#### After extensively developing the design of our genetic circuits, we mathematically modelled the population dynamics of *L. reuteri* and *C. difficile* to determine the time at which probiotic should be administered to obtain optimal therapeutic effect.

### Regulatory System Primer

*C. difficile* employs the use of two transmembrane proteins AgrB and AgrD to produce its AIP quorum sensing molecule, as seen below. This occurs via the insertion of cytosolic AgrB and AgrD into the cell membrane, followed by the conversion of AgrD protein into AIP.

### Assumptions

- Populations of
*C. difficile*and*L. reuteri*are perfectly mixed. - The gut provides sufficient nutrients that there is no interspecific or intraspecific competition for nutrients.
- The delay between
*C. difficile*reaching its threshold density, and its subsequent release of toxins is negligible on the timescale of hours.

#### Species Key

### Translation of AgrB and AgrD Proteins

Kinetic Rate Equation | Kinetic Rate Constant | Kinetic Rate Constant Value Modelled |
---|---|---|

$$M \to M + B$$ | $$K_b$$ | $$5.00 \times 10^{-14} \ hr^{-1}$$ |

$$M \to M + D$$ | $$K_d$$ | $$5.00 \times 10^{-11} \ hr^{-1}$$ |

### Membrane Insertion of Cytosolic AgrB and AgrD Proteins

Kinetic Rate Equation | Kinetic Rate Constant | Kinetic Rate Constant Value Modelled |
---|---|---|

$$B \to Bt$$ | $$\alpha_{Bt}$$ | $$3.334 \ hr^{-1}$$ |

$$D \to Dt$$ | $$\alpha_{Dt}$$ | $$3.334 \ hr^{-1}$$ |

### Conversion of AgrD to AIP

Kinetic Rate Equation | Kinetic Rate Constant | Kinetic Rate Constant Value Modelled |
---|---|---|

$$Dt \overset{\mbox{Bt}}{\to} AIP$$ | $$k_S$$ | $$4.00 \times 10^8 \ M^{-1} hr^{-1}$$ |

### Endolysin Transcription & Translation by *L. reuteri*

Kinetic Rate Equation | Kinetic Rate Constant | Kinetic Rate Constant Value Modelled |
---|---|---|

$$NULL \to M_e$$ | $$v$$ | $$1 \times 10^{2} \ hour^{-1}$$ |

$$M_e \to M_e + E$$ | $$K_e$$ | $$4.65 \times 10^{-3} \ hour^{-1}$$ |

### Degradation of Molecules

Kinetic Rate Equation | Kinetic Rate Constant | Kinetic Rate Constant Value Modelled |
---|---|---|

$$X \to NULL \ (where X=B, D, Bt, Dt, S, me, E)$$ | $$\delta_X$$ | $$3.334 \ hr^{-1}$$ |

*C. difficile*'s AgrAC two-component detection & regulatory system can be described by the same differential equations as developed in previous sections.

#### Additionally, *C. difficile* will produce the quorum signalling AIP molecules and *L. reuteri* will produce endolysin. These processes are modelled as the following differential equations:

Concentration Modelled | Differential Equation | Process Modelled |
---|---|---|

Cytosolic AgrB Protein | $$\frac {dB}{dt} = Kb \times M - (\alpha_{Bt} + \delta_{B}) \times B$$ | Translation + Degradation + Membrane Insertion |

Cytosolic AgrD Protein | $$\frac {dD}{dt} = Kd \times M - (\alpha_{Dt} + \delta_{D}) \times D$$ | Translation + Degradation + Membrane Insertion |

Transmembrane AgrB Protein | $$\frac {dB_t}{dt} = \alpha_B \times B - \delta_{Bt} \times Bt $$ | Membrane Insertion + Degradation |

Transmembrane AgrD Protein | $$\frac {dD_t}{dt} = \alpha_Dt \times D - \delta_{Dt} - k_S \times Bt \times Dt$$ | Membrane Insertion + Degradation + Conversion into AIP |

AIP | $$\frac {dS}{dt} =k_S \times Bt \times Dt + k_{ibind} \times R_{b_{cd}} - k_{bind} \times R_{cd} \times S + k_{ibind} \times R_{b_{lr}} - k_{bind} \times R_{lr} \times S - \delta_S \times S$$ | Conversion into AIP + Binding/Unbinding of AIP on C. difficile + Binding/Unbinding on L. reuteri + Binding + Degradation |

Endolysin mRNA | $$\frac {dM_e}{dt} = v \times P - \delta_{M_e} \times M_e$$ | Transcription + Degradation |

Endolysin Protein | $$Ke \times M_e - \delta_E \times E$$ | Translation + Degradation |

#### Now that we have a cellular mathematical model of both *C. difficile* and *L. reuteri*, it is straightforward to convert it to a population model by using the following differential equations:

Concentration Modelled | Differential Equation | Process Modelled |
---|---|---|

Population of L. reuteri |
$$\frac {dN_l}{dt} = r_{lr} \times N_{t} \times (1- \frac {N_t}{N_{lr}})$$ | Logistic Growth of L. reuteri |

Population of C. difficile |
$$\frac {dN_c}{dt} = r_{cd} \times N_{c} \times (1- \frac {N_c}{N_{cd}})$$ | Logistic Growth of C. difficile |

Total mRNA concentration in Population of L. reuteri |
$$\frac {dM}{dt} = m \times N_{cd} - \delta_M \times M$$ | Transcription in Entire Population + Degradation |

Total Endolysin concentration in Population of L. reuteri |
$$\frac {dM_e}{dt} = v \times P \times N_{lr} - \delta_{M_e} \times M_e$$ | Transcription in Entire Population + Degradation |

### Results

Fig. 1 *L. reuteri* Growth Curve Assuming Initial Population = 1

Fig. 2 *C. difficile* Growth Curve Assuming Initial Population = 1

#### As seen in the above figures, *L. reuteri* grows normally to its carrying capacity of 10^{6} CFU. Meanwhile, *C. difficile* grows normally to its carrying capacity of 10^{8} CFU and starts to die when *L. reuteri* begins exponential growth at approximately 36 hours. Although, the probiotic fails to kill the entire population of *C. difficile*, the remaining population remains downregulated and no longer produces toxins, as shown in Figure 3 below.

#### The figure below shows the upregulation profiles of both colonies to determine whether the *L. reuteri* and *C. difficile* colonies are producing endolysin and toxins respectively.

Fig. 3 Upregulation Profile of *C. difficile* and *L. reuteri*

#### Although *L. reuteri* keeps the upregulated population below 0.5% in the steady state, there is a period of time for which *C. difficile* produces toxins. This is caused by the slower growth rate of *L. reuteri* compared to the growth rate of *C. difficile*. The above result suggests that if the *L. reuteri* colony grew before the *C. difficile* infection started flourishing, the probiotic could ensure that the percentage of upregulated *C. difficile* doesn’t cross 0.5%.

#### To test this hypothesis, we ran simulations where *C. difficile* growth was delayed by a certain amount of time. This is equivalent to a prophylactic approach whereby our probiotic is introduced into the gut before *C. difficile* colonies flourish. This is shown in Figure 5 below.

Fig. 4 Growth Curves for Delayed *C. difficile* Growth

#### As seen in the graph, the steady state populations for all delayed curves is about 10^{6} CFU.

Fig. 5 Upregulation Profile of *C. difficile* and *L. reuteri* For Different Delay Times