Biomedical Engineering Reference
In-Depth Information
appropriately defined in relation to the case of interest. In particular, if f
is an increasing (respectively, decreasing) function of the component is
;j
,
where j = 1;:::;m, we can therefore say that s
;j
enhances (respectively,
inhibits) . In some cases it is useful to order the components of s
, to
write it as s
= (s
;A
; s
;I
), where s
;A
2R
k
(respectively, s
;I
2R
mk
,
with k m) consists of the activators (respectively, the inhibitors) of the
biophysical property dened by .
According to the same notation, if
2fJ
int
(
);(
0
)
;J
ext
();(
0
)
g;
for each local interface between neighboring objects (i.e., (@x 2 @
)\(@x
0
2
@
0
) or (@x 2 @) \ (@x
0
2 @
0
)), we have
((@x \@x
0
);t) = (s
(x;t); s
0
(x
0
;t)) = g
(s
(x;t); s
0
(x
0
;t)); (4.10)
where g
:R
m
R
m
7!R. In particular, the local adhesive strengths are
determined by the (local) internal state of both elements, as they are not only
a property of each single individual. Moreover, differentiating the components
of the internal state vector that either enhance or downregulate the relative
contact forces, we have that, if g
depends only on activators then g
=
g
(s
;A
; s
;A
0
) is a decreasing function of its components, while, if vice versa
g
= g
(s
;I
; s
;I
0
), then it is an increasing function.
Indeed, the Potts parameters that can locally vary (such as the adhesive
interactions or the effective strengths of specific forces) require that the rela-
tive functions of the internal state vector s be local (i.e., they need to take into
account the local concentration of the internal factors of interest), whereas the
Potts parameters characterizing an entire individual (such as the motility or
the elasticity) require that the relative functions of s be global (i.e., they need
to take into account the overall level of the internal factor of interest).
Equations (4.9) and (4.10) state that the variation of the Potts coe-
cients of the element
(as usual, either an entire individual or one of its
compartments) is due to the evolution of its internal state: in this way, the
mesoscopic biophysical properties of
are no longer given a priori (or varied
with prescribed rules), but are autonomously and continuously inherited from
the flow of information coming from the microscopic molecular level, which in
turn may be affected, for instance, by exchanges of signals with the external
environment (i.e., as in a typical hybrid approach). They therefore assume a
biologically more realistic and accurate characterization and, in principle, can
be more easily compared with experimentally measurable quantities.
As explained in more detail in Appendix A, procedurally, at every simula-
tion time step t, the microscopic model of each object
is run. The outcome
is then used to modulate the values of the relative Potts coecients, which
in turn rescale the pattern energy H. After the subsequent spin flip, the mi-
croscopic model is rederived, based on the new position of the object. The
Search WWH ::
Custom Search