A hypothesis is presented for the universal properties of operators evolving under Hamiltonian dynamics in many-body systems. The hypothesis states that successive Lanczos coefficients in the continued fraction expansion of the Green's functions grow linearly with rate $\alpha$ in generic systems, with an extra logarithmic correction in 1d. The rate $\alpha$ --- an experimental observable --- governs the exponential growth of operator complexity in a sense that is made precise. This exponential growth prevails beyond semiclassical or large-$N$ limits. Moreover, $\alpha$ upper bounds a large class of operator complexity measures, including the out-of-time-order correlator. As a result, a sharp bound is obtained on Lyapunov exponents $\lambda_L \leq 2 \alpha$, which complements and improves the known universal low-temperature bound $\lambda_L \leq 2 \pi T$. Our results are illustrated in paradigmatic examples such as non-integrable spin chains, the Sachdev-Ye-Kitaev model, and classical models. Finally hypothesis is used in in conjunction with the recursion method to develop a technique for computing diffusion constants.

Characterizing single cells has become an important field in understanding human structure and function, both in healthy and diseased individuals. Recently, this characterization has included multi-omics methods, which use complementary data sets to better understand cellular function from several angles. RNA-seq is a staple for characterizing cell phenotypes, but fails to cover the entire picture of the cells phenotype. Protein expression has long been considered as a primary driver in characterizing cells, but inefficiencies in capturing meaningful protein data has long been a problem. Recently, antibodies conjugated to oligonucleotides have helped address these inefficiencies, bringing the scale of single-cell proteomics assays more on par with single-cell RNA assays. However, these assays have characterized only cell-surface marker proteins, and have not yet shown an ability to capture proteins with more meaningful intracellular data. Of deeper interest are intranuclear transcription factors, which are directly related to RNA expression patterns. This dissertation describes strategies of single-cell protein and RNA capture, including design of probe oligonucleotides and antibody conjugation methods. It also describes use of antibody fragments and the small molecule pitstop 2 as a method of intranuclear protein assaying. Finally, the dissertation describes dual RNA-protein characterization of cell cycle in thousands of cultured cells, and additional conclusions drawn from this combined data.

Chapter 1 will provide an overview of the biosynthetic origins of paxilline indole diterpenes, followed by a thorough account of their known bioactivities, structure-activity relationships, and studies in relation to their development for human medicine, veterinary medicine, and agricultural uses. A brief summary of prior synthetic efforts from our laboratory will preface the discussion of two phases of a single project. The first phase will cover the development of single-pot, indium-mediated alkenylation-alkylation for the synthesis of trisubstituted alkenes from enoxysilanes, alkynes, and alpha-iodocarbonyl compounds. A short discussion on radical reactions of organoindium reagents with relevance to this project will be included. The second phase will discuss the application of this novel alkenylation-alkylation towards the synthesis of paxilline indole diterpene-derived biochemical probes bearing unprecedented (natural or synthetic) functionality pendant to the terpene core of the parent natural products at C26. Successful indium-mediated alkenylation-alkylation was realized by taking advantage of physical properties of the alkyne coupling partner. While the desired product of HAT-initiated radical-polar crossover cyclization cascade was observed, low reaction efficiencies hampered completion of the natural product derivatives. These probes would represent the first examples of modifications that are not otherwise possible through known total syntheses or semisynthetic derivatization.Chapter 2 will discuss the development of an iron-catalyzed, directed HAT-initiated hydrofunctionalization for selective reactions of polyenols. A short account of the modern landscape of metal-mediated hydrogen atom transfer-initiated hydrofunctionalizations will be included, followed by a brief discussion of known strategies for site-selective HAT based on innate alkene reactivity. A detailed analysis of empirical and computational investigations undertaken by other laboratories that informed our research direction will be followed by a discussion of the experiments our laboratory conducted to identify the origin of a directing effect in iron-catalyzed HAT processes involving polyenols as substrates. Strong mechanistic evidence suggests that the reaction is directed by the allylic hydroxy groups via complexation between the reactive iron(III) hydride and the polyenol. Exploratory work with polyenols bearing diverse alkene substitution patterns, degrees of unsaturation, and steric environments were able to provide a reliable guide for the behavior of directed HAT for different substrates, allowing for reasonable predictions for how a substrate will perform. This reaction was applied to the synthesis of novel beta- and gamma-amino alcohols via a directed hydrohydrazonation. This represents the first directed HAT-initiated hydrofunctionalization in the literature.