ZX-Calculus: Cyclotomic Supplementarity and Incompleteness for Clifford+T quantum mechanics

The ZX-Calculus is a powerful graphical language for quantum mechanics and quantum information processing. The completeness of the language -- i.e. the ability to derive any true equation -- is a crucial question. In the quest of a complete ZX-calculus, supplementarity has been recently proved to be necessary for quantum diagram reasoning (MFCS 2016). Roughly speaking, supplementarity consists in merging two subdiagrams when they are parameterized by antipodal angles. We introduce a generalised supplementarity -- called cyclotomic supplementarity -- which consists in merging n subdiagrams at once, when the n angles divide the circle into equal parts. We show that when n is an odd prime number, the cyclotomic supplementarity cannot be derived, leading to a countable family of new axioms for diagrammatic quantum reasoning.We exhibit another new simple axiom that cannot be derived from the existing rules of the ZX-Calculus, implying in particular the incompleteness of the language for the so-called Clifford+T quantum mechanics. We end up with a new axiomatisation of an extended ZX-Calculus, including an axiom schema for the cyclotomic supplementarity.Comment: Mathematical Foundations of Computer Science, Aug 2017, Aalborg, Denmar

Aurora-A-mediated AP-2α degradation depends on Aurora-A kinase activity.

<p>(A) KYSE150/GFP-Aurora-A cells were exposed to 1 µM of Aurora-A kinase inhibitor and cellular proteins were collected 2 hours later. Western Blot was performed with antibodies to pThr288 on Aurora-A or total Aurora-A. DMSO was used as a negative control. (B) KYSE150/GFP-Aurora-A cells were exposed to Aurora-A kinase inhibitor or DMSO for 2 hours prior to treatment with CHX. Cells were harvested at the indicated time points and analyzed by Western Blot. (C) The amounts of AP-2α were calculated by densitometry and normalized to corresponding actin levels. The column diagram represents the amount of normalized AP-2α at each time point comparing with the original levels (0 h).</p

Overexpression of Aurora-A enhances the degradation of AP-2α.

<p>(A) KYSE150/GFP-Aurora-A and KYSE150/GFP cells were treated with CHX respectively, and then harvested at the indicated time points. AP-2α protein level at each time point was determined by Western Blot. (B) The amounts of AP-2α were calculated by densitometry and normalized to corresponding actin levels. The column diagram represents the amount of normalized AP-2α at each time point comparing with the original levels (0 h). (C) Aurora-A enhances the proteasome-dependent degradation of AP-2α. KYSE150/GFP-Aurora-A and KYSE150/GFP cells were treated with CHX alone or combination with MG-132. AP-2α protein level was determined by Western Blot.</p

Transcription activity of AP-2α is downregulated by Aurora-A.

<p>(A) The summary of the transcription factors with significant changes in their activities by transcription factors array when Aurora-A was overexpressed. (B) Fluorescence images of the transcription factors arrays hybridized with nuclear extracts from KYSE150/GFP-Aurora-A and KYSE150/GFP cells. The analysis of arrays was replicated twice, exchanging the Cy3- and Cy5- label. Magnified panels show the spots representing the activity of AP-2 transcription factor that was down-regulated upon Aurora-A overexpression. (C) The summary of the AP-2α-regulated genes with significant down-expression when Aurora-A was overexpressed by mRNA expression profiling array. The black bars represent the AP-2α-regulated genes, and the gray bars represent the genes which were not the AP-2α-regulated genes. (D) The nucleoprotein extraction of KYSE150/GFP-Aurora-A and KYSE150/GFP cells were incubated with AP-2α binding DNA double-stranded oligonucleotidic probe on multi-well plates, then anti-AP-2α antibodies and the peroxidase-conjugated anti-mouse IgG were subsequently added. At last, the samples could be read for the colorimetric detection. (E) KYSE150/GFP-Aurora-A and KYSE150/GFP cells were cotransfected with (AP-2)₃-TK-LUC expression vector and a pRL-SV40 expression vector. Transcription activities were expressed as luciferase values after normalization. All experiments were performed three times and described as mean ± SD.</p

Figure 2

<p>A. Cell growth curve measured with MTT assay indicated siRNA mediated knockdown of CHRNA3 increased the growth rate of KYSE510 cells. B. Transwell assay indicated that knockdown of CHRNA3 increased the invasion and migration of KYSE30 cells. C. Western blot analysis indicated 3 Stealth siRNA constructs effectively knocked down CHRNA3. Depletion of CHRNA3 led to a decrease of phosphorylated YAP1 and an increased of dephosphorylated YAP1, particularly a decreased S127 phosphorylation of YAP1. D. Observation of YAP1 subcellular localization with confocal fluorescence microscopy following depletion of CHRNA3. Translocation of YAP1 (green) from the cytoplasm to the nucleus was observed after siRNA mediated knockdown of CHRNA3 in KYSE510 cells for 48 h. E. Real-time PCR test indicated YAP1 targeted genes were induced by CHRNA3 knockdown in KYSE510 cell.</p

Protein Immunoprecipitation indicated a decreased physical association between YAP1 with 14-3-3, α-catenin, β-catenin and p63 upon nicotine administration at the concentration of 80 nM for 48 h in KYSE510 cells.

<p>B. KYSE510 cells exposed to nicotine for more than 4 days led to upregulation of total protein level of YAP1, including both the phosphorylated YAP1 and dephosphorylated YAP1 indicated by western blot analysis.</p

KYSE510 cells were treated with or without nicotine for 48(PI) were analyzed by flow cytometry.

<p>Nicotine treatment decreased the parent of apoptotic cells in the lower right quadrant.</p

Figure 1

<p>A. Nicotine administration stimulates the growth of esophageal cancer KYSE510 cell measured by E-Plateofx CELLigence RTCA MP system. B. Nicotine administration increased the invasion and migration of esophageal cancer KYSE30 cells in transwell assays. C. Subcellular localization of YAP1 examined with confocal fluorescence microscope. Translocation of YAP1 (green) from the cytoplasm to the nucleus was observed after nicotine administration in KYSE510 cells for 48 h. D. KYSE510 cells were treated with nicotine for 48 hs, decreased phosphorylation of YAP1 and increased dephosphorylated YAP1 was observed by Western blot analysis. E. Real-time PCR verification of induction of mRNAs of genes transcriptionally activated by YAP1 upon nicotine administration. F. PKC inhibitor Enzastaurin blocked nicotine induced upregualtion of YAP1 protein level, and resulted in reduction of YAP protein level, particularly the dephosphorylated form of YAP1 by Western blot.</p

Figure 3

<p>A. Endogenous IP test showed positive protein interactions between YAP1 and CHRNA3/CHRNB4/CHRNA5 in KYSE510 cells. B. Positive interaction between tGFP-CHRNA3 and YAP1 was detected in cells transfected with tGFP-CHRNA3. C. GST Pull assay indicated positive interaction of exogenously expressed GST-YAP1 with CHRNA3 in KYSE510 cell. D. Confocal immunofluorescence microscopy observed colocalization of YAP1 and CHRNA3 in KYSE510 cell. The KYSE510 cells were transiently expressed tGFP labeled CHRNA3 of TureClone vector (OriGene). YAP1 was labeled with red-FITC conjugated goat anti rabbit IgG. Cell nuclei were visualized with DAPI. E. endogenous IP detected positive interactions between 14-3-3 and CHRNA3/CHRNB4/CHRNA5.</p

YAP1 expression is positively correlated with smoking status of ESCC patients¹.

<p>(<i>P</i><0.05, χ2 = 6.74).</p>1<p>83 ESCC clinical samples were examined for YAP1 expression using immunohistochemical approach as described in Materials and Methods.</p>2<p>Strong and median YAP1 staining in Non-smoker ESCC patients (8/29, 27.6%).</p>3<p>Strong and median YAP1 staining in smoker ESCC patients (31/54, 57.4%).</p><p>Correlations between the subgroups of staining and cigarette smoking were calculated using the Pearson and χ2 test.</p