In this paper, several weak Orlicz–Hardy martingale spaces associated with concave functions are introduced, and some weak atomic decomposition theorems for them are established. With the help of weak atomic decompositions, a sufficient condition for a sublinear operator defined on the weak Orlicz–Hardy martingale spaces to be bounded is given. Further, we investigate the duality of weak Orlicz–Hardy martingale spaces and obtain a new John–Nirenberg type inequality when the stochastic basis is regular. These results can be regarded as weak versions of the Orlicz–Hardy martingale spaces due to Miyamoto, Nakai and Sadasue.

Grain size has an important effect on rice yield. Although several key genes that regulate seed size have been reported in rice, their molecular mechanisms remain unclear. In this study, a rice small grain size 2 (sms2) mutant was identified, and MutMap resequencing analysis results showed that a 2 bp insertion in the second exon of the LOC_Os02g01590 gene resulted in a grain length and width lower than those of the wild-type Teqing (TQ). We found that SMS2 encoded vacuolar acid invertase, a novel allele of OsINV3, which regulates grain size. GO and KEGG enrichment analyses showed that SMS2 was involved in endoplasmic reticulum protein synthesis, cysteine and methionine metabolism, and propionic acid metabolism, thereby regulating grain size. An analysis of sugar content in young panicles showed that SMS2 reduced sucrose, fructose, and starch contents, thus regulating grain size. A haplotype analysis showed that Hap2 of SMS2 had a longer grain and was widely present in indica rice varieties. Our results provide a new theoretical basis for the molecular and physiological mechanisms by which SMS2 regulates grain size.

The DA1-like gene family plays a crucial role in regulating seed and organ size in plants. The DA1 gene family has been identified in several species but has not yet been reported in sweet potatoes. In this study, nine, eleven, and seven DA1s were identified in cultivated sweet potato (Ipomoea batatas, 2n = 6x = 90) and its two diploid wild relatives, I. trifida (2n = 2x = 30) and I. triloba (2n = 2x = 30), respectively. The DA1 genes were classified into three subgroups based on their phylogenetic relationships with Arabidopsis thaliana and Oryza sativa (rice). Their protein physiological properties, chromosomal localization, phylogenetic relationships, gene structure, promoter cis-elements, and expression patterns were systematically analyzed. The qRT-PCR results showed that the expression levels of four genes, IbDA1-1, IbDA1-3, IbDA1-6, and IbDA1-7, were higher in the sweet potato leaves than in the roots, fiber roots, and stems. In our study, we provide a comprehensive comparison and further the knowledge of DA1-like genes in sweet potatoes, and provide a theoretical basis for functional studies.

Seed size is an important agronomic trait determining crop yield. Identifying key genes involved in seed size regulation and elucidating their molecular mechanisms are of great significance for crop breeding. Recent studies in crops have uncovered numerous genes that control seed size and weight, many of which function by modulating phytohormone biosynthesis, metabolism, or signaling pathways. This review provides a comprehensive overview of the genetic and molecular mechanisms by which phytohormones regulate seed size and weight and their cross-talks in modulating seed size. We highlight the functional conservation and divergence of homologous genes that control seed size across species. A particular focus is placed on those genes that have promising potential for yield improvement. Finally, we discuss current challenges in phytohormone regulation of seed size and molecular design breeding strategies for translating this knowledge into crop improvement.

Grain size significantly affects rice yield and quality. Although several genes that regulate grain size have been identified, their mechanisms remain unclear. In this study, we characterized the swg5 mutant, which has a smaller plant height, shorter panicles, and smaller grains compared to the wild type (WT). MutMap resequencing and gene knockout analysis identified SWG5, a gene encoding the kinesin-13a protein, a new allele of SRS3 that positively regulates grain length and weight. RNA sequencing analyses revealed that the SWG5 allele is involved in diterpenoid biosynthesis, amino sugar metabolism, and pentose-glucuronate interconversions. Furthermore, young panicles of the swg5 mutant exhibited decreased sucrose invertase activity as well as reduced sugar and starch content. These findings indicate that SWG5/SRS3 plays a significant role in sugar metabolism, influencing grain size and weight in rice. This research provides valuable insights into breeding rice varieties with improved yield and grain quality.

The Mediator complex (MED) functions as a co-activator in plants, transmitting transcriptional signals to regulate gene expression, including responses to environmental stresses. While the MED gene family has been identified in several species, it has not yet been reported in cassava. In this study, we identified 32 members of the MeMED gene family in cassava (Manihot esculenta Crantz) distributed across 13 chromosomes. These genes were categorized into distinct Mediator subunits based on their similarity to Arabidopsis modules. Promoter analysis revealed the presence of various cis-regulatory elements, which likely play key roles in regulating plant growth, development, and stress responses. RNA-seq data showed tissue-specific expression patterns for the MeMED genes, with significant expression observed in leaves, roots, petioles, stems, friable embryogenic callus, and shoot apical meristems. Further RT-qPCR analysis under various abiotic stress conditions—including drought, exogenous hydrogen peroxide, cold, heat, and salt—demonstrated that 10 selected MeMED genes exhibited significant differential expression, indicating their potential functional involvement in stress adaptation. These findings offer insights into the biological roles of the MeMED gene family in cassava, with implications for improving stress tolerance in future breeding programs.