Nonrepetitive Colouring via Entropy Compression

A vertex colouring of a graph is \emph{nonrepetitive} if there is no path whose first half receives the same sequence of colours as the second half. A graph is nonrepetitively $k$-choosable if given lists of at least $k$ colours at each vertex, there is a nonrepetitive colouring such that each vertex is coloured from its own list. It is known that every graph with maximum degree $\Delta$ is $c\Delta^2$-choosable, for some constant $c$. We prove this result with $c=1$ (ignoring lower order terms). We then prove that every subdivision of a graph with sufficiently many division vertices per edge is nonrepetitively 5-choosable. The proofs of both these results are based on the Moser-Tardos entropy-compression method, and a recent extension by Grytczuk, Kozik and Micek for the nonrepetitive choosability of paths. Finally, we prove that every graph with pathwidth $k$ is nonrepetitively $O(k^{2})$-colourable.Comment: v4: Minor changes made following helpful comments by the referee

Differential Expression of Cytokines in Response to Respiratory Syncytial Virus Infection of Calves with High or Low Circulating 25-Hydroxyvitamin D3

Deficiency of serum levels of 25-hydroxyvitamin D3 has been related to increased risk of lower respiratory tract infections in children. Respiratory syncytial virus (RSV) is a leading cause of low respiratory tract infections in infants and young children. The neonatal calf model of RSV infection shares many features in common with RSV infection in infants and children. In the present study, we hypothesized that calves with low circulating levels of 25-hydroxyvitamin D3 (25(OH)D3) would be more susceptible to RSV infection than calves with high circulating levels of 25(OH)D3. Calves were fed milk replacer diets with different levels of vitamin D for a 10 wk period to establish two treatment groups, one with high (177 ng/ml) and one with low (32.5 ng/ml) circulating 25(OH)D3. Animals were experimentally infected via aerosol challenge with RSV. Data on circulating 25(OH)D3 levels showed that high and low concentrations of 25(OH)D3 were maintained during infection. At necropsy, lung lesions due to RSV were similar in the two vitamin D treatment groups. We show for the first time that RSV infection activates the vitamin D intracrine pathway in the inflamed lung. Importantly, however, we observed that cytokines frequently inhibited by this pathway in vitro are, in fact, either significantly upregulated (IL-12p40) or unaffected (IFN-γ) in the lungs of RSV-infected calves with high circulating levels of 25(OH)D3. Our data indicate that while vitamin D does have an immunomodulatory role during RSV infection, there was no significant impact on pathogenesis during the early phases of RSV infection. Further examination of the potential effects of vitamin D status on RSV disease resolution will require longer-term studies with immunologically sufficient and deficient vitamin D levels

Cancer Biomarker Discovery: The Entropic Hallmark

Background: It is a commonly accepted belief that cancer cells modify their transcriptional state during the progression of the disease. We propose that the progression of cancer cells towards malignant phenotypes can be efficiently tracked using high-throughput technologies that follow the gradual changes observed in the gene expression profiles by employing Shannon's mathematical theory of communication. Methods based on Information Theory can then quantify the divergence of cancer cells' transcriptional profiles from those of normally appearing cells of the originating tissues. The relevance of the proposed methods can be evaluated using microarray datasets available in the public domain but the method is in principle applicable to other high-throughput methods. Methodology/Principal Findings: Using melanoma and prostate cancer datasets we illustrate how it is possible to employ Shannon Entropy and the Jensen-Shannon divergence to trace the transcriptional changes progression of the disease. We establish how the variations of these two measures correlate with established biomarkers of cancer progression. The Information Theory measures allow us to identify novel biomarkers for both progressive and relatively more sudden transcriptional changes leading to malignant phenotypes. At the same time, the methodology was able to validate a large number of genes and processes that seem to be implicated in the progression of melanoma and prostate cancer. Conclusions/Significance: We thus present a quantitative guiding rule, a new unifying hallmark of cancer: the cancer cell's transcriptome changes lead to measurable observed transitions of Normalized Shannon Entropy values (as measured by high-throughput technologies). At the same time, tumor cells increment their divergence from the normal tissue profile increasing their disorder via creation of states that we might not directly measure. This unifying hallmark allows, via the the Jensen-Shannon divergence, to identify the arrow of time of the processes from the gene expression profiles, and helps to map the phenotypical and molecular hallmarks of specific cancer subtypes. The deep mathematical basis of the approach allows us to suggest that this principle is, hopefully, of general applicability for other diseases