Department of Mathematics

It is an important problem in algebraic combinatorics to deduce the Schur function expansion of a symmetric function whose expansion in terms of the fundamental quasisymmetric function is known. For example, formulas are known for the fundamental expansion of a Macdonald symmetric function and for the plethysm of two Schur functions, while the Schur expansions of these expressions are still elusive. Based on work of Egge, Loehr and Warrington, Garsia and Remmel provided a method to obtain the Schur expansion from the fundamental expansion by replacing each quasisymmetric function by a Schur function (not necessarily indexed by a partition) and using straightening rules to obtain the Schur expansion. Here we provide a new method that only involves the coefficients of the quasisymmetric functions indexed by partitions and the quasi-Kostka matrix. As an application, we identify the lexicographically largest term in the Schur expansion of the plethysm of two Schur functions. We provide the Schur expansion of sw [sh](x, y) for w = 2, 3, 4 using novel symmetric chain decompositions of Young’s lattice for partitions in a w × h box. For w = 4, this is the first known combinatorial expression for the coefficient of sλ in sw [sh] for two-row partitions λ, and for w = 3 the combinatorial expression is new.

Abstract: The classical Klein–Maskit combination theorems provide sufficient conditions to construct new Kleinian groups using old ones. There are two distinct but closely related combination theorems: the first deals with amalgamated free products, whereas the second deals with HNN extensions. This article gives analogs of both combination theorems for Anosov subgroups.

Light microscopy methods have continued to advance allowing for unprecedented analysis of various cell types in tissues including the brain. Although the functional state of some cell types such as microglia can be determined by morphometric analysis, techniques to perform robust, quick, and accurate measurements have not kept pace with the amount of imaging data that can now be generated. Most of these image segmentation tools are further burdened by an inability to assess structures in three-dimensions. Despite the rise of machine learning techniques, the nature of some biological structures prevents the training of several current day implementations. Here we present PrestoCell, a novel use of persistence-based clustering to segment cells in light microscopy images, as a customized Python-based tool that leverages the free multidimensional image viewer Napari. In evaluating and comparing PrestoCell to several existing tools, including 3DMorph, Omipose, and Imaris, we demonstrate that PrestoCell produces image segmentations that rival these solutions. In particular, our use of cell nuclei information resulted in the ability to correctly segment individual cells that were interacting with one another to increase accuracy. These benefits are in addition to the simplified graphically based user refinement of cell masks that does not require expensive commercial software licenses. We further demonstrate that PrestoCell can complete image segmentation in large samples from light sheet microscopy, allowing quantitative analysis of these large datasets. As an open-source program that leverages freely available visualization software, with minimum computer requirements, we believe that PrestoCell can significantly increase the ability of users without data or computer science expertise to perform complex image analysis.

The function of the smooth muscle cells lining the walls of mammalian systemic arteries and arterioles is to regulate the diameter of the vessels to control blood flow and blood pressure. Here, we describe an in silico model, which we call the 'Hernandez-Hernandez model', of electrical and Ca²⁺ signaling in arterial myocytes based on new experimental data indicating sex-specific differences in male and female arterial myocytes from murine resistance arteries. The model suggests the fundamental ionic mechanisms underlying membrane potential and intracellular Ca²⁺ signaling during the development of myogenic tone in arterial blood vessels. Although experimental data suggest that K_V1.5 channel currents have similar amplitudes, kinetics, and voltage dependencies in male and female myocytes, simulations suggest that the K_V1.5 current is the dominant current regulating membrane potential in male myocytes. In female cells, which have larger K_V2.1 channel expression and longer time constants for activation than male myocytes, predictions from simulated female myocytes suggest that K_V2.1 plays a primary role in the control of membrane potential. Over the physiological range of membrane potentials, the gating of a small number of voltage-gated K⁺ channels and L-type Ca²⁺ channels are predicted to drive sex-specific differences in intracellular Ca²⁺ and excitability. We also show that in an idealized computational model of a vessel, female arterial smooth muscle exhibits heightened sensitivity to commonly used Ca²⁺ channel blockers compared to male. In summary, we present a new model framework to investigate the potential sex-specific impact of antihypertensive drugs.

Mexican, Puerto Rican, and Central American Ancestry (MPRCA) individuals represent 82% of US Latinos. An intergenerational group of MPRCA women and allies met to discuss persistent underrepresentation of MPRCA women in STEM, identifying multi-level challenges and solutions. Implementation of these solutions is important and will benefit MPRCA women and the entire academic community.

Polydiacetylene (PDA) Langmuir films are well known for their blue to red chromatic transitions in response to a variety of stimuli, including UV light, heat, bio-molecule bindings and mechanical stress. In this work, we detail the ability to tune PDA Langmuir films to exhibit discrete chromatic transitions in response to applied mechanical stress. Normal and shear-induced transitions were quantified using the Surface Forces Apparatus and established to be binary and tunable as a function of film formation conditions. Both monomer alkyl tail length and metal cations were used to manipulate the chromatic transition force threshold to enable discrete force sensing from ~50 to ~500 nN μm^-2 for normal loading and ~2 to ~40 nN μm^-2 for shear-induced transitions, which are appropriate for biological cells. The utility of PDA thin-film sensors was demonstrated with the slime mold Physarum polycephalum. The fluorescence readout of the films enabled: the area explored by Physarum to be visualized, the forces involved in locomotion to be quantified, and revealed novel puncta formation potentially associated with Physarum sampling its environment.

Dense innervation of the heart by the sympathetic nervous system (SNS) allows cardiac output to respond appropriately to the needs of the body under varying conditions, but occasionally the abrupt onset of SNS activity can trigger cardiac arrhythmias. Sympathetic activity leads to the release of norepinephrine (NE) onto cardiomyocytes, activating β₁-adrenergic receptors (β₁-ARs) and leading to the production of the second messenger cyclic AMP (cAMP). Upon sudden activation of β₁-ARs in experiments, intracellular cAMP can transiently rise to a high concentration before converging to a steady state level. Although changes to cellular cAMP concentration are important in modulating the overall cardiovascular response to sympathetic tone, the underlying mechanisms of the cAMP transients and the parameters that control their magnitude are unclear. We reduce a detailed computational model of the β₁-adrenergic signaling cascade to a system of two differential equations by eliminating extraneous variables and applying quasi-steady state approximation. The structure of the reduced model reveals that the large cAMP transients associated with abrupt β₁-AR activation are generated by the interplay of production/degradation of cAMP and desensitization/resensitization of β₁-ARs. The reduced model is used to predict how the dynamics of intracellular cAMP depend on the concentrations of norepinephrine (NE), phosphodiesterases 3 and 4 (PDE3,4), G-protein coupled receptor kinase 2 (GRK2), and β₁-AR, in healthy conditions and a simple model of early stages of heart failure. The key findings of the study are as follows: 1) Applying a reduced model of the dynamics of cardiac sympathetic signaling we show that the concentrations of two variables, cAMP and non-desensitized β₁-AR, capture the overall dynamics of sympathetic signaling; 2) The key factors influencing cAMP production are AC activity and PDE3,4 activity, while those that directly impact β₁-AR phosphorylation are GRK2 and PKA₁. Thus, disease states that affect sympathetic control of the heart can be thoroughly assessed by studying AC activity, PDE3,4, GRK2 and PKA activity, as these factors directly impact cAMP production/degradation and β₁-AR (de) phosphorylation and are therefore predicted to comprise the most effective pharmaceutical targets in diseases affecting cardiac β₁-adrenergic signaling.

State-dependent sodium channel blockers are often prescribed to treat cardiac arrhythmias, but many sodium channel blockers are known to have pro-arrhythmic side effects. While the anti and proarrhythmic potential of a sodium channel blocker is thought to depend on the characteristics of its rate-dependent block, the mechanisms linking these two attributes are unclear. Furthermore, how specific properties of rate-dependent block arise from the binding kinetics of a particular drug is poorly understood. Here, we examine the rate-dependent effects of the sodium channel blocker lidocaine by constructing and analyzing a novel drug-channel interaction model. First, we identify the predominant mode of lidocaine binding in a 24 variable Markov model for lidocaine-sodium channel interaction by Moreno et al. Specifically, we find that (1) the vast majority of lidocaine bound to sodium channels is in the neutral form, i.e., the binding of charged lidocaine to sodium channels is negligible, and (2) neutral lidocaine binds almost exclusively to inactivated channels and, upon binding, immobilizes channels in the inactivated state. We then develop a novel 3-variable lidocaine-sodium channel interaction model that incorporates only the predominant mode of drug binding. Our low-dimensional model replicates an extensive amount of the voltage-clamp data used to parameterize the Moreno et al. model. Furthermore, the effects of lidocaine on action potential upstroke velocity and conduction velocity in our model are similar to those predicted by the Moreno et al. model. By exploiting the low-dimensionality of our model, we derive an algebraic expression for level of rate-dependent block as a function of pacing frequency, restitution properties, diastolic and plateau potentials, and drug binding rate constants. Our model predicts that the level of rate-dependent block is sensitive to alterations in restitution properties and increases in diastolic potential, but it is insensitive to variations in the shape of the action potential waveform and lidocaine binding rates.

Geometrically infinite Kleinian groups have non-conical limit sets with the cardinality of the continuum. In this paper, we construct a geometrically infinite Fuchsian group such that the Hausdorff dimension of the non-conical limit set equals zero. For finitely generated, geometrically infinite Kleinian groups, we prove that the Hausdorff dimension of the non-conical limit set is positive.