Crossings and nestings in colored set partitions

Chen, Deng, Du, Stanley, and Yan introduced the notion of $k$-crossings and $k$-nestings for set partitions, and proved that the sizes of the largest $k$-crossings and $k$-nestings in the partitions of an $n$-set possess a symmetric joint distribution. This work considers a generalization of these results to set partitions whose arcs are labeled by an $r$-element set (which we call \emph{$r$-colored set partitions}). In this context, a $k$-crossing or $k$-nesting is a sequence of arcs, all with the same color, which form a $k$-crossing or $k$-nesting in the usual sense. After showing that the sizes of the largest crossings and nestings in colored set partitions likewise have a symmetric joint distribution, we consider several related enumeration problems. We prove that $r$-colored set partitions with no crossing arcs of the same color are in bijection with certain paths in \NN^r, generalizing the correspondence between noncrossing (uncolored) set partitions and 2-Motzkin paths. Combining this with recent work of Bousquet-M\'elou and Mishna affords a proof that the sequence counting noncrossing 2-colored set partitions is P-recursive. We also discuss how our methods extend to several variations of colored set partitions with analogous notions of crossings and nestings.Comment: 25 pages; v2: material revised and condensed; v3 material further revised, additional section adde

Phylogenetic mixtures: Concentration of measure in the large-tree limit

The reconstruction of phylogenies from DNA or protein sequences is a major task of computational evolutionary biology. Common phenomena, notably variations in mutation rates across genomes and incongruences between gene lineage histories, often make it necessary to model molecular data as originating from a mixture of phylogenies. Such mixed models play an increasingly important role in practice. Using concentration of measure techniques, we show that mixtures of large trees are typically identifiable. We also derive sequence-length requirements for high-probability reconstruction.Comment: Published in at http://dx.doi.org/10.1214/11-AAP837 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

A non-conserved amino acid variant regulates differential signalling between human and mouse CD28

CD28 superagonistic antibodies (CD28SAb) can preferentially activate and expand immunosuppressive regulatory T cells (Treg) in mice. However, pre-clinical trials assessing CD28SAbs for the therapy of autoimmune diseases reveal severe systemic inflammatory response syndrome in humans, thereby implying the existence of distinct signalling abilities between human and mouse CD28. Here, we show that a single amino acid variant within the C-terminal proline-rich motif of human and mouse CD28 (P212 in human vs. A210 in mouse) regulates CD28-induced NF-κB activation and pro-inflammatory cytokine gene expression. Moreover, this Y209APP212 sequence in humans is crucial for the association of CD28 with the Nck adaptor protein for actin cytoskeleton reorganisation events necessary for CD28 autonomous signalling. This study thus unveils different outcomes between human and mouse CD28 signalling to underscore the importance of species difference when transferring results from preclinical models to the bedside