Characterizing spatial patterns and distributions

.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[
### 2024-12-13
]

---

# Shape has data, and data has shape

---

# We use Topological Data Analysis (TDA)!

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

---

background-image: url("../../demat/figs/fam9_3.png")
background-size: 100px
background-position: 98% 2%

# From MX to MI to MO at MU

.right-column[
- 2013 - 2018 : Licenciatura (Bachelor): Math @ at the Universidad de Guanajuato and CIMAT. Thesis focused on Topological Data Analysis.

- 2018 - 2023 : PhD: Computational Mathematics, Science, and Engineering @ Michigan State University. **Came for the math. Stayed for the plants.**

- 2023 - Present : PFFIE Postdoc Fellow @ Division of Plant Science & Department of Mathematics at Mizzou

]

---

# Topological Data Analysis

## A primer

---

# 1st TDA Ingredient: Complexes

- Think the data as a collection of elementary building blocks ( _simplices_ )

Vertices | Edges | Faces | Tetrahedra
---------|-------|-------|-------
  0-dim  | 1-dim | 2-dim | 3-dim

- A collection of cells is a _simplicial complex_

- Count the number of topologically invariant features ( _holes_ ):

Connected components | Loops | Voids
---------------------|-------|-------
       0-dim         | 1-dim | 2-dim

- Example with 3 connected components, 1 loop, 1 voids

---

# 2nd TDA Ingredient: Filters

- Each cell is assigned a real value which defines how the complex is constructed.

- Observe how the number of topological features change as the complex grows.

.pull-left[
<img src="../../barley/figs/eigcurv_filter.gif" width="250px" style="display: block; margin: auto;" /><img src="../../barley/figs/gaussian_density_filter.gif" width="250px" style="display: block; margin: auto;" />
]

.pull-right[
<img src="../../barley/figs/eccentricity_filter.gif" width="250px" style="display: block; margin: auto;" /><img src="../../barley/figs/vrips_ver2.gif" width="250px" style="display: block; margin: auto;" />
]

---

# Ye olde simplicial homology *with* time

- A *filtration* of simplicial complex `$\mathbf K$` is a collection of nested subcomplexes `$\mathbf K_0\subset\mathbf K_0\subset\ldots\subset\mathbf K_m=\mathbf K$`.

- We have canonical inclusions `$\iota_{i,j}:\mathbf K_i\to\mathbf K_j$`.

- We see that `$\iota_{i,k} = \iota_{j,k}\circ\iota_{i,j}$` for any `$0\leq i\leq j\leq k\leq m$`.

- (Abusing notation) These correspond to inclusions `$\iota_{i,j}:H_q(\mathbf K_i)\to H_q(\mathbf K_j)$` for `$q=0,1,2,\ldots$`.

- The `$(i,j)$`-th persistent `$q$`-th homology group is `$H_{i,j;q}(\mathbf K) = \text{Im}\,(\iota_{i,j})$`.

- It's dimension is the `$(i,j)$`-th persistent `$q$`-th Betti number, `$\beta_{i,j;q}(\mathbf K)$`.

- A class `$\alpha\in H_q(\mathbf K_i)$` *is born* at time `$i$` if `$\alpha\notin H_{i-1,i;q}(\mathbf K)$`.

- This class `$\alpha\in H_q(\mathbf K_j)$` *dies* at time `$j+1$` if `$\iota_{j,j+1}(\alpha)\in H_{i,j;q}(\mathbf K)$`.

---

# An example later will make this much clearer

---

# TDA + Biology in the literature

<div class="row">
  <div class="column" style="max-width:25%">
    <img style="padding: 0 0 0 0;" src="https://ars.els-cdn.com/content/image/1-s2.0-S1361841518302688-gr1.jpg">
    <img style="padding: 0 0 0 0;" src="https://ars.els-cdn.com/content/image/1-s2.0-S1361841518302688-gr2.jpg">
    <p style="font-size: 10px; text-align: right; color: Grey;"> Credits: <a href="https://doi.org/10.1016/j.media.2019.03.014">Qaiser <em>et al.</em> (2019)</a></p>
  </div>
  <div class="column" style="max-width:25%">
    <img style="padding: 0 0 0 0;" src="https://www.pnas.org/cms/10.1073/pnas.1313480110/asset/ce30e9df-d595-4520-a68a-74f3c6f151d1/assets/graphic/pnas.1313480110fig01.jpeg">
    <img style="padding: 0 0 0 0;" src="https://www.pnas.org/cms/10.1073/pnas.1313480110/asset/b733c3a3-00f2-43d5-b372-9da25f3d33c9/assets/graphic/pnas.1313480110fig02.jpeg">
    <p style="font-size: 10px; text-align: right; color: Grey;"> Credits: <a href="https://doi.org/10.1073/pnas.1313480110">Chan <em>et al.</em> (2013)</a></p>
  </div>
  <div class="column" style="max-width:22%">
    <img style="padding: 0 0 0 0;" src="https://www.degruyter.com/document/doi/10.1515/sagmb-2015-0057/asset/graphic/j_sagmb-2015-0057_fig_001.jpg">
    <img style="padding: 0 0 0 0;" src="https://www.degruyter.com/document/doi/10.1515/sagmb-2015-0057/asset/graphic/j_sagmb-2015-0057_fig_012.jpg">
    <img style="padding: 0 0 0 0;" src="https://www.degruyter.com/document/doi/10.1515/sagmb-2015-0057/asset/graphic/j_sagmb-2015-0057_fig_010.jpg">
    <p style="font-size: 10px; text-align: right; color: Grey;"> Credits: <a href="https://doi.org/10.1515/sagmb-2015-0057">Kovacev-Nikolic <em>et al.</em> (2016)</a></p>
  </div>
  <div class="column" style="max-width:28%">
    <img style="padding: 0 0 0 0;" src="https://media.springernature.com/original/springer-static/image/chp%3A10.1007%2F978-3-030-20867-7_7/MediaObjects/484957_1_En_7_Fig1_HTML.png">
    <p style="font-size: 10px; text-align: right; color: Grey;"> Credits: <a href="https://doi.org/10.1007/978-3-030-20867-7_7">Chitwood <em>et al.</em> (2019)</a></p>
  </div>
</div>

<div class="row" style="font-family: 'Yanone Kaffeesatz'; font-size:25px;">
  <div class="column" style="width:25%;">
    <p style="text-align: center;">Holes</p>
    <p style="text-align: center;">&darr;</p>
    <p style="text-align: center;">Cancerous tissue</p>
  </div>
  <div class="column" style="width:25%;">
    <p style="text-align: center;">Holes</p>
    <p style="text-align: center;">&darr;</p>
    <p style="text-align: center;">Horizontal Reassortment</p>
  </div>
  <div class="column" style="width:25%;">
    <p style="text-align: center;">Holes</p>
    <p style="text-align: center;">&darr;</p>
    <p style="text-align: center;">Open/closed conformations</p>
  </div>
  <div class="column" style="width:25%;">
    <p style="text-align: center;">Components</p>
    <p style="text-align: center;">&darr;</p>
    <p style="text-align: center;">Panicle structure</p>
  </div>
</div>

---

# [1] mRNA localization

# Molecular Cartography&trade;

### &amp;

# Kernel Density Estimators (KDEs)

[2] Quantify these heatmaps via Topological Data Analysis (TDA)

[3] Link back to biological context

---

# 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.
]

- 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
]

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# Molecular Cartography&trade;

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

- Soybean nodule 10&micro;m thick cross-sections.
- (X,Y,Z) coordinates for 3.7M+ cytosolic transcripts.
- 97 genes (including 10 bacterial ones) &rarr; 2 genes
- 2938 cells &rarr; 918 infected ones.
- Considered 918 &times; 2 = 1836 cell-gene pairs

---

# Kernel Density Estimations (KDEs)

![](../figs/kernel_density_estimate_2x4.png)

- **Normalize by cell AND gene:** The sum of all transcripts OF ALL genes add to 100%
    - Each gene adds to a certain percentage
    - The sum of all genes add to 100%
    - Compares absolute concentrations 
---

[1] Molecular Cartography&trade; &amp; Kernel Density Estimations

# [2] Quantify the shape of these heatmaps

## Topological Data Analysis (TDA) and persistent homology

[3] Link back to biological context

---

# Keep track of blobs `$H_0$` and holes `$H_1$`

![](../figs/GLYMA_05G092200_TDA_c00282.gif)

---

# Walking through an example

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