Factorization of Primes Primes Primes: Elements Ideals and in Extensions

It is often taken it for granted that all positive whole numbers except 0 and 1 can be factored uniquely into primes. However, if K is a finite extension of the rational numbers, and OK its ring of integers, it is not always the case that non-zero, non-unit elements of OK factor uniquely. We do find, though, that the proper ideals of OK do always factor uniquely into prime ideals. This result allows us to extend many properties of the integers to these rings. If we a finite extension L of K and OL of OK , we find that prime ideals of OK need not remain prime when they are extended into OL; instead, they can split into a product of prime ideals of OL in a very structured way. If L is a normal extension of K, we can use Galois theory to further study this splitting by considering the intermediate fields of K and L, as well as quotient rings of the associated rings of integers. In this paper, we will introduce these topics of algebraic number theory, prove that unique factorization of ideals holds using two different methods, and observe the patterns that arise in the splitting of prime ideals

Cytosolic calcium and protein kinase C reduce complement-mediated glomerular epithelial injury

Cytosolic calcium and protein kinase C reduce complement-mediated glomerular epithelial injury. In rat membranous nephropathy, proteinuria is due to formation of the C5b-9 membrane attack complex of complement (C), and is associated with morphological evidence of glomerular epithelial cell (GEC) injury. Analogous morphological changes are induced by C5b-9 in cultured GEC. In addition, in cultured GEC C5b-9 induces Ca2+ influx, as well as Ca2+ mobilization and increased 1,2-diacylglycerol due to the activation of phospholipase C. In this study we investigated how this GEC activation pattern might influence C-mecliated GEC injury. We demonstrate that the C5b-9-induced increase in cytosolic Ca2+ concentration ([Ca2+]i) did not impair ATP generation by mitochondria, suggesting that it does not contribute to cytotoxicity. Moreover, this increase in [Ca2+]i protected GEC from C-mediated cytolysis. However, a large increase in [Ca2+]i (produced by the Ca2+ ionophore A23187) impaired ATP generation and aggravated C-mediated cytotoxicity, suggesting that intact mitochondrial activity is necessary for GEC to withstand C attack. Activation of protein kinase C (PKC) by phorbol myristate acetate (PMA) also decreased C-mediated cytolysis. Conversely, C lysis was enhanced in GEC that had been pretreated for 18 hours with a high dose of PMA to deplete PKC, and following PKC inhibition with H-7. Therefore, PKC activation, possibly resulting from C5b-9-induced increase in 1,2-diacylglycerol, triggered mechanisms that protected GEC from C-mediated injury. Thus, as a consequence of C5b-9-induced phospholipase activation, the amount of C-induced GEC injury is diminished

Cytosolic Phospholipase A2α and Eicosanoids Regulate Expression of Genes in Macrophages Involved in Host Defense and Inflammation

Acknowledgments: We thank Dr. Robert Barkley and Charis Uhlson for mass spectrometry analysis. Funding: This work was supported by grants from the National Institutes of Health HL34303 (to C.C.L., R.C.M. and D.L.B), DK54741 (to J.V.B.), GM5322 (to D.L.W.) and the Wellcome Trust (to N.A.R.G. and G.D.B.). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.Peer reviewedPublisher PD

Kidney injury molecule-1 is an early biomarker of cadmium nephrotoxicity

Cadmium (Cd) exposure results in injury to the proximal tubule characterized by polyuria and proteinuria. Kidney injury molecule-1 (Kim-1) is a transmembrane glycoprotein not normally detected in the mature kidney, but is upregulated and shed into the urine following nephrotoxic injury. In this study, we determine if Kim-1 might be a useful early biomarker of Cd nephrotoxicity. Male Sprague–Dawley rats were given daily injections of Cd for up to 12 weeks. Weekly urine samples were analyzed for Kim-1, protein, creatinine, metallothionein, and Clara cell protein CC-16. Significant levels of Kim-1 were detected in the urine by 6 weeks and continued to increase throughout the treatment period. This appearance of Kim-1 occurred 4–5 weeks before the onset of proteinuria, and 1–3 weeks before the appearance of metallothionein and CC-16. Higher doses of Cd gave rise to higher Kim-1 excretion. Reverse transcriptase-polymerase chain reaction (RT-PCR) expression analysis showed that Kim-1 transcript levels were increased after 6 weeks at the low dose of Cd. Immunohistochemical analysis showed that Kim-1 was present in proximal tubule cells of the Cd-treated rats. Our results suggest that Kim-1 may be a useful biomarker of early stages of Cd-induced proximal tubule injury

Nitric Oxide-Releasing Nanoparticles Prevent Propionibacterium acnes-Induced Inflammation by Both Clearing the Organism and Inhibiting Microbial Stimulation of the Innate Immune Response.

Propionibacterium acnes induction of IL-1 cytokines through the NLRP3 (NLR, nucleotide oligomerization domain-like receptor) inflammasome was recently highlighted as a dominant etiological factor for acne vulgaris. Therefore, therapeutics targeting both the stimulus and the cascade would be ideal. Nitric oxide (NO), a potent biological messenger, has documented broad-spectrum antimicrobial and immunomodulatory properties. To harness these characteristics to target acne, we used an established nanotechnology capable of generating/releasing NO over time (NO-np). P. acnes was found to be highly sensitive to all concentrations of NO-np tested, although human keratinocyte, monocyte, and embryonic zebra fish assays revealed no cytotoxicity. NO-np significantly suppressed IL-1β, tumor necrosis factor-α (TNF-α), IL-8, and IL-6 from human monocytes, and IL-8 and IL-6 from human keratinocytes, respectively. Importantly, silencing of NLRP3 expression by small interfering RNA did not limit NO-np inhibition of IL-1 β secretion from monocytes, and neither TNF-α nor IL-6 secretion, nor inhibition by NO-np was found to be dependent on this pathway. The observed mechanism by which NO-np impacts IL-1β secretion was through inhibition of caspase-1 and IL-1β gene expression. Together, these data suggest that NO-np can effectively prevent P. acnes-induced inflammation by both clearing the organism and inhibiting microbial stimulation of the innate immune response

M1 muscarinic receptors inhibit L-type Ca2+ current and M-current by divergent signal transduction cascades

Ion channels reside in a sea of phospholipids. During normal fluctuations in membrane potential and periods of modulation, lipids that directly associate with channel proteins influence gating by incompletely understood mechanisms. In one model, M(1)-muscarinic receptors (M(1)Rs) may inhibit both Ca(2+) (L- and N-) and K(+) (M-) currents by losing a putative interaction between channels and phosphatidylinositol-4,5-bisphosphate (PIP(2)). However, we found previously that M(1)R inhibition of N-current in superior cervical ganglion (SCG) neurons requires loss of PIP(2) and generation of a free fatty acid, probably arachidonic acid (AA) by phospholipase A(2) (PLA(2)). It is not known whether PLA(2) activity and AA also participate in L- and M-current modulation in SCG neurons. To test whether PLA(2) plays a similar role in M(1)R inhibition of L- and M-currents, we used several experimental approaches and found unanticipated divergent signaling. First, blocking resynthesis of PIP(2) minimized M-current recovery from inhibition, whereas L-current recovered normally. Second, L-current inhibition required group IVa PLA(2) [cytoplasmic PLA(2) (cPLA(2))], whereas M-current did not. Western blot and imaging studies confirmed acute activation of cPLA(2) by muscarinic stimulation. Third, in type IIa PLA(2) [secreted (sPLA(2))](-/-)/cPLA(2)(-/-) double-knock-out SCG neurons, muscarinic inhibition of L-current decreased. In contrast, M-current inhibition remained unaffected but recovery was impaired. Our results indicate that L-current is inhibited by a pathway previously shown to control M-current over-recovery after washout of muscarinic agonist. Our findings support a model of M(1)R-meditated channel modulation that broadens rather than restricts the roles of phospholipids and fatty acids in regulating ion channel activity

Water and Sodium in Heart Failure: A Spotlight on Congestion.

Despite all available therapies, the rates of hospitalization and death from heart failure (HF) remain unacceptably high. The most common reasons for hospital admission are symptoms related to congestion. During hospitalization, most patients respond well to standard therapy and are discharged with significantly improved symptoms. Post-discharge, many patients receive diligent and frequent follow-up. However, rehospitalization rates remain high. One potential explanation is a persistent failure by clinicians to adequately manage congestion in the outpatient setting. The failure to successfully manage these patients post-discharge may represent an unmet need to improve the way congestion is both recognized and treated. A primary aim of future HF management may be to improve clinical surveillance to prevent and manage chronic fluid overload while simultaneously maximizing the use of evidence-based therapies with proven long-term benefit. Improvement in cardiac function is the ultimate goal and maintenance of a ‘‘dry’’ clinical profile is important to prevent hospital admission and improve prognosis. This paper focuses on methods for monitoring congestion, and strategies for water and sodium management in the context of the complex interplay between the cardiac and renal systems. A rationale for improving recognition and treatment of congestion is also proposed