The vertex model uses polygons, composed of vertices and edges, to illustrate the mechanics in formation of polygon sheets. This model has successfully been applied to fly wings and eyes in 2D.
Simulations compute the energy and forces in the system using the following equations:
E = 0.5 Σα Kα (A α - Aα)2 + Σ i,j Λi,j l i,j + 0.5 Σ α Γα Pα2
where A is the area and P is the perimeter.
⃗Fi = - ⃗∇ iEi
The terms represent cell elasticity, line tension and contraction, respectively.
Farhadifar, R., Röper, J.-C., Aigouy, B., Eaton, S. & Jülicher, F. The Influence of Cell Mechanics, Cell-Cell Interactions, and Proliferation on Epithelial Packing. Current Biology 17, 2095–2104 (2007).
| Parameter | Definition | Range |
|---|---|---|
| lx | length of box x-axis | positive float |
| ly | length of box y-axis | positive float |
| kA | elasticity | |
| Λ | line tension | |
| Γ | contraction | |
| lmin | minimum bond length | positive float |
| Δ t | time step | small positive float |
vertices.txt (x,y) coordinates for every vertex in network
edges.txt (index1, index2) indices for every edge between two vertices in the network
cells.txt (index0, index1, ... indexN) indices in counter-clockwise order that form every cell in the network, a cell is defined as a polygon, does not assume number of sides
- Periodic boundary conditions