Discrete and Continuum Multiscale Behaviour in Bacterial Communication - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 309

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0.8

0.6

0.4

0.2

0

0 10 20 30 40 50

λ

Fig. 2 Steady-state diagram for P when the model is well-mixed (e.g. when D !1 and there is
no spatial variation in the initial condition). Stable steady states are given by solid lines and
unstable by the dashed line. Protein degradation rate is given by k. A range of k exists where the
system is bistable: the cells will either reach an active or inactive state, depending upon the initial
conditions of all variables. Outside this region, either activation (low k) or inactivation (high k)is
guaranteed

between, there exists an interval of k in which the cells can achieve either state,
depending upon the initial conditions. For the following numerical simulations we
consider two values to examine what can happen as you near either extreme of this
region. All other parameters (except D which is varied for each simulation and
stated in the corresponding caption) are taken to be unity and scaled with the small
parameter ¼ 0 : 1 [as written in ( 26 )-( 35 )] where necessary, as per [ 7 ].

4.4 k ¼ 25: Slower Protein Degradation Renders the Achievement
of Quorum Sensing Activity Easier

We begin by considering k ¼ 25. On examination of Fig. 2 , one can see that this
falls to the lower end of the bistable region, meaning that the cells are more likely to
be drawn towards an active state in the well-mixed scenario (this being initial
condition dependent). In Fig. 3 , several solutions are illustrated with varying initial
conditions and diffusion rates. We depict the steady-state value of P across the full
region, i.e. the final level of quorum sensing activity. For each subfigure, the
number of compartments, n, initially selected to be quorum sensing active increases
as we move through the plots (the exact number is given above each graph).
In Fig. 3 a, the rate of signal molecule diffusion is so low that the whole interval
of cells simply retain their initial states: insufficient ''communication'' occurs
between them and they act as single entities. Increasing D in Fig. 3 b (so that the
signal molecules will spread out more readily) yields a different pattern which
enables the visualisation of cell communication: the central cells are able to drag