Other Applications

Biological Background

Species in the plant genus Arabidopsis are of high interest in plant morphology studies, since 1) its genome was fully sequenced in 1996, relatively early (Kaul et al. 2000), and 2) its structure makes it relatively easy to capture images of the area of active cell division: the shoot apical meristem (SAM). Recent work (Schaefer et al. 2017) has found that mutant Arabidopsis specimens with a decreased level of expression of genes trm6,trm7, and trm8 demonstrate more variance in the placement of new cell walls during cell division.

We prepare a dataset of Arabidopsis images with the following procedure:

1. Two varieties of Arabidopsis (wild type as well as “trm678 mutants”: mutants with decreased expression of all three of TRM6,TRM7,and TRM8) were planted and kept in the short-day condition (8 hours of light, 16 hours of dark) for 6 weeks.

2. Plants were transferred to long-day conditions and kept there until the SAM had formed. This took two weeks for wild-type plants and three weeks for TRM678 mutants.

3. The SAM of each pant was then was then dissected and observed with a confocal microscope, Leica SP8 upright scanning confocal microscope equipped with a water immersion objective (HC FLUOTAR L 25x/0.95 W VISIR).

4. This resulted in 3D images of the SAM from above (e.g. perpendicular to the plane of cell division) collected for both types of specimens.

5. Each 3D image was converted to a 2D image showing only the cell wall of the top layer of cells in the SAM.

We take 20 confocal microscope images (13 from wild-type plants and 7 from trm678 mutants) of shoot apical meristems, and process them to extract graphs representing local neighborhoods of cells. Each graph consists of a cell and its 63 closest neighbors (64 cells total). Cell neighborhood selection was limited to the central region of each SAM image, since the primordia surrounding the SAM are known to have different morphological properties. For each cell neighborhood, we produce a graph by connecting two cells iff their shared boundary is 30 pixels or longer. For each edge, we save the length of this shared boundary, as well as the angle of the edge from horizontal and the edge length. We extracted 600 cell neighborhoods for each type, for a total of 1200 graphs. See Figure [fig:cell_lines] for an example SAM image and resulting graph, and see Figure [fig:cell_extracted].

[fig:cell_lines]

[fig:cell_extracted]

Species in the plant genus Arabidopsis are of high interest in plant morphology studies, owing to the facts that 1) its genome was fully sequenced in 1996, relatively early (Kaul et al. 2000), and 2) its structure makes it relatively easy to capture images of the area of active cell division: the shoot apical meristem (SAM). Cell-scale interaction of microtubules (as described in Chapter 7) and microtubule bundles are thought to influence the orientation of cell division planes in the SAM, and thereby influence the morphology of the organism as a whole. However, this connection is not yet entirely well-understood. Recent work (Dumais 2009; Schaefer et al. 2017) has found that mutations which decrease the level of expression of genes TRM6,TRM7,and TRM8 inhibit the formation of the preprophase band (PPB). Arabidopsis mutants with inhibited PPB formation demonstrate more variance in the placement of new cell walls during cell division.

Dataset

The dataset used in this analysis was prepared as follows:

1. Two varieties of Arabidopsis (wild type as well as “trm678 mutants”: mutants with decreased expression of all three of TRM6,TRM7,and TRM8) were planted and kept in the short-day condition (8 hours of light, 16 hours of dark) for 6 weeks.

2. Plants were transferred to long-day conditions and kept there until the SAM had formed. This took two weeks for wild-type plants and three weeks for TRM678 mutants.

3. The SAM of each pant was then was then dissected and observed with a confocal microscope, Leica SP8 upright scanning confocal microscope equipped with a water immersion objective (HC FLUOTAR L 25x/0.95 W VISIR).

4. This resulted in 3D images of the SAM from above (e.g. perpendicular to the plane of cell division) collected for both types of specimens.

5. Each 3D image was converted to a 2D image showing only the cell wall of the top layer of cells in the SAM.

6. Each image was segmented into individual cells, by finding connected components separated by cell walls.

7. Each segmented image was used to produce several graphs: a cell and its 63 nearest neighbors were taken as vertices (so that each graph had 64 vertices total). Vertices were connected if their corresponding cells shared a cell wall. We will refer to the resulting graph as a “morphological graph". Only cells which were within 128 pixels of the center of the image were used, to ensure that each morphological graph was only composed of cells in the central region of the SAM. This is important because cell morphology is radically different at the boundary between the SAM and the surrounding primordia. See Figure [fig:cell_images] for examples of cell segmentations and extracted graphs.

This procedure produced a population of 1062 morphological graphs of 64 vertices each: 513 graphs coming from wild-type cell images, and 549 graphs coming from trm678 mutants. See Figure [fig:cell_lines] for examples. Graph Diffusion Distance was used to compute a $1062 \times 1062$ distance matrix comparing each pair of graphs. Figure [fig:cell_line_embeddings] shows an embedding of these distances, produced by plotting each point according to its projection onto the first two principal components of the distance matrix. This technique, multidimensional scaling (MDS), produces a 2D embedding such that inter-point distances in the embedding are as close as possible to those in the original distance matrix. We see that GDD appears to detect a trend: the embedding shows a clear clustering of each type of cell with cells of the same type.

[fig:cell_images]

[fig:cell_line_embeddings]

GDD is a differentiable function of $t$ and edge weights

Once all of the eigenvalues $\lambda_i$ and eigenvectors $v_i$ (of a matrix $L$) are computed, we may backpropagate through the eigendecomposition as described in (Nelson 1976) and (Andrew, Chu, and Lancaster 1993). If our edge weights (and therefore the values in the Laplacian matrix $L$) are parametrized by some value $\theta$, and our loss function $\mathcal{L}$ is dependent on the eigenvalues of $L$, then we can collect the gradient $\frac{\partial \mathcal{L}}{\partial \theta}$ as: $$\label{eqn:derivative_eigen} \frac{\partial \mathcal{L}}{\partial \theta} = \sum_k \left( \frac{\partial \mathcal{L}}{\partial \lambda_k} \frac{\partial \lambda_k}{\partial \theta} \right) = \sum_k \left( \frac{\partial \mathcal{L}}{\partial \lambda_k} v_k^T \frac{\partial L}{\partial \theta} v_k \right).$$ In practice, if the entries of $L$ are computed as a function of $\theta$ using an automatic differentiation package (such as PyTorch (Paszke et al. 2019)) the gradient matrix $\frac{\partial L}{\partial \theta}$ is already known before eigendecomposition. We note here that for any fixed value of $t$, all of the operations needed to compute GDD are either simple linear algebra or continuous or both. Therefore, for any loss function $\mathcal{L}$ which takes the GDD between two graphs as input, we may optimize $\mathcal{L}$ by backpropagation through the calculation of GDD using Equation [eqn:derivative_eigen].

Weighted Diffusion Distance

We make two main changes to GDD to make it capable of being tuned to specific graph data. First, we replace the real-valued optimization over $t$ with a maximum over an explicit list of $t$ values ${t_1, t_2, \ldots t_p}$. This removes the need for an optimization step inside the GDD calculation. Second, we re-weight the Frobenius norm in the GDD calculation with a vector of weights $\beta_j$ which is the same length as the list of eigenvalues (these weights are normalized to sum to 1). The resulting GDD calculation is then: $$\label{eqn:diff_dist_tunable_v1} D(G_1,G_2) = \max_{t \in t_1, t_2, \ldots t_p} \sqrt{\sum_{j=1}^n {\beta_j \left(e^{t \lambda^{(1)}_j} - e^{t \lambda^{(2)}_j} \right)}^2 } .$$ This distance calculation may then be explicitly included in the computation graph (e.g. in PyTorch) of a machine learning model, without needing to invoke some external optimizer to find the supremum over all $t$. $t_n$ and $\beta_j$ may be tuned by gradient descent or some other optimization algorithm to minimize a loss function which takes $d$ as input. Tuning the $t_n$ values results in a list of values of $t$ for which GDD is most informative for a given dataset, while tuning $\beta_j$ reweights GDD to pay most attention to the eigenvalues which are most discriminative. In the experiments in Section [sec:numeric] we demonstrate the efficacy of tuning these parameters using contrastive loss.

Learning Edge Weighting Functions

Here, we note that if graph edge weights are determined by some function $f$ parametrized by $\theta$, we may still apply all of the machinery of Sections 8.4.1 and 8.4.2. A common edge weighting function for graphs embedded in Euclidean space is the Gaussian Distance Kernel, $w_{ij} = \exp \left( \frac{-1}{2\sigma^2} d_{ij}\right)$, where $d_{ij}$ is the distance between nodes $i$ and $j$ in the embedding. $\sigma$ is the ‘radius’ of the distance kernel and can be tuned in the same way as $\beta$ and $t$. In cases like the data discussed in this section, our edge weights are vector-valued, and it is therefore advantageous to replace this hand-picked edge weight with weights chosen by a general function approximator, e.g. an artificial neural network (Fukushima and Miyake 1982). As before, the parameters of this ANN could be tuned using gradients backpropagated through the GDD calculation and eigendecomposition. Example weights learned by an ANN, trained with the constrastive loss function, can be seen in Figure [fig:learned_edge_weights].

[fig:learned_edge_weights]

Data Description

In this brief section we describe the two datasets used in preparation of this manuscript. For more details, please see the README accompanying each dataset.