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Abstract The enzymes performing protein post-translational modifications (PTMs) form a critical post-translational regulatory circuitry that orchestrates literally all cellular processes in the organism. In particular, the balance between cellular stemness and differentiation is crucial for the development of multicellular organisms. Importantly, the fine-tuning of this balance on the genetic level is largely mediated by specific PTMs of histones including lysine methylation. Lysine methylation is carried out by special enzymes (lysine methyltransferases) that transfer the methyl group from S-adenosyl-L-methionine to the lysine residues of protein substrates. Set7/9 is one of the exemplary protein methyltransferases that however, has not been fully studied yet. It was originally discovered as histone H3 lysine 4-specific methyltransferase, which later was shown to methylate a number of non-histone proteins that are crucial regulators of stemness and differentiation, including p53, pRb, YAP, DNMT1, SOX2, FOXO3, and others. In this review we summarize the information available to date on the role of Set7/9 in cellular differentiation and tissue development during embryogenesis and in adult organisms. Finally, we highlight and discuss the role of Set7/9 in pathological processes associated with aberrant cellular differentiation and self-renewal, including the formation of cancer stem cells.

Abstract  We describe recent results on attainability of sharp constants in the Sobolev inequality, the Sobolev–Poincaré inequality, the Hardy–Sobolev inequality and related inequalities. This gives us the solvability of boundary value problems to critical Emden–Fowler equations. [ABSTRACT FROM AUTHOR]