Tensor Completion for Multidimensional Inverse Problems with Applications to Magnetic Resonance Relaxometry

This thesis deals with tensor completion for the solution of multidimensional inverse problems. We study the problem of reconstructing an approximately low rank tensor from a small number of noisy linear measurements. New recovery guarantees, numerical algorithms, non-uniform sampling strategies, and parameter selection algorithms are developed. We derive a fixed point continuation algorithm for tensor completion and prove its convergence. A restricted isometry property (RIP) based tensor recovery guarantee is proved. Probabilistic recovery guarantees are obtained for sub-Gaussian measurement operators and for measurements obtained by non-uniform sampling from a Parseval tight frame. We show how tensor completion can be used to solve multidimensional inverse problems arising in NMR relaxometry. Algorithms are developed for regularization parameter selection, including accelerated k-fold cross-validation and generalized cross-validation. These methods are validated on experimental and simulated data. We also derive condition number estimates for nonnegative least squares problems. Tensor recovery promises to significantly accelerate N-dimensional NMR relaxometry and related experiments, enabling previously impractical experiments. Our methods could also be applied to other inverse problems arising in machine learning, image processing, signal processing, computer vision, and other fields.

ENZYME-FREE UBIQUITIN LIGATION. FROM NATIVE CHEMICAL LIGATION AND SPIN LABELING TO DIMERIZATION BY INTER-UBIQUITIN MIMICKING LINKAGE AND PH-DEPENDENT CONFORMATIONAL SWITCH

The delicate balance between the production and disposal of proteins is vital for the changes required in the cell to respond to given stimulus. Ubiquitination is a protein modification with a range of signaling outcomes when ubiquitin is attached to a protein through a highly ordered enzymatic cascade process. Understanding ubiquitination is a growing field and nowadays the application of chemical reactions allows the isolation of quantitative materials for structural studies. Therefore, in this dissertation it is described some of these suitable chemical methodologies to produce an isopeptide bond toward the polymerization of ubiquitin bypassing the enzymatic control with the purpose of showing if these chemical modifications have a direct impact on the structure of ubiquitin. First, the possibility of incorporating non-natural lysine analogs known as mercaptolysines into the polypeptide chain of Ubiquitin was explored when they were attached to ubiquitin by native chemical ligation at its C terminus. The sulfhydryl group was used for the attachment of a paramagnetic label to map the surface of ubiquitin. Second, the condensation catalyzed by silver nitrate was used for the dimer assembly. In particular, the main focus was on examining whether orthogonal protection and deprotection of each monomer have an impact on the reaction yield, since the synthetic strategy has been previously attempted successfully. Third, the formation of ubiquitin dimers was approached by building an inter-ubiquitin linkage mimicking the isopeptide bond with two approaches, the classic disulfide exchange as well as the thiol-ene click reaction by thermal initiation in aqueous conditions. After assembling the dimeric units, they were studied by Nuclear Magnetic Resonance, in order to establish a conformational state profile which depends on the pH conditions. The latter is a very important concept since some ligands have a preferred affinity when the protein-protein hydrophobic patches are in close proximity.