On dominant contractions and a generalization of the zero–two law

Zaharopol proved the following result: let \({T,S:L^1(X,{\mathcal{F}},\mu)\to L^1(X, {\mathcal{F}},\mu)}\) be two positive contractions such that T ≤ S. If \({\|S-T\| <1 }\) then \({\left\|S^n-T^n\right\| <1 }\) for all \({n\in\mathbb{N}}\). In the present paper we generalize this result to multi-parameter contractions acting on L ¹. As an application of that result we prove a generalization of the “zero–two” law.     

Saffron and Its Major Ingredients’ Effect on Colon Cancer Cells with Mismatch Repair Deficiency and Microsatellite Instability

Background: Colorectal cancer (CRC) is one of the most common cancers worldwide. One of its subtypes is associated with defective mismatch repair (dMMR) genes. Saffron has many potentially protective roles against colon malignancy. However, these roles in the context of dMMR tumors have not been explored. In this study, we aimed to investigate the effects of saffron and its constituents in CRC cell lines with dMMR. Methods: Saffron crude extracts and specific compounds (safranal and crocin) were used in the human colorectal cancer cell lines HCT116, HCT116+3 (inserted MLH1), HCT116+5 (inserted MSH3), and HCT116+3+5 (inserted MLH1 and MSH3). CDC25b, p-H2AX, TPDP1, and GAPDH were analyzed by Western blot. Proliferation and cytotoxicity were analyzed by MTT. The scratch wound assay was also performed. Results: Saffron crude extracts restricted (up to 70%) the proliferation in colon cells with deficient MMR (HCT116) compared to proficient MMR. The wound healing assay indicates that deficient MMR cells are doing better (up to 90%) than proficient MMR cells when treated with saffron. CDC25b and TDP1 downregulated (up to 20-fold) in proficient MMR cells compared to deficient MMR cells, while p.H2AX was significantly upregulated in both cell types, particularly at >10 mg/mL saffron in a concentration-dependent manner. The reduction in cellular proliferation was accompanied with upregulation of caspase 3 and 7. The major active saffron compounds, safranal and crocin reproduced most of the saffron crude extracts’ effects. Conclusions: Saffron’s anti-proliferative effect is significant in cells with deficient MMR. This novel effect may have therapeutic implications and benefits for MSI CRC patients who are generally not recommended for the 5-fluorouracil-based treatment.