Basic CPM - Cellular Potts Models: Multiscale Extensions and Biological Applications - page 12

FIGURE 1.2: Illustrative simulations of the biological role of the geometrical
constraints. In a 100 100-site domain , an initially round cell of 75 sites of
diameter (i.e., initial area of 4415 sites 2 and initial perimeter of 235.5 sites)
is placed. The cell target measures A surface

C and A perimete C coincide with the
initial dimensions. Top line: If the surface constraint surfac C is low (i.e., 0.2
site 2 ), the cell shrinks. Middle line: If both the surface and the perimeter
elastic moduli are high (i.e., surface

C = perimete C = 50), the cell freezes as
soon as it reaches its target measure. Bottom line: If the cell surface constraint
is high (i.e., surface

C = 50 site 2 ), whereas the perimeter constraint is low (i.e.,
perimete C = 0:2 site 1 ), the cell continuously remodels while preserving its
overall area.

to their target states/values. As already sketched, the most relevant parame-
ters of this family are the geometrical constraint. To provide a simple descrip-
tion of their role, we reproduce the random movement of a single cell = C
placed in a homogeneous medium = M, which may represent the uid com-
ponent of the extracellular matrix; see Figure 1.2. We do not consider other
terms in the Hamiltonian. If surfac C is low (i.e., 1), the cell shrinks, re-
gardless of its initial dimension and of the value of perimeter

C ; Figure 1.2 (top
line). On the contrary, if surfac C is high (i.e., 1), the cell rapidly reaches
its target measures. After this transient, the cell freezes if also perimete C 1,
while it undergoes random fluctuations that, however, preserve its area, if
surface
C

1; Figure 1.2 (middle and bottom lines). It is straightforward to