Vector Functions and their Differentiation Formulas in 3-dimensional Euclidean Spaces

In this article, we first extend several basic theorems of the operation of vector in 3-dimensional Euclidean spaces. Then three unit vectors: e1, e2, e3 and the definition of vector function in the same spaces are introduced. By dint of unit vector the main operation properties as well as the differentiation formulas of vector function are shown [12].Liang Xiquan - Qingdao University of Science and Technology, ChinaZhao Piqing - Qingdao University of Science and Technology, ChinaBai Ou - University of Science and Technology of China, Hefei, ChinaGrzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Jarosław Kotowicz. Partial functions from a domain to the set of real numbers. Formalized Mathematics, 1(4):703-709, 1990.Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.Konrad Raczkowski and Paweł Sadowski. Real function differentiability. Formalized Mathematics, 1(4):797-801, 1990.Murray R. Spiegel. Vector Analysis and an Introduction to Tensor Analysis. McGraw-Hill Book Company, New York, 1959.Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445-449, 1990

Impacts of Protein from High-Protein Rice on Gelatinization and Retrogradation Properties in High- and Low-Amylose Reconstituted Rice Flour

High-protein rice is nutritional, but its taste attributes are inferior to low-protein rice. Many documents correlate its taste attributes with its gelatinization and retrogradation properties. This study investigated the changes in gelatinization and retrogradation properties of low- and high-amylose reconstituted rice flour (RRF) added with different fractions of proteins extracted from high-protein rice. The addition of protein decreased the RVA (rapid viscosity analyzer) viscosity parameters of the RRF but increased the peak time. The high amylose fractions in the RRF mainly increased the parameters PV, FV, SB, and peak times, and scarcely affected the parameters BD and PaT. The interaction between amylose and protein determined the pasting temperature. Protein addition in RRF significantly decreased gelatinization enthalpies but increased the onset temperature (To) and peak temperature (Tp), while the amylose in RRF increased the gelatinization enthalpies, To and Tp. Protein additions decreased the gel hardness and the pore size, while the amylose increased the gel hardness but decreased pore size. Our findings may be potentially useful in breeding and cultivating high-protein rice

Development of the PARMS Markers of the Waxy Gene and Utilization in Discriminating Wild Accessions, and Cultivated Rice (Oryza sativa L.) with Different Eating and Cooking Quality

Amylose content (AC) is the major indicator of rice eating and cooking quality (ECQ). Its synthesis in rice endosperm is mainly regulated by the protein, granule-bound starch synthase 1, which is encoded by the waxy gene (Os06g0133000, LOC_Os06g04200). The diversity of AC is largely attributable to the allelic variation at the Wx locus and the development of effective and accurate functional molecular markers to target rice variant alleles is crucial in the breeding strategy. In the present study, we developed six pairs of penta-primer amplification refractory mutation system (PARMS) markers to distinguish between Wxlv, Wxa, Wxin, Wxb, Wxmp, and Wxop,hp. These markers were successfully used to screen the genotype of large assets of genetic resources including 98 wild accessions, 55 cultivars, and 22 parental lines. Our results showed that Wxb in a low AC type was predominant in Guangxi cultivated rice as a result of cultural preference, while Wxlv in the wild accessions. Moreover, our findings surprisingly revealed the presence of Wxb in wild accession, which is a new outcome that may contribute to understanding the origin, selection and domestication processes of rice. These functional markers could be effectively used in marker-assisted breeding to improve selection efficiency of cultivars with desired AC in the early generation

Computational and Transcriptomic Analysis Unraveled OsMATE34 as a Putative Anthocyanin Transporter in Black Rice (Oryza sativa L.) Caryopsis

Anthocyanin is a flavonoid compound with potential antioxidant properties beneficial to human health and sustains plant growth and development under different environmental stresses. In black rice, anthocyanin can be found in the stems, leaves, stigmas, and caryopsis. Although the anthocyanin biosynthesis in rice has been extensively studied, limited knowledge underlying the storage mechanism and transporters is available. This study undertook the complementation of computational and transcriptome analysis to decipher a potential multidrug and toxic compound extrusion (MATE) gene candidate for anthocyanin transportation in black rice caryopsis. The phylogenetic analysis showed that OsMATE34 has the same evolutionary history and high similarities with VvAM1, VvAM3, MtMATE2, SlMATE/MTP77, RsMATE8, AtFFT, and AtTT12 involved in anthocyanin transportation. RNA sequencing analysis in black caryopsis (Bc; Bc11, Bc18, Bc25) and white caryopsis (Wc; Wc11, Wc18, Wc25), respectively, at 11 days after flowering (DAF), 18 DAF, and 25 DAF revealed a total of 36,079 expressed genes, including 33,157 known genes and 2922 new genes. The differentially expressed genes (DEGs) showed 15,573 genes commonly expressed, with 1804 and 1412 genes uniquely expressed in Bc and Wc, respectively. Pairwise comparisons showed 821 uniquely expressed genes out of 15,272 DEGs for Wc11 vs. Bc11, 201 uniquely expressed genes out of 16,240 DEGs for Wc18 vs. Bc18, and 2263 uniquely expressed genes out of 16,240 DEGs for Wc25 vs. Bc25. Along with anthocyanin biosynthesis genes (OsPAL, OsCHS, OsCHI, OsF3H, OsDFR, OsANS, and OsUFGT/Os3GT), OsMATE34 expression was significantly upregulated in all Bc but not in Wc. OsMATE34 expression was similar to OsGSTU34, a transporter of anthocyanin in rice leaves. Taken together, our results highlighted OsMATE34 (Os08g0562800) as a candidate anthocyanin transporter in rice caryopsis. This study provides a new finding and a clue to enhance the accumulation of anthocyanin in rice caryopsis

Development of the PARMS Markers of the Waxy Gene and Utilization in Discriminating Wild Accessions, and Cultivated Rice (<i>Oryza sativa</i> L.) with Different Eating and Cooking Quality

Amylose content (AC) is the major indicator of rice eating and cooking quality (ECQ). Its synthesis in rice endosperm is mainly regulated by the protein, granule-bound starch synthase 1, which is encoded by the waxy gene (Os06g0133000, LOC_Os06g04200). The diversity of AC is largely attributable to the allelic variation at the Wx locus and the development of effective and accurate functional molecular markers to target rice variant alleles is crucial in the breeding strategy. In the present study, we developed six pairs of penta-primer amplification refractory mutation system (PARMS) markers to distinguish between Wxlv, Wxa, Wxin, Wxb, Wxmp, and Wxop,hp. These markers were successfully used to screen the genotype of large assets of genetic resources including 98 wild accessions, 55 cultivars, and 22 parental lines. Our results showed that Wxb in a low AC type was predominant in Guangxi cultivated rice as a result of cultural preference, while Wxlv in the wild accessions. Moreover, our findings surprisingly revealed the presence of Wxb in wild accession, which is a new outcome that may contribute to understanding the origin, selection and domestication processes of rice. These functional markers could be effectively used in marker-assisted breeding to improve selection efficiency of cultivars with desired AC in the early generation

Engineered Dwarf Male-Sterile Rice: A Promising Genetic Tool for Facilitating Recurrent Selection in Rice

Rice is a crop feeding half of the world’s population. With the continuous raise of yield potential via genetic improvement, rice breeding has entered an era where multiple genes conferring complex traits must be efficiently manipulated to increase rice yield further. Recurrent selection is a sound strategy for manipulating multiple genes and it has been successfully performed in allogamous crops. However, the difficulties in emasculation and hand pollination had obstructed efficient use of recurrent selection in autogamous rice. Here, we report development of the dwarf male-sterile rice that can facilitate recurrent selection in rice breeding. We adopted RNAi technology to synergistically regulate rice plant height and male fertility to create the dwarf male-sterile rice. The RNAi construct pTCK-EGGE, targeting the OsGA20ox2 and OsEAT1 genes, was constructed and used to transform rice via Agrobacterium-mediated transformation. The transgenic T0 plants showing largely reduced plant height and complete male-sterile phenotypes were designated as the dwarf male-sterile plants. Progenies of the dwarf male-sterile plants were obtained by pollinating them with pollens from the wild-type. In the T1 and T2 populations, half of the plants were still dwarf male-sterile; the other half displayed normal plant height and male fertility which were designated as tall and male-fertile plants. The tall and male-fertile plants are transgene-free and can be self-pollinated to generate new varieties. Since emasculation and hand pollination for dwarf male-sterile rice plants is no longer needed, the dwarf male-sterile rice can be used to perform recurrent selection in rice. A dwarf male-sterile rice-based recurrent selection model has been proposed

Analysis of the agronomic traits of 6 homozygous T₂ mutant lines.

<p>Analysis of the agronomic traits of 6 homozygous T₂ mutant lines.</p

Ratios of mutant genotype and mutation type at the target site (ERF922-S2) in T₀ mutant plants.

<p>Ratios of mutant genotype and mutation type at the target site (ERF922-S2) in T₀ mutant plants.</p