QR-DOSY

At the beginning of the month we saw that there are cases where the diffusion coefficients can be measured with the same mathematical tools used to measure relaxation. We have seen DOSY spectra of pure compounds where each signal decays as a pure exponential. Even a mixture can behave in the same way, if the signals don't overlap. In summary, calculating the diffusion coefficients is often easy.
What I was curious to discover was: is it possible to recalculate the components of a mixture if the diffusion coeffients are known? From a pure mathematical point of view the answer was already: "yes", but I wanted to verify it in practice.
As far as I know, something like: "QR-DOSY" has never been mentioned. So many DOSY methods already exist and I don't mind to increase the Babel with yet another acronym. Do you?

THEORY
If we know the diffusion coefficient D(j) we can calculate that the NMR signal in the spectrum i will be proportional to a value A(i,j) = exp(-D(j)F(i)) where F is a function of the gyromagnetic ratio, the gradient strength, the diffusion delay, etc.. but not a function of the chemical shift nor of the diffusion coefficient.
For each column of the DOSY spectrum we have a system of equations: Ax = b.
x = intensity of the spectrum of the pure compounds at the chemical shift that coresponds to the given column.
b = intensity of the DOSY along the same column.
We know A and b, therefore we can calculate x. A is the same for all the columns and this is a great advantage. We can apply a well known decomposition: Rx = Qb.
Calculating Q and R from A takes time, but we need to do it only once. Then then computer can swiftly solve all the systems in the form Rx = Qb.

METHODS
Obviously, we will find that some values of the x will be negative. In such a case, we can choose a subset of A (omitting the component with negative intensity) and solve the reduced problem. This simplifying mechanism can be applied iteratively, even when the value of x is positive yet small.

RESULTS

This is the same old spectrum we are familiar with, processed with the new QR-DOSY.

DISCUSSION
Two components are completely separated. The third component is not, although at this point it becomes easy to recognize its peaks. Other cases I have studied yield similar results, maybe not as nice. Advantages of the QR-DOSY method:
- easy to understand;
- easy to use WITH THE ASSISTANCE of a software for the book-keeping activity (like measuring the diffusion coefficients);
- fast;
- the user can play with a few parameters, trying to improve the results;
- the final spectra are clean from artifacts.
Cons:
- not all the components are always resolved;
- it's a problem if two diffusion coefficients are similar (of course the program itself can easily detect this circumstance).

Jacek Stawinski

Q. What's your position and where are you working?
A. I am Professor of Organic Chemistry at Stockholm University, Stockholm, Sweden, and at the Institute of Bioorganic Chemistry, Polish Academy of Science, Poznan, Poland.
Q. Where have you been working before?
A. Adam Mickiewicz University, Poznan, Poland, and Institute of Organic Chemistry, Polish Academy of Science, Warsaw, Poland
Q. Briefly describe your research.
A. My field of expertise is bioorganic phosphorus chemistry, nucleic acids chemistry, and lipid and phospholipid chemistry (http://www.organ.su.se/js).
Q. What do you use NMR for?
A. Characterization of synthetic intermediates, structure determination, spin simulations, NMR dynamic processes.
Q. Which NMR software are you using now?
iNMR, the latest version.
Q. Which other NMR software have you used in the past?
A. Swan NMR, Topspin, MestreC, MNOVA
Q. How do you rate iNMR?
A. iNMR is superior, by far, to all NMR software I have used. It provides a powerful, intuitive and professional environment for processing and plotting NMR data. The software is very fast, has scripting ability, and a lot of keyboard shortcuts and useful extras. No doubts, iNMR is right on the cutting edge on the NMR processing software development. Due to simple interface, iNMR is a user friendly application, but it hides a lot of powerful tools for advanced tasks. And last, but not least, the support from the programmer is prompt, competent, and friendly.
Q. Is it enough for your needs?
A. I have never faced a situation when iNMR could not do, what other software can. For me, it is the tool of choice for dynamic NMR. Also very useful during teaching NMR courses.

Bull's-Eyes

Here are some caffeine peaks (below) and EthoxyEthanol peaks (above). The display is DOSY-like, yet the numerical treatment is a simpler and faster mono-exponential fit. Each column has been processed independently from the rest, and each column yields a different result:

Now let's apply a Whittaker Smoother, with a small value of lambda (100), along each ROW:

or a big lambda (500):

or a huge value (8000):

The smoother averages the results obtained from the different columns. The peaks are perfectly aligned.

Wandering

Jean Marc Nuzillard writes:

The example that is provided by Carlos in his blog would benefit from a processing trick I use when there is no multiplet superimposition:
I simply integrate the multiplets that are recorded at different gradient intensities and I perform a monoexponential fit on integral values.
I suspect the noise on integral values is lower than the one in individual columns of the 1D spectra set, thus making D values more accurate.
Bye bye butterflies. This idea is absolutely trivial but maybe it would be interesting
to implement it.

I agree with the idea. Actually I had arrived at the same conclusion for a different reason. Look at this spectrum (a DOSY after FT):
The signal/noise ratio is low yet acceptable, the compound is pure, the decay mono-exponential. No concern about phase and baseline. Alas, this peak is too nasty for my tastes. The frequency is not constant over time. The "trivial" trick by Jean-Marc should work. I give my preference to binning, because the output of binning is not a numerical table (like with integration) but a new spectrum, so the same NMR program can be used for the subsequent exponential fit.
Before Carlos corrects me, let me stress a few details. We have processed the same experiment with two different algorithms. I got the butterflies and I am not proud of it. Carlos' pictures are a little smaller and cannot be compared:
(taken from his blog). Anyway, we are really confusing the matter here. From what I understand, the purpose of Carlos was to show how Bayesian DOSY can effectively separate the components of a mixture. I can't express any opinion, because as I said I only have two experiments to work with. In both cases there is no superposition of peaks, therefore there is nothing to separate. I normally like simple examples like these, yet I acknowledge they are not enough.
Today I could repeat the same processing of Carlos, because he himself has very kindly given me both the raw data and the software. But 1) I am not terribly interested into this comparison 2) He doesn't give me these things for free for me to criticize him 3) If I really want to do such a thing I'll post a comment directly on his blog.
Now let's go on: ho to make the butterflies go away? The next post shows how to transform a butterfly into bull's-eyes.

Pictures

In my fourth year of blogging I have started publishing pictures of spectra. I like my pictures, yet this is not the point. I don't want to convince you that my pictures are beautiful. I want to convince you that I am a spectroscopist and not a programmer.
The programmers have always stated that "in the near future" it will be possible to obtain a completely automatic analysis: from the sample directly to the response (meaning a chemical formula or a list of values), by-passing the plotted spectrum. I have found the same concept and expression "near future", scattered in the literature of all the decades, from the 60s onward.
I really believe it: someday in the future we'll arrive at the completely automatic analysis.
Being that, at this writing time, I am working as a programmer and not as a spectroscopist, I should adhere to this belief and be happy. It happens, instead, that I always think like a spectroscopist scared of remaining unemployed. I really hate to design and write programs for automatic processing and reporting. I like creating programs to display the spectra.
Here comes the difference between "diffusion" and "DOSY". The mere idea of DOSY is to make a troble-free program: push this button and you'll have everything, the diffusion coefficients and the individual components of the mixture. This is what I have understood up to now. In my whole life I have only worked with 2 DOSY experiments, which have not been carried out by me. I have already shown the first one and today I am going to show the second one (please wait a few minutes..).

Ghost of a Butterfly

Sometimes the butterfly disappears when the picture is scaled down, like into this web page. Click it!

Butterflies

Carlos kindly gave me a copy of the DOSY spectrum he showed on his blog last year.
I am experimenting alternative processing routes. Here is a detail of the caffeine peaks, after applying the rudest (and probably simplest) treatment. The decays have been linearized (by taking the logarithm). The slope of the line is proportional to the diffusion coefficient. The final results are reported as a normal DOSY spectrum.
Click on the thumbnail to see the image at natural scale.

There is less signal/noise in the tails of the peaks, obviously, therefore the error increases: graphically we see the wings of a butterfly.
Being that it's impossible to correct the phase perfectly, some butterflies are asymmetric.