Application of the sparse regularization in NMR Diffusometry measurements

Application of the sparse

regularization in NMR Diffusometry measurements.

Mateusz Urbańczyk

Wydział Chemii

Centrum Nauk Biologiczno-Chemicznych Uniwersytet Warszawski

Promotor: prof. dr hab. Wiktor Koźmiński

Promotor pomocniczy: dr hab. Krzysztof Kazimierczuk

Acknowledgements

The Foundation for Polish Science TEAM Project:

“Towards new applications of NMR spectroscopy in chemical and biomolecular structural studies”, (2010-2014) is gratefully acknowledged.

The National Science Centre Preludium grant UMO-2014/13/N/ST4/04085

“Sparse regularization in NMR Relaxometry” is gratefully acknowledged.

The National Science Centre Harmonia grant UMO-2013/08/M/ST4/00975

“Regularization algorithms for the processing of NMR spectra of metabolite mixtures” is gratefully acknowledged.

The National Centre for Research and Development grant PBS2/A27/09/2013 is

gratefully acknowledged.

Streszczenie

Dyfuzyjna spektroskopia magnetycznego rezonansu jądrowego (NMR) jest tech- niką o bardzo szerokim zastosowaniu. Technika jest używana w badaniach mie- szanin (włącznie z płynami ustrojowymi), badaniach polimerów, monitorowaniu reakcji oraz w wielu innych problemach.

Zadaniem metod dyfuzyjnych NMR jest odkodowanie współczynników dyfuzji z mierzonego sygnału. Do tego celu używa się odwrotnej transformaty Laplace’a, której natura generuje wiele problemów. Główną przeszkodą jest niestabilność numeryczna procedury.

Celem niniejszej pracy jest wprowadzenie regularyzacji rzadkiej oraz prawie- rzadkiej do ILT. Dodatkowo wprowadzono pierwszą na świecie szybką metodę pomiaru wielowymiarowych widm NMR z jednym wymiarem Laplace’owskim (dyfuzyjnym, albo relaksacyjnym).

Praca składa się z trzech części:

• Omówienie Metodologii

• Narzędzia oparte na więzach rzadkości dla NMRu

• Zastosowania.

Pierwsza część zawiera trzy rozdziały. W pierwszym rozdziale zostają omó-

wione podstawowy metodologii NMR. W szczególności stany energetyczne spinu

w polu magnetycznym, przesunięcie chemiczne, sprzężenie, podstawy Fourie-

rowskiej spektroskopii NMR oraz zjawisko relaksacji.

Drugi rozdział poświęcony jest zjawisku dyfuzji. Po wprowadzeniu definicji i pod- stawowych pojęć związanych z tym zjawiskiem, praca skupia się na omówieniu wybranych metod mierzenie współczynnika dyfuzji.

Szczególna uwaga zwrócona jest na dyfuzyjny NMR. Omówione są podstawy tej metody, sposoby pomiaru, ograniczenia związane z pomiarem. Wprowadzone jest pojęcie odrotnej transformaty Laplace’a (ILT), która jest głównym proble- mem numerycznym w pomiarach współczynników dyfuzji za pomocą NMRu.

W dalszej części zostają omówione zostają metody na obliczanie odwrotnej trans- formaty Laplace’a dla eksperymentów dyfuzyjnych z podziałem na dwie główne grupy: „Univariate“ oraz „Multivariate”.

Rozdział trzeci skupia się na omówieniu NMR w o więcej niż jednym wymiarze Fourierowskim. Rozdizał rozpoczyna się opisem zasad pomiaru wielowymiaro- wego eksperymentu NMR na przykładzie pomiaru HSQC. Następnie pojawia się rozszerzenie tego eksperymentu na widma trójwymiarowe, gdzie jeden z wy- miarów jest wymiarem Laplace’owskim, czyli na trójwymiarowe eksperymenty dyfuzyjne oraz relaksacyjne.

Druga część składa się z dwóch rozdziałów. W pierwszym z nich zostaje przed- stawiona zostaje klasyczna teoria próbkowania Nyquista-Shannona. W opozycji do niej i jej ograniczeń zostaje omówiona teoria Oszczędnego Próbkowania (CS) oraz jej podstawowe założenia.

Następnie wprowadzonej są zastosowania teorii CS w spektroskopii NMR.

Kolejnym podrozdziałem w pracy jest omówienie wprowadzenia więzów rzad-

kości do dyfuzyjnych pomiarów NMR na podobieństwo teorii CS. Omówione

zostają przyczyny dla których nie można ściśle zastosować klasycznej teorii CS

W oparciu o założenia regularyzacji Tikhonova oraz więzy rzadkości wprowa- dzony zostaje nowy algorytm ITAMeD (Iteracyjny progowy algorytm dla zaników wielowykładniczych). Pokazane zostają wyniki testów tegoż algorytmu na trzech różnych symulacjach oraz dla porównania wyniki konkurencyjnych metod ta- kich jak Maksimum Entropii (MaxEnt) oraz CONTIN.

W kolejnym podrozdziale metoda ITAMeD zostaje rozszerzona na układ trójwy- miarowych losowo próbkowanych widm z jednym z wymiarów Laplace’owskim.

Na początku pokazane zostaje podobieństwo metod CS dla losowo próbkowa- nych widm NMR oraz regularyzacji Tikhonova z więzami rzadkości. Obie me- tody zostają połączone dla losowo próbkowanego sygnału będącego iloczynem wymiaru Fourierowskiego oraz Laplace’owskiego. Przedstawione zostają teore- tyczne zalety takiego rozwiązania oraz możliwe zastosowania w widmach typu 3D HSQC-iDOSY oraz serii relaksacyjnych HSQC. Dodatkowo pokazane zostają wyniki symulacji dla widm 3D HSQC-iDOSY.

Drugi rozdział tej części dotyczy regularyzacji dopasowanej, czyli procedury, która automatycznie dobiera balans między więzem rzadkości, a „gładkością”

rozwiązania.

Rozdział zaczyna się od przedstawienia motywacji dla takiego rodzaju regulary- zacji. Następnie wprowadza koncepcję regularyzacji dopasowej (Tailored norm) oraz przedstawia algorytm, który przeprowadza regularyzację dla ILT z auto- matycznie dobieraną normą `

oraz pokazuje potencjalne zastosowanie takiej regularyzacji w symulacjach próbek o różnej polidyspersyjności.

Trzecia część przedstawia przykłady zastosowań nowych metod omówionych w części drugiej. Część ta składa się z sześciu rozdziałów.

Pierwszy z tych rozdziałów przedstawia zastosowanie algorytmu ITAMeD do ana-

lizy mieszanin polimerów PEG o różnej masie cząsteczkowej. Pokazane wyniki

oraz porównanie z konkurencyjnymi metodami wskazuje na zdecydowaną prze- wagę metody dla układów o wielowykładniczym zaniku.

W dalszej części rozdziału pokazane zostaje zastosowanie dopasowanych norm do analizy polimerów o wysokiej dyspersji. Zademonstrowana zostaje przydatność tej metody do wyznaczenia stopnia dysperyjności (PDI) polimeru za pomocą eksperymentu NMR.

W drugim rozdziale pokazane zostaje zastosowanie dopasowanej regularyzacji do badań płynów ustrojowych (w szczególności osocza krwi). Pokazano analizę próbek osocza krwi dziesięciu ochotników. Krew pobrano przed i po posiłku.

Następnie zmierzono widmo dyfuzyjne NMR oraz przetworzono je. W widmie poddano analizie rejon odpowiadający lipidom demonstrując możliwość jasnego podziału na różne frakcje cholesterolu w takim widmie. Dodatkowo całkując rejony widma odpowiadające cholesterolowi o wysokiej gęstości oraz o niskiej gęstości udało się uzyskać prostą metodę rozróżniającą próbki osocza pobrane przed i po posiłku. Wskazuje to, że metodę można użyć do badań chorób zwią- zanych z frakcją lipidową osocza.

W rozdziale trzecim pokazano zastosowanie dyfuzyjnych pomiarów NMR do monitorowania reakcji. Po omówieniu koncepcji „kroczącej klatki” zapropono- wano adaptację tej metody dla eksperymentów dyfuzyjnych oraz przedstawiono jej zastosowanie na przykładzie monitorowanie rozkładu heparyny przez enzym heparynazę.

Kolejny rozdział przedstawia zastosowanie metod dla losowo próbkowanych widm

3D HSQC-iDOSY. Pokazano wyniki dwóch eksperymentów, wraz z porównaniem

z klasycznym pomiarem. Pierwszą badaną próbką była mieszanina alaniny, cy-

trynianiu, TMA oraz tauryny, która służy za prosty model płynu ustrojowego,

durga próbka zaś była mieszaniną kwercetyny oraz rutyny. Ta próbka została

W przedostatnim rozdziale przedstawiono modyfikację wyżej opisanej metody dla widm relaksacyjnych

N T1-HSQC dla trzech różnych białek SH3, GB1 oraz ubikwityny. Pokazano porównanie uzyskanych wyników z wynikami uzy- skanymi metodą klasyczną, sprawdzono wpływ ilości zmierzonych punktów na dokładność rekonstrukcji oraz dla ubikwityny dokonano porównania z wyni- kami uzyskanymi niedawno przedstawioną konkurencyjną metodą analizy lo- sowo próbkowanych widm relaksacyjnych z użyciem przetwarzania co-MDD.

Ostatni rozdział niniejszej pracy zawiera opis wpływu przedstawionych tu badań

na dziedzinę spektroskopii NMR oraz plany dalszego wykorzystania tych metod

w przyszłości.

Abstract

This work’s main focus is on the application of sparse regularization in to NMR diffusometry experiments. The thesis is divided into three main parts:

• Methodology overview, where the basic theory of methods is described on which the methodology introduced in further parts is built. This part contains the discussion on the basic NMR phenomena, the idea of multi- dimensional NMR, NMR diffusometry and relaxometry. Additionally the part is focused on the process of diffusion and available methods allowing to study this phenomenon.

• Sparsity-based toolbox for NMR, where the discussion on the sparsity constraint in NMR is presented. The part starts from discussing the cur- rent state of the art sparsity constraint in NMR - the Compressed Sensing methods. It is followed by the discussion on the use of this constraint in NMR diffusometry where the new methods and algorithms, utilizing such constraint, are presented. Finally the work goes beyond the sparsity con- straint by introducing mixed constraint that optimizes the balance between sparseness and smoothness.

• Applications, where the application of the methodology introduced in pre-

vious part are shown. The part starts by showing the examples of using

both sparsity and mixed constraint in the measurements of polymer mix-

cations of sparse-sampled 3D experiments, where one dimension is either

encoding diffusion coefficient, or relaxation constant.

The list of the author's articles

• Articles directly associated with the thesis:

1. M. Urbańczyk, D. Bernin, W. Koźmiński, and K. Kazimierczuk, “Iter- ative Thresholding Algorithm for Multiexponential Decay applied to PGSE NMR data,” Anal. Chem., vol. 85, no. 3, pp. 1828–1833, 2013.

2. M. Urbańczyk and K. Kazimierczuk, “A method for joint sparse sam- pling of time and gradient domains in diffusion-ordered NMR spec- troscopy,” in Signal Processing Symposium (SPS) 2013, , pp. 1–6, 2013.

3. M. Urbańczyk, W. Koźmiński, and K. Kazimierczuk, “Accelerating Diffusion- Ordered NMR Spectroscopy by Joint Sparse Sampling of Diffusion and Time Dimensions.,” Angew. Chemie Int. Ed., vol. 53, no. 25, pp.

6464–6467, 2014.

1 M. Urbańczyk, D. Bernin, A. Czuroń, and K. Kazimierczuk, “Monitoring polydispersity by NMR diffusometry with tailored norm regularisation and moving-frame processing,” Analyst, vol. 141, no. 5, pp. 1745–1752, 2016.

• Other articles:

1. M. Pecul, M. Urbańczyk, A. Wodyński, and M. Jaszuński, “DFT cal-

culations of 31P spin-spin coupling constants and chemical shift in

2. P. Horeglad, M. Cybularczyk, A. Litwińska, A. M. Dąbrowska, M. Dranka,

G. Z. Żukowska, M. Urbańczyk, M. Michalak, G. Zukowska, M. Ur-

bańczyk, and M. Michalak, “Controlling the stereoselectivity of rac-LA

polymerization by chiral recognition induced formation of homochiral

dimeric metal alkoxides,” Polym. Chem., vol. 7, no. 11, 2016.

Contents

List of Figures xix

List of Tables xxv

Glossary xxvii

I Methodology Overview 1

1 Basics of NMR 3

1.1 Spin and Energy . . . . 3

1.1.1 Chemical Shift . . . . 5

1.1.2 Scalar Coupling . . . . 5

1.2 Fourier NMR . . . . 6

1.3 Relaxation . . . . 7

2 Diffusion 9 2.1 Non-NMR methods . . . 11

2.1.1 Electrochemistry . . . 11

2.1.2 Dynamic Light scattering . . . 12

2.1.3 Closed capillary methods . . . 12

2.1.4 Attenuated total reflection infrared spectroscopy (ATRIR) . . . 13

2.2 Diffusion NMR . . . 14

2.2.1 Processing . . . 17

CONTENTS

2.2.1.1 Univariate . . . 17

2.2.1.2 Multivariate . . . 20

3 Multidimensional NMR 21 3.1 Fourier-based Multidimensional NMR . . . 21

3.2 Inverse Laplace Transformion multidimensional NMR . . . 23

3.2.1 Diffusometry . . . 23

3.2.2 Relaxometry . . . 24

II Sparsity-based toolbox for NMR 25 4 NMR and sparsity 27 4.1 Basics of Compressed Sensing . . . 28

4.2 Compressed Sensing in NMR . . . 31

4.3 Sparsity constraint in non-Fourier world . . . 33

4.3.1 Tikhonov regularization with sparsity constraint . . . 33

4.3.1.1 Iterative Thresholding Algorithm for multi-exponential decay 34 4.4 Where the two worlds meet. Sparsity constraint for joint Fourier-Inverse Laplace Transform . . . 41

4.4.1 Introduction . . . 41

4.4.2 3D HSQC-iDOSY . . . 42

4.4.3 NMR Relaxomtery with NUS . . . 46

5 When the Sparsity is not an answer - Tailored Norms 49

CONTENTS

III Applications 55

6 Diffusometry studies of polymers 57

6.1 Mixtures of mono-dispersed polymers . . . 57

6.2 Polydisperse polymer studies . . . 60

6.2.1 Log-normal fitting versus Tailored norm regularization . . . 60

7 Diffusion in metabolite studies 65 7.1 Lipoprotein profiling using PGSE NMR . . . 65

8 Diffusion based reaction monitoring 69 8.1 Moving frame . . . 69

8.2 Heparin depolymerization . . . 71

8.2.1 Depolymerization studies with tailored norm and moving frame meth- ods . . . 71

9 When 2D DOSY resolution is not enough. Mixture analysis with 3D HSQC- DOSY 75 9.1 Metabolomic model- Mixture of Alanine, Citrate, TMA and Taurine . . . 75

9.2 Sample of high overlap of signals- Quercetin and Rutin . . . 77

10 Protein studies with NUS Relaxometry 81 10.1 SH3 . . . 82

10.2 GB1 . . . 85

10.3 Ubiquitin . . . 87

10.4 Comparison with co-MDD . . . 90

11 Impact of the research and future plans 93

References 95

CONTENTS

IV Appendixes 105

A Matlab code of ITAMeD 107

B Matlab Code of Tailored Norm regularization 109

C Publications 113

C.1 Urbańczyk, M.; Bernin, D.; Koźmiński, W.; Kazimierczuk, K. Anal. Chem.

2013, 85 (3), 1828. . . 113 C.2 Urbańczyk, M.; Kazimierczuk, K. In Signal Processing Symposium (SPS),

2013; 2013; pp 1–6. . . 130 C.3 Urbańczyk, M.; Koźmiński, W.; Kazimierczuk, K. Angew. Chemie Int. Ed.

2014, 53 (25), 6464. . . 137

C.4 Urbańczyk, M.; Bernin, D.; Czuroń, A.; Kazimierczuk, K. Analyst 2016, 141. . 150

List of Figures

1.1 The one pulse NMR experiment . . . . 7 2.1 A- Presents a model of the diffusion of particles in a vacuum as a random

thermal motion. B- presents the model of diffusion in liquid. In this model the molecules from solvent are negligibly smaller than diffusing particle.

Therefore they can be treated as a continuum rather than separate molecules 10 2.2 The idea of NMR-based diffusion measurement. The first π/2 pulse puts

the sample magnetization to x-y axis. Then, short magnetic field gradient is applied. During gradient the local magnetic field varies along the sample.

Because of that, the magnetic vector in each position of the sample rotate with different speed. Therefore, after the gradient pulse the magnetization distribution forms as a helix. During diffusion delay ∆ the nuclei are diffus- ing and therefore the coherence of the helix is decreasing. After π pulse and repetition of gradient field pulse the overall magnetization is weaken by the incoherence. Because of that, the observed signal is decaying in the function of gradient strength, diffusion delay and diffusion coefficients. . . . 15 2.3 Stimulated Spin Echo sequence. In this sequence if the condition: 2τ  T is

fulfilled (that the spin relaxation depends mostly on longitudinal relaxation,

which is slower than transverse relaxation, in the case of larger molecules)

and if τ  1/J the J-modulation effects are negligible. Therefore this method

is far more practical and popular in the NMR diffusometry than Hahn spin-

echo presented in Figure 2.2. . . 16

LIST OF FIGURES

3.1 HSQC pulse sequence. The narrow blocks refer to π/2 pulses, wide ones to π pulse, the gray pulse symbolizes the decoupling during acquisition, t1 is a evolution delay for

C. . . 21 3.2 The idea of 2D acquisition in NMR . . . 22 4.1 The graphical depiction of the determined system of Equation 4.1 . . . 29 4.2 The graphical depiction of the system of Equation 4.1 with CS conditions

applied . . . 31 4.3 The graphical depiction of CS processing of NMR NUS data . . . 32 4.4 Comparison of the convergence rate between ITAMeD and ISTA. . . 34 4.5 Comparison between the results of ITAMeD (A), CONTIN (B), MaxEnt (C) and

NNLS (D) processing of the simulated data sets with two peaks at varying ratio of diffusion coefficients (Test 1). Red dashed line represents the reference value set in the simulation. . . 37 4.6 Comparison between the results of ITAMeD (A), CONTIN (B), MaxEnt (C)

and NNLS (D) processing results of the simulated data sets (Test 2). For each method, 100 diffusion coefficient distributions were calculated (marked with differently colored lines). Black dashed line represents the simulated signal. 38 4.7 The idea of joint sparse processing of Fourier and Inverse Laplace Transform 42 4.8 The simulation of HSQC DOSY

H crossection . . . 43 4.9 The simulation of HSQC DOSY

H and indirect frequency dimension crossec-

tion - Diffusion profile for different noise level: A:0.1 %, B:0.3%, C:1% and D:

1.5% . . . 45 4.10 HSQC-iDOSY sequence with NUS . . . 46

5.1 Result of ITAMeD processing for a simulated signal having ’broad’ distribu-

LIST OF FIGURES

5.3 Comparison between the results of the optimal norm (dashed yellow line),

`

-norm (dashed green line) and `

-norm (dashed red line) used in the pro- cessing of the simulated data set of various diffusion distributions. The widths of the diffusion profiles (σ) and the corresponding optimal norms are: A - 0.5, 1.6, B - 0.4,1.45, C - 0.3,1.3, D - 0.2,1.15, E - 0.1,1.1, F - 0.05,1. . . 52 5.4 The fit residuum for the simulation (described in Figure 5.3) in the function

of different `

-norms . . . 53

6.1 Comparison between the ITAMeD (A), CONTIN (B) and MaxEnt (C) for the experimental data: Sample 4: red line, Sample 3: gray line, Sample 2 dark blue line, Sample 1:light blue line. The dotted line represents the reference data obtained by the mono-exponential fitting of single component solutions with following diffusion coefficients: 1.6·10

with sum of squared residu- als (ssq) 1.1·10

for PEG124700; 6.3·10

, ssq = 2.5·10

for PEG11840;

2.3 · 10

, ssq = 2.8 · 10

for PEG1080 . . . 59 6.2 Comparison between the CONTIN (green dashed line), MaxEnt (blue dotted

line) and ITAMeD (red line) for the Sample 5. Reference diffusion coeffi- cients marked with dotted lines were obtained by mono-exponential fitting of signals obtained for single-compound samples. . . 60 6.3 The dependence between peak width (σ) and Polydispersity index (PDI) for

the group of PEO polymers. Diffusion spectra reconstructed with Tailored Norm (A), and log-normal fitting (B).) . . . 62

7.1 The concept of studies of diffusion profiling of the lipoprotein fraction in blood for discriminating subjects after and before English breakfast. . . 66 7.2 DOSY spectra of lipoprotein region of 20 blood samples. The highlighted

samples corresponds to patients after English breakfast . . . 67

LIST OF FIGURES

7.3 The ratio between integration of HDL and VLDL regions for 20 blood samples.

The black lines shows the threshold that allows to differentiate the sample according to existence, or lack of breakfast . . . 68 8.1 The concept of moving frame experiment . . . 70 8.2 The result of monitoring the reaction of depolymerization of heparin by

heparinase. A: the monitoring based on the Diffusion coefficient of the peak, B: the polydispersity measured by Tailored norm and C: classic approach of monitoring the intensity of the peak correlated with reaction products. R

represents the fitting error for calculation the kinetic constant of the reaction (k) . . . 73 9.1 3D HSQC-DOSY spectrum of mixture of L-alanine, citrate, TMA and taurine.

A- projection over

C dimension, B-E Crossection of Diffusion dimension for different diffusion coefficient mark on A. . . 77 9.2 HSQC-iDOSY spectra of mixture of quercetin and rutin in deuturated DMSO.

A represents the spectrum from “classic” sampling. In the speech balloons upper value is diffusion coefficient calculated for the peak from 5 gradients and lower value from 2 gradients, B- Diffusion Crossection from the sparse sample HSQC-iDOSY for the region of Rutin, C- Diffusion Crossection for Quercetin. In the speech balloons position of the center of the peak in diffusion dimension. All diffusion values are in 10

m

s

. . . 78 10.1 Projection over R of the sparse FTILT

N-T1-HSQC spectrum of SH3 protein.

The numbers are arbitrary labels for the protein residuues . . . 82 10.2 Pairs of the zoomed regions from the Figure 10.1 with blue dot at the center

of the peak and the relaxation profile of this peak bellow. Numbering as in

the Figure 10.1 . . . 83

LIST OF FIGURES

10.4 Projection over R of the sparse FTILT

N-T1-HSQC spectrum of GB1 protein.

The numbers are arbitrary labels for the protein residues . . . 85 10.5 Pairs of the zoomed regions from the Figure 10.4 with blue dot at the center

of the peak and the relaxation profile of this peak bellow. Numbering in accordance with the Figure 10.4 . . . 86 10.6 Comparison between classic R values from classic experiment (green line)

and from sparse data (blue line) for GB1 protein . . . 87 10.7 Projection over R of the sparse FTILT

N-T1-HSQC spectrum of Ubiquitin

protein. The numbers are arbitrary labels for the protein residuues . . . 88 10.8 Comparison between classic R values from classic experiment (green line)

and from sparse data (blue line) . . . 88 10.9 Pairs of the zoomed regions from the Figure 10.7 with blue dot at the center

of the peak and the relaxation profile of this peak bellow. Numbering as in the Figure 10.7 . . . 89 10.10 Sparse

N-T1-HSQC spectrum of Ubiquitin. A- projection over relaxation

dimension, B- projection of

N, C- projection over

H . . . 90

LIST OF FIGURES

List of Tables

4.1 Results of simulations (Test number 3) for different noise levels. Diffusion coefficients and amplitudes are displayed as a difference between the median values obtained from 100 simulations and reference, expressed in percent of reference value. Standard deviation of peak position and amplitudes in 100 simulations are also presented. . . 39 4.2 Results of simulations (Test number 3) for different noise levels . Diffusion

coefficients and amplitudes are displayed as a difference between the median values obtained from 100 simulations and reference, expressed in m

/s . Standard deviations of peak positions and amplitudes in 100 simulations are also presented. . . 40 4.3 Parameters of 10 peaks used in the simulation for FTILT . . . 44 4.4 The influence of a noise . . . 44 4.5 Phase cycling . . . 46 6.1 Polymers used in the experiment with decoding of the sample codes pre-

sented in Figure 6.3. D corresponds to the diffusion coefficient at the center of the peak obtained from both methods. . . 63 8.1 Kinetic constants calculated for different frame sizes and spectral parameters. 72 9.1 Peak coordinates and integrals for the sparse sampled HSQC-iDOSY exper-

iment of mixture of citrate, L-alanine, TMA, taurine . . . 76

LIST OF TABLES

10.1 Residual of relaxation fit for different methods of sparse sampling in NMR

relaxation . . . 91

Glossary

ADC Analog-to-digital converter

ATRIR Attenuated total reflection infrared spectroscopy

BSS Blind Source Separation

CPMG Carr-Purcell-Meiboom-Gill

CS Compressed Sensing

CT-HSQC-iDOSY Constant Time Heteronuclear Single Quantum Coherence internal Diffusion Ordered Spectroscopy

CW Continuous Wave

DECRA Direct Exponential Curve Resolution DOSY Diffusion Order Spectroscopy

FID Free Induction Decay

FISTA Fast Iterative Shrinkage Algorithm

FT Fourier Transform

HDL High Density Lipoproteins

hmsIST Harvard Medical School implementation of the iterative soft thresholding ap- proach

HSQC Heteronuclear Single Quantum Corelation

HSQC-iDOSY Heteronuclear Single Quantum Coherence internal Diffusion Ordered Spec- troscopy

iDOSY internal Diffusion Ordered Spectroscopy

ILT Inverse Laplace Transform

INEPT Insensitive Nuclei Enhanced by Polarization Transfer IRLS Iterative Reweighted Least Squares

ISTA Iterative Soft Thresholding Algorithm

ITAMeD Iterative Thresholding Algorithm for Multi-exponential Decay

GLOSSARY

LDL Low Density Lipoproteins

LPMP Lorentzian peak matching pursuit

MaxEnt Maximum Entropy

MCR Multivariate Curve Resolution

MDD Multidimensional Decomposition

MRI Magnetic Resonance Imaging

NMR Nuclear Magnetic Resonance

NNLS Non-negative Least Squares

NUS Non Uniform Sampling

p-DOSY permuted DOSY

PALMA Proximal Algorithm for L

combined with MaxEnt prior

PDI Polydispersity Index

PEG Polyethylene Glycol

PEO Poly-ethyleno oxide

PGSE Pulse Gradient Spin Echo

SCORE Speedy Component Resolution SSA Suppression of sampling artefacts

ST Stimulated Echo

TRAIn Trust Region Algorithm for Inversion

VLDL Very Low Density Lipoproteins

Part I

Methodology Overview

Chapter 1

Basics of NMR

1.1 Spin and Energy

Nuclear Magnetic Resonance (NMR) is one of the most useful techniques available to chemists. It allows to gather a variety of information about sample such as identification of compounds, their structure, concentration, thermodynamic parameters and many others.

Therefore, NMR is one of the basic methods in chemists’ toolboxes and ongoing effort to extract more information using this method is made.

Since 71 years from its discovery by two independently working groups lead by Bloch (1) and Purcell (2) NMR has changed dramatically. It took only few years to spread from the field of physics to the chemistry (3) and few another to be used in structural biology (4).

Even with such big progress of the method, its basics remain the same.

As the name Nuclear Magnetic Resonance suggests, the technique is based on the reso- nance of the nuclei in magnetic field. The nuclei to be measured using this technique have to have a non-zero spin. The most commonly used nuclei are those with spin I =

such as

H,

C,

N,

F, or

P. As for spin I = 1, the most important nuclei is deuteron

H.

The non-zero spin generates the magnetic moment defined as:

µ = γI, (1.1)

where γ is a gyromagnetic ratio of nucleus, and I is a nuclear spin angular momentum,

1. BASICS OF NMR

whose magnitude is defined as:

kIk = ~ h

I (I + 1)

i

(1.2)

Therefore, the energy of the spin state in magnetic field B can be defined as:

E = µ · B (1.3)

and in case of magnetic field vector being directed in only one direction, the energy can be described as

E = −γI

B

= −m~γB

(1.4)

Where m is the magnetic quantum number. For the spin I =

m can be either −

, or +

and therefore the energy levels allowed for such state are‘:

E

= − 1

2~ γB

(1.5)

E

= 1

2~ γB

. (1.6)

Then the energy required for the transition between states is defined as:

∆E

= ~γB

. (1.7)

The aforementioned magnetic moments are precessing along the z axis with the fre- quency defined as:

ω

= −γB

. (1.8)

The description presented is based on a single-state system. In case of the real life

sample the spins in the equilibrium are rotating incoherently. The resulting bulk magnetic

vector is approximately zero and therefore the system is unobservable, until a pulse that will

force the coherence of the system is applied.

1.1 Spin and Energy

1.1.1 Chemical Shift

Atoms by their nature contain not only nuclei, but also electrons. Those electrons by their motion induce magnetic field, which changes the field affecting the nuclei. The effect of this field is called nuclear shielding and can be described as:

B = (1 − bσ)B

(1.9)

Where bσ is a shielding tensor. In case of liquid state NMR (which will be discussed in this work) due to rapid isotropic reorientation of molecules the average shielding constant tensor is symmetrical. Therefore the trace of shielding tensor can be defined as:

σ = (σ

+ σ

+ σ

) /3. (1.10) Than the resonance frequency is:

ω = −γ(1 − σ)B

. (1.11)

The value of σ depends on the surrounding electrons and therefore is a “probe” for nuclei chemical environment. As the value of σ is not varying very distinctively between different environments and to make the scale of frequency independent from the used magnetic field the chemical shift (δ) concept was introduced:

δ ≈ (σ

− σ ) · 10

≈ ω − ω

ω

· 10

(1.12)

Where subscription

means the value from reference molecule.

1.1.2 Scalar Coupling

Another phenomenon that has great influence on NMR spectrum is scalar coupling which is the interaction between different spins in the molecule mediated by electrons forming chemical bond between interacting nuclei (5) (in contrary to the direct dipolar couplings, whose effect in isotropic liquid NMR is averaged to zero dueo to fast motions).

The strength of this interaction is reflected in the scalar coupling constant

J

, where

n is a number of covalent bond between interacting nuclei a and b.

1. BASICS OF NMR

The energy of the system of two coupled

spins I and S can reach four different levels:

E

=

~ω

+

~ω

+

π ~J

E

=

~ω

−

~ω

−

π ~J

E

= −

~ω

+

~ω

−

π ~J

E

= −

~ω

−

~ω

+

π ~J

(1.13)

In case of measuring the resonance of spin I there are two possible transitions: E

Ï E

, E

Ï E

For these transitions the energy difference is:

∆E

= ~ω

+π~J

∆E

= ~ω

−π ~J

(1.14)

Therefore, the resonance peak is split into doublet.

The scalar coupling phenomenon is utilized in the technique called Insensitive Nuclei Enhanced by Polarization Transfer (INEPT) proposed by Morris and Freeman in 1979 (6).

This method utilizes the scalar coupling to transfer magnetization from one between nuclei connected by chemical bonds. The method is widely used in multidimensional heteronuclear NMR (See Chapter 3.1).

1.2 Fourier NMR

The classical approach to NMR experiment was to sweep the magnetic field or the frequency of the radio waves exciting the sample and measure the absorption. Such straightforward approach known as Continuous Wave (CW) had many limitations such as: acquisition time, resolution, sensitivity and limitation to only one dimensional experiments.

To circumvent this limitation Richard Ernst and Weston Anderson proposed Fourier based NMR spectroscopy (7). The idea behind this concept was that instead of sweeping through different frequency in quest of the resonance, one should irradiate the sample with the electromagnetic pulse that would excite all spins in the system at once and then measure the return to the equilibrium.

This signal is called Free Induction Decay (FID) and is described as:

1.3 Relaxation

So FID is a sum of oscillating functions from all nuclei that were excited by the pulse and the decay of FID is defined by relaxation rate (R)

Figure 1.1: The one pulse NMR experiment

Of course the signal in the time domain is not easily interpretable. Therefore, the signal is later transformed to the frequency domain using Fourier Transform (FT):

S (ω) = Z

−∞

S (t) exp(−jωt)dt. (1.16)

The idea of Fourier NMR enables to circumvent many limitations of CW and is a standard method in modern NMR. With many advantages, come also some disadvantages. Most of them come directly from the nature of FT. As the FID decay is asymptotically decaying to zero, the measured signal has to be truncated (As it is impossible to wait infinite amount of time to acquire full FID). That means that signal is multiplied by the box function, whose FT is a sinc function and therefore the FT of truncated signal has sinc sidebands, as one can prove using Fourier Transform Convolution theorem. To circumvent this problem the FID before FT is multiplied usually by a weighting function which suppresses the final part of FID.

1.3 Relaxation

The R symbol in equation 1.15 describes the Relaxation phenomenon which is responsible for the return of magnetization to the equilibrium. The phenomenon can be quantified either by using the relaxation constant R, or relaxation time T, related as:

R = 1

T (1.17)

This mechanism can be divided into two processes:

1. BASICS OF NMR

• Longitudinal relaxation (R

)

• Transverse relaxation (R

)

The first process is the return of the magnetization along z-axis to its equilibrium value.

The second process describes the loss of coherence in the x-y plane - decay of transverse bulk magnetization. The source of relaxation in liquid phase can be described based on the main two mechanisms:

• The dipolar mechanism

• Chemical shift anisotropy.

The first mechanism is based on dipole-dipole interaction between two magnetic moments

(spins). Interestingly this mechanism is very sensitive to the distance between spins as it

is decreasing with the distance (r) as

. The second mechanism is related to the local

magnetic fields created by the electron density of the atoms.

Chapter 2

Diffusion

Diffusion, in general, describes the motion. It can be a spread of the infection in the population (8, 9), the movement of people and cultures in some region (10), the spread of information in social media (11, 12), or more familiar to NMR spectroscopists the spin- diffusion, the exchange of magnetization between spins (13). For a chemist the most inter- esting is the movement of molecules. However, even in chemistry, the diffusion has very wide meaning. It can describe the movement of molecules in the flow, or transport induced by concentration gradient, or (in case of ions) by electric field. There is also other type of diffusion: self-diffusion. This type of diffusion describes the stochastic thermal movement of the molecules. The very general idea of such movement is presented in the Figure 2.1.A where the group of molecules in vacuum are randomly moving in space. In such a case, the diffusion coefficient (which describes the diffusion) can defined as:

D = lim

tÏ∞

1 6t

D r

(t) − r

(0) 

E

, (2.1)

where r

(t) is the position of molecule i at time t.

Usually however, chemists are more focused on liquid phase than on molecules in vacuum. In liquid, the situation is a little different. The first description of diffusion was proposed by Adolf Fick in 1855 (14):

J = −D∇c(r, t), (2.2)

2. DIFFUSION

where ∇c(r, t) is a concentration gradient. Equation 2.2 is known as first Fick’s law. Together with Fick’s second law:

∂c

∂t = D∇

c, (2.3)

In case of sample of uniform concentration the Fick’s Law is still applicable. However, the concentration gradient exists only in the microstate of local environment. The sample itself is uniform. Diffusion in such system is described as self diffusion

In such case the diffusion can be described as Brownian movement as shown in the Fig- ure 2.1.B. The diffusion coefficient of such system is described by Stokes-Einstein-Sutherland equation:

D = k

T

6πηR (2.4)

where k

is Boltzman constant, T is temperature, η is a viscosity of the solvent and R is an radius of sphere. Equation 2.4 is valid for the molecules whose shape can be approximated with a sphere.

Figure 2.1: A- Presents a model of the diffusion of particles in a vacuum as a random thermal

motion. B- presents the model of diffusion in liquid. In this model the molecules from solvent

2.1 Non-NMR methods

2.1 Non-NMR methods

The diffusion phenomenon is extremely important in chemistry. It provides better insight into mass transport, or even describes the reaction. The importance of diffusion caused the development of various techniques to measure it. In this section, I will describe some of methods that are not based on Nuclear Magnetic Resonance.

2.1.1 Electrochemistry

When chemist thinks about diffusion usually the first thought is related to electrochem- istry methods. However, electrochemical methods are not describing the self-diffusion in equilibrium, but rather the concentration-gradient-induced diffusion. Nevertheless, such an important technique should be mentioned in any review of diffusometric methods. The aforementioned Fick’s description of diffusion is based on the existence of the concentra- tion gradient.

The cyclic voltamperometry (15) is generating the concentration gradient in the red-ox reaction. The Randles-Sevcik equation describes the connection of parameters measured by voltamperometry (i

- peak current and v -scan rate V/s) with diffusion parameter:

i

= 0.4463 nFAc

 nF vD RT



, (2.5)

where n is a number of electrons transfer in the reaction, F is Faraday constant, A is the area of electrode, c is a concentration, D diffusion coefficient, R gas constant and t is temperature.

By plotting the function of i

for √

v one can determine the diffusion coefficient from the slope of the line.

The cyclic voltamperometry is very fast and cheap method, but can be applied only to the systems where the red-ox reaction occurs.

Electrochemistry in its variety allows to measure the diffusion coefficient with many

different methods, even in solids (16).

2. DIFFUSION

2.1.2 Dynamic Light scattering

Dynamic Light Scattering (17) is based on the setup where the coherent monochromatic light passes through the sample. Then the scattered light is measured in the form of unique speckle pattern similar to the diffraction pattern. If the same experiment is repeated in short time the pattern will differ due to Brownian motion of molecules. Therefore, one can calculate the second order autocorrelation function (g

(q; τ)):

g

(q; τ) = hI (t)I(t + τ)i

hI (t)i

(2.6)

Where q is a wave vector, τ is delay between two measurements. I denotes the intensity of signal, and operator hi denotes expected value. For the monodisperse samples, the Siegert equation introduces the connection between second and first order autocorrelation:

g

(q; τ) = 1 + β h

g

(q; τ) i

(2.7) Where β is a correction factor for setup. Also one can connect the first order autocorrelation with the translational diffusion coefficient by:

g

(q; τ) = exp(−q

Dτ ) (2.8)

And in case of polydisperse samples:

g

(q; τ) = X

i=1

G (i)q

D (i) exp(−q

D (i)τ). (2.9) To decode the diffusion coefficient from Equation 2.9 one employs Inverse Laplace Trans- form, which will be described in details in later parts of the thesis.

2.1.3 Closed capillary methods

Method is based on tracing the diffusion of radioactive material (e.g. tritium-enriched water)

inside the close tube. The setup consist of the tube filled in the bottom with radioactive

2.1 Non-NMR methods

for the radioactivity in the upper part of the capillary. By plotting the Count Per Minute value in the function of time one can determine the diffusion coefficient. The method’s biggest limitation is an experimental time usually hundreds of hours for methods waiting for equilibrium (18) to tens of hours for ”fast” methods (19).

2.1.4 Attenuated total reflection infrared spectroscopy (ATRIR)

Attenuated total reflection infrared spectroscopy is a technique used for diffusion studies inside the polymers. The measurement system contains ATR crystal (e.g. ZnSe 60° reflection angle), on the crystal polymer film, radiation source and detector. In the system, the radiation that propagates through crystal of high refractive index (n

) is subject to total internal reflection at the interface with polymer of lower refractive index. The energy absorbed by polymer generates the information about the molecules of polymer. The wave that vanishes in the polymer is called evanescent wave (E) and it is exponentially decaying as:

E = E

exp



−2π λ

sin

Θ −

 n

n



!

x



 , (2.10)

where λ

is a wavelength in a crystal, Θ is a incident angle, and x is a depth of polymer.

In 1989 Vorenkamp et al. (20) proved that by controlling the time(t) of irradiation of the sample one can determine the diffusion coefficient (D) from the intensity (A

) of the peak of interest, using such equation:

A

A

= 1 − 4 π exp



− π

Dt 4x



. (2.11)

Later studies proved the usefulness of the method for diffusion studies in polymer with the

condition of none or small polymer swelling (21, 22)

2. DIFFUSION

2.2 Diffusion NMR

Unique feature of NMR diffusometry is its ability to measure the self-diffusion coefficient in equilibrium.

The idea of NMR-based diffusion measurements was conceived when Erwin Hahn in- vented the spin-echo in 1950 (23). However, it took 15 years to introduce the first experimen- tal NMR diffusion experiment by Edward Stejskal and John Tanner (24). The researchers used the Hahn spin-echo in the version proposed by Herman Carr and Edward Purcell (2) and run it several times with different gradient strength levels (as described in Figure 2.2).

Due to the problems with transverse relaxation during diffusion delay and J-modulation (which can severely distort measured decaying signal) other pulse sequence was proposed:

Stimulated Echo (STE) (25) (Showed in Figure 2.3).

To analyze the results,the Stejskal-Tanner (S-T) equation can be used, which describes the signal attenuation:

S(g) = S(0)e

, (2.12) where S(g) is the signal intensity for a given magnetic field gradient amplitude g, γ is the gyromagnetic ratio, δ is the duration of the magnetic field gradient pulse and ∆

is the effective diffusion time (Comprehensive review of modification of S-T equation for different pulse sequences can be found in (26)). However, the equation 2.12 describes the situation where the diffusion is fully isotropic and therefore one can use diffusion coefficient in form of scalar. Therefore, such equation for cases of multi-exponential decay, or in case of polydisperse shape of diffusion coefficient profile has to be rewritten for continuous form of D:

Ψ = S (G) S(0) =

Z

Dmin

A (D)e

dD, (2.13)

where A(D) is a distribution of diffusion coefficients. For calculation purposes one can

rewrite the equation in the matrix form:

2.2 Diffusion NMR

Figure 2.2: The idea of NMR-based diffusion measurement. The first π/2 pulse puts the sample

magnetization to x-y axis. Then, short magnetic field gradient is applied. During gradient the

local magnetic field varies along the sample. Because of that, the magnetic vector in each

position of the sample rotate with different speed. Therefore, after the gradient pulse the

magnetization distribution forms as a helix. During diffusion delay ∆ the nuclei are diffusing

and therefore the coherence of the helix is decreasing. After π pulse and repetition of gradient

field pulse the overall magnetization is weaken by the incoherence. Because of that, the observed

signal is decaying in the function of gradient strength, diffusion delay and diffusion coefficients.

2. DIFFUSION

Figure 2.3: Stimulated Spin Echo sequence. In this sequence if the condition: 2τ  T is fulfilled (that the spin relaxation depends mostly on longitudinal relaxation, which is slower than transverse relaxation, in the case of larger molecules) and if τ  1/J the J-modulation effects are negligible. Therefore this method is far more practical and popular in the NMR diffusometry than Hahn spin-echo presented in Figure 2.2.

Where Φ is a matrix of transformation.

This model is valid for non disturbed system. The system can be disturbed for example by the flow due to temperature gradient inside the sample. Such gradient is generated by the way the NMR spectrometer is controlling the temperature of the sample. The temperature regulation is done by blowing the gas with stable temperature on the sample. Often, such setup creates the temperature gradient across the sample and therefore the convection. The probability of convection can be determined for the Rayleigh number (27):

R

= gα

kν a

∆T, (2.14)

where g is acceleration due to gravity, α - thermal expansion, k liquid thermal diffusivity, ν- kinematic viscosity, a is a radius of the NMR tube, and ∆T is a temperature gradient.

There are many tricks to avoid convection problems during experiment. The easiest

one is to use tubes of smaller diameter, which will decrease the Rayleigh number, other

method are isolating sample either by putting glass wool around it or by surrounding it

with the fluid of higher heat capacity (28), using tube made from material of higher thermal

conductivity than the sample (29), or by using the special pulse sequences (30).

2.2 Diffusion NMR

is called a restricted diffusion. In such case, the displacement of the molecule during diffusion delay will be a function of ∆, diffusion coefficient, and the size and shape of the space confining molecule (e.g. pore). If this is neglected the measured diffusion coefficient, will be different than real one and will be called apparent diffusion coefficient (D

). The relationship between D

and D can be established from the equation derived by Mitra and coworkers (31) :

D

(∆) = D



1 − 4 3d √

π S

V D

∆

+ · · ·



, (2.15)

where d - represents the number of spatial dimensions and

is the surface to volume ratio.

The equation 2.15 shows that the deviation from diffusion coefficient is linear with respect to ∆

. So the longer diffusion delay the effect is larger.

In this thesis I will concentrate on the “free diffusion”. However, restricted diffusion is a valuable tool for studying porous, or inhomogeneous samples as shown in (32, 33, 34, 35, 36, 37, 38, 39)

2.2.1 Processing

Inverse Laplace Transform (ILT) which is applied to obtain the diffusion coefficient from Pulse gradient spin echo (PGSE) experiments, is mathematically unstable procedure. It is extremely prone to noise. To circumvent this limitation, researchers have proposed a big variety of methods. Those methods are based on certain assumptions about the sample, like number of components in the mixture, if a sample is poly-, or monodisperse etc. For the sake of this work I have divided this methods into two groups:

1. Univariate 2. Multivariate

2.2.1.1 Univariate

The first group is based on independent processing of the diffusion decay corresponding to

each point of the frequency spectrum. The calculations can be easily parallelized because

2. DIFFUSION

each spectral point is completely independent from the rest. On the other hand, such methods cannot utilize the information from other spectral points that could enhance the result. Nevertheless, methods of this group are widely used thanks to the simplicity on one hand and often due to the fact that many methods from this group are calculating the ILT creating pseudo 2D spectrum called Diffusion Order Spectroscopy (DOSY). Such sequential manner of processing is more natural for NMR spectroscopist used to similar processing of FT in the indirect dimension.

Monoexponential fitting The simplest possible way of such processing is to fit for every diffusion decay the monoexponential function. Such method is widely used for samples without peak overlapping. The monoexponential fitting can be done with many different algorithms such like Levenberg-Marquardt (40), Conjugate Gradient Search (41) etc.

Log-normal fitting Log-normal fitting is an extension of monoexponential fitting (42), where, instead of fitting the single diffusion coefficients one fits the Gaussian function reflecting the distribution A(D). Such method can be used to evaluate the polydispersity of the sample. However, the model works only for the distributions that follow the model of single Gaussian distribution.

Gamma distribution fitting In 2012 Röding et al. (43) suggested another model for fitting the polydisperse samples - the Gamma distribution fitting, which uses the gamma function whose distribution is closer to the real distributions of the polymers and thus provides better model for real samples.

CONTIN In contrary to aforementioned methods CONTIN proposed by Provencher in

1982 (44), is not using any model of distribution. It is based on the Tikhonov regularization

2.2 Diffusion NMR

Where L is a priori matrix which introduces additional assumptions about distribution. The functional used in CONTIN by using the penalty function in the form of `

-norm favors the result with smooth distributions of the diffusion coefficients. Such approach guarantees good stability of the result together with proper description of polydisperse samples. However, the ”smooth constraint” causes the problems with some multiexponential decays, where the distribution is non-smooth.

Maximum Entropy Maximum Entropy (MaxEnt) method introduced into NMR diffusom- etry by Delsuc et al (45) in 1998 is very similar to CONTIN. It is also based on the Tikhonov regularization, but instead of using `

-norm as a penalty function it looks for the result with the highest entropy:

min

k ΦA − Ψk

− λ X

i

A

P

j

A

log A

P

j

A

. (2.17)

Such approach guarantees less noise vulnerability and better resolution, but for the cost of suppressing smaller peaks.

Trust Region Algorithm for Inversion (TRAIn) For the cases of non-symmetrical dis- tribution Xu et al. (46) proposed Trust-Region Algorithm for the Inversion of Molecu- lar Diffusion NMR data. This method is in someway similar to the two aforementioned Tikhonov-based approaches, but is based on other group of regularizers which minimize the functional:

min

k Φ · diag(η) · η − Ψk

subject to: kη

− η

k

≤ r

, (2.18)

where η

= A and r

is a trust-region which is updated for every iteration to guarantee

the most effective inversion. Method performs extremely well for multiexponential or

asymmetric distributions, but only at low noise levels (47).

2. DIFFUSION

2.2.1.2 Multivariate

This group utilizes the fact, that sample contains only a limited number of different com- pounds and therefore should have a limited number of components in diffusion dimension.

Methods from this group try to find the spectrum with either defined number of diffusion components, or the smallest number that fits the data. Such methods usually guarantee bet- ter stability of the result but at the expense of no information about the diffusion coefficient distribution. The name refers to the multivariate statistics, so the simultaneous observation of many outcome variables.

Additionally, as recently reported by Zhou et. al. (48) the usage of such methods for polydisperse samples biases the obtained diffusion coefficient.

Methods such as Direct Exponential Curve Resolution (DECRA) (49), Multivariate Curve

Resolution (MCR) Speedy Component Resolution (SCORE) (50) Blind Source Separation

(BSS) (51) belong to this group. All these methods perform very well in case on mixtures

(with limited number of species) where all diffusion profiles are monodisperse.

Chapter 3

Multidimensional NMR

3.1 Fourier-based Multidimensional NMR

The Fourier-based NMR experiments mentioned in the Chapter 1.2 can be extended to many spectral dimensions. In such case the magnetization from one group of nuclei can be transfered to the other, evolve there, and be transfered back. The most basic tool for such magnetization is called Insensitive nuclei enhanced by polarization transfer (INEPT).

The method utilizes J coupling to transfer magnetization from one type of nucleus to other (e.g. from

H to

C).

The pulse sequence on which all Multidimensional experiments presented in this work are based is Heteronuclear Single Quantum Corelation (HSQC). Which is a basic heteronu- clear two dimensional sequence.

Sequence is presented in the Figure 3.1

Figure 3.1: HSQC pulse sequence. The narrow blocks refer to π/2 pulses, wide ones to π pulse, the gray pulse symbolizes the decoupling during acquisition, t1 is a evolution delay for

C.

In case of HSQC sequence the experiment measures two-dimensional FID of both

H

and

C in a sequential manner, as illustrated in the Figure 3.2. For each point of the indirect

3. MULTIDIMENSIONAL NMR

Figure 3.2: The idea of 2D acquisition in NMR

dimension FID one has to record the whole direct dimension FID, so the pulse sequence is repeated with varying value of t

(indirect dimension evolution time).

The acquisition according to the scheme presented in Figure 3.2 generates the FID from the presented in the Equation 1.15 changes to the function of the evolution of two types of nuclei:

S

(t

, t

) = X

i=1

s

(t

) ⊗ s

(t

) (3.1)

3.2 Inverse Laplace Transformion multidimensional NMR

3.2 Inverse Laplace Transformion multidimensional NMR

The experiment presented presented in the Chapter 3.1 can be also extended in to Non- Fourier dimension. Such dimension can be for example time which corresponds to the reaction progress (52), the temperature change to control protein unfold (53), Pulsed Field Gradient strength to encode diffusion dimension (54, 55, 56, 57), or CPMG delay encoding relaxation rate (58, 59, 60, 61).

In this work, the key interest will be put in the last two of those dimensions.

3.2.1 Diffusometry

The multidimensional diffusometry experiments will be refereed in the thesis to the exper- iments where at least two Fourier dimensions.

The first 3D DOSY experiment was designed by Wu et al in 1997 (56) and it was the extension of proton-proton correlation spectrum (COSY) with one of the basic PGSE pulse se- quences (bipolar longitudinal delay). The sequence consisted of two blocks PGSE and COSY which where executed one after another with both t

and gradient strengths incremented in separate loops.

According to authors, the extension of DOSY experiment to more dimensions helped with the peak overlap in complex mixture allowing to decode the diffusion coefficients even with simple monoexponential fit.

Soon, new methods 3D DOSY methods were introduced like DOSY-NOESY (62), DOSY- HMQC (63), DOSY-TOCSY (64).

Also, the design of pulse sequences was improved by introducing the iDOSY sequences (i from internal), where the PGSE block was executed inside the sequence on which the method was based (The pulse sequence from this group is presented in the Chapter 4.4.2). Such design shortened the length of the pulse sequence and therefore allowed better sensitivity.

The experiments utilizing this idea are e.g. HSQC-iDOSY (54, 55) and COSY-iDOSY (65).

The classic processing approach for such experiments is to perform FT on all Fourier

dimensions and then perform peak picking of each generated 2D plane. With the list of

3. MULTIDIMENSIONAL NMR

peaks and their intensity in the function of PFG strength one can than perform ILT for each peak and get the diffusion coefficient related to this peak.

3.2.2 Relaxometry

The multidimensional relaxation experiments provide a signal that is mathematically similar to the diffusometry signal described in Chapter 3.2.1. However their physical basics which generates the signal are completely different and were discussed in Chapter 1.3. As in the Chapter 1.3 here the focus will be on two types of relaxation:

• Longitudinal relaxation (R

)

• Transverse relaxation (R

)

For both mechanisms the modification of

N-HSQC pulse sequence was proposed by Farrow et al. in 1994 (58). The HSQC has additional block that encodes the relaxation decay described as:

I(t) = I

exp −R

t

(3.2) The block that encodes the R

is basically the delay t while the magnetization is transfered to the

N nuclei and in case of R

this block is based on Carr-Purcell-Meiboom-Gill (CPMG) pulse sequence (66, 67, 68). In the first case the value of t is incrementally changed for each HSQC recorded for each point in the third dimension which is a exponential decay of the signal and in case of CPMG block the t is controlled by the number of the π pulses in the pulse train.

The method of processing such relaxation data is in fact identical with the diffusometry

data described in Chapter 3.2.1.

Part II

Sparsity-based toolbox for NMR

Chapter 4

NMR and sparsity

4. NMR AND SPARSITY

4.1 Basics of Compressed Sensing

According to Nyquist-Shannon theorem (69, 70) to discretely sample any continous-time, oscillating signal one has to do this with at least twice of the maximum frequency of the signal. Such sampling for the spectrum of finite bandwidth will guarantee to capture all the information of the contains-time signal. This theorem is widely used in almost all field of signal processing such as image processing, radio, medical imaging, radarology etc.

Almost every analog-to-digital converter (ADC) is based on this principle. Nyquist-Shannon theorem is extremely useful as it allows to design detectors that will be able to sample the signal. However, in the case of signals which every points takes a lot of time to acquire (e.g.

multidimensional NMR, MRI, Radars) such theorem leads to very long acquisitions.

The acquisitions of the signal can be described in the form of the matrix M which can be for example a Fourier transform matrix.

Then the processing of a signal can look like:

y = Mx (4.1)

Where y is a measured signal, and x is an data of interest. In case of NMR y would be a FID and x would be a spectrum.

In the terms of classic algebra y and x should be a vectors of the same size (y ∈ C

; x ∈ C

)and M should be a square matrix x ∈ C

) as shown in Figure 4.1. Only on such conditions the equation will be determined.

If the vector x will be longer than y the system will be underdetermined and will have more than one solution.

For many years the only possibility to guarantee the reconstruction of spectrum with

discrete sampling was to follow the Nyquist-Shannon theorem and ensure that the vectors

x and y are of the same length.

4.1 Basics of Compressed Sensing

Figure 4.1: The graphical depiction of the determined system of Equation 4.1

that one can possibly recover the data from much fewer measurements than required by Nyquist-Shannon theorem. (So the vector x is much longer than y).

The CS theory is based on two principles:

1. Sparsity 2. Incoherence

The first term means that the data (or its certain representation) contains only a small number of important points. The most of the elements of vector are insignificant. This is common in many fields (e.g. NMR spectrum usually contains only few peaks and most of space between peaks is insignificant).

The second term means that the signal representing a sparse data has to be spread out in measured domain. (As a inverse Fourier transform of single point (Dirac delta function) is just a line spread at the whole time domain). This means that each sampled points should provide as much as possible information. Such situation is, for example in the case of random sampling of NMR experiment.

The number of sampling points required to reconstruct K-sparse spectrum is:

m ≥ CK log (N/K) (4.2)

Where x ∈ C

and has K non-zero elements, y ∈ C

and C is a constant (independent of

m , K and N). As shown in the Figure 4.2.