Abstract. We give a new explicit construction of n × N matrices satisfying the Restricted Isometry Property (RIP). Namely, for some ε > 0, large N and any n satisfying N 1−ε ≤ n ≤ N , we construct RIP matrices of order k ≥ n 1/2+ε and constant δ = n −ε . This overcomes the natural barrier k = O(n 1/2 ) for proofs based on small coherence, which are used in all previous explicit constructions of RIP matrices. Key ingredients in our proof are new estimates for sumsets in product sets and for exponential sums with the products of sets possessing special additive structure. We also give a construction of sets of n complex numbers whose k-th moments are uniformly small for 1 ≤ k ≤ N (Turán's power sum problem), which improves upon known explicit constructions when (log N ) 1+o(1) ≤ n ≤ (log N ) 4+o(1) . This latter construction produces elementary explicit examples of n × N matrices that satisfy RIP and whose columns constitute a new spherical code; for those problems the parameters closely match those of existing constructions in the range (log N ) 1+o(1) ≤ n ≤ (log N ) 5/2+o(1) .
Protein phosphatase 2A (PP2A) is a family of multifunctional serine/threonine phosphatases all composed of a catalytic C, a structural A, and a regulatory B subunit. Assembly of the complex with the appropriate B subunit forms the key to the functional specificity and regulation of PP2A. Emerging evidence suggests a crucial role for methylation and phosphorylation of the PP2A C subunit in this process. In this study, we show that PP2A C subunit methylation was not absolutely required for binding the PR61/B and PR72/B؆ subunit families, whereas binding of the PR55/B subunit family was determined by methylation and the nature of the C-terminal amino acid side chain. Moreover mutation of the phosphorylatable Tyr 307 or Thr 304 residues differentially affected binding of distinct B subunit family members. Down-regulation of the PP2A methyltransferase LCMT1 by RNA interference gradually reduced the cellular amount of methylated C subunit and induced a dynamic redistribution of the remaining methylated PP2A C between different PP2A trimers consistent with their methylation requirements. Persistent knockdown of LCMT1 eventually resulted in specific degradation of the PR55/B subunit and apoptotic cell death. Together these results establish a crucial foundation for understanding PP2A regulatory subunit selection.Protein phosphatase 2A (PP2A) 2 represents a family of heterotrimeric serine/threonine phosphatases implicated in the regulation of a plethora of cellular processes such as apoptosis, transcription, translation, DNA replication, signal transduction, protection against tumorigenesis, and cell division (for reviews, see Refs. 1 and 2). It is estimated that, depending on the cell type, PP2A holoenzymes are responsible for 30 -50% of the total cellular serine/threonine dephosphorylating activity, demonstrating the importance of this enzyme system for almost any aspect of life.The basis of this broad functional importance is formed by the diversity of specific PP2A subunit compositions. Typically the PP2A core enzyme exists as a dimer (PP2A D ) consisting of a catalytic subunit (C subunit/PP2A C ) and a scaffolding A subunit (PR65/A subunit). Proper functioning and regulation of PP2A is achieved by the association of regulatory B subunits 3 with the PP2A core enzyme, resulting in the formation of heterotrimeric PP2A holoenzymes with specific catalytic properties, subcellular locations, and substrate specificities. At present, three multigene families of B-type subunits have been described, PR55/B, PR61/BЈ, and PR72/BЉ, all with specific cellular functions. Therefore, the assembly of the complex with the appropriate B-type subunit is the key to specificity and regulation of PP2A (2). In this process, the highly conserved C-terminal PP2A C tail seems to play a crucial role (3, 4). Recently a major breakthrough has been achieved by elucidating the crystal structure of a heterotrimeric PP2A T61␥ holoenzyme (5, 6). It was shown that the C-terminal PP2A C tail recognizes a surface groove at the interface between the PR65 a...
The Raf-1 kinase plays a key role in relaying proliferation signals elicited by mitogens or oncogenes. Raf-1 is regulated by complex and incompletely understood mechanisms including phosphorylation. A number of studies have indicated that phosphorylation of serines 259 and 621 can inhibit the Raf-1 kinase. We show that both serines are hypophosphorylated during early mitogenic stimulation and that hypophosphorylation correlates with peak Raf-1 activation. Concentrations of okadaic acid that selectively inhibit protein phosphatase 2A (PP2A) induce phosphorylation of these residues and prevent maximal activation of the Raf-1 kinase. This effect is mediated via phosphorylation of serine 259. The PP2A core heterodimer forms complexes with Raf-1 in vivo and in vitro. These data identify PP2A as a positive regulator of Raf-1 activation and are the first indication that PP2A may support the activation of an associated kinase.The Raf-1 kinase is an important intermediate in the transduction of proliferative signals, and its activation may be a key event in the development of a wide range of tumors (1). Activated Raf-1 can regulate the mitogen-activated protein kinase network by phosphorylating and activating MEK 1 ; within the mitogen-activated protein kinase cascade, Raf interacts physically with MEK-1 via its kinase domain and with GTP-loaded Ras via its N terminus (2). Activated Ras is the best studied activator of Raf-1. It binds to Raf-1 with high affinity and mediates its translocation from the cytosol to the plasma membrane, where activation takes place (3,4). Artificial tethering of Raf-1 to the cell membrane results in partial activation, which can be further enhanced by mitogenic stimulation, suggesting that at the cell membrane Raf-1 is exposed to both constitutive and mitogen-regulated activators (5-8).Mitogenic stimulation of cells typically induces hyperphosphorylation of Raf-1 and a retardation of its migration on SDS gels. This hyperphosphorylation correlates with the down-regulation of Raf-1 kinase activity (9, 10) and may be implemented by a negative feedback mechanism depending on MEK activity (10, 11). Serines 43, 621, and 259 are phosphorylated in resting fibroblasts, albeit to different degrees (12). Phosphorylation of all three residues has been implicated in the negative regulation of Raf-1. Phosphorylation of serine 43 interferes with Ras binding and consequently with Ras-mediated activation (3). Phosphorylated serine 259 and serine 621 represent binding sites for 14-3-3 adaptor proteins (13,14), whose function in Raf-1 activation is controversial. While bivalent binding to Ser 259 and Ser 621 has been suggested to maintain Raf-1 in an inactive conformation (15, 16), reversible association with 14-3-3 facilitates Ras-dependent activation in vivo and in vitro (17). In particular, binding to the Ser(P) 621 site appears to be necessary for kinase activity (16, 18), a finding that contrasts with the studies indicating that phosphorylation of this site by PKA in vitro is inhibitory (19). Therefore, ...
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