Iterative Thresholding Algorithm for Multiexponential Decay Applied to PGSE NMR Data

Simple MATLAB and Python scripts for multi-exponential analysis

Multi-exponential decay is prevalent in magnetic resonance spectroscopy, relaxation, and imaging. This paper describes simple MATLAB and Python functions and scripts for regularized multi-exponential analysis methods for 1D and 2D data and example test problems and experiments. Regularized least-squares solutions provide production-quality outputs with robust stopping rules in ~5 and ~20 lines of code for 1D and 2D inversions, respectively. The software provides an open-architecture simple solution for transforming exponential decay data to the distribution of their decay lifetimes. Examples from magnetic resonance relaxation of a complex fluid, a Danish North Sea crude oil, and fluid mixtures in porous materials-brine/crude oil mixture in North Sea reservoir chalk-are presented. Developed codes may be incorporated in other software or directly used by other researchers, in magnetic resonance relaxation, diffusion, and imaging or other physical phenomena that require multi-exponential analysis.

Thermoplastic starch/polyvinyl alcohol blends modification by citric acid-glycerol polyesters

Thermoplastic starch/polyvinyl alcohol (TPS/PVA) films have limitations for being used in long-term applications due to starch retrogradation. This leads to plasticizer migration, especially when low molecular weight plasticizers such as glycerol, are used. In this work, we employed mixtures of oligomers based on glycerol citrates with higher molecular weight than glycerol as plasticizers for potato-based TPS/PVA blends obtained by melt-mixing. This constitutes an alternative to reduce plasticizer migration while keeping high swelling degree, and to provide high mechanical performance. The novelty lies in the usage of these oligomers by melt-mixing technique, aspect not deeply explored previously and that represents the first step towards industrial scalability. Prior to the blending process, oligomers mixtures were prepared with different molar ratios of citric acid (0-40 mol%) and added them. This minimizes the undesirable hydrolysis effect of free carboxylic groups on starch chains. The results demonstrated that the migration of plasticizers in TPS/PVA blends decreased by up to 70 % when the citric acid content increased. This reduction was attributed to the higher molecular weight (the majority in the range 764-2060 Da) and the 3D structure of the oligomers compared to using raw glycerol. Furthermore, the films exhibited a 150 % increase in Young's modulus and tensile strength without a reduction in elongation at break, while maintaining a high gel content, due to a moderate crosslinking.

Monitoring cheese ripening by single-sided nuclear magnetic resonance

The noninvasive, longitudinal study of products and food processing is of interest for the dairy industry. Here, we demonstrated that single-sided nuclear magnetic resonance (NMR) can be used for noninvasive monitoring of the cheese ripening process. The maturation of soft-ripened Camembert-like molded cheese samples was monitored for 20 d measuring 1-dimensional and 2-dimensional NMR relaxation and diffusion data at various depths, ranging from the hard surface layer to the soft center. Gelation and gel shrinkage were observed throughout ripening, and a complete loss of free water signal was observed at the cheese rind. Transversal (T₂) relaxation distributions include 3 components that evolve with ripening time and position, corresponding to water inside the casein gel network, water trapped in casein, and fat. Two-dimensional T₁-T₂ relaxation experiments provided enhanced resolution of the 3 components, allowing quantification of the relative proportions of each phase. Furthermore, diffusion (D)-T₂ relaxation correlation experiments revealed the bimodal size distribution of fat globules. The study demonstrated that single-sided NMR can provide spatially resolved signal intensity, relaxation, and diffusion parameters that reflect structural changes during the ripening process and can be exploited to understand and monitor the ripening of cheeses.

Ultrafast transverse relaxation exchange NMR spectroscopy

Molecular exchange between different physical or chemical environments occurs due to either diffusion or chemical transformation. Nuclear magnetic resonance (NMR) spectroscopy provides a means of understanding the molecular exchange in a noninvasive way and without tracers. Here, we introduce a novel two dimensional, single-scan ultrafast Laplace NMR (UF LNMR) method to monitor molecular exchange using transverse relaxation as a contrast. The UF T2-T2 relaxation exchange spectroscopy (REXSY) method shortens the experiment time by one to two orders of magnitude compared to its conventional counterpart. Contrary to the conventional EXSY, the exchanging sites are distinguished based on T2 relaxation times instead of chemical shifts, making the method especially useful for systems including physical exchange of molecules. Therefore, the UF REXSY method offers an efficient means for quantification of exchange processes in various fields such as cellular metabolism and ion transport in electrolytes. As a proof of principle, we studied a halogen-free orthoborate based ionic liquid system and followed molecular exchange between molecular aggregates and free molecules. The results are in good agreement with the conventional exchange studies. Due to the single-scan nature, the method potentially significantly facilitates the use of modern hyperpolarization techniques to boost the sensitivity by several orders of magnitude.

Deeper Insight into Photopolymerization: The Synergy of Time-Resolved Nonuniform Sampling and Diffusion NMR

The comprehensive real-time in situ monitoring of chemical processes is a crucial requirement for the in-depth understanding of these processes. This monitoring facilitates an efficient design of chemicals and materials with the precise properties that are desired. This work presents the simultaneous utilization and synergy of two novel time-resolved NMR methods, i.e., time-resolved diffusion NMR and time-resolved nonuniform sampling. The first method allows the average diffusion coefficient of the products to be followed, while the second method enables the particular products to be monitored. Additionally, the average mass of the system is calculated with excellent resolution using both techniques. Employing both methods at the same time and comparing their results leads to the unequivocal validation of the assignment in the second method. Importantly, such validation is possible only via the simultaneous combination of both approaches. While the presented methodology was utilized for photopolymerization, it can also be employed for any other polymerization process, complexation, or, in general, chemical reactions in which the evolution of mass in time is of importance.

Neural Network Method for Diffusion-Ordered NMR Spectroscopy

Diffusion-ordered NMR spectroscopy (DOSY) presents an essential tool for the analysis of compound mixtures by revealing intrinsic diffusion behaviors of the mixed components. For the interpretation of the diffusion information, intrinsically designed algorithms for a DOSY spectrum reconstruction are required. The estimated diffusion coefficients are desired to have consistency for all the spectral signals from the same molecule and good separation of signals from different molecules. For this purpose, we propose a novel method that adopts a coordinated multiexponential fitting to ensure the consistency of diffusion coefficients and apply a sparse constraint to enhance the robustness. A lightweight neural network is applied as an optimizer to solve this highly nonlinear and nonconvex optimization problem. The proposed method provides estimated diffusion coefficients with excellent distinguishment between species and outperforms the state-of-the-art reconstruction algorithms, such as the Laplacian inversion and the multivariate fitting methods.

A General Reconstruction Method for Multidimensional Sparse Sampling Nuclear Magnetic Resonance Spectroscopy

Multidimensional NMR spectroscopy provides a powerful tool for structure elucidation and dynamic analysis of complex samples, particularly for biological macromolecules. Multidimensional sparse sampling effectively accelerates NMR experiments while an efficient reconstruction method is generally required for unraveling spectra. Various reconstruction methods were proposed for pure Fourier NMR (only involving chemical shifts and J couplings detection). However, reconstruction concerned with Laplace-related NMR (i.e., involving relaxation or diffusion detection) is more challenging due to its ill-posed property. The existing Laplace-related NMR sparse sampling reconstruction methods suffer from poor resolution and possible artifacts in the resulting spectra owing to the pitfalls of the optimization algorithms. Herein, we propose a general approach for fast high-resolution reconstruction of multidimensional sparse sampling NMR, including pure Fourier, mixed Fourier-Laplace, and pure Laplace NMR, benefiting from the comprehensive sparse constraint and effective optimization algorithm and thus showing the promising prospects of multidimensional NMR.

Ultrafast methods for relaxation and diffusion

Relaxation and diffusion NMR measurements offer an approach to studying rotational and translational motion of molecules non-invasively, and they also provide chemical resolution complementary to NMR spectra. Multidimensional experiments enable the correlation of relaxation and diffusion parameters as well as the observation of molecular exchange phenomena through relaxation or diffusion contrast. This review describes how to accelerate multidimensional relaxation and diffusion measurements significantly through spatial encoding. This so-called ultrafast Laplace NMR approach shortens the experiment time to a fraction and makes even single-scan experiments possible. Single-scan experiments, in turn, significantly facilitate the use of nuclear spin hyperpolarization methods to boost sensitivity. The ultrafast Laplace NMR method is also applicable with low-field, mobile NMR instruments, and it can be exploited in many disciplines. For example, it has been used in studies of the dynamics of fluids in porous materials, identification of intra- and extracellular metabolites in cancer cells, and elucidation of aggregation phenomena in atmospheric surfactant solutions.

Tracking oxidation-induced alterations in fibrin clot formation by NMR-based methods

Plasma fibrinogen is an important coagulation factor and susceptible to post-translational modification by oxidants. We have reported impairment of fibrin polymerization after exposure to hypochlorous acid (HOCl) and increased methionine oxidation of fibrinogen in severely injured trauma patients. Molecular dynamics suggests that methionine oxidation poses a mechanistic link between oxidative stress and coagulation through protofibril lateral aggregation by disruption of AαC domain structures. However, experimental evidence explaining how HOCl oxidation impairs fibrinogen structure and function has not been demonstrated. We utilized polymerization studies and two dimensional-nuclear magnetic resonance spectrometry (2D-NMR) to investigate the hypothesis that HOCl oxidation alters fibrinogen conformation and T₂ relaxation time of water protons in the fibrin gels. We have demonstrated that both HOCl oxidation of purified fibrinogen and addition of HOCl-oxidized fibrinogen to plasma fibrinogen solution disrupted lateral aggregation of protofibrils similarly to competitive inhibition of fibrin polymerization using a recombinant AαC fragment (AαC 419–502). DOSY NMR measurement of fibrinogen protons demonstrated that the diffusion coefficient of fibrinogen increased by 17.4%, suggesting the oxidized fibrinogen was more compact and fast motion in the prefibrillar state. 2D-NMR analysis reflected that water protons existed as bulk water (T₂) and intermediate water (T_2i) in the control plasma fibrin. Bulk water T₂ relaxation time was increased twofold and correlated positively with the level of HOCl oxidation. However, T₂ relaxation of the oxidized plasma fibrin gels was dominated by intermediate water. Oxidation induced thinner fibers, in which less water is released into the bulk and water fraction in the hydration shell was increased. We have confirmed that T₂ relaxation is affected by the self-assembly of fibers and stiffness of the plasma fibrin gel. We propose that water protons can serve as an NMR signature to probe oxidative rearrangement of the fibrin clot.

High-Resolution Reconstruction for Multidimensional Laplace NMR

As a perfect complement to conventional NMR that aims for chemical structure elucidation, Laplace NMR constitutes a powerful technique to study spin relaxation and diffusion, revealing information on molecular motions and spin interactions. Different from conventional NMR adopting Fourier transform to deal with the acquired data, Laplace NMR relies on specially designed signal processing and reconstruction algorithms resembling the inverse Laplace transform, and it generally faces severe challenges in cases where high spectral resolution and high spectral dimensionality are required. Herein, based on the tensor technique for high-dimensional problems and the sparsity assumption, we propose a general method for high-resolution reconstruction of multidimensional Laplace NMR data. We show that the proposed method can reconstruct multidimensional Laplace NMR spectra in a high-resolution manner for exponentially decaying relaxation and diffusion data acquired by commercial NMR instruments. Therefore, it would broaden the scope of multidimensional Laplace NMR applications.

Computing and visualising intra-voxel orientation-specific relaxation-diffusion features in the human brain

Diffusion MRI techniques are used widely to study the characteristics of the human brain connectome in vivo. However, to resolve and characterise white matter (WM) fibres in heterogeneous MRI voxels remains a challenging problem typically approached with signal models that rely on prior information and constraints. We have recently introduced a 5D relaxation-diffusion correlation framework wherein multidimensional diffusion encoding strategies are used to acquire data at multiple echo-times to increase the amount of information encoded into the signal and ease the constraints needed for signal inversion. Nonparametric Monte Carlo inversion of the resulting datasets yields 5D relaxation-diffusion distributions where contributions from different sub-voxel tissue environments are separated with minimal assumptions on their microscopic properties. Here, we build on the 5D correlation approach to derive fibre-specific metrics that can be mapped throughout the imaged brain volume. Distribution components ascribed to fibrous tissues are resolved, and subsequently mapped to a dense mesh of overlapping orientation bins to define a smooth orientation distribution function (ODF). Moreover, relaxation and diffusion measures are correlated to each independent ODF coordinate, thereby allowing the estimation of orientation-specific relaxation rates and diffusivities. The proposed method is tested on a healthy volunteer, where the estimated ODFs were observed to capture major WM tracts, resolve fibre crossings, and, more importantly, inform on the relaxation and diffusion features along with distinct fibre bundles. If combined with fibre-tracking algorithms, the methodology presented in this work has potential for increasing the depth of characterisation of microstructural properties along individual WM pathways.

Multidimensional correlation MRI

Multidimensional correlation spectroscopy is emerging as a novel MRI modality that is well suited for microstructure and microdynamic imaging studies, especially of biological specimens. Conventional MRI methods only provide voxel-averaged and mostly macroscopically averaged information; these methods cannot disentangle intra-voxel heterogeneity on the basis of both water mobility and local chemical interactions. By correlating multiple MR contrast mechanisms and processing the data in an integrated manner, correlation spectroscopy is able to resolve the distribution of water populations according to their chemical and physical interactions with the environment. The use of a non-parametric, phenomenological representation of the multidimensional MR signal makes no assumptions about tissue structure, thereby allowing the study of microscopic structure and composition of complex heterogeneous biological systems. However, until recently, vast data requirements have confined these types of measurement to non-localized NMR applications and prevented them from being widely and successfully used in conjunction with imaging. Recent groundbreaking advancements have allowed this powerful NMR methodology to be migrated to MRI, initiating its emergence as a promising imaging approach. This review is not intended to cover the entire field of multidimensional MR; instead, it focuses on pioneering imaging applications and the challenges involved. In addition, the background and motivation that have led to multidimensional correlation MR development are discussed, along with the basic underlying mathematical concepts. The goal of the present work is to provide the reader with a fundamental understanding of the techniques developed and their potential benefits, and to provide guidance to help refine future applications of this technology.

Statistical learning of NMR tensors from 2D isotropic/anisotropic correlation nuclear magnetic resonance spectra

Many linear inversion problems involving Fredholm integrals of the first kind are frequently encountered in the field of magnetic resonance. One important application is the direct inversion of a solid-state nuclear magnetic resonance (NMR) spectrum containing multiple overlapping anisotropic subspectra to obtain a distribution of the tensor parameters. Because of the ill-conditioned nature of this inverse problem, we investigate the use of the truncated singular value decomposition and the smooth least absolute shrinkage and selection operator based regularization methods, which (a) stabilize the solution and (b) promote sparsity and smoothness in the solution. We also propose an unambiguous representation for the anisotropy parameters using a piecewise polar coordinate system to minimize rank deficiency in the inversion kernel. To obtain the optimum tensor parameter distribution, we implement the k-fold cross-validation, a statistical learning method, to determine the hyperparameters of the regularized inverse problem. In this article, we provide the details of the linear-inversion method along with numerous illustrative applications on purely anisotropic NMR spectra, both synthetic and experimental two-dimensional spectra correlating the isotropic and anisotropic frequencies.

Accelerating Restricted Diffusion NMR Studies with Time-Resolved and Ultrafast Methods

Restricted diffusion of fluids in porous materials can be studied by pulsed field gradient nuclear magnetic resonance (NMR) non-invasively and without tracers. If the experiment is repeated many times with varying diffusion delays, detailed information about pore sizes and tortuosity can be recorded. However, the measurements are very time-consuming because numerous repetitions are needed for gradient ramping and varying diffusion delays. In this paper, we demonstrate two different strategies for acceleration of the restricted diffusion NMR measurements: time-resolved diffusion NMR and ultrafast Laplace NMR. The former is based on time-resolved non-uniform sampling, while the latter relies on spatial encoding of two-dimensional data. Both techniques allow similar 1–2 order of magnitude acceleration of acquisition, but they have different strengths and weaknesses, which we discuss in detail. The feasibility of the methods was proven by investigating restricted diffusion of water inside tracheid cells of thermally modified pine wood.

Spatial molecular-dynamically ordered NMR spectroscopy of intact bodies and heterogeneous systems

Noninvasive evaluation of the spatial distribution of chemical composition and diffusion behavior of materials is becoming possible by advanced nuclear magnetic resonance (NMR) pulse sequence editing. However, there is room for improvement in the spectral resolution and analytical method for application to heterogeneous samples. Here, we develop applications for comprehensively evaluating compounds and their dynamics in intact bodies and heterogeneous systems from NMR data, including spatial z-position, chemical shift, and diffusion or relaxation. This experiment is collectively named spatial molecular-dynamically ordered spectroscopy (SMOOSY). Pseudo-three-dimensional (3D) SMOOSY spectra of an intact shrimp and two heterogeneous systems are recorded to evaluate this methodology. Information about dynamics is mapped onto two-dimensional (2D) chemical shift imaging spectra using a pseudo-spectral imaging method with a processing tool named SMOOSY processor. Pseudo-2D SMOOSY spectral images can non-invasively assess the different dynamics of the compounds at each spatial z-position of the shrimp's body and two heterogeneous systems.

Ultrafast diffusion exchange nuclear magnetic resonance

The exchange of molecules between different physical or chemical environments due to diffusion or chemical transformations has a crucial role in a plethora of fundamental processes such as breathing, protein folding, chemical reactions and catalysis. Here, we introduce a method for a single-scan, ultrafast NMR analysis of molecular exchange based on the diffusion coefficient contrast. The method shortens the experiment time by one to four orders of magnitude. Consequently, it opens the way for high sensitivity quantification of important transient physical and chemical exchange processes such as in cellular metabolism. As a proof of principle, we demonstrate that the method reveals the structure of aggregates formed by surfactants relevant to aerosol research.

Analysis of exchange processes is time consuming by two-dimensional exchange NMR spectroscopy. Here the authors demonstrate a single-scan ultrafast Laplace NMR approach based on spatial encoding to measure molecular diffusion, with an increase by a factor six in the sensitivity per unit time.

Choice of Force Field for Proteins Containing Structured and Intrinsically Disordered Regions

Biomolecular force fields optimized for globular proteins fail to properly reproduce properties of intrinsically disordered proteins. In particular, parameters of the water model need to be modified to improve applicability of the force fields to both ordered and disordered proteins. Here, we compared performance of force fields recommended for intrinsically disordered proteins in molecular dynamics simulations of three proteins differing in the content of ordered and disordered regions (two proteins consisting of a well-structured domain and of a disordered region with and without a transient helical motif and one disordered protein containing a region of increased helical propensity). The obtained molecular dynamics trajectories were used to predict measurable parameters, including radii of gyration of the proteins and chemical shifts, residual dipolar couplings, paramagnetic relaxation enhancement, and NMR relaxation data of their individual residues. The predicted quantities were compared with experimental data obtained within this study or published previously. The results showed that the NMR relaxation parameters, rarely used for benchmarking, are particularly sensitive to the choice of force-field parameters, especially those defining the water model. Interestingly, the TIP3P water model, leading to an artificial structural collapse, also resulted in unrealistic relaxation properties. The TIP4P-D water model, combined with three biomolecular force-field parameters for the protein part, significantly improved reliability of the simulations. Additional analysis revealed only one particular force field capable of retaining the transient helical motif observed in NMR experiments. The benchmarking protocol used in our study, being more sensitive to imperfections than the commonly used tests, is well suited to evaluate the performance of newly developed force fields.

Non-uniform sampling in quantitative assessment of heterogeneous solid-state NMR line shapes

Non-uniform sampling has been successfully used for solution and solid-state NMR of homogeneous samples. In the solid state, protein samples are often dominated by inhomogeneous contributions to the homogeneous line widths. In spite of different technical strategies for peak reconstruction by different methods, we validate that NUS can generally be used also for such situations where spectra are made up of complex peak shapes rather than Lorentian lines. Using the RMSD between subsampled and reconstructed data and those spectra obtained with uniform sampling for a sample comprising a wide conformational distribution, we quantitatively evaluate the identity of inhomogeneous peak patterns. The evaluation comprises Iterative Soft Thresholding (hmsIST implementation) as a method explicitly not assuming Lorentian lineshapes, as well as Sparse Multidimensional Iterative Lineshape Enhanced (SMILE) algorithm and Signal Separation Algorithm (SSA) reconstruction, which do work on the basis of Lorentian lineshape models, with different sampling densities. Even though individual peculiarities are apparent, all methods turn out principally viable to reconstruct the heterogeneously broadened peak shapes.

Comprehensive NMR Analysis of Pore Structures in Superabsorbing Cellulose Nanofiber Aerogels

Highly porous cellulose nanofiber (CNF) aerogels are promising, environmentally friendly, reusable, and low-cost materials for several advanced environmental, biomedical, and electronic applications. The aerogels have a complex and hierarchical 3D porous network structure with pore sizes ranging from nanometers to hundreds of micrometers. The morphology of the network has a critical role on the performance of aerogels, but it is difficult to characterize thoroughly with traditional techniques. Here, we introduce a combination of nuclear magnetic resonance (NMR) spectroscopy techniques for comprehensive characterization of pore sizes and connectivity in the CNF aerogels. Cyclohexane absorbed in the aerogels was used as a probe fluid. NMR cryoporometry enabled us to characterize the size distribution of nanometer scale pores in between the cellulose nanofibers in the solid matrix of the aerogels. Restricted diffusion of cyclohexane revealed the size distribution of the dominant micrometer scale pores as well as the tortuosity of the pore network. T ₂ relaxation filtered microscopic magnetic resonance imaging (MRI) method allowed us to determine the size distribution of the largest, submillimeter scale pores. The NMR techniques are nondestructive, and they provide information about the whole sample volume (not only surfaces). Furthermore, they show how absorbed liquids experience the complex 3D pore structure. Thorough characterization of porous structures is important for understanding the properties of the aerogels and optimizing them for various applications. The introduced comprehensive NMR analysis set is widely usable for a broad range of different kinds of aerogels used in different applications, such as catalysis, batteries, supercapacitors, hydrogen storage, etc.

Nonstationary Two-Dimensional Nuclear Magnetic Resonance: A Method for Studying Reaction Mechanisms in Situ

Nuclear magnetic resonance spectroscopy (NMR) is a versatile tool of chemical analysis allowing one to determine structures of molecules with atomic resolution. Particularly informative are two-dimensional (2D) experiments that directly identify atoms coupled by chemical bonds or a through-space interaction. Thus, NMR could potentially be powerful tool to study reactions in situ and explain their mechanisms. Unfortunately, 2D NMR is very time-consuming and thus often cannot serve as a "snapshot" technique for in situ reaction monitoring. Particularly difficult is the case of spectra, in which resonance frequencies vary in the course of reaction. This leads to resolution and sensitivity loss, often hindering the detection of transient products. In this paper we introduce a novel approach to correct such nonstationary 2D NMR signals and raise the detection limits over 10 times. We demonstrate success of its application for studying the mechanism of the reaction of AgSO₄-induced synthesis of diphenylmethane-type compounds. Several reactions occur in the studied mixture of benzene and toluene, all with rather low yield and leading to compounds with similar chemical shifts. Nevertheless, with the use of a proposed 2D NMR approach we were able to describe complex mechanisms of diphenylmethane formation involving AgSO₄-induced toluene deprotonation and formation of benzyl carbocation, followed by nucleophilic attacks.

Regularized inversion of the Laplace transform for series of experiments

Not only in low-field nuclear magnetic resonance, Laplace inversion is a relevant and challenging topic. Considerable conceptual and technical progress has been made, especially for the inversion of data encoding two decay dimensions. Distortion of spectra by overfitting of even moderate noise is counteracted requiring a priori smooth spectra. In this contribution, we treat the case of simple and fast one-dimensional decay experiments that are repeated many times in a series in order to study the evolution of a sample or process. Incorporating the a priori knowledge that also in the series dimension evolution should be smooth, peak position can be stabilized and resolution improved in the decay dimension. It is explained how the standard one-dimensional regularized Laplace inversion can be extended quite simply in order to include regularization in the series dimension. Obvious improvements compared with series of one-dimensional inversions are presented for simulated as well as experimental data. For the latter, comparison with multiexponential fitting is performed.

Insight into human insulin aggregation revisited using NMR derived translational diffusion parameters

The NMR derived translational diffusion coefficients were performed on unlabeled and uniformly labeled ¹³C,¹⁵N human insulin in water, both in neat, with zinc ions only, and in pharmaceutical formulation, containing only m-cresol as phenolic ligand, glycerol and zinc ions. The results show the dominant role of the pH parameter and the concentration on aggregation. The diffusion coefficient Dav was used for monitoring the overall average state of oligomeric ensemble in solution. The analysis of the experimental data of diffusion measurements, using the direct exponential curve resolution algorithm (DECRA) allows suggesting the two main components of the oligomeric ensemble. The 3D HSQC-iDOSY, (diffusion ordered HSQC) experiments performed on ¹³C, ¹⁵N-fully labeled insulin at the two pH values, 4 and 7.5, allow for the first time a more detailed experimental observation of individual components in the ensemble. The discussion involves earlier static and dynamic laser light scattering experiments and recent NMR derived translational diffusion results. The results bring new informations concerning the preparation of pharmaceutical formulation and in particular a role of Zn²⁺ ions. They also will enable better understanding and unifying the results of studies on insulin misfolding effects performed in solution by diverse physicochemical methods at different pH and concentration.

The GNAT: A new tool for processing NMR data

The GNAT (General NMR Analysis Toolbox) is a free and open-source software package for processing, visualising, and analysing NMR data. It supersedes the popular DOSY Toolbox, which has a narrower focus on diffusion NMR. Data import of most common formats from the major NMR platforms is supported, as well as a GNAT generic format. Key basic processing of NMR data (e.g., Fourier transformation, baseline correction, and phasing) is catered for within the program, as well as more advanced techniques (e.g., reference deconvolution and pure shift FID reconstruction). Analysis tools include DOSY and SCORE for diffusion data, ROSY T₁ /T₂ estimation for relaxation data, and PARAFAC for multilinear analysis. The GNAT is written for the MATLAB® language and comes with a user-friendly graphical user interface. The standard version is intended to run with a MATLAB installation, but completely free-standing compiled versions for Windows, Mac, and Linux are also freely available.

Molecular weight prediction in polystyrene blends. Unprecedented use of a genetic algorithm in pulse field gradient spin echo (PGSE) NMR

A genetic algorithm that uses boxcar functions (diffGA) has been applied for the first time in PGSE NMR. It reconstructs accurate diffusion coefficients for all the components of the mixture, and therefore predicts correct weight-average molecular weights for all of them. The results reported herein complement those obtained with established methods such as ITAMeD, CONTIN and TRAIn algorithms, and provide a detailed solution picture. Its robustness and limits have been stretched in order to ascertain the minimum separation within diffusion coefficients or relative proportion between components. In addition, the new genetic algorithm has been also applied to a mixture of small molecules, providing excellent results at very low computational times.

Obtaining sparse distributions in 2D inverse problems

The mathematics of inverse problems has relevance across numerous estimation problems in science and engineering. L₁ regularization has attracted recent attention in reconstructing the system properties in the case of sparse inverse problems; i.e., when the true property sought is not adequately described by a continuous distribution, in particular in Compressed Sensing image reconstruction. In this work, we focus on the application of L₁ regularization to a class of inverse problems; relaxation-relaxation, T₁-T₂, and diffusion-relaxation, D-T₂, correlation experiments in NMR, which have found widespread applications in a number of areas including probing surface interactions in catalysis and characterizing fluid composition and pore structures in rocks. We introduce a robust algorithm for solving the L₁ regularization problem and provide a guide to implementing it, including the choice of the amount of regularization used and the assignment of error estimates. We then show experimentally that L₁ regularization has significant advantages over both the Non-Negative Least Squares (NNLS) algorithm and Tikhonov regularization. It is shown that the L₁ regularization algorithm stably recovers a distribution at a signal to noise ratio<20 and that it resolves relaxation time constants and diffusion coefficients differing by as little as 10%. The enhanced resolving capability is used to measure the inter and intra particle concentrations of a mixture of hexane and dodecane present within porous silica beads immersed within a bulk liquid phase; neither NNLS nor Tikhonov regularization are able to provide this resolution. This experimental study shows that the approach enables discrimination between different chemical species when direct spectroscopic discrimination is impossible, and hence measurement of chemical composition within porous media, such as catalysts or rocks, is possible while still being stable to high levels of noise.

Reconstructing diffusion ordered NMR spectroscopy by simultaneous inversion of Laplace transform

2D diffusion-ordered NMR spectroscopy (DOSY) has been widely recognized as a powerful tool for analyzing mixtures and probing inter-molecular interactions in situ. But it is difficult to differentiate molecules with similar diffusion coefficients in presence of overlapped spectra. Its performance is susceptible to the number of chemical components, and usually gets worse when the number of components increases. Here, to alleviate the problem, numerical simultaneous inversion of Laplace transform (SILT) of many related variables is proposed for reconstructing DOSY spectrum (SILT-DOSY). The advantage of the proposed method in comparison to other methods is that it is capable of estimating the number of analytes more accurately and deriving corresponding component spectra, which in turn leads to the more reliable identification of the components.

PALMA, an improved algorithm for DOSY signal processing

NMR is a tool of choice for the measurement of diffusion coefficients of species in solution.

Multi-Timescale Dynamics in Intrinsically Disordered Proteins from NMR Relaxation and Molecular Simulation

Intrinsically disordered proteins (IDPs) access highly diverse ensembles of conformations in their functional states. Although this conformational plasticity is essential to their function, little is known about the dynamics underlying interconversion between accessible states. Nuclear magnetic resonance (NMR) relaxation rates contain a wealth of information about the time scales and amplitudes of motion in IDPs, but the highly dynamic nature of IDPs complicates their interpretation. We present a novel framework in which a series of molecular dynamics (MD) simulations are used in combination with experimental (15)N relaxation measurements to characterize the ensemble of dynamic processes contributing to the observed rates. By accounting for the distinct dynamic averaging present in the different conformational states sampled by the equilibrium ensemble, we are able to accurately describe both dynamic time scales and local and global conformational sampling. The method is robust, systematically improving agreement with independent experimental relaxation data, irrespective of the actively targeted rates, and suggesting interdependence of motions occurring on time scales varying over 3 orders of magnitude.

The CONTIN algorithm and its application to determine the size distribution of microgel suspensions

We review a powerful regularization method, known as CONTIN, for obtaining the size distribution of colloidal suspensions from dynamic light scattering data. We show that together with the so-called L-curve criterion for selecting the optimal regularization parameter, the method correctly describes the average size and size distribution of microgel suspensions independently characterized using small-angle neutron scattering. In contrast, we find that when using the default regularization process, where the regularizer is selected via the "probability to reject" method, the results are not as satisfactory.

Molecular weight prediction with no dependence on solvent viscosity. A quantitative pulse field gradient diffusion NMR approach

To progress on the practical issues of molecular weight prediction via diffusion NMR, the first log(Dη) vs. log(M_w) calibration curve is provided, allowing the easy and fast determination of weight-average molecular weights with no matter of the solvent used.

Controlling the stereoselectivity of rac-LA polymerization by chiral recognition induced the formation of homochiral dimeric metal alkoxides

Chiral recognition of monomeric Me₂MOR units resulting in the formation of homochiral dimeric species [Me₂M(μ-OR)]₂(M = Ga, In), leads to heteroselective catalysts for the ring opening polymerization ofrac-lactide (rac-LA).

Monitoring polydispersity by NMR diffusometry with tailored norm regularisation and moving-frame processing

A new way of processing diffusion NMR data is proposed that exploits _p-norm regularization of the inverse Laplace transform tailored to polydisperse macromolecular samples.

Dereplication: racing to speed up the natural products discovery process

Covering: 1993-2014 (July)To alleviate the dereplication holdup, which is a major bottleneck in natural products discovery, scientists have been conducting their research efforts to add tools to their "bag of tricks" aiming to achieve faster, more accurate and efficient ways to accelerate the pace of the drug discovery process. Consequently dereplication has become a hot topic presenting a huge publication boom since 2012, blending multidisciplinary fields in new ways that provide important conceptual and/or methodological advances, opening up pioneering research prospects in this field.

Systematic bias in NMR diffusion measurements on polydisperse systems

Least-squares fitting of the Stejskal-Tanner equation is a routine process in the measurement of molecular diffusion coefficient (MDC) using Nuclear Magnetic Resonance (NMR) Spectroscopy. It is simple and elegant. However, a bias of the MDC is noticed when the system is polydispersed. This is due to improper accounts of the diffusion coefficient distribution. Eventually, it leads to a discrepancy between the observed MDC and the statistical mean value of the distribution. To reveal the discrepancy, an analytical solution is derived when the diffusion data is taken a logarithmic linearization. Computer simulation is also applied to obtain a non-linear regression result. For a Gaussian distribution of the MDCs, the bias is proportional to the square of the distribution width (linear regression), but it is also inversely proportional to the statistical mean value of the distribution (non-linear regression). This indicates that the MDC derived from Stejskal-Tanner equation only holds well for narrow distribution of MDCs. Otherwise, molecular radius derived from the Stokes-Einstein equation needs to be reconsidered due to the incorrect estimation of the MDC.

l-Lactide polymerization studied by 1H NMR with diffusion-ordered spectroscopy (DOSY): a “One NMR Tube Experiment” providing data on monomer conversion, polymer structure, Mn and Mw

The standard NMR and DOSY techniques allowed for the first time the on-line study of the polymerization kinetics together with the determination of number average molar mass M_n and mass average molar mass M_w.

Solving 2D Fredholm Integral from Incomplete Measurements Using Compressive Sensing

We present an algorithm to solve the two-dimensional Fredholm integral of the first kind with tensor product structure from a limited number of measurements, with the goal of using this method to speed up nuclear magnetic resonance spectroscopy. This is done by incorporating compressive sensing-type arguments to fill in missing measurements, using a priori knowledge of the structure of the data. In the first step we recover a compressed data matrix from measurements that form a tight frame, and establish that these measurements satisfy the restricted isometry property. Recovery can be done from as few as 10% of the total measurements. In the second and third steps, we solve the zeroth-order regularization minimization problem using the Venkataramanan-Song-Hürlimann algorithm. We demonstrate the performance of this algorithm on simulated data and show that our approach is a realistic approach to speeding up the data acquisition.