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  1. Abstract

    Persian walnuts (Juglans regiaL.) are the second most produced and consumed tree nut, with over 2.6 million metric tons produced in the 2022–2023 harvest cycle alone. The United States is the second largest producer, accounting for 25% of the total global supply. Nonetheless, producers face an ever‐growing demand in a more uncertain climate landscape, which requires effective and efficient walnut selection and breeding of new cultivars with increased kernel content and easy‐to‐open shells. Past and current efforts select for these traits using hand‐held calipers and eye‐based evaluations. Yet there is plenty of morphology that meets the eye but goes unmeasured, such as the volume of inner air or the convexity of the kernel. Here, we study the shape of walnut fruits based on X‐ray computed tomography three‐dimensional reconstructions. We compute 49 different morphological phenotypes for 1264 individual nuts comprising 149 accessions. These phenotypes are complemented by traits of breeding interest such as ease of kernel removal and kernel‐to‐nut weight ratio. Through allometric relationships, relative growth of one tissue to another, we identify possible biophysical constraints at play during development. We explore multiple correlations between all morphological and commercial traits and identify which morphological traits can explain the most variability of commercial traits. We show that using only volume‐ and thickness‐based traits, especially inner air content, we can successfully encode several of the commercial traits.

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  2. Chen, Tsu-Wei ; Long, Stephen P (Ed.)
    Abstract Shape plays a fundamental role in biology. Traditional phenotypic analysis methods measure some features but fail to measure the information embedded in shape comprehensively. To extract, compare and analyse this information embedded in a robust and concise way, we turn to topological data analysis (TDA), specifically the Euler characteristic transform. TDA measures shape comprehensively using mathematical representations based on algebraic topology features. To study its use, we compute both traditional and topological shape descriptors to quantify the morphology of 3121 barley seeds scanned with X-ray computed tomography (CT) technology at 127 μm resolution. The Euler characteristic transform measures shape by analysing topological features of an object at thresholds across a number of directional axes. A Kruskal–Wallis analysis of the information encoded by the topological signature reveals that the Euler characteristic transform picks up successfully the shape of the crease and bottom of the seeds. Moreover, while traditional shape descriptors can cluster the seeds based on their accession, topological shape descriptors can cluster them further based on their panicle. We then successfully train a support vector machine to classify 28 different accessions of barley based exclusively on the shape of their grains. We observe that combining both traditional and topological descriptors classifies barley seeds better than using just traditional descriptors alone. This improvement suggests that TDA is thus a powerful complement to traditional morphometrics to comprehensively describe a multitude of ‘hidden’ shape nuances which are otherwise not detected. 
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  3. Societal Impact Statement

    Citrus are intrinsically connected to human health and culture, preventing human diseases like scurvy and inspiring sacred rituals. Citrus fruits come in a stunning number of different sizes and shapes, ranging from small clementines to oversized pummelos, and fruits display a vast diversity of flavors and aromas. These qualities are key in both traditional and modern medicine and in the production of cleaning and perfume products. By quantifying and modeling overall fruit shape and oil gland distribution, we can gain further insight into citrus development and the impacts of domestication and improvement on multiple characteristics of the fruit.


    Citrus come in diverse sizes and shapes, and play a key role in world culture and economy. Citrus oil glands in particular contain essential oils which include plant secondary metabolites associated with flavor and aroma. Capturing and analyzing nuanced information behind the citrus fruit shape and its oil gland distribution provide a morphology‐driven path to further our insight into phenotype–genotype interactions.

    We investigated the shape of citrus fruit of 51 accessions based on 3D X‐ray computed tomography (CT) scan reconstructions. Accessions include members of the three ancestral citrus species as well as related genera, and several interspecific hybrids. We digitally separate and compare the size of fruit endocarp, mesocarp, exocarp, and oil gland tissue. Based on the centers of the oil glands, overall fruit shape is approximated with an ellipsoid. Possible oil gland distributions on this ellipsoid surface are explored using directional statistics.

    There is a strong allometry along fruit tissues; that is, we observe a strong linear relationship between the logarithmic volume of any pair of major tissues. This suggests that the relative growth of fruit tissues with respect to each other follows a power law. We also observe that on average, glands distance themselves from their nearest neighbor following a square root relationship, which suggests normal diffusion dynamics at play.

    The observed allometry and square root models point to the existence of biophysical developmental constraints that govern novel relationships between fruit dimensions from both evolutionary and breeding perspectives. Understanding these biophysical interactions prompts an exciting research path on fruit development and breeding.

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  4. Abstract

    Shape is data and data is shape. Biologists are accustomed to thinking about how the shape of biomolecules, cells, tissues, and organisms arise from the effects of genetics, development, and the environment. Less often do we consider that data itself has shape and structure, or that it is possible to measure the shape of data and analyze it. Here, we review applications of topological data analysis (TDA) to biology in a way accessible to biologists and applied mathematicians alike. TDA uses principles from algebraic topology to comprehensively measure shape in data sets. Using a function that relates the similarity of data points to each other, we can monitor the evolution of topological features—connected components, loops, and voids. This evolution, a topological signature, concisely summarizes large, complex data sets. We first provide a TDA primer for biologists before exploring the use of TDA across biological sub‐disciplines, spanning structural biology, molecular biology, evolution, and development. We end by comparing and contrasting different TDA approaches and the potential for their use in biology. The vision of TDA, that data are shape and shape is data, will be relevant as biology transitions into a data‐driven era where the meaningful interpretation of large data sets is a limiting factor.

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