Abstract. circadian rhythms are daily oscillations in various biological processes, generated by the feedback loops of eight core circadian genes: Period1 (Per1), Period2 (Per2), Period3 (Per3), Cryptochrome1 (Cry1), Cryptochrome2 (Cry2), Clock, Bmal1 and Casein Kinase I ε (CKIε). recent studies have suggested that circadian genes participate in the growth and development of various cancers. this study examined the relations of circadian gene expression to clinicopathological factors and outcomes in patients with colorectal cancer. We studied surgical specimens of cancer tissue and adjacent normal mucosa obtained from 202 patients with untreated colorectal cancer. the relative expression levels of the circadian genes in the specimens were measured by quantitative real-time, reverse-transcription polymerase chain reaction. expression of the Clock gene and the CKIε gene in cancer tissue were significantly higher compared to that in adjacent normal mucosa. expression of the Per1 and Per3 genes in cancer tissue was significantly lower compared to that in adjacent normal mucosa. Analysis of the relations between clinicopathological features and expression of the eight circadian genes in cancer tissue showed that high expression of the Bmal1 gene and low expression of the Per1 gene correlated with liver metastasis. on analysis of the relations between outcomes and gene expression, high expression of the Per2 gene was associated with significantly better outcomes than low expression of the Per2 gene. overexpression of the Bmal1 gene and reduced expression of the Per1 gene may thus be useful predictors of liver metastasis. Moreover, reduced expression of the Per2 gene may be a predictor of outcomes in patients with colorectal cancer. Introductioncircadian rhythms are daily oscillations in various biologic processes. In mammals, the master circadian pacemaker is located in the suprachiasmatic nuclei (scn) (1). the master circadian clock coordinates peripheral circadian clocks within virtually every cell in the body (2). this coordination is accomplished directly through autonomic nervous system innervation and indirectly through daily rhythmic synthesis and release of an array of hypothalamic, pituitary, and dispersed endocrine hormones (3-6). the molecular mechanism of circadian oscillation in the scn and peripheral cells is based on the feedback loops of eight core circadian genes (3,7,8). these eight genes are Period1 (Per1), Period2 (Per2), Period3 (Per3), Cryptochrome1 (Cry1), Cryptochrome2 (Cry2), Clock, Bmal1, and Casein Kinase I ε (CKIε). the feedback loops of the eight core circadian genes are as follows. the Clock gene remains steady throughout the 24-h day. High levels of Bmal1 promote the formation of Bmal1/clock heterodimers. these heterodimers bind to e-box sequences in the promoters of the Cry and Per genes to activate transcription. Bmal1/clock heterodimers can also inhibit Bmal1 transcription. After transcription and translation, the per proteins accumulate in the cytoplasm and are phosphorylated by...

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Contents § 0. Introduction § 1. Construction of a compact imbedding § 2. Invariant differential operators § 3. Boundary value maps § 4. Principal series § 0. Introduction

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