Free Motile Vertex Model
Vertex models are a very popular choice in describing multicellular systems. They are actively being used in great variety such as to describe mechanical properties of plant cells1 or organoid structures of epithelial cells2,3.
Mathematical Description
In this model, we are only concerned with cellular forces and their representation in space. One single cell-agent can be described by a collection of (ordered) vertices which in turn also allows for a dual description in terms of edges.
$$\begin{align} \{\vec{v}_i\}_{i=0\dots n}\\ \vec{v}_i = \begin{bmatrix}v_{i,0}\\v_{i,1}\end{bmatrix} \end{align}$$
In the following text, we assume that vertices are always ordered (clockwise or anti-clockwise) and this ordering is identical for every cell in our simulation.
Mechanics
Every vertex is connected to its next neighbours in order via springs with an associated length $d$ and spring constant $\gamma$. The potential used to calculate the force $F_i$ acting along the edges of the cell between vertex $i$ and $i+1$ is given by
$$\begin{align} \vec{F}_{\text{edges},i} &= - \gamma \left(|\vec{v}_i - \vec{v}_{i+1}| - d\right) \frac{\vec{v}_i - \vec{v}_{i+1}}{|\vec{v}_i - \vec{v}_{i+1}|}\\ %V_\text{edges} &= \sum\limits_{i=0}^n \frac{\gamma}{2}\left(d_i - d\right)^2 \end{align}$$
where $d_i = |\vec{v}_i - \vec{v}_{i+1}|$ is the distance between individual vertices.
From the length of the individual edges, we can determine the total 2D volume $V$ of the cell when the equilibrium configuration of a perfect hexagon is reached.
$$\begin{equation} V = d^2\sum\limits_{i=0}^{n-1}\frac{1}{2\sin(\pi/n)} \end{equation}$$
However, since the individual vertices are mobile, we require an additional mechanism which simulates a central pressure $P$ depending on the currently measured volume $\tilde{V}$. This area can be calculated by summing over the individual areas of the triangles given by two adjacent vertices and the center point $\vec{c}=\sum_i\vec{v}_i/(n+1)$. They can be calculated by using the parallelogramm formula
$$\begin{align} \tilde{V}_i &= \det\begin{vmatrix} \vec{v}_{i+1} - \vec{c} & \vec{v}_i - \vec{c} \end{vmatrix}\\ &= \det\begin{pmatrix} (\vec{v}_{i+1} - \vec{c})_0 & (\vec{v}_{i} - \vec{c})_0\\ (\vec{v}_{i+1} - \vec{c})_1 & (\vec{v}_{i} - \vec{c})_1 \end{pmatrix}\\ \tilde{V} &= \sum\limits_{i=0}^{n-1}\tilde{V}_i \end{align}$$
The resulting force then points from the center of the cell $\vec{c}$ towards the individual vertices $\vec{v}_i$.
$$\begin{align} \vec{F}_{\text{pressure},i} = P\left(V-\tilde{V}\right)\frac{\vec{v}_i - \vec{c}}{|\vec{v}_i - \vec{c}|} \end{align}$$
These mechanical considerations alone are enough to yield perfect hexagonal configurations for individual cells without any interactions. If we also take into account an external force acting on the cell, the total force acting on the individual vertices $\vec{v}_i$ can be calculated via
$$\begin{equation} \vec{F}_{\text{total},i} = \vec{F}_{\text{external},i} + \vec{F}_{\text{edges},i} + \vec{F}_{\text{pressure},i} \end{equation}$$
Interaction
Cell-agents are interacting via forces $\vec{F}(\vec{p},\vec{q})$ which are dependent on two points $\vec{p}$ and $\vec{q}$ in either cell. The mechanical model we are currently using does not fully capture the essence of these cellular interactions. In principle, we would have to calculate the total force $\vec{F}’$ by integrating over all points either inside the cell or on its boundary but for the sake of simplicity we consider a different approach. Let us denote the vertices of the two cells in question with $\{\vec{v}_i\}$ and $\{\vec{w}_j\}$.
Case 1: Outside Interaction
In this case, we assume that the vertex $\vec{v}_i$ in question is not inside the other cell. We make the simplified assumption that each vertex $\vec{v}_i$ is interacting with the closest point on the outer edge of the other cell. Given these sets of vertices, we calculate for each vertex $\vec{v}_i$ the closest point $$\begin{equation}\vec{p} = (1-q)\vec{w}_j + q\vec{w}_{j+1}\end{equation}$$ (assuming that we set $\vec{w}_{j+1}=\vec{w}_1$ when $j=N_\text{vertices}$) on the edge and then the force acting on this vertex can be calculated $$\begin{equation}\vec{F}_{\text{outside},i} = \vec{F}(\vec{v}_i, \vec{p})\end{equation}$$ by applying $\vec{F}$ on them. The force acting on the other cell acts on the vertices $j$ and $j+1$ with relative strength $1-q$ and $q$ respectively. $$\begin{alignat}{5} &\vec{F}_{\text{outside},j} &=& - &(1-q)&\vec{F}(\vec{v}_i,\vec{p})\\ &\vec{F}_{\text{outside},j+1} &=& &-q&\vec{F}(\vec{v}_i,\vec{p}) \end{alignat}$$
Case 2: Inside Interaction
In the second case, a vertex of the other cell $\vec{w}_j$ has managed to move inside. Here, a different force $\vec{W}$ acts which is responsible for pushing the vertex outwards. The force is calculated between the center of our cell $$\begin{equation}\vec{v}_c = \frac{1}{N_\text{vertices}}\sum\limits_i \vec{v}_i\end{equation}$$ and the external vertex in question. The force which is calculated this way acts in equal parts on all vertices. $$\begin{alignat}{5} &\vec{F}_{\text{inside},j} &=& \frac{1}{N_\text{vertices}}\vec{W}(\vec{v}_c,\vec{w}_j)\\ &\vec{F}_{\text{inside},i} &=& \end{alignat}$$
Parameters
| Parameter | Symbol | Value |
|---|
Initial State
Results & Movie
Code
The code for this simulation and the visualization can be found in the
examples
folder of cellular_raza.
References
[1] R. M. H. Merks, M. Guravage, D. Inzé, and G. T. S. Beemster, “VirtualLeaf: An Open-Source Framework for Cell-Based Modeling of Plant Tissue Growth and Development ,” Plant Physiology, vol. 155, no. 2. Oxford University Press (OUP), pp. 656–666, Feb. 01, 2011. doi: 10.1104/pp.110.167619.
[2] A. G. Fletcher, M. Osterfield, R. E. Baker, and S. Y. Shvartsman, “Vertex Models of Epithelial Morphogenesis,” Biophysical Journal, vol. 106, no. 11. Elsevier BV, pp. 2291–2304, Jun. 2014. doi: 10.1016/j.bpj.2013.11.4498.
[3] D. L. Barton, S. Henkes, C. J. Weijer, and R. Sknepnek, “Active Vertex Model for cell-resolution description of epithelial tissue mechanics,” PLOS Computational Biology, vol. 13, no. 6. Public Library of Science (PLoS), p. e1005569, Jun. 30, 2017. doi: 10.1371/journal.pcbi.1005569.