A Characterization Theorem for the K-functional Associated with the Algebraic Version of Trigonometric Jackson Integrals

2000 Mathematics Subject Classification: 41A25, 41A36.

A New Characterization of Weighted Peetre K-Functionals (II)

2000 Mathematics Subject Classification: 46B70, 41A25, 41A17, 26D10. ∗Part of the results were reported at the Conference “Pioneers of Bulgarian Mathematics”, Sofia, 2006.

Equivalence Between K-functionals Based on Continuous Linear Transforms

2000 Mathematics Subject Classification: 46B70, 41A10, 41A25, 41A27, 41A35, 41A36, 42A10.

On an Extremal Problem concerning Bernstein Operators

* The second author is supported by the Alexander-von-Humboldt Foundation. He is on leave from: Institute of Mathematics, Academia Sinica, Beijing 100080, People’s Republic of China.

Direct and Converse Theorems for Generalized Bernstein-Type Operators

2000 Mathematics Subject Classification: 41A25, 41A27, 41A36.

Теглови апроксимации с оператори от тип на Гудман–Шарма

P. E. Parvanov - The uniform weighted approximation errors of the Goodman–Sharma operators are characterized for functions.

Best Approximation and Moduli of Smoothness

2010 Mathematics Subject Classification: 41A25, 41A10.

On the approximation by convolution operators in homogeneous Banach spaces on R^d

AMS Subject Classification 2010: 41A25, 41A35, 41A40, 41A63, 41A65, 42A38, 42A85, 42B10, 42B20

The Lower Estimate for Bernstein Operator

MSC 2010: 41A10, 41A15, 41A25, 41A36

Weighted Approximation by a Class of Bernstein-Type Operators

AMS classification: 41A36, 41A10, 41A25, 41Al7.

On the Approximation by Convolution Operators in Homogeneous Banach Spaces of Periodic Functions

AMS Subject Classification 2010: 41A25, 41A27, 41A35, 41A36, 41A40, 42Al6, 42A85.

The fourth extracellular loop of the alpha subunit of Na,K-ATPase. Functional evidence for close proximity with the second extracellular loop.

Na,K-ATPase is the main active transport system that maintains the large gradients of Na(+) and K(+) across the plasma membrane of animal cells. The crystal structure of a K(+)-occluding conformation of this protein has been recently published, but the movements of its different domains allowing for the cation pumping mechanism are not yet known. The structure of many more conformations is known for the related calcium ATPase SERCA, but the reliability of homology modeling is poor for several domains with low sequence identity, in particular the extracellular loops. To better define the structure of the large fourth extracellular loop between the seventh and eighth transmembrane segments of the alpha subunit, we have studied the formation of a disulfide bond between pairs of cysteine residues introduced by site-directed mutagenesis in the second and the fourth extracellular loop. We found a specific pair of cysteine positions (Y308C and D884C) for which extracellular treatment with an oxidizing agent inhibited the Na,K pump function, which could be rapidly restored by a reducing agent. The formation of the disulfide bond occurred preferentially under the E2-P conformation of Na,K-ATPase, in the absence of extracellular cations. Using recently published crystal structure and a distance constraint reproducing the existence of disulfide bond, we performed an extensive conformational space search using simulated annealing and showed that the Tyr(308) and Asp(884) residues can be in close proximity, and simultaneously, the SYGQ motif of the fourth extracellular loop, known to interact with the extracellular domain of the beta subunit, can be exposed to the exterior of the protein and can easily interact with the beta subunit.

Functional effects of Na+,K+-ATPase gene mutations linked to familial hemiplegic migraine.

Familial hemiplegic migraine type 2, an autosomal dominant form of migraine with aura, has been associated with four distinct mutations in the alpha2-subunit of the Na+,K+-ATPase. We have introduced these mutations in the alpha2-subunit of the human Na+,K+-ATPase and the corresponding mutations in the Bufo marinus alpha1-subunit and studied these mutants by expression in Xenopus oocyte. Metabolic labeling studies showed that the mutants were synthesized and associated with the beta-subunit, except for the alpha2HW887R mutant, which was poorly synthesized, and the alpha1BW890R, which was partially retained in the endoplasmic reticulum. [3H]ouabain binding showed the presence of the alpha2HR689Q and alpha2HM731T at the membrane, whereas the alpha2HL764P and alpha2HW887R could not be detected. Functional studies with the mutants of the B. marinus Na+,K+-ATPase showed a reduced or abolished electrogenic activity and a low K+ affinity for the alpha1BW890R mutant. Through different mechanisms, all these mutations result in a strong decrease of the functional expression of the Na+,K+-pump. The decreased activity in alpha2 isoform of the Na+,K+-pump expressed in astrocytes seems an essential component of hemiplegic migraine pathogenesis and may be responsible for the cortical spreading depression, which is one of the first events in migraine attacks.

Structural and functional interaction sites between Na,K-ATPase and FXYD proteins.

Several members of the FXYD protein family are tissue-specific regulators of Na,K-ATPase that produce distinct effects on its apparent K(+) and Na(+) affinity. Little is known about the interaction sites between the Na,K-ATPase alpha subunit and FXYD proteins that mediate the efficient association and/or the functional effects of FXYD proteins. In this study, we have analyzed the role of the transmembrane segment TM9 of the Na,K-ATPase alpha subunit in the structural and functional interaction with FXYD2, FXYD4, and FXYD7. Mutational analysis combined with expression in Xenopus oocytes reveals that Phe(956), Glu(960), Leu(964), and Phe(967) in TM9 of the Na,K-ATPase alpha subunit represent one face interacting with the three FXYD proteins. Leu(964) and Phe(967) contribute to the efficient association of FXYD proteins with the Na,K-ATPase alpha subunit, whereas Phe(956) and Glu(960) are essential for the transmission of the functional effect of FXYD proteins on the apparent K(+) affinity of Na,K-ATPase. The relative contribution of Phe(956) and Glu(960) to the K(+) effect differs for different FXYD proteins, probably reflecting the intrinsic differences of FXYD proteins on the apparent K(+) affinity of Na,K-ATPase. In contrast to the effect on the apparent K(+) affinity, Phe(956) and Glu(960) are not involved in the effect of FXYD2 and FXYD4 on the apparent Na(+) affinity of Na,K-ATPase. The mutational analysis is in good agreement with a docking model of the Na,K-ATPase/FXYD7 complex, which also predicts the importance of Phe(956), Glu(960), Leu(964), and Phe(967) in subunit interaction. In conclusion, by using mutational analysis and modeling, we show that TM9 of the Na,K-ATPase alpha subunit exposes one face of the helix that interacts with FXYD proteins and contributes to the stable interaction with FXYD proteins, as well as mediating the effect of FXYD proteins on the apparent K(+) affinity of Na,K-ATPase.

Functional roles of Na,K-ATPase subunits

PURPOSE OF REVIEW: Na,K-ATPase is an oligomeric protein composed of alpha subunits, beta subunits and FXYD proteins. The catalytic alpha subunit hydrolyzes ATP and transports the cations. Increasing experimental evidence suggest that beta subunits and FXYD proteins essentially contribute to the variable physiological needs of Na,K-ATPase function in different tissues. RECENT FINDINGS: Beta subunits have a crucial role in the structural and functional maturation of Na,K-ATPase and modulate its transport properties. The chaperone function of the beta subunit is essential, for example, in the formation of tight junctions and cell polarity. Recent studies suggest that beta subunits also have inherent functions, which are independent of Na,K-ATPase activity and which may be involved in cell-cell adhesiveness and in suppression of cell motility. As for FXYD proteins, they modulate Na,K-ATPase activity in a tissue-specific way, in some cases in close cooperation with posttranslational modifications such as phosphorylation. SUMMARY: A better understanding of the multiple functional roles of the accessory subunits of Na,K-ATPase is crucial to appraise their influence on physiological processes and their implication in pathophysiological states