Theories of fuzzy sets and rough sets are powerful mathematical tools for modelling various types of uncertainty. Dubois and Prade investigated the problem of combining fuzzy sets with rough sets. Soft set theory was proposed by Molodtsov as a general framework for reasoning about vague concepts. The present paper is devoted to a possible fusion of these distinct but closely related soft computing approaches. Based on a Pawlak approximation space, the approximation of a soft set is proposed to obtain a hybrid model called rough soft sets. Alternatively, a soft set instead of an equivalence relation can be used to granulate the universe. This leads to a deviation of Pawlak approximation space called a soft approximation space, in which soft rough approximations and soft rough sets can be introduced accordingly. Furthermore, we also consider approximation of a fuzzy set in a soft approximation space, and initiate a concept called soft-rough fuzzy sets, which extends Dubois and Prade's rough fuzzy sets. Further research will be needed to establish whether the notions put forth in this paper may lead to a fruitful theory.

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Trichomes originate from the epidermal cells of nearly all terrestrial plants, which are specialized unicellular or multicellular structures. Although the molecular mechanism regulating unicellular trichome formation has been extensively characterized, most of the genes essential for multicellular trichome formation remain unknown. In this study, we identified an associated locus on the long arm of chromosome 10 using a genome-wide association study (GWAS) on type-I trichomes of 180 diverse Solanum lycopersicum (tomato) accessions. Using map-based cloning we then cloned the key gene controlling the initiation of this type of trichome, named Hair (H), which encodes a single C2H2 zinc-finger protein. Transgenic experiments showed that hair-absent phenotype is caused by the deletion of the entire coding region of H. We identified three alleles of H containing several missense mutations and a nucleotide deletion, which result in amino acid substitutions and a reading frame shift, respectively. In addition, knockdown of H or Woolly (Wo) represses the formation of type-I trichomes, suggesting that both regulators may function as a heterodimer. Direct protein-protein interaction between them was further detected through pull-down and yeast two-hybrid assays. In addition, ectopic expression of H in Nicotiana tabacum (tobacco) and expression of its homologs from Capsicum annuum (pepper) and tobacco in tomato can trigger trichome formation. Taken together, these findings suggest that the H gene may be functionally conserved in multicellular trichome formation in Solanaceae species.

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Tomato ( Solanum lycopersicum ) is one of the highest-value vegetable crops worldwide. Understanding the genetic regulation of primary metabolite levels can inform efforts aimed toward improving the nutrition of commercial tomato cultivars, while maintaining key traits such as yield and stress tolerance. We identified 388 suggestive association loci (including 126 significant loci) for 92 metabolic traits including nutrition and flavor-related loci by genome-wide association study from 302 accessions in two different environments. Among them, an ascorbate quantitative trait locus TFA9 ( T OMATO F RUIT A SCORBATEON CHROMOSOME 9 ) co-localized with SlbHLH59 , which promotes high ascorbate accumulation by directly binding to the promoter of structural genes involved in the D-mannose/L-galactose pathway. The causal mutation of TFA9 is an 8-bp InDel, named InDel_8, located in the promoter region of SlbHLH59 and spanned a 5’UTR Py-rich stretch motif affecting its expression. Phylogenetic analysis revealed that differentially expressed SlbHLH59 alleles were selected during tomato domestication. Our results provide a dramatic illustration of how ascorbate biosynthesis can be regulated and was selected during the domestication of tomato. Furthermore, the findings provide novel genetic insights into natural variation of metabolites in tomato fruit, and will promote efficient utilization of metabolite traits in tomato improvement.