Stein's Method and Characters of Compact Lie Groups

Stein's method is used to study the trace of a random element from a compact Lie group or symmetric space. Central limit theorems are proved using very little information: character values on a single element and the decomposition of the square of the trace into irreducible components. This is illustrated for Lie groups of classical type and Dyson's circular ensembles. The approach in this paper will be useful for the study of higher dimensional characters, where normal approximations need not hold.Comment: 22 pages; same results, but more efficient exposition in Section 3.

The central limit problem for random vectors with symmetries

Motivated by the central limit problem for convex bodies, we study normal approximation of linear functionals of high-dimensional random vectors with various types of symmetries. In particular, we obtain results for distributions which are coordinatewise symmetric, uniform in a regular simplex, or spherically symmetric. Our proofs are based on Stein's method of exchangeable pairs; as far as we know, this approach has not previously been used in convex geometry and we give a brief introduction to the classical method. The spherically symmetric case is treated by a variation of Stein's method which is adapted for continuous symmetries.Comment: AMS-LaTeX, uses xy-pic, 23 pages; v3: added new corollary to Theorem

Pervasive Adaptive Protein Evolution Apparent in Diversity Patterns around Amino Acid Substitutions in Drosophila simulans

In Drosophila, multiple lines of evidence converge in suggesting that beneficial substitutions to the genome may be common. All suffer from confounding factors, however, such that the interpretation of the evidence—in particular, conclusions about the rate and strength of beneficial substitutions—remains tentative. Here, we use genome-wide polymorphism data in D. simulans and sequenced genomes of its close relatives to construct a readily interpretable characterization of the effects of positive selection: the shape of average neutral diversity around amino acid substitutions. As expected under recurrent selective sweeps, we find a trough in diversity levels around amino acid but not around synonymous substitutions, a distinctive pattern that is not expected under alternative models. This characterization is richer than previous approaches, which relied on limited summaries of the data (e.g., the slope of a scatter plot), and relates to underlying selection parameters in a straightforward way, allowing us to make more reliable inferences about the prevalence and strength of adaptation. Specifically, we develop a coalescent-based model for the shape of the entire curve and use it to infer adaptive parameters by maximum likelihood. Our inference suggests that ∼13% of amino acid substitutions cause selective sweeps. Interestingly, it reveals two classes of beneficial fixations: a minority (approximately 3%) that appears to have had large selective effects and accounts for most of the reduction in diversity, and the remaining 10%, which seem to have had very weak selective effects. These estimates therefore help to reconcile the apparent conflict among previously published estimates of the strength of selection. More generally, our findings provide unequivocal evidence for strongly beneficial substitutions in Drosophila and illustrate how the rapidly accumulating genome-wide data can be leveraged to address enduring questions about the genetic basis of adaptation

A narrative review of the potential pharmacological influence and safety of ibuprofen on coronavirus disease 19 (COVID-19), ACE2, and the immune system: a dichotomy of expectation and reality

The coronavirus disease 19 (COVID-19) pandemic is currently the most acute healthcare challenge in the world. Despite growing knowledge of the nature of Severe Acute Respiratory Syndrome coronavirus-2 (SARS-CoV-2), treatment options are still poorly defined. The safety of non-steroidal anti-inflammatory drugs (NSAIDs), specifically ibuprofen, has been openly questioned without any supporting evidence or clarity over dose, duration, or temporality of administration. This has been further conflicted by the initiation of studies to assess the efficacy of ibuprofen in improving outcomes in severe COVID-19 patients. To clarify the scientific reality, a literature search was conducted alongside considerations of the pharmacological properties of ibuprofen in order to construct this narrative review. The literature suggests that double-blind, placebo-controlled study results must be reported and carefully analysed for safety and efficacy in patients with COVID-19 before any recommendations can be made regarding the use of ibuprofen in such patients. Limited studies have suggested: (i) no direct interactions between ibuprofen and SARS-CoV-2 and (ii) there is no evidence to suggest ibuprofen affects the regulation of angiotensin-converting-enzyme 2 (ACE2), the receptor for COVID-19, in human studies. Furthermore, in vitro studies suggest ibuprofen may facilitate cleavage of ACE2 from the membrane, preventing membrane-dependent viral entry into the cell, the clinical significance of which is uncertain. Additionally, in vitro evidence suggests that inhibition of the transcription factor nuclear factor-κB (NF-kB) by ibuprofen may have a role in reducing excess inflammation or cytokine release in COVID-19 patients. Finally, there is no evidence that ibuprofen will aggravate or increase the chance of infection of COVID-19

Generalizations of the General Lotto and Colonel Blotto Games

In this paper, we generalize the General Lotto game (budget constraints satisfied in expectation) and the Colonel Blotto game (budget constraints hold with probability one) to allow for battlefield valuations that are heterogeneous across battlefields and asymmetric across players, and for the players to have asymmetric resource constraints. We completely characterize Nash equilibrium in the generalized version of the General Lotto game and then show how this characterization can be applied to identify equilibria in the Colonel Blotto version of the game. In both games, we find that there exist sets of non-pathological parameter configurations of positive Lebesgue measure with multiple payoff nonequivalent equilibria

Stochastic loss and gain of symmetric divisions in the C. elegans epidermis perturbs robustness of stem cell number

Biological systems are subject to inherent stochasticity. Nevertheless, development is remarkably robust, ensuring the consistency of key phenotypic traits such as correct cell numbers in a certain tissue. It is currently unclear which genes modulate phenotypic variability, what their relationship is to core components of developmental gene networks, and what is the developmental basis of variable phenotypes. Here, we start addressing these questions using the robust number of Caenorhabditis elegans epidermal stem cells, known as seam cells, as a readout. We employ genetics, cell lineage tracing, and single molecule imaging to show that mutations in lin-22, a Hes-related basic helix-loop-helix (bHLH) transcription factor, increase seam cell number variability. We show that the increase in phenotypic variability is due to stochastic conversion of normally symmetric cell divisions to asymmetric and vice versa during development, which affect the terminal seam cell number in opposing directions. We demonstrate that LIN-22 acts within the epidermal gene network to antagonise the Wnt signalling pathway. However, lin-22 mutants exhibit cell-to-cell variability in Wnt pathway activation, which correlates with and may drive phenotypic variability. Our study demonstrates the feasibility to study phenotypic trait variance in tractable model organisms using unbiased mutagenesis screens