Biomedical Engineering Reference
In-Depth Information
where, as seen, x and x
0
are two neighboring sites and
and
0
two neigh-
boring objects. The coecients J
(
);(
0
)
2Rare the binding forces per
unit area, the first type of the so-called Potts parameters, and are obviously
symmetric w.r.t. the indices. In basic CPMs, they only depend on the type
of discrete objects that are in contact (i.e., (
) and (
0
) in Equation
(1.5)), as they are not a characteristic of each single unit; see, for example,
[261, 268, 334, 342]. Moreover, they are uniformly distributed over the whole
surface of the discrete objects, neglecting microscopic inhomogenities, such as
a clusterization or a different strength of adhesion in well-localized parts of
their membranes. This issue will be discussed in more detail in Section 4.3, as
the model will be extended in this respect.
In the case of cells, such contact strengths give a qualitative measure of
the expression of adhesion molecules in the individuals on either side of the
common border, whose activity defines their binding properties. At least two
classes of J
0
s can be identified: those relative to the adhesion between cells and
extracellular material (and thus modeling the activity of cell{matrix adhesion
molecules, such as integrins), and those that mediate the adhesion between
cells of either the same or of different populations (and thus related to the ex-
pression of cell{cell adhesion molecules, such as cadherins). A surface contact
force can also be defined with an external undifferentiated medium (for exam-
ple, culture medium, air, generic substrate), but it is biologically meaningless
and therefore negligible.
The term H
constraint
, whose use also comes from the physics of classical
mechanics, sums the energetic components that describe the object attributes.
They are written as energetic penalties that increase as the objects deviate
from a designed state. In a characteristically elastic form, it is classically
written as
H
constraint
(t) =
X
X
i
(t)
a
i
(t) A
i
(t)
2
;
(1.6)
iconstraint
where a
i
(t) is the actual value of the i-attribute of individual
, and A
i
(t)
is its target value that usually characterizes an object type and that can
also vary in time. The Potts parameters
i
2R
+
take the role of elastic
moduli, which determine the weight of the relative energetic constraint, and
thus the importance of the relative attribute. Low values of
i
, in fact, allow
the discrete unit
to deviate more from the configuration that satisfies
the constraint. Indeed, since the energetic contributions given in Equation
(1.6) smoothly decrease to a minimum when the attributes are satisfied, the
modified Metropolis algorithm automatically drives any configuration toward
one that satisfies the constraints. Obviously, the simulated system is not able
to exactly satisfy all the constraints of all the objects
at any given time t,
since multiple attributes may be in conflict: this leads to lattice configurations
characterized by local energetic minima.
Among others, the energetic components relative to geometrical attributes
of discrete objects, such as their volume and surface, are of particular rele-
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