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

The life and works of Christopher Dock,

2 p.l., 3-272 p

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

Transfer RNA–Mediated Editing in Threonyl-tRNA Synthetase The Class II Solution to the Double Discrimination Problem

AbstractThreonyl-tRNA synthetase, a class II synthetase, uses a unique zinc ion to discriminate against the isosteric valine at the activation step. The crystal structure of the enzyme with an analog of seryl adenylate shows that the noncognate serine cannot be fully discriminated at that step. We show that hydrolysis of the incorrectly formed ser-tRNAThr is performed at a specific site in the N-terminal domain of the enzyme. The present study suggests that both classes of synthetases use effectively the ability of the CCA end of tRNA to switch between a hairpin and a helical conformation for aminoacylation and editing. As a consequence, the editing mechanism of both classes of synthetases can be described as mirror images, as already seen for tRNA binding and amino acid activation

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

Biophysical modeling of the SARS-CoV-2 viral cycle reveals ideal antiviral targets

AbstractEffective therapies for COVID-19 are urgently needed. Presently there are more than 800 COVID-19 clinical trials globally, many with drug combinations, resulting in an empirical process with an enormous number of possible combinations. To identify the most promising potential therapies, we developed a biophysical model for the SARS-CoV-2 viral cycle and performed a sensitivity analysis for individual model parameters and all possible pairwise parameter changes (162 = 256 possibilities). We found that model-predicted virion production is fairly insensitive to changes in most viral entry, assembly, and release parameters, but highly sensitive to some viral transcription and translation parameters. Furthermore, we found a cooperative benefit to pairwise targeting of transcription and translation, predicting that combined targeting of these processes will be especially effective in inhibiting viral production.</jats:p