The mathematical shape of plants

.title[
# The mathematical shape of plants
]
.subtitle[
## Topological Data Analysis and plant science
]
.author[
### Erik Amézquita<br>—<br>Division of Plant Science & Technology<br>Department of Mathematics<br>University of Missouri<br>—
]
.date[
### 2025-11-24
]

---

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# About me: From MX to MI to MO at MU

## I work across multiple disciplines and countries

.right-column[
- 2013 - 2018 : Licenciatura (Bachelor): Mathematics @ Universidad de Guanajuato and CIMAT. 
- 2018 - 2023 : PhD: Computational Mathematics, Science &amp; Engineering @ Michigan State University 
- 2023 - 2025 : ~~PFFFD~~ ~~PFFIE~~ PFF Postdoctoral Fellow: Plant Science / Mathematics @ University of Missouri (MU).
- 2025 - present: Assistant Professor: Plant Science / Math @ MU. 
]

---

background-image: url("../../demat/figs/ecg_results.png")
background-size: 700px
background-position: 50% 80%

# College &rarr; TDA &oplus; Archaeology

Topological Data Analysis for pre-Columbian mask classification

<img src="../../demat/figs/barrett_poster-1.jpg" width="310" style="display: block; margin: auto;" />
]

``` r
*- Lessons learned for better 
* future interdisciplinary 
* collaborations
  
*- Always include the other
* discipline experts at
* every stage of the analysis

*- Learn the basics of other
* people disciplines
```
]

---

# TDA &oplus; Archaeology: retold

![](../../demat/figs/hairdo.png)
<p style="font-size: 10px; text-align: center; color: Grey;">Credits: <a href="http://mediateca.inah.gob.mx/repositorio/islandora/object/tesis%3A4497">Olmedo Vera and Gonz&aacute;lez (1986)</a></p>
]

---

# A mathematician by training, a data scientist by trade, and a plant biologist by ~~collaboration~~ osmosis

---

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# Shape quantification &oplus; Plant Science

- Quantifying the intuition
- Intuifying the quantities

---

# Come for the math, stay for the plants

<div class="row" style="font-family: 'Yanone Kaffeesatz'; font-size:22px;">
  <div class="column" style="max-width:33%">
    <p style="line-height:0;text-align: center; font-size:28px">The shape of adaptability</p>
    <img style="padding: 0 70px;" src="../../barley/figs/S017_L3_1.gif"></img>
    <img style="padding: 0 55px;" src="../../barley/figs/ecc_X.gif"></img>
    <p style="text-align: center;">Topological Data Analysis</p>
    <p style="text-align: center;">Euler Characteristic Transform</p>
  </div>
  <div class="column" style="max-width:33%">
    <p style="line-height:0;font-size:28px;text-align: center;">The shape of development</p>
    <img style="padding: 0 60px;" src="../../citrus/figs/SR01_L01_black_exocarp.gif"></img>
    <img style="padding: 0 27px;" src="../../citrus/figs/SW03_CRC1241-B_12B-4-3_L00_lambproj_v.jpg"></img>
    <p style="text-align: center;">Ellipsoidal modeling</p>
    <p style="text-align: center;">Directional statistics</p>
  </div>
  <div class="column" style="max-width:33%">
    <p style="line-height:0;font-size:28px;text-align: center;">The shape of domestication</p>
    <img style="padding: 0 35px;" src="../../walnuts/figs/2014SBa_R5_T81_meat_2.gif"></img>
    <img style="padding: 0 10px;" src="../../walnuts/figs/kernel_volume_vs_nut_volume.png"></img>
    <p style="text-align: center;">Allometry of multiple tissues</p>
    <p style="text-align: center;">Convexity indices</p>
  </div>
</div>

---

# Phenotyping the shape of things to come

<div class="row" style="font-family: 'Yanone Kaffeesatz'; font-size:22px;">
  <div class="column" style="max-width:33%">
    <p style="line-height:0;text-align: center; font-size:28px">Phenotyping patterns</p>
    <img style="padding: 0 30px;" src="../../mcarto/figs/molecular_cartography_2x2.jpg"></img>
    <img style="padding: 0 30px;" src="../../mcarto/figs/persistence_images_1x1.svg"></img>
    <p style="text-align: center;">mRNA sub-cellular localization in soybean nodule cells.</p>
  </div>
  <div class="column" style="max-width:33%">
    <p style="line-height:0;font-size:28px;text-align: center;">Phenotyping movement</p>
    <img style="padding: 0 10px;" src="../../cuscuta/figs/qmlphv.gif"></img>
    <img style="padding: 35px 30px;" src="../../cuscuta/figs/avg_pl.svg"></img>
    <p style="text-align: center;">Tracking and describing <i>Cuscuta campestris</i> circumnutation</p>
  </div>
  <div class="column" style="max-width:33%">
    <p style="line-height:0;font-size:28px;text-align: center;">Phenotyping data</p>
    <img style="padding: 0 0px;" src="../../nasrin/figs/fpkm_raw_3.png"></img>
    <img style="padding: 30px 0px;" src="../../nasrin/figs/lung_fpkm_meancorr_eps1.0e+06_r40_g40.png"></img>
    <p style="text-align: center;">Reducing <strong>and</strong> clustering high-dimensional omics data</p>
  </div>
</div>

---

# The culture shock of switching disciplines has been greater than that of switching countries

# But it is fun

---

background-image: url("../../barley/figs/seed.png")
background-size: 325px
background-position: 99% 99%

# Roadmap for today

1. Phenotyping shapes: barley seeds
   - Euler Characteristic Transform

1. Phenotyping patterns: mRNA localization in soybean
   - Sublevel set persistence
   - Persistence Images

1. Phenotyping data: lung tissue RNAseq 
   - Mapper
   
1. Current plant data and TDA opportunities

---

# Phenotyping shapes

## How _shape-y_ &nbsp; is a shape?

---

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# A crash course in plant breeding

<br>&nbsp;<br>&nbsp;<br>&nbsp;<br>&nbsp;<br>&nbsp;<br>&nbsp;<br>&nbsp;<br>&nbsp;<br>&nbsp;<br>&nbsp;<br>&nbsp;<br>&nbsp;<br>&nbsp;<br>&nbsp;<br>&nbsp;<br>&nbsp;<br>&nbsp;<br>&nbsp;

<p style="font-size: 10px; text-align: center; color: Grey;">Image Credits: <a href="https://doi.org/10.1016/j.pbi.2020.03.003">Turner-Hissong <em>et al.</em> (2020)</a></p>

- Genetics are **hard**: Looking for a needle in a haystacks: lots of genes
    - E.g. [barley](https://doi.org/10.1105/tpc.110.082537) (32k), [rice](https://doi.org/10.1093/aob/mcf218) (45k), [maize](https://doi.org/10.1104/pp.105.068718) (50k) &oplus; transposons, polyploidy, oh my
- Use shape (phenotype) to direct the search

---

<div class="row">
  <div class="column" style="max-width:44%">
    <a href="https://cereal.interreg-npa.eu/subsites/CEREAL/Barley_cultivation-Norway_Presentation_2018.pdf" target="_blank"><img style="padding: 0 0 0 0;" src="../../barley/figs/barley_norway.jpg"></a>
    <a href="https://www.resilience.org/stories/2020-03-09/the-last-crop-before-the-desert/" target="_blank"><img style="padding: 0 0 0 0;" src="../../barley/figs/barley_morocco.jpg"></a>
    <a href="https://www.doi.org/10.1007/978-1-4419-0465-2_2168" target="_blank"><img style="padding: 0 0 0 0;" src="../../barley/figs/barley_world.jpg"></a>
  </div>
  <div class="column" style="max-width:44%">
    <a href="https://www.bloomberg.com/news/articles/2020-09-13/iraq-to-offer-first-ever-barley-exports-as-rains-yield-surplus" target="_blank"><img style="padding: 0 0 0 0;" src="../../barley/figs/barley_iraq.jpg"></a>
    <a href="https://www.tibettravel.org/tibetan-culture/highland-barley.html" target="_blank"><img style="padding: 0 0 0 0;" src="../../barley/figs/barley_seed_tibet.jpg"></a>
    <div class="row">
      <div class="column" style="max-width:46%">
        <img src="../../barley/figs/S019_L0_1.gif">
      </div>
      <div class="column" style="max-width:55%">
        <img src="../../barley/figs/S017_L0_seed_10_0.gif">
      </div>
    </div>
  </div>
  <div class="column" style="max-width:8%; font-size: 15px;">
    <p style="text-align: center; font-size: 30px; line-height: 1em;"> <strong> Barley from across the world </strong></p>
    <p>28 collected barley varieties</p>
    <p>Brought to California in 1929</p>
    <p>Artificial evolution experiment for 58 generations</p>
    <p>975 panicles scanned</p>
    <p>38,000 seeds isolated</p>
  </div>
</div>

---

## The Euler Characteristic Curve (ECC)

## The Euler Characteristic Transform (ECT)

---

## The Euler Characteristic Curve (ECC)

## The Euler Characteristic Transform (ECT)

---

# Quantify the shape of barley

**Goal:** Classify the **28** founding barley varieties solely by grain morphology information.

### What does topology actually measure?

.pull-left[
<img src="../../barley/figs/discerning_directions.png" width="225" style="display: block; margin: auto;" />
]

---

# Into semi-supervised territory

- Train an SVM with 100% of the founders `\((F_0)\)`

- Classify the progeny `\(F_{58}\)`: DNA (genotype) &harr; shape (phenotype) enrichment

---

background-image: url("../figs/amezquita_etal_2021.png")
background-size: 300px
background-position: 98% 2%

# Summary

.left-column[
<img src="../../barley/figs/S017_L3_1.gif" width="150" style="display: block; margin: auto;" /><img src="../../barley/figs/ecc_X.gif" width="150" style="display: block; margin: auto;" />
]

.right-column[
- **TDA** captures **nuanced** morphological information that is not obvious to the naked eye.

- We get the best results by **combining** both traditional and topological morphological descriptors.

- We can tell **similar** population genetic dynamics by using combined **morphological** descriptors.

- Seed phenotyping can be **cheaper** than genotyping.
]

---

# Color me rooted and phenotyped!

.pull-left[
![](../../yeast2hybrid/figs/plate_02_p53_plate_02_raw.pdf.png)
![](../../yeast2hybrid/figs/p53_plate_pipeline_ver04.jpg)

Persistent intensity peak color
]

![](../../root_necrosis/figs/geodesic_comparison_-_222M_side1_030117006.png)

![](../../root_necrosis/figs/1-CAL-1-M212_-_geodesic_distance_transform.png)

Soybean root phenotyping across years

]

---

# Phenotyping patterns

## How *pattern-y* is a pattern?

(w/ 
    Sutton Tennant,
    Sandra Thibivillers,
    Sai Subhash,
    Benjamin Smith,
    Samik Bhattacharya,
    Jasper Kläver,
    Marc Libault
)

---

# The Central Dogma and Gene Expression

![](https://images.squarespace-cdn.com/content/v1/52668d02e4b0f593739ec2b6/1533240020250-N1ISWHD91G3VPFJAN7YP/1.png)
<p style="font-size: 10px; text-align: right; color: Grey;">Credits: <a href="https://doi.org/10.1242/bio.017178">Xu <em>et al.</em> (2016)</a></p>
]

- Every cell in an organism has the same* DNA.
- But cells will only express a subset of genes depending on their type. 
- By knowing which genes are over/under-expressed, we can gather what makes each cell type unique and what are their exact functions.

![](https://www.researchgate.net/publication/303559263/figure/fig3/AS:366370148634637@1464360858751/A-heat-map-showing-SmARF-gene-expression-patterns-in-different-organs-The-red-color.png)
]

---

# mRNA localization at a sub-cellular level

- Beyond gene expression counts: Spatial segregation and asymmetrical distribution of mRNA across the cytosol in the soybean nodule.

- Molecular Cartography&trade; data provided by the Libault Lab

.pull-left[
![](../../mcarto/figs/Infected_Cells_05G216000_red_05G092200_green.jpg)
Infected soybean nodule cells. Glyma.05G092200 in green. Glyma.05G216000 in red.
]

**Goals**: "How patterny is a pattern?"

- Quantify the spatial patterns followed by mRNA within individual cells.
- Mathematically model all observed mRNA sub-cellular distributions.
- *Use this mathematical model to differentiate cell types and genotypes.*

**Challenge**

- Develop a mathematical model that works for any cell size, orientation, shape, and dimension.

]

---

# Same expression levels, different patterns

![](../../mcarto/figs/molecular_cartography_2x4.png)

**Subcellular transcript patterns &harr; spatial location of the cell within the nodule**

---

# How *pattern-y* is a pattern?

![](../../mcarto/figs/cell_sample.jpg)

Patterns are subject to a large variety of cell shapes, sizes, orientations, and mRNA quantities.
- 97 genes (including 10 bacterial ones) &rarr; 6 genes
- 2938 cells &rarr; 918 infected ones.

---

background-image: url("../../mcarto/figs/GLYMA_05G092200_TDA_c00541_transp.gif")
background-size: 800px
background-position: 50% 85%

# KDEs &oplus; Sublevel sets &oplus; Pers. images

---

# Focus on `\(H_1\)` and `\(H_2\)`

---

background-image: url("../../mcarto/figs/bw25_scale32_-_PI_1_1_1_H1+2_cell_sample.png")
background-size: 620px
background-position: 75% 99%

# PCA on all topological descriptors

---

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# PC 1
- Related to the number of distinct hotspots
- Correlated to transcript number and cell size

# PC 2
- Related to the heterogeneity of hotspots
- Correlated to transcript density

]

---

# Connecting PC 02 to the biological context

![](../../mcarto/figs/GLYMA_17G195900_bw15_PI_scale24_fulleccentricity_correlations.png)

- Senescent cells exhibit a distinct transcriptomic spatial pattern compared to the rest of population.
- Loss of mRNA localization may be a lesser known contributor to cell senescence.

---

# We define a morphospace of transcriptomic patterns

![](../../mcarto/figs/bw25_both_scale16_-_PI_1_1_1_pca_H1+2_gridded.png)

# We then work "backward"

---

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background-size: 900px
background-position: 50% 1%

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background-image: url("../../mcarto/figs/scale32_-_PI_1_1_1_H1+2_synthetic_varclusters.jpg")
background-size: 900px
background-position: 50% 1%

---

# Summary

**Biologically speaking**

- Senescent cells exhibit a distinct transcriptomic spatial pattern compared to the rest of population.
- Loss of mRNA localization may be a lesser known contributor to cell senescence.
- *How does the morphospace of patterns change if we take into account more genes, more cell types, more tissues, and more mutants?*

**Mathematically speaking**

- Topological Data Analysis offers a robust way to encode the shape of patterns.
- Robust to differences in scale, underlying boundaries, or orientation.
- The framework is open to any number of cells, genes, and dimensions.

---

# Gerrymandering tears and battlefields

Voting districts &asymp; pavement cells.

]

Spatial transcriptomics of a battlefield

]

---

# Phenotyping data

## The shape of omics data and molecular biology

---

# The Central Dogma and Gene Expression

![](https://www.researchgate.net/publication/303559263/figure/fig3/AS:366370148634637@1464360858751/A-heat-map-showing-SmARF-gene-expression-patterns-in-different-organs-The-red-color.png)
]

---

background-image: url("../../nasrin/figs/mapper_vs_tsne_half.png")
background-size: 450px
background-position: 10% 90%

# Problem: Data is very high-dimensional

- FPKM counts of RNAseq data from human lung tissue &rarr; 19,648 genes
    - 314 healthy samples (GTEx)
    - 500 cancerous samples (TCGA)

- tSNE (or UMAP) separates healthy vs cancerous samples (blue vs red)
   
.pull-right[

**Question**:

"Is the RNAseq data arranged into a specific shape?"

- Are there subgroups that we are ignoring?
- Can we go from clusters to continua?
- What is the biological characterization of such continua?

]

---

background-image: url("../../tda/figs/mapper_b_00.svg")
background-size: 725px
background-position: 50% 95%

# Mapper

## Topological summary: exploration and visualization

- We start with **lots** of data points in a **high-dimensional** space.

- We want just a **handful** of points in a **low-dimensional** space that roughly preserve the original **shape**.

---

background-image: url("../../tda/figs/mapper_c_complete.svg")
background-size: 525px
background-position: 50% 99%

# Mapper in a single picture

---

# Mapper and lung cancer data

![](../../nasrin/figs/lung_t0_meancorr_eps7.0e+02_r40_g30.png)
]

- Healthy subjects tend to stay at the center

- Cancerous samples are distributed at both ends

- Healthy subjects that land in between might be at risk

- **Predictive model**: Take a new patient sample and you can assess its cancer risk based on where they land in this continuum.
]

---

# Biological significance

![](https://journals.plos.org/plosone/article/figure/image?size=large&id=10.1371/journal.pone.0284820.g009)

---

# Summary

- Mapper finds novel sub-clusters that reveal important nuances.

- Agnostic to any kind of -omics data

- Mapper remains underused and there is plenty of untapped potential in plant genomics

![](https://www.pnas.org/cms/10.1073/pnas.1102826108/asset/43af8964-ab82-4cbb-bbcb-68f5db437862/assets/graphic/pnas.1102826108fig03.jpeg)
]

![](../../tda/figs/tissue_leaf_lens.png)

![](https://media.springernature.com/full/springer-static/image/art%3A10.1038%2Fnbt.3854/MediaObjects/41587_2017_Article_BFnbt3854_Fig3_HTML.jpg?as=webp)

]

---

background-image: url("../../nasrin/figs/rice_mapper_2025-11-20.png")
background-size: 905px
background-position: 50% 90%

# Leading the Resistance with mapper

- RNASeq data from rice samples: control and bacterial infection
- Look for differentially expressed genes to understand disease resistance

---

background-image: url("../../tutorials/figs/knehansIPG.jpg")
background-size: 470px
background-position: 0% 85%

# Since we're talking about networks

- Academic social network analysis inspired by the IPG (Interdisciplinary Plant Group) at the University of Missouri

- Expanded data collection for all 51 land-grant institutions

- Weighted graph:
    - Plant faculty: nodes
    - No. of papers together: edge
    
- Does collaboration/interdisciplinarity lead to better academic outcomes?

- How to quantify collaboration/interdisciplinarity?

W/ Roberto Herrera, Sophia Knehans, David Braun
]

---

background-image: url("https://plantsandpython.github.io/PlantsAndPython/_images/plants_python_logo.jpg")
background-size: 180px
background-position: 99% 1%

# Shaping the next generation of interdisciplinary scientists

## Large amounts of data require large amounts of people

(Ask me later about my undergrad course on Data Science to Life Sciences)

---

## Pers. Diagrams &oplus; distances &oplus; MDS &oplus; PCA

---

## Mapper without the need to code

---

background-image: url("https://upload.wikimedia.org/wikipedia/commons/4/4a/University_of_Missouri_logo.svg")
background-size: 60px
background-position: 99% 1%

## Thank you!

**Barley seeds &oplus; ECT**

- Liz Munch
- Dan Chitwood
- Michelle Quigley
- Tim Ophelders
- Jacob Landis
- Dan Koenig

**mRNA sub-cellular localization**

- Sutton Tennant
- Sandra Thibivillers
- Sai Subhash
- Benjamin Smith
- Samik Bhattacharya
- Jasper Kläver
- Marc Libault

**Mapper for lung cancer**

- Farzana Nasrain
- Katie Storey
- Masato Yoshizawa

]

**Collaboration of the IPG network**

- Ethan Lenhardt
- Sophia Knehans
- Roberto Herrera
- David Braun

**Other ongoing TDA projects**

- Laura Martins
- Mather Khan
- David Mendoza-Cozatl
- Jie Zhu
- Felix Fritschi
- Jose Costa Netto
- Tim Duff
- Olivia Fisk
- Gloria Asare
- Ajay Gupta
- Bing Yang
- Tyler Mitts
- Colin Nichols

**More details**

<p style="font-size: 20px; text-align: center; color: Blue;">ejamezquita.github.io/</p>
<p style="font-size: 20px; text-align: center; color: Blue;">eah4d@missouri.edu</p>

]