Insect-Development-Model

Code by Jordan Hoffmann Work by Seth Donoughe and Jordan Hoffmann Done in the lab of Cassandra Extavour

All code will be uploaded with the manuscript preprint. Feel fee to contact me if you have any questions.

Based on experimental data from a light-sheet microscope, I developed a model that recapitulates the observed motion of nuclei.

Below, we show a schematic representation of a computational method for simulating syncytial development. Nucleus movements are simulated in a 3D egg, but it is schematized here in 2D.

The model is implemented in Python. By solving the Eikonal equation, . I do this using the scikit-fmm library. For each nucleus, a shell of a given size is simulated, where each voxel in the shell exerts a pull on the nucleus. This pull is normalized by a factor 1/R^2. Each shell is also given an asymmetry. This approach naturally captures many aspects of development and results in a minimal number of free parameters, almost all of which can be directly extracted from the data.

Putting it Together

A single shell will move the nucleus due to the asymmetry that is incorporated . As a result, the shell will pull the nucleus towards a wall. Next, we need to allow for nuclei to divide. When they do so, their shells are oriented in opposite directions. We are able to extract the division rate from the data. Lastly, we need nuclei to divide and move, allowing them to interact with one another and the boundary. With an arbitrarily chosen division rate, we get rapid expansion to naturally occur, due to steric effects, and as density increases nuclei slow down. Lastly, we can tune individual parameters from the data to get something that appears more realistic. The bottom image is merely a reflection of the upper image to show the full 3D nature.

Geometry

We are able to deploy our model over a wide range of different geometries. Below, we show an example in a slightly bent embryo shape as well as an ellipsoidal shape. The bent shape is more reminiscent of a true Gryllus egg shape. Nuclei move to the surface in development, and therefore in a bent geometry we employ a more sophisticated root-finding method. Below, I show an ellipsoidal geometry. In the above two plots, the colors are random for the different nuclei. What is rendered are the shells associated with each nucleus.

Insect eggs adopt a very large space of shapes and sizes. Many eggs are curved into U shapes, have large asymmetries, and eggs range in volume by more than 8 orders of magnitude. An advantage of having an in silico model is that by altering the geometry, we are able to ask what would happen if a model, tuned on one organism, where to operate in the egg of another. This allows us to make potential hypothesis and targeted experimental predictions. In the above plot, the color represents the instantaneous speed of motion at the snapshot that is shown. Warmer colors represent faster nuclei and cooler colors represent nuclei that are moving less quickly.

Comparison to Experimental Data

By extract parameters from the data, we are able to match every quantifiable aspect of embryonic development that we are able to measure.

Fate

In the development of Gryllus bimaculatus, only a small fraction of the entire egg volume comprises the region where the embryo will coalesce. By using our developmental model, we are able to perfectly track nuclei through time. This allows us to hypothesis answers related to fate determination of nuclei (whether they will become embryonic or extra embryonic). A rough example of the capabilities are shown below. For each nuclei, we compute which fraction of the nuclei end up in a preselected embryonic region, versus what fraction of nuclei end up in the extra-embryonic region.

Rendering Code

A few people have asked me for the code I used to render the images. I used povray Link which can be installed easily on Ubuntu. On a Mac, it is easily installed with either MacPorts or Brew. At the moment, I added a random file I had. It is a bit messy, but feel free to modify it to suit your needs or contact me. The code is included in the Render folder. Examples are shown below, as well as all other images in this document The code is structured to render convex objects at the moment, but this is easily modified for any shape of Polygon[] objects. The printvertex[vec_] function will convert the vertices into povray format. The majority of the heavy lifting is this code, which structures a mesh2 object.

triangles = Table[
   triangles[[j]][[1]]
   , {j, 1, Length[triangles]}];
Len = Length[triangles];
getnormals = Table[
   Table[
    {triangles[[k]][[j]], unitnormal[triangles[[k]]] // N}
    , {j, 1, 3}]
   , {k, 1, Length[triangles]}];
getnormals = Partition[Flatten[getnormals], 6];
getnormals = Table[
   Partition[getnormals[[j]], 3]
   , {j, 1, Length[getnormals]}];
corrds = getnormals[[All, 1]];
normals = getnormals[[All, 2]];
NORMALS = Table[
   Table[Mean[
      Extract[normals, Position[corrds, triangles[[j]][[i]]]]] + 
     0.0001 {RandomReal[{-1, 1}], RandomReal[{-1, 1}], 
       RandomReal[{-1, 1}]}, {i, 1, 3}]
   , {j, 1, Length[triangles]}];
sets = Partition[Range[3*Len] - 1, 3];
ps = Position[Abs[NORMALS], _?(# < 0.0001 &)];
Table[
  NORMALS[[ps[[j]][[1]], ps[[j]][[2]], ps[[j]][[3]]]] = 0;
  , {j, 1, Length[ps]}];
ps = Position[Abs[triangles], _?(# < 0.0001 &)];
Table[
  triangles[[ps[[j]][[1]], ps[[j]][[2]], ps[[j]][[3]]]] = 0;
  , {j, 1, Length[ps]}];

Where it takes in a list of triangles. The code below it just generates a .pov file in the proper format.