MdMYB1 is a crucial regulator of light-induced anthocyanin biosynthesis and fruit coloration in apple (Malus domestica). In this study, it was found that MdMYB1 protein accumulated in the light but degraded via a ubiquitin-dependent pathway in the dark. Subsequently, the MdCOP1-1 and MdCOP1-2 genes were isolated from apple fruit peel and were functionally characterized in the Arabidopsis (Arabidopsis thaliana) cop1-4 mutant. Yeast (Saccharomyces cerevisiae) two-hybrid, bimolecular fluorescence complementation, and coimmunoprecipitation assays showed that MdMYB1 interacts with the MdCOP1 proteins. Furthermore, in vitro and in vivo experiments indicated that MdCOP1s are necessary for the ubiquitination and degradation of MdMYB1 protein in the dark and are therefore involved in the light-controlled stability of the MdMYB1 protein. Finally, a viral vectorbased transformation approach demonstrated that MdCOP1s negatively regulate the peel coloration of apple fruits by modulating the degradation of the MdMYB1 protein. Our findings provide new insight into the mechanism by which light controls anthocyanin accumulation and red fruit coloration in apple and even other plant species.

In this paper, a novel compact operator is derived for the approximation of the Riesz derivative with order α ∈ (1, 2]. The compact operator is proved with fourth-order accuracy. Combining the compact operator in space discretization, a linearized difference scheme is proposed for a two-dimensional nonlinear space fractional Schrödinger equation. It is proved that the difference scheme is uniquely solvable, stable, and convergent with order O(τ 2 + h 4 ), where τ is the time step size, h = max{h 1 , h 2 }, and h 1 , h 2 are space grid sizes in the x direction and the y direction, respectively. Based on the linearized difference scheme, a compact alternating direction implicit scheme is presented and analyzed. Numerical results demonstrate that the compact operator does not bring in extra computational cost but improves the accuracy of the scheme greatly. Introduction.Fractional quantum mechanics is a theory used to discuss quantum phenomena in fractal environments. In quantum physics the first successful attempt to apply the fractality concept was the Feynman path integral approach to quantum mechanics. Feynman and Hibbs reformulated the nonrelativistic quantum mechanics as a path integral over Brownian paths [1,2]. Thus the Feynman-Hibbs fractional background leads to standard (nonfractional) quantum mechanics. Laskin [3] extended the fractality concept in quantum physics to construct a fractional path integral and formulate the fractional quantum mechanics as a path integral over the paths of the Lévy flights. It is shown that if the fractality of the Brownian trajectories leads to standard quantum and statistical mechanics, then the fractality of the Lévy paths leads to fractional quantum mechanics and fractional statistical mechanics, namely, if the path integral over Brownian trajectories leads to the well-known Schrödinger equation, then the path integral over Lévy trajectories leads to the fractional Schrödinger equation.The space fractional Schrödinger equation includes a space fractional derivative of order α (1 < α < 2) instead of the Laplacian in the standard Schrödinger equation [4]. Assumption that an anomalous relation between energy and the angular frequency is of fractional Planck quantum energy relation leads to a time fractional Schrödinger equation [5], which replaces iu t in the classical time dependent Schrödinger equation by i α ∂ α u ∂t α , where i 2 = −1 and α is the order of fractional derivative.

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Summary The spatial heterogeneity of limiting soil resources is an essential factor determining ecosystem processes and function. It has been reported that large herbivores can strongly impact the variation and spatial distribution pattern of soil nitrogen (N). However, it remains unclear how large herbivores affect soil spatial heterogeneity and whether this influence is dependent on plant community diversity. Here we examined effects of different herbivore assemblages [no grazing; cattle grazing (CG); sheep grazing (SG); and mixed grazing (MG) of cattle and sheep] on soil N spatial heterogeneity in grasslands with high and low pre‐grazing plant diversity in an eastern Eurasian steppe. We found that herbivore grazing generated and maintained spatial patterns of soil nutrients, depending on herbivore assemblage and the level of pre‐grazing plant diversity. CG increased the spatial heterogeneity of soil available N in Leymus chinensis‐dominated steppe meadows, which were independent of pre‐grazing plant diversity. However, the effects of SG and MG strongly depended on grassland plant diversity, with an increased spatial heterogeneity of soil available N in the high‐diversity grassland, but not in the low‐diversity grassland. Synthesis and applications. We concluded that in a L. chinensis‐dominated eastern Eurasian steppe, cattle ranching could be considered as an optimal grazing management protocol to improve soil spatial heterogeneity because cattle grazing (CG) consistently increased soil spatial heterogeneity in the context of both low and high plant diversity. Nevertheless, soil spatial heterogeneity could be improved by any herbivore grazing regime [CG and/or sheep grazing (SG)] when high plant diversity is maintained. These findings highlight the importance of conserving plant diversity to maintain grassland structure and ecosystem function. In grassland systems with high plant diversity, herbivore grazing and plant diversity would jointly improve soil spatial heterogeneity, thus feeding back to maintain higher plant diversity. Therefore, high plant diversity could generate a positive feedback loop of herbivore–plant–soil interactions in grazed grassland systems. Our findings indicate the importance of herbivore assemblages in maintaining spatial heterogeneity in low‐ and high‐diversity grassland systems.

We consider the nonrotating isolated horizon as an inner boundary of a four-dimensional asymptotically flat spacetime region. Due to the symmetry of the isolated horizon, it turns out that the boundary degrees of freedom can be described by a SO(1,1) BF theory with sources. This provides a new alternative approach to the usual one using Chern-Simons theory to study the black hole entropy. To count the microscopical degrees of freedom with the boundary BF theory, the entropy of the isolated horizon can also be calculated in the framework of loop quantum gravity. The leading-order contribution to the entropy coincides with the Bekenstein-Hawking area law only for a particular choice of the Barbero-Immirzi parameter, which is different from its value in the usual approach using Chern-Simons theory. Moreover, the quantum correction to the entropy formula is a constant term rather than a logarithmic term. *

MdCRY2 was isolated from apple fruit skin, and its function was analyzed in MdCRY2 transgenic Arabidopsis. The interaction between MdCRY2 and AtCOP1 was found by yeast two-hybrid and BiFC assays. Cryptochromes are blue/ultraviolet-A (UV-A) light receptors involved in regulating various aspects of plant growth and development. Investigations of the structure and functions of cryptochromes in plants have largely focused on Arabidopsis (Arabidopsis thaliana), tomato (Solanum lycopersicum), pea (Pisum sativum), and rice (Oryza sativa). However, no data on the function of CRY2 are available in woody plants. In this study, we isolated a cryptochrome gene, MdCRY2, from apple (Malus domestica). The deduced amino acid sequences of MdCRY2 contain the conserved N-terminal photolyase-related domain and the flavin adenine dinucleotide (FAD) binding domain, as well as the C-terminal DQXVP-acidic-STAES (DAS) domain. Relationship analysis indicates that MdCRY2 shows the highest similarity to the strawberry FvCRY protein. The expression of MdCRY2 is induced by blue/UV-A light, which represents a 48-h circadian rhythm. To investigate the function of MdCRY2, we overexpressed the MdCRY2 gene in a cry2 mutant and wild type (WT) Arabidopsis, assessed the phenotypes of the resulting transgenic plants, and found that MdCRY2 functions to regulate hypocotyl elongation, root growth, flower initiation, and anthocyanin accumulation. Furthermore, we examined the interaction between MdCRY2 and AtCOP1 using a yeast two-hybrid assay and a bimolecular fluorescence complementation assay. These data provide functional evidence for a role of blue/UV-A light-induced MdCRY2 in controlling photomorphogenesis in apple.

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