alsignoriello / active_vertex_model

Simulates vertex model with an additional motility force

Vertex Model

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 Σ_α K_α (A _α - A_α)² + Σ _i,j Λ_i,j l _i,j + 0.5 Σ _α Γ_α P_α²

where A is the area and P is the perimeter.

⃗F_i = - ⃗∇ _iE_i

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).

Parameters

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

Input

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

Notes

• Periodic boundary conditions