In our mathematical model we developed a system to characterize each of the two (las and rhl) plasmids. Simple detection is fairly straightforward. The engineered E. coli cells will be saturated with R-proteins (LasR and RhlR) due to constitutive production. In the presence of PAI-1 and PAI-2, the R-proteins and the autoinducers will dimerize which results in the induction of the induced promoter. Upon induction the induced promoters will express the reporter genes.

On the other hand, quorum sensing is somewhat unusual. The las system is independent of all other cell signaling systems; however, the rhl system is not. PAI-1 can competitively bind to the RhlR/PAI-2 induced promoter, deactivating the induced promoter. Therefore, at quorum, the rhl system is regulated, while simultaneously induced owing to the presence of both PAI-1 and PAI-2. Our modeling approach will express the concentration of the relevant molecules as a system of first-order, nonlinear, ordinary differential equations using the variable specified in the table below.

The first system of differential equations is centered on the las plasmid. This entails the production of LasR from the plasmid, diffusion of the autoinducer into the cell, and finally, the transcriptional activation and fluorescent reporting. A graphical representation of the biochemical system can be found below in Figure 1.

Our first assumption is that LasR/PAI-1 dimer (D_L) is produced at a rate r1 and degrades at rate r2. Additionally, D_L can act as a transcriptional factor and bind to the induced promoter (LasP), which induces the expression of the reporter at the rate r6,

LasR is produced by the translation of the LasR mRNA (LRmRNA) at a rate r5 and degrades at the rate r10. Moreover, LasR can forward dimerize at the rate r1 and reverse at rate r2. LRmRNA is transcribed at the rate r3 by the constitutive promoter (CP) and degrades at the rate r4,

Upon the binding of DL to LasP at the rate r6, GFP mRNA (GmRNA) is transcribed. GmRNA degrades at the rate r9 and is translated to GFP at the rate r7. GFP degrades at the rate r8,

The autoinducer PAI-1 diffuses passively into the cell as a result of the concentration gradient, cell volume, surface area and membrane thickness which establish the equation mass transfer1. The intracellular (A1i) and extracellular (A1e) PAI-1 degrade at the rate r11 and r12,

The Rhl Plasmid System

Conversely, the second system of differential equations focuses on the rhl plasmid. This plasmid constitutively expresses RhlR. Similar to the Las system, a fluorescent reporter protein is expressed upon the induction of the (RhlR/PAI-2) induced promoter. Moreover, these equations also account for the passive diffusion of the autoinducer PAI-2 into and out of cell to facilitate the transcriptional activation of the reporter genes. A graphical representation of the biochemical system can be found below in Figure 2.

Figure 2: The Rhl plasmid system

The rhl characterization is similar to that of the las system. Mathematically speaking, when both las and rhl plasmids are characterized independently of the other, they are quite similar. Quorum signaling only enters the model when both las and rhl plasmids are characterized simultaneously. In the rhl plasmid system, the rate at which the RhlR/PAI-2 dimer (D_R) is formed is assumed as r13 and degraded at rate r14. Moreover, the protein-autoinducer dimer D_R can activate the inducible RhlR/PAI-2 promoter (RhlP) to facilitate the expression of the reporter at the rate r19,

Free RR molecules can dimerize to form DR at the rate r13 and degrade back to RR at the rate r14. The translation of RhlR mRNA (RRmRNA) leads to the production of the protein RhlR (RR) at the rate r18. RR degrades at the rate r23. The rate at which RRmRNA is transcribed by the constitutive promoter (CP) is r16, whereas the rate of RRmRNA degeneration is r17.

RFP mRNA (RmRNA) is transcribed at the rate r19 when RhlP is induced by DR. RmRNA either degrades at the rate r22 or is translated to RFP at the rate r20. Additionally, RFP undergoes degradation at the rate r21,

Extracellular and intracellular autoinducers passively diffuse across the cell membrane at a rate influenced by the concentration gradient, cell volume, surface area and membrane thickness which establish the equation mass transfer1. The intracellular concentration of PAI-2 increases when DR breaks down, and decreases when DR forms. The intracellular (A2i) and extracellular (A2e) PAI-2 degrade at the rate r24 and r25,