A Reduction from Unbounded Linear Mixed Arithmetic Problems into Bounded Problems

We present a combination of the Mixed-Echelon-Hermite transformation and the Double-Bounded Reduction for systems of linear mixed arithmetic that preserve satisfiability and can be computed in polynomial time. Together, the two transformations turn any system of linear mixed constraints into a bounded system, i.e., a system for which termination can be achieved easily. Existing approaches for linear mixed arithmetic, e.g., branch-and-bound and cuts from proofs, only explore a finite search space after application of our two transformations. Instead of generating a priori bounds for the variables, e.g., as suggested by Papadimitriou, unbounded variables are eliminated through the two transformations. The transformations orient themselves on the structure of an input system instead of computing a priori (over-)approximations out of the available constants. Experiments provide further evidence to the efficiency of the transformations in practice. We also present a polynomial method for converting certificates of (un)satisfiability from the transformed to the original system

Expression of the embryonic stem cell marker SOX2 in early-stage breast carcinoma

<p>Abstract</p> <p>Background</p> <p>The SRY-related HMG-box family of transcription factors member <it>SOX2 </it>has been mainly studied in embryonic stem cells as well as early foregut and neural development. More recently, SOX2 was shown to participate in reprogramming of adult somatic cells to a pluripotent stem cell state and implicated in tumorigenesis in various organs. In breast cancer, SOX2 expression was reported as a feature of basal-like tumors. In this study, we assessed SOX2 expression in 95 primary tumors of postmenopausal breast cancer patients.</p> <p>Methods</p> <p>Samples from 95 patients diagnosed and treated at the University of Tuebingen Institute of Pathology and Women's Hospital were analyzed by immunohistochemistry for SOX2 expression in the primary tumor samples and in corresponding lymph node metastasis, where present. Furthermore, SOX2 amplification status was assessed by FISH in representative samples. In addition, eighteen fresh frozen samples were analyzed for <it>SOX2</it>, <it>NANOG </it>and <it>OCT4 </it>gene expression by real-time PCR.</p> <p>Results</p> <p>SOX2 expression was detected in 28% of invasive breast carcinoma as well as in 44% of ductal carcinoma in situ (DCIS) lesions. A score of SOX2 expression (score 0 to 3) was defined in order to distinguish SOX2 negative (score 0) from SOX2 positive samples (score 1-3) and among latter the subgroup of SOX2 high expressors (score 3 > 50% positive cells). Overall, the incidence of SOX2 expression (score 1-3) was higher than previously reported in a cohort of lymph node negative patients (28% versus 16.7%). SOX2 expression was detected across different breast cancer subtypes and did not correlate with tumor grading. However, high SOX2 expression (score 3) was associated with larger tumor size (p = 0.047) and positive lymph node status (0.018). Corresponding metastatic lymph nodes showed higher SOX2 expression and were significantly more often SOX2 positive than primary tumors (p = 0.0432).</p> <p>Conclusions</p> <p>In this report, we show that the embryonic stem cell factor SOX2 is expressed in a variety of early stage postmenopausal breast carcinomas and metastatic lymph nodes. Our data suggest that SOX2 plays an early role in breast carcinogenesis and high expression may promote metastatic potential. Further studies are needed to explore whether SOX2 can predict metastatic potential at an early tumor stage.</p