Valid plane trees: Combinatorial models for RNA secondary structures with Watson-Crick base pairs

The combinatorics of RNA plays a central role in biology. Mathematical biologists have several commonly-used models for RNA: words in a fixed alphabet (representing the primary sequence of nucleotides) and plane trees (representing the secondary structure, or folding of the RNA sequence). This paper considers an augmented version of the standard model of plane trees, one that incorporates some observed constraints on how the folding can occur. In particular we assume the alphabet consists of complementary pairs, for instance the Watson-Crick pairs A-U and C-G of RNA. Given a word in the alphabet, a valid plane tree is a tree for which, when the word is folded around the tree, each edge matches two complementary letters. Consider the graph whose vertices are valid plane trees for a fixed word and whose edges are given by Condon, Heitsch, and Hoos's local moves. We prove this graph is connected. We give an explicit algorithm to construct a valid plane tree from a primary sequence, assuming that at least one valid plane tree exists. The tree produced by our algorithm has other useful characterizations, including a uniqueness condition defined by local moves. We also study enumerative properties of valid plane trees, analyzing how the number of valid plane trees depends on the choice of sequence length and alphabet size. Finally we show that the proportion of words with at least one valid plane tree goes to zero as the word size increases. We also give some open questions.Comment: 15 pages, 10 figure

Geometric combinatorics and computational molecular biology: branching polytopes for RNA sequences

Questions in computational molecular biology generate various discrete optimization problems, such as DNA sequence alignment and RNA secondary structure prediction. However, the optimal solutions are fundamentally dependent on the parameters used in the objective functions. The goal of a parametric analysis is to elucidate such dependencies, especially as they pertain to the accuracy and robustness of the optimal solutions. Techniques from geometric combinatorics, including polytopes and their normal fans, have been used previously to give parametric analyses of simple models for DNA sequence alignment and RNA branching configurations. Here, we present a new computational framework, and proof-of-principle results, which give the first complete parametric analysis of the branching portion of the nearest neighbor thermodynamic model for secondary structure prediction for real RNA sequences.Comment: 17 pages, 8 figure

Splines in geometry and topology

This survey paper describes the role of splines in geometry and topology, emphasizing both similarities and differences from the classical treatment of splines. The exposition is non-technical and contains many examples, with references to more thorough treatments of the subject.Comment: 18 page

Chevalley-Monk and Giambelli formulas for Peterson Varieties

A Peterson variety is a subvariety of the flag variety $G/B$ defined by certain linear conditions. Peterson varieties appear in the construction of the quantum cohomology of partial flag varieties and in applications to the Toda flows. Each Peterson variety has a one-dimensional torus $S^1$ acting on it. We give a basis of Peterson Schubert classes for $H_{S^1}^*(Pet)$ and identify the ring generators. In type A Harada-Tymoczko gave a positive Monk formula, and Bayegan-Harada gave Giambelli's formula for multiplication in the cohomology ring. This paper gives a Chevalley-Monk rule and Giambelli's formula for all Lie types

Air quality and error quantity: pollution and performance in a high-skilled, quality-focused occupation

We provide the first evidence that short-term exposure to air pollution affects the work performance of a group of highly-skilled, quality-focused employees. We repeatedly observe the decision-making of individual professional baseball umpires, quasi-randomly assigned to varying air quality across time and space. Unique characteristics of this setting combined with high-frequency data disentangle effects of multiple pollutants and identify previously under-explored acute effects. We find a 1 ppm increase in 3-hour CO causes an 11.5% increase in the propensity of umpires to make incorrect calls and a 10 mg/m3 increase in 12-hour PM2.5 causes a 2.6% increase. We control carefully for a variety of potential confounders and results are supported by robustness and falsification checks

Pressão intra-abdominal em cães

Sob consenso recente em pacientes humanos, os valores basais da pressão intra-abdominal (PIA) e seus prováveis limiares em casos mórbidos, foram estipulados com intuito de favorecer a homogeneidade de estudos científicos, além de fornecer diretrizes para conduta diagnóstica e terapêutica destes pacientes. Valores basais e alterações na pressão intra-abdominal em animais não são ainda suficientemente conhecidos e cientificamente determinados. Médicos veterinários necessitam conhecer melhor os estados mórbidos que cursam com Hipertensão Intra-Abdominal (HIA) ou Síndrome de Compartimento Abdominal (SCA). O presente estudo objetivou testar em cães a técnica já descrita sob consenso na medicina como modelo de mensuração da pressão intra-abdominal, assim como determinar os valores normais para a espécie. Foram utilizados 15 cães hígidos, machos e fêmeas, homogêneos e todos sem raça definida. A mensuração da pressão intra-abdominal foi realizada por meio da técnica indireta de sondagem vesical e utilização de coluna de água com régua graduada em cm de H2O, sendo seu valor final convertido para a unidade de mm de Hg. Foi observado valor mínimo subatmosférico (abaixo de zero mm de Hg) até o valor máximo de 3,75 mmHg. Houve importante variação entre os valores encontrados individualmente em determinados cães, colocando sob discussão questões como massa corporal, freqüência/volume total de solução fisiológica a ser infundida e agitação destes pacientes no momento do exame. Os valores limítrofes encontrados são considerados fisiológicos, indicando confiabilidade da técnica e possibilidade do seu emprego clínico. A ausência de sedação e utilização de sonda uretral não inviabilizou a realização da mensuração, porém podendo culminar numa sub ou superestimativa dos valores encontrados