Discrete and Continuum Multiscale Behaviour in Bacterial Communication - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 307

Table 1 Variables
representing elements of the
agr operon. In addition, P
represents agr activity levels
(see text)

Variable agr element

M mRNA

A AgrA (response regulator)

B AgrB (processes the signal molecule)

S Transmembrane AgrD (signal precursor)

T Transmembrane AgrB

a AIP (signal molecule)

R Transmembrane AgrC (receptor)

R Phosphorylated AgrC (AIP-bound)

A P Phosphorylated AgrA (activated)

oR i

os ¼ R i a i ð k þ c Þ R i ;

ð 33 Þ

o A Pi

os ¼ A i R i ð k þ l Þ A Pi ;

ð 34 Þ

o P i
os ¼ A Pi ð 1 P i Þ uP i ;

ð 35 Þ

in 0\x\1, where i ¼ 0 ; ... ; p. All variables except P i represent subcellular
processes in a population of cells on a one-dimensional interval with the region
divided into p equally spaced (of dimensionless size h ¼ 1 = p) compartments; we
treat each compartment as containing a subpopulation of identically behaving
cells. In the current context it is preferable to view i as labelling distinct sub-
populations rather than individual cells; since it is not our intention here to explore
the detailed biological implications of the model results, the distinction is in any
case somewhat moot. In ( 31 ), D = h 2 captures the role played by t 2 in Sect. 3 ,
governing the rate of transport between compartments. P i ð x ; s Þ defines the quo-
rum-sensing activity level of the population of cells in compartment i (P 1is
quorum-sensing inactive, while P 1 is fully active); see Table 1 for definitions of
the other variables and Table 2 for the rates represented by the nondimensional
parameters.
Focusing on signal accumulation, we assume there is no flux across the
boundaries (i.e. signal molecules cannot be lost to, or gained from, the surrounding
environment) yielding the conditions:

a 1 ¼ a 1 and a p þ 1 ¼ a p 1 ð 36 Þ

(see [ 9 ]). For each simulation we vary the initial conditions of the system, with
different sections of cells beginning in an inactive (non-quorum-sensing) state, and
others in an active (quorum-sensing) state. Hence we monitor diffusion of the
signal molecules across the region, envision the resulting cell communication and
investigate the ability of these cells either to retain their initial state or to pull the