Contact Plugin

Relevant Examples:

• Cell Sorting

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The contact plugin implements computations of adhesion energy between neighboring cells.

Together with volume constraint, contact energy is one of the most commonly used plugins. In essence, it describes how cells “stick” to each other.

Contact energy is contrived. It is merely a way to replicate the properties of a cell’s membrane, the bindings of the nano-structures on its surface, and its environment (the Medium).

Two cell types that have high contact energy will not “want to” adhere to each other. If possible, those cells may separate, effectively reducing the total energy to stay in that position. Conversely, low contact energy “encourages” cell types to bind. As contact energy is lowered, it also increases the surface of the contact.

Contact energy is constantly re-calculated each time a cell’s surface changes.

We define contact energy as a matrix that compares each cell type against each other cell type. In CC3DML, try changing the contact energy between CellA and CellB to 2. Keep everything else the same, then click Play to see the result.

<Plugin Name="Contact">
    <Plugin Name="Contact">
    <Energy Type1="CellA" Type2="CellA">10</Energy>
    <Energy Type1="CellB" Type2="CellB">10</Energy>
    <Energy Type1="CellA" Type2="CellB">2</Energy> <!--Here!-->
    <Energy Type1="CellA" Type2="Medium">10</Energy>
    <Energy Type1="CellB" Type2="Medium">10</Energy>
    <Energy Type1="Medium" Type2="Medium">10</Energy> <!--Medium-Medium energy doesn't really matter-->
    <NeighborOrder>2</NeighborOrder>
</Plugin>

This should stimulate cell mixing, the opposite of cell Sorting.

What Does Energy Mean?

When tweaking your parameters, it’s most important to pay attention to their relative values.

However, the actual value of each parameter—whether you set 10 or 1—affects the contact behaviors, too, because of temperature.

Each time a cell flexes its surface, a decision is made about which pixels in the lattice will be copied over to new lattice sites. We discussed this in the Potts page.

The probability of accepting a lattice site copy is:

\begin{eqnarray} P = e^{-\bigtriangleup H/T} \end{eqnarray}

where \(\bigtriangleup H = H_f - H_i\). \(H_f\) is the new energy after a potential lattice site copy. \(H_i\) is the current energy of the system at step i. The new energy, \(H_i\), may be higher, lower, or the same after a site copy. Meanwhile, \(T\) is the temperature (or fluctuation amplitude or membrane fluctuation level or noise level). A high temperature lowers the probability that site copy attempts will be made.

Calculating Contact Energy

The explicit formula for contact energy is:

\begin{eqnarray} E_{contact} = \sum_{i,j,neighbors} J\left ( \tau_{\sigma(i)},\tau_{\sigma(j)} \right )\left ( 1-\delta_{\sigma(i), \sigma(j)} \right ) \end{eqnarray}

where i and j label two neighboring lattice sites \(\sigma\)’s denote cell IDs, and each \(\tau\) denotes a cell type. Note that i and j are are at separate coordinates; for example, i = (0,4) and j = (1,4).

In the above formula, we need to differentiate between cell types and cell IDs. This formula shows that cell types and cell IDs are not the same and the expression in parentheses involving delta function ensures that we do not include contact surfaces of voxels that belong to the same cell. The Contact plugin in the .xml file, defines the energy per unit area of contact between cells of different types (\(J\left ( \tau_{\sigma(i)},\tau_{\sigma(j)} \right )\)) and the interaction range NeighborOrder of the contact:

<Plugin Name="Contact">
    <Energy Type1="Foam" Type2="Foam">3</Energy>
    <Energy Type1="Medium" Type2="Medium">0</Energy>
    <Energy Type1="Medium" Type2="Foam">0</Energy>
    <NeighborOrder>2</NeighborOrder>
</Plugin>

In this case, the interaction range is 2. Thus, only up to second-nearest neighbor pixels of a given pixel undergoing a change will be used to calculate contact energy change. Foam cells have contact energy per unit area of 3 and Foam and Medium as well as Medium and Medium have contact energy of 0 per unit area.

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Suppose we have 8 cells on the checkboard and 4 sides each. If the cells are in a checkerboard pattern, they have the most possible contact energy with the Medium.

So, the total contact energy is:

\begin{eqnarray} H_{contact} = 4 \times 8 \times J_{Cell-to-Medium} \end{eqnarray}

where \(J_{Cell-to-Medium}\) denotes the contact energy between the base cell type, Cell, and the Medium.

How would you calculate the contact energy if all the cells were touching the medium as little as possible?

\begin{eqnarray} H_{contact} = 12 \times J_{Cell-to-Medium} + 10 \times J_{Cell-to-Cell} \end{eqnarray}

Again, count the number of contact surfaces and multiply them by the respective contact energies for those cell types.

Note

CompuCell3D never computes total contact energy. Such calculation would be costly. Instead, CC3D computes a change in Contact energy from pixel copies, and such calculations are local because we are only examining pixels in a local neighborhood of the change pixel.