Characterizing spatial patterns and distributions

class: center, middle, inverse, title-slide

.title[
# Characterizing spatial patterns and distributions
]
.subtitle[
## with Topological Data Analysis (TDA)
]
.author[
### <strong>Erik Amézquita</strong>, Sutton Tennant, Sandra Thibivillers, Sai Subhash<br>Benjamin Smith, Samik Bhattacharya, Jasper Kläver, Marc Libault<br>—<br>Division of Plant Science & Technology<br>Department of Mathematics<br>University of Missouri<br>—
]
.date[
### 2025-03-20
]

---

# mRNA localization FTW

- **Another regulational mechanism**: Spatial segregation and asymmetrical distribution of mRNA.

- Observed various spatial patterns across animal and plant species.

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

.pull-right[

- Energetically more efficient to transport mRNA than a whole protein

- Efficient protein complex assembly

- Prevent proteins from reaching the wrong cellular compartment

- Subcellular localization influences proteome architecture and adaptation

- mRNA can exert fuctions beyond the protein they encode
]

---

# Grand goal with Molecular Cartography&trade;

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

- 97 genes (including 10 bacterial ones) &rarr; 2 genes
- 2938 cells &rarr; 918 infected ones.

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

---

# Traditional model: Density of transcripts

![](../figs/sublevel_both_bw3_S25_rho_cyto_GLYMA_05G092200.png)
![](../figs/sublevel_both_bw3_S25_rho_cyto_GLYMA_17G195900.png)

But these characterization discard subcellular information!

---

class: inverse, middle, center

# Alternate model: Topological Data Analysis

![](../figs/GLYMA_05G092200_TDA_c00541_transp.gif)

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# TDA: Keep track of blobs and holes

![](../figs/GLYMA_05G092200_TDA_c02352_00.png)

---
# TDA: Keep track of blobs and holes

![](../figs/GLYMA_05G092200_TDA_c02352_01.png)

---
# TDA: Keep track of blobs and holes

![](../figs/GLYMA_05G092200_TDA_c02352_02.png)

---
# TDA: Keep track of blobs and holes

![](../figs/GLYMA_05G092200_TDA_c02352_03.png)

---
# TDA: Keep track of blobs and holes

![](../figs/GLYMA_05G092200_TDA_c02352_04.png)

---
# TDA: Keep track of blobs and holes

![](../figs/GLYMA_05G092200_TDA_c02352_05.png)

---
# TDA: Keep track of blobs and holes

![](../figs/GLYMA_05G092200_TDA_c02352_06.png)

---
# TDA: Keep track of blobs and holes

![](../figs/GLYMA_05G092200_TDA_c02352_07.png)

---
# TDA: Keep track of blobs and holes

![](../figs/GLYMA_05G092200_TDA_c02352_08.png)

---
# TDA: Keep track of blobs and holes

![](../figs/GLYMA_05G092200_TDA_c02352_09.png)

---
# TDA: Keep track of blobs and holes

![](../figs/GLYMA_05G092200_TDA_c02352_10.png)

---
# TDA: Keep track of blobs and holes

![](../figs/GLYMA_05G092200_TDA_c02352_11.png)

---

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

# PCA on all topological descriptors

<img src="../figs/bw25_both_scale16_-_PI_1_1_1_pca_H1+2_gridded.png" width="350" style="display: block; margin: auto auto auto 0;" />

---

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

# Show me

<img src="../figs/bw25_both_scale16_-_PI_1_1_1_pca_H1+2_gridded.png" width="350" style="display: block; margin: auto auto auto 0;" />

---

# Connecting PC 02 to the biological context

![](../figs/sublevel_both_bw3_S25_PCA02_GLYMA_05G092200.png)
![](../figs/sublevel_both_bw3_S25_PCA02_GLYMA_17G195900.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

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

---

class: bottom

background-image: url("../figs/scale32_-_PI_1_1_1_H1+2_synthetic_30_clusters.jpg")
background-size: 900px
background-position: 50% 1%

# Working "backward"

--

<img src="../figs/scale32_-_PI_1_1_1_H1+2_synthetic_pca_30_clusters.jpg" width="600" style="display: block; margin: auto;" />

---

class: bottom

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

# Working "backward"

--

<img src="../figs/scale32_-_PI_1_1_1_H1+2_synthetic_pca_varclusters.jpg" width="600" style="display: block; margin: auto;" />

---

# Discussion

**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.

**Mathematically speaking**

- Topological Data Analysis offers a robust way to encode the *shape of patterns*
- Robust to differences in scale, underlying boundaries, or orientation
- Adaptable to other patterns: Canopy patterns, Cover crop patterns, Species spatial distributions?

![](../figs/D2_GLYMA_05G092200_z_kde_pd_suplevel_by_both_00512.jpg)

---

class: inverse

# Thank you!

<div class="row">
  <div class="column" style="max-width:60%; font-size: 15px;">
    <img style="padding: 0 0 0 0;" src="https://bondlsc.missouri.edu/wp-content/uploads/2023/11/libault-returns-to-bond-lsc-as-plant-scientist-with-single-cell-focus-1.jpg">
  </div>
  <div class="column" style="max-width:40%; font-size: 24px; line-height:1.25">
  <p style="text-align: center;"><strong>Contact and slides:</strong></p>
  <p style="text-align: center;color:Blue">eah4d@missouri.edu</p>
  <p style="text-align: center;color:Blue">ejamezquita.github.io</p>
  </div>
</div>

<div class="row">
  <div class="column" style="max-width:35%; font-size: 20px;">
  <p style="font-size: 25px; text-align: center;"> Libault Lab (MU) </p>
    <ul>
      <li><strong>Marc Libault</strong></li>
      <li><strong>Sandra Thibivillers</strong></li>
      <li>Hengping Xu</li>
      <li>Sahand Amini</li>
      <li>Hong Fu</li>
      <li><strong>Sutton Tennant</strong></li>
      <li>Md Sabbir Hossain</li>
    <ul>
  </div>
  <div class="column" style="max-width:65%; font-size: 20px;">
  <p style="font-size: 25px; text-align: center;"> With help from:</p>
    <ul>
      <li>Sai Subash (Nebraska-Lincoln)</li>
      <li>Benjamin Smith (UC Berkeley)</li>
      <li>Samik Bhattacharya (Resolve Biosciences)</li>
      <li>Jasper Kläver (Resolve Biosciences)</li>
    <ul>
  </div>
</div>

---

# Software used

.pull-left[
- All the work has been done in python with mostly standard libries (`numpy`, `scikit-learn`, `matplotlib`, `pandas`, etc.)

- KDEs computed efficiently with [`KDEpy`](https://kdepy.readthedocs.io/en/latest/)

- Sublevel set filtration of images and persistence diagrams done with [`gudhi`](https://gudhi.inria.fr/)

![](https://gudhi.inria.fr/assets/img/logo.png)
]

.pull-right[
- Persistence Images computed with [`persim`](https://persim.scikit-tda.org/en/latest/)

![](https://persim.scikit-tda.org/en/latest/_static/logo.png)
]

---

# Second model: Peripherality

![](../figs/peripherality.png)

---

# This characterization has some issues

<img src="../figs/peripherality_nodule_spatial.png" width="600" style="display: block; margin: auto;" /><img src="../figs/peripherality_eccentricity_spatial.png" width="600" style="display: block; margin: auto;" />

---

# Rotate 45 degrees for ML ammenability

![](../figs/GLYMA_05G092200_TDA_c02352_12.png)

---

# From patterns to numbers

![](../figs/GLYMA_05G092200_TDA_c02352_13.png)

---

# We actually work with 3D data

- We keep track of blobs `\(H_0\)`, holes `\(H_1\)`, and voids `\(H_2\)`

![](https://upload.wikimedia.org/wikipedia/commons/d/db/Earth_Internal_Structure.svg)
<p style="font-size: 10px; text-align: right; color: Grey;"> Credits: <a href="https://commons.wikimedia.org/wiki/File:Earth_Internal_Structure.svg">Wikipedia</a></p>

---

# Mathematical justifications

- **Definition:** Given two persistence diagrams `\(D_1, D_2\)`, for `\(1\leq p<\infty\)`, we define the *p-Wasserstein* distance between them as
`$$W_p(D_1, D_2) := \inf_{\gamma:D_1\to D_2}\left(\sum_{u\in D_1} \left\| u-\gamma(u) \right\|_\infty^p\right)^{1/p},$$`
where the infimum is over all possible bijections `\(\gamma: D_1\to D_2\)`.

- **Theorem [[Mileyko *et al* (2011)](https://doi.org/10.1088/0266-5611/27/12/124007)]:** For nice filtrations, the persistence diagrams are Wasserstein-stable under small perturbations of the data they summarize.

- **Theorem [[Adams *et al.* (2017)](http://jmlr.org/papers/v18/16-337.html)]:** The persistence image `\(I(D)\)` of a persistence diagram `\(D\)` with Gaussian distributions is stable with respect to the 1-Wasserstein distance between diagrams.

### If the overall shape/pattern is perturbed a little bit, then the resulting persistence images are perturbed only a little bit as well

---

# Do TDA for all cell-gene combinations

<img src="../figs/molecular_cartography_2x4.png" width="500" style="display: block; margin: auto;" />

<img src="../figs/persistence_images_2x4.png" width="600" style="display: block; margin: auto;" />

---

background-image: url("../figs/scale32_-_PI_1_1_1_H1+2_density13_pc02.png")
background-size: 600px
background-position: 50% 99%

# The PCs are capturing other information