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

Combinatorics of Equivariant Cohomology: Flags and Regular Nilpotent Hessenberg Varieties

The field of Schubert Calculus deals with computations in the cohomology rings of certain algebraic varieties, including flag varieties and Schubert varieties. In the equivariant setting, GKM theory turns multiplication in the cohomology ring of certain varieties into a combinatorial computation. This dissertation uses combinatorial tools, including Billey’s formula, to do Schubert calculus computations in several varieties. First we address do computations in the equivariant cohomology of full and partial flag varieties, the classical spaces in Schubert calculus. We then do computations in the equivariant cohomology of a family of non-classical spaces: regular nilpotent Hessenberg varieties. The final chapter gives a complete presentation for the cohomology ring of the Peterson variety, a type of regular nilpotent Hessenberg variety

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

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

The containment poset of type A Hessenberg varieties

Flag varieties are well-known algebraic varieties with many important geometric, combinatorial, and representation theoretic properties. A Hessenberg variety is a subvariety of a flag variety identified by two parameters: an element X of the Lie algebra g and a Hessenberg subspace H ⊆ g. This paper considers when two Hessenberg spaces define the same Hessenberg variety when paired with X. To answer this question we present the containment poset Px of type A Hessenberg varieties with a fixed first parameter X and give a simple and elegant proof that if X is not a multiple of the element 1 then the Hessenberg spaces containing the Borel subalgebra determine distinct Hessenberg varieties. Lastly we give a natural involution on Px that induces a homeomorphism of varieties and prove additional properties of Px when X is a regular nilpotent element