Accelerated acquisition in pure-shift spectra based …Accelerated acquisition in pure-shift spectra...

Accelerated acquisition in pure-shift spectra based on priorknowledge from 1H NMR

Electronic Supplementary Information

Alexandra Shchukina,a Magdalena Kazmierczak,b Paweł Kasprzak, a,c MatthewDavy, d Geoffrey Akien, e Craig P. Butts, d and Krzysztof Kazimierczuk∗a

Fig.1, top panel, presents a burst sampling schedule example: ones denoteacquiring measurement points, and zeros denote their omission. Unlike conven-tional non-uniform sampling in NMR, not separate measurement points, but theirchunks are either acquired or omitted. Fig.1, bottom panel, is the sampling sched-ule Fourier transform, or the point-spread function. When the sampling sched-ule is applied in an experiment, the spectrum is convolved with the point-spreadfunction as a result of the undersampling. In Fig.2, two simulated spectra arepresented. Fully “measured” ones are plotted with a red line, and the undersam-pled ones are plotted with a blue line. One of the spectra (top panel) yields thena strong artefact, while the other (bottom panel) does not. Thus, a “good” or a“bad” schedule is spectrum-specific.

We measured 1H and full pure-shift spectra of the following samples: α-asarone,(R)-(+)-limonene, L-menthol, α-pinene, quinine in CDCl3; D-glucose and sucrosein D2O; β -estradiol in CD3OD; all 50mM solutions; and a mixture of 0.5M (R)-(+)-limonene and 50mM α-pinene in CDCl3. We introduced the mixture sampleto have an example with a large difference in peak heights.

We acquired the spectra with the standard PSYCHE pulse sequence on Varian700MHz spectrometer equipped with HCN room-temperature probe. We set the“tau delay” parameter corresponding to the length of the chunk to 4 ms (the stan-dard one is 8 ms) to ensure better decoupling. It also required a change of severalother parameters for the sequence to function properly: rof1, rof2, gstab and gt1.

The raw data along with the “procpar” files with the acquisition parameterscan be downloaded from http://nmr.cent.uw.edu.pl/downloads/, sec-tion "Pure Shift" −> "Pseudo-2D pure shift sampling optimizer".

For each 1H spectrum, we performed automated multiplet analysis and auto-mated peak picking with MestreNova software (11.0 version). The results are in“multiplets1.txt” and “peaks1.txt” files in the corresponding folders. The peak-picking procedure was carried out to have peak widths at our disposal.

We used the following python script to find the best and the worst samplingsaccording to two different criteria: the maximum criterion (corresponding to the

a Centre of New Technologies, University of Warsaw, Banacha 2C, 02-097 Warsaw, Poland.b Faculty of Chemistry, Warsaw University of Technology, Noakowskiego 3, 00-664 Warsaw, Poland.c Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland.d School of Chemistry, University of Bristol, Cantock’s Close, Clifton, BS8 1TS Bristol, United Kingdom.e Department of Chemistry, Lancaster University, Lancaster, LA1 4YB, UK

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Electronic Supplementary Material (ESI) for ChemComm.This journal is © The Royal Society of Chemistry 2019

http://nmr.cent.uw.edu.pl/downloads/

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Figure 1 A simulated burst sampling schedule (top panel) and its point-spread function (bottom panel).

`∞ norm of the reconstruction procedure input, that is why the name “Linf” isused) and the sum criterion (“L1”, as corresponding to the `1 norm). The samescript then artificially undersamples the pure-shift data according to the “best”and “worst” samplings of both criteria and then performs the reconstruction. Inpractical applications, the user will only need the sampling schedules. It is statedin the script which part of it should be omitted in this case.

#!/usr/bin/python

import numpy as npimport matplotlib.pyplot as pltimport nmrglue as ngimport mathimport osimport itertoolsimport scipyimport time

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Figure 2 Two simulated spectra (red lines, divided by 4) and the results of their undersampling (blue lines,original scale) with the sampling schedule given in Fig.1. Both spectra are undersampled with the sameschedule, but the one in the top panel yields a strong artefact, while the one in the bottom panel does not.

import struct

###### General parameters and Mnova data importfactor = 0.2 # proportion of chunks measuredsamp_limit = int(1e4)

multiplets = np.loadtxt(’multiplets1.txt’)shift_ppm = multiplets[:, 0]a = multiplets[:, 1]

peakwidths = np.loadtxt(’peaks1.txt’)fwhm_Hz = np.mean(peakwidths)*math.pi

###### PSYCHE parametersdirs = os.listdir(os.getcwd())for i in range(len(dirs)):

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if "PSYCHE" in dirs[i]:psyche_dir = os.getcwd() + ’/’ + dirs[i]break

dic, data = ng.varian.read(dir=psyche_dir, fid_file=’fid’,procpar_file=’procpar’)

dic1 = ng.varian.read_procpar(psyche_dir+’/procpar’)npo = int(dic1[’np’][’values’][0])/2at = float(dic1[’at’][’values’][0])reffrq = float(dic1[’reffrq’][’values’][0])sfrq = float(dic1[’sfrq’][’values’][0])sw = float(dic1[’sw’][’values’][0])sw1 = float(dic1[’sw1’][’values’][0])rp = float(dic1[’rp’][’values’][0])lp = float(dic1[’lp’][’values’][0])

D = len(data)data = data*np.exp(-1j*rp*math.pi/180) # p0 phasingdata = np.flipud(np.fft.fftshift(np.fft.fft(data)))data = data*np.exp(-1j*np.arange(D)*lp*math.pi/(D*180)) # p1 phasingdata = np.fft.ifft(np.flipud(np.fft.fftshift(data)))

######chunk = int(math.floor(sw/sw1 + 0.5))print ’\nChunk = ’, chunk, ’points’t = np.linspace(0, at, npo)shift_Hz = (sfrq -(shift_ppm*1e-6*reffrq + reffrq))*1e6shift = np.round(shift_Hz*npo/sw + npo/2)fwhm_t = fwhm_Hz*npo/swfwhm = fwhm_t*at/npo

###### Simulating singlet spectrumfid = np.zeros((npo,), dtype=complex)n = np.linspace(0, npo-1, npo);for i in range(len(a)):

fid = fid + a[i]*np.exp((2*math.pi*1j*shift[i]/npo - fwhm)*n)s = np.fft.fft(fid)peak_for_normalization = np.argmax(np.abs(s))s = s/np.abs(s[peak_for_normalization])

###### Generating samplingsfull_number_of_chunks = int(math.floor(npo/chunk))number_of_measured_chunks = int(math.floor(full_number_of_chunks*factor))print ’Full number of chunks -’, full_number_of_chunks, ’, number of

measured chunks -’, number_of_measured_chunksnumber_of_samplings =

int(math.floor(scipy.misc.comb(full_number_of_chunks-1,number_of_measured_chunks-1)))

print ’Number of all samplings: ’, number_of_samplings

print ’\nGenerating sampling schedules...’start = time.time()np.random.seed(1)if number_of_samplings > samp_limit:

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print ’Using randomly selected samplings!’Chunks_measured = np.ones((samp_limit+2, number_of_measured_chunks))for i in range(samp_limit):

Chunks_measured[i, 1:] = np.sort(np.random.permutation(np.arange(2,full_number_of_chunks)) \[:number_of_measured_chunks - 1])

Chunks_measured[samp_limit, :] = np.linspace(1,number_of_measured_chunks, number_of_measured_chunks) # truncation

Chunks_measured[samp_limit+1, :] = Chunks_measured[samp_limit,:]*math.floor(1/factor) - \(math.floor(1/factor) - 1) # uniform distribution

else:print ’Using all samplings!’C = np.array(list(itertools.combinations(np.linspace(2,

full_number_of_chunks, full_number_of_chunks-1), \number_of_measured_chunks-1)))

Chunks_measured = np.hstack((np.ones((np.shape(C)[0], 1)), C))Chunks_measured = Chunks_measured.astype(int)print ’\nSamplings: ’, Chunks_measured, np.shape(Chunks_measured)

###### Comparing samplingss_nus = np.zeros((np.shape(Chunks_measured)[0], len(s)), dtype=complex)max_artefact = np.zeros((np.shape(Chunks_measured)[0],))L1 = np.zeros((np.shape(Chunks_measured)[0],))L2 = np.zeros((np.shape(Chunks_measured)[0],))

if os.path.isfile(’s_nus.npy’):s_nus = np.load(’s_nus.npy’)for j in range(np.shape(Chunks_measured)[0]):

max_artefact[j] = np.max(np.abs(s_nus[j, :]-s))L1[j] = np.sum(np.abs(s_nus[j, :]-s))

else:for j in range(np.shape(Chunks_measured)[0]):

a = Chunks_measured[j, :]fid_nus = np.zeros(np.shape(fid), dtype=complex)for k in range(np.shape(Chunks_measured)[1]):

fid_nus[(a[k]-1)*chunk:a[k]*chunk] = fid[(a[k]-1)*chunk:a[k]*chunk]s_nus[j, :] = np.fft.fft(fid_nus)s_nus[j, :] = s_nus[j, :]/np.abs(s_nus[j, peak_for_normalization])

max_artefact[j] = np.max(np.abs(s_nus[j, :]-s))L1[j] = np.sum(np.abs(s_nus[j, :]-s))

end = time.time()print ’done!’print ’Time: ’, end - start, ’s’

criterion = {}criterion[’Linf’] = max_artefactcriterion[’L1’] = L1

main_folder = os.getcwd()

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def best_and_worst_samplings(c, bw):global sos.chdir(main_folder)if bw == ’best’:

C = np.argmin(criterion[c])if bw == ’worst’:

C = np.argmax(criterion[c])selected_spectrum = s_nus[C, :]selected_fid = np.fft.ifft(s_nus[C, :])*np.abs(s_nus[C,

peak_for_normalization])selected_sampling = Chunks_measured[C, :]print selected_sampling

print ’Saving figures of simulated FIDs...’plt.figure()plt.plot(t, np.real(selected_fid))plt.title(bw + ’ sampling sampling - ’ + c + ’ criterion (simulated FID)’)plt.savefig(bw + ’_’ + c + ’_FID.eps’)

plt.figure()plt.plot(np.abs(selected_spectrum))plt.plot(np.abs(s))plt.title(bw + ’ sampling - ’ + c + ’ criterion \n(simulated spectrum

with artefacts)’)plt.savefig(bw + ’_’ + c + ’_spec0.eps’)print ’done!’

### From this point forward, the script performs the artificialundersampling of the fully measured data and the subsequentreconstruction. In practical applications, the following section ofthe function should be commented out for the script to yield only thesampling schedules ###

print ’Preparing files for reconstruction...’folder_r = main_folder + ’/’ + cif not os.path.exists(folder_r):

os.makedirs(folder_r)os.chdir(folder_r)

folder_r_bw = folder_r + ’/’ + c + ’_’ + bwif not os.path.exists(folder_r_bw):

os.makedirs(folder_r_bw)os.chdir(folder_r_bw)

np.savetxt(bw + ’_sampling.txt’, selected_sampling, fmt=’%d’)

folder_r_bw0 = folder_r_bw + ’/0th_iteration_’ + c + ’_’ + bwif not os.path.exists(folder_r_bw0):

os.makedirs(folder_r_bw0)ng.fileio.varian.write(folder_r_bw0, dic, selected_fid, overwrite=True)

I = len(selected_sampling)*chunkind = np.zeros(I,)for k in range(len(selected_sampling)):

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ind[chunk*k:chunk*(k+1)] = np.arange((selected_sampling[k] - 1)*chunk,selected_sampling[k]*chunk, 1)

ind = ind.astype(int)

k = 0ind_alternated = np.zeros(2*I)for i in range(I):

ind_alternated[k] = 2*ind[i]ind_alternated[k + 1] = ind_alternated[k] + 1k = k + 2

ind_alternated = ind_alternated.astype(int)

regions = 1dimensions = 2line1 = str(dimensions) + ’ 1 ’ + str(regions*2*I) + ’\n’line2 = str(regions) + ’ ’ + str(2*D)s1 = ’’for i in range(2*I):

s1 = s1 + ’\n’ + str(0) + ’ ’ + str(ind_alternated[i])

folder = folder_r_bw + ’/MDD’if not os.path.exists(folder):

os.makedirs(folder)os.chdir(folder)

fid_alternated = np.empty(2*I)fid_alternated[0::2] = np.real(data[ind])fid_alternated[1::2] = np.imag(data[ind])f = ’’for j in range(2*I):

f = f + ’\n’ + str(fid_alternated[j])contents = ’mdd asc sparse f180.0 \n ./MDD/region01.mdd \n MDD sparse\n $

\n’ + line1 + line2 + s1 + f # contains the data to be written to .mddfile

name = str(1)file = open(name + ’.mdd’, ’w’)file.write(contents)file.close()

command = ’cssolver ’ + name + ’ CS_alg=IST CS_niter=300 CS_VE=nMDD_NOISE=1 > ./’ + name + ’.log’

command = "tcsh -c ’" + command + "’"os.system(command) # executes .mdd fileprint ’done!’

print ’Reading reconstruction results...’file = open(name + ’.cs’, ’rb’)cs = np.empty(2*D)for j in range(2*D):

cs[j] = float(struct.unpack(’f’, file.read(4))[0])file.close()fid_rec = cs[0::2].astype(np.float32) + 1j*cs[1::2].astype(np.float32)print ’done!’

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print ’Writing reconstructed FIDs...’ng.fileio.varian.write(folder_r_bw, dic, fid_rec, overwrite=True)print ’done!’

return None

bw = [’best’, ’worst’]c = [’Linf’, ’L1’]for i in c:

for j in bw:print ’\n’, j, ’sampling for’, i, ’criterion’best_and_worst_samplings(i, j)

plt.show()

The script yields:

• the best and worst samplings for both criteria (text files and figures)• the reconstrcution results for the best and worst samplings for both criteria

(raw spectrometer output format).

The comparisons for the best and worst samplings for each criterion (the recon-struction results), as well the comparisons of the different criteria best results, areshown below for each sample. The corresponding samplings are presented as wellin the form of the convolution with the simulated spectrum of singlets (step 3 ofthe workflow described in the main text).

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Max criterion

Worst sampling

Best sampling

Fully sampled

4.684.694.704.714.724.73

ppm

1.6251.6501.6751.7001.7251.7501.7751.800

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Figure 3 (R)-(+)-limonene, 50mM in CDCl3. Maximum criterion: reconstructions for the best sampling, theworst sampling and the original fully sampled pure-shift spectrum.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

t , s

Best sampling - Max criterion (simulated FID)

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Worst sampling - Max criterion (simulated FID)

Figure 4 Maximum criterion for (R)-(+)-limonene: best and worst samplings applied to simulated FID ofsinglets (point 3 of the workflow).

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Sum criterion

Worst sampling

Best sampling

Fully sampled

1.6251.6501.6751.7001.7251.7501.7751.800

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5.385.395.405.415.425.435.445.45

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Figure 5 (R)-(+)-limonene, 50mM in CDCl3. Sum criterion: reconstructions for the best sampling, the worstsampling and the original fully sampled pure-shift spectrum.

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t , s

Best sampling - Sum criterion (simulated FID)

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Worst sampling - Sum criterion (simulated FID)

Figure 6 Sum criterion for (R)-(+)-limonene: best and worst samplings applied to simulated FID of singlets(point 3 of the workflow).

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Different criteria - best sam plings

Max criterion

Sum criterion

Fully sampled

1.851.901.952.002.052.102.152.20

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1.601.651.701.751.80

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Figure 7 (R)-(+)-limonene, 50mM in CDCl3. Maximum and sum criterion comparison and the original fullysampled pure-shift spectrum.

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Max criterion

Worst sampling

Best sampling

Fully sampled

3.783.803.823.843.863.883.903.92

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6.466.476.486.496.506.516.526.536.54

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Figure 8 α-asarone, 50mM in CDCl3. Maximum criterion: reconstructions for the best sampling, the worstsampling and the original fully sampled pure-shift spectrum.

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Figure 9 Maximum criterion for α-asarone: best and worst samplings applied to simulated FID of singlets(point 3 of the workflow).

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Sum criterion

Worst sampling

Best sampling

Fully sampled

3.783.803.823.843.863.883.903.92

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L1 criterium

worst sam pling

best sam pling

original

6.466.486.506.526.546.566.58

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Figure 10 α-asarone, 50mM in CDCl3. Sum criterion: reconstructions for the best sampling, the worstsampling and the original fully sampled pure-shift spectrum.

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Figure 11 Sum criterion for α-asarone: best and worst samplings applied to simulated FID of singlets (point3 of the workflow).

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Max criterion

Sum criterion

Fully sampled

3.803.823.843.863.883.90

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5.956.006.056.106.156.20

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Figure 12 α-asarone, 50mM in CDCl3. Maximum and sum criterion comparison and the original fullysampled pure-shift spectrum.

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Max criterion

Worst sampling

Best sampling

Fully sampled

1.921.931.941.951.961.971.981.992.00

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1.561.581.601.621.641.661.681.701.72

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Figure 13 L-menthol, 50mM in CDCl3. Maximum criterion: reconstructions for the best sampling, the worstsampling and the original fully sampled pure-shift spectrum.

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Figure 14 Maximum criterion for L-menthol: best and worst samplings applied to simulated FID of singlets(point 3 of the workflow).

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Sum criterion

Worst sampling

Best sampling

Fully sampled

3.363.383.403.423.443.46

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0.750.800.850.900.951.00

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Figure 15 L-menthol, 50mM in CDCl3. Sum criterion: reconstructions for the best sampling, the worstsampling and the original fully sampled pure-shift spectrum.

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Figure 16 Sum criterion for L-menthol: best and worst samplings applied to simulated FID of singlets (point3 of the workflow).

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Max criterion

Sum criterion

Fully sampled

1.561.581.601.621.641.661.681.70

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0.800.850.900.951.00

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Figure 17 L-menthol, 50mM in CDCl3. Maximum and sum criterion comparison and the original fullysampled pure-shift spectrum.

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Max criterion

Worst sampling

Best sampling

Fully sampled

2.282.302.322.342.362.38

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1.051.101.151.201.251.30

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Figure 18 α-pinene, 50mM in CDCl3. Maximum criterion: reconstructions for the best sampling, the worstsampling and the original fully sampled pure-shift spectrum.

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Figure 19 Maximum criterion for α-pinene: best and worst samplings applied to simulated FID of singlets(point 3 of the workflow).

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Sum criterion

Worst sampling

Best sampling

Fully sampled

5.105.125.145.165.185.205.225.245.26

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1.101.151.201.251.30

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Figure 20 α-pinene, 50mM in CDCl3. Sum criterion: reconstructions for the best sampling, the worstsampling and the original fully sampled pure-shift spectrum.

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Figure 21 Sum criterion for α-pinene: best and worst samplings applied to simulated FID of singlets (point3 of the workflow).

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Max criterion

Sum criterion

Fully sampled

1.92.02.12.22.32.4

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0.8150.8200.8250.8300.8350.840

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Figure 22 α-pinene, 50mM in CDCl3. Maximum and sum criterion comparison and the original fully sam-pled pure-shift spectrum. In the bottom zoomed panel on the right, the results of the maximum criterionfollow the fully sampled spectrum.

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Max criterion

Worst sampling

Best sampling

Fully sampled

1.41.51.61.71.81.9

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5.455.505.555.605.655.705.755.80

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Figure 23 Quinine, 50mM in CDCl3. Maximum criterion: reconstructions for the best sampling, the worstsampling and the original fully sampled pure-shift spectrum.

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Figure 24 Maximum criterion for quinine: best and worst samplings applied to simulated FID of singlets(point 3 of the workflow).

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Sum criterion

Worst sampling

Best sampling

Fully sampled

8.548.568.588.608.628.64

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2.62.72.82.93.03.1

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Figure 25 Quinine, 50mM in CDCl3. Sum criterion: reconstructions for the best sampling, the worst sam-pling and the original fully sampled pure-shift spectrum.

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Figure 26 Sum criterion for quinine: best and worst samplings applied to simulated FID of singlets (point 3of the workflow).

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Max criterion

Sum criterion

Fully sampled

1.41.51.61.71.81.9

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2.622.632.642.652.662.672.682.692.70

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Figure 27 Quinine, 50mM in CDCl3. Maximum and sum criterion comparison and the original fully sampledpure-shift spectrum. In the bottom zoomed panels, the results of the maximum criterion follow the fullysampled spectrum.

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Max criterion

Worst sampling

Best sampling

Fully sampled

4.604.624.644.664.684.704.72

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3.303.353.403.453.503.553.60

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Figure 28 Sucrose, 50mM in D2O. Maximum criterion: reconstructions for the best sampling, the worstsampling and the original fully sampled pure-shift spectrum.

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t , s


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t , s


Figure 29 Maximum criterion for sucrose: best and worst samplings applied to simulated FID of singlets(point 3 of the workflow).

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Sum criterion

Worst sampling

Best sampling

Fully sampled

5.245.265.285.305.32

ppm

3.33.43.53.63.73.8

ppm

Figure 30 Sucrose, 50mM in D2O. Sum criterion: reconstructions for the best sampling, the worst samplingand the original fully sampled pure-shift spectrum.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

t , s


0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

t , s


Figure 31 Sum criterion for sucrose: best and worst samplings applied to simulated FID of singlets (point3 of the workflow).

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3.03.54.04.55.05.5

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Max criterion

Sum criterion

Fully sampled

3.3003.3253.3503.3753.4003.4253.4503.475

ppm

3.5203.5253.5303.5353.5403.5453.550

ppm

Figure 32 Sucrose, 50mM in D2O. Maximum and sum criterion comparison and the original fully sampledpure-shift spectrum. In the bottom zoomed panel on the left, the results of the maximum criterion follow thefully sampled spectrum.

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2.53.03.54.04.55.05.5

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Linf criterium

L1 criterium

original

4.454.554.654.75

ppm

Worst sampling

Best sampling

Fully sampled

3.263.283.303.323.343.36

Max criterion

Figure 33 D-glucose, 50mM in D2O. Maximum criterion: reconstructions for the best sampling, the worstsampling and the original fully sampled pure-shift spectrum.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

t , s


0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

t , s


Figure 34 Maximum criterion for d-glucose: best and worst samplings applied to simulated FID of singlets(point 3 of the workflow).

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2.53.03.54.04.55.05.5

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Linf criterium

L1 criterium

original

Sum criterion

4.454.554.654.75

ppm

Worst sampling

Best sampling

Fully sampled

3.263.283.303.323.343.36

Figure 35 D-glucose, 50mM in D2O. Sum criterion: reconstructions for the best sampling, the worstsampling and the original fully sampled pure-shift spectrum.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

t , s


0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

t , s


Figure 36 Sum criterion for D-glucose: best and worst samplings applied to simulated FID of singlets (point3 of the workflow).

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2.53.03.54.04.55.05.5

ppm


Max criterion

Sum criterion

Fully sampled

3.553.603.653.70 3.753.80 3.7453.7503.7553.7603.765

ppm ppm

Figure 37 D-glucose, 50mM in D2O. Maximum and sum criterion comparison and the original fully sampledpure-shift spectrum.

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123456

ppm

Max criterion

Worst sampling

Best sampling

Fully sampled

1.61.71.81.92.02.12.2

ppm

4.644.664.684.704.724.744.764.78

ppm

Figure 38 Mixture of 0.5M (R)-(+)-limonene and 50mM α-pinene in CDCl3. Maximum criterion: reconstruc-tions for the best sampling, the worst sampling and the original fully sampled pure-shift spectrum.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

t , s


0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

t , s


Figure 39 Maximum criterion for the mixture: best and worst samplings applied to simulated FID of singlets(point 3 of the workflow).

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123456

ppm

Sum criterion

Worst sampling

Best sampling

Fully sampled

4.664.684.704.724.744.76

ppm

1.601.651.701.751.801.85

ppm

Figure 40 Mixture of 0.5M (R)-(+)-limonene and 50mM α-pinene in CDCl3. Sum criterion: reconstructionsfor the best sampling, the worst sampling and the original fully sampled pure-shift spectrum.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

t , s


0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

t , s


Figure 41 Sum criterion for the mixture: best and worst samplings applied to simulated FID of singlets(point 3 of the workflow).

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123456

ppm


Max criterion

Sum criterion

Fully sampled

1.241.251.261.271.281.29

ppm

1.851.901.952.002.052.102.152.202.25

ppm

Figure 42 Mixture of 0.5M (R)-(+)-limonene and 50mM α-pinene in CDCl3. Maximum and sum criterioncomparison and the original fully sampled pure-shift spectrum.

The results for the β -estradiol sample (50mM in CD3OD), maximum criterion,are given in the main text, Fig.1. Here, we present the results for the sum criterionof the same sample:

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0.700.720.740.760.780.80

ppm

1.21.41.61.82.02.22.4

ppm

012345678

ppm

Sum criterion

Worst sampling

Best sampling

Fully sampled

Figure 43 Sum criterion: reconstructions for the best sampling, the worst sampling and the original fullysampled pure-shift spectrum. β -estradiol, 50mM in CD3OD.

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0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

t , s

Best sampling - Sum criterion (simulated FID))

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

t , s


Figure 44 Sum criterion: best and worst samplings applied to simulated FID of singlets (step 3 of thework-flow).

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1.21.41.61.82.02.22.4

ppm

6.4006.4256.4506.4756.5006.5256.5506.5756.600

ppm

012345678

ppm


Max criterion

Sum criterion

Fully sampled

Figure 45 Maximum and sum criterion comparison and the original fully sampled pure-shift spectrum.β -estradiol, 50mM in CD3OD. See the results for other samples in ESI†.

The sum criterion often yields FID truncation as the best sampling. We also ex-plored if it is worth to reconstruct this truncated FID with linear prediction insteadof compressed sensing, as we normally do. It turns out that linear prediction doesnot cope with the task, as is illustrated in Figs.46.

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1.52.02.53.03.54.04.55.05.56.06.57.07.5f1 (ppm)

00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70t1 (sec)

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 6500 7000 7500 8000t1 (pt)

Figure 46 (R)-(+)-limonene: reconstruction of the truncated FID (best sampling according to the sumcriterion) with iterative soft thresholding algorithm belonging to the compressed sensing family (red) andwith linear prediction (blue). Top panel: reconstructed spectra. Bottom panels: reconstructed FIDs. Thelinear prediction reconstruction is carried out in MestreNova 11.0 software with standard parameters. Thecompressed sensing reconstruction is carried out with mddnmr software with standard parameters.

We also explored the issue of decreasing the number of measured chunks onthe example of D-glucose and β -estradiol. In the main text, as well as in all theother examples here, the number of measured chunks was 18 out of 93. For thesetwo examples, we performed the same sampling optimization and reconstruction

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procedure for 17, 16, ... etc. chunks. The results for the maximum criterion andthe sum criterion are shown in Figs.47, 49, 51, 53. Figs.48, 50, 52, 54 present thecorresponding best samplings.

3.513.533.553.573.593.613.633.653.673.693.713.733.753.773.793.813.833.85f1 (ppm)

18 chunks

17 chunks

16 chunks

15 chunks

14 chunks

13 chunks

12 chunks

11 chunks

10 chunks

9 chunks

8 chunks

7 chunks

6 chunks

5 chunks

4 chunks

3 chunks

Figure 47 Decreasing the number of measured chunks: sampling optimization with the maximum criterionand reconstruction of the best sampling results. 50 mM glucose.

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00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70t1 (sec)

18 chunks

17 chunks

16 chunks

15 chunks

14 chunks

13 chunks

12 chunks

11 chunks

10 chunks

9 chunks

8 chunks

7 chunks

6 chunks

5 chunks

4 chunks

3 chunks

Figure 48 Decreasing the number of measured chunks: best samplings according to the maximum crite-rion.

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3.473.493.513.533.553.573.593.613.633.653.673.693.713.733.753.773.793.813.833.85f1 (ppm)

18 chunks

17 chunks

16 chunks

15 chunks

14 chunks

13 chunks

12 chunks

11 chunks

10 chunks

9 chunks

8 chunks

7 chunks

6 chunks

5 chunks

Figure 49 Decreasing the number of measured chunks: sampling optimization with the sum criterion andreconstruction of the best sampling results. 50 mM glucose.

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0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70t1 (sec)

18 chunks

17 chunks

16 chunks

15 chunks

14 chunks

13 chunks

12 chunks

11 chunks

10 chunks

9 chunks

8 chunks

7 chunks

6 chunks

5 chunks

Figure 50 Decreasing the number of measured chunks: best samplings according to the sum criterion.

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1.151.201.251.301.351.401.451.501.551.601.651.701.751.801.851.901.952.002.052.102.152.202.252.30f1 (ppm)

18 chunks

17 chunks

16 chunks

15 chunks

14 chunks

13 chunks

12 chunks

11 chunks

10 chunks

9 chunks

8 chunks

7 chunks

6 chunks

5 chunks

Figure 51 Decreasing the number of measured chunks: sampling optimization with the maximum criterionand reconstruction of the best sampling results. 50 mM β -estradiol.

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0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70t1 (sec)

18 chunks

17 chunks

16 chunks

15 chunks

14 chunks

13 chunks

12 chunks

11 chunks

10 chunks

9 chunks

8 chunks

7 chunks

6 chunks

5 chunks

Figure 52 Decreasing the number of measured chunks: best samplings according to the maximum crite-rion.

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1.151.201.251.301.351.401.451.501.551.601.651.701.751.801.851.901.952.002.052.102.152.202.252.30f1 (ppm)

18 chunks

17 chunks

16 chunks

15 chunks

14 chunks

13 chunks

12 chunks

11 chunks

10 chunks

9 chunks

8 chunks

7 chunks

6 chunks

5 chunks

Figure 53 Decreasing the number of measured chunks: sampling optimization with the sum criterion andreconstruction of the best sampling results. 50 mM β -estradiol.

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0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70t1 (sec)

18 chunks

17 chunks

16 chunks

15 chunks

14 chunks

13 chunks

12 chunks

11 chunks

10 chunks

9 chunks

8 chunks

7 chunks

6 chunks

5 chunks

Figure 54 Decreasing the number of measured chunks: best samplings according to the sum criterion.

In all the cases presented above, the sum criterion yielded a simple truncationof the FID as the best sampling. However, this is not always the case: whenthe FID does not fully decay during the acquisition time, the sum criterion willyield another type of sampling, as is illustrated in Figs.56 (samplings) and 57(results). This may occur in practice when one does not need a long FID, as allpeaks are already resolved earlier. Among our examples, quinine demonstratedsuch a behaviour. Fig 55 shows that a short acquisition of 0.1835 s (16 chunks of128 points) is enough in this case.

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0.51.01.52.02.53.03.54.04.55.05.56.06.57.07.58.08.5f1 (ppm)

Figure 55 Quinine, 50mM in CDCl3, PSYCHE shorter acquisition: 0.1835 s (16 chunks of 128 points). Allpeaks are resolved.

0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175t, s

best sampling - L1 criterion (simulated FID)

0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175t, s

worst sampling - L1 criterion (simulated FID)

Figure 56 Sum criterion for quinine, shorter acquisition: 0.1835 s (16 chunks of 128 points): best and worstsamplings applied to simulated FID of singlets (point 3 of the workflow). 6 measured chunks out of 16. Thebest sampling is not a truncated one.

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3.03.23.43.63.8

ppm

1.41.61.82.02.22.4

1234578

ppm

Sum criterion

Worst sampling

Best sampling

Fully sampled

9 6

2.8

Figure 57 Quinine, 50mM in CDCl3, shorter acquisition: 0.1835 s. 6 measured chunks out of 16. Sumcriterion: reconstructions for the best sampling, the worst sampling and the original fully sampled pure-shiftspectrum.

To demonstrate how the results of the proposed optimization procedure dependon the multiplet analysis quality, we artificially removed some multiplets from thelist and performed the whole procedure otherwise normally. We chose the sampleof sucrose for this aim, as it has a dense region at about 3.7 ppm. This region ishighly sensitive to such changes, as is illustrated in Figs.59, 61, 63 (best samplingreconstruction results, maximum criterion, superimposed with the full PSYCHEspectrum). Figs.58, 60, 62 present the spectra simulated on the basis of the corre-sponding multiplet lists superimposed with the sucrose proton spectrum. Figs.59and 58 present the results for the full list of multiplets. Figs.61 and 60 illustratethe removal of two multiplets in the region of interest. This results in a worse

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reconstruction, though still fairly acceptable. Figs.63 and 62 present the removalof two other multiplets, which are outside the dense region. Such a removal, in itsturn, does not lead to a considerable contamination of the reconstruction results.

3.33.43.53.63.73.83.94.04.14.24.34.44.54.64.74.84.95.05.15.25.3f1 (ppm)

Spectrum simulated from multiplet list

PROTON spectrum

Figure 58 1H spectrum of β -estradiol, 50mM in CD3OD, red line, and spectrum simulated from full multipletlist, blue line.

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3.203.253.303.353.403.453.503.553.603.653.703.753.803.853.903.954.004.054.104.15f1 (ppm)

Full PSYCHE

Best sampling (maximum criterion) reconstruction results

Figure 59 Full PSYCHE spectrum of β -estradiol, 50mM in CD3OD, red line, and reconstruction form thebest sampling scheme (maximum criterion) simulated from correct full multiplet list, green line.

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3.23.33.43.53.63.73.83.94.04.14.24.34.44.54.64.74.84.95.05.15.25.35.4f1 (ppm)


PROTON spectrum

Figure 60 1H spectrum of β -estradiol, 50mM in CD3OD, red line, and spectrum simulated from corruptedmultiplet list, blue line, with multiplets removed in the region of interest.

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3.203.253.303.353.403.453.503.553.603.653.703.753.803.853.903.954.004.054.104.15f1 (ppm)

Full PSYCHE

Best sampling (maximum criterion)

Figure 61 Full PSYCHE spectrum of β -estradiol, 50mM in CD3OD, red line, and reconstruction form thebest sampling scheme (maximum criterion) simulated from corrupted multiplet list (multiplets removed inthe region of interest), green line.

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3.23.33.43.53.63.73.83.94.04.14.24.34.44.54.64.74.84.95.05.15.25.35.4f1 (ppm)


PROTON spectrum

Figure 62 1H spectrum of β -estradiol, 50mM in CD3OD, red line, and spectrum simulated from corruptedmultiplet list, blue line, with multiplets removed outside the region of interest.

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3.203.253.303.353.403.453.503.553.603.653.703.753.803.853.903.954.004.054.104.15f1 (ppm)

Full PSYCHE

Best sampling (maximum criterion)reconstruction results

Figure 63 Full PSYCHE spectrum of β -estradiol, 50mM in CD3OD, red line, and reconstruction form thebest sampling scheme (maximum criterion) simulated from corrupted multiplet list (multiplets removed out-side the region of interest), green line.

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Accelerated acquisition in pure-shift spectra based …Accelerated acquisition in pure-shift spectra...

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