Semidefinite Representation of the kk-Ellipse

The $k$-ellipse is the plane algebraic curve consisting of all points whose sum of distances from $k$ given points is a fixed number. The polynomial equation defining the $k$-ellipse has degree $2^k$ if $k$ is odd and degree $2^k{-}\binom{k}{k/2}$ if $k$ is even. We express this polynomial equation as the determinant of a symmetric matrix of linear polynomials. Our representation extends to weighted $k$-ellipses and $k$-ellipsoids in arbitrary dimensions, and it leads to new geometric applications of semidefinite programming.Comment: 16 pages, 5 figure

Pseudomonas aeruginosa virulence genes identified in a Dictyostelium host model.

The human pathogen Pseudomonas aeruginosa has been shown previously to use similar virulence factors when infecting mammalian hosts or Dictyostelium amoebae. Here we randomly mutagenized a clinical isolate of P. aeruginosa, and identified mutants with attenuated virulence towards Dictyostelium. These mutant strains also exhibited a strong decrease in virulence when infecting Drosophila and mice, confirming that P. aeruginosa makes use of similar virulence traits to confront these very different hosts. Further characterization of these bacterial mutants showed that TrpD is important for the induction of the quorum-sensing circuit, while PchH and PchI are involved in the induction of the type III secretion system. These results demonstrate the usefulness and the relevance of the Dictyostelium host model to identify and analyse new virulence genes in P. aeruginosa