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\begin{document}

\title{The shape and volume of air, kernels, and cracks, in a nutshell}



\author[1]{Erik  Amézquita}%
\author[1]{Michelle Quigley}%
\author[1]{Pat Brown}%
\author[1]{Daniel H. Chitwood}%
\author[1]{Elizabeth Munch}%
\affil[1]{Affiliation not available}%


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  \date{May 18, 2024}


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\selectlanguage{english}
\begin{abstract}
Walnuts are the second most produced and consumed tree nut, with over
2.6 million metric tons produced in the 2022-23 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 CT (Computed Tomography)
3D reconstructions. We compute 49 different morphological phenotypes for
1264 individuals comprising 149 accessions. These phenotypes are
complemented by traits of breeding interest such as ease of kernel
removal and kernel weight. 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.%
\end{abstract}%



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\selectlanguage{ngerman}\emph{\textbf{Erik Amézquita\textsuperscript{1}}, Michelle
Quigley\textsuperscript{2}, Pat Brown\textsuperscript{3}, Elizabeth
Munch\textsuperscript{4}, Dan Chitwood\textsuperscript{4}}

\emph{\textsuperscript{1}University of Missouri, Columbia, MO, USA}

\emph{\textsuperscript{2}Pennsylvania State University, University Park,
PA, USA}

\emph{\textsuperscript{3}University of California, Davis, CA, USA}

\emph{\textsuperscript{4}Michigan State University, East Lansing, MI,
USA}

\textbf{ORCiD: 0000-0002-9837-0397}

\textbf{Keywords:} walnuts; shell crack; mathematical biology; data
science; shape.

\selectlanguage{english}
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