Team:Oxford/Model/Therapeutics-Insight

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. 2 L. reuteri Growth Curve

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

Fig. 2 C. difficile Growth Curve

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 106 CFU. Meanwhile, C. difficile grows normally to its carrying capacity of 108 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<i>C. difficile</i>> and <i>L. reuteri</i>

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 106 CFU.

Fig. 4 Upregulation Profile of<i>C. difficile</i>> and <i>L. reuteri</i>

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

Up until the 20 hour delay point, about 80% of the C. difficile infection was seen to upregulate before the probiotic rendered it dormant. After the 20 hour delay point, the maximum percentage of upregulated C. difficile population dropped dramatically with every 2 hour increase in delay.

At the 26 hour delay point, the maximum upregulated population of C. difficile was less than 1%.


Discussion

The population-wide deterministic mathematical model of the interaction between C. difficile and L. reuteri colonies demonstrated that while our probiotic does not kill the entire C. difficile infection, it is successful in rendering it dormant.

It was further demonstrated that to ensure toxin synthesis is not triggered in C. difficile for any period of time, the probiotic should be introduced approximately 26 hours before C. difficile colonies flourish. This effectively means that our probiotic works best as a preventative measure to the C. difficile infection.