Global Volume and Surface Constraints

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

<Plugin Name="Volume">

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

<Plugin Name="Surface">

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

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:

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


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