Hi,
The following is a copy of my post to the NESSY mailing list at
https://mail.gna.org/public/nessy-users/2014-03/msg00001.html. This
replicated here on the relax-users mailing list to be used as a
permanent relax reference (parts of it might be copied into the relax
wiki or manual in the future). If you are interested in relaxation
dispersion, you might find the concepts useful for understanding the
Hertz vs. 1/s vs. rad/s vs. radian Hertz base units, especially for
the kex exchange parameter. Below is the verbatim copy of the text.
Regards,
Edward
P. S. Here is the text with a few minor corrections:
Hi Henriette,
Welcome to the NESSY mailing lists! The problem you are seeing here
is not very well understood - even by the top experts in the field.
Many people are confused by the concept and it often results in
mistakes in papers. To explain this, I will use the following
resources which I recommend that you read:
1) The relaxation dispersion chapter of the user manual for the
software relax (http://www.nmr-relax.com) has a comprehensive table
listing all conceivable dispersion parameters and their base units
(http://download.gna.org/relax/manual/relax.pdf). An online version
of the table can be seen at
http://www.nmr-relax.com/manual/Dispersion_model_summary.html.
2) The hidden radian unit concept which I have explained at
http://wiki.nmr-relax.com/Hidden_radian_units.
3) The unit analysis for the relaxation equations at
http://thread.gmane.org/gmane.science.nmr.relax.devel/1023/focus=1034.
A similar unit analysis can be performed for kex, which I have
performed but not published or written anywhere. This analysis works
for all models showing kex to have rad/s units, except for the IT99
model (http://wiki.nmr-relax.com/IT99).
4) The software comparison table I produced for relax which compares
numerical results from most of the dispersion software available in
the field. This is available with relax in the file
./test_suite/shared_data/
dispersion/software_comparison or online at
http://svn.gna.org/viewcvs/*checkout*/relax/trunk/test_suite/shared_data/dispersion/software_comparison?revision=21873.
So, to begin, rad/s and 1/s are almost always the same thing in NMR as
almost everything we look at is rotational. This is due to the hidden
radian unit concept (ref. 2). Note that 1/s and Hz in NMR are most
definitely not the same thing and you need a factor of 2pi to convert
between them. Just as a side note, 1/s is properly called radian
Hertz though this is not often used in the field. By convention,
parameters with rad/s units are often called omega whereas parameter
with Hertz units are called nu. Chemical exchange relates to chemical
shift, which is the change in how a nucleus spins in a magnetic field.
The spinning means that chemical shift is a rotational process (which
interacts with the non-rotational external magnetic field and induced
magnetic fields for the electrons in the system, though these have
rotational fluctuations in the structural reference frame due to
Brownian and internal motions). The Rex term which contributes to
relaxation hence has the units of rad/s. This matches the units of
R1, R2, and sigma_NOE (see ref. 3), as well as the spectral density
functions governing standard relaxation.
In the case of kex, this is where a lot of confusion arises. The
reason is because kex has both rad/s and Hertz units - there are
actually two kex parameters. The first is the one which influences
the chemical shift - this has the radian unit. The second is the
translational component of the mechanical process causing exchange -
the kinetic parameter of the process - this is non-rotational and
hence has Hertz units. There is no factor of 2pi when converting
between the two! A rotational process in rad/s or radian Hertz can
interact directly with a non-rotational process in Hertz with no
conversion factors. For relaxation dispersion we assume both kex are
the same. Therefore it is technically correct to present the single
numeric value with both 1/s and Hertz units (as well as rad/s and
radian Hertz units). There is a case where kex can have rad^2 units
but I won't get into that (the rad^2/s units are mentioned at
https://en.wikipedia.org/wiki/Rotational_diffusion).
For a quick unit analysis, the CR72 dispersion model for CPMG-type
data is a simple starting point (http://wiki.nmr-relax.com/CR72).
From section 10.3.3 of the relax manual:
R2eff = 1/2 (R20A + R02B + kex - 2*nu_CPMG*cosh^-1 (D+*cosh(eta+) -
D-*cos(eta-))).
The units of R2eff, R20A, and R20B are by definition rad/s. Therefore
to add kex directly to these, kex must also have rad/s units (unit
analysis of the last term also produces rad/s, but I won't go into
that as it's too long). For the M61 model for R1rho-type data
(http://wiki.nmr-relax.com/M61 and section 10.7.1 of the relax
manual):
R1rho = R1rho' + phi_ex*kex / (kex^2 + omega_e^2),
where:
phi_ex = pA*pB*delta_omega^2.
Here both phi_ex and omega_e have the units of rad^2/s^2. To add
kex^2 to omega_e^2, it must also have rad^2/s^2 units. The
denominator therefore has rad^2/s^2 units. The numerator has
(rad^2/s^2 * rad/s) units, or rad^3/s^3 units. Dividing the numerator
and denominator, we have (rad^3/s^3 / (rad^2/s^2)). This can be seen
to cancel to rad/s units. This matches the base units for R1rho' and
R1rho which are also rad/s units by definition. Such a unit analysis
can be performed for all analytic and numeric dispersion models and
kex will always have rad/s units (IT99 is a bizarre exception).
I hope some of this information helps. But if you look at ref. 4, you
will see all dispersion software compared produce the same numerical
kex value. There are no errors in NESSY, relax, CPMGFit, cpmg_fit, or
ShereKhan due to missing factors of 2pi. There are errors/differences
due to other problems, but that's a different story.
Regards,
Edward
P. S. For a more detailed comparison of different relaxation
dispersion software, see section 10.12 "Comparison of dispersion
analysis software" of the relax manual
(http://download.gna.org/relax/manual/relax.pdf).
On 26 March 2014 09:49, Henriette Molinari <henriette.molinari@xxxxxxxx>
wrote:
Dear Nessy authors,
we have an enormous and important doubt.
Looking at your equations and to the supplementary files found in your
"help" menu,
it appears that kex units, deriving from NESSY fitting analysis, are rad/sec
AND NOT 1/sec.
This means that all the extracted kex should be multiplied by 2pi.
Could you confirm this point?
It is crucial as we are publishing a review comparing different kex values
reported in the literature
for a class of protein, and this point is extremely important.
I apologise if my question is trivial, however I find literature very
confusing on this subject
Thank you so much
With best regards
Henriette Molinari
Prof. Henriette Molinari
ISMAC CNR Milano, Via Bassini 15,
20133 Milano
cell. 3290594246 phone 0223699724
President GIDRM
The hidden radian unit in relaxation dispersion.