A \"morphogenetic action\" principle for 3D shape formation by the growth of thin sheets

How does growth encode form in developing organisms? Many different spatiotemporal growth profiles may sculpt tissues into the same target 3D shapes, but only specific growth patterns are observed in animal and plant development. In particular, growth profiles may differ in their degree of spatial variation and growth anisotropy, however, the criteria that distinguish observed patterns of growth from other possible alternatives are not understood. Here we exploit the mathematical formalism of quasiconformal transformations to formulate the problem of ``growth pattern selection'' quantitatively in the context of 3D shape formation by growing 2D epithelial sheets. We propose that nature settles on growth patterns that are the `simplest' in a certain way. Specifically, we demonstrate that growth pattern selection can be formulated as an optimization problem and solved for the trajectories that minimize spatiotemporal variation in areal growth rates and deformation anisotropy.  The result i..., Analysis of growing appendage in Parhyale hawaiensis

The recording of the transgenic Parhyale embryo with a construct for heat-inducible expression of a nuclear marker (H2B-mRFPruby) was generated using multi-view lightsheet fluorescence microscopy (LSFM) with 7.5 minute time intervals beginning 3 days after egg lay (AEL). More details regarding data acquisition and pre-processing can be found in [1].
Our analysis focused on a period of dramatic outgrowth in the T2 appendage from 95 − 109h AEL and utilized tissue cartography methods to generate coarse-grained flow patterns on cells on the growing limb [2, 3]. Down-sampled data volumes were effectively denoised using Ilastik [4] by training a classifier to distinguish tissue from background. The result of this step was a pixel probability map for each time point (with high values in tissue regions and low values in background regions).
Segmented nuclei positions from [1] were then used to help distinguish the limb from surrounding tissu..., , ## OptimalGrowthData

### External Code

This repository relies on a modest amount of external code in order to run
properly. Unfortunately, due to licensing concerns, that code cannot be
directly included in this repository. If you want run the various scripts in
this repository, you must first download the open source MATLAB software
package [`gptoolbox`](https://github.com/alecjacobson/gptoolbox.git). Among a
great deal of additional functionality, this package contains the `readOFF`
function which enables users to open the many `.off` mesh files. Add these
functions to the MATLAB path using:

```
addpath(genpath('/Path/To/gptoolbox'));
```

Please note that you do not actually need to install `gptoolbox`. You merely
have to clone the GitHub repository and add it to your path. If you would like
to install it, please consult the extensive installation instructions on that
GitHub page.

### Mesh Data Format

Most of the data is surface mesh data stored in `.off` format. You can view
th...