Discrete and Continuum Multiscale Behaviour in Bacterial Communication - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 313

Fig. 6 Time dependent solutions when k ¼ 40 (so these correspond with the steady state plots in
Fig. 5 ), identical initial conditions for each are used (n ¼ 39 central compartments begin in an
active state) and with various diffusion rates: 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 . As the diffusion rate is increased, the signal molecules
spread out further and, since greater signal accumulation is required than in Fig. 4 because the
protein degradation rate is higher, when this arises the cells are all dragged into an inactive state
signal will diffuse out too quickly and any activity is lost right across the full
interval, see Fig. 5 d (in medical terms, this might represent a successful treatment).
5 Discussion
We have sought in the analysis above to demonstrate some of the explicitly
multiscale behaviour that can arise in even the simplest spatio-temporal models
describing gene-regulatory and signalling processes: we emphasise that capturing
true biological complexity requires much more detailed, and hence involved,
models, but contend that the phenomena which arise in our model problems are
both of mathematical interest in their own right and instructive to gaining insight
into the qualitative properties that are shared by more realistic models arising in
integrative systems biology, but which may be obscured by the complexity of such
models. Given the breadth of the field, we have chosen to focus on two particular
classes of effect that are absent from the PDE models most commonly explored in
classical mathematical biology, namely delay effects and, most extensively, the
implications of the discreteness that is intrinsically associated with populations