(Total) Vector Domination for Graphs with Bounded Branchwidth

Given a graph $G=(V,E)$ of order $n$ and an $n$-dimensional non-negative vector $d=(d(1),d(2),\ldots,d(n))$, called demand vector, the vector domination (resp., total vector domination) is the problem of finding a minimum $S\subseteq V$ such that every vertex $v$ in $V\setminus S$ (resp., in $V$) has at least $d(v)$ neighbors in $S$. The (total) vector domination is a generalization of many dominating set type problems, e.g., the dominating set problem, the $k$-tuple dominating set problem (this $k$ is different from the solution size), and so on, and its approximability and inapproximability have been studied under this general framework. In this paper, we show that a (total) vector domination of graphs with bounded branchwidth can be solved in polynomial time. This implies that the problem is polynomially solvable also for graphs with bounded treewidth. Consequently, the (total) vector domination problem for a planar graph is subexponential fixed-parameter tractable with respectto $k$, where $k$ is the size of solution.Comment: 16 page

Minimal vertex covers on finite-connectivity random graphs - a hard-sphere lattice-gas picture

The minimal vertex-cover (or maximal independent-set) problem is studied on random graphs of finite connectivity. Analytical results are obtained by a mapping to a lattice gas of hard spheres of (chemical) radius one, and they are found to be in excellent agreement with numerical simulations. We give a detailed description of the replica-symmetric phase, including the size and the entropy of the minimal vertex covers, and the structure of the unfrozen component which is found to percolate at connectivity $c\simeq 1.43$. The replica-symmetric solution breaks down at $c=e\simeq 2.72$. We give a simple one-step replica symmetry broken solution, and discuss the problems in interpretation and generalization of this solution.Comment: 32 pages, 9 eps figures, to app. in PRE (01 May 2001

The helper NLR immune protein NRC3 mediates the hypersensitive cell death caused by the cell-surface receptor Cf-4

Cell surface pattern recognition receptors (PRRs) activate immune responses that can include the hypersensitive cell death. However, the pathways that link PRRs to the cell death response are poorly understood. Here, we show that the cell surface receptor-like protein Cf-4 requires the intracellular nucleotide-binding domain leucine-rich repeat containing receptor (NLR) NRC3 to trigger a confluent cell death response upon detection of the fungal effector Avr4 in leaves of Nicotiana benthamiana. This NRC3 activity requires an intact N-terminal MADA motif, a conserved signature of coiled-coil (CC)-type plant NLRs that is required for resistosome-mediated immune responses. A chimeric protein with the N-terminal α1 helix of Arabidopsis ZAR1 swapped into NRC3 retains the capacity to mediate Cf-4 hypersensitive cell death. Pathogen effectors acting as suppressors of NRC3 can suppress Cf-4-triggered hypersensitive cell-death. Our findings link the NLR resistosome model to the hypersensitive cell death caused by a cell surface PRR

Two NLR immune receptors acquired high-affinity binding to a fungal effector through convergent evolution of their integrated domain

A subset of plant NLR immune receptors carry unconventional integrated domains in addition to their canonical domain architecture. One example is rice Pik-1 that comprises an integrated heavy metal-associated (HMA) domain. Here, we reconstructed the evolutionary history of Pik-1 and its NLR partner, Pik-2, and tested hypotheses about adaptive evolution of the HMA domain. Phylogenetic analyses revealed that the HMA domain integrated into Pik-1 before Oryzinae speciation over 15 million years ago and has been under diversifying selection. Ancestral sequence reconstruction coupled with functional studies showed that two Pik-1 allelic variants independently evolved from a weakly binding ancestral state to high-affinity binding of the blast fungus effector AVR-PikD. We conclude that for most of its evolutionary history the Pik-1 HMA domain did not sense AVR-PikD, and that different Pik-1 receptors have recently evolved through distinct biochemical paths to produce similar phenotypic outcomes. These findings highlight the dynamic nature of the evolutionary mechanisms underpinning NLR adaptation to plant pathogens

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

The signal sequence influences post-translational ER translocation at distinct stages

The metazoan Sec61 translocon transports polypeptides into and across the membrane of the endoplasmic reticulum via two major routes, a well-established co-translational pathway and a post-translational alternative. We have used two model substrates to explore the elements of a secretory protein precursor that preferentially direct it towards a co- or post-translational pathway for ER translocation. Having first determined the capacity of precursors to enter ER derived microsomes post-translationally, we then exploited semi-permeabilized mammalian cells specifically depleted of key membrane components using siRNA to address their contribution to the membrane translocation process. These studies suggest precursor chain length is a key factor in the post-translational translocation at the mammalian ER, and identify Sec62 and Sec63 as important components acting on this route. This role for Sec62 and Sec63 is independent of the signal sequence that delivers the precursor to the ER. However, the signal sequence can influence the subsequent membrane translocation process, conferring sensitivity to a small molecule inhibitor and dictating reliance on the molecular chaperone BiP. Our data support a model where secretory protein precursors that fail to engage the signal recognition particle, for example because they are short, are delivered to the ER membrane via a distinct route that is dependent upon both Sec62 and Sec63. Although this requirement for Sec62 and Sec63 is unaffected by the specific signal sequence that delivers a precursor to the ER, this region can influence subsequent events, including both Sec61 mediated transport and the importance of BiP for membrane translocation. Taken together, our data suggest that an ER signal sequence can regulate specific aspects of Sec61 mediated membrane translocation at a stage following Sec62/Sec63 dependent ER delivery.Nicholas Johnson, Sarah Haßdenteufel, Melanie Theis, Adrienne W. Paton, James C. Paton, Richard Zimmermann, Stephen Hig

1α,25(OH)2-3-Epi-Vitamin D3, a Natural Physiological Metabolite of Vitamin D3: Its Synthesis, Biological Activity and Crystal Structure with Its Receptor

Background: The 1 alpha,25-dihydroxy-3-epi-vitamin-D(3) (1 alpha,25(OH)(2)-3-epi-D(3)), a natural metabolite of the seco-steroid vitamin D(3), exerts its biological activity through binding to its cognate vitamin D nuclear receptor (VDR), a ligand dependent transcription regulator. In vivo action of 1 alpha,25(OH)(2)-3-epi-D(3) is tissue-specific and exhibits lowest calcemic effect compared to that induced by 1 alpha,25(OH)(2)D(3). To further unveil the structural mechanism and structure-activity relationships of 1 alpha,25(OH)(2)-3-epi-D3 and its receptor complex, we characterized some of its in vitro biological properties and solved its crystal structure complexed with human VDR ligand-binding domain (LBD). Methodology/Principal Findings: In the present study, we report the more effective synthesis with fewer steps that provides higher yield of the 3-epimer of the 1 alpha,25(OH)(2)D(3). We solved the crystal structure of its complex with the human VDR-LBD and found that this natural metabolite displays specific adaptation of the ligand-binding pocket, as the 3-epimer maintains the number of hydrogen bonds by an alternative water-mediated interaction to compensate the abolished interaction with Ser278. In addition, the biological activity of the 1 alpha,25(OH)(2)-3-epi-D(3) in primary human keratinocytes and biochemical properties are comparable to 1 alpha,25(OH)(2)D(3). Conclusions/Significance: The physiological role of this pathway as the specific biological action of the 3-epimer remains unclear. However, its high metabolic stability together with its significant biologic activity makes this natural metabolite an interesting ligand for clinical applications. Our new findings contribute to a better understanding at molecular level how natural metabolites of 1 alpha,25(OH)(2)D(3) lead to significant activity in biological systems and we conclude that the C3-epimerization pathway produces an active metabolite with similar biochemical and biological properties to those of the 1 alpha,25(OH)(2)D(3)