Summary Background Altered expression of apicobasal polarity factors is associated with cancer in vertebrates and tissue overgrowth in invertebrates, yet mechanisms by which these factors affect growth regulatory pathways are not well defined. We have tested the basis of an overgrowth phenotype driven by the Drosophila protein Crumbs (Crb), which nucleates an apical membrane complex that functionally interacts with the Par6/Par3/aPKC and Scrib/Dlg/Lgl apicobasal polarity complexes. Results We find that Crb-driven growth is dependent upon the Salvador-Warts-Hippo (SWH) pathway and its transcriptional effector Yorkie (Yki). Expression of the Crb intracellular domain elevates Yki activity and this correlates in tissues and cultured cells with loss of Expanded (Ex), an apically localized SWH component that inhibits Yki. Reciprocally, loss of crb elevates Ex levels, although this excess Ex does not concentrate to its normal location at apical junctions. The Ex-regulatory domain of Crb maps to the juxtamembrane FERM-domain binding motif (JM), a cytoskeletal interaction domain distinct from the PDZ-binding motif (PBM) through which Crb binds polarity factors. Expression of Crb-JM drives Yki activity and organ growth with little effect on tissue architecture, while reciprocally Crb-PBM produces tissue architectural defects without significant effect on Yki activity. Conclusions These studies identify Crb as a novel SWH regulator via JM-dependent effects on Ex levels and localization, and show that discrete domains within Crb may allow it to integrate junctional polarity signals with a conserved growth pathway.

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In this paper, we consider an unknown source problem for the modified Helmholtz equation. The Tikhonov regularization method in Hilbert scales is extended to deal with ill-posedness of the problem. An a priori strategy and an a posteriori choice rule have been present to obtain the regularization parameter and corresponding error estimates have been obtained. The smoothness parameter and the a priori bound of exact solution are not needed for the a posteriori choice rule. Numerical results are presented to show the stability and effectiveness of the method.