A Shape Optimization Problem on Planar Sets with Prescribed Topology

We consider shape optimization problems involving functionals depending on perimeter, torsional rigidity and Lebesgue measure. The scaling free cost functionals are of the form \(P(\Omega )T^q(\Omega )|\Omega |^{-2q-1/2}\), and the class of admissible domains consists of two-dimensional open sets \(\Omega \) satisfying the topological constraints of having a prescribed number k of bounded connected components of the complementary set. A relaxed procedure is needed to have a well-posed problem, and we show that when \(q<1/2\) an optimal relaxed domain exists. When \(q>1/2\), the problem is ill-posed, and for \(q=1/2\), the explicit value of the infimum is provided in the cases \(k=0\) and \(k=1\).

     

Phenotyping Reveals Targets of a Pseudo-Natural-Product Autophagy Inhibitor

Pseudo-natural-product (NP) design combines natural product fragments to provide unprecedented NP-inspired compounds not accessible by biosynthesis, but endowed with biological relevance. Since the bioactivity of pseudo-NPs may be unprecedented or unexpected, they are best evaluated in target agnostic cell-based assays monitoring entire cellular programs or complex phenotypes. Here, the Cinchona alkaloid scaffold was merged with the indole ring system to synthesize indocinchona alkaloids by Pd-catalyzed annulation. Exploration of indocinchona alkaloid bioactivities in phenotypic assays revealed a novel class of azaindole-containing autophagy inhibitors, the azaquindoles. Subsequent characterization of the most potent compound, azaquindole-1, in the morphological cell painting assay, guided target identification efforts. In contrast to the parent Cinchona alkaloids, azaquindoles selectively inhibit starvation- and rapamycin-induced autophagy by targeting the lipid kinase VPS34.