On a system of qq-partial differential equations with applications to qq-series

Using the theory of functions of several variables and $q$-calculus, we prove an expansion theorem for the analytic function in several variables which satisfies a system of $q$-partial differential equations. Some curious applications of this expansion theorem to $q$-series are discussed. In particular, an extension of Andrews' transformation formula for the $q$-Lauricella function is given.Comment: 15 pages, accepted for the proceedings of the Alladi 60th birthday conferenc

Nephron development and extrarenal features in a child with congenital nephrotic syndrome caused by null LAMB2 mutations

Abstract Background Congenital nephrotic syndrome (CNS) is a rare disorder caused by various structural and developmental defects of glomeruli. It occurs typically as an isolated kidney disorder but associates sometimes with other systemic, extrarenal manifestations. Case Presentations An infant presented with severe CNS, which progressed rapidly to renal failure at age of 3 months and death at 27 months. The clinical phenotypes and genetic causes were studied, including the renal pathology at autopsy. Besides the CNS, the affected child had remarkable right-side predominant eye-ball hypoplasia with bilateral anterior chamber dysgenesis (microcoria). Brain MRI revealed grossly normal development in the cerebrum, cerebellum, and brain stem. Auditory brainstem responses were bilaterally blunted, suggesting a defective auditory system. At autopsy, both kidneys were mildly atrophied with persistent fetal lobulation. Microscopic examination showed a diffuse global sclerosis. However, despite of the smaller size of glomeruli, the nephron number remained similar to that of the age-matched control. Whole-exome sequencing revealed that the affected child was compound heterozygous for novel truncating LAMB2 mutations: a 4-bp insertion (p.Gly1693Alafs*8) and a splicing donor-site substitution (c.1225 + 1G > A), presumably deleting the coiled-coil domains that form the laminin 5–2-1 heterotrimer complex. Conclusions Our case represents a variation of Pierson syndrome that accompanies CNS with unilateral ocular hypoplasia. The average number but smaller glomeruli could reflect either mal-development or glomerulosclerosis. Heterogeneous clinical expression of LAMB2 defects may associate with the difference in fetal β1 subtype compensation among affected tissues. Further study is necessary to evaluate incidence and features of auditory defect under LAMB2 deficiency

Mechanism of cystogenesis by Cd79a-driven, conditional mTOR activation in developing mouse nephrons

Abstract Polycystic kidney disease (PKD) is a common genetic disorder arising from developmental and postnatal processes. Defects in primary cilia and their signaling (eg, mTOR) underlie the pathogenesis. However, how mTOR regulates tubular integrity remains unclear. The paucity of faithful models has limited our understanding of pathogenesis and, therefore, the refinement of therapeutic targets. To understand the role of mTOR in early cystogenesis, we studied an in-house mouse model, Cd79a-Cre;Tsc1ff. (Cd79a-Tsc1 KO hereafter), recapitulating human autosomal-dominant PKD histology. Cre-mediated Tsc1 depletion driven by the promoter for Cd79a, a known B-cell receptor, activated mTORC1 exclusively along the distal nephron from embryonic day 16 onward. Cysts appeared in the distal nephron at 1 weeks of age and mice developed definite PKD by 4 weeks. Cd79a-Tsc1 KO tubule cells proliferated at a rate comparable to controls after birth but continued to divide even after postnatal day 14 when tubulogenesis is normally completed. Apoptosis occurred only after 9 weeks. During postnatal days 7–11, pre-cystic Cd79a-Tsc1 KO tubule cells showed cilia elongation, aberrant cell intercalation, and mitotic division, suggesting that defective cell planar polarity (PCP) may underlie cystogenesis. mTORC1 was activated in a portion of cyst-lining cells and occasionally even when Tsc1 was not depleted, implying a non-autonomous mechanism. Our results indicate that mTORC1 overactivation in developing distal tubules impairs their postnatal narrowing by disrupting morphogenesis, which orients an actively proliferating cell toward the elongating axis. The interplay between mTOR and cilium signaling, which coordinate cell proliferation with PCP, may be essential for cystogenesis