Heat flow and calculus on metric measure spaces with Ricci curvature bounded below - the compact case

We provide a quick overview of various calculus tools and of the main results concerning the heat flow on compact metric measure spaces, with applications to spaces with lower Ricci curvature bounds. Topics include the Hopf-Lax semigroup and the Hamilton-Jacobi equation in metric spaces, a new approach to differentiation and to the theory of Sobolev spaces over metric measure spaces, the equivalence of the L^2-gradient flow of a suitably defined "Dirichlet energy" and the Wasserstein gradient flow of the relative entropy functional, a metric version of Brenier's Theorem, and a new (stronger) definition of Ricci curvature bound from below for metric measure spaces. This new notion is stable w.r.t. measured Gromov-Hausdorff convergence and it is strictly connected with the linearity of the heat flow.Comment: To the memory of Enrico Magenes, whose exemplar life, research and teaching shaped generations of mathematician

Seeing the forest for the trees: using the Gene Ontology to restructure hierarchical clustering

Motivation: There is a growing interest in improving the cluster analysis of expression data by incorporating into it prior knowledge, such as the Gene Ontology (GO) annotations of genes, in order to improve the biological relevance of the clusters that are subjected to subsequent scrutiny. The structure of the GO is another source of background knowledge that can be exploited through the use of semantic similarity

Generalized reduced basis methods and n-width estimates for the approximation of the solution manifold of parametric PDEs

The set of solutions of a parameter-dependent linear partial differential equation with smooth coefficients typically forms a compact manifold in a Hilbert space. In this paper we review the generalized reduced basis method as a fast computational tool for the uniform approximation of the solution manifold. We focus on operators showing an affine parametric dependence, expressed as a linear combination of parameter-independent operators through some smooth, parameter-dependent scalar functions. In the case that the parameter-dependent operator has a dominant term in its affine expansion, one can prove the existence of exponentially convergent uniform approximation spaces for the entire solution manifold. These spaces can be constructed without any assumptions on the parametric regularity of the manifold -- only spatial regularity of the solutions is required. The exponential convergence rate is then inherited by the generalized reduced basis method. We provide a numerical example related to parametrized elliptic equations confirming the predicted convergence rate

Identification of Arx transcriptional targets in the developing basal forebrain

Mutations in the aristaless-related homeobox (ARX) gene are associated with multiple neurologic disorders in humans. Studies in mice indicate Arx plays a role in neuronal progenitor proliferation and development of the cerebral cortex, thalamus, hippocampus, striatum, and olfactory bulbs. Specific defects associated with Arx loss of function include abnormal interneuron migration and subtype differentiation. How disruptions in ARX result in human disease and how loss of Arx in mice results in these phenotypes remains poorly understood. To gain insight into the biological functions of Arx, we performed a genome-wide expression screen to identify transcriptional changes within the subpallium in the absence of Arx. We have identified 84 genes whose expression was dysregulated in the absence of Arx. This population was enriched in genes involved in cell migration, axonal guidance, neurogenesis, and regulation of transcription and includes genes implicated in autism, epilepsy, and mental retardation; all features recognized in patients with ARX mutations. Additionally, we found Arx directly repressed three of the identified transcription factors: Lmo1, Ebf3 and Shox2. To further understand how the identified genes are involved in neural development, we used gene set enrichment algorithms to compare the Arx gene regulatory network (GRN) to the Dlx1/2 GRN and interneuron transcriptome. These analyses identified a subset of genes in the Arx GRN that are shared with that of the Dlx1/2 GRN and that are enriched in the interneuron transcriptome. These data indicate Arx plays multiple roles in forebrain development, both dependent and independent of Dlx1/2, and thus provides further insights into the understanding of the mechanisms underlying the pathology of mental retardation and epilepsy phenotypes resulting from ARX mutations

Using C. elegans to decipher the cellular and molecular mechanisms underlying neurodevelopmental disorders

Prova tipográfica (uncorrected proof)Neurodevelopmental disorders such as epilepsy, intellectual disability (ID), and autism spectrum disorders (ASDs) occur in over 2 % of the population, as the result of genetic mutations, environmental factors, or combination of both. In the last years, use of large-scale genomic techniques allowed important advances in the identification of genes/loci associated with these disorders. Nevertheless, following association of novel genes with a given disease, interpretation of findings is often difficult due to lack of information on gene function and effect of a given mutation in the corresponding protein. This brings the need to validate genetic associations from a functional perspective in model systems in a relatively fast but effective manner. In this context, the small nematode, Caenorhabditis elegans, presents a good compromise between the simplicity of cell models and the complexity of rodent nervous systems. In this article, we review the features that make C. elegans a good model for the study of neurodevelopmental diseases. We discuss its nervous system architecture and function as well as the molecular basis of behaviors that seem important in the context of different neurodevelopmental disorders. We review methodologies used to assess memory, learning, and social behavior as well as susceptibility to seizures in this organism. We will also discuss technological progresses applied in C. elegans neurobiology research, such as use of microfluidics and optogenetic tools. Finally, we will present some interesting examples of the functional analysis of genes associated with human neurodevelopmental disorders and how we can move from genes to therapies using this simple model organism.The authors would like to acknowledge Fundação para a Ciência e Tecnologia (FCT) (PTDC/SAU-GMG/112577/2009). AJR and CB are recipients of FCT fellowships: SFRH/BPD/33611/2009 and SFRH/BPD/74452/2010, respectively