Biomedical Engineering Reference
In-Depth Information
H
ext
adhesion
is formed instead by the effective adhesion energies between dif-
ferent compartmentalized individuals that interact with their external mem-
branes:
X
H
ext
J
ext
adhesion
(t) =
();(
0
)
(t):
(4.3)
x
;
x
0
2
0
x
:
(@
x
2@)\(@
x
0
2@
0
)6=;
The strengths J
ext
();(
0
)
2R
+
depend, as a simple extension of the basic
CPM, on the types of the respective interacting clusters.
To further increase the realism of the model, the bond energies J
ext
s may
be spatially inhomogeneous throughout the object membranes and may specif-
ically depend on the single units in contact, and not only on their types, i.e.,
J
ext
();(
0
)
= J
ext
();(
0
)
(@x \@x
0
;t):
These issues will be addressed in the following section. It is noteworthy that, if
the objects in contact represent standard noncompartmentalized individuals,
the relative energetic contributions are in the classical form of Equation (1.5).
The compartmentalized approach requires then to define the attributes,
and the relative energetic contributions, of all the units and those that regulate
their mutual interactions. A straightforward generalization of Equations (1.6)
and (1.7) results in
X
i
;
(t)
a
i
;
(t) A
i
;
(t)
2
;
H
constraint
(t) =
(4.4)
;
iconstraint
H
constraint
(t) =
(;
0
);(
;
0
)
(t)
h
a
(;
0
);(
;
0
)
(t) A
(;
0
);(;
0
)
(t)
i
2
X
=
;
(;
0
);(
;
0
)
jconstraint
(4.5)
where a
i
;
and a
(;
0
);(;
0
)
are the actual values of the attributes, A
i
;
and A
j
(;
0
);(
;
0
)
are their target values, and
i
;
and
j
(;
0
);(;
0
)
2R
+
are the relative Potts parameters .
The
form
of
(4.4)
and
(4.5)
is
the
simplest
quadratic
potential
quantities
a
i
;
(t) A
i
;
(t)
2
one
can
think
of.
However,
the
and
h
a
(;
0
);(
;
0
)
(t) A
(;
0
);(;
0
)
(t)
i
2
can be easily replaced by more sophis-
ticated potentials U(aiη,Σσ
i
;
;A
i
;
) and U(a
(;
0
);(
;
0
)
;A
(;
0
);(
;
0
)
) with
the property
U(A
i
;
;A
i
;
) = U(A
(;
0
);(
;
0
)
;A
(;
0
);(
;
0
)
) = 0:
In particular, the form of (4.4) and (4.5) has the disadvantage that a finite
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