Ancestral genome estimation reveals the history of ecological diversification in Agrobacterium

Horizontal gene transfer (HGT) is considered as a major source of innovation in bacteria, and as such is expected to drive adaptation to new ecological niches. However, among the many genes acquired through HGT along the diversification history of genomes, only a fraction may have actively contributed to sustained ecological adaptation. We used a phylogenetic approach accounting for the transfer of genes (or groups of genes) to estimate the history of genomes in Agrobacterium biovar 1, a diverse group of soil and plant-dwelling bacterial species. We identified clade-specific blocks of cotransferred genes encoding coherent biochemical pathways that may have contributed to the evolutionary success of key Agrobacterium clades. This pattern of gene coevolution rejects a neutral model of transfer, in which neighboring genes would be transferred independently of their function and rather suggests purifying selection on collectively coded acquired pathways. The acquisition of these synapomorphic blocks of cofunctioning genes probably drove the ecological diversification of Agrobacterium and defined features of ancestral ecological niches, which consistently hint at a strong selective role of host plant rhizospheres

Jack vertex operators and realization of Jack functions

We give an iterative method to realize general Jack functions from Jack functions of rectangular shapes. We first show some cases of Stanley's conjecture on positivity of the Littlewood-Richardson coefficients, and then use this method to give a new realization of Jack functions. We also show in general that vectors of products of Jack vertex operators form a basis of symmetric functions. In particular this gives a new proof of linear independence for the rectangular and marked rectangular Jack vertex operators. Thirdly a generalized Frobenius formula for Jack functions was given and was used to give new evaluation of Dyson integrals and even powers of Vandermonde determinant.Comment: Expanded versio

Explicit formulas for the generalized Hermite polynomials in superspace

We provide explicit formulas for the orthogonal eigenfunctions of the supersymmetric extension of the rational Calogero-Moser-Sutherland model with harmonic confinement, i.e., the generalized Hermite (or Hi-Jack) polynomials in superspace. The construction relies on the triangular action of the Hamiltonian on the supermonomial basis. This translates into determinantal expressions for the Hamiltonian's eigenfunctions.Comment: 19 pages. This is a recasting of the second part of the first version of hep-th/0305038 which has been splitted in two articles. In this revised version, the introduction has been rewritten and a new appendix has been added. To appear in JP

GC-Content evolution in bacterial genomes: The biased gene conversion hypothesis expands

The characterization of functional elements in genomes relies on the identification of the footprints of natural selection. In this quest, taking into account neutral evolutionary processes such as mutation and genetic drift is crucial because these forces can generate patterns that may obscure or mimic signatures of selection. In mammals, and probably in many eukaryotes, another such confounding factor called GC-Biased Gene Conversion (gBGC) has been documented. This mechanism generates patterns identical to what is expected under selection for higher GC-content, specifically in highly recombining genomic regions. Recent results have suggested that a mysterious selective force favouring higher GC-content exists in Bacteria but the possibility that it could be gBGC has been excluded. Here, we show that gBGC is probably at work in most if not all bacterial species. First we find a consistent positive relationship between the GC-content of a gene and evidence of intra-genic recombination throughout a broad spectrum of bacterial clades. Second, we show that the evolutionary force responsible for this pattern is acting independently from selection on codon usage, and could potentially interfere with selection in favor of optimal AU-ending codons. A comparison with data from human populations shows that the intensity of gBGC in Bacteria is comparable to what has been reported in mammals. We propose that gBGC is not restricted to sexual Eukaryotes but also widespread among Bacteria and could therefore be an ancestral feature of cellular organisms. We argue that if gBGC occurs in bacteria, it can account for previously unexplained observations, such as the apparent non-equilibrium of base substitution patterns and the heterogeneity of gene composition within bacterial genomes. Because gBGC produces patterns similar to positive selection, it is essential to take this process into account when studying the evolutionary forces at work in bacterial genomes

Diversity and evolution of surface polysaccharide synthesis loci in Enterobacteriales.

Bacterial capsules and lipopolysaccharides are diverse surface polysaccharides (SPs) that serve as the frontline for interactions with the outside world. While SPs can evolve rapidly, their diversity and evolutionary dynamics across different taxonomic scales has not been investigated in detail. Here, we focused on the bacterial order Enterobacteriales (including the medically relevant Enterobacteriaceae), to carry out comparative genomics of two SP locus synthesis regions, cps and kps, using 27,334 genomes from 45 genera. We identified high-quality cps loci in 22 genera and kps in 11 genera, around 4% of which were detected in multiple species. We found SP loci to be highly dynamic genetic entities: their evolution was driven by high rates of horizontal gene transfer (HGT), both of whole loci and component genes, and relaxed purifying selection, yielding large repertoires of SP diversity. In spite of that, we found the presence of (near-)identical locus structures in distant taxonomic backgrounds that could not be explained by recent exchange, pointing to long-term selective preservation of locus structures in some populations. Our results reveal differences in evolutionary dynamics driving SP diversity within different bacterial species, with lineages of Escherichia coli, Enterobacter hormaechei and Klebsiella aerogenes most likely to share SP loci via recent exchange; and lineages of Salmonella enterica, Citrobacter sakazakii and Serratia marcescens most likely to share SP loci via other mechanisms such as long-term preservation. Overall, the evolution of SP loci in Enterobacteriales is driven by a range of evolutionary forces and their dynamics and relative importance varies between different species

On Poincare and logarithmic Sobolev inequalities for a class of singular Gibbs measures

This note, mostly expository, is devoted to Poincar{\'e} and log-Sobolev inequalities for a class of Boltzmann-Gibbs measures with singular interaction. Such measures allow to model one-dimensional particles with confinement and singular pair interaction. The functional inequalities come from convexity. We prove and characterize optimality in the case of quadratic confinement via a factorization of the measure. This optimality phenomenon holds for all beta Hermite ensembles including the Gaussian unitary ensemble, a famous exactly solvable model of random matrix theory. We further explore exact solvability by reviewing the relation to Dyson-Ornstein-Uhlenbeck diffusion dynamics admitting the Hermite-Lassalle orthogonal polynomials as a complete set of eigenfunctions. We also discuss the consequence of the log-Sobolev inequality in terms of concentration of measure for Lipschitz functions such as maxima and linear statistics.Comment: Minor improvements. To appear in Geometric Aspects of Functional Analysis -- Israel Seminar (GAFA) 2017-2019", Lecture Notes in Mathematics 225

Diversity and evolution of surface polysaccharide synthesis loci in Enterobacteriales.

Bacterial capsules and lipopolysaccharides are diverse surface polysaccharides (SPs) that serve as the frontline for interactions with the outside world. While SPs can evolve rapidly, their diversity and evolutionary dynamics across different taxonomic scales has not been investigated in detail. Here, we focused on the bacterial order Enterobacteriales (including the medically relevant Enterobacteriaceae), to carry out comparative genomics of two SP locus synthesis regions, cps and kps, using 27,334 genomes from 45 genera. We identified high-quality cps loci in 22 genera and kps in 11 genera, around 4% of which were detected in multiple species. We found SP loci to be highly dynamic genetic entities: their evolution was driven by high rates of horizontal gene transfer (HGT), both of whole loci and component genes, and relaxed purifying selection, yielding large repertoires of SP diversity. In spite of that, we found the presence of (near-)identical locus structures in distant taxonomic backgrounds that could not be explained by recent exchange, pointing to long-term selective preservation of locus structures in some populations. Our results reveal differences in evolutionary dynamics driving SP diversity within different bacterial species, with lineages of Escherichia coli, Enterobacter hormaechei and Klebsiella aerogenes most likely to share SP loci via recent exchange; and lineages of Salmonella enterica, Citrobacter sakazakii and Serratia marcescens most likely to share SP loci via other mechanisms such as long-term preservation. Overall, the evolution of SP loci in Enterobacteriales is driven by a range of evolutionary forces and their dynamics and relative importance varies between different species

Weak forms of unconditionality of bases in greedy approximation

We study a new class of bases, weaker than quasi-greedy bases, which retain their unconditionality properties and can provide the same optimality for the thresholding greedy algorithm. We measure how far these bases are from being unconditional and use this concept to give a new characterization of nearly unconditional bases.F. Albiac acknowledges the support of the Spanish Ministry for Science and Innovation under Grant PID2019-107701GB-I00 for Operators, lattices, and structure of Banach spaces. F. Albiac and J. L. Ansorena acknowledge the support of the Spanish Ministry for Science, Innovation, and Universities under Grant PGC2018-095366-B-I00 for Analisis Vectorial, Multilineal y Aproximacion. M. Berasategui and S. Lassalle were supported by ANPCyT PICT-2018-04104 and CONICET PIP 1609. P. Berna by Grant PID2019-105599GB-I00 (Agencia Estatal de Investigacion, Spain) and 20906/PI/18 from Fundacion Seneca (Region de Murcia, Spain). S. Lassalle was also supported in part by CONICET PIP 0483 and PAI UdeSA 2020-2021

Orthogonal basis for the energy eigenfunctions of the Chern-Simons matrix model

We study the spectrum of the Chern-Simons matrix model and identify an orthogonal set of states. The connection to the spectrum of the Calogero model is discussed.Comment: 11 pages, LaTeX, minor typo corrections, section 6 slightly extended to include more information on Jack polynomial

Quantum Calogero-Moser Models: Integrability for all Root Systems

The issues related to the integrability of quantum Calogero-Moser models based on any root systems are addressed. For the models with degenerate potentials, i.e. the rational with/without the harmonic confining force, the hyperbolic and the trigonometric, we demonstrate the following for all the root systems: (i) Construction of a complete set of quantum conserved quantities in terms of a total sum of the Lax matrix (L), i.e. (\sum_{\mu,\nu\in{\cal R}}(L^n)_{\mu\nu}), in which ({\cal R}) is a representation space of the Coxeter group. (ii) Proof of Liouville integrability. (iii) Triangularity of the quantum Hamiltonian and the entire discrete spectrum. Generalised Jack polynomials are defined for all root systems as unique eigenfunctions of the Hamiltonian. (iv) Equivalence of the Lax operator and the Dunkl operator. (v) Algebraic construction of all excited states in terms of creation operators. These are mainly generalisations of the results known for the models based on the (A) series, i.e. (su(N)) type, root systems.Comment: 45 pages, LaTeX2e, no figure