Fractional Hardy-Sobolev type inequalities for half spaces and John domains

As our main result we prove a variant of the fractional Hardy-Sobolev-Maz'ya inequality for half spaces. This result contains a complete answer to a recent open question by Musina and Nazarov. In the proof we apply a new version of the fractional Hardy-Sobolev inequality that we establish also for more general unbounded John domains than half spaces

MOESM1 of Vitamin C induces specific demethylation of H3K9me2 in mouse embryonic stem cells via Kdm3a/b

Additional file 1: Figure S1. Evaluation of changes in H3 PTMs following vitamin C treatment. A) Western blot for several H3 PTMs in ES cells ± vitamin C. B) Immunofluorescence for H3K9me2 and corresponding DAPI staining in untreated and vitamin C-treated ES cells. Merged images show H3K9me2 in green and DAPI staining in red. H3K9me2 immunofluorescence is also shown in Fig. 1e. Scale bar represents 20 μm

MOESM2 of Vitamin C induces specific demethylation of H3K9me2 in mouse embryonic stem cells via Kdm3a/b

Additional file 2: Figure S2. Analysis of H3K9me2 in G9a and GLP knockout ES cells treated with vitamin C. A) Western blot for H3K9me2 in wild-type parental TT2, G9a knockout, and GLP knockout ES cells ± vitamin C. B) Immunofluorescence for H3K9me2 in GiP ES cells ± vitamin C and untreated wild-type TT2, G9a knockout, and GLP knockout ES cells. GiP ES cells treated with vitamin C show a H3K9me2 staining pattern that is similar to G9a and GLP knockout ES cells. Scale bar represents 20 μm

MOESM4 of Vitamin C induces specific demethylation of H3K9me2 in mouse embryonic stem cells via Kdm3a/b

Additional file 4: Figure S4. Analysis of H3K9me2 at repetitive elements in ES cells treated with vitamin C. ChIP-qPCR for H3K9me2 in ES cells ± vitamin C at the repetitive element families indicated. ChIP for IgG was performed as a negative control. Data are mean ± SD. Asterisks represent P < 0.05 by t test

Pcl3 promotes ESC self-renewal.

<p>(A) <i>Pcl3</i> expression levels measured by qRT-PCR in ESC clones transduced with scramble or multiple <i>Pcl3</i> shRNAs. Graph represents average expression from 3–6 different clones. (B) A portion of cells transduced with <i>Pcl3</i> shRNA, but not scramble shRNA, are larger, flatter, and less dense, signifying a decrease in ESC cell morphology. Scale bar 25 µm. These pictures are representative of 2–3 different clones of scramble and <i>Pcl3</i> shRNA cells taken at three different time points. (C) Expression levels of <i>Oct4</i> and <i>Nanog</i> in scramble, <i>Pcl3</i> shRNA, and <i>Pcl3</i> overexpressing cells. (D) Quantification of Oct4 and Nanog staining in scramble and <i>Pcl3</i> shRNA treated ESCs. +++indicates bright staining, ++indicates less bright staining, and+indicates little or no staining as assessed by eye. Graphs are representative of two clones and between 5–10 fields of view at 10× magnification. (E) Alkaline phosphatase activity in scramble and <i>Pcl3</i> shRNA cells. Graph represents average activity from 3–6 different clones in three experiments assayed in duplicate. (F) Quantification of the number of colonies formed per well from scramble and <i>Pcl3</i> shRNA cells plated at 100 cells/well in a 6-well plate. Experiment was performed four times in duplicate with two clones each of scramble and <i>Pcl3</i> shRNA ESCs. (G) Quantification of colonies formed by plating 100 cells/well of wild type and Pcl3 overexpressing cells in a 6-well plate. LIF was reduced to 5% and was performed four times in duplicate. (H) Images of teratomas derived from scramble or <i>Pcl3</i> shRNA ESCs containing all three germ layers stained with hematoxylin and eosin. Abbreviations: EN-endoderm, NE-neuroectoderm, B-bone, C-cartilage, M-muscle, N-neural tissue. Scale bar 25 µm. Error bars indicate standard deviation. Expression analysis experiments represent 3–4 experiments assayed in quadruplet. For all experiments, asterisk denotes statistical significance of <i>p</i><0.05. Staining was performed 2–3 times in two or more clones.</p

Pcl3 regulates Suz12, but not complex stability.

<p>(A) PRC2 components associate in the absence of Pcl3 based on co-immunoprecipitation. β-actin used as a loading control. (B) Protein levels of Suz12, Ezh2, and Eed in scramble and <i>Pcl3</i> shRNA clones as measured by immunoblot. Experiments were performed with 3–6 scramble and <i>Pcl3</i> shRNA clones each. (C) Expression levels of <i>Suz12</i>, <i>Ezh2</i>, and <i>Eed</i> in scramble and <i>Pcl3</i> shRNA measured by qRT-PCR. (D) <i>Suz12</i> mRNA levels measured by qRT-PCR in wild type and <i>Pcl3</i> overexpressing cells. (E) Suz12 protein levels in wild type cells and wild type cells overexpressing <i>Pcl3-TAP</i>. All immunoblot and co-immunoprecipitation experiments were performed 2–5 times. All expression analysis represents three experiments assayed in quadruplicate. Error bars represent standard deviation, and asterisk indicates statistical significance of <i>p</i><0.005.</p

Pcl2 and Pcl3 localize to CpG islands.

<p>(A) The 500 bp central regions of Pcl3 ChIP-seq peaks were scanned for enriched motifs by using a 9th order Markov background dependence model <a href="http://www.plosgenetics.org/article/info:doi/10.1371/journal.pgen.1002576#pgen.1002576-Carroll1" target="_blank">[86]</a>. Two examples of 10- and 14-mer enriched motifs are shown. (B–C) Smoothed scatter plots of maximum position specific-scoring matrix (PSSM) scores for the two motifs and CpG density are shown for (B) Suz12 binding sites depleted upon <i>Pcl3</i> knockdown overlapping with Pcl2 and Pcl3 and (C) Suz12 binding sites unaffected upon <i>Pcl3</i> knockdown and that do not overlap with Pcl2 and Pcl3. (D) Shown are the decision boundaries of a support vector machine classifier using these three features, where the purple regions correspond to Suz12 co-localizing with Pcl2 and Pcl3. The predictor had a cross validation accuracy of 75%. (E) A model of Pcl3 and Pcl2 regulation of PRC2 binding and activity. In wild type ESCs, Pcl3 promotes PRC2 binding and H3K27me3. Pcl2 antagonizes Pcl3-mediated Suz12 binding at sites bound by both but promotes PRC2 function at sites solely regulated by Pcl2. Knockdown of <i>Pcl3</i> causes decreased PRC2 binding and H3K27me3. Pcl2 does not compensate at Pcl2 and Pcl3 targets and continues to inhibit or promote PRC2 function depending on the gene.</p

Pcl3 is a component of PRC2.

<p>(A) Protein levels of Suz12 and Suz12-TAP were measured in wild type, <i>Suz12^Gt/+</i>, <i>Suz12^Rev/+</i>, and <i>Suz12^Suz12TAP/+</i> cell lines by immunoblot. (B) Proteins detected by mass spectrometry that specifically co-purified with Suz12-TAP, their symbol, unique hits, and percent coverage. (C) Pcl3-V5 binds to Suz12-TAP. <i>Suz12^Suz12TAP/+</i> ESCs were transfected with empty vector, Pcl3-V5, or Mks1-V5 (control), and lysates were immunoprecipitated with FlagM2 and probed with anti-V5. (D) Pcl3-V5 binds Suz12, Ezh2, and Eed. Lysates from ESCs transfected with empty vector, Pcl3-V5, and Mks1-V5 (control) vectors were subjected to immunoprecipitation with anti-V5. Samples were then probed with anti-Suz12, anti-Ezh3, and anti-Eed. All westerns and co-immunoprecipitations were performed three times.</p

Pcl3 promotes PRC2 function.

<p>(A) Immunoblot showing levels of H3K27me3 in multiple clones of scramble and <i>Pcl3</i> shRNA ESCs and EBs. (B) H3K27me3 levels in <i>Suz12</i> and <i>Pcl3</i> siRNA treated cells. (C) Increased levels of H3K27me3 as measured by immunoblot in cells overexpressing Pcl3. (D) Immunoblot of H3K27me3, H2AK119Ub, H3K9me3, H3K4me3, and H3K27ac levels in histones from scramble and <i>Pcl3</i> shRNA-expressing cells. (E) Pcl3-TAP resistant to <i>Pcl3</i> shRNA was reintroduced into <i>Pcl3</i> shRNA cells, immunoprecipitated, and detected with anti-FlagM2. <i>Suz12^Suz12TAP/+</i> cells were used as a positive control. (F) Pcl3-TAP binds Suz12, Eed, and Ezh2. Lysates from scramble and <i>Pcl3</i> shRNA cells containing Pcl3-TAP were immunoprecipitated with FlagM2 and immunoblotted for Suz12, Eed, and Ezh2. (G) qRT-PCR shows partial rescue of <i>Pcl3</i> expression in <i>Pcl3</i> shRNA clones expressing <i>Pcl3-TAP</i>. Error bars indicate standard deviation. Graph represents average expression from 3–6 different clones in three experiments assayed in quadruplet. (H) Immunoblot showing restoration of H3K27me3 levels in <i>Pcl3</i> shRNA cells transduced with Pcl3-TAP. Histone H3 and α-tubulin were used as loading controls. All westerns and immunoprecipitations were performed three or more times with 2–6 clones.</p

Pcl3 localizes to PRC2 targets.

<p>(A) Heatmaps showing Pcl3 and Suz12 ChIP-seq read density in counts per 100 bp around Pcl3 peak centers. Approximately 44% of Suz12 targets are bound by Pcl3-TAP. Each row corresponds to a Pcl3 ChIP-seq peak with rows ranked by Pcl3 peak significance (assessed by the Skellam distribution p-values). (B) Graph of −log p-values indicating that the most significant Suz12 ChIP-seq peaks overlap with Pcl3, while regions containing less significant Suz12 ChIP-seq peaks are not bound by Pcl3. Wilcoxon p-value<1e⁻³⁰⁶ for the difference in the distribution of ChIP-seq −log p-values. (C) Pcl3 co-localizes with Suz12 depletion sites (blue). Regions containing only Suz12 depletion or only Pcl3 binding are indicated in cyan and orange respectively. (D) 84% of sites bound by Suz12 and Pcl3 show decreased Suz12 binding upon <i>Pcl3</i> (Fisher test p-value = 1.5×10⁻³⁰⁶). Two binding sites were considered to be overlapping if their peak centers were within 2 kb from each other. (E) Suz12 (Wilcoxon p-value<1e⁻³⁰⁶) and H3K27me3 (Wilcoxon p-value = 3.7e⁻³⁶) co-localizing with Pcl3 show much more significant depletion compared to Suz12 and H3K27me3 targets not bound by Pcl3. Depletion score = −log p-value (read counts before/after <i>Pcl3</i> KD). (F) Genes and microRNAs bound by Pcl3-TAP measured by FlagM2 ChIP-qRT-PCR. The graph depicts average fold enrichment over control levels for three different clones in three different experiments assayed in quadruplet. Error bars indicate standard deviation. All increases are statistically significant, <i>p</i><0.04.</p