Agent-Based Modeling

At the group level, considering the complexity of the colonization process, we choose Agent-Based Modeling (ABM) for the simulation of complex systems. NetLogo[1] is a Agent-Based Modeling and simulation platform that can create and simulate interactions and behaviors between individuals for studying and exploring the behavior of complex systems. It is particularly suitable for simulating complex systems that evolve over time. Based on Netlogo, we have developed a model for the colonization process.

  • The initial position of bacteria is randomly generated in space
  • Bacteria move randomly in space
  • Movement consumes energy, and there is a fundamental energy loss in both movement and colonization states
  • When the energy of bacteria reaches a certain value, they replicate, produce new individuals, and lose some energy
  • Bacterial will die when energy is less than 0
  • The initial nutrient values in the space were randomly distributed and increased at a certain rate
  • When there are eight other individuals within a radius of 1, the bacteria stop moving and complete colonization, which is irreversible
  • When there are 6 engineering bacteria (blue) within a radius of 1 for Streptococcus mutans (red), it causes Streptococcus mutans to die

The initial state is shown in the figure below. The blue dots represent engineering bacteria, the red dots represent Streptococcus mutans, and the darker the green grid, the more nutrients the location contains.


Figure 1. Initial bacterial and nutrient distribution.

The changes in the number and process of S. mutans and engineering bacteria when nutrition is sufficient are shown as follows

Figure 2. Growth curves for S. mutans (left) and engineering bacteria (right) under nutrient sufficiency.

Figure 3. The simulation under nutrient sufficiency. (gif)

The changes in the number and process of Streptococcus mutans and engineering bacteria under insufficient nutrition are shown below


Figure 4. Growth curves for Streptococcus mutans (left) and engineering bacteria (right) under nutrient insufficiency.

Figure 5. The simulation under nutrient insufficiency. (gif)

It can be found that only when the nutritional conditions are sufficient, can engineering bacteria reproduce in large numbers and achieve a high population density, thereby achieving the effect of engineering bacterial colonization and complete elimination of Streptococcus mutans. This reminds us of the potential issues in our project, namely whether the nutritional conditions in the cat's mouth can meet the desired results.


Dynamic Simulation

In order to more accurately characterize the colonization process, ordinary differential equations are used to describe the expression levels of various bacterial components, and then the macroscopic bacterial quantity is described at the microscopic level.

$$\begin{array}{c} P_{V q m A} \stackrel{\beta_{1}}{\rightarrow} V q m A \\ Engineering Bacteria \stackrel{K}{\rightarrow} DPO \\ D P O+V q m A \stackrel{k_{1} k_{2}}{\longleftrightarrow} V q m A \cdot D P O \\ P_{\text {tet }} \stackrel{\beta_{2}}{\rightarrow} \text { tet } R \\ P_{CsgA\text{-}Hsa} \stackrel{\beta_{3}}{\rightarrow} CsgA\text{-}Hsa \\ V q m A \stackrel{\alpha_{1}}{\rightarrow} \Phi \\ D P O \stackrel{\alpha_{2}}{\rightarrow} \Phi \\ V q m A \cdot D P O \stackrel{\alpha_{3}}{\rightarrow} \Phi \\ \operatorname{tetR} \stackrel{\alpha_{4}}{\rightarrow} \Phi \\ CsgA\text{-}Hsa \stackrel{\alpha_{5}}{\rightarrow} \Phi \\ \end{array} $$

The following ODE equation describes the variation of each quantity over time.

$$\begin{array}{c} \frac{d V q m A}{d t}=\beta_{1}+k_{2}[\mathrm{VqmA} \cdot \mathrm{DPO}]-k_{1}[\mathrm{DPO}][\mathrm{VqmA}]-\alpha_{1}[\mathrm{VqmA}] \\ \frac{d D P O}{d t}=K N-\alpha_{2}[\mathrm{DPO}] \\ \frac{d V q m A \cdot D P O}{d t}=k_{1}[\mathrm{DPO}][\mathrm{VqmA}]-k_{2}[\mathrm{VqmA} \cdot \mathrm{DPO}]-\alpha_{3}[\mathrm{VqmA} \cdot \mathrm{DPO}] \\ \frac{d t e t R}{d t}=\beta_{2}\left(\frac{[\mathrm{VqmA}]}{[\mathrm{VqmA}]+X_{m}}+s \frac{[\mathrm{VamA} \cdot \mathrm{DPO}]}{[\mathrm{VqmA} \cdot \mathrm{DPO}]+X_{m}}\right)-\alpha_{4}[\mathrm{tet} R] \\ \frac{dCsgA\text{-}Hsa}{d t}=\beta_{3}\left(\frac{K d_{P T e t}}{K d_{P T e t}+[\text { tetR }]}\right)-\alpha_{5}[\mathrm{CsgA\text{-}Hsa}] \end{array} $$

The meaning of the parameters is as follows.


Table 1. List of parameters involved in colonization model


Solve the equation and draw it as shown in the following figure.

Figure 6. The simulation of DPO-VqmA system.

Based on the theoretical situation in the oral cavity, the following equation can be obtained. In the formula, kon and koff are other bacteria's kon and koff.

$$\begin{array}{c} {\left[B_{T}\right]=[B]+[B A]+[B E]} \\ \frac{d([B E]+[B A])}{d t}=k_{o n}[B]\left([A]+c_{1}[E]\right)-k_{o f f}\left([B A]+c_{2}[B E]\right)=0 \\ K_{x}=\frac{k_{o f f}}{k_{o n}} \\ K_{x}=\frac{\left(\left[B_{T}\right]-[B A]-[B E]\right)\left([A]+c_{1}[E]\right)}{[B A]+c_{2}[B E]} \end{array} $$

Table 2. Equation description

To solve the above model, using the Lotka Volterra model,Solve the following equation and bring the steady-state parameters into the above equation.

$$\begin{array}{c} \frac{d([E]+[E B])}{d t}=r_{1}([E]+[E B])(1-\frac{[E]+[E B]}{K_{1}}-\frac{[A]+[A B]}{K_{2}}) \\ \frac{d([A]+[A B])}{d t}=r_{2}([A]+[A B])(1-\frac{[E]+[E B]}{K_{1}}-\frac{[A]+[A B]}{K_{2}}) \end{array} $$

Table 3. Equation description

In order to better describe the colonization process in this circuit, the bacterial colonization model was simplified, and the migration and colonization of bacteria were recorded as a simple probability combination. Simplifying engineering bacteria into a specific rectangular body of length and width, where P represents the probability of contact between each surface and the bottom surface, and K is assumed to be a dimensional constant of 1. This model may be contrary to biological colonization, but we can obtain a more accurate Kd by taking residuals from two equations.

$$K_{o n}{ }^{\prime}=\sum_{i=1}^{3} P_{i}(\frac{B_{T}-B A-B E}{S})(\frac{CsgA\text{-}Hsa}{E B})(\frac{S_{i}}{S}) K $$

By fitting the different CsgA-Hsa content and Kd of each bacterium at different times, the following curve is obtained.

$$\begin{array}{c} \mathrm{K}_{\mathrm{d}}=\frac{\mathrm{k}_{\text {off }}{ }^{\prime}}{\mathrm{k}_{\mathrm{on}}{ }^{\prime}} \\ k_{\text {off }}{ }^{\prime}=c_{2} k_{\text {off }} \\ k_{\text {on }}{ }^{\prime}=c_{1} k_{\text {on }} \\ \end{array} $$
Figure 7. Description the quantitative relationship between CsgA-Hsa and Kd.

By fitting the above equation, assuming that there is a certain amount of miscellaneous bacteria entering the oral cavity of cats and cats while eating, we can fit the competitive relationship between engineering bacteria and miscellaneous bacteria, as shown in the following figure.

$$\begin{array}{c} \frac{d E}{d t}=P[E B]-\operatorname{die}[E]-\frac{T[E]}{\mathrm{K}_{\mathrm{d}}} \\ \frac{d E B}{d t}=r[E B](1-\frac{[E B]}{K}-\frac{[A B]}{K})-P[E B]+\frac{T[E]}{\mathrm{K}_{\mathrm{d}}} \\ \frac{d A}{d t}=P[A B]-\operatorname{die}[A]-T[A]+F_{1}(a, t) \\ \frac{d A B}{d t}=r[A B](1-\frac{[E B]}{K}-\frac{[A B]}{K})-P[A B]+T[A] \\ \frac{d K_{d}}{d t}=F_{2}(E+E B, CsgA\text{-}Hsa) \\ \end{array} $$

F1 represents the number of miscellaneous bacteria obtained by cats per feeding;F2 represents the relationship between engineering bacterial count and CsgA-Hsa and Kd.

Figure 8. Colonization and interaction model

It can be found that engineering bacteria still maintain a relatively stable value during continuous feeding, providing model support for our use of CsgA-Hsa for adhesion.



Reference

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