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Abstract In this paper, we provide an elementary and simple proof of the Calabi holomorphic extension theorem, a result which plays an important role in complex analysis and complex geometry. 

Abstract The paper studies complex manifolds whose Bergman metrics are incomplete but have constant holomorphic sectional curvature.We will construct a real analytic unbounded domain in ${\mathbb{C}}^2$\mathbb{C}^{2}whose Bergman metric is well-defined and has a positive constant holomorphic sectional curvature, which appears to be the first example of this kind.We will answer a long standing folklore conjecture that a Stein manifold has a negative constant holomorphic sectional curvature if and only if it is biholomorphic to a ball with a pluripolar set removed.Together with the uniqueness of a moment problem in the appendix of the paper provided by John Treuer, we will show that, under natural assumptions, there is no complex manifold whose Bergman metric is flat. 

Synthetic Notch (synNotch) receptors are genetically encoded, modular synthetic receptors that enable mammalian cells to detect environmental signals and respond by activating user-prescribed transcriptional programs. Although some materials have been modified to present synNotch ligands with coarse spatial control, applications in tissue engineering generally require extracellular matrix (ECM)-derived scaffolds and/or finer spatial positioning of multiple ligands. Thus, we develop here a suite of materials that activate synNotch receptors for generalizable engineering of material-to-cell signaling. We genetically and chemically fuse functional synNotch ligands to ECM proteins and ECM-derived materials. We also generate tissues with microscale precision over four distinct reporter phenotypes by culturing cells with two orthogonal synNotch programs on surfaces microcontact-printed with two synNotch ligands. Finally, we showcase applications in tissue engineering by co-transdifferentiating fibroblasts into skeletal muscle or endothelial cell precursors in user-defined micropatterns. These technologies provide avenues for spatially controlling cellular phenotypes in mammalian tissues. 

We investigate how matrix stiffness regulates chromatin reorganization and cell reprogramming and find that matrix stiffness acts as a biphasic regulator of epigenetic state and fibroblast-to-neuron conversion efficiency, maximized at an intermediate stiffness of 20 kPa. ATAC sequencing analysis shows the same trend of chromatin accessibility to neuronal genes at these stiffness levels. Concurrently, we observe peak levels of histone acetylation and histone acetyltransferase (HAT) activity in the nucleus on 20 kPa matrices, and inhibiting HAT activity abolishes matrix stiffness effects. G-actin and cofilin, the cotransporters shuttling HAT into the nucleus, rises with decreasing matrix stiffness; however, reduced importin-9 on soft matrices limits nuclear transport. These two factors result in a biphasic regulation of HAT transport into nucleus, which is directly demonstrated on matrices with dynamically tunable stiffness. Our findings unravel a mechanism of the mechano-epigenetic regulation that is valuable for cell engineering in disease modeling and regenerative medicine applications.