Spectral statistics for quantized skew translations on the torus

We study the spectral statistics for quantized skew translations on the torus, which are ergodic but not mixing for irrational parameters. It is shown explicitly that in this case the level--spacing distribution and other common spectral statistics, like the number variance, do not exist in the semiclassical limit.Comment: 7 pages. One figure, include

Parabolic maps with spin: Generic spectral statistics with non-mixing classical limit

We investigate quantised maps of the torus whose classical analogues are ergodic but not mixing. Their quantum spectral statistics shows non-generic behaviour, i.e.it does not follow random matrix theory (RMT). By coupling the map to a spin 1/2, which corresponds to changing the quantisation without altering the classical limit of the dynamics on the torus, we numerically observe a transition to RMT statistics. The results are interpreted in terms of semiclassical trace formulae for the maps with and without spin respectively. We thus have constructed quantum systems with non-mixing classical limit which show generic (i.e. RMT) spectral statistics. We also discuss the analogous situation for an almost integrable map, where we compare to Semi-Poissonian statistics.Comment: 29 pages, 20 figure

Nos2 Inactivation Promotes the Development of Medulloblastoma in Ptch1+/− Mice by Deregulation of Gap43–Dependent Granule Cell Precursor Migration

Medulloblastoma is the most common malignant brain tumor in children. A subset of medulloblastoma originates from granule cell precursors (GCPs) of the developing cerebellum and demonstrates aberrant hedgehog signaling, typically due to inactivating mutations in the receptor PTCH1, a pathomechanism recapitulated in Ptch1+/− mice. As nitric oxide may regulate GCP proliferation and differentiation, we crossed Ptch1+/− mice with mice lacking inducible nitric oxide synthase (Nos2) to investigate a possible influence on tumorigenesis. We observed a two-fold higher medulloblastoma rate in Ptch1+/− Nos2−/− mice compared to Ptch1+/− Nos2+/+ mice. To identify the molecular mechanisms underlying this finding, we performed gene expression profiling of medulloblastomas from both genotypes, as well as normal cerebellar tissue samples of different developmental stages and genotypes. Downregulation of hedgehog target genes was observed in postnatal cerebellum from Ptch1+/+ Nos2−/− mice but not from Ptch1+/− Nos2−/− mice. The most consistent effect of Nos2 deficiency was downregulation of growth-associated protein 43 (Gap43). Functional studies in neuronal progenitor cells demonstrated nitric oxide dependence of Gap43 expression and impaired migration upon Gap43 knock-down. Both effects were confirmed in situ by immunofluorescence analyses on tissue sections of the developing cerebellum. Finally, the number of proliferating GCPs at the cerebellar periphery was decreased in Ptch1+/+ Nos2−/− mice but increased in Ptch1+/− Nos2−/− mice relative to Ptch1+/− Nos2+/+ mice. Taken together, these results indicate that Nos2 deficiency promotes medulloblastoma development in Ptch1+/− mice through retention of proliferating GCPs in the external granular layer due to reduced Gap43 expression. This study illustrates a new role of nitric oxide signaling in cerebellar development and demonstrates that the localization of pre-neoplastic cells during morphogenesis is crucial for their malignant progression

Microarray-based CGH analysis of MB samples.

<p>(A–B) Alteration frequency plots of <i>Ptch1^+/− Nos2^−/−</i> and <i>Ptch1^+/− Nos2^+/+</i> tumors show a common trisomy of chromosome 6 and deleted regions around the <i>Ptch1</i> locus (chromosome 13), green: gains, red: losses. Arrows denote frequent aberrations, black: common, white: differential. Color shading indicates percentage of available signals for a respective oligonucelotide (% of samples), light: high percentage, dark: low percentage. Concurrent gains and losses of X and Y chromosomes, or <i>vice versa</i>, reflect sample against reference hybridizations of different gender. The most consistent difference between MBs of both genotypes affected a small region on chromosome 14 harboring the <i>Entpd4</i> gene. (C) QRT-PCR of <i>Entpd4</i> showed no significant difference in an expanded set of tumor samples. Relative expression values are normalized to housekeeping genes. Error bars reflect SEM (standard error of the mean).</p

Characteristics and functional implication of Gap43 expression in cell culture.

<p>(A–B) Gene expression of <i>Gap43</i> was reduced upon inhibition of NO synthases. Expression values were obtained from qRT-PCR measurements and indicate linear expression values normalized to a pool of housekeeping genes. Error bars reflect SEM of three replicates. (A) In c17.2 cells, <i>Gap43</i> expression is significantly decreased upon L-NAME treatment after 120 hours (*p = 0.0234). (B) In D458 cells, <i>Gap43</i> expression is significantly decreased upon L-NAME treatment after 72 hours (***p<0.0001). (C–D) Functional analyses were performed after knockdown of <i>Gap43</i> in neuronal progenitor cells (c17.2). (C) <i>Gap43</i> and <i>Ptch1</i> showed inverse gene expression behavior measured by qRT-PCR and normalized to non-target control. (D) Cells exhibited reduced migration upon decrease of Gap43 protein levels. The percentage of migrated cells was normalized to non-target control. Significant decrease of migration in knockdown samples is indicated by asterisks (*p = 0.013, **p = 0.007). Sh39, sh42: anti-<i>Gap43</i> target shRNA, shGFP: control shRNA against GFP, shNT: non-target control shRNA.</p

<i>Nos2</i> deficiency increases the incidence of MB in <i>Ptch1</i>^+/− mice.

<p>(A) Kaplan-Meier analysis of MB incidence in 215 <i>Ptch1</i>^+/−<i>Nos2</i>^+/+ mice (orange line) versus 221 <i>Ptch1</i>^+/−<i>Nos2</i>^−/− mice (red line). <i>Ptch1</i>^+/−<i>Nos2</i>^−/− mice demonstrated an approximately two-fold increase in MB incidence (p = 0.0007, Logrank test). None of 315 wild-type (green line) and 412 <i>Ptch1</i>^+/+<i>Nos2</i>^−/−mice (blue line) (control littermates) developed any MB. Vertical ticks represent censored mice. (B–E) Histological features of MBs from <i>Ptch1</i>^+/−<i>Nos2</i>^+/+ and <i>Ptch1</i>^+/−<i>Nos2</i>^−/− mice. MBs in both genotypes (B–C, <i>Ptch1</i>^+/−<i>Nos2</i><b>^+/+</b>; D–E, <i>Ptch1</i>^+/−<i>Nos2</i>^−/−) were densely cellular primitive neuroectodermal tumors of the cerebellum corresponding histologically to the classic subtype of human MB. Hematoxylin eosin-stained sections showed no obvious differences between the genotypes concerning gross growth pattern, with well demarcated growth in the cerebellar cortex (B, D, scale bar = 500 µm), and cellular morphology (C, E, scale bar = 50 µm).</p

Identification of <i>Nos2</i>-regulated candidate genes.

<p>(A) The overlap between three group comparisons of expression profiles revealed <i>Gap43</i> and <i>Stmn1</i> as commonly deregulated by <i>Nos2</i> inactivation according to the microarray data. (B) <i>Gap43</i> gene expression was <i>Nos2</i>-dependent during different developmental stages of cerebellar development. <i>Nos2</i>-sufficient: wild-type and <i>Ptch1^+/− Nos2^+/+</i>, <i>Nos2</i>-deficient: <i>Ptch1^+/+ Nos2^−/−</i> and <i>Ptch1^+/− Nos2^−/−</i>. (C) <i>Gap43</i> was differentially expressed between <i>Ptch1^+/− Nos2^+/+</i> and <i>Ptch1^+/− Nos2^−/−</i> MB samples. Values in (B) and (C) were obtained from the microarray data and indicate log2 ratios of sample against Universal Reference RNA (Stratagene). (D) Differential <i>Gap43</i> gene expression was confirmed by qRT-PCR in an expanded tumor sample set (n = 7 per genotype). Linear expression values are normalized to housekeeping genes. Significant expression differences between groups are indicated by asterisks (**p<0.01). Error bars reflect SEM. (E) Immunofluorescent co-staining of Gap43 (green) and Ki-67 (red) on FFPE sections from P9 cerebella. Blue: DAPI-stained nuclei. Overview sections (upper left corner) were acquired by wide-field microscopy and detail sections represent confocal laser scanning microscopy images. Intense Gap43 signal was observed in wild-type and <i>Ptch1^+/− Nos2^+/+</i> cerebella, in particular at the outer ML, scale bar = 50 µM. (F) Quantification of Gap43 staining showing intensity histograms of the EGL and ML area (upper panel), as well as intensity distribution of the different genotypes with subtracted wild-type signal (lower panel). Black arrow indicates a common peak of nucleus background signal. EGL: external granule layer, ML: molecular layer, and IGL: internal granule layer.</p

Microarray-based gene expression profiling.

<p>(A) Hierarchical cluster analysis of normal cerebellar tissue and MB samples showed a clear separation of <i>Ptch1^+/+ Nos2^−/−</i> P9 cerebella. (B) Heat map of proliferation-associated genes downregulated in P9 cerebella of <i>Ptch1^+/+ Nos2^−/−</i> versus wild-type mice. Depicted samples include postnatal cerebellar tissue samples of each genotype and all MB cases. Values are normalized to gene-wise average for better visualization. (C) Gene expression of <i>Ptch1</i> was increased in P9 cerebellar tissue samples of <i>Nos2</i>-deficient mice. <i>Ptch2</i> was increased over <i>Ptch1</i> expression only in <i>Ptch1^+/− Nos2^−/−</i> P9 cerebella and MBs. Values are indicated as log2 ratios of sample against Universal Reference RNA (Stratagene). Error bars reflect SEM. AU: approximately unbiased p-values by multiple bootstrap resampling [%], BP: boostrap probability p-values by normal bootstrap resampling [%]. MB: medulloblastoma, CB: cerebellum, P9: postnatal day 9, 6W: 6 weeks after birth, 1Y: 1 year after birth.</p

Proliferation and accumulation of GCPs in postnatal cerebellum.

<p>(A) Immunofluorescent co-staining of NeuN (green) and Ki-67 (red) on FFPE sections from P9 cerebella showed an increased accumulation of proliferating GCP in the EGL of <i>Ptch1^+/− Nos2^−/−</i> mice. Overview sections were acquired by wide-field microscopy and detail images by confocal laser scanning microscopy. Blue: DAPI-stained nuclei. White arrows denote proliferating granule cells in the IGL. (B) High magnification images of the EGL and ML displayed altered morphologies especially in <i>Ptch1^+/− Nos2^−/−</i> and <i>Ptch1^+/+ Nos2^−/−</i> mice. (C) Cell counts from immunofluorescence images. Numbers of proliferating (Ki-67+) and non-proliferating (Ki67−) cells normalized to the length of the EGL edge show significant enrichment of dividing cells in <i>Ptch1^+/− Nos2^−/−</i> mice and correspondingly low numbers for <i>Ptch1^+/+ Nos2^−/−</i> mice. Ratios of dividing to non-dividing cells (Ki-67+, Ki-67− NeuN+) are indicated for each genotype. Significant differences are indicated by asterisks (*p<0.05). Scale bars = 50 µM. EGL: external granule layer, ML: molecular layer, IGL: internal granule layer.</p