Section: New Results

Modeling of neurogenesis and brain development

The timing of cortical neurogenesis has a major effect on the size and organization of the mature cortex. The deletion of the LIM-homeodomain transcription factor Lhx2 in cortical progenitors by Nestin-cre leads to a dramatically smaller cortex. In [19] we report that Lhx2 regulates the cortex size by maintaining the cortical progenitor proliferation and delaying the initiation of neurogenesis. The loss of Lhx2 in cortical progenitors results in precocious radial glia differentiation and a temporal shift of cortical neurogenesis. We further investigated the underlying mechanisms at play and demonstrated that in the absence of Lhx2, the Wnt/$\beta$-catenin pathway failed to maintain progenitor proliferation. We developed and applied a mathematical model that reveals how precocious neurogenesis affected cortical surface and thickness. Thus, we concluded that Lhx2 is required for $\beta$-catenin function in maintaining cortical progenitor proliferation and controls the timing of cortical neurogenesis.

We analyze in [22] an inhomogeneous system of coupled reaction-diffusion equations representing the dynamics of gene expression during differentiation of nerve cells. The outcome of this developmental phase is the formation of distinct functional areas separated by sharp and smooth boundaries. It proceeds through the competition between the expression of two genes whose expression is driven by monotonic gradients of chemicals, and the products of gene expression undergo local diffusion and drive gene expression in neighboring cells. The problem therefore falls in a more general setting of species in competition within a non-homogeneous medium. We show that in the limit of arbitrarily small diffusion, there exists a unique monotonic stationary solution, which splits the neural tissue into two winner-take-all parts at a precise boundary point: on both sides of the boundary, different neuronal types are present. In order to further characterize the location of this boundary, we use a blow-up of the system and define a traveling wave problem parametrized by the position within the monotonic gradient: the precise boundary location is given by the unique point in space at which the speed of the wave vanishes.

Boundary formation in the developing neuroepithelium decides on the position and size of compartments in the adult nervous system. In [23] , we started from the French Flag model proposed by Lewis Wolpert, in which boundaries are formed through the combination of morphogen diffusion and of thresholds in cell responses. In contemporary terms, a response is characterized by the expression of cell-autonomous transcription factors, very often of the homeoprotein family. Theoretical studies suggest that this sole mechanism results in the formation of boundaries of imprecise shapes and positions. Alan Turing, on the other hand, proposed a model whereby two morphogens that exhibit self-activation and reciprocal inhibition, and are uniformly distributed and diffuse at different rates lead to the formation of territories of unpredictable shapes and positions but with sharp boundaries (the 'leopard spots'). Here, we have combined the two models and compared the stability of boundaries when the hypothesis of local homeoprotein intercellular diffusion is, or is not, introduced in the equations. We find that the addition of homeoprotein local diffusion leads to a dramatic stabilization of the positioning of the boundary, even when other parameters are significantly modified. This novel Turing/Wolpert combined model has thus important theoretical consequences for our understanding of the role of the intercellular diffusion of homeoproteins in the developmental robustness of and the changes that take place in the course of evolution.