On arithmetic and asymptotic properties of up-down numbers

Let $\sigma=(\sigma_1,..., \sigma_N)$, where $\sigma_i =\pm 1$, and let $C(\sigma)$ denote the number of permutations $\pi$ of $1,2,..., N+1,$ whose up-down signature $\mathrm{sign}(\pi(i+1)-\pi(i))=\sigma_i$, for $i=1,...,N$. We prove that the set of all up-down numbers $C(\sigma)$ can be expressed by a single universal polynomial $\Phi$, whose coefficients are products of numbers from the Taylor series of the hyperbolic tangent function. We prove that $\Phi$ is a modified exponential, and deduce some remarkable congruence properties for the set of all numbers $C(\sigma)$, for fixed $N$. We prove a concise upper-bound for $C(\sigma)$, which describes the asymptotic behaviour of the up-down function $C(\sigma)$ in the limit $C(\sigma) \ll (N+1)!$.Comment: Recommended for publication in Discrete Mathematics subject to revision

The Cyclohedron Test for Finding Periodic Genes in Time Course Expression Studies

The problem of finding periodically expressed genes from time course microarray experiments is at the center of numerous efforts to identify the molecular components of biological clocks. We present a new approach to this problem based on the cyclohedron test, which is a rank test inspired by recent advances in algebraic combinatorics. The test has the advantage of being robust to measurement errors, and can be used to ascertain the significance of top-ranked genes. We apply the test to recently published measurements of gene expression during mouse somitogenesis and find 32 genes that collectively are significant. Among these are previously identified periodic genes involved in the Notch/FGF and Wnt signaling pathways, as well as novel candidate genes that may play a role in regulating the segmentation clock. These results confirm that there are an abundance of exceptionally periodic genes expressed during somitogenesis. The emphasis of this paper is on the statistics and combinatorics that underlie the cyclohedron test and its implementation within a multiple testing framework.Comment: Revision consists of reorganization and further statistical discussion; 19 pages, 4 figure

Young Tableaux for Gene Expressions

Young tableaux are certain tabulararrangements of integers. Alfred Young introducedthem to describe irreducible representations of thesymmetric group at the end of the 19th century. We willuse combinatorial algorithms of permutations andYoung tableaux to describe a modification of theresearch method of Ahnert et al for identifyingsignificant genes in the biological processes studied inmicroarray experiments. In the last decade, DNAmicroarrays (DNA chips) have been used to study geneexpressions in many diseases such as cancer anddiabetes. To analyze data of microarray expressioncurves of genes, Ahnert et al associated permutations tothe data points of the microarray curves. Using Monte-Carlo simulation they established boundscorresponding to various maps of permutations for anymicroarray curve’s algorithmic compressibility whichmeasures its significance in the underlying biologicalprocess. Using the Robinson- Schensted-Knuthprocedure, we will associate Young tableaux topermutations corresponding to the data points ofmicroarray curves. We will calculate the bound ofAhnert et al corresponding to the map which gives thelength of the longest increasing or decreasingsubsequence of a permutation

Up-Down Sequences of Permutations for Gene Expressions

We will describe modifications of theresearch methods of Willbrand et al and Ahnert etal for identifying significant genes in the biologicalprocesses studied in microarray experiments.Willbrand et al introduced a new method ofidentifying significant genes by analyzingprobabilities of up-down signatures of microarrayexpression curves of genes. Ahnert et al generalizedthe method of Willbrand et al and establishedvarious bounds on any microarray curve’salgorithmic compressibility which measures itssignificance in underlying biological process. Wewill compute the probabilities of up-downsignatures of microarray curves defined byWillbrand et al by using Foulkes’ method forenumeration of permutations with prescribed updownsequences and the hook length formula ofFrame et al. Moreover, we will compute the boundof Ahnert et al corresponding to the map whichgives the number of permutations with the samepattern of rises and falls for any microarraycurve’s algorithmic compressibility. It isfascinating to see that how combinatorialalgorithms of permutations and Young tableauxare useful in analyzing data of gene expressionsand identifying significant genes in biologicalprocesses

Cyclebase.org—a comprehensive multi-organism online database of cell-cycle experiments

The past decade has seen the publication of a large number of cell-cycle microarray studies and many more are in the pipeline. However, data from these experiments are not easy to access, combine and evaluate. We have developed a centralized database with an easy-to-use interface, Cyclebase.org, for viewing and downloading these data. The user interface facilitates searches for genes of interest as well as downloads of genome-wide results. Individual genes are displayed with graphs of expression profiles throughout the cell cycle from all available experiments. These expression profiles are normalized to a common timescale to enable inspection of the combined experimental evidence. Furthermore, state-of-the-art computational analyses provide key information on both individual experiments and combined datasets such as whether or not a gene is periodically expressed and, if so, the time of peak expression. Cyclebase is available at http://www.cyclebase.org

Convex Rank Tests and Semigraphoids

Convex rank tests are partitions of the symmetric group which have desirable geometric properties. The statistical tests defined by such partitions involve counting all permutations in the equivalence classes. Each class consists of the linear extensions of a partially ordered set specified by data. Our methods refine existing rank tests of non-parametric statistics, such as the sign test and the runs test, and are useful for exploratory analysis of ordinal data. We establish a bijection between convex rank tests and probabilistic conditional independence structures known as semigraphoids. The subclass of submodular rank tests is derived from faces of the cone of submodular functions, or from Minkowski summands of the permutohedron. We enumerate all small instances of such rank tests. Of particular interest are graphical tests, which correspond to both graphical models and to graph associahedra

Bayesian meta-analysis for identifying periodically expressed genes in fission yeast cell cycle

The effort to identify genes with periodic expression during the cell cycle from genome-wide microarray time series data has been ongoing for a decade. However, the lack of rigorous modeling of periodic expression as well as the lack of a comprehensive model for integrating information across genes and experiments has impaired the effort for the accurate identification of periodically expressed genes. To address the problem, we introduce a Bayesian model to integrate multiple independent microarray data sets from three recent genome-wide cell cycle studies on fission yeast. A hierarchical model was used for data integration. In order to facilitate an efficient Monte Carlo sampling from the joint posterior distribution, we develop a novel Metropolis--Hastings group move. A surprising finding from our integrated analysis is that more than 40% of the genes in fission yeast are significantly periodically expressed, greatly enhancing the reported 10--15% of the genes in the current literature. It calls for a reconsideration of the periodically expressed gene detection problem.Comment: Published in at http://dx.doi.org/10.1214/09-AOAS300 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Extracting binary signals from microarray time-course data

This article presents a new method for analyzing microarray time courses by identifying genes that undergo abrupt transitions in expression level, and the time at which the transitions occur. The algorithm matches the sequence of expression levels for each gene against temporal patterns having one or two transitions between two expression levels. The algorithm reports a P-value for the matching pattern of each gene, and a global false discovery rate can also be computed. After matching, genes can be sorted by the direction and time of transitions. Genes can be partitioned into sets based on the direction and time of change for further analysis, such as comparison with Gene Ontology annotations or binding site motifs. The method is evaluated on simulated and actual time-course data. On microarray data for budding yeast, it is shown that the groups of genes that change in similar ways and at similar times have significant and relevant Gene Ontology annotations

Extracting binary signals from microarray time-course data

This article presents a new method for analyzing microarray time courses by identifying genes that undergo abrupt transitions in expression level, and the time at which the transitions occur. The algorithm matches the sequence of expression levels for each gene against temporal patterns having one or two transitions between two expression levels. The algorithm reports a P-value for the matching pattern of each gene, and a global false discovery rate can also be computed. After matching, genes can be sorted by the direction and time of transitions. Genes can be partitioned into sets based on the direction and time of change for further analysis, such as comparison with Gene Ontology annotations or binding site motifs. The method is evaluated on simulated and actual time-course data. On microarray data for budding yeast, it is shown that the groups of genes that change in similar ways and at similar times have significant and relevant Gene Ontology annotations

Hierarchical coordination of periodic genes in the cell cycle of Saccharomyces cerevisiae

<p>Abstract</p> <p>Background</p> <p>Gene networks are a representation of molecular interactions among genes or products thereof and, hence, are forming causal networks. Despite intense studies during the last years most investigations focus so far on inferential methods to reconstruct gene networks from experimental data or on their structural properties, e.g., degree distributions. Their structural analysis to gain functional insights into organizational principles of, e.g., pathways remains so far under appreciated.</p> <p>Results</p> <p>In the present paper we analyze cell cycle regulated genes in <it>S. cerevisiae</it>. Our analysis is based on the transcriptional regulatory network, representing causal interactions and not just associations or correlations between genes, and a list of known periodic genes. No further data are used. Partitioning the transcriptional regulatory network according to a graph theoretical property leads to a hierarchy in the network and, hence, in the information flow allowing to identify two groups of periodic genes. This reveals a novel conceptual interpretation of the working mechanism of the cell cycle and the genes regulated by this pathway.</p> <p>Conclusion</p> <p>Aside from the obtained results for the cell cycle of yeast our approach could be exemplary for the analysis of general pathways by exploiting the rich causal structure of inferred and/or curated gene networks including protein or signaling networks.</p