Overview

To further understand, predict and control the behavior of our engineered microbial quorum sensing, we developed the model of quorum sensing by researching the entire process of CrpP enzyme production and degradation. Quorum sensing involves complex biological and physical processes, such as the diffusion of signal molecules and the binding process of complexes to promoters. We rationally simplified and abstracted the quorum sensing process and established a differential equation system to obtain the connection between the maximal production of CrpP enzyme and the concentration of external AHL with a certain number of cells.

By solving the model, we found optimal external AHL concentration (after reaching this concentration, as the concentration of AHL continues to increase, the amount of CrpP enzyme production varies scarcely)in the case of different cell numbers. In the future, based on this data, the experiment will be used to find the optimal ratio of bacteria piGEM2019-01 to piGEM2019-02 placed in the device. The model is instructive for the experiment, allowing us to obtain greater benefits for the project with relatively less cost. According to the function of different cells, we refer to piGEM2019-01 as the detection cell, and piGEM2019-02 as the processing cell below.

Goals

1. Find the connection between the maximal production of CrpP enzyme and the concentration of external AHL at a certain cell number.

2. Find the optimal external AHL concentration at different processing cell numbers as a basis for finding the optimal ratio of detection cell and processing cell placed in the device.

2. Find the optimal external AHL concentration at different processing cell numbers as a basis for finding the optimal ratio of detection cell and processing cell placed in the device.

Observations

For piGEM2019-01, SOS response happens caused by ciprofloxacin stimulation and then P

_{tisAB}promoter responds. AHL can be found outside the cells because of the diffusion through cell membrane.For piGEM2019-02, AHL molecules enter the cell and combine with LuxR chemically to form a dimer. Binding to the promoter P_{LuxR}, the complex can activate the promoter that enable downstream CrpP genes to produce enzymes which degrades ciprofloxacin in water.Model Assumptions

1. Assume that the volume of the system is constant

2.Assume that the environmental factors of the system do not vary with time(eg.PH, temperature)

3. Assume that cells are evenly distributed in the system

2.Assume that the environmental factors of the system do not vary with time(eg.PH, temperature)

3. Assume that cells are evenly distributed in the system

Symbol and variable description

Variable | Description |
---|---|

A | Concentration of AHL in a processing cell |

A_{ex} |
Concentration of external AHL |

A' | Concentration of AHL in a detection cell |

R | Concentration of LuxR in a processing cell |

C | Concentration of AHL-LuxR in a processing cell |

ρ_{1} |
Number of detection cells |

ρ_{2} |
Number of processing cells |

P | Concentration of CIP |

C_{CrpP} |
Concentration of CrpP |

Model establishment

## Detection cells

The differential equation for the changes of AHL concentration in detection cells:

\begin{equation} \frac{dA'}{dt}=A'_{0}+f(P)-n'(A'-A_{ex})- \gamma _{1}A' \end{equation}

\begin{equation} \frac{dA'}{dt}=A'_{0}+f(P)-n'(A'-A_{ex})- \gamma _{1}A' \end{equation}

A' is intracellular AHL concentration of detection cell and A'

_{0}is the basal expression of the P_{tisAB}promoter. P is the concentration of ciprofloxacin, and f(P) is the rate of AHL production affected by external CIP. (f(P) indicates the effect of SOS response induced by CIP on the promoter of P_{tisAB}, and the degree of influence is related to CIP concentration.) The kinetic constant n' is internal transport constant and γ_{1}is the degradation rate of internal AHL. A_{ex}is the concentration of extracellular AHL.
The meaning of the equation: The rate of AHL change in the processing cells is determined by the rate of basal production, the effect of CIP concentration, the rate of AHL released into the environment, and the rate of self-degradation of internal AHL.

## External AHL concentration

(1)Considering detection cells only, the rate of external AHL concentration change:
\[
\frac{dA_{ex}}{dt}=\rho_{1}n'(A'-A_{ex})-\gamma_{2}A_{ex}
\]

ρ

_{1}is the number of detection cells, γ_{2}is the degradation rate of external AHL.
The meaning of the equation: in this case, the rate of the concentration of external AHL change depends on the rate at which AHL is emitted to the outside by the detection cells and its own degradation rate externally.

(2)Considering processing cells and detection cells, the rate of external AHL concentration change:
\[ \frac{dA_{ex}}{dt}=\rho_{1}n'(A'-A_{ex})-\rho_{2}n(A_{ex}-A)-\gamma_{2}A_{ex}
\]

ρ

_{2}is the number of processing cells, and the kinetic constant n is external transport constant.
The meaning of the equation: In this case, the rate of external AHL concentration change depends on the rate at which AHL is emitted to the outside by the detection cells, the rate at which AHL enters the processing cells and its own degradation rate externally.

(3)Considering processing cells only, we gave the extracellular AHL concentration. The rate of external AHL concentration change:
\[
\frac{dA_{ex}}{dt}=-\rho_{2}n(A_{ex}-A)-\gamma_{2}A_{ex}
\]

The meaning of the equation: In this case, the rate of external AHL concentration change depends on the rate at which AHL enters the processing cells and its own degradation rate externally.

## Processing cells

(1)Process of AHL-LuxR complex production:
\[
A+R \rightleftharpoons C
\]

It is now generally accepted that AHL binds directly to the LuxR protein to form a complex, and this is represented in the model by the interaction between AHL(A) and LuxR(R) form an active complex(C). The dynamics of this reaction is described by the binding rate constant k

_{1}and dissociation rate k_{2}[1].
The rate of complex production reaction:
\[
v_{complexation}=k_{1}AR
\]
Thus, the binding rate is proportional to the product of the concentration of A and R.

The rate of complex dissociation reaction: \[ v_{dissociation}=k_{2}C \] The rate of the dissociation reaction is proportional to the concentration of C only.

The rate of complex dissociation reaction: \[ v_{dissociation}=k_{2}C \] The rate of the dissociation reaction is proportional to the concentration of C only.

(2)The rate of internal AHL concentration change in processing cells:
\[
\frac{dA}{dt}=k_{2}C-k_{1}AR+n(A_{ex}-A)-\gamma_{1}A
\]

The meaning of the equation: the rate of internal AHL concentration change in processing cells depends on the rate at which the complex decomposes to produce AHL, the rate at which complex synthesis consumes AHL, the rate at which AHL enters the processing cells and its own degradation rate internally.

(3)The rate of internal LuxR concentration change in processing cells:
\[
\frac{dR}{dt}=k_{2}C-k_{1}AR-bR+R_{0}
\]

The meaning of the equation: the rate of internal LuxR concentration change depends on the rate at which the complex is decomposed to produce LuxR, the rate at which the complex synthesis consumes LuxR, the rate of regular expression of LuxR in the processing cells and its own degradation rate internally.

(4)The rate of internal AHL-LuxR concentration change in processing cells:
\[
\frac{dC}{dt}=k_{1}AR-k_{2}C
\]

The meaning of the equation: the rate of internal AHL-LuxR concentration change depends on the rate at which AHL and LuxR bind together and the rate at which the complex decomposes.

(5)The rate of CrpP production in processing cells:
\[
\frac{dC_{CrpP}}{dt}=C_{0}+a\frac{C^{\beta}}{k_{m}^{\beta}+C^{\beta}}-\gamma_{d}C_{CrpP}
\]

C

_{CrpP}is the concentration of CrpP. C_{0}is the basal expression of the promoter, and coefficient a is the dynamic rang of the promoter[2]. The regulatory effect of the promoter is modelled as a Hill-like function, whereby β is the Hill coefficient and k_{m}is binding affinity of complex to LuxR promoter. The kinetic constant γ_{d}is the degradation rate of CrpP.
The meaning of the equation: the rate of CrpP concentration change in processing cells depends on the constant expression rate of the promoter, the effect of the complex and the rate of degradation of CrpP internally.

the Values of Parameter

Description | Name | Value | Units | Ref |
---|---|---|---|---|

Rate constant of complex formation | k_{1} |
0.1 | nM^{-1}∙min^{-1} |
[1] |

Rate constant of complex dissociation | k_{2} |
10 | min^{-1} |
[3] |

External transport constant of AHL | n | 0.3 | min^{-1} |
[2] |

Internal transport constant of AHL | n' | 0.006 | min^{-1} |
[2] |

Rate constant of degradation of LuxR | b | 0.02 | min^{-1} |
Team:Tsinghua2018 |

Efficiency of the constant promoter | R_{0} |
1 | nM∙min^{-1} |
Estimated by ourselves |

Rate constant of AHL degradation outside the cell | γ_{2} |
0.0004 | min^{-1} |
Estimated by ourselves |

Rate constant of AHL degradation inside the cell | γ_{1} |
2.82×10^{-3} |
min^{-1} |
[2] |

Rate constant of CrpP formation | C_{0} |
0.02 | nM∙min^{-1} |
Estimated by ourselves |

Enhanced expression efficiency of P_{LuxR} when bound to the complex |
a | 5 | nM∙min^{-1} |
Team:Tsinghua2018 |

Binding affinity of the complex to P_{LuxR} promoter |
k_{m} |
10 | nM | Team:Tsinghua2018 |

Rate constant of degradation of CrpP | k_{d} |
0.005 | min^{-1} |
Estimated by ourselves |

The Hill coefﬁcient | β | 1.5 | [2] |

Model Solving

Since SOS response in detection cells is complicated and it is difficult to establish quantitative equation, we considered the processing cells only and set the initial concentration of external AHL to get the relationship between time and the yield of each substance in the processing cells. Later, through experiments we are ready to find the relationship between the number of detection cells and the amount of AHL produced and then find the optimal ratio of detection cells and processing cells, obtaining the optimum degradation rate of CIP with low experimental cost.

If we give the initial concentration of AHL outside the cells and the number of processing cells:

\[
A_{ex0}=10^{-6}mol/L
\]
\[
\rho_{2}=1\times10^{7}
\]

Thus we got a set of simultaneous equations related to the processing cells and the initial values (except for the two initial values given above, the initial values of the other variables are all zero) are given.

\[
\frac{dA_{ex}}{dt}=-\rho_{2}n(A_{ex}-A)-\gamma_{2}A_{ex}
\]
\[
\frac{dA}{dt}=k_{2}C-k_{1}AR+n(A_{ex}-A)-\gamma_{1}A
\]
\[
\frac{dR}{dt}=k_{2}C-k_{1}AR-bR+R_{0}
\]
\[
\frac{dC}{dt}=k_{1}AR-k_{2}C
\]
\[
\frac{dC_{CrpP}}{dt}=C_{0}+a\frac{C^{\beta}}{k_{m}^{\beta}+C^{\beta}}-\gamma_{d}C_{CrpP}
\]

Running the model with relevant parameters set on MATLAB2018a can provide us with diagrams showing the result of simulation, using ode15s function. The results are as follows:

The curve of internal AHL concentration changing with time in processing cells:

Fig. 1. The curve of internal AHL concentration changing with time in processing cells

It can be seen from the Fig. 1 that the external AHL enters the processing cells in a short time and the rate of internal AHL concentration increase is less than the rate of its decrease. So there is a tendency that the concentration of AHL inside the processing cells gradually decreases with time.

The curve of internal LuxR concentration changing with time in processing cells:

Fig. 2. The curve of internal LuxR concentration changing with time in processing cells

It can be seen from the Fig. 2 that the rate of LuxR production in the cells is greater than the total rate of reduction before 245 min. After 245 min the rate of LuxR production is almost equal to the rate of complex formation and its own degradation rate. So the concentration of LuxR is almost stable.

The curve of internal AHL-LuxR concentration changing with time in processing cells:

Fig. 3. The curve of internal AHL-LuxR concentration changing with time in processing cells

It can be seen from the Fig. 3 that as AHL enters the cell and the LuxR concentration increases, the concentration of the synthesized complex gradually increases. AHL-LuxR concentration reaches a peak at 88 min. Then due to the decrease of AHL concentration in the cell, the concentration of the complex also decreases.

The curve of internal CrpP concentration changing with time in processing cells:

Fig. 4. The curve of internal CrpP concentration changing with time in processing cells

It can be seen from the Fig. 4 that the concentration of CrpP increases rapidly, because the concentration of the complex increases with time, promoting the expression of the promoter. The concentration of CrpP reaches a peak at 308 min and then begins to decrease due to the decreased complex concentration. The rate of CrpP production is less than its degradation rate. So the concentration of CrpP is gradually reduced.

Through our calculations, it is found that if we give the external AHL concentration of different initial values, there is a maximal CrpP concentration correspondingly. The more external initial AHL concentration is, the more corresponding maximal CrpP concentration is. However, since LuxR is controlled by a normal promoter and its production rate is constant, we guess that the corresponding maximal CrpP production concentration will not change significantly when the initial concentration of external AHL increases to a certain value, even if the higher external AHL concentration is used as the initial concentration. If this optimal external AHL concentration is found, the optimal number of detection cells can be found based on the relationship between the number of detection cells and the concentration of AHL produced. Finally we can find the optimal ratio of the two cells.

We gave the number of cells, different initial AHL concentration and the initial concentration of external AHL in the interval was traversed from 0 to 5×10

^{10}in steps of 1×10^{7}.
The curve of the maximal CrpP concentration changing with initial external AHL concentration:

Fig. 5. The curve of CrpP concentration changing with external AHL concentration

As can be seen from the Fig. 5, it can be found that increasing external AHL concentration will greatly increase the yield of CrpP before the curve inflection point. However, after the curve inflection point, even if the initial concentration of external AHL is greatly increased, CrpP maximal concentration will not change too much. If we increase the initial AHL concentration to obtain a larger CrpP concentration after the curve inflection point, it is obviously uneconomical. Therefore, it can be reasonably inferred that the external AHL concentration corresponding to the inflection point is optimal. Due to the high price of AHL, this finding can greatly reduce the cost of our projects and obtain a more suitable CrpP production.

In order to verify that this is not an accidental situation and find the optimal external AHL initial concentration corresponding to different numbers of detection cells, we found the relationship curve. The number of cells was traversed from 1×10

^{3}to 1×10^{7}in steps of 1×10^{3}. We found the corresponding optimal external AHL concentration and the relationship curve.
The figure is as follows:

Fig. 6. The curve of optimum concentration of AHL changing with number of the processing cells

It can be seen from the Fig. 6 that the initial external AHL concentration is proportional to the number of detection cells. When we design experiments with different cell numbers, the optimal external AHL initial concentration can be given according to the curve to find the best detection cell concentration.

Due to the complexity of CIP activation of SOS response[4], it is difficult to establish and solve the model. But we can skip the theoretical analysis and then find out the relationship between CIP concentration and corresponding AHL concentration produced by different numbers of detection cells. Also, we can find the relationship curve by fitting, so that we can deduce the required number of detection cells according to the previously calculated optimal external AHL concentration and initial CIP concentration. Then we can find the optimal detection cell and processing cell ratio for the project. It is of great significance for the optimization of the project and configuration optimization of the device[5].

Annex: Click on the link below to download our code.

References

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2013 European Control Conference (ECC)

(pp. 3633-3639). IEEE.
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, 91-99.
[4] Dörr, T., Vulić, M., & Lewis, K. (2010). Ciprofloxacin causes persister formation by inducing the TisB toxin in Escherichia coli.

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[5] Pérez-Velázquez, J., Gölgeli, M., & García-Contreras, R. (2016). Mathematical modelling of bacterial quorum sensing: a review.

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