# Quorum Sensing¶

## Intro¶

**Quorum Sensing** is a well-known systems biology example of a communication mechanism of bacteria, working based on some signaling molecules, developed by Jack J. Dockery and James P. Keener [Dockery2000].

Quorum sensing is an ability expressed by certain bacterial species where a shift in population density leads to changes in gene expression. Once the Quorum Sensing process is activated, the concentration of the signalling molecule is an indicator of the number of cells in the colony. More specifically, *Pseudomonas aeruginosa* relies on a quorum sensing system comprised of two genes, *lasR* and *lasI*. *lasR* codes for a transcriptional activator protein that is activated by *3-oxo-C12-HSL*, an autoinducer synthesized by *lasI*. Their dimer promotes both *lasI* and *lasR* activity and *3-oxo-C12-HSL*
also diffuses between cells. At higher cell densities, the concentration of intracellular *3-oxo-C12-HSL* rises, which leads to changes in *Pseudomonas aeruginosa* associated with population density.

## IBL Modelling¶

Outlined below is the IBL model for the Quorum Sensing system, while the corresponding IBL file can be downloaded here.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 | ```
define QuorumCell typeof CELL()
{
LasR = PROTEIN()
HSL_inside = MOLECULE(amount=5)
PlasR = PROMOTER()
PlasI = PROMOTER()
RULE LasR_production: <-> LasR
LasR_production.forwardRate = 50 min-1
LasR_production.backwardRate = 0.7 min-1
RULE HSL_production: <-> HSL_inside
HSL_production.forwardRate = 50 min-1
HSL_production.backwardRate = 0.02 min-1
RULE PlasR_binding: LasR~HSL_inside + PlasR <-> LasR~HSL_inside~PlasR
PlasR_binding.forwardRate = 10 min-1
PlasR_binding.backwardRate = 10000 min-1
RULE PlasR_expression: LasR~HSL_inside~PlasR -> PlasR + LasR~HSL_inside + LasR
PlasR_expression.forwardRate = 2000 min-1
RULE PlasI_binding: LasR~HSL_inside + PlasI <-> LasR~HSL_inside~PlasI
PlasI_binding.forwardRate = 10 min-1
PlasI_binding.backwardRate = 10000 min-1
RULE PlasI_expression: LasR~HSL_inside~PlasI -> PlasI + LasR~HSL_inside + HSL_inside
PlasI_expression.forwardRate = 2000 min-1
RULE LasR_HSL_dimerization: HSL_inside + LasR <-> LasR~HSL_inside
LasR_HSL_dimerization.forwardRate = 5 min-1
LasR_HSL_dimerization.backwardRate = 5000 min-1
VERIFY [LasR] EVENTUALLY INCREASES WITH PROBABILITY ?
VERIFY [HSL_inside] EVENTUALLY INCREASES WITH PROBABILITY ?
VERIFY [LasR >= 0.1 uM] EVENTUALLY HOLDS WITHIN [0,1] s WITH PROBABILITY ?
VERIFY [HSL_inside >= 3 uM] EVENTUALLY HOLDS WITHIN [0,1] s WITH PROBABILITY ?
VERIFY [[LasR >= 0.005 uM] AND [LasR <= 0.15 uM]] HOLDS IN STEADY-STATE WITH PROBABILITY ?
VERIFY [[HSL_inside >= 0.05 uM] AND [HSL_inside <= 4.2 uM]] HOLDS IN STEADY-STATE WITH PROBABILITY ?
}
define site typeof REGION() {
CELL cell = new QuorumCell()
}
``` |

## Simulation¶

In order to analyse the evolution in the concentration of *LasR* (blue), stochastic simulation has been performed, using the following parameters:

- Max Time: 0.00002 seconds;
- Interval: 0.0000001 seconds;
- Number of runs: 5;
- Gillespie algorithm: Direct Method

## Verification¶

The property below has been verified in order to validate the system behaviour. The verification has been performed using the embedded MC2 model checker, using the following parameters:

- Max Time: 0.00002 seconds;
- Interval: 0.0000001 seconds;
- Number of runs: 1000;
- Gillespie algorithm: Direct Method

- The
LasRconcentration eventually increases, with a probability of 100%;- The
HSLconcentration eventually increases, with a probability of 100%;- The
LasRconcentration will eventually be greater than 0.1 uM within the 1st second, with a probability of 100%;- The
HSLconcentration will eventually be greater than 3 uM within the 1st second, with a probability of 100%;- The
LasRconcentration will continuously be ranged within the [0.005 uM, 0.15 uM] interval, with a probability of 98%;- The
HSLconcentration will continuously be ranged within the [0.05 uM, 4.2 uM] interval, with a probability of 92%.

# | Verification Statement | Result |

1 | VERIFY [LasR] EVENTUALLY INCREASES WITH PROBABILITY ? | 100% |

2 | VERIFY [HSL_inside] EVENTUALLY INCREASES WITH PROBABILITY ? | 100% |

3 | VERIFY [LasR >= 0.1 uM] EVENTUALLY HOLDS WITHIN [0,1] s WITH PROBABILITY ? | 100% |

4 | VERIFY [HSL_inside >= 3 uM] EVENTUALLY HOLDS WITHIN [0,1] s WITH PROBABILITY ? | 100% |

5 | VERIFY [[LasR >= 0.005 uM] AND [LasR <= 0.15 uM]] HOLDS IN STEADY-STATE WITH PROBABILITY ? | 98% |

6 | VERIFY [[HSL_inside >= 0.05 uM] AND [HSL_inside <= 4.2 uM]] HOLDS IN STEADY-STATE WITH PROBABILITY ? | 92% |