Compressed Sensing for Reconstructing Coherent Multidimensional Spectra

We apply two sparse reconstruction techniques, the least absolute shrinkage and selection operator (LASSO) and the sparse exponential mode analysis (SEMA), to two-dimensional (2D) spectroscopy. The algorithms are first tested on model data, showing that both are able to reconstruct the spectra using only a fraction of the data required by the traditional Fourier-based estimator. Through the analysis of a sparsely sampled experimental fluorescence detected 2D spectra of LH2 complexes, we conclude that both SEMA and LASSO can be used to significantly reduce the required data, still allowing to reconstruct the multidimensional spectra. Of the two techniques, it is shown that SEMA offers preferable performance, providing more accurate estimation of the spectral line widths and their positions. Furthermore, SEMA allows for off-grid components, enabling the use of a much smaller dictionary than the LASSO, thereby improving both the performance and lowering the computational complexity for reconstructing coherent multidimensional spectra

Development and applications of NMR relaxation methods to study protein and ligand dynamics.

This thesis involves the study of proteins. These molecules exist in many variants, and perform most of our bodily functions. This includes signaling (for example the presence of capsaicin, which makes food spicy), molecular transport (including oxygen), catalysis (including food digestion) and part of our body structure (such as muscles). So it is perhaps not too strange that an overwhelming fraction of the medicines used and developed today target different proteins. The building blocks of proteins are 20 different amino acids. These amino acids can be linked in any conceivable combination and length to chains. Each protein is defined by their unique chain of amino acids. During the creation of each chain, it spontaneously folds due to the electrochemical forces that are always present between amino acids. The result of this folding process is called the protein structure, which help proteins to perform their tasks. To fully understand protein function, knowledge of the protein structure needs to be complemented with knowledge of protein dynamics, the inherent flexibility, and movements of the structure. The dynamic properties of proteins is the focus of this thesis, and the pharmaceutical industry its likeliest area of application. In the thesis I studied three different proteins. Galectin-3 is a protein that binds to sugar molecules, which can be found both inside and outside of cells. Problems with this protein are connected to a variety of diseases including cancer, fibrosis, and various forms of dementia. The protein BRD4 helps to control protein production in cells. This gives it a large impact on cancer progression and several types of cardiovascular diseases. When it comes to Ubiquitin, one of its main purposes is to signal which proteins have served their purpose and are to be degraded and recycled. In the thesis I study Ubiquitin only indirectly, as I use it as a model system given the wealth of previously published data to compare my results with. The main experimental method that I used is called Nuclear Magnetic Resonance spectroscopy (NMR), which share the same underlying technology as Magnetic Resonance Imaging (MRI) used in hospitals. My work also utilises results from other experimental methods, as different methods often complement each other. Important methods include Isothermal Calorimetry (ITC), which allows for measurements of the energy difference when two molecules bind to each other. Molecular Dynamics (MD) simulations, are a form of computer calculations that studies protein dynamics and interactions in atomistic detail. X-ray crystallography, uses high energy X-ray beams to discover protein structures. A total of five articles are presented in this thesis. The topic of the first article is the protein BRD4. It is a very large protein, with an amino acid chain of up to 1362 residues long. BRD4 is considered to have a large potential as a target for cancer treatments, but its length makes it hard to study. In the article we look at the impact on the protein dynamics from restricting studies of BRD4 to protein fragments of different lengths. The second article is a study on the interaction between Galectin-3 and a small molecule (similar to a pharmaceutical) from the perspective of the small molecule. This is accomplished by directly measuring on the fluorine atoms located on the small molecule. The third article is a study of the impact on interaction between Galectin-3 and a small molecule in the presence of Dimethyl Sulfoxide (DMSO). The largest effect of DMSO is an increased viscosity, which makes the interaction slower. The fourth and fifth articles presents a method for faster measurements of protein dynamics using NMR, which is especially useful for unstable and low-concentration samples. My hope is that this thesis work helps us gain an improved understanding on how different parts of proteins affect each other and on how proteins interact with small molecules (especially regarding BRD4 and Galectin-3). In addition, I hope that this work has provided future scientists additional tools for continued studies of proteins and their dynamics

Designing sampling schemes for multi-dimensional data

In this work, we propose a method for determining a non-uniform sampling scheme for multi-dimensional signals by solving a convex optimization problem reminiscent of the sensor selection problem. The resulting sampling scheme minimizes the sum of the Cramér–Rao lower bounds for the parameters of interest, given a desired number of sampling points. The proposed framework allows for selecting an arbitrary subset of the parameters detailing the model, as well as weighing the importance of the different parameters. Also presented is a scheme for incorporating any imprecise a priori knowledge of the locations of the parameters, as well as defining estimation performance bounds for the parameters of interest. Numerical examples illustrate the efficiency of the proposed scheme