On the Geometry of the Nodal Lines of Eigenfunctions of the Two-Dimensional Torus

The width of a convex curve in the plane is the minimal distance between a pair of parallel supporting lines of the curve. In this paper we study the width of nodal lines of eigenfunctions of the Laplacian on the standard flat torus. We prove a variety of results on the width, some having stronger versions assuming a conjecture of Cilleruelo and Granville asserting a uniform bound for the number of lattice points on the circle lying in short arcs.Comment: 4 figures. Added some comments about total curvature and other detail

Poly(ADP-Ribose) Polymerase 1 (PARP-1) Regulates Ribosomal Biogenesis in Drosophila Nucleoli

Poly(ADP-ribose) polymerase 1 (PARP1), a nuclear protein, utilizes NAD to synthesize poly(AD-Pribose) (pADPr), resulting in both automodification and the modification of acceptor proteins. Substantial amounts of PARP1 and pADPr (up to 50%) are localized to the nucleolus, a subnuclear organelle known as a region for ribosome biogenesis and maturation. At present, the functional significance of PARP1 protein inside the nucleolus remains unclear. Using PARP1 mutants, we investigated the function of PARP1, pADPr, and PARP1-interacting proteins in the maintenance of nucleolus structure and functions. Our analysis shows that disruption of PARP1 enzymatic activity caused nucleolar disintegration and aberrant localization of nucleolar-specific proteins. Additionally, PARP1 mutants have increased accumulation of rRNA intermediates and a decrease in ribosome levels. Together, our data suggests that PARP1 enzymatic activity is required for targeting nucleolar proteins to the proximity of precursor rRNA; hence, PARP1 controls precursor rRNA processing, post-transcriptional modification, and pre-ribosome assembly. Based on these findings, we propose a model that explains how PARP1 activity impacts nucleolar functions and, consequently, ribosomal biogenesis