Harvey J Williams 

Mechanical Forces Tuning Tissue Compartmentalization During Early Avian Development

Keywords: Collective cell motion computational model

I set out to answer the question “is there a causal relationship between convergent-extension motion, which forms the primitive streak, and an anterior-posterior asymmetry in multicellular finger-like protrusions observed at the edge (area opaca) of the epiblast cell monolayer during chicken embryo gastrulation?” Since primitive streak formation is still not fully understood, an answer would potentially offer new insight.

To address this question, I reproduced the work of Tarle, Ravasio, Hakim and Gov (2015) which simulates the motion of cell-centroids as point-particles influenced by effective potentials, corresponding to various cellular processes. However, I made substantial modifications (improvements?) to better suit the biological problem at hand. Tarle kindly shared some existing code, but this was not complete enough to run and most of the project time was spent re-developing the simulation code.

The most substantial change was the choice to replace Tarle’s algorithm for detecting cell nearest neighbours with the Bowyer-Watson algorithm which calculates the Delaunay triangulation (which in turn is a valid nearest-neighbour network). This was inspired by conversations with Dr Susie McLaren about what the model physically represents and my own experience with the Wigner-Seitz cell/Brillouin zone duality in condensed-matter physics. The Delaunay triangulation is dual to the Voronoi diagram, which matches the cell outlines seen in microscopy images.

An additional benefit of using the Delaunay triangulation to find the nearest-neighbour network is that it solves the second non-trivial problem of defining the border (“coastline”) of the cell monolayer: all edges (and their vertices) which are boundaries of only one triangle are part of the coastline, as opposed to the rest of the edges (and their vertices) which are necessarily boundaries of exactly two triangles and reside in the bulk.

Whilst all parts of Tarle’s model were considered theoretically, and checked against the existing code and literature, the relatively short duration of the project meant that I got so far as actually running simulations with only a cell-cell interaction “turned on”. Without viscosity, we see typical orbit-like behaviour, which isn’t much like the collective motion of cells seen in nature. Nonetheless, this is ideal for validating our numerical integration method, which is simply a naïve first-order Euler integrator. I used Noether’s theorem to derive that angular momentum is an invariant of the equations used, which is especially useful since convergent-extension features high local vorticities (which we therefore expect to cancel). The integrator passed this test.

Code was written in the Go programming language. I chose to use this as it is more than one order of magnitude faster than Python when running standard benchmarking tests. Other benefits include automatic memory management (“garbage collection”) which does not exist in other “fast” languages such as C++ and a very minimal design. Notably, using Python code to convert the CSV simulation output from Go into an animation was an order of magnitude slower than actually running the simulation code, vindicating the choice to use Go.

Graphical abstract: a) From Michaut et al. (2024). Is there a link between primitive streak formation and edge finger instabilities in the gastrulating avian embryo? b) From Rozbicki et al. (2015). We see that the biological cells are densely packed. c) To model this, we compute the Delaunay triangulation (black) which is dual to the Voronoi diagram (red). d) Still frame from a simulation where only a cell-cell interaction force is modelled. e) Verifying that the numerical integration conserves momentum (the centre of mass remains stationary). f) Verifying that the numerical integration conserves angular momentum.

Please see the attached document: “Particle-based Simulation of Convergent-Extension Modifying Finger Instabilities in Chicken Embryo Gastrulation” for a detailed account of the work. This will also be made available at https://github.com/harveywilliams56

For source code, please email bothharvey.john.williams2000@gmail.comandharvey.williams562@gmail.com