A Harnack's inequality for mixed type evolution equations

We define a homogeneous parabolic De Giorgi classes of order 2 which suits a mixed type class of evolution equations whose simplest example is $\mu (x) \frac{\partial u}{\partial t} - \Delta u = 0$ where $\mu$ can be positive, null and negative, so in particular elliptic-parabolic and forward-backward parabolic equations are included. For functions belonging to this class we prove local boundedness and show a Harnack inequality which, as by-products, gives H\"older-continuity, in particular in the interface $I$ where $\mu$ change sign, and a maximum principle

Harnack's Inequality for Parabolic De Giorgi Classes in Metric Spaces

In this paper we study problems related to parabolic partial differential equations in metric measure spaces equipped with a doubling measure and supporting a Poincare' inequality. We give a definition of parabolic De Giorgi classes and compare this notion with that of parabolic quasiminimizers. The main result, after proving the local boundedness, is a scale and location invariant Harnack inequality for functions belonging to parabolic De Giorgi classes. In particular, the results hold true for parabolic quasiminimizers

An inhomogeneous Harnack inequality for parabolic equations and applications to elliptic-parabolic and forward-backward parabolic equations

We define a homogeneous De Giorgi class of order $p = 2$ that contains the solutions of evolution equations of the types \uprho (x,t) u_t + A u = 0 and (\uprho (x,t) u)_t + A u = 0, where \uprho > 0 almost everywhere and $A$ is a suitable elliptic operator. \\ For functions belonging to this class we prove an inhomogeneous Harnack inequality, i.e. a Harnack inequality that takes into account the mean value of \uprho in different regions of $\Omega \times (0,T)$. \\ As a consequence we get a Harnack inequality for solutions, and in these case only for solutions, of elliptic-parabolic equations and of forward-backward parabolic equations. \\ As a byproduct one obtains H\"older continuity for solutions of a subclass the first equation: in particular the solutions of this subclass are H\"older continuous in the interface where \uprho changes its sign

GG-convergence of elliptic and parabolic operators depending on vector fields

We consider sequences of elliptic and parabolic operators in divergence form where the operators depend on a family of vector fields. We show, using a unitary approach, i.e., taking suitable elliptic-parabolic operators, compactness results with respect to G-convergence, or H-convergence, by assuming the existence of affine functions, with respect to the family of vector fields

G-convergence of elliptic and parabolic operators depending on vector fields

We consider sequences of elliptic and parabolic operators in divergence form and depending on a family of vector fields. We show compactness results with respect to G-convergence, or H-convergence, by means of the compensated compactness theory, in a setting in which the existence of affine functions is not always guaranteed, due to the nature of the family of vector fields

The RNA Binding Protein SAM68 Transiently Localizes in the Chromatoid Body of Male Germ Cells and Influences Expression of Select MicroRNAs

The chromatoid body (CB) is a unique structure of male germ cells composed of thin filaments that condense into a perinuclear organelle after meiosis. Due to the presence of proteins involved in different steps of RNA metabolism and of different classes of RNAs, including microRNAs (miRNAs), the CB has been recently suggested to function as an RNA processing centre. Herein, we show that the RNA binding protein SAM68 transiently localizes in the CB, in concomitance with the meiotic divisions of mouse spermatocytes. Precise staging of the seminiferous tubules and co-localization studies with MVH and MILI, two well recognized CB markers, documented that SAM68 transiently associates with the CB in secondary spermatocytes and early round spermatids. Furthermore, although SAM68 co-immunoprecipitated with MVH in secondary spermatocytes, its ablation did not affect the proper localization of MVH in the CB. On the other hand, ablation of the CB constitutive component MIWI did not impair association of SAM68 with the CB. Isolation of CBs from Sam68 wild type and knockout mouse testes and comparison of their protein content by mass spectrometry indicated that Sam68 ablation did not cause overall alterations in the CB proteome. Lastly, we found that SAM68 interacts with DROSHA and DICER in secondary spermatocytes and early round spermatids and that a subset of miRNAs were altered in Sam68−/−germ cells. These results suggest a novel role for SAM68 in the miRNA pathway during spermatogenesis

The centrosomal kinase NEK2 is a novel splicing factor kinase involved in cell survival

NEK2 is a serine/threonine kinase that promotes centrosome splitting and ensures correct chromosome segregation during the G2/M phase of the cell cycle, through phosphorylation of specific substrates. Aberrant expression and activity of NEK2 in cancer cells lead to dysregulation of the centrosome cycle and aneuploidy. Thus, a tight regulation of NEK2 function is needed during cell cycle progression. In this study, we found that NEK2 localizes in the nucleus of cancer cells derived from several tissues. In particular, NEK2 co-localizes in splicing speckles with SRSF1 and SRSF2. Moreover, NEK2 interacts with several splicing factors and phosphorylates some of them, including the oncogenic SRSF1 protein. Overexpression of NEK2 induces phosphorylation of endogenous SR proteins and affects the splicing activity of SRSF1 toward reporter minigenes and endogenous targets, independently of SRPK1. Conversely, knockdown of NEK2, like that of SRSF1, induces expression of pro-apoptotic variants from SRSF1-target genes and sensitizes cells to apoptosis. Our results identify NEK2 as a novel splicing factor kinase and suggest that part of its oncogenic activity may be ascribed to its ability to modulate alternative splicing, a key step in gene expression regulation that is frequently altered in cancer cells

SAM68 is a physiological regulator of SMN2 splicing in spinal muscular atrophy

Spinal muscular atrophy (SMA) is a neurodegenerative disease caused by loss of motor neurons in patients with null mutations in the SMN1 gene. The almost identical SMN2 gene is unable to compensate for this deficiency because of the skipping of exon 7 during pre-messenger RNA (mRNA) processing. Although several splicing factors can modulate SMN2 splicing in vitro, the physiological regulators of this disease-causing event are unknown. We found that knockout of the splicing factor SAM68 partially rescued body weight and viability of SMAΔ7 mice. Ablation of SAM68 function promoted SMN2 splicing and expression in SMAΔ7 mice, correlating with amelioration of SMA-related defects in motor neurons and skeletal muscles. Mechanistically, SAM68 binds to SMN2 pre-mRNA, favoring recruitment of the splicing repressor hnRNP A1 and interfering with that of U2AF65 at the 3' splice site of exon 7. These findings identify SAM68 as the first physiological regulator of SMN2 splicing in an SMA mouse model