Multivariate orthogonal polynomials and integrable systems

Multivariate orthogonal polynomials in D real dimensions are considered from the perspective of the Cholesky factorization of a moment matrix. The approach allows for the construction of corresponding multivariate orthogonal polynomials, associated second kind functions, Jacobi type matrices and associated three term relations and also Christoffel-Darboux formulae. The multivariate orthogonal polynomials, their second kind functions and the corresponding Christoffel-Darboux kernels are shown to be quasi-determinants as well as Schur complements of bordered truncations of the moment matrix; quasi-tau functions are introduced. It is proven that the second kind functions are multivariate Cauchy transforms of the multivariate orthogonal polynomials. Discrete and continuous deformations of the measure lead to Toda type integrable hierarchy, being the corresponding flows described through Lax and Zakharov-Shabat equations; bilinear equations are found. Varying size matrix nonlinear partial difference and differential equations of the 2D Toda lattice type are shown to be solved by matrix coefficients of the multivariate orthogonal polynomials. The discrete flows, which are shown to be connected with a Gauss-Borel factorization of the Jacobi type matrices and its quasi-determinants, lead to expressions for the multivariate orthogonal polynomials and their second kind functions in terms of shifted quasi-tau matrices, which generalize to the multidimensional realm, those that relate the Baker and adjoint Baker functions to ratios of Miwa shifted tau-functions in the 1D scenario. In this context, the multivariate extension of the elementary Darboux transformation is given in terms of quasi-determinants of matrices built up by the evaluation, at a poised set of nodes lying in an appropriate hyperplane in R^D, of the multivariate orthogonal polynomials. The multivariate Christoffel formula for the iteration of m elementary Darboux transformations is given as a quasi-determinant. It is shown, using congruences in the space of semi-infinite matrices, that the discrete and continuous flows are intimately connected and determine nonlinear partial difference-differential equations that involve only one site in the integrable lattice behaving as a Kadomstev-Petviashvili type system. Finally, a brief discussion of measures with a particular linear isometry invariance and some of its consequences for the corresponding multivariate polynomials is given. In particular, it is shown that the Toda times that preserve the invariance condition lay in a secant variety of the Veronese variety of the fixed point set of the linear isometry.

Studying Coxiella burnetii Type IV Substrates in the Yeast Saccharomyces cerevisiae: Focus on Subcellular Localization and Protein Aggregation.

Coxiella burnetii is a Gram-negative obligate parasitic bacterium that causes the disease Q-fever in humans. To establish its intracellular niche, it utilizes the Icm/Dot type IVB secretion system (T4BSS) to inject protein effectors into the host cell cytoplasm. The host targets of most cognate and candidate T4BSS-translocated effectors remain obscure. We used the yeast Saccharomyces cerevisiae as a model to express and study six C. burnetii effectors, namely AnkA, AnkB, AnkF, CBU0077, CaeA and CaeB, in search for clues about their role in C. burnetii virulence. When ectopically expressed in HeLa cells, these effectors displayed distinct subcellular localizations. Accordingly, GFP fusions of these proteins produced in yeast also decorated distinct compartments, and most of them altered cell growth. CaeA was ubiquitinated both in yeast and mammalian cells and, in S. cerevisiae, accumulated at juxtanuclear quality-control compartments (JUNQs) and insoluble protein deposits (IPODs), characteristic of aggregative or misfolded proteins. AnkA, which was not ubiquitinated, accumulated exclusively at the IPOD. CaeA, but not AnkA or the other effectors, caused oxidative damage in yeast. We discuss that CaeA and AnkA behavior in yeast may rather reflect misfolding than recognition of conserved targets in the heterologous system. In contrast, CBU0077 accumulated at vacuolar membranes and abnormal ER extensions, suggesting that it interferes with vesicular traffic, whereas AnkB associated with the yeast nucleolus. Both effectors shared common localization features in HeLa and yeast cells. Our results support the idea that C. burnetii T4BSS effectors manipulate multiple host cell targets, which can be conserved in higher and lower eukaryotic cells. However, the behavior of CaeA and AnkA prompt us to conclude that heterologous protein aggregation and proteostatic stress can be a limitation to be considered when using the yeast model to assess the function of bacterial effectors.

Well posedness of an integrodifferential kinetic model of Fokker-Planck type for angiogenesis.

Tumor induced angiogenesis processes including the effect of stochastic motion and branching of blood vessels can be described coupling a (nonlocal in time) integrodifferential kinetic equation of Fokker–Planck type with a diffusion equation for the tumor induced ingiogenic factor. The chemotactic force field depends on the flux of blood vessels through the angiogenic factor. We develop an existence and uniqueness theory for this system under natural assumptions on the initial data. The proof combines the construction of fundamental solutions for associated linearized problems with comparison principles, sharp estimates of the velocity integrals and compactness results for this type of kinetic and parabolic operators

The magnetically-active, low-mass, triple system WDS 19312+3607 (Research Note)

Aims. We investigated in detail the system WDS 19312+3607, whose primary is an active M4.5Ve star previously inferred to be young (τ ~ 300–500 Ma) based on its high X-ray luminosity. Methods. We collected intermediate- and low-resolution optical spectra taken with 2 m-class telescopes, photometric data from the B to 8 μm bands, and data for eleven astrometric epochs with a time baseline of over 56 years for the two components in the system, G 125–15 and G 125–14. Results. We derived the M4.5V spectral types of both stars, confirmed their common proper motion, estimated their heliocentric distance and projected physical separation, determined their Galactocentric space velocities, and deduced a most-probable age of older than 600 Ma. We discovered that the primary, G 125–15, is an inflated, double-lined, spectroscopic binary with a short period of photometric variability of 1.6 d, which we associated with orbital synchronisation. The observed X-ray and Hα emissions, photometric variability, and abnormal radius and effective temperature of G 125–15 AB are indicative of strong magnetic activity, possibly because of the rapid rotation. In addition, the estimated projected physical separation between G 125–15 AB and G 125–14 of about 1200 AU ensures that WDS 19312+3607 is one of the widest systems with intermediate M-type primaries. Conclusions. G 125–15 AB is a nearby (d ≈ 26 pc), bright (J ≈ 9.6 mag), active spectroscopic binary with a single proper-motion companion of the same spectral type at a wide separation. They are thus ideal targets for specific follow-ups to investigate wide and close multiplicity or stellar expansion and surface cooling because of the lower convective efficiency.