The Berry phases of bosonic and fermionic coherent sates and the special cases with a 1D Hilbert space are summarized in the Appendix. The classical approach to the central limit theorem and the accuracy of approximations for independent random variables rely heavily on Fourier transform methods. However, the use of Fourier methods is highly limited without an independence structure, which makes it far less possible to capture the explicit bounds for the accuracy of approximations. In 1972, Charles Stein introduced a novel technique, now known as Stein’s method, for normal approximation. The method works for both independent and dependent random variables. The method also provides bounds of approximation accuracy. Extensive applications of Stein’s method to obtain uniform and non-uniform Berry–Esseen-type bounds for independent and dependent random variables can be found in, for example, Diaconis , Baldi et al. , Barbour , Dembo and Rinott , Goldstein and Reinert , Chen and Shao , Chatterjee , Nourdin and Peccati and Chen and Fang . In addition to the traditional study of Berry–Esseen bounds, container growing raspberries new developments to Stein’s method have triggered a series of research on Cramér-type moderate deviations, which address the relative error of two tail probabilities. See, for example, Raič , Chen et al. and Shao and Zhou , among others.
Various extensions of Stein’s idea have been applied to many other probability approximations, most notably to Poisson, Poisson process, compound Poisson, binomial approximations and more recently to multivariate, combinatorial and discretized normal approximations. Stein’s method has also found diverse applications in a wide range of fields, see for example,Arratia et al. , Barbour et al. and Chen . Expositions of Stein’s method and its applications in normal and other distributional approximations can be found in Diaconis and Holmes , Barbour and Chen . We also refer to Chen et al. a thorough coverage of the method’s fundamentals and recent developments in both theory and applications. The paper is organized as follows. In the next section, we give a brief review on recent developments on Stein’s method. In Section 3, we present the main results in this paper, the Berry–Esseen bounds and Cramér type moderate deviations for Studentized nonlinear statistics. Applications to Studentized U-statistics and L-statistics are discussed in Section 4. The proofs of the main results are in Section 5, while other technical proofs are postponed to Appendix. Vitis vinifera grapevines originated approximately 65 million years ago from Eurasia and have been cultivated for at least the last 8000 years for its fruits that are crushed to make wine. Grapevines are now grown throughout the world in many kinds of environments. Grape berry development is a complex process involving three developmental phases and multiple hormones.
It is in the latter ripening phase that many compounds involved in flavor and aromas are synthesized, conjugated or catabolized. Most of these compounds reside in the skin of the berry and seem to develop in the very last stages of berry development. Aroma and flavor are important sensory components of wine. They are derived from multiple classes of compounds in grapes including important volatile compounds from the grape and from yeast metabolism during grape fermentation. Each grape cultivar produces a unique set of volatile and flavor compounds at varying concentration that represents its wine typicity or typical cultivar characteristics. Esters and terpenes are volatile compound chemical classes largely responsible for the fruity and floral aromas in wines. Esters are largely produced during yeast fermentation from grape-derived products such as aliphatic alcohols and aldehydes. Grape lipoxygenases are thought to provide the six carbon precursors from fatty acids for the synthesis of the fruity aroma, hexyl acetate, in yeast during wine fermentation. Terpenes mostly originate from the grapes and are found in both the free and bound forms. Both plant fatty acid and terpenoid metabolism pathways are very sensitive to the environment. Climate has large effects on berry development and composition.
Besides grape genetics other factors may influence metabolite composition including the local grape berry microbiome, the soil type and the rootstock. While there is evidence that rootstock can affect fruit composition and transcript abundance, this effect appears to be minor relative to other environmental factors. Many cultural practices used by the grape grower may directly or indirectly affect the environment sensed by the grapevine . Temperature and light are major contributors to “terroir”. Terroir refers to the environmental effects on grapes and how it contributes distinctive characteristics to the typicity of a wine . The terroir term includes biotic and abiotic factors, soil environments as well as the viticultural practices. In the present work, we will use the term “place” to address all of the above except for the viticultural practices. Recently, a transcriptomic approach was used to elucidate the common gene subnetworks of the late stages of berry development when grapes are normally harvested at their peak maturity. One of the major sub-networks associated with ripening involved autophagy, catabolism, RNA splicing, proteolysis, chromosome organization and the circadian clock. An integrated model was constructed to link light sensing with the circadian clock highlighting the importance of the light environment on berry development. In this report, in order to get a better understanding of how much of the gene expression in Cabernet Sauvignon berry skin could be attributed to environmental influences, we tested the hypothesis that there would be significant differences in gene expression during the late stages of Cabernet Sauvignon berry ripening between two widely different locations: one in Reno, NV, USA and the other in Bordeaux, France . The analysis revealed a core set of genes that did not depend on location, climate, vineyard management, grafting and soil properties. Also, the analysis revealed key genes that are differentially expressed between the two locations. Some of these differences were linked to the effects of temperature and other environmental factors known to affect aromatic and other quality-trait-associated pathways. Many gene families were differentially expressed and may provide useful levers for the vine grower to adjust berry composition. Among others, these families encompassed genes involved in amino acid and phenylpropanoid metabolism, as well as aroma and flavor synthesis.To test the hypothesis that the transcript abundance of grape berries during the late stages of ripening differed in two locations with widely different environmental conditions, we compared the transcript abundance of grape berry skins in BOD and RNO. The vineyards were originally planted in RNO in 2004 and in BOD in 2009. The RNO vines were grown on their own roots, whereas the BOD vines were grafted on to SO4 rootstock. A vertical shoot positioning trellis design was used in both locations. There were a number environmental variables that differed between the two locations. BOD is located at a slightly more northern latitude than RNO. This resulted in slightly longer day lengths in BOD at the beginning of harvest and slightly shorter at the end of harvest . On the final harvest dates, blueberries in pots the day length differed between RNO and BOD by about 30 min. RNO had warmer average monthly maximum temperatures than that in BOD, but minimum September temperatures were cooler in RNO . Thus, RNO had a larger average daily day/night temperature differential of 20 °C, whereas BOD had a smaller average daily day/ night temperature differential of 10 °C during the harvest periods. RNO had warmer day temperatures by about6 °C and cooler night temperatures by about 4 °C than that of BOD. The RNO vineyard location was much drier than the BOD vineyard location . The monthly precipitation totals for RNO in September were 2.03 mm whereas it was 65.5 mm in BOD; the average relative humidities were 34 and 74% for RNO and BOD, respectively.
The soil at the RNO vineyard was a deep sandy loam with a pH of 6.7; the BOD vineyard was a gravelly soil with a pH of 6.2. No pathogens, nutrient deficiencies or toxicity symptoms were observed on or in the vines at either site.The analysis of transcript profiles of Cabernet Sauvignon grapes harvested in RNO in September of 2012 was previously described. Individual berry skins were separated immediately from the whole berry and the individual total soluble solids level of the berry, which is mostly composed of sugars, was determined. The Cabernet Sauvignon berry skins from BOD were harvested in a similar manner as the RNO berry skins. The berry skins in BOD were harvested from midway in September, 2013 until the first week of October . The berry skins were separated and the °Brix analyzed in the same manner as that in RNO. Grapes were harvested at a lower °Brix range in BOD than in RNO because fruit maturity for making wine is typically reached in the BOD region at a lower sugar level. Transcript abundance of the RNA-Seq reads from both RNO and BOD was estimated using Salmon software with the assembly and gene model annotation of Cabernet Sauvignon. The TPM were computed for each gene from each experimental replicate from berry skins at different sugar levels ranging from 19 to 26°Brix . Principal component analysis of the transcriptomic data showed clear grouping of experimental replicates with the largest separation by location = 51% variance and then °Brix = 22% variance of the berry skin samples . To get different perspectives of the data, three approaches were used to further analyze the transcriptomic data. One focused on expression at one similar sugar level in both locations. Another identified a common set of genes whose transcript abundance changed in both locations. And the third one was a more comprehensive network analysis using all of the sugar levels and the two locations. We chose two very similar sugar levels to determine the differential gene expression between the two locations, since sugar levels were not exactly the same at harvest. We identified 5528 differentially expressed genes between the two locations in approach 1 at the sugar level closest to the 22°Brix level using DESeq2. DEGs will refer to this set of differentially expressed genes throughout this manuscript. Gene set enrichment analysis using topGO determined the top gene ontology categories for biological processes for these 5528 genes . Based on the number of genes identified, the top GO categories were cellular metabolic process , biosynthetic process , and response to stimulus . Other important and highly significant categories were response to stress and developmental process . The relationship between the top 25 GO categories can be seen in Additional file 4. We use the term “significantly” throughout this text to mean statistically significant at or below a padj-value of 0.05. Amongst the top stimulus subcategories with the largest number of genes were response to abiotic stimulus , response to endogenous stimulus , response to external stimulus , and biotic stimulus . Some other significant environmental stimuli GO categories included response to light stimulus , response to osmotic stress , and response to temperature stimulus . In approach 2, we examined which gene expression was changing with °Brix level in both locations to identify a common set of genes differentially expressed during berry development with very different environmental conditions. The significant differences in transcript abundance in each location was determined with DESeq2 using the lowest °Brix sampling as the control. For example, the control sample in RNO was the lowest sugar sampling at 20 °Brix; the transcript abundance of the three higher °Brix samplings were compared to the transcript abundance of the control. The genes that had significantly different transcript abundance relative to control in at least one of the comparisons were identified in RNO and BOD. These gene lists were compared and the common gene set consisting of 1985 genes for both locations was determined . Comparing this common gene list to the DEGs from approach 1 identified 907 genes that were common to both sets, indicating that this subset was differentially expressed between the locations at 22°Brix. The other 1078 genes did not differ significantly between locations. This 1078 gene subset list can be found in Additional file 5 . The GO categories most enriched in this gene set included response to inorganic substance, response to abiotic stimulus and drug metabolic process. In approach 3, using a more powerful approach to finely distinguish the expression data for all sugar levels, Weighted Gene Coexpression Network Analysis identified gene sets common to and different gene expression profiles between BOD and RNO. All expressed genes for all °Brix levels were used in this analysis.