Biomedical Engineering Reference
In-Depth Information
energy is sucient by a discrete unit to achieve a vanishing value of one of its
constraints, for example, to shrink a cell to a point, a situation that should be
avoided and that would in principle require infinite energy. For this reason, we
prefer to use potentials that blow up in the case of a
i
;
;a
(;
0
);(
;
0
)
! 0.
In particular, among the possible different forms, we define the constraint
contributions to the Hamiltonian as
p
X
1
A
i
;
(t)
a
i
;
(t)
i
;
(t)
H
constraint
(t) =
;
(4.6)
;
iconstraint
p
1
A
(;
0
);(
;
0
)
(t)
a
(;
0
);(
;
0
)
(t)
X
(;
0
);(
;
0
)
(t)
H
constraint
(t) =
;
(;
0
);(
;
0
)
jconstraint
(4.7)
with p 2R
+
. In this way, in addition to the just-stated advantages, all the
components of H
constraint
are nondimensional, and thus all the relative Potts
coecients are coherently scaled to units of energy.
Finally, we characterize the forces acting on each object with
H
force
(t) =
X
x
2
X
k
;
(
x
)
(t) F
k
(t) r
x
(t);
(4.8)
kforce
where
k
;
is the Potts coecient that measures the effective strength of the
force F
k
sensed by the unit
of the individual .
The compartmentalization technique is not entirely new in the CPM: it
was in fact first introduced in [374], where the authors have subdivided a
Myxococcus xanthus into strings of subcellular domains, in order to give the
bacterium a particular geometry and to control its overall length. Another
example is reported in [254], where a keratocyte has been represented with a
set of undifferentiated hexagonal subunits, which have facilitated the repro-
duction of its polarization during motion.
Although these approaches are correct, the fact that the proposed subcel-
lular compartments do not have an immediate or direct correspondence with
real subcellular elements has limited the realism and the usefulness of the
related models. Indeed, although each simulated element can in principle be
compartmentalized in a variety of ways (for example, along symmetry planes,
or in a fixed number of equivalent and undifferentiated subunits), a biologically
plausible compartmentalization (which agrees with the compartmentalization
suggested in nature), is preferable. In this way, the specific subunits, and the
relative attributes, assume, in fact, experimentally relevant meanings and al-
low a detailed description of the microscopic biochemical and biomechanical
mechanisms, which typically are strongly localized within well-defined indi-
vidual subcompartments (as we will comment in the next section).
For example, referring to the ideal compartmentalization of a cell repro-
duced in Figure 4.2, an explicit representation of the plasmamembrane (PM),
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