The Discrepancy of the Lex-Least De Bruijn Sequence

We answer the following question of R. L. Graham: What is the discrepancy of the lexicographically-least binary de Bruijn sequence? Here, "discrepancy" refers to the maximum (absolute) difference between the number of ones and the number of zeros in any initial segment of the sequence. We show that the answer is $\Theta(2^n \log n/n)$.Comment: 11 pages, 0 figure

RNAprofiling 2.0: Enhanced cluster analysis of structural ensembles

Understanding the base pairing of an RNA sequence provides insight into its molecular structure.By mining suboptimal sampling data, RNAprofiling 1.0 identifies the dominant helices in low-energy secondary structures as features, organizes them into profiles which partition the Boltzmann sample, and highlights key similarities/differences among the most informative, i.e. selected, profiles in a graphical format. Version 2.0 enhances every step of this approach. First, the featured substructures are expanded from helices to stems. Second, profile selection includes low-frequency pairings similar to featured ones. In conjunction, these updates extend the utility of the method to sequences up to length 600, as evaluated over a sizable dataset. Third, relationships are visualized in a decision tree which highlights the most important structural differences. Finally, this cluster analysis is made accessible to experimental researchers in a portable format as an interactive webpage, permitting a much greater understanding of trade-offs among different possible base pairing combinations.Comment: 9 pages, 2 figures; supplement 6 pages, 3 figures, 1 tabl

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

Meander Graphs

We consider a Markov chain Monte Carlo approach to the uniform sampling of meanders. Combinatorially, a meander $M = [A:B]$ is formed by two noncrossing perfect matchings, above $A$ and below $B$ the same endpoints, which form a single closed loop. We prove that meanders are connected under appropriate pairs of balanced local moves, one operating on $A$ and the other on $B$. We also prove that the subset of meanders with a fixed $B$ is connected under a suitable local move operating on an appropriately defined meandric triple in $A$. We provide diameter bounds under such moves, tight up to a (worst case) factor of two. The mixing times of the Markov chains remain open

Large Deviations for Random Trees

We consider large random trees under Gibbs distributions and prove a Large Deviation Principle (LDP) for the distribution of degrees of vertices of the tree. The LDP rate function is given explicitly. An immediate consequence is a Law of Large Numbers for the distribution of vertex degrees in a large random tree. Our motivation for this study comes from the analysis of RNA secondary structures.Comment: 10 page