Biomedical Engineering Reference
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(a)
n=1
n=19
n=31
n=49
1
1
1
1
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0
0
0
0
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
x
x
x
x
(b)
n=1
n=5
n=45
n=49
1
1
1
1
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0
0
0
0
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
x
x
x
x
(c)
n=1
n=11
n=39
n=49
1
1
1
1
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0
0
0
0
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
x
x
x
x
(d)
n=1
n=19
n=31
n=49
1
1
1
1
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0
0
0
0
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
x
x
x
x
Fig. 5 The steady state of P, representing the final quorum sensing activity levels across the
interval, with k
¼
40. Compared with Fig.
3
, this higher value of k renders more signal
accumulation required to achieve upregulation. n central compartments begin in an active state
(red circles), as given by (
37
). 51
n outer compartments have zero initial conditions for all
variables (blue crosses). In a D
¼
1
10
3
, b D
¼
8
10
3
, c D
¼
2
:
2
10
2
and
d D
¼
2
:
3
10
2
. Thus as we move down the plots, the diffusion rate increases and as we
move across, the number of compartments initially starting active increases. Complementary
time-dependent solutions are illustrated in Fig.
6
Contrasting Fig.
5
a and b we see that low diffusion (cutting off communication
with adjacent cells) or sufficiently large n (making the initially active population
larger) is required to retain (localised) upregulation. See also the time-dependent
solutions in Fig.
6
. These solutions are clean-cut reflections of the essence of
quorum sensing: a small population in its own right can be active in the correct
environment, but if this population migrates to a more open environment con-
taining inactive cells, the associated signal dilution could imply that the subcel-
lular mechanisms will force the smaller population to react to its neighbours and
transition to near-identical behaviour.
For values of D within a certain range, we obtain some particularly interesting
behaviour, however, in which the distinction between the central and outer cells can
become much more blurred than in
Sect. 4.4
. In Fig.
5
c it is evident that some of the
central cells can lose their quorum-sensing activity, with those left in the centre
effectively forming a new smaller population of active cells. Increasing the diffusion
rate beyond this range means that (unless all but two compartments begin active) the
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