Two classes of metric spaces

The class of metric spaces (X,d) known as small-determined spaces, introduced by Garrido and Jaramillo, are properly defined by means of some type of real-valued Lipschitz functions on X. On the other hand, B-simple metric spaces introduced by Hejcman are defined in terms of some kind of bornologies of bounded subsets of X. In this note we present a common framework where both classes of metric spaces can be studied which allows us to see not only the relationships between them but also to obtain new internal characterizations of these metric properties.

Linear non-local diffusion problems in metric measure spaces

The aim of this paper is to provide a comprehensive study of some linear non-local diffusion problems in metric measure spaces. These include, for example, open subsets in ℝN, graphs, manifolds, multi-structures and some fractal sets. For this, we study regularity, compactness, positivity and the spectrum of the stationary non-local operator. We then study the solutions of linear evolution non-local diffusion problems, with emphasis on similarities and differences with the standard heat equation in smooth domains. In particular, we prove weak and strong maximum principles and describe the asymptotic behaviour using spectral methods.

Fitness cost and interference of Arm/Rmt aminoglycoside resistance with the RsmF housekeeping methyltransferases

Arm/Rmt methyltransferases have emerged recently in pathogenic bacteria as enzymes that confer high-level resistance to 4,6-disubstituted aminoglycosides through methylation of the G1405 residue in the 16S rRNA (like ArmA and RmtA to -E). In prokaryotes, nucleotide methylations are the most common type of rRNA modification, and they are introduced posttranscriptionally by a variety of site-specific housekeeping enzymes to optimize ribosomal function. Here we show that while the aminoglycoside resistance methyltransferase RmtC methylates G1405, it impedes methylation of the housekeeping methyltransferase RsmF at position C1407, a nucleotide that, like G1405, forms part of the aminoglycoside binding pocket of the 16S rRNA. To understand the origin and consequences of this phenomenon, we constructed a series of in-frame knockout and knock-in mutants of Escherichia coli, corresponding to the genotypes rsmF(+), ΔrsmF, rsmF(+) rmtC(+), and ΔrsmF rmtC(+). When analyzed for the antimicrobial resistance pattern, the ΔrsmF bacteria had a decreased susceptibility to aminoglycosides, including 4,6- and 4,5-deoxystreptamine aminoglycosides, showing that the housekeeping methylation at C1407 is involved in intrinsic aminoglycoside susceptibility in E. coli. Competition experiments between the isogenic E. coli strains showed that, contrary to expectation, acquisition of rmtC does not entail a fitness cost for the bacterium. Finally, matrix-assisted laser desorption ionization (MALDI) mass spectrometry allowed us to determine that RmtC methylates the G1405 residue not only in presence but also in the absence of aminoglycoside antibiotics. Thus, the coupling between housekeeping and acquired methyltransferases subverts the methylation architecture of the 16S rRNA but elicits Arm/Rmt methyltransferases to be selected and retained, posing an important threat to the usefulness of aminoglycosides worldwide.