Euler’s beta integral in Pietro Mengoli’s works

Beta integrals for several non-integer values of the exponents were calculated by Leonhard Euler in 1730, when he was trying to find the general term for the factorial function by means of an algebraic expression. Nevertheless, 70 years before, Pietro Mengoli (1626–1686) had computed such integrals for natural and half-integer exponents in his Geometriae Speciosae Elementa (1659) and Circolo(1672) and displayed the results in triangular tables. In particular, his new arithmetic–algebraic method allowed him to compute the quadrature of the circle. The aim of this article is to show how Mengoli calculated the values of these integrals as well as how he analysed the relation between these values and the exponents inside the integrals. This analysis provides new insights into Mengoli’s view of his algorithmic computation of quadratures.     

Ectopic beta-chain of ATP synthase is an apolipoprotein A-I receptor in hepatic HDL endocytosis

The effect of high-density lipoprotein (HDL) in protecting against atherosclerosis is usually attributed to its role in 'reverse cholesterol transport'. In this process, HDL particles mediate the efflux and the transport of cholesterol from peripheral cells to the liver for further metabolism and bile excretion. Thus, cell-surface receptors for HDL on hepatocytes are chief partners in the regulation of cholesterol homeostasis. A high-affinity HDL receptor for apolipoprotein A-I (apoA-I) was previously identified on the surface of hepatocytes. Here we show that this receptor is identical to the beta-chain of ATP synthase, a principal protein complex of the mitochondrial inner membrane. Different experimental approaches confirm this ectopic localization of components of the ATP synthase complex and the presence of ATP hydrolase activity at the hepatocyte cell surface. Receptor stimulation by apoA-I triggers the endocytosis of holo-HDL particles (protein plus lipid) by a mechanism that depends strictly on the generation of ADP. We confirm this effect on endocytosis in perfused rat liver ex vivo by using a specific inhibitor of ATP synthase. Thus, membrane-bound ATP synthase has a previously unsuspected role in modulating the concentrations of extracellular ADP and is regulated by a principal plasma apolipoprotein.     

Intracellular protein degradation in mammalian cells: recent developments

In higher organisms, dietary proteins are broken down into amino acids within the digestive tract but outside the cells, which incorporate the resulting amino acids into their metabolism. However, under certain conditions, an organism loses more nitrogen than is assimilated in the diet. This additional loss was found in the past century to come from intracellular proteins and started an intensive research that produced an enormous expansion of the field and a dispersed literature. Therefore, our purpose is to provide an updated summary of the current knowledge on the proteolytic machinery involved in intracellular protein degradation and its physiological and pathological relevance, especially addressed to newcomers in the field who may find further details in more specialized reviews. However, even providing a general overview, this is an extremely wide field and, therefore, we mainly focus on mammalian cells, while other cells will be mentioned only for comparison purposes.