Corollary and Lemma Finally Corollary and Lemma show

(Corollary 3 and Lemma 9). Finally, Corollary 5 and Lemma 10 show that if then the support of is wide enough and so must contain the points of maximum of the attached. However, the support of is too narrow and so does not contain the points of maximum of the remote. Hence, the remote has at most two points of maximum and so, the attached has a single point of maximum. Thus, the equilibrium is the one where the remote pins down the attached.
Future work
Conclusion
Introduction Postnatal development of the porcine intestine is accompanied by fundamental structural and functional changes. Maintenance of an integer epithelial barrier and adequate mucosal immunity is mandatory to prevent intestinal infections and disease. The critical phase in postnatal development of the porcine gut is associated with weaning of piglets. During this phase, piglets are massively exposed to novel dietary and microbial antigens. The transition is not carried out gradually but abrupt, therefore, slow adaption of the intestine to new feed residue is not possible. In many cases, this leads to reduced feed intake, diarrhea and higher susceptibility to infection [1]. The postnatal development and associated environmental challenges require a complex interplay of endogenous molecular mechanisms controlling processes such as proliferation and differentiation. An endogenous key regulator of postnatal intestinal development is the hedgehog signaling pathway [2]. Its dysregulation has been implicated in several developmental disorders and cancers [3]. The hedgehog signaling cascade plays an important role in the establishment and maintenance of a balanced homeostasis of embryonic and adult intestine. It controls mesenchymal growth and equilibrium in the mammalian digestive tract [2]. Out of the three Hedgehog molecules Indian Hedgehog (IHH), Sonic Hedgehog (SHH) and Desert Hedgehog (DHH), IHH is the predominant molecule expressed in the small intestine and colon [4]. Signaling takes place in a paracrine manner to the mesenchymal cells. The anti-proliferative effect of Hedgehog peptides competes against the Wnt signaling pathway which promotes cell-proliferation [5]. Regulatory RNAs such as microRNAs (miRNAs) were shown to have a strong influence on processes such as development, proliferation and differentiation by binding and silencing their target Ciclopirox ethanolamine australia [6]. Interestingly, miRNAs were demonstrated to regulate cell differentiation and proliferation in intestinal postnatal development [7] and were also reported to impact several factors of hedgehog signaling [8,9]. We were one of the first groups to create an intestinal miRNA map of a mammal using the model of weaned piglets [10]. This study revealed subsets of miRNAs that are specific to distinct parts of the intestine pointing out their impact on distinct cellular signaling networks. For example, miRNA miR-15b belonged to a cluster which showed upregulated expression in ileum and distal jejunum in 31 days old piglets in comparison to duodenum and colon. The miR-15 family consists of 6 miRNAs (miR-15a/15b/16/195/424/497) which are highly conserved but possessing binding capabilities of a given target with deviating affinities. In silico prediction identified miR-15 family as a potential regulator of hedgehog signaling. Thus, the aim of the present study was to analyze the role of miR-15 family on postnatal development of the porcine intestine and its potential impact on the hedgehog signaling pathway. Expression of key hedgehog factors as well as miR-15 family members was followed up in ileum and colon of 7–56 days old piglets.
Materials and methods
Results
Discussion Postnatal development of mammalian intestine requires a well-balanced coordination between proliferation and differentiation. Several signaling pathways are involved to ensure proper organogenesis and tissue homeostasis of the digestive tract during embryogenesis but also in the postnatal development, especially during the sensitive phase around weaning when the gut is exposed to novel antigens and microbial challenges. The hedgehog signaling pathway is crucial to regulate expansion and homeostasis of intestinal mesenchymal cells [2]. Hedgehog molecules are produced by the superficial epithelium and signal to underlying mesenchymal cells such as smooth-muscle cells and myofibroblast-like cells. The activated hedgehog signaling pathway controls mesenchymal factors that negatively regulate epithelial proliferation and supports differentiation. Regulation of IHH expression itself is poorly understood. However, miRNAs were reported to be involved in processes such as proliferation and differentiation and are also known to influence several factors of hedgehog signaling [8,9].