Biomedical Engineering Reference
In-Depth Information
ological cell sorting consists of a cellular aggregate formed by two types of
randomly positioned individuals, namely, light = L and dark = D, and
placed in the standard extracellular medium = M; see again [165, 169].
The target measures of all cells are the same as in the previous simula-
tions, as well as their geometrical constraints (i.e., we use high values of
surface
L
L
=
perimete
D
). If the ad-
hesive energy between the two cell types is higher than the two self{contact
interactions (i.e., J
L;D
< J
L;L
= J
D;D
), cells heterogeneously mix to form an
experimentally observed checkerboard (Figure 1.4 (left panel)). Again, this is
due to the fact that making heterotypic bonds is more convenient from the
energetic point of view than making homotypic bonds with cells of the same
type.
If the homotypic adhesions are stronger than the heterotypic ones (i.e.,
J
L;D
> J
L;L
= J
D;D
), we find a spontaneous cell sorting, with the formation of
small clusters of cells of the same type within the spheroid; Figure 1.4 (middle
panel). In particular, if further, the edge adhesion between a cell type and the
medium is significantly low (i.e., J
L;D
> J
L;L
= J
D;D
and J
L;M
< J
D;M
), a
single cluster of this cell type forms in the center of the spheroid, surrounded
by a crew of individuals of the other type (Figure 1.4 (bottom right panel)):
this phenomenon is called engulfment and was first investigated in a CPM in
[169]. This is due to the fact that after clustering, the cell type that is able to
build stronger bonds with the medium (i.e., lower energetic costs) will tend
to stay in the exterior layers of the aggregate.
As previously seen, the chemical responses are the most important and
commonly used forces described by the term (1.8) of the Hamiltonian. To
provide a useful analysis of how it works, we simulate the chemotactic move-
ment of a single cell, = C, inuenced by the chemical concentration gradients
created in the extracellular medium, = M, by a punctual source; see Figure
1.5. The chemical substance evolves as the following:
=
surface
D
and low values for
perimeter
@c
@t
=
D
c
r
2
c
c
c + S
;
(1.15)
|{z}
source
| {z }
diffusion
where, as seen, c = c(x;t) denotes the actual concentration of the peptide at
the medium site x. The coecients of diffusivity, D
c
, and of degradation,
c
,
are assumed to be constant throughout the domain. S describes the production
of the chemical at a constant rate per unit of time by the discrete source.
If the cell chemical strength
che
C
is positive, the cell undergoes a gradual
transition from the initial symmetric stationary state to a polarized migratory
state characterized by clearly distinguishable leading and trailing edges, and
moves in the direction of the source up the concentration gradients. On the
contrary, if
che
C
is negative, the cell again somehow elongates, but starts to
move in the opposite direction, down the concentration gradients. Finally, if
chem
C
= 0, the cell remains more or less hemispheric and starts a slow random
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