Small doubling in groups

Let A be a subset of a group G = (G,.). We will survey the theory of sets A with the property that |A.A| <= K|A|, where A.A = {a_1 a_2 : a_1, a_2 in A}. The case G = (Z,+) is the famous Freiman--Ruzsa theorem.Comment: 23 pages, survey article submitted to Proceedings of the Erdos Centenary conferenc

Metabolic networking in Brunfelsia calycina petals after flower opening

Brunfelsia calycina flowers change colour from purple to white due to anthocyanin degradation, parallel to an increase in fragrance and petal size. Here it was tested whether the production of the fragrant benzenoids is dependent on induction of the shikimate pathway, or if they are formed from the anthocyanin degradation products. An extensive characterization of the events taking place in Brunfelsia flowers is presented. Anthocyanin characterization was performed using ultraperfomance liquid chromatography–quadrupole time of flight–tandem mass specrometry (UPLC-QTOF-MS/MS). Volatiles emitted were identified by headspace solid phase microextraction–gas chromatography–mass spectrometry (HS-SPME-GC-MS). Accumulated proteins were identified by 2D gel electrophoresis. Transcription profiles were characterized by cross-species hybridization of Brunfelsia cDNAs to potato cDNA microarrays. Identification of accumulated metabolites was performed by UPLC-QTOF-MS non-targeted metabolite analysis. The results include characterization of the nine main anthocyanins in Brunfelsia flowers. In addition, 146 up-regulated genes, 19 volatiles, seven proteins, and 17 metabolites that increased during anthocyanin degradation were identified. A multilevel analysis suggests induction of the shikimate pathway. This pathway is the most probable source of the phenolic acids, which in turn are precursors of both the benzenoid and lignin production pathways. The knowledge obtained is valuable for future studies on degradation of anthocyanins, formation of volatiles, and the network of secondary metabolism in Brunfelsia and related species

Tissue-Specific Transcriptome and Hormonal Regulation of Pollinated and Parthenocarpic Fig (Ficus carica L.) Fruit Suggest that Fruit Ripening is Coordinated by the Reproductive Part of the Syconium

In the unconventional climacteric fig (Ficus carica) fruit, pollinated and parthenocarpic fruit of the same genotype exhibit different ripening characteristics. Integrative comparative analyses of tissue-specific transcript and of hormone levels during fruit repining from pollinated vs. parthenocarpic fig fruit were employed to unravel the similarities and differences in their regulatory processes during fruit repining. Assembling tissue-specific transcripts into 147,000 transcripts with 53,000 annotated genes provided new insights into the spatial distribution of many classes of regulatory and structural genes, including those related to color, taste and aroma, storage, protein degradation, seeds and embryos, chlorophyll, and hormones. Comparison of the pollinated and parthenocarpic tissues during fruit ripening showed differential gene expression, especially in the fruit inflorescence. The distinct physiological green phase II and ripening phase III differed significantly in their gene-transcript patterns in both pulp and inflorescence tissues. Gas chromatographic analysis of whole fruits enabled the first determination of ripening-related hormone levels from pollinated and non-pollinated figs. Ethylene and auxin both increased during fruit ripening, irrespective of pollination, whereas no production of active gibberellins or cytokinins was found in parthenocarpic or pollinated ripening fruit. Tissue-specific transcriptome revealed apparent different metabolic gene patterns for ethylene, auxin and ABA in pollinated vs. parthenocarpic fruit, mostly in the fruit inflorescence. Our results demonstrate that the production of abscisic acid (ABA), non-active ABA–GE conjugate and non-active indoleacetic acid (IAA)–Asp conjugate in pollinated fruits is much higher than in parthenocarpic fruits. We suggest that fruit ripening is coordinated by the reproductive part of the syconium and the differences in ABA production between pollinated and parthenocarpic fig fruit might be the key to their different ripening characteristics