + kEG - kGE)
But when I compare to:
Supplementary Section 4. Relation to Carver Richards equation. Equation 70.
h1 = zeta = 2 dw alpha_ = 2 dw (dR2 + kGE - kEG)
Is there a typo here? The GE and EG has been swapped.
2) Missing zeta in equation 70
There is missing \zeta instead of z in equation 70, which is probably
a typesetting error.
3) Definition of " Delta R2" is opposite convention?
Near equation 10, 11 you define: delta R2 = R2e - R2g
if we let e=B and g=A, then delta R2 = R2B - R2A
And in equation 70:
alpha_ = delta R2 + kGE - kEG
but i CR original work, A8,
alpha_ = R2A - R2B + kA - kB = - delta R2 + kGE - kEG
So, that doesn't match?
Sorry if these questions are trivial.
But I hope you can help clear them up for me. :-)
Best
Troels
2014-05-01 10:07 GMT+02:00 Andrew Baldwin <andrew.baldwin@xxxxxxxxxxxxx>:
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.
Re: [sr #3154] Implementation of Baldwin (2014) B14 model - 2-site exact solution model for all time scales