Image-derived Models of Cell Organization Changes During Differentiation of PC12 Cells

Cellular differentiation is a complex process requiring the coordination of many cellular components. PC12 cells are a popular model system to study changes driving and accompanying neuronal differentiation. While significant attention has been paid to changes in transcriptional regulation and protein signaling, much less is known about the changes in cell organization that accompany PC12 differentiation. Fluorescence microscopy can provide extensive information about this, although photobleaching and phototoxicity frequently limit the ability to continuously observe changes in single cells over the many days that differentiation occurs. Here we describe a generative model of differentiation-associated changes in cell and nuclear shape and their relationship to mitochondrial distribution constructed from images of different cells at discrete time points. We show that our spherical harmonic-based model can accurately represent cell and nuclear shapes by measuring reconstruction errors. We then learn a regression model that relates cell and nuclear shape and mitochondrial distribution and observe that the predictive accuracy generally increases during differentiation. Most importantly, we propose a method, based on cell matching and linear interpolation in the shape space, to model the dynamics of cell differentiation using only static images. Without any prior knowledge, the method produces a realistic shape evolution process. Author Summary Cellular differentiation is an important process that is challenging to study due to the number of organizational changes it includes and the different time scales over which it occurs. Fluorescent microscopy is widely used to study cell dynamics and differentiation, but photobleaching and phototoxicity often make it infeasible to continuously observe a single cell undergoing differentiation for several days. In this work, we described a method to model aspects of the dynamics of PC12 cell differentiation without continuous imaging. We constructed accurate representations of cell and nuclear shapes and quantified the relationships between shapes and mitochondrial distributions. We used these to construct a generative model and combined it with a matching process to infer likely sequences of the changes in single cells undergoing differentiation.


Introduction
48 Cellular differentiation is a highly complex process that is incompletely understood. While 49 fluorescence microscopy provides a widely-used tool for investigating the organization of cell 50 components, given the number and complexity of the resulting images it is clear that there exists 51 a need for automated methods for their analysis [1]. Tools are needed not just for describing 52 these images, but also for creating models of cell organization that incorporate information from 53 many cells [2].

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Due to the intimate relationship between neuron morphology and function, particular attention 55 has been paid to how to model and represent cell shapes. Tools

Relationship between mitochondrial localization and cell and nuclear shape
136 For each cell in the collection, the distribution of mitochondrial localization was described as the 137 probability of a mitochondrial object occurring at a position inside of the cell according to a 138 standardized coordinate system relative to the cell and nuclear membranes. We used the 139 CellOrganizer implementation of the previously described method [16] in which each object is 140 represented by its relative distance from the nucleus and the azimuth and angle from the major 141 axis and the positions of all objects are fit using a logistic model (see Methods). The 142 mitochondrial distribution for each cell is thus represented by the 6 parameters of the model.
143 Given these parameters, we asked how the relationship between the mitochondrial location 144 pattern and the cell shape changes as a function of differentiation.

145
To evaluate this relationship, we used multi-response regression to predict the mitochondria 146 localization model given the cell and nuclear shapes, as described in Methods. We used nested 147 leave-one-out cross validation to first determine the optimal regularization parameters 1 , 2 and 148 and the corresponding model parameters , . The parameters of the held-out cell were 329 Joint models for cell and nuclear shapes were constructed using the spherical harmonic 330 framework as described [15]. To make different shapes comparable, this framework aligns 331 shapes using the first-order ellipse before creating the model. As an alternative we also did 332 alignment using the major axis. For this, the primary direction was obtained by projection of 333 surface points to the XY-plane, followed by PCA to find the major axis. The cell shape was then 334 aligned to this axis. After that, if the skewness along the x-axis was negative, the shape was 335 flipped in the XY-plane. 360 The accuracy of shape reconstruction was measured using Hausdorff distance, which is defined 361 as 362 where X and Y are two sets of points, and is a metric of distance between two points ( , ) 363 (Euclidean distance in our case). The 3D volume images of shapes were converted to surface 364 meshes, and vertices in the meshes for the original and reconstructed surfaces were used to 365 calculate the Hausdorff distance. An additional error metric, peak signal-to-noise ratios (PSNR) 366 between the original and reconstructed shape, was also included to evaluate the reconstruction 367 quality. PSNR is calculated based on the Hausdorff distance with the following form: 1 + -( 0 + 1 + 2 2 + 3 381 The spatial probability distribution for each cell was parameterized by the 6-element vector .

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Regression model between shape and mitochondrial distribution 384 We used a multi-response regression to model and predict the mitochondrial localization model 385 given the cell and nuclear shapes: where is a matrix of joint shape-space positions of dimension , with each row ∈ × 388 corresponding to a cell and nuclear shape, and each column a dimension of the shape space (in 389 this case 300 dimensions without scale, 301 dimensions including scale factors as an additional 390 feature).
is a matrix of mitochondrial localization models, with each row being a ∈ × 391 model corresponding to the cell at the same row in .
is the n-dimensional column vector × 392 with all elements as 1. and are model parameters, where is the parameter for the 0 0 ∈ 1 × 393 intercept and is the regression matrix describing the relationship between the shape ∈ × 394 space and mitochondria localization models.
is a matrix of random noise following ∈ × 395 multivariate Gaussian distribution with zero-mean (the residual variation in the localization