Quantifying barley morphology

.title[
# Quantifying barley morphology
]
.subtitle[
## using Euler characteristic curves
]
.author[
### <strong>Erik Amézquita</strong>, Michelle Quigley, Tim Ophelders <br> Elizabeth Munch, Dan Chitwood <br> Dan Koenig, Jacob Landis <br> -
]
.institute[
### Computational Mathematics, Science and Engineering <br> Michigan State University <br> -
]
.date[
### July 31st 2020
]

---

# Plant morphology

---

# Topological Data Analysis

<div class="row">
  <div class="column" style="max-width:25%; font-size: 15px;">
    <img style="padding: 25px 0 35px 0;" src="../figs/S019_L0_1.gif">
    <p style="font-size: 25px; text-align: center; color: DarkRed;"> Raw Data </p>
    <ul>
      <li> X-ray CT </li>
      <li> Point clouds </li>
      <li> Time series </li>
    <ul>
  </div>
  <div class="column" style="max-width:40%; padding: 0 25px 0 25px; font-size: 15px;">
    <img src="../figs/ecc_X.gif">
    <p style="font-size: 23px; text-align: center; color: DarkRed;"> Topological Summary </p>
    <ul>
      <li> Euler Characteristic </li>
      <li> Persistence diagrams </li>
      <li> Mapper/Reeb graphs </li>
    <ul>
  </div>
  <div class="column" style="max-width:35%; font-size: 15px;">
    <img src="../figs/svm_mds_canberra_algerian_everest.png">
    <img src="../figs/svm_mds_canberra_multan_white-smyrna.png">
    <p style="font-size: 25px; text-align: center; color: DarkRed;"> Analysis </p>
    <ul>
      <li> Statistics </li>
      <li> Machine learning </li>
      <li> Classification/prediction </li>
    <ul>
  </div>
</div>

---

# 1. Plant Biology: Barley

---

# Barley: since the dawn of agriculture

- It is the 4th most cultivated grain in the world, behind rice, wheat, and maize.

<div class="row">
  <div class="column" style="max-width:35%">
    <img src="../figs/barley_grains.png">
    <img src="../figs/barley_scotland.jpg">
  </div>
  <div class="column" style="max-width:30%">
  <p style="font-size: 15px; text-align: center; color: DimGrey;"> Cuneiform tablets from Mesopotamia depicting barley </p>
    <img src="../figs/Ancient-Mesopotamia-tablet.jpg">
    <img src="../figs/mesopotamian-tablet.jpg">
  </div>
  <div class="column" style="max-width:35%">
    <img src="../figs/beerancientegypt.jpg">
    <p style="font-size: 15px; text-align: center; color: DimGrey;"> Beer consumption in ancient Egypt </p>
  </div>
</div>

---

background-image: url("../figs/barley_domestication.jpg")
background-size: 910px
background-position: 50% 70%

# Diversification of floral morphology

---

background-image: url("../figs/composite_hybrid_mixture.jpg")
background-size: 400px
background-position: 95% 5%

# Composite Cross (CC II)

<div class="row">
  <div class="column" style="max-width:36%">
    <img src="../figs/BarleyEars_2_vs_6.jpg">
    <p style="font-size: 15px; text-align: center; color: DimGrey;"> 2-row vs 6-row barley </p>
  </div>
  <div class="column" style="max-width:64%">
    <img src="../figs/barley_world.jpg">
    <p style="font-size: 15px; text-align: center; color: DimGrey;"> 28 founder lines </p>
  </div>
</div>

- Experiment started in 1929, Aberdeen, Idaho
- Maintenance by a number of people at UC Riverside.

---

background-image: url("../figs/composite_cross_v_05.svg")
background-size: 450px
background-position: 95% 50%

# Experimental design

.pull-left[
- Cross all possilbe 28 parent combinations (F1s)
- Self-fertilize the resulting 379 hybrids (F2s)
- Plant the progeny of hybrids in different plots and let nature do the rest
- Mostly self-fertilization was carried out throughout almost 58 generations
- We have records of all plots throughout generations
    - Measure morphology
    - DNA sequencing
    - Which genes are selected for?
    - How did morphology change?
]

---

# 2. Genetics: Zygosity

---

background-image: url("../../biology/figs/focus_DNA.jpg")
background-size: 610px
background-position: 97% 55%

# All we are is DNA

.left-column[
The DNA is composed of 4 nucleotides
- Adenine (A)
- Cytosine (C)
- Thymine (T)
- Guanine (G)

Complementarity
- `$A \leftrightarrow T$`
- `$C \leftrightarrow G$`

Natural replication mechanism 
]

---

background-image: url("../../biology/figs/alleles.jpg")
background-size: 800px
background-position: 50% 60%

# Molecular basis of mutations

**Alelle:** Variation of a known gene. Difference could be one or more nucleotides.

**Mutation:** Change of nucleotides in an allele.
 
 Add pineapple to a pizza _(mutation)_ vs calling such monster "Hawaiian" _(allele)_

---

background-image: url("../../biology/figs/mutant_phenotypes.jpg")
background-size: 600px
background-position: 97% 55%

# Rarely, one gene = on trait

**Phenotype:** Shape of a trait

**Cross:** Controlled mating

**Wild type:** The phenotype observed in nature

**Mutant:** Individual with an altered trait
]
 
---

background-image: url("../../biology/figs/how-are-chromosomes-inherited1.png")
background-size: 700px
background-position: 65% 95%
 
# Dominance vs recesiveness

Chromosomes, and thus **genes**, come in pairs: one copy from each parent.

- **Dominant:** One copy of the allele is enough to express a phenotype (A)
- **Recessive:** Allele needs two copies to be expressed (a)
- **Homozygous:** Organism with a pair of identical alleles (line: A/A or a/a)
- **Heterozygous:** Organism with a pair of different alleles (A/a)
- **Genotype:** Allelic combination of an organism (A/A, A/a, a/a)

Each copy of every chromosome **has the same probability** of being inherited.

---

background-image: url("../../biology/figs/mendel_single_gene_ratios.jpg")
background-size: 790px
background-position: 50% 75%

# Selfing an heterozygote
 
---

background-image: url("../../biology/figs/punnett_2_alleles_a.jpg")
background-size: 370px
background-position: 95% 35%

# Same idea for two alleles

.pull-left[
- Assuming independence, the phenotypic ratio is `$9:3:3:1$`
- The genotypic ratio is much more different
- *Recall:* Being dominant/recessive does not influence the chance of being inherited
- *Mendel's second law:* Gene pairs in **different chromosomes** are grouped independently during the formation of gametes.

![](../../biology/figs/punnett_2_alleles_b.jpg)
]

---

background-image: url("../../biology/figs/tomato_varieties.jpg")
background-size: 350px
background-position: 90% 85%

# Synthetising pure lines

.pull-left[
- If we keep selfing heterozygotes, half of their progeny will be homozygous.
- Selfing an homozygous produces a genotypic (and phenotypic) invariant.
- Not all homozygous are viable.
- Multiple genes:
    - A/A b/b c/c D/D
    - a/a B/B C/C d/d
    - Both are homozygous
- If left to their own devices, which homozygous combinations are the most successful?
]

---

# 3. Image procesing

## X-ray CT scan 3D images

---

background-image: url("../figs/barley_lab_composition.jpg")
background-size: 750px
background-position: 99% 99%

# Raw data: X-ray CT scans

`$\sim30\mu m$` resolution

Batch scans - 4 per tray

2Gb+ per raw scan
]

---

# _Ad-hoc_ image processing

<div class="row">
  <div class="column" style="max-width:12%; color: DimGrey; font-size: 15px;">
    <img src="../figs/S017_0_original.gif">
    <p style="text-align: center"> Original </p>
  </div>
  <div class="column" style="max-width:12%; color: DimGrey; font-size: 15px;">
    <img src="../figs/S017_1_normal.gif">
    <p style="text-align: center"> Normalized </p>
  </div>
  <div class="column" style="max-width:12%; color: DimGrey; font-size: 15px;">
    <img src="../figs/S017_2_unair.gif">
    <p style="text-align: center"> Clean </p>
  </div>
  <div class="column" style="max-width:12%; color: DimGrey; font-size: 15px;">
    <img src="../figs/S017_3_denoise.gif">
    <p style="text-align: center"> Prunned </p>
  </div>
  <div class="column" style="max-width:27%; color: DimGrey; font-size: 15px;">
    <img src="../figs/S015_alignment.jpg">
    <p style="text-align: center"> Labeled </p>
  </div>
  <div class="column" style="max-width:21%; color: DimGrey; font-size: 15px;">
    <img src="../figs/S019_L0_1.gif">
    <p style="text-align: center"> Analysis! </p>
  </div>
</div>

- 224 raw scans in total
- 875 individual barley spikes

---

# Analyze seed distribution

<div class="row">
  <div class="column" style="max-width:63%; padding: 0px 50px;">
    <img src="../figs/seeds_batch1.png">
    <img src="../figs/seeds_batch2.png">
  </div>
  <div class="column" style="max-width:12%; background-color: DarkRed; padding: 5px; border-radius: 15px 0px 0px 30px;">
    <img src="../figs/seed_outlier1.png">
    <img src="../figs/seed_outlier2.png">
  </div>
  <div class="column" style="max-width:12%; color: Yellow; text-align: center; font-size: 15px; background-color: DarkRed; padding: 5px;">
    <img src="../figs/seed_outlier4.png">
    <img src="../figs/seed_outlier6.png">
    <p><strong> Outliers </strong></p>
  </div>
  <div class="column" style="max-width:12%; background-color: DarkRed; padding: 5px; border-radius: 0px 15px 30px 0px;">
    <img src="../figs/S102_1_44_0.png">
    <img src="../figs/S129_0_37_0.png">
  </div>
  
</div>

- `$\sim38,000$` clean seeds across 3 generations.
- Gen 0 (4K), Gen 18 (27K), Gen 58 (7K)

---

background-image: url("../figs/seed_mesh.png")
background-size: 250px
background-position: 80% 60%

# Traditional measures

All seeds are oblong: align them based on SVD

.pull-left[
- Length
- Width
- Height
- Surface area
- Volume
- Convex surface area
- Convex volume
![](../figs/seed_orientation3.png)
]

---

# Allometry and Morphological evolution

---

background-image: url("../figs/boxplot_founders_Length.png")
background-size: 850px
background-position: 90% 50%

# Differences between founder seeds

---

# 3. Topology

## Euler Characteristic Transform (ECT)

---

background-image: url("../../tda/figs/euler_characteristic_2.png")
background-size: 450px
background-position: 97% 50%

# The Euler characteristic

`$$\chi = V_0 - V_1 + V_2$$`
**Betti numbers:** Number of homologically different holes.
- `$\beta_0$`: Number of component components
- `$\beta_1$`: Number of cycles
- `$\beta_2$`: Number of voids

`$\chi = \beta_0 - \beta_1 + \beta_2$`

In general:

`$\chi = \sum_{i=0}^n (-1)^iV_i = \sum_{i=0}^n (-1)^i\beta_i$`
]

---

class: center
background-image: url("../../tda/figs/euler_characteristic_variety.jpg")
background-size: 900px
background-position: 50% 80%

# Euler characteristic for different shapes

The Euler characteristic is a topological invariant: invariant under smooth transformations.

.pull-left[
If the Euler characteristic is **different**, the two spaces must be topologically distinct
]

.pull-right[
Two spaces can be topologically distintic but their Euler characteristic **can be the same.**
]

---

background-image: url("../figs/ecc_X.gif")
background-size: 300px
background-position: 90% 60%

# Euler characteristic curve (ECC)

- A direction `$\nu\in S^{d-1}$`

- Height filtration
`$M(\nu)_r = \{x\in M: \langle x,\nu\rangle \leq r\}$` 
`$\simeq 	\{\Delta\in M : \langle x, \nu\rangle\leq r\:\forall	\,x\in\Delta\}$`

- Define the ECC as `$\chi(M,\nu):\mathbb{R}\to\mathbb{Z}$`

`$$\chi(M,\nu)(r) = \chi(M(\nu)_r).$$`
]

---

background-image: url("../figs/ect.gif")
background-size: 700px
background-position: 50% 90%

# Euler characteristic transform (ECT)

- Define `$ECT(M): S^{d-1}\to\mathbb{Z}^{\mathbb{R}}$` with `$\nu\mapsto\chi(M,\nu)$`
- Concatenate an infinite number of ECCs.

---

# Why choose the ECT?

- Easy to compute: a quick alternating sum.

[**Theorem _(Turner, Mukherjee, Boyer 2014)_**](https://doi.org/10.1093/imaiai/iau011): The ECT is injective for finite simplicial complexes in 3D.

[**Theorem _(ibid)_**](https://arxiv.org/abs/1310.1030): The ECT is a sufficient statistic for finite simplicial complexes in 3D.

*Translation:*

- Given all the (infinite) ECCs corresponding to all possible directions,

- *Different* simplicial complexes correspond to *different* ECTs.

- The ECT effectively summarizes all possible information related to shape.

---

# 5. Machine Learning

## Results on barley seeds

---

background-image: url("../figs/pole_directions_102.png")
background-size: 350px
background-position: 90% 50%

# Computing the ECT of seeds

- 74 directions

- 32 thresholds per direction

- Every seed is associated a `$74\times32=2368$`-dimensional vector.

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

---

background-image: url("../../demat/figs/pca_figure.jpg")
background-size: 290px
background-position: 99% 90%

## Unsupervised: Principal Component Analysis (PCA)

- Consider `$\mathbf x_1,\ldots,\mathbf x_n\in\mathbb R^d$` (say `$d=2368$`)

- Let `$1\leq k \leq d$` fixed. Say `$k=2$`.

- We want to find the _best_ __affine__ `$k$`-dimensional approximation `$U\beta + \mu$` such that the columns form an `$U = [u_1, \ldots, u_k]\in\mathbb R^{d\times k}$` orthonormal basis.

- Optimization problem:
`$$\min_{\beta,\mu, U} \sum_{i=1}^n\|x_i - (\mu + U\beta_i)\|^2$$` 
such that `$U^\top U=I$` and `$\sum_i\beta_i=0$`.

---

## Unsupervised: MultiDimensional Scaling (MDS)

- Consider a symmetric matrix `$D = (d_{ij}) \in\mathbb R^{n\times n}$` corresponding to all pairwise **distances** of a data set  `$\mathbf x_1,\ldots,\mathbf x_n\in\mathbb R^d$` (say `$d=2368$`)

--
- Let `$1\leq k \leq d$` fixed. Say `$k=2$`.

- We want to find points `$\mathbf z_1,\ldots,\mathbf z_n\in\mathbb R^k$` such that their Euclidean distances preserves the original distances as _best_ as possible.

- Optimization problem
`$$\min_{\mathbf z_i\in\mathbb R^k} \sum_{i,j}(\|\mathbf z_i-\mathbf z_j\|_2 - d_{ij})^2$$`
such that `$\sum_{i}\mathbf z_i = 0$`

- MDS can be used for dimension reduction.

---

background-image: url("../../demat/figs/separable-svm.svg")
background-size: 450px
background-position: 50% 99%

## Supervised: Support Vector Machines (SVM)

- `$n$` labeled points `$\{\mathbf x_i,y_i\}_{i=1}^n \subset \mathbb R^d$` with `$y_i\in\{-1,+1\}$`.

- We want the __hyperplane H__ that splits the data the _best_, with `$\mathbf H = \{\mathbf x\,:\,\langle\mathbf{x,w}\rangle+b=0\}$`.

- Optimization problem
`$$\begin{align}
	\min_{(\mathbf w,b)\in\mathbb R^d\times\mathbb R}\;\; &\frac12||\mathbf w||^2, \\
	\textrm{tal que }\;\;& y_i(\langle \mathbf x_i,\mathbf w\rangle + b)\geq 1 \textrm{ para todo } i=1,\ldots,n.
\end{align}$$`

---

background-image: url("../figs/confusion_matrix_traditional_crop.png")
background-size: 450px
background-position: 97% 95%

# SVM with traditional measures

- How different are the founder seeds from each other?

- SVM to distinguish 28 classes simultaneously.

- Each seeds is associated to an `$11$`-dimensional vector.

- (Take 80% as training and 20% as test) `$\times$` 50 times.

- Roughly `$54\%$` overall classification accuracy.

- Confusion happens with hybrids and geographical relatives.
]

background-image: url("../figs/confusion_matrix_combined_crop.png")
background-size: 450px
background-position: 97% 95%

# Same idea with topology

- The curse of dimensionality is real.

- Repeat SVM procedure with only topological information only.

- Actually we get just `$46\%$` accuracy this time.

- Overall confusion structure remains the same

- Confusion happens between 2-row founders
]

---

# MDS + SVM

- Focus on two founders at a time

- Project each ECT to 2D with a metric MDS.
    - `$d(\mathbf x, \mathbf z) = \sum_i\frac{|x_i-y_i|}{|x_i|+|y_i|}$`

- Compute a linear SVM to gauge the separability of the ECT+MDS signals.

- Used 100% of the data set as training.
]

.pull-right[
![](../figs/svm_mds_canberra_algerian_everest.png)
![](../figs/svm_mds_canberra_multan_white-smyrna.png)
]

---

background-image: url("../figs/heatmap_svm_linear_C1_mds_canberra_k4.png")
background-size: 450px
background-position: 98% 50%

# Accuracy heatmap

Idea:

- If two founders are easy to separate, then their morphology must be significantly different.

- Thus, such founders ought to be distant in the phylogenetic tree.

- The reverse must be true as well.
]

---

background-image: url("../figs/hclust_ward.D2_svm_radial_C1_mds_manhattan_k4.png")
background-size: 450px
background-position: 99% 50%

# Hierarchical clustering

- The behavior is mantained as we modify:
    - Number of projected dimensions with MDS: `$k=2,4,8$` 
    - Canberra or Manhattan distance for metric MDS
    - Radial or linear SVM
    - Different hierarchical clustering criteria (complete, average, ward, mcquitty)
]

---

background-image: url("../figs/founders_rownums2_pca_d74_T32.png")
background-size: 450px
background-position: 99% 60%

# PCA suggests a slight asymmetry.

- Hypothesis: The lateral seeds in the 6-row barley present a natural asymmetry/handedness.

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

---
class: inverse, center, middle

# 6. Bonus

## Wearing a morphometrician hat

---

background-image: url("../figs/BarleySeed.svg")
background-size: 450px
background-position: 99% 60%

# Procrustes shape analysis

- Define landmarks corresponding to interesting points on every individual.

- Rotate, translate, and scale so that overlap difference is minimized

- Actually, the collection of landmarks can be thought as representing a "cannonical" shape in a high-dimensional non-Euclidean manifold.

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

---

# Asymmetry between 2-row and 6-row

---

background-image: url("../figs/gpa_founders_yproj.png")
background-size: 450px
background-position: 99% 55%

# There is something going on with the Y projection

- The first two PCs explain more than 2/3rds of the total variance

- The second PC only accounts for 17% of variance, but it seems to split 2-vs-6 row barley

- The other 2 projections didn't show anything similar
]

---

# Procrustes analysis only with one row at a time

- PC1 distinguishes founders.

- Each of the 2-row founders is split into two clusters along PC2.

- Unclear on how interpret the results.

---

# Future work

- Combine traditional and topological measures to increase classification accuracy

- Transfer learning: train on founders to predict progeny

- Do a 3D Procrustes analysis to analyze seed handedness

- Figure out how to use the barley spike as a whole

- Find a more concrete link between morphology and genomics

- Many more plant morphology data sets to be analyzed!

---

background-image: url("../figs/acknowledgments.jpg")
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