A broader range of studies from different cultivars, locations and environments are needed to determine a common set of genes involved in berry and flavor development. A similar study was conducted on the production of volatile aromas in Cabernet Sauvignon berries across many developmental stages, including a detailed analysis of the °Brix levels that was surveyed in this study. They found that the production of alcohol volatiles from the lipoxygenase pathway dominated in the later stages of berry ripening and suggested that the activity of alcohol dehydrogenases also could play an important role. The abundance of the transcript of VviOMT1 decreased in the pulp with increasing °Brix level and was correlated with IBMP concentrations in the late stages of berry development in this study. Both OMT1 and OMT3 have been shown to synthesize IBMP. Furthermore, the transcript abundance of each gene has been correlated with IBMP concentration, but the transcript abundance of each gene cannot fully account for the total IBMP present in all genotypes and conditions. OMT3 was found to be the major genetic determinant for this trait in two independent studies. Nevertheless, it is possible that OMT1 may contribute to the IBMP concentration, blueberries in containers growing because OMT1 can synthesize IBMP and it is located at the edge of a QTL significantly contributing to this trait.
Furthermore, the majority of IBHP , the precursor for the OMT1 and OMT3 biosynthesis of IBMP, is produced in the pulp of the berry complicating the factors that influence IBMP concentration. Our results raise questions that require additional research to clarify this relationship of transcript abundance to IBMP concentration, including determination of the rates of biosynthesis and catabolism, enzyme activities, volatilization of IBMP from the berry, as well as the concentrations of substrates for the enzymes involve. There are a number of other transcriptomic ripening studies in grapes and other fruit species. Many of these have compared broad developmental stages with partial genome microarrays. One study compared transcriptomic responses of the lates stages of ripening of whole berries of Chardonnay. This study used a different microarray platform with only about half of the genome represented on the array. In this study, 12 genes were found to be differentially expressed in each of the 3 different stages investigated. There were approximately another 50 genes that were differentially expressed at one stage versus another. Several genes were proposed as good candidates for markers of ripeness and these were also examined in Cabernet Sauvignon berries using qPCR. Several of these candidate genes are consistent with our results in the present study. They include CCD4a , a late embryogenesis abundant protein , a dirigent-like protein , and an S-adenosyl-L-methionine:salicylic acid carboxyl methyltransferase . Of these, the transcript expression of SAMT was found to be temperature insensitive. Like the previous study, the present study focused on very close stages in the mature berry when fruit flavors are known to develop. In contrast to the previous study on Chardonnay, there were massive changes in the transcript abundance in hundreds of GO categories over this narrow window of ripening.
This may in part be due to using six biological replicates rather than the standard three, which probably improved the detection of significantly changing transcripts. In addition, we used a different threshold level for statistical significance and an improved microarray platform, which was able to detect double the number of transcripts. In the present study, many differences were found between the skin and the pulp, °Brix levels and the interaction of tissue and °Brix. Important fruit ripening processes were affected including ethylene signaling, senescence, volatile aroma production, lipid metabolism and cell wall softening. These data indicate that fruit ripening in the late stages of maturity is a very dynamic and active process.Ethylene is involved in climacteric fruit ripening with a CO2 burst preceding the rise in ethylene. In tomato, this occurs at the time the seeds become mature in the mature green fruit stage. At this stage, tomato fruits become sensitive to ethyene and can continue through the ripening stage. Prior to the mature breaker stage, ethylene cannot promote tomato ripening to full ripeness. In non-climacteric fruit, there is no respiratory burst of CO2 and the ripening of most non-climacteric fruits was thought not to respond significantly to an extra application of ethylene. However, recently some non-climacteric fruit such as strawberry, bell pepper and grape have been found to produce a small amount of ethylene and appear to have responses to ethylene at certain stages. In the study of grapes, this peak was observed just before the start of veraison, followed by decreases in ethylene concentrations for several weeks afterwards; the late mature stages of ripening were not examined. Ethylene action is dependent upon ethylene concentration and ethylene sensitivity or signaling. In this study, there were clear and significant changes in transcript abundance of genes involved in ethylene signaling and biosynthesis in the late stages of berry ripening. Seeds become fully mature at this time .
Perhaps there is a signal from the seeds when they become mature that allows the fruit to ripen and senesce? Perhaps small amounts of ethylene are produced or there is a change in sensitivity to ethylene? Seymour et al. suggested the response of EIN3 might be a common signaling mechanism for both climacteric and non-climacteric fruit. The responses of VviEIN3 in this study and in a pepper fruit ripening study are consistent with this hypothesis. In addition, the transcript abundance of VviEIN3 in grape is very responsive to ethylene and the ethylene inhibitor, MCP. There are many other factors other than fruit development that can influence ethylene signaling. Could chilling of the fruit or other aspects of the processing of the grapes influence these responses? Could there be some influence of other abiotic or biotic stresses? These are questions that can only be addressed in future studies with additional experiments that are designed to answer these questions.The Berry phase reveals geometric information of quantum wave functions via their phases acquired after an adiabatic cyclic process, and its concept has laid the foundation for understanding many topological properties of materials. For example, the ground state fits the description as the limit of a statistical ensemble at zero temperature. At finite temperatures, the density matrix describes thermal properties of a quantum system by associating a thermal distribution to all the states of the system. Therefore, it is an important task to generalize the Berry phase to the realm of mixed quantum states. There have been several approaches to address this problem, among which the Uhlmann phase has attracted much attention recently since it has been shown to exhibit topological phase transitions at finite temperatures in several 1D, 2D, and spin-j systems. A key feature of those systems is the discontinuous jumps of the Uhlmann phase at the critical temperatures, signifying the changes of the underlying Uhlmann holonomy as the system traverses a loop in the parameter space. However, due to the complexity of the mathematical structure and physical interpretation, the knowledge of the Uhlmann phase is far less than that of the Berry phase in the literature. Moreover, planting blueberries in containers only a handful of models allow analytical results of the Uhlmann phase to be obtained. The Berry phase is purely geometric in the sense that it does not depend on any dynamical effect during the time evolution of the quantum system of interest. Therefore, the theory of the Berry phase can be constructed in a purely mathematical manner. As a generalization, the Uhlmann phase of density matrices was built in an almost parallel way from a mathematical point of view and shares many geometric properties with the Berry phase. We will first summarize both the Berry and Uhlmann phases using a fiber-bundle language to highlight their geometric properties. Next, we will present the analytic expressions of the Uhlmann phases of bosonic and fermionic coherent states and show that their values approach the corresponding Berry phases as temperature approaches zero. Both types of coherent states are useful in the construction of path integrals of quantum fields. While any number of bosons are allowed in a single state, the Pauli exclusion principle restricts the fermion number of a single state to be zero or one. Therefore, complex numbers are used in the bosonic coherent states while Grassmann numbers are used in the fermionic coherent states. The bosonic coherent states are also used in quantum optics to describe radiation from a classical source . Moreover, the Berry phases of coherent states can be found in the literature , and we summarize the results in Appendix A. Our exact results of the Uhlmann phases of bosonic and fermionic coherent states suggest that they indeed carry geometric information, as expected by the concept of holonomy and analogy to the Berry phase.
We will show that the Uhlmann phases of both cases decrease smoothly with temperature without a finite-temperature transition, in contrast to some examples with finite-temperature transitions in previous studies. As temperature drops to zero, the Uhlmann phases of bosonic and fermionic coherent state approach the corresponding Berry phases. Our results of the coherent states, along with earlier observations , suggest the Uhlmann phase reduce to the corresponding Berry phase in the zero-temperature limit. The correspondence is nontrivial because the Uhlmann phase requires full-rank density matrices, which cannot be satisfied only by the ground state at zero temperature. Moreover, the fiber bundle for density matrices in Uhlmann’s theory is a trivial one, but the fiber bundle for wavevfunctions in the theory of Berry phase needs not be trivial. A similar question on why the Uhlmann phase agrees with the Berry phase in certain systems as temperature approaches zero was asked in Ref. without an answer. In the last part of the paper, we present a detailed analysis of the Uhlmann phase at low temperatures to search for direct relevance with the Berry phase. With the clues from the previous examples, we present a conditional proof of the correspondence by focusing on systems allowing analytic treatments of the path-ordering operations. Before showing the results, we present a brief comparison between the Uhlmann phase and another frequently mentioned geometrical phase for mixed quantum states proposed in Refs., which was originally introduced for unitary evolution but later extended to nonunitary evolution. This geometrical phase was inspired by a generalization of the Mach-Zehnder interferometry in optics and was named accordingly as the interferometric phase. It has a different formalism with a more intuitive physical picture and has been measured in experiments. In general situations, the interferometric phase can be expressed as the argument of a weighted sum of the Berry phase factors from each individual eigenstate. Thus, its relation to the Berry phase is obvious. However, the concise topological meaning of the interferometric phase is less transparent since it is not directly connected to the holonomy of the underlying bundle as the Uhlmann phase does. The reason has been discussed in a previous comparison between the two geometrical phases. The interferometric phase relies solely on the evolution of the system state while the Uhlmann phase is influenced by the changes of both the system and ancilla, which result in the Uhlmann holonomy. Although Uhlmann’s approach can be cast into a formalism parallel to that of the Berry phase as we will explain shortly, its exact connection to the Berry phase is still unclear. The Uhlmann-Berry correspondence discussed below will offer an insight into this challenging problem. The rest of the paper is organized as follows. In Sec. II, we first present concise frameworks based on geometry for the Berry and Uhlmann phases, using a fiber-bundle language. In Sec. III, we derive the analytic expressions of theUhlmann phases of bosonic and fermionic coherent states and analyze their temperature dependence. Additionally, the Uhlmann phase of a three-level system is also presented. Importantly, the Uhlmann phases of both types of coherent states and the three-level system are shown to approach the respective Berry phases as temperature approaches zero. In Sec. IV, we propose the generality of the correspondence between the Uhlmann and Berry phases in the zerotemperature limit and give a conditional proof. In Sec. V, we discuss experimental implications and propose a protocol for simulating and measuring the Uhlmann phase of bosonic coherent states. Sec. VI concludes out work.