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We define the vector-valued, matrix-weighted function spacesḞ αq p (W ) (homogeneous) and F αq p (W ) (inhomogeneous) on R n , for α ∈ R, 0 < p < ∞, 0 < q ≤ ∞, with the matrix weight W belonging to the A p class. For 1 < p < ∞, we show that L p (W ) =Ḟ 02 p (W ), and, for k ∈ N, that F k2 p (W ) coincides with the matrix-weighted Sobolev space L p k (W ), thereby obtaining Littlewood-Paley characterizations of L p (W ) and L p k (W ). We show that a vectorvalued function belongs toḞ αq p (W ) if and only if its wavelet or ϕ-transform coefficients belong to an associated sequence spaceḟ αq p (W ). We also characterize these spaces in terms of reducing operators associated to W .

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Specific small deletions within the rpoC gene encoding the β′-subunit of RNA polymerase (RNAP) are found repeatedly after adaptation of Escherichia coli K-12 MG1655 to growth in minimal media. Here we present a multiscale analysis of these mutations. At the physiological level, the mutants grow 60% faster than the parent strain and convert the carbon source 15-35% more efficiently to biomass, but grow about 30% slower than the parent strain in rich medium. At the molecular level, the kinetic parameters of the mutated RNAP were found to be altered, resulting in a 4-to 30-fold decrease in open complex longevity at an rRNA promoter and a ∼10-fold decrease in transcriptional pausing, with consequent increase in transcript elongation rate. At a genome-scale, systems biology level, gene expression changes between the parent strain and adapted RNAP mutants reveal large-scale systematic transcriptional changes that influence specific cellular processes, including strong down-regulation of motility, acid resistance, fimbria, and curlin genes. RNAP genome-binding maps reveal redistribution of RNAP that may facilitate relief of a metabolic bottleneck to growth. These findings suggest that reprogramming the kinetic parameters of RNAP through specific mutations allows regulatory adaptation for optimal growth in new environments.kinetics | stringent response | transcription

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