Models for transcript quantification from RNA-Seq

RNA-Seq is rapidly becoming the standard technology for transcriptome analysis. Fundamental to many of the applications of RNA-Seq is the quantification problem, which is the accurate measurement of relative transcript abundances from the sequenced reads. We focus on this problem, and review many recently published models that are used to estimate the relative abundances. In addition to describing the models and the different approaches to inference, we also explain how methods are related to each other. A key result is that we show how inference with many of the models results in identical estimates of relative abundances, even though model formulations can be very different. In fact, we are able to show how a single general model captures many of the elements of previously published methods. We also review the applications of RNA-Seq models to differential analysis, and explain why accurate relative transcript abundance estimates are crucial for downstream analyses

Selecting universities: personal preference and rankings

Polyhedral geometry can be used to quantitatively assess the dependence of rankings on personal preference, and provides a tool for both students and universities to assess US News and World Report rankings

Combinatorics of least squares trees

A recurring theme in the least squares approach to phylogenetics has been the discovery of elegant combinatorial formulas for the least squares estimates of edge lengths. These formulas have proved useful for the development of efficient algorithms, and have also been important for understanding connections among popular phylogeny algorithms. For example, the selection criterion of the neighbor-joining algorithm is now understood in terms of the combinatorial formulas of Pauplin for estimating tree length. We highlight a phylogenetically desirable property that weighted least squares methods should satisfy, and provide a complete characterization of methods that satisfy the property. The necessary and sufficient condition is a multiplicative four point condition that the the variance matrix needs to satisfy. The proof is based on the observation that the Lagrange multipliers in the proof of the Gauss--Markov theorem are tree-additive. Our results generalize and complete previous work on ordinary least squares, balanced minimum evolution and the taxon weighted variance model. They also provide a time optimal algorithm for computation

MAVID: Constrained ancestral alignment of multiple sequences

We describe a new global multiple alignment program capable of aligning a large number of genomic regions. Our progressive alignment approach incorporates the following ideas: maximum-likelihood inference of ancestral sequences, automatic guide-tree construction, protein based anchoring of ab-initio gene predictions, and constraints derived from a global homology map of the sequences. We have implemented these ideas in the MAVID program, which is able to accurately align multiple genomic regions up to megabases long. MAVID is able to effectively align divergent sequences, as well as incomplete unfinished sequences. We demonstrate the capabilities of the program on the benchmark CFTR region which consists of 1.8Mb of human sequence and 20 orthologous regions in marsupials, birds, fish, and mammals. Finally, we describe two large MAVID alignments: an alignment of all the available HIV genomes and a multiple alignment of the entire human, mouse and rat genomes

The Mathematics of Phylogenomics

The grand challenges in biology today are being shaped by powerful high-throughput technologies that have revealed the genomes of many organisms, global expression patterns of genes and detailed information about variation within populations. We are therefore able to ask, for the first time, fundamental questions about the evolution of genomes, the structure of genes and their regulation, and the connections between genotypes and phenotypes of individuals. The answers to these questions are all predicated on progress in a variety of computational, statistical, and mathematical fields. The rapid growth in the characterization of genomes has led to the advancement of a new discipline called Phylogenomics. This discipline results from the combination of two major fields in the life sciences: Genomics, i.e., the study of the function and structure of genes and genomes; and Molecular Phylogenetics, i.e., the study of the hierarchical evolutionary relationships among organisms and their genomes. The objective of this article is to offer mathematicians a first introduction to this emerging field, and to discuss specific mathematical problems and developments arising from phylogenomics.Comment: 41 pages, 4 figure

Reconstructing Trees from Subtree Weights

The tree-metric theorem provides a necessary and sufficient condition for a dissimilarity matrix to be a tree metric, and has served as the foundation for numerous distance-based reconstruction methods in phylogenetics. Our main result is an extension of the tree-metric theorem to more general dissimilarity maps. In particular, we show that a tree with n leaves is reconstructible from the weights of the m-leaf subtrees provided that n \geq 2m-1

Towards the Human Genotope

The human genotope is the convex hull of all allele frequency vectors that can be obtained from the genotypes present in the human population. In this paper we take a few initial steps towards a description of this object, which may be fundamental for future population based genetics studies. Here we use data from the HapMap Project, restricted to two ENCODE regions, to study a subpolytope of the human genotope. We study three different approaches for obtaining informative low-dimensional projections of this subpolytope. The projections are specified by projection onto few tag SNPs, principal component analysis, and archetypal analysis. We describe the application of our geometric approach to identifying structure in populations based on single nucleotide polymorphisms