In this paper we present the construction of a group Hopf algebra on the class of rational tangles. A locally finite partial order on this class is introduced and a topology is generated. An interval coalgebra structure associated with the locally finite partial order is specified. Irrational and real tangles are introduced and their relation with rational tangles are studied. The existence of the maximal real tangle is described in detail.

Zebrafish provide a highly versatile model in which to study vertebrate development. Many recent studies have elucidated early events in the organogenesis of the zebrafish pancreas; however, several aspects of early endocrine pancreas formation in the zebrafish are not homologous to the mammalian system. To better identify mechanisms of islet formation in the zebrafish, with true homology to those observed in mammals, we have temporally and spatially characterized zebrafish secondary islet formation. As is the case in the mouse, we show that Notch inhibition leads to precocious differentiation of endocrine tissues. Furthermore, we have used transgenic fish expressing fluorescent markers under the control of a Notch-responsive element to observe the precursors of these induced endocrine cells. These pancreatic Notch-responsive cells represent a novel population of putative progenitors that are associated with larval pancreatic ductal epithelium, suggesting functional homology between secondary islet formation in zebrafish and the secondary transition in mammals. We also show that Notch-responsive cells persist in the adult pancreas and possess the classical characteristics of centroacinar cells, a cell type believed to be a multipotent progenitor cell in adult mammalian pancreas.

Multicellular aggregates of circulating tumor cells (CTC clusters) are potent initiators of distant organ metastasis. However, it is currently assumed that CTC clusters are too large to pass through narrow vessels to reach these organs. Here, we present evidence that challenges this assumption through the use of microfluidic devices designed to mimic human capillary constrictions and CTC clusters obtained from patient and cancer cell origins. Over 90% of clusters containing up to 20 cells successfully traversed 5-to 10-μm constrictions even in whole blood. Clusters rapidly and reversibly reorganized into single-file chain-like geometries that substantially reduced their hydrodynamic resistances. Xenotransplantation of human CTC clusters into zebrafish showed similar reorganization and transit through capillary-sized vessels in vivo. Preliminary experiments demonstrated that clusters could be disrupted during transit using drugs that affected cellular interaction energies. These findings suggest that CTC clusters may contribute a greater role to tumor dissemination than previously believed and may point to strategies for combating CTC cluster-initiated metastasis.microfluidics | cancer metastasis | CTC clusters | circulating tumor cell cluster microemboli | capillary microhemodynamics