Biomedical Engineering Reference
In-Depth Information
t
¼
t
c
þ
1
1
t
2
ln t
T
z
;
u
i
¼
t
2
ln tU
i
t
2
T
c
þ
giving
U
i
1
1
dU
z
i
64
T
z
þ
ð
i
i
c
Þ
2
2
dT
z
U
i
;
ð
25
Þ
16
the diffusion term being logarithmically negligible; the extent of the analytical
progress that is possible with this problem is thus striking.
The conclusions of this t
!þ1
analysis are of limited practical significance,
but are instructive in terms of the general issues associated with discrete to con-
tinuous limits. Firstly, the blow up time in (
25
) T
z
¼
T
c
depends on i
c
with
T
c
2
64
;
, the lower bound arising when the blow up point of the continuous
1
32
limit coincides with a cell location (i.e. with integer i) and the upper bound when it
is 'half way between' cells (integer i
þ
1
=
2), i.e. (and not surprisingly) upregu-
lation is fastest when the peak of the initial profile (when specified throughout R)
can be regarded as coinciding with a cell. Secondly, the signal molecule distri-
bution at the blow up time is given as t
!þ1
with i
i
c
¼
O
ð
1
Þ
by
16
ðð
i
i
c
Þ
2
min
ð
i
i
c
Þ
2
Þ
u
i
t
2
ln t
for integer i; this represents single point blow up, but for the two neighbouring
cells there are additional contributions to U
i
of the form
ln
ð
T
c
T
z
Þ=
ln t, the
associated further non-uniformity manifesting itself in the three-point blow up
described above.
We now turn to a more complicated differential-difference formulation that was
developed (see [
9
]) specifically to investigate quorum-sensing behaviour in S.
aureus, again focusing our discussion of phenomena that are explicitly associated
with discreteness and are multiscale in the sense that they reflect both subcellular
genetic regulation and population-level transport of signalling molecules.
4 A Quorum-Sensing Case-Study: The agr Operon
4.1 Background
We now move away from the generic and highly simplified representations of
processes such as quorum sensing given in the previous sections and investigate a
specific quorum sensing system using distinct equations to represent each relevant
component. The result is a spatially structured model of a population of bacteria
utilising an agr operon. This quorum-sensing system is found in a number of
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