Curves on Brill-Noether special K3 surfaces

Mukai showed that projective models of Brill-Noether general polarized K3 surfaces of genus $6-10$ and $12$ are obtained as linear sections of projective homogeneous varieties, and that their hyperplane sections are Brill-Noether general curves. In general, the question, raised by Knutsen, and attributed to Mukai, of whether the Brill-Noether generality of any polarized K3 surface $(S,H)$ is equivalent to the Brill-Noether generality of smooth curves $C$ in the linear system $|H|$, is still open. Using Lazarsfeld-Mukai bundle techniques, we answer this question in the affirmative for polarized K3 surfaces of genus $\leq 19$, which provides a new and unified proof even in the genera where Mukai models exist.Comment: 14 pages. Comments welcome! arXiv admin note: text overlap with arXiv:2206.0461

Brill--Noether theory via K3 surfaces

Brill--Noether theory studies the different projective embeddings that an algebraic curve admits. For a curve with a given projective embedding, we study the question of what other projective embeddings the curve can admit. Our techniques use curves on K3 surfaces. Lazarsfeld\u27s proof of the Gieseker--Petri theorem solidified the role of K3 surfaces in the Brill--Noether theory of curves. In this thesis, we further the study of the Brill--Noether theory of curves on K3 surfaces. We prove results concerning lifting line bundles from curves to K3 surfaces. Via an analysis of the stability of Lazarsfeld--Mukai bundles, we deduce a bounded version of a conjecture of Donagi--Morrison concerning when a Brill--Noether special line bundle of rank 3 on a curve on a polarized K3 surface lifts to a line bundle on the K3 surface. In joint work with Asher Auel, we also present a strategy for distinguishing Brill--Noether loci by studying the lifting of linear systems on curves in polarized K3 surfaces, which motivates a conjecture identifying the maximal Brill--Noether loci. Using our new lifting results, we prove cases of the maximal Brill--Noether loci conjecture. We also investigate the Brill--Noether theory of K3 surfaces and verify cases of a conjecture of Knutsen and Mukai concerning the Picard groups of K3 surfaces with Brill--Noether special curves

ACR-12 ionotropic acetylcholine receptor complexes regulate inhibitory motor neuron activity in Caenorhabditis elegans

Heterogeneity in the composition of neurotransmitter receptors is thought to provide functional diversity that may be important in patterning neural activity and shaping behavior (Dani and Bertrand, 2007; Sassoe-Pognetto, 2011). However, this idea has remained difficult to evaluate directly because of the complexity of neuronal connectivity patterns and uncertainty about the molecular composition of specific receptor types in vivo. Here we dissect how molecular diversity across receptor types contributes to the coordinated activity of excitatory and inhibitory motor neurons in the nematode Caenorhabditis elegans. We show that excitatory and inhibitory motor neurons express distinct populations of ionotropic acetylcholine receptors (iAChRs) requiring the ACR-12 subunit. The activity level of excitatory motor neurons is influenced through activation of nonsynaptic iAChRs (Jospin et al., 2009; Barbagallo et al., 2010). In contrast, synaptic coupling of excitatory and inhibitory motor neurons is achieved through a second population of iAChRs specifically localized at postsynaptic sites on inhibitory motor neurons. Loss of ACR-12 iAChRs from inhibitory motor neurons leads to reduced synaptic drive, decreased inhibitory neuromuscular signaling, and variability in the sinusoidal motor pattern. Our results provide new insights into mechanisms that establish appropriately balanced excitation and inhibition in the generation of a rhythmic motor behavior and reveal functionally diverse roles for iAChR-mediated signaling in this process

Biocatalytic self-assembly cascades

The properties of supramolecular materials are dictated by both kinetic and thermodynamic aspects, providing opportunities to dynamically regulate morphology and function. Herein, we demonstrate time-dependent regulation of supramolecular self-assembly by connected, kinetically competing enzymatic reactions. Starting from Fmoc-tyrosine phosphate and phenylalanine amide in the presence of an amidase and phosphatase, four distinct self-assembling molecules may be formed which each give rise to distinct morphologies (spheres, fibers, tubes/tapes and sheets). By varying the sequence or ratio in which the enzymes are added to mixtures of precursors, these structures can be (transiently) accessed and interconverted. The approach provides insights into dynamic self-assembly using competing pathways that may aid the design of soft nanostructures with tunable dynamic properties and life times

Monoaminergic Orchestration of Motor Programs in a Complex C. elegans Behavior

Monoamines provide chemical codes of behavioral states. However, the neural mechanisms of monoaminergic orchestration of behavior are poorly understood. Touch elicits an escape response in Caenorhabditis elegans where the animal moves backward and turns to change its direction of locomotion. We show that the tyramine receptor SER-2 acts through a $G\alpha_o$ pathway to inhibit neurotransmitter release from GABAergic motor neurons that synapse onto ventral body wall muscles. Extrasynaptic activation of SER-2 facilitates ventral body wall muscle contraction, contributing to the tight ventral turn that allows the animal to navigate away from a threatening stimulus. Tyramine temporally coordinates the different phases of the escape response through the synaptic activation of the fast-acting ionotropic receptor, LGC-55, and extrasynaptic activation of the slow-acting metabotropic receptor, SER-2. Our studies show, at the level of single cells, how a sensory input recruits the action of a monoamine to change neural circuit properties and orchestrate a compound motor sequence.Physic

Structural basis for the antiarrhythmic blockade of a potassium channel with a small molecule

Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/154620/1/fsb2fj201700349r.pd

Maximal Brill-Noether Loci via K3 Surfaces

We explain a strategy for distinguishing Brill-Noether loci in the moduli space of curves by studying the lifting of linear systems on curves in polarized K3 surfaces, which motivates a conjecture identifying the maximal Brill-Noether loci with respect to containment. Via an analysis of the stability of Lazarsfeld-Mukai bundles, we obtain new lifting results for linear systems of rank 3 which suffice to prove the maximal Brill-Noether loci conjecture in genus 9-19, 22, and 23.Comment: 29 pages, 1 figure. Comments welcome

Ligand–Receptor Interaction Modulates the Energy Landscape of Enzyme-Instructed Self-Assembly of Small Molecules

The concurrence of enzymatic reaction and ligand–receptor interactions is common for proteins, but rare for small molecules and has yet to be explored. Here we show that ligand–receptor interaction modulates the morphology of molecular assemblies formed by enzyme-instructed assembly of small molecules. While the absence of ligand–receptor interaction allows enzymatic dephosphorylation of a precursor to generate the hydrogelator that self-assembles to form long nanofibers, the presence of the ligand–receptor interaction biases the pathway to form precipitous aggregates containing short nanofibers. While the hydrogelators self-assemble to form nanofibers or nanoribbons that are unable to bind with the ligand (i.e., vancomycin), the addition of surfactant breaks up the assemblies to restore the ligand–receptor interaction. In addition, an excess amount of the ligands can disrupt the nanofibers and result in the precipitates. As the first example of the use of ligand–receptor interaction to modulate the kinetics of enzymatic self-assembly, this work not only provides a solution to evaluate the interaction between aggregates and target molecules but also offers new insight for understanding the emergent behavior of sophisticated molecular systems having multiple and parallel processes

Supramolecular Detoxification of Neurotoxic Nanofibrils of Small Molecules via Morphological Switch

Insoluble amyloid plagues are likely cytoprotective, but the cellular mechanism remains less known. To model β-amyloid we use a small peptide derivative to generate cytotoxic nanofibrils that cause the death of model neuron cells (i.e., PC12). The use of supramolecular interaction effectively converts the nanofibrils to nanoparticles that are innocuous to cells. This approach also removes the cytotoxicity of the fibrils to other mammalian cells (e.g., HeLa). Preliminary mechanistic study reveals that, in contrast to the fibrils, the particles promote the expression of TNFR2, a cell survival signal, and decrease the expression of TNFR1 and DR5, two extrinsic cell death receptors. As the first use of ligand–receptor interaction to abrogate the cytotoxicity of nanoscale assemblies of small molecules, this work illustrates an effective way to use supramolecular interaction to control the morphology of supramolecular assemblies for modulating their biological activity