U(R) PHASE RETRIEVAL, LOCAL NORMALIZING FLOWS, AND HIGHER ORDER FOURIER TRANSFORMS

The classical phase retrieval problem arises in contexts ranging from speech recognition to x-ray crystallography and quantum state tomography. The generalization to matrix frames is natural in the sense that it corresponds to quantum tomography of impure states. Chapter 1 provides computable global stability bounds for the quasi-linear analysis map $\beta$ and a path forward for understanding related problems in terms of the differential geometry of key spaces. In particular, Chapter 1 manifests a Whitney stratification of the positive semidefinite matrices of low rank which allows us to ``stratify'' the computation of the global stability bound. We show that for the impure state case no such global stability bounds can be obtained for the non-linear analysis map $\alpha$ with respect to certain natural distance metrics. Finally, our computation of the global lower Lipschitz constant for the $\beta$ analysis map provides novel conditions for a frame to be generalized phase retrievable. In Chapter 2 we develop the concept of local normalizing flows. Normalizing flows provide an elegant approach to generative modeling that allows for efficient sampling and exact density evaluation of unknown data distributions. However, current techniques have significant limitations in their expressivity when the data distribution is supported on a low-dimensional manifold or has a non-trivial topology. We introduce a novel statistical framework for learning a mixture of local normalizing flows as ``chart maps'' over the data manifold. Our framework augments the expressivity of recent approaches while preserving the signature property of normalizing flows, that they admit exact density evaluation. We learn a suitable atlas of charts for the data manifold via a vector quantized auto-encoder (VQ-AE) and the distributions over them using a conditional flow. We validate experimentally that our probabilistic framework enables existing approaches to better model data distributions over complex manifolds. In Chapter 3 we examine higher order Fourier transforms in both discrete and continuous contexts. We demonstrate a connection to a matrix time variant of the free Schr\"{o}dinger equation, as well as a potential application to magnetic resonance imaging. In the discrete case we show that the reconstruction properties of higher order Fourier frames are intricately related to quadratic Gauss sums

Aminoacylation at the atomic level in class IIa aminoacyl-tRNA synthetases

The crystal structures of histidyl- (HisRS) and threonyl-tRNA synthetase (ThrRS) from E. coli and glycyl-tRNA synthetase (GlyRS) from T. thermophilus, all homodimeric class IIa enzymes, were determined in enzyme-substrate and enzyme-product states corresponding to the two steps of aminoacylation. HisRS was complexed with the histidine analog histidinol plus ATP and with histidyl-adenylate, while GlyRS was complexed with ATP and with glycyl-adenylate; these complexes represent the enzyme-substrate and enzyme-product states of the first step of aminoacylation, i.e. the amino acid activation. In both enzymes the ligands occupy the substrate-binding pocket of the N-terminal active site domain, which contains the classical class II aminoacyl-tRNA synthetase fold. HisRS interacts in the same fashion with the histidine, adenosine and α-phosphate moieties of the substrates and intermediate, and GlyRS interacts in the same way with the adenosine and α-phosphate moieties in both states. In addition to the amino acid recognition, there is one key mechanistic difference between the two enzymes: HisRS uses an arginine whereas GlyRS employs a magnesium ion to catalyze the activation of the amino acid. ThrRS was complexed with its cognate tRNA and ATP, which represents the enzyme-substrate state of the second step of aminoacylation, i.e. the transfer of the amino acid to the 3′-terminal ribose of the tRNA. All three enzymes utilize class II conserved residues to interact with the adenosine-phosphate. ThrRS binds tRNA^Thr so that the acceptor stem enters the active site pocket above the adenylate, with the 3′-terminal OH positioned to pick up the amino acid, and the anticodon loop interacts with the C-terminal domain whose fold is shared by all three enzymes. We can thus extend the principles of tRNA binding to the other two enzymes

Zinc ion mediated amino acid discrimination by threonyl-tRNA synthetase

Accurate translation of the genetic code depends on the ability of aminoacyl-tRNA synthetases to distinguish between similar amino acids. In order to investigate the basis of amino acid recognition and to understand the role played by the zinc ion present in the active site of threonyl-tRNA synthetase, we have determined the crystal structures of complexes of an active truncated form of the enzyme with a threonyl adenylate analog or threonine. The zinc ion is directly involved in threonine recognition, forming a pentacoordinate intermediate with both the amino group and the side chain hydroxyl. Amino acid activation experiments reveal that the enzyme shows no activation of isosteric valine, and activates serine at a rate 1,000-fold less than that of cognate threonine. This study demonstrates that the zinc ion is neither strictly catalytic nor structural and suggests how the zinc ion ensures that only amino acids that possess a hydroxyl group attached to the β-position are activated

The α-Amino Group of the Threonine Substrate as The General Base During tRNA Aminoacylation: A New Version of Substrate-Assisted Catalysis Predicted by Hybrid DFT

Density functional theory-based methods in combination with large chemical models have been used to investigate the mechanism of the second half-reaction catalyzed by Thr-tRNA synthetase; aminoacyl transfer from Thr-AMP onto the (A76)3'OH of the cognate tRNA. In particular, we have examined pathways in which an active site His309 residue is either protonated or neutral (i.e., potentially able to act as a base). In the protonated His309-assisted mechanism, the rate-limiting step is formation of the tetrahedral intermediate. The barrier for this step is 155.0 kJ mol(−1) and thus, such a pathway is concluded to not be enzymatically feasible. For the neutral His309-assisted mechanism two models were used with the difference being whether Lys465 was included. For either model the barrier of the rate-limiting step is below the upper-thermodynamic enzymatic limit of ∼125 kJ mol(−1). Specifically, without Lys465 the rate-limiting barrier is 122.1 kJ mol(−1) and corresponds to a rotation about the tetrahedral intermediates C(carb)—OH bond. For the model with Lys465 the rate-limiting barrier is slightly lower and corresponds to the formation of the tetrahedral intermediate. Importantly, for both neutral His309’ models the neutral amino group of the threonyl substrate directly acts as the proton accepter; in the formation of the tetrahedral intermediate the (A76)3'OH proton is directly transferred onto the Thr-NH(2). Therefore, the overall mechanism follows a general substrate assisted catalytic mechanism

Subretinal Hyperreflective Material in the Comparison of Age-Related Macular Degeneration Treatments Trials

PurposeTo evaluate the association of subretinal hyperreflective material (SHRM) with visual acuity (VA), geographic atrophy (GA), and scar in the Comparison of Age-Related Macular Degeneration Treatments Trials (CATT).DesignProspective cohort study within a randomized clinical trial.ParticipantsThe 1185 CATT participants.MethodsMasked readers graded scar and GA on fundus photography and fluorescein angiography and graded SHRM on time-domain and spectral-domain (SD) optical coherence tomography (OCT) throughout 104 weeks. Measurements of SHRM height and width in the fovea, within the center 1 mm(2), or outside the center 1mm(2) were obtained on SD OCT images at 56 (n = 76) and 104 (n = 66) weeks.Main outcome measuresPresence of SHRM, as well as location and size, and associations with VA, scar, and GA.ResultsAmong CATT participants, the percentage with SHRM at enrollment was 77%, decreasing to 68% at 4 weeks after treatment and to 54% at 104 weeks. At 104 weeks, scar was present more often in eyes with persistent SHRM than in eyes with SHRM that resolved (64% vs. 31%; P < 0.0001). Among eyes with detailed evaluation of SHRM at weeks 56 (n = 76) and 104 (n = 66), mean VA letter score was 73.5 (standard error [SE], 2.8), 73.1 (SE, 3.4), 65.3 (SE, 3.5), and 63.9 (SE, 3.7) when SHRM was absent, present outside the central 1 mm(2), present within the central 1 mm(2) but not the foveal center, or present at the foveal center (P = 0.02), respectively. When SHRM was present, the median maximum height under the fovea, within the central 1 mm(2) including the fovea and anywhere within the scan, was 86 μm, 120 μm, and 122 μm, respectively. Visual acuity was decreased with greater SHRM height and width (P < 0.05).ConclusionsIn eyes with neovascular age-related macular degeneration (AMD), SHRM is common and often persists after anti-vascular endothelial growth factor treatment. At 2 years, eyes with scar were more likely to have SHRM than other eyes. Greater SHRM dimensions were associated with worse VA. In eyes with neovascular AMD, SHRM is an important morphologic biomarker