A note on primes in certain residue classes

Given positive integers $a_1,\ldots,a_k$, we prove that the set of primes $p$ such that $p \not\equiv 1 \bmod{a_i}$ for $i=1,\ldots,k$ admits asymptotic density relative to the set of all primes which is at least $\prod_{i=1}^k \left(1-\frac{1}{\varphi(a_i)}\right)$, where $\varphi$ is the Euler's totient function. This result is similar to the one of Heilbronn and Rohrbach, which says that the set of positive integer $n$ such that $n \not\equiv 0 \bmod a_i$ for $i=1,\ldots,k$ admits asymptotic density which is at least $\prod_{i=1}^k \left(1-\frac{1}{a_i}\right)$

On the greatest common divisor of nn and the nnth Fibonacci number

Let $\mathcal{A}$ be the set of all integers of the form $\gcd(n, F_n)$, where $n$ is a positive integer and $F_n$ denotes the $n$th Fibonacci number. We prove that $\#\left(\mathcal{A} \cap [1, x]\right) \gg x / \log x$ for all $x \geq 2$, and that $\mathcal{A}$ has zero asymptotic density. Our proofs rely on a recent result of Cubre and Rouse which gives, for each positive integer $n$, an explicit formula for the density of primes $p$ such that $n$ divides the rank of appearance of $p$, that is, the smallest positive integer $k$ such that $p$ divides $F_k$

Practical numbers among the binomial coefficients

A practical number is a positive integer n such that every positive integer less than n can be written as a sum of distinct divisors of n. We prove that most of the binomial coefficients are practical numbers. Precisely, letting f(n) denote the number of binomial coefficients (nk), with 0≤k≤n, that are not practical numbers, we show that f(n

On the p-adic valuation of Stirling numbers of the first kind

For all integers n≥k≥1, define H(n,k):=∑1/(i1⋯ik), where the sum is extended over all positive integers i1<⋯−klogpn+Ok(1) and we conjecture that there exists a positive constant c = c(p, k) such that νp(H(n,k))<−clogn for all large n. In this respect, we prove the conjecture in the affirmative for all n≤x whose base p representations start with the base p representation of k − 1, but at most 3x0.835 exceptions. We also generalize a result of Lengyel by giving a description of ν2(H(n,2)) in terms of an infinite binary sequence

Down-regulation of the Lamin A/C in neuroblastoma triggers the expansion of tumor initiating cells

Tumor-initiating cells constitute a population within a tumor mass that shares properties with normal stem cells and is considered responsible for therapy failure in many cancers. We have previously demonstrated that knockdown of the nuclear envelope component Lamin A/C in human neuroblastoma cells inhibits retinoic acid-mediated differentiation and results in a more aggressive phenotype. In addition, Lamin A/C is often lost in advanced tumors and changes in the nuclear envelope composition occur during tumor progression. Based on our previous data and considering that Lamin A/C is expressed in differentiated tissues, we hypothesize that the lack of Lamin A/C could predispose cells toward a stem-like phenotype, thus influencing the development of tumor-initiating cells in neuroblastoma. This paper demonstrates that knockdown of Lamin A/C triggers the development of a tumor-initiating cell population with self-renewing features in human neuroblastoma cells. We also demonstrates that the development of TICs is due to an increased expression of MYCN gene and that in neuroblastoma exists an inverse relationship between LMNA and MYCN expression

Pharmacological activation of SIRT6 triggers lethal autophagy in human cancer cells

Sirtuin 6 (SIRT6) is a member of the NAD+-dependent class III deacetylase sirtuin family, which plays a key role in cancer by controlling transcription, genome stability, telomere integrity, DNA repair, and autophagy. Here we analyzed the molecular and biological effects of UBCS039, the first synthetic SIRT6 activator. Our data demonstrated that UBCS039 induced a time-dependent activation of autophagy in several human tumor cell lines, as evaluated by increased content of the lipidated form of LC3B by western blot and of autophagosomal puncta by microscopy analysis of GFP-LC3. UBCS039-mediated activation of autophagy was strictly dependent on SIRT6 deacetylating activity since the catalytic mutant H133Y failed to activate autophagy. At the molecular level, SIRT6-mediated autophagy was triggered by an increase of ROS levels, which, in turn, resulted in the activation of the AMPK-ULK1-mTOR signaling pathway. Interestingly, antioxidants were able to completely counteract UBCS039-induced autophagy, suggesting that ROS burst had a key role in upstream events leading to autophagy commitment. Finally, sustained activation of SIRT6 resulted in autophagy-related cell death, a process that was markedly attenuated using either a pan caspases inhibitor (zVAD-fmk) or an autophagy inhibitor (CQ). Overall, our results identified UBCS039 as an efficient SIRT6 activator, thereby providing a proof of principle that modulation of the enzyme can influence therapeutic strategy by enhancing autophagy-dependent cell death

In vitro and in vivo evaluation of NCX 4040 cytotoxic activity in human colon cancer cell lines

BACKGROUND: Nitric oxide-releasing nonsteroidal antiinflammatory drugs (NO-NSAIDs) are reported to be safer than NSAIDs because of their lower gastric toxicity. We compared the effect of a novel NO-releasing derivate, NCX 4040, with that of aspirin and its denitrated analog, NCX 4042, in in vitro and in vivo human colon cancer models and investigated the mechanisms of action underlying its antitumor activity. METHODS: In vitro cytotoxicity was evaluated on a panel of colon cancer lines (LoVo, LoVo Dx, WiDr and LRWZ) by sulforhodamine B assay. Cell cycle perturbations and apoptosis were evaluated by flow cytometry. Protein expression was detected by Western blot. In the in vivo experiments, tumor-bearing mice were treated with NCX 4040, five times a week, for six consecutive weeks. RESULTS: In the in vitro studies, aspirin and NCX 4042 did not induce an effect on any of the cell lines, whereas NCX 4040 produced a marked cytostatic dose-related effect, indicating a pivotal role of the -NO(2 )group. Furthermore, in LoVo and LRWZ cell lines, we observed caspase-9 and -3-mediated apoptosis, whereas no apoptotic effect was observed after drug exposure in WiDr or LoVo Dx cell lines. In in vivo studies, both NCX 4040 and its parental compound were administered per os. NCX 4040 induced a 40% reduction in tumor weight. Conversely, aspirin did not influence tumor growth at all. CONCLUSIONS: NCX 4040, but not its parental compound, aspirin, showed an in vitro and in vivo antiproliferative activity, indicating its potential usefulness to treat colon cancer

From single gene analysis to single cell profiling: a new era for precision medicine

Molecular profiling of DNA and RNA has provided valuable new insights into the genetic basis of non-malignant and malignant disorders, as well as an increased understanding of basic mechanisms that regulate human disease. Recent technological advances have enabled the analyses of alterations in gene-based structure or function in a comprehensive, high-throughput fashion showing that each tumor type typically exhibits distinct constellations of genetic alterations targeting one or more key cellular pathways that regulate cell growth and proliferation, evasion of the immune system, and other aspects of cancer behavior. These advances have important implications for future research and clinical practice in areas as molecular diagnostics, the implementation of gene or pathway-directed targeted therapy, and the use of such information to drive drug discovery. The 1st international and 32nd Annual Conference of Italian Association of Cell Cultures (AICC) conference wanted to offer the opportunity to match technological solutions and clinical needs in the era of precision medicine

Cathepsin B inhibition interferes with metastatic potential of human melanoma: an in vitro and in vivo study

<p>Abstract</p> <p>Background</p> <p>Cathepsins represent a group of proteases involved in determining the metastatic potential of cancer cells. Among these are cysteinyl- (e.g. cathepsin B and cathepsin L) and aspartyl-proteases (e.g. cathepsin D), normally present inside the lysosomes as inactive proenzymes. Once released in the extracellular space, cathepsins contribute to metastatic potential by facilitating cell migration and invasiveness.</p> <p>Results</p> <p>In the present work we first evaluated, by <it>in vitro </it>procedures, the role of cathepsins B, L and D, in the remodeling, spreading and invasiveness of eight different cell lines: four primary and four metastatic melanoma cell lines. Among these, we considered two cell lines derived from a primary cutaneous melanoma and from a supraclavicular lymph node metastasis of the same patient. To this purpose, the effects of specific chemical inhibitors of these proteases, i.e. CA-074 and CA-074Me for cathepsin B, Cathepsin inhibitor II for cathepsin L, and Pepstatin A for cathepsin D, were evaluated. In addition, we also analyzed the effects of the biological inhibitors of these cathepsins, i.e. specific antibodies, on cell invasiveness. We found that i) cathepsin B, but not cathepsins L and D, was highly expressed at the surface of metastatic but not of primary melanoma cell lines and that ii) CA-074, or specific antibodies to cathepsin B, hindered metastatic cell spreading and dissemination, whereas neither chemical nor biological inhibitors of cathepsins D and L had significant effects. Accordingly, <it>in vivo </it>studies, i.e. in murine xenografts, demonstrated that CA-074 significantly reduced human melanoma growth and the number of artificial lung metastases.</p> <p>Conclusions</p> <p>These results suggest a reappraisal of the use of cathepsin B inhibitors (either chemical or biological) as innovative strategy in the management of metastatic melanoma disease.</p