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[
### 2026-04-07
]

---

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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. 
]

# 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"><mark>Phenotyping patterns</mark></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;"><mark>Phenotyping data</mark></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>

---

# Phenotyping patterns

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

---

# 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.

]

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# Same expression levels, different patterns

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

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

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

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# 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

]

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# 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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# Gerrymandering tears and battlefields

(w/Colin Nichols and Tyler Mitts)

]

Spatial transcriptomics of a battlefield

]

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# Phenotyping data

## The shape of omics data and molecular biology

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# 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?

]

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# 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**.

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# Mapper in a single picture

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# 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.
]

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# Biological significance

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

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# Leading the Resistance with mapper

- RNASeq data from rice samples: control and bacterial infection
- w/ Olivia Fisk and Gloria Asare

---

# Other related research interests

.pull-left[
![](../../tutorials/figs/knehansIPG.jpg)
**Studying the people that study plants**
w/ Sophia Knehans and Roberto Herrera
]

.pull-right[
**The landscape of data science education in biology programs**
![](../../tutorials/figs/Catalog_Freq_data.png)

]

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## 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>

]