class: center, middle, inverse, title-slide .title[ # The shape of the aroma ] .subtitle[ ## modeling the citrus oil gland distribution ] .author[ ### <strong>Erik Amézquita</strong>, Michelle Quigley, Tim Ophelders <br> Danelle Seymour, Elizabeth Munch, Dan Chitwood <br> - ] .institute[ ### Computational Mathematics, Science and Engineering <br> Michigan State University <br> - ] .date[ ### 2022-10-18<br> - <br> Published in <a href="https://doi.org/10.1002/ppp3.10333">Plants, People, Planet (2022)</a> ] --- class: inverse, center # When life gives you lemons... <div class="row"> <div class="column" style="max-width:50%"> <iframe width="375" height="210" src="https://www.youtube-nocookie.com/embed/khRS5AuleIM?controls=0" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe> <iframe width="375" height="210" src="https://www.youtube-nocookie.com/embed/HAIaektvV3Q?controls=0" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe> </div> <div class="column" style="max-width:50%"> <iframe width="375" height="210" src="https://www.youtube-nocookie.com/embed/Xo2HULkkm3s?controls=0" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe> <iframe width="375" height="210" src="https://www.youtube-nocookie.com/embed/EwTwSNrBuIo?controls=0" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe> </div> </div> --- # Citrus origins are... complicated - 10* _fundamental_ citrus gave rise to all citrus diversity we see today - Originated in South and South East Asia, and Australia. - Still some debate as to what exactly is a citrus .pull-left[  <p style="font-size: 8px; text-align: right; color: Grey;"> Credits: <a href="https://doi.org/10.1038/nature25447">Wu <em>et al.</em> (2018)</a></p> ] .pull-right[  <p style="font-size: 8px; text-align: right; color: Grey;"> Credits: <a href="https://doi.org/10.1038/nature25447">Wu <em>et al.</em> (2018)</a></p> ] --- ## Think of citrus as lego blocks  <p style="font-size: 8px; text-align: right; color: Grey;"> Credits: <a href="https://doi.org/10.1038/nature25447">Wu <em>et al.</em> (2018)</a></p> --- ## Oil glands are closely linked to fruit development .pull-left[  <p style="font-size: 8px; text-align: right; color: Grey;"> Credits: <a href="https://www.boredpanda.com/life-cycles-pics/">BoredPanda</a></p> Developing cycle of a lemon ] .pull-right[ <iframe width="560" height="300" src="https://static-movie-usa.glencoesoftware.com/webm/10.1073/956/d916befc88029defb1ecef6c4a2fd83db89428d9/pnas.1720809115.sm02.webm" frameborder="0" allowfullscreen></iframe> <p style="font-size: 8px; text-align: right; color: Grey;"> Credits: <a href="https://doi.org/10.1073/pnas.1720809115">Smith <em>et al.</em> (2018)</a></p> - Cross-sectional view of a navel orange peel bending to the point of jetting. - Huge perfume and food industry behind essential oils ] --- ## Previous work on nucleome-based phylogeny - 360K + SNPs in non-genic, non-pericentromeric regions - **SNP**: Single Nucleotide Polymorphism - Do a PCA of pairwise nuclear genome distance .pull-left[  ] .pull-right[  <p style="font-size: 8px; text-align: right; color: Grey;"> Credits: <a href="https://doi.org/10.1038/nature25447">Wu <em>et al.</em> (2018)</a></p> ] --- ### Relationships can be stronger amongst certain groups <img src="https://media.springernature.com/full/springer-static/image/art%3A10.1038%2Fnature25447/MediaObjects/41586_2018_Article_BFnature25447_Fig3_HTML.jpg?as=webp" width="450" style="display: block; margin: auto;" /> <p style="font-size: 8px; text-align: right; color: Grey;"> Credits: <a href="https://doi.org/10.1038/nature25447">Wu <em>et al.</em> (2018)</a></p> --- class: inverse, center, middle # Scan it all! --- ## Raw Data: X-rays &rarr; Image Processing <div class="row"> <div class="column" style="max-width:38%; color: Navy; font-size: 15px;"> <img style="padding: 2px 0 2px 0;" src="../../citrus/crc_pics/crc_diversity.jpg"> <p style="text-align: center;"> UCR Collaboration </p> </div> <div class="column" style="max-width:38%; color: Navy; font-size: 15px;"> <img style="padding: 2px 0 2px 0;" src="../../citrus/crc_pics/citrus_xrayct_scanning.jpeg"> <p style="text-align: center;"> 3D X-Ray CT scan </p> </div> <div class="column" style="max-width:23%; color: Navy; font-size: 15px;"> <img src="../../citrus/figs/SR01_CRC3289_12B-19-9_L01_raw.gif"> <p style="text-align: center;"> Raw </p> </div> </div> <div class="row" style="margin: 0 auto;"> <div class="column" style="max-width:20%; color: Navy; font-size: 15px;"> <img src="../figs/SR01_L01_black_spine.gif"> <p style="text-align: center;"> Spine </p> </div> <div class="column" style="max-width:20%; color: Navy; font-size: 15px;"> <img src="../figs/SR01_L01_black_endocarp.gif"> <p style="text-align: center;"> Endocarp </p> </div> <div class="column" style="max-width:20%; color: Navy; font-size: 15px;"> <img src="../figs/SR01_L01_black_rind.gif"> <p style="text-align: center;"> Rind </p> </div> <div class="column" style="max-width:20%; color: Navy; font-size: 15px;"> <img src="../figs/SR01_L01_black_exocarp.gif"> <p style="text-align: center;"> Exocarp </p> </div> <div class="column" style="max-width:20%; color: Navy; font-size: 15px;"> <img src="../figs/SR01_L01_oil_black_glands.gif"> <p style="text-align: center;"> Oil glands</p> </div> </div> --- background-image: url("../figs/citrus_diagnostic.gif") background-size: 650px background-position: 50% 70% ## Rich tissue collection --- background-image: url("https://insideucr.ucr.edu/sites/default/files/2020-02/givaudan%20taste%20trek_0.jpg") background-size: 850px background-position: 50% 50% ### Citrus provided by the Givaudan Citrus Variety Collection @ UC Riverside --- class: inverse, center, middle # Looking to strike some oil ## Phenotyping all the citrus --- # Allometry + Linear relationship between size-related measurements. <div class="row" style="margin: 0 auto;"> <div class="column" style="max-width:49%; color: Navy; font-size: 15px;"> <img src="../figs/exocarp_vs_endocarp.jpg"> <img src="../figs/whole_vs_oil_glands.jpg"> </div> <div class="column" style="max-width:49%; color: Navy; font-size: 15px;"> <img src="../figs/whole_vs_rind.jpg"> <img src="../figs/oil_glands_vs_num_glands.jpg"> </div> </div> + Total number of glands seems to be decoupled --- ## Isolating the oil glands of a sweet orange .pull-left[  ] .pull-right[  ] - Explore gland sizes, distribution, locations, patterns, etc. - 500,000 voxels per orange --- ## The centers will be just fine .pull-left[  - About 6K&rarr;20K individual oil glands detected - Number in the right ballpark - **We have a typical point cloud in `\(\mathbb{R}^3\)`.** ] .pull-right[  <p style="font-size: 8px; text-align: right; color: Grey;"> Credits: <a href="https://doi.org/10.1006/anbo.2001.1546">Knight <em>et al.</em> (2001)</a></p> ] --- ## Size and average distance between closest oil glands  - Smaller fruits report higher density of oil glands - Similar results when looking at the average distance between each oil gland and its 2nd, 3rd, ..., 25th nearest neighbors --- ## Average distance from each gland to its `\(k\)`-th nearest neighbor  + `\(\text{Avg.dist}(k) = \sqrt{Mk + B}\)` + Oil glands might be distributing themselves following **normal diffusion** mechanics + The **outliers** for citrus groups usually correspond to hybrids. --- ## Hybrids show interesting behavior  - Carrizo citranges are a hybrid of Trifoliates and Washington sweet orange. - Oil glands seem to diffuse faster in the citranges than in the parents --- class: inverse, center, middle # Citrus are quite spherical ## Ellipsoidal to be more precise --- background-image: url("../ellipsoids/Li_and_Griffiths_2004.png") background-size: 250px background-position: 95% 3% ## Best ellipsoid fitting &mdash; least squares - A general quadratic surface is defined by the equation `$$\eqalignno{ & ax^{2}+by^{2}+cz^{2}+2fxy+2gyz+2hzy\ \ \ \ \ \ \ \ \ &\hbox{(1)}\cr &+2px+2qy+2rz+d=0.}$$` With our observations `\(\{(x_i,y_i,z_i)\}_i\)`, $$ `\begin{pmatrix} x_1^2 & y_1^2 & z_1^2 & 2x_1y_1 & 2y_1z_1 & 2x_1z_1 & x_1 & y_1 & z_1 & 1\\ x_2^2 & y_2^2 & z_2^2 & 2x_2y_2 & 2y_2z_2 & 2x_2z_2 & x_2 & y_2 & z_2 & 1\\ \vdots& \vdots& \vdots& \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ x_n^2 & y_n^2 & z_n^2 & 2x_ny_n & 2y_nz_n & 2x_nz_n & x_n & y_n & z_n & 1 \end{pmatrix}` `\begin{pmatrix} a \\ b \\ \vdots \\ d \end{pmatrix}` = `\begin{pmatrix} 0 \\ 0 \\ \vdots \\ 0 \end{pmatrix}` $$ or $$ \mathbf{D}\mathbf{v} = 0 $$ The solution to the system above can be obtained via Lagrange multipliers `$$\min_{\mathbf{v}\in\mathbb{R}^{10}}\left\|\mathbf{D}\mathbf{v}\right\|^2, \quad \mathrm{s.t.}\; \text{ some ellipsoid-specific conditions hold}$$` --- background-image: url("../ellipsoids/DiazToca_etal_2020.png") background-size: 250px background-position: 99% 6% # Projecting points to the ellipsoid ## Geocentric vs Geodetic <img src="https://ars.els-cdn.com/content/image/1-s2.0-S0098300420305410-gr1_lrg.jpg" width="400" style="display: block; margin: auto;" /> <p style="font-size: 8px; text-align: right; color: Grey;"> Credits: <a href="https://doi.org/10.1016/j.cageo.2020.104551">Diaz-Toca <em>et al.</em> (2020)</a></p> - Geocentric projections preserve better the original shape --- ## Model the whole fruit as an ellipsoid .pull-left[   ] .pull-right[   ] --- ## How spherical _is_ a round orange?  .pull-left[ - No unique way to compute sphericity - Most formulas in terms of the longest, intermediate, and shortest `\(A,B,C\)` semi-axes lengths of the best-fit ellipsoid. - **Most citrus are highly spherical** ] .pull-right[ + `\(\sqrt[3]{\frac{BC}{A^2}}\)` (Krumbein, 1941) + `\(\frac{C}{\sqrt{AB}}\)` (Corey, 1949) + `\(\sqrt[3]{\frac{C^2}{AB}}\)` (Sneed and Folk, 1958) + `\(\frac{C}{\sqrt{\frac13 (A^2+B^2+C^2)}}\)` (Janke, 1966) ] --- # Statistics on a ellipsoid is hard ## Project geocentrically, coordinate geodetically - An ellipsoid is just an sphere under an affine transformation. <img src="https://upload.wikimedia.org/wikipedia/commons/0/0e/Ellipsoid-affin.svg" width="550" style="display: block; margin: auto;" /> <p style="font-size: 8px; text-align: right; color: Grey;"> Credits: <a href="https://en.wikipedia.org/wiki/Ellipsoid">Wikipedia: Ellipsoid</a></p> - Geodetic coordinates are a deformation of the geocentric (spherical) coordinates. ### Now we can think of the oil glands as if lying on a sphere --- # Lambert equal-area projection .pull-left[  <p style="font-size: 8px; text-align: right; color: Grey;"> Credits: <a href="https://en.wikipedia.org/wiki/Lambert_azimuthal_equal-area_projection">Wikipedia</a></p> ] .pull-right[  <p style="font-size: 8px; text-align: right; color: Grey;"> Credits: <a href="https://en.wikipedia.org/wiki/Lambert_azimuthal_equal-area_projection">Wikipedia</a></p> ] Helps to visualize better a sphere in 2D --- # Citrus modeling: Sweet orange  --- # Citrus modeling: Pummelo   --- # Citrus modeling: Pummelo  --- # Citrus modeling: Mandarin   --- # Citrus modeling: Mandarin  --- # Citrus modeling: Kumquat   --- # Citrus modeling: Kumquat  --- # Citrus modeling: Finger lime   --- # Citrus modeling: Finger lime  --- class: inverse, center, middle # Dancing the macarena with directional statistics ## Perform statistics directly on a sphere surface ### Most of the citrus are nearly spherical, anyway --- # Statistics, but on the sphere surface - It is possible to compute - Mean direction and variance - Kernel density estimation (KDE) - Parameter estimation for especific spherical distributions - Hypothesis testing - Test for uniformity - Test for symmetry - Test if two samples come from the same distribution - Directional statistics was established circa 1975 - Spherical study cases didn't really take off until very recently - Very recent support to do testing in R - Nothing advanced available in python --- background-image: url("../ellipsoids/GarciaPortugues_etal_2020a.png") background-size: 200px background-position: 95% 3% # Uniformity tests - A battery of tests all **reject strongly** the hypothesis that oil glands are uniformly distributed .pull-left[  ] .pull-right[  ] - Another battery of tests **rejected strongly** the hypothesis that oil gland distribution is rotationally symmetric --- # Citrus projections  --- # Uniform and rotationally symmetric spherical distributions  --- # Kernel Density Estimators  <p style="font-size: 8px; text-align: right; color: Grey;"> Credits: <a href="https://www.statsmodels.org/dev/examples/notebooks/generated/kernel_density.html">Statsmodels</a></p> - The choice of bandwidth is crucial - Large bandwidths tend to oversmooth - Small bandwidths tend to overfit - Plenty of literature on `\(\mathbb{R}\)` on how to choose an adequate value --- background-image: url("../ellipsoids/GarciaPortugues_2013.png") background-size: 300px background-position: 95% 3% # Spherical KDE  - Oil glands of a papeda - Red arrows indicate the most significant gradient values for the density function - Concentration parameter `\(k\)` is chosen to be optimal --- # Lateral view of a papeda  - The original has holes throughout its skin - We have densities to compare citrus! --- background-image: url("../ellipsoids/Vuollo_and_Holmstrom_2018.png") background-size: 300px background-position: 95% 3% # More exploratory analysis ## SphereSiZer  --- # Side view of a pummelo  --- # Polar view of a sweet orange  --- # Side view of a sweet orange  --- # Polar view of a mandarin  --- # Side view of a mandarin  --- # Polar view of a kumquat  --- # Side view of a kumquat  --- # Polar view of a finger lime  --- # Side view of a finger lime  --- # Future directions .pull-left[  <p style="font-size: 12px; text-align: right; color: Grey;"> Credits: <a href="https://citrusaustralia.com.au/wp-content/uploads/Citrus-picking-guide.pdf">Citrus Australia</a></p> ] .pull-right[ - Compare differences in oil gland density distribution - Between "species" - Between wild types and domesticated fruits - Propose a distribution model - Insights into fruit development - Think of smarter ways to go from an ellipsoid to a sphere - And thus exploit more tools from directional statistics ] --- background-image: url("../../img/phd_institutional_logos.jpg") background-size: 500px background-position: 95% 5% class: inverse # Thank you! <div class="row"> <div class="column" style="max-width:23%; font-size: 15px;"> <img style="padding: 0 0 0 0;" src="https://i1.rgstatic.net/ii/profile.image/607374528233472-1521820780707_Q128/Elizabeth-Munch.jpg"> <p style="text-align: center; color: White">Liz Munch (MSU)</p> <img style="padding: 0 0 0 0;" src="https://alga.win.tue.nl/images/staff/tim.jpg"> <p style="text-align: center; color: White">Tim Ophelders (Utrecht)</p> </div> <div class="column" style="max-width:23%; font-size: 15px;"> <img style="padding: 0 0 0 0;" src="https://www.canr.msu.edu/contentAsset/image/9ae9777d-157c-46e6-9f12-d062ad35671e/fileAsset/filter/Jpeg,Resize,Crop/jpeg_q/80/resize_w/400/crop_x/0/crop_y/45/crop_w/400/crop_h/400"> <p style="text-align: center; color: White">Dan Chitwood<br>(MSU)</p> <img style="padding: 0 0 0 0;" src="https://i1.rgstatic.net/ii/profile.image/926632407748609-1597937790454_Q128/Michelle-Quigley.jpg"> <p style="text-align: center; color: White">Michelle Quigley<br>(MSU)</p> </div> <div class="column" style="max-width:16%; font-size: 15px;"> <img style="padding: 0 0 0 0;" src="https://iigb.ucr.edu/sites/g/files/rcwecm5716/files/styles/scale_225/public/Seymour.jpg?itok=pnUK_orh"> <p style="text-align: center; color: White">Danelle Seymour<br>(UC Riverside)</p> <img style="padding: 0 0 0 0;" src="../figs/SR01_L01_black_exocarp.gif"> </div> <div class="column" style="width:10%; font-size: 24px;"> </div> <div class="column" style="max-width:30%; font-size: 24px; line-height:1.25"> <p style="text-align: center; color: White"><strong>Email</strong></p> <p style="text-align: center; color: Yellow">amezqui3@msu.edu</p> <p style="text-align: center; color: White"><strong>Website and slides</strong></p> <p style="text-align: center; color: Yellow">egr.msu.edu/~amezqui3</p> <img style="padding: 0 0 0 0;" src="../figs/ppp3.png"> </div> </div>