Locally quasi-nilpotent elementary operators

Let A be a unital dense algebra of linear mappings on a complex vector space X. Let φ = Σⁿ _i=1 M_ai,_bi be a locally quasi-nilpotent elementary operator of length n on A. We show that, if {a1, . . . , an} is locally linearly independent, then the local dimension of V (φ) = span{b_ia_j : 1 ≤ i, j ≤ n} is at most n(n−1) 2 . If ldim V (φ) = n(n−1) 2 , then there exists a representation of φ as φ = Σⁿ _i=1 M_ui,_vi with v_iu_j = 0 for i ≥ j. Moreover, we give a complete characterization of locally quasinilpotent elementary operators of length 3.

Prognostic role of the LCS6 KRAS variant in locally advanced rectal cancer: results of the EXPERT-C trial.

Background: Lethal-7 (let-7) is a tumour suppressor miRNA which acts by down-regulating several oncogenes including KRAS. A single-nucleotide polymorphism (rs61764370, T > G base substitution) in the let-7 complementary site 6 (LCS-6) of KRAS mRNA has been shown to predict prognosis in early-stage colorectal cancer (CRC) and benefit from anti-epidermal growth factor receptor monoclonal antibodies in metastatic CRC. Patients and methods: We analysed rs61764370 in EXPERT-C, a randomised phase II trial of neoadjuvant CAPOX followed by chemoradiotherapy, surgery and adjuvant CAPOX plus or minus cetuximab in locally advanced rectal cancer. DNA was isolated from formalin-fixed paraffin-embedded tumour tissue and genotyped using a PCR-based commercially available assay. Kaplan–Meier method and Cox regression analysis were used to calculate survival estimates and compare treatment arms. Results: A total of 155/164 (94.5%) patients were successfully analysed, of whom 123 (79.4%) and 32 (20.6%) had the LCS-6 TT and LCS-6 TG genotype, respectively. Carriers of the G allele were found to have a statistically significantly higher rate of complete response (CR) after neoadjuvant therapy (28.1% versus 10.6%; P = 0.020) and a trend for better 5-year progression-free survival (PFS) [77.4% versus 64.5%: hazard ratio (HR) 0.56; P = 0.152] and overall survival (OS) rates (80.3% versus 71.9%: HR 0.59; P = 0.234). Both CR and survival outcomes were independent of the use of cetuximab. The negative prognostic effect associated with KRAS mutation appeared to be stronger in patients with the LCS-6 TT genotype (HR PFS 1.70, P = 0.078; HR OS 1.79, P = 0.082) compared with those with the LCS-6 TG genotype (HR PFS 1.33, P = 0.713; HR OS 1.01, P = 0.995). Conclusion: This analysis suggests that rs61764370 may be a biomarker of response to neoadjuvant treatment and an indicator of favourable outcome in locally advanced rectal cancer possibly by mitigating the poor prognosis of KRAS mutation. In this setting, however, this polymorphism does not appear to predict cetuximab benefit.

RAS mutations and cetuximab in locally advanced rectal cancer: results of the EXPERT-C trial.

Background: RAS mutations predict resistance to anti-epidermal growthfactor receptor (EGFR) monoclonal antibodies in metastatic colorectal cancer. We analysed RAS mutations in 30 non-metastatic rectal cancer patients treated with or without cetuximab within the 31 EXPERT-C trial.

Methods: Ninety of 149 patients with tumours available for analysis were KRAS/BRAF wild-type, and randomly assigned to capecitabine plus oxaliplatin (CAPOX) followed by chemoradiotherapy, surgery and adjuvant CAPOX or the same regimen plus cetuximab (CAPOX-C). Of these, four had a mutation of NRAS exon 3, and 84 were retrospectively analysed for additional KRAS (exon 4) and NRAS (exons 2/4) mutations by using bi-directional Sanger sequencing. The effect of cetuximab on study end-points in the RAS wild-type population was analysed.

Results: Eleven (13%) of 84 patients initially classified as KRAS/BRAF wild-type were found to have a mutation in KRAS exon 4 (11%) or NRAS exons 2/4 (2%). Overall, 78/149 (52%) assessable patients were RAS wild-type (CAPOX, n = 40; CAPOX-C, n = 38). In this population, after a median follow-up of 63.8 months, in line with the initial analysis, the addition of cetuximab was associated with numerically higher, but not statistically significant, rates of complete response (15.8% versus 7.5%, p = 0.31), 5-year progression-free survival (75.5% versus 67.5%, hazard ratio (HR) 0.61, p = 0.25) and 5-year overall survival (83.8% versus 70%, HR 0.54, p = 0.20).

Conclusions: RAS mutations beyond KRAS exon 2 and 3 were identified in 17% of locally advanced rectal cancer patients. Given the small sample size, no definitive conclusions on the effect of additional RAS mutations on cetuximab treatment in this setting can be drawn and further investigation of RAS in larger studies is warranted.