Flux norm approach to finite dimensional homogenization approximations with non-separated scales and high contrast

We consider divergence-form scalar elliptic equations and vectorial equations for elasticity with rough ($L^\infty(\Omega)$, $\Omega \subset \R^d$) coefficients $a(x)$ that, in particular, model media with non-separated scales and high contrast in material properties. We define the flux norm as the $L^2$ norm of the potential part of the fluxes of solutions, which is equivalent to the usual $H^1$-norm. We show that in the flux norm, the error associated with approximating, in a properly defined finite-dimensional space, the set of solutions of the aforementioned PDEs with rough coefficients is equal to the error associated with approximating the set of solutions of the same type of PDEs with smooth coefficients in a standard space (e.g., piecewise polynomial). We refer to this property as the {\it transfer property}. A simple application of this property is the construction of finite dimensional approximation spaces with errors independent of the regularity and contrast of the coefficients and with optimal and explicit convergence rates. This transfer property also provides an alternative to the global harmonic change of coordinates for the homogenization of elliptic operators that can be extended to elasticity equations. The proofs of these homogenization results are based on a new class of elliptic inequalities which play the same role in our approach as the div-curl lemma in classical homogenization.Comment: Accepted for publication in Archives for Rational Mechanics and Analysi

Expansion and mechanistic insights into de novo DEAF1 variants in DEAF1-associated neurodevelopmental disorders

De novo deleterious and heritable biallelic mutations in the DNA binding domain (DBD) of the transcription factor deformed epidermal autoregulatory factor 1 (DEAF1) result in a phenotypic spectrum of disorders termed DEAF1-associated neurodevelopmental disorders (DAND). RNA-sequencing using hippocampal RNA from mice with conditional deletion of Deaf1 in the central nervous system indicate that loss of Deaf1 activity results in the altered expression of genes involved in neuronal function, dendritic spine maintenance, development, and activity, with reduced dendritic spines in hippocampal regions. Since DEAF1 is not a dosage-sensitive gene, we assessed the dominant negative activity of previously identified de novo variants and a heritable recessive DEAF1 variant on selected DEAF1-regulated genes in 2 different cell models. While no altered gene expression was observed in cells over-expressing the recessive heritable variant, the gene expression profiles of cells over-expressing de novo variants resulted in similar gene expression changes as observed in CRISPR-Cas9-mediated DEAF1-deleted cells. Altered expression of DEAF1-regulated genes was rescued by exogenous expression of WT-DEAF1 but not by de novo variants in cells lacking endogenous DEAF1. De novo heterozygous variants within the DBD of DEAF1 were identified in 10 individuals with a phenotypic spectrum including autism spectrum disorder, developmental delays, sleep disturbance, high pain tolerance, and mild dysmorphic features. Functional assays demonstrate these variants alter DEAF1 transcriptional activity. Taken together, this study expands the clinical phenotypic spectrum of individuals with DAND, furthers our understanding of potential roles of DEAF1 on neuronal function, and demonstrates dominant negative activity of identified de novo variants

Expansion and mechanistic insights into de novo DEAF1 variants in DEAF1-associated neurodevelopmental disorders

De novo deleterious and heritable biallelic mutations in the DNA binding domain (DBD) of the transcription factor deformed epidermal autoregulatory factor 1 (DEAF1) result in a phenotypic spectrum of disorders termed DEAF1-associated neurodevelopmental disorders (DAND). RNA-sequencing using hippocampal RNA from mice with conditional deletion of Deaf1 in the central nervous system indicate that loss of Deaf1 activity results in the altered expression of genes involved in neuronal function, dendritic spine maintenance, development, and activity, with reduced dendritic spines in hippocampal regions. Since DEAF1 is not a dosage-sensitive gene, we assessed the dominant negative activity of previously identified de novo variants and a heritable recessive DEAF1 variant on selected DEAF1-regulated genes in 2 different cell models. While no altered gene expression was observed in cells over-expressing the recessive heritable variant, the gene expression profiles of cells over-expressing de novo variants resulted in similar gene expression changes as observed in CRISPR-Cas9-mediated DEAF1-deleted cells. Altered expression of DEAF1-regulated genes was rescued by exogenous expression of WT-DEAF1 but not by de novo variants in cells lacking endogenous DEAF1. De novo heterozygous variants within the DBD of DEAF1 were identified in 10 individuals with a phenotypic spectrum including autism spectrum disorder, developmental delays, sleep disturbance, high pain tolerance, and mild dysmorphic features. Functional assays demonstrate these variants alter DEAF1 transcriptional activity. Taken together, this study expands the clinical phenotypic spectrum of individuals with DAND, furthers our understanding of potential roles of DEAF1 on neuronal function, and demonstrates dominant negative activity of identified de novo variants.Genetics of disease, diagnosis and treatmen