Biomedical Engineering Reference
In-Depth Information
Table 2 Interpretations of the nondimensional parameters
Nondimensional parameter Interpretation
a AgrB becoming transmembrane
b AIP binding to receptors
g Receptor loss through AIP binding
c Spontaneous separation of AIP and receptors
k Natural protein degradation
k
a
Natural AIP degradation
l Housekeeping dephosphorylation of AgrA
/ Activation of AgrA
D Diffusion coefficient of AIP
k AIP production
k
S
AgrD loss through AIP production
u Unbinding of active AgrA from the DNA binding site
v Ratio of activated agr transcription to basal transcription
Ratio of basal to QS-induced transcription
other cells into the same state. To represent an already active cell, we use the
following (dimensionless) initial conditions:
M
i
ð
x
;
0
Þ¼
B
i
ð
x
;
0
Þ¼
S
i
ð
x
;
0
Þ¼
T
i
ð
x
;
0
Þ¼
10
;
A
i
ð
x
;
0
Þ¼
R
i
ð
x
;
0
Þ¼
A
Pi
ð
x
;
0
Þ¼
5
;
a
i
ð
x
;
0
Þ¼
100
;
ð
37
Þ
R
i
ð
x
;
0
Þ¼
30
;
P
i
ð
x
;
0
Þ¼
0
:
9
;
where x
is the region(s) containing these cells. These initial conditions loosely
characterise the up-regulated steady state of a cell in the spatially homogeneous
case (this varies with diffusion rate). For inactive cells, we simply assume zero
initial conditions for all variables in the relevant compartment(s).
4.3 Numerical Solutions
The discrete system is solved in Matlab v7.14 using the ODE15s solver. In the
numerical solutions displayed, the interval is broken into p
¼
50 compartments
(chosen for ease of illustration). To verify that the results are qualitatively re-
produceable in the continuum limit, the equations have also been solved with
larger numbers of discretisations.
To investigate multi-stability, we first consider the well-mixed non-spatial
version of (
26
)-(
35
). We find that a key parameter, k, can generate bistability in
the system, see Fig.
2
. k captures the degradation rate of all intracellular proteins
in the agr system. Low values of this parameter result in the preservation of the
agr machinery to the extent that the response regulator (A
P
) will always build-up,
reach a critical level and trigger activation (P
!
1). Increasing k renders this less
and less likely to occur, eventually resulting in guaranteed downregulation. In
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