Global Volume and Surface Constraints

One of the most commonly used energy terms in the GGH Hamiltonian is a term that restricts variation of single cell volume. Its simplest form can be coded like this:

<Plugin Name="Volume">
    <VolumeEnergyParameters CellType="CellA" LambdaVolume="2.0" TargetVolume="25"/>
</Plugin>

By analogy, we may define a term which will put a similar constraint regarding the surface of the cell:

<Plugin Name="Surface">
    <SurfaceEnergyParameters CellType="CellA" LambdaSurface="1.5" TargetSurface="20.0"/>
</Plugin>

These two plugins inform CompuCell that the Hamiltonian will have two additional terms associated with volume and surface conservation. That is, when a pixel copy is attempted, one cell will increase its volume and another cell will decrease. Thus, the overall energy of the system changes. Volume constraint essentially ensures that cells maintain the volume which close to (this depends on thermal fluctuations) its target volume. The role of the surface plugin is analogous to volume; it “preserves” the total length of the cell’s surface.

The examples shown below apply volume and surface constraints to all cells in the simulation. You can, however, apply volume and surface constraints to individual cell types or individual cells.

Energy terms for volume and surface constraints have the form:

\[\begin{eqnarray} E_{volume} = \lambda_{volume} \left ( V_{cell} - V_{target} \right )^2 \end{eqnarray}\]

\[\begin{eqnarray} E_{surface} = \lambda_{surface} \left ( S_{cell} - S_{target} \right )^2 \end{eqnarray}\]

Note

Copying a single pixel may cause surface change in more than two cells – this is especially true in 3D.

Note

The Volume and Surface plugins handle their respective behaviors automatically as long as they are added to the XML. However, if your particular use case needs to avoid the built-in constraints brought by these plugins, you can still track the volume and surface manually.