Some inequalities for kk-colored partition functions

Motivated by a partition inequality of Bessenrodt and Ono, we obtain analogous inequalities for $k$-colored partition functions $p_{-k}(n)$ for all $k\geq2$. This enables us to extend the $k$-colored partition function multiplicatively to a function on $k$-colored partitions, and characterize when it has a unique maximum. We conclude with one conjectural inequality that strengthens our results.Comment: 11 pages, 1 tabl

The two possible workflows of the <i>1</i><i>mm</i> approach.

<p>The <i>1 </i><i>mm</i> approach can be used in two different ways: For cross-species microarray analyses or for comparative transcriptomics studies. Each of the two workflows results in a probe-masked cdf colored in green. The tips of the colored pieces show the flow of information. The blue colored pieces show the input data provided by the user, whereas the yellow, orange, and red pieces show the two or three steps of the <i>1 </i><i>mm</i> approach leading to a probe mask. The <i>species-specific module</i> consists of the sequence similarity step with the microarray-specific probe sequences and the species-specific transcript sequences as input, and the probe selection step that results in a list of probe sets containing only reliable probes. The <i>species-specific module</i> can be used for generating a probe-masked cdf for cross-species microarray analyses of non-model species. Two different <i>species-specific modules</i> can be used with an orthologous gene list for generating a probe-masked cdf for comparative transcriptomics studies.</p

Optimized Probe Masking for Comparative Transcriptomics of Closely Related Species

<div><p>Microarrays are commonly applied to study the transcriptome of specific species. However, many available microarrays are restricted to model organisms, and the design of custom microarrays for other species is often not feasible. Hence, transcriptomics approaches of non-model organisms as well as comparative transcriptomics studies among two or more species often make use of cost-intensive RNAseq studies or, alternatively, by hybridizing transcripts of a query species to a microarray of a closely related species. When analyzing these cross-species microarray expression data, differences in the transcriptome of the query species can cause problems, such as the following: (i) lower hybridization accuracy of probes due to mismatches or deletions, (ii) probes binding multiple transcripts of different genes, and (iii) probes binding transcripts of non-orthologous genes. So far, methods for (i) exist, but these neglect (ii) and (iii). Here, we propose an approach for comparative transcriptomics addressing problems (i) to (iii), which retains only transcript-specific probes binding transcripts of orthologous genes. We apply this approach to an <i>Arabidopsis lyrata</i> expression data set measured on a microarray designed for <i>Arabidopsis thaliana</i>, and compare it to two alternative approaches, a <i>sequence-based</i> approach and a genomic DNA <i>hybridization-based</i> approach. We investigate the number of retained probe sets, and we validate the resulting expression responses by qRT-PCR. We find that the proposed approach combines the benefit of sequence-based stringency and accuracy while allowing the expression analysis of much more genes than the alternative sequence-based approach. As an added benefit, the proposed approach requires probes to detect transcripts of orthologous genes only, which provides a superior base for biological interpretation of the measured expression responses.</p></div

A table of verified candidate genes.

*<p>: , ae: array element, At: <i>Arabidopsis thaliana</i>, Al: <i>Arabidopsis lyrata</i>.</p><p>For each gene the corresponding array element name and the orthologous gene pair (locus <i>A. thaliana</i> by the TAIR id and locus <i>A. lyrata</i> by the Phytozome gene id) are listed. Additionally, the composition of the probe set in the 1 mm mask, the 0 mm mask, and the gDNA mask are shown. Originally, each probe set consists of 11 probes. The “–” represents a masked probe, “0” a perfectly matching probe, “1” a probe matching with one mismatch, and “x” represents a transcript-unspecific probe. The labeled column contains the expression values of the 1 h treatment versus no treatment experiments derived from qRT-PCR. The next four columns contain the expression values of the 1 h treatment versus no treatment experiments derived from the three probe masking approaches and the non-masking approach. The last column contains the category used for computation of the Pearson correlation coefficient.</p

Assignment of a probe set to one of five groups.

<p>The assignment of a probe set to a specific group depends on the characteristics of the matching probes in the probe sets. The term <i>isoform</i> refers to one possible transcript of a gene.</p

Number of probe sets obtained by the three masking approaches and the naive approach.

<p>The height of each bar shows the number of probe sets falling in one of the following categories: <i>transcript-specific</i>: retained probe sets targeting orthologs, not affected by cross hybridization, and containing at least 3 probes; <i>no match</i>: probe sets matching no transcript in <i>A. thaliana</i> or <i>A. lyrata</i>; <i>cross hybridization</i>: probe sets affected by cross hybridization; <i>non-ortholog</i>: probe sets targeting non-orthologs, and <i>less than 3 probes</i>: probe sets containing less than 3 matching probes in the 1 mm approach but at least 3 probes in the other approach. The naive approach, the gDNA approach, and the 1 mm approach retain approximately 16000 transcript-specific probe sets, and the 0 mm approach retains approximately 10500 transcript-specific probe sets.</p

qRT-PCR verification of masked and non-masked microarray data.

<p>Pearson correlation coefficients of (i) the expression responses resulting from the three masking approaches and the naive approach, and (ii) the expression responses resulting from qRT-PCR of the genes of category A, B, C, and D (Methods Candidate selection). We find that the 1 mm approach and the 0 mm approach yield similar Pearson correlation coefficients that are higher than those of the gDNA approach and the naive approach.</p

Additional file 18: of Ambient temperature and genotype differentially affect developmental and phenotypic plasticity in Arabidopsis thaliana

Correlations among temperature responses in individual accessions. (PDF 12069 kb