Mathematical analysis of a model of chemotaxis arising from morphogenesis

We consider non-negative solution of a chemotaxis system with non constant chemotaxis sensitivity function X. This system appears as a limit case of a model formorphogenesis proposed by Bollenbach et al. (Phys. Rev. E. 75, 2007).Under suitable boundary conditions, modeling the presence of a morphogen source at x=0, we prove the existence of a global and bounded weak solution using an approximation by problems where diffusion is introduced in the ordinary differential equation. Moreover,we prove the convergence of the solution to the unique steady state provided that ? is small and ? is large enough. Numerical simulations both illustrate these results and give rise to further conjectures on the solution behavior that go beyond the rigorously proved statements.

Dyrk1A Influences Neuronal Morphogenesis Through Regulation of Cytoskeletal Dynamics in Mammalian Cortical Neurons

Down syndrome (DS) is the most frequent genetic cause of mental retardation. Cognitive dysfunction in these patients is correlated with reduced dendritic branching and complexity, along with fewer spines of abnormal shape that characterize the cortical neuronal profile of DS. DS phenotypes are caused by the disruptive effect of specific trisomic genes. Here, we report that overexpression of dual-specificity tyrosine phosphorylation-regulated kinase 1A, DYRK1A, is sufficient to produce the dendritic alterations observed in DS patients. Engineered changes in Dyrk1A gene dosage in vivo strongly alter the postnatal dendritic arborization processes with a similar progression than in humans. In cultured mammalian cortical neurons, we determined a reduction of neurite outgrowth and synaptogenesis. The mechanism underlying neurite dysgenesia involves changes in the dynamic reorganization of the cytoskeleton.