Monomer-dimer tatami tilings of square regions

We prove that the number of monomer-dimer tilings of an $n\times n$ square grid, with $m<n$ monomers in which no four tiles meet at any point is $m2^m+(m+1)2^{m+1}$, when $m$ and $n$ have the same parity. In addition, we present a new proof of the result that there are $n2^{n-1}$ such tilings with $n$ monomers, which divides the tilings into $n$ classes of size $2^{n-1}$. The sum of these tilings over all monomer counts has the closed form $2^{n-1}(3n-4)+2$ and, curiously, this is equal to the sum of the squares of all parts in all compositions of $n$. We also describe two algorithms and a Gray code ordering for generating the $n2^{n-1}$ tilings with $n$ monomers, which are both based on our new proof.Comment: Expanded conference proceedings: A. Erickson, M. Schurch, Enumerating tatami mat arrangements of square grids, in: 22nd International Workshop on Combinatorial Al- gorithms (IWOCA), volume 7056 of Lecture Notes in Computer Science (LNCS), Springer Berlin / Heidelberg, 2011, p. 12 pages. More on Tatami tilings at http://alejandroerickson.com/joomla/tatami-blog/collected-resource

Statistical models for RNA-seq data derived from a two-condition 48-replicate experiment

High-throughput RNA sequencing (RNA-seq) is now the standard method to determine differential gene expression. Identifying differentially expressed genes crucially depends on estimates of read count variability. These estimates are typically based on statistical models such as the negative binomial distribution, which is employed by the tools edgeR, DESeq and cuffdiff. Until now, the validity of these models has usually been tested on either low-replicate RNA-seq data or simulations. A 48-replicate RNA-seq experiment in yeast was performed and data tested against theoretical models. The observed gene read counts were consistent with both log-normal and negative binomial distributions, while the mean-variance relation followed the line of constant dispersion parameter of ~0.01. The high-replicate data also allowed for strict quality control and screening of bad replicates, which can drastically affect the gene read-count distribution. RNA-seq data have been submitted to ENA archive with project ID PRJEB5348.Comment: 15 pages 6 figure

How many biological replicates are needed in an RNA-seq experiment and which differential expression tool should you use?

An RNA-seq experiment with 48 biological replicates in each of 2 conditions was performed to determine the number of biological replicates ($n_r$) required, and to identify the most effective statistical analysis tools for identifying differential gene expression (DGE). When $n_r=3$, seven of the nine tools evaluated give true positive rates (TPR) of only 20 to 40 percent. For high fold-change genes ($|log_{2}(FC)|\gt2$) the TPR is $\gt85$ percent. Two tools performed poorly; over- or under-predicting the number of differentially expressed genes. Increasing replication gives a large increase in TPR when considering all DE genes but only a small increase for high fold-change genes. Achieving a TPR $\gt85$% across all fold-changes requires $n_r\gt20$. For future RNA-seq experiments these results suggest $n_r\gt6$, rising to $n_r\gt12$ when identifying DGE irrespective of fold-change is important. For $6 \lt n_r \lt 12$, superior TPR makes edgeR the leading tool tested. For $n_r \ge12$, minimizing false positives is more important and DESeq outperforms the other tools.Comment: 21 Pages and 4 Figures in main text. 9 Figures in Supplement attached to PDF. Revision to correct a minor error in the abstrac

Paired domination in prisms of graphs

The paired domination number $γ_{pr}(G)$ of a graph G is the smallest cardinality of a dominating set S of G such that ⟨S⟩ has a perfect matching. The generalized prisms πG of G are the graphs obtained by joining the vertices of two disjoint copies of G by |V(G)| independent edges. We provide characterizations of the following three classes of graphs: $γ_{pr}(πG) = 2γ_{pr}(G)$ for all πG; $γ_{pr}(K₂☐ G) = 2γ_{pr}(G)$; $γ_{pr}(K₂☐ G) = γ_{pr}(G)$

The depression of a graph and k-kernels

An edge ordering of a graph G is an injection f : E(G) → R, the set of real numbers. A path in G for which the edge ordering f increases along its edge sequence is called an f-ascent ; an f-ascent is maximal if it is not contained in a longer f-ascent. The depression of G is the smallest integer k such that any edge ordering f has a maximal f-ascent of length at most k. A k-kernel of a graph G is a set of vertices U ⊆ V (G) such that for any edge ordering f of G there exists a maximal f-ascent of length at most k which neither starts nor ends in U. Identifying a k-kernel of a graph G enables one to construct an infinite family of graphs from G which have depression at most k. We discuss various results related to the concept of k-kernels, including an improved upper bound for the depression of trees

Wild-type yeast gene read counts from 48 replicate experiment

<p>These data are RNA-seq read counts for 48 biological replicates of wild-type yeast.</p> <p>The reads were aligned to the Saccharomyces cerevisiae genome with Tophat and gene counts determined with htseq-count. Reference genome and gene annotations were taken from Ensembl.</p> <p>See the linked papers for more details and the source for the raw data.</p