Orthogonalities and functional equations

In this survey we show how various notions of orthogonality appear in the theory of functional equations. After introducing some orthogonality relations, we give examples of functional equations postulated for orthogonal vectors only. We show their solutions as well as some applications. Then we discuss the problem of stability of some of them considering various aspects of the problem. In the sequel, we mention the orthogonality equation and the problem of preserving orthogonality. Last, but not least, in addition to presenting results, we state some open problems concerning these topics. Taking into account the big amount of results concerning functional equations postulated for orthogonal vectors which have appeared in the literature during the last decades, we restrict ourselves to the most classical equations

Publisher Correction: Genetic tool development in marine protists: emerging model organisms for experimental cell biology.

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Genetic tool development in marine protists: emerging model organisms for experimental cell biology

Abstract: Diverse microbial ecosystems underpin life in the sea. Among these microbes are many unicellular eukaryotes that span the diversity of the eukaryotic tree of life. However, genetic tractability has been limited to a few species, which do not represent eukaryotic diversity or environmentally relevant taxa. Here, we report on the development of genetic tools in a range of protists primarily from marine environments. We present evidence for foreign DNA delivery and expression in 13 species never before transformed and for advancement of tools for eight other species, as well as potential reasons for why transformation of yet another 17 species tested was not achieved. Our resource in genetic manipulation will provide insights into the ancestral eukaryotic lifeforms, general eukaryote cell biology, protein diversification and the evolution of cellular pathways

A complex of genes involved in adaptation of Leptinotarsa decemlineata larvae to induced potato defence

The Colorado potato beetle (Leptinotarsa decemlineata) is the most important pest of potato in many areas of the world. One of the main reasons for its success lies in the ability of its larvae to counteract plant defense compounds. Larvae adapt to protease inhibitors (PIs) produced in potato leaves through substitution of inhibitor-sensitive digestive cysteine proteases with inhibitor-insensitive cysteine proteases. To get a broader insight into the basis of larval adaptation to plant defenses, we created a "suppression subtractive hybridisation" library using cDNA from the gut of L. decemlineata larvae fed methyl jasmonate-induced or uninduced potato leaves. Four hundred clones, randomly selected from the library, were screened for their relevance to adaptation with DNA microarray hybridizations. Selected enzyme systems of beetle digestion were further inspected for changes in gene expression using quantitative PCR and enzyme activity measurements. We identified two new groups of digestive cysteine proteases, intestains D and intestains E. Intestains D represent a group of structurally distinct digestive cysteine proteases, of which the tested members are strongly upregulated in response to induced plant defenses. Moreover, we found that other digestive enzymes also participate in adaptation, namely, cellulases, serine proteases, and an endopolygalacturonase. In addition, juvenile hormone binding protein-like (JHBP-like) genes were upregulated. All studied genes were expressed specifically in larval guts. In contrast to earlier studies that reported experiments based on PI-enriched artificial diets, our results increase understanding of insect adaptation under natural conditions. (C) 2012 Wiley Periodicals, Inc