Biomedical Engineering Reference
In-Depth Information
(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=3
n=47
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=3
n=47
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. 3 The steady state of P, representing the final quorum sensing activity levels across the
interval, with k
¼
25. Higher values of P
2½
0
;
1
represent compartments of cells which have
been induced into an active state by the quorum sensing machinery and signal accumulation. 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
4
10
5
,
b D
¼
8
:
12
10
5
, c D
¼
8
:
2
10
5
and d D
¼
8
:
4
10
4
. 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.
4
the surrounding cells into an up-regulated state. Interestingly, for 3
n
45, all
compartments except two at either extremity become active, with the remaining
four receiving insufficient signal for them to become active: the very outer com-
partments receive signal from only one direction; that these are therefore not up-
regulated has a knock-on effect on the adjacent cells. The population is effectively
divided into active and inactive populations at steady state because the subcellular
processes are unable to even out sufficiently across the interval for the entire
population to upregulate (i.e. to achieve quorum).
Successively increasing D first leads to only the very outer compartments
remaining down-regulated and then to all the cells becoming quorum-sensing
active, see Fig.
3
c and d. Thus while the machinery is the same at the subcellular
level, the phenotype of particular cells can be strikingly different, the effect being
associated with the well-known phenomenon of wave pinning (failure of signal
propagation) that occurs in such discrete systems (e.g. [
15
,
18
]) but not their
corresponding continuous limits.
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