Zeros of analytic functions, with or without multiplicities

The classical Mason-Stothers theorem deals with nontrivial polynomial solutions to the equation $a+b=c$. It provides a lower bound on the number of distinct zeros of the polynomial $abc$ in terms of the degrees of $a$, $b$ and $c$. We extend this to general analytic functions living on a reasonable bounded domain $\Omega\subset\mathbb C$, rather than on the whole of $\mathbb C$. The estimates obtained are sharp, for any $\Omega$, and a generalization of the original result on polynomials can be recovered from them by a limiting argument.Comment: This is a retitled and slightly revised version of my paper arXiv:1004.359

Aquaporin 2 promotes cell migration and epithelial morphogenesis

The aquaporin 2 (AQP2) water channel, expressed in kidney collecting ducts, contributes critically to water homeostasis in mammals. Animals lacking or having significantly reduced levels of AQP2, however, have not only urinary concentrating abnormalities but also renal tubular defects that lead to neonatal mortality from renal failure. Here, we show that AQP2 is not only a water channel but also an integrin-binding membrane protein that promotes cell migration and epithelial morphogenesis. AQP2 expression modulates the trafficking and internalization of integrin beta 1, facilitating its turnover at focal adhesions. In vitro, disturbing the interaction between AQP2 and integrin beta 1 by mutating the RGD motif led to reduced endocytosis, retention of integrin beta 1 at the cell surface, and defective cell migration and tubulogenesis. Similarly, in vivo, AQP2-null mice exhibited significant retention of integrin beta 1 at the basolateral membrane and had tubular abnormalities. In summary, these data suggest that the water channel AQP2 interacts with integrins to promote renal epithelial cell migration, contributing to the structural and functional integrity of the mammalian kidney