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Let G be a graph with vertex set V(G) and edge set E(G), and let g and f be two nonnegative integer-valued functions defined on V(G) such that g(x)⩽f(x) for every vertex x of V(G). We use dG(x) to denote the degree of a vertex x of G. A graph G is called a (g,f)-graph if g(x)⩽dG(x)⩽f(x) for each x∈V(G). Then a spanning subgraph F of G is said to be a (g,f)-factor of G if F itself is a (g,f)-graph. A (g,f)-factorization of G is a partition of E(G) into edge disjoint (g,f)-factors. Let F={F1,F2,…,Fm} be a factorization of G and H be a subgraph of G with m edges. If Fi,1⩽i⩽m, has exactly one edge in common with H, we say that F is orthogonal to H. In this paper it is proved that every (mg+k,mf−k)-graph contains a subgraph R such that R has a (g,f)-factorization orthogonal to a given subgraph with k edges, where m and k are positive integers with 1⩽k and g(x)⩾0. This result has been conjectured by Yan in Yan (Sci. China Ser. A 41(1) (1998) 48). [Copyright &y& Elsevier]

Background: Our previous study showed that calreticulin (CRT) promoted EGF-induced epithelial-mesenchymal transition (EMT) in pancreatic cancer (PC) via Integrin/EGFR-ERK/MAPK signaling. We next investigated the novel signal pathway and molecular mechanism involving the oncogenic role of CRT in PC. Methods: We investigated the potential role and mechanism of CRT in regulating intracellular free Ca2+ dependent acute and chronic endoplasmic reticulum stress (ERS)-induced EMT in PC in vitro and vivo. Results: Thapsigargin (TG) induced acute ERS via increasing intracellular free Ca2+ in PC cells, which was reversed by CRT silencing. Additionally, CRT silencing inhibited TG-induced EMT in vitro by reversing TG-induced changes of the key proteins in EMT signaling (ZO-1, E-cadherin and Slug) and ERK/MAPK signaling (pERK). TG-promoted cell invasion and migration was also rescued by CRT silencing but enhanced by IRE1α silencing (one of the key stressors in unfolded protein response). Meanwhile, CRT was co-immunoprecipitated and co-localized with IRE1α in vitro and its silencing led to the chronic ERS via upregulating IRE1α independent of IRE1-XBP1 axis. Moreover, CRT silencing inhibited IRE1α silencing-promoted EMT, including inhibiting the activation of EMT and ERK/MAPK signaling and the promotion of cell mobility. In vivo, CRT silencing decreased subcutaneous tumor size and distant liver metastasis following with the increase of IRE1α expression. A negative relationship between CRT and IRE1α was also observed in clinical PC samples, which coordinately promoted the advanced clinical stages and poor prognosis of PC patients. Conclusions: CRT promotes EMT in PC via mediating intracellular free Ca2+ dependent TG-induced acute ERS and IRE1α-mediated chronic ERS via Slug and ERK/MAPK signaling. [ABSTRACT FROM AUTHOR]

Musashi2-Numb interaction plays a vital role in the progression of myeloid leukemia. However, its potential role in solid cancers has rarely been reported. We investigated the coordinate function of Musashi2-Numb in the development of pancreatic cancer (PC) in vitro and vivo. Both Musashi2 protein and mRNA levels were higher in PC tissues than that in paired normal pancreas (P<0.05). Musashi2 overexpression and Numb positive expression were positively and negatively associated with tumor size and UICC stage, respectively (P<0.05). Multivariate analysis identified Musashi2 and Numb as adverse and favorable independent indicators for the survival of PC patients. Moreover, patients with high Musashi2 expression combining with negative Numb expression had a significantly worse overall survival (P=0.001). The negative relationship between Musashi2 and Numb was found at both PC tissue and cell levels. These two endogenous proteins can be co-immunoprecipitated from PC cell lines, and Musashi2 silence up-regulated Numb protein in vitro and vivo. Meanwhile, its silence decreased cell invasion and migration in vitro and inhibited the growth of subcutaneous tumors and the frequency of liver metastasis in vivo. However, Numb knockdown significantly reversed the decrease of cell invasion and migration induced by Musashi2 silence. Musashi2 promotes the development and progression of pancreatic cancer by down-regulating Numb protein. The interaction of Musashi2-Numb plays a significant role in the development and progression of PC.