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The spectrum of normalized Laplacian matrix of a network has attracted more and more attention because it is related to the structural properties and dynamical aspects of the network, specially in random walks. In this paper, we study the spectra and their applications of normalized Laplacian matrices for weighted level-3 Sierpiński graphs that are constructed in an iterative way. We analytically obtain all the spectra from two successive generations by applying the decimation method. Using the obtained spectra, we then derive closed-form expressions for their eigentime identity and number of spanning trees.

To study the molecular mechanism of cisplatin to enhance the ability of tumor necrosis factor-related apoptosis-inducing ligand (TRAIL) in reversing multidrug resistance in vincristine-resistant human gastric cancer SGC7901/VCR cells.MTT assay was used to measure the 50% inhibiting concentration (IC₅₀) and cell survival in SGC7901 and SGC7901/VCR cells after different treatments. SGC7901/VCR cells were treated with different concentrations of DDP, different concentrations of TRAIL alone or in combination, and then the mRNA and protein levels of several genes were determined by RT-PCR, RT-qPCR and Western-blot analysis. After targeted silencing with specific siRNA and transfection of recombinant plasmid c-myc into the SGC7901/VCR cells, the mRNA and protein levels of DR4, DR5 and c-myc were determined by RT-PCR and Western-blot analysis.After combined treatment with TRAIL and DDP of the SGC7901/VCR cells, the IC₅₀ of VCR, DDP, ADM, and 5-Fu treatment was significantly decreased compared with the control group or TRAIL-treated group (P0.05). After treatment with 0, 10, 50 ng/ml TRAIL in combination with 0.4 µg/ml DDP, the SGC7901/VCR cells showed significantly higher activation of caspase 3, down-regulation of DNA-PKcs/Akt/GSK-3β signaling pathway, and higher inhibition of MDR1(P-gp) and MRP1 than those treated with TRAIL alone (P0.01 for all). The mRNA and protein levels of DR4, DR5, c-myc were significantly decreased after silencing c-myc with specific siRNA in the SGC7901/VCR cells (P0.01 for all), and were significantly increased after transfection of recombinant plasmid c-myc into the SGC7901/VCR cells (P0.01 foe all). After the treatment with 10 ng/ml TRAIL, 0.25 µg/ml DDP + 10 ng/ml TRAIL and 0.5 µg/ml DDP + 10 ng/ml TRAIL, the relative expression level of c-myc protein in the SGC7901/VCR cells was 0.314 ± 0.012, 0.735 ± 0.026, and 0.876 ± 0.028, respectively, and the relative expression of cytochrome C was 0.339 ± 0.036, 0.593 ± 0.020 and 0.735 ± 0.031, respectively, and the relative expression levels of DR4, DR5, active-caspase 3 and active-caspase 9 in the SGC7901/VCR cells were also increased along with increasing DDP concentrations.The activation of DNA-PKcs/Akt/GSK-3β signaling pathway and high expression of MDR1 and MRP1 play an important role in the multi-drug resistance properties of SGC7901/VCR cells. After combining with TRAIL, DDP can enhance the expression of DR4 and DR5 through up-regulating c-myc and enhancing the activation of caspase 3 and caspase 9 by facilitating mitochondrial release of cytochrome C. It may be an important molecular mechanism of DDP-induced sensitization of TRAIL to reverse the multidrug resistancein SGC7901/VCR cells.

Tumor necrosis factor-related apoptosis-inducing ligand (TRAIL) reverses multidrug resistance (MDR) and induces apoptosis in MDR gastric carcinoma cells. In our previous study, cisplatin proved to be a sensitizing agent for TRAIL. To study the synergistic effects of cisplatin and TRAIL, we investigated the mechanism by which TRAIL reverses multidrug resistance, the role of c-myc in modulating the death receptors DR4 and DR5 and the relationship between cisplatin and cytochrome c (cyt c) release in SGC7901/VCR and SGC7901/DDP cells. We found that after treatment with TRAIL, the DNA-PKcs/Akt/GSK-3β pathway, which is positively correlated with the levels of MDR1 and MRP1, was significantly inhibited and that this tendency can be abolished by Z-DEVD-FMK (a specific caspase 3 inhibitor). We also found that suppression of c-myc by siRNA reduced the expression of DR4 and DR5 and that transfection with a pAVV-c-myc expression vector increased the expression of DR4 and DR5. Moreover, cisplatin increased the expression of c-myc in the presence of TRAIL, and there is a clear increase in cyt c release from mitochondria with the increasing concentrations of cisplatin. Meanwhile, the intrinsic death receptor pathway of caspase 9, as well as the common intrinsic and extrinsic downstream target, caspase 3, was potently activated by the release of cyt c. Together, we conclude that in TRAIL-treated MDR gastric carcinoma cells, cisplatin induces the death receptors DR4 and DR5 through the up-regulation of c-myc and strengthens the activation of caspases via promoting the release of cyt c. These effects would then be responsible for the TRAIL sensitization effect of cisplatin.