Quorum Sensing

Therefore, a middle ground of abiotic degradation must be present.

If we also assume there is AIs degradation through lactonase (an

enzyme actively degrading AIs) we obtained a time course more

similar to the one observed experimentally (solid line).P. putida

QS system has further been studied for continuous cultures [12].

Surprisingly, the mathematical analysis showed constant and similar

values to those reported for batch cultures, indicating the stability

of the system under different environmental conditions.

3 Partial Differential Equations Models of QS

Koeber and coworkers developed a PDE model for the early stages

of an infection caused byPseudomonas aeruginosain burn wounds

[13]. They modeled only the primary QS system inP. aeruginosa,

i.e., AIs are 3-oxo-C12-HSL. Their equations involve the concen-

tration of AIs (3-oxo-C12-HSL) in time and space,A(x,t), the

density of up-regulated cellsNu(x,t) and down-regulated cellsNd

(x,t). They basically used a previously described modeling frame-

work [4] to include space. Population growth is modeled by a

logistic expression. They assume that bacteria are in a zone of

colonization within the wound (seeFig. 7a). Note how their equa-

tion describes changes in space and time through the partial deriv-

ative sign∂, instead of a d (like in previous section), this sign means

160

140

120

100

80

60

40

20

0

0 5 10 15 20 25

Time

Time course of AHL

Moderate abiotic degradation

Increased abiotic degradation

Upregulated lactonase production

AHL

30 35 40 45 50

Fig. 6 A model-predicted time course of AIs, from [6]. Reproduced with

permission

264 Judith Pe ́rez-Vela ́zquez and Burkhard A. Hense