Use Math to Explain and Simulate!
This year, we constructed two models to simulate the bacteria quorum sensing process and Kill Switch. For the quorum-sensing model, we used the Gillespie algorithm in random simulation to simulate the physiological processes at macro and micro levels. Specifically, we described Acyl-homoserine lactone's concentration accurately by monitoring both production and diffusion, which is first-of-its-kind to reveal details of the process, and developed originally by us. We further improved the model from 19Fudan team to better model the quorum sensing process for various bacteria growing stages. For the stage change caused by the anti-microbial peptide (mcbA), we provided explanations to its observed dynamic stability. For the Kill Switch model, based on the experimental data from 19BNU team, we improved the logistic equation, adjusted the promoter properties according to more suitable literature references, and made sensitivity analysis – and we obtain better and more insightful results.
Model one:Bacteria Quorum Sensing Process
In the first model, we describe the quorum sensing process in three parts. In Model 1.1, we discuss the change of AHL concentration in a single cell. AHL is produced in bacteria and transported to the surrounding environment through free diffusion. Since the number of molecules in a single section is small and random, we adopted Gillespie algorithm to simulate the trend of molecule numbers under the random process. In model 1.2, we followed the discussion in 1.1. To know when the quorum sensing switch is turned on, it is necessary to know polymer concentration formed by AHL and LuxR when AHL concentration is high enough. Our results demonstrate that polymer concentration remains dynamically stable even in a single cell, which guarantees our system's stable operation. In Model 1.3, we discuss the role of antimicrobial peptide circuits. Since the loop will open when the concentration of AHL increases and will close when the concentration is low enough, it forms a dynamic equilibrium. We use a phase diagram to describe this steady state.
1.1 AHL in a Cell
In this model, we simulate changes in the number of AHL in a cell. As mentioned above, AHL is generated inside the cell and transported out by free diffusion. In the initial stage, for AHL's concentration outside the bacteria is approximately 0, free diffusion is basically not inhibited and only related to AHL's concentration inside the cell. We believe that the main physiological processes about AHL are as follows:
Firstly,the expression of LuxI and LuxR:
Then,the degradation of LuxI and LuxR:
The production of AHL and its diffusion outside the cell:
Lastly,the combination and dissociationof AHL and LuxR:
A key point of Gillespie Algorithm is to describe the system in matrix.
In this matrix, each row is one substance, and each column is one reaction. The reactants per molecule are represented by -1, and the products per molecule are represented by 1. Set the order of the reactants to LuxI, LuxR AHLprecursor, AHL, RA, reaction sequence in the order listed above, then the matrix should be:
After each reaction, the number of molecules changes by the number of a column. What column is that? The probability is proportional to the rate of reaction. So let's figure out the rates of all the reactions.
So the probability is as follows:
According to probability theory, the time interval between each reaction follows the exponential distribution, and the parameter is the sum rate.
The current molecule number will change after each reaction, so that this algorithm can be repeated.In this certain progress,we can see that AHL molecule number is changed as follows:(We provide 2 results to gain better understanding.)
The result shows that at the early stage of colonization, AHL will be rapidly produced and diffused to a high concentration, leading to the generation of RA and the initiation of quorum sensing.
Noted:this is just AHL in a single cell.We should keep in mind that E. coli also completes the colonization process at the same time.
1.2 RA,the Switch
In the previous model, we have discussed the rapid emergence and spread of AHL during breeding. When the concentration of AHL is high enough, the net diffusion rate can be almost zero. At this point, the intracellular reaction occurs that is identical to model 1.1 except for diffusion. Therefore, we used the same simulation method and considered the situation before the secretion of antimicrobial peptides. We obtain the changes of RA in a cell number. Since the process is still random, we provide the results of two simulations.
As you can see, although the number of molecules varies, it ends up being relatively stable. This proves that our system has a self-steady state regulating quorum sensing state.
1.3 Engineering Bacteria Number Circulation
As the quorum sensing switch is turned on, the antimicrobial peptides begin to kill engineering bacteria, and the number of engineering bacteria decreases. Subsequently, the concentration of AHL decreases, resulting in a decrease in RA concentration and a partial shutdown of the quorum sensing switch. Our system ensures that engineering bacteria are neither too many to affect the gut environment nor too few to function properly. We will describe this process in terms of differential equations and explain it in terms of phase diagrams.
For AHL, the increase in the number of engineered bacteria will slow down the net degradation rate of AHL, which can be described as follows:
As for engineering bacteria, there is a net growth under no other conditions. However, the increase in AHL concentration will eventually lead to a rise in antimicrobial peptides. Furthermore, antimicrobial peptides will increase the survival pressure of engineering bacteria. So we can describe the process as follows:
We set kA-Nto 1, and obtain the following two results.The first is a phase diagram and the second is a concentration-time diagram.
From the two phases, we can see our system can operate dynamically.
From the second phase, we can see that the phase change of AHL lags behind the number of engineering bacteria. This tells us that the concentration of AHL has a time-delay effect, which is the essential reason why the system remains steady.
Module two: Kill Switch
In the modeling of kill Switch, we adopted the following ideas: First, we cooperated with the 19BNU team and adopted the data they tested. We found that without kill-switch, the colony number curve was similar to the logistic equation. Through data fitting, we obtained the parameters of the logistic equation. When kill-switch exists, in vitro, the toxin will increase the competitive pressure, causing parameters' change. Due to the improvement of a more efficient promoter, the competitive pressure caused by toxin can further increase when our design is adopted, and we performed sensitivity analysis here. Due to the abnormal part of the experimental data, we also selected the experimental data by referring to the literature and will explain the reasons for these anomalies.
2.1 Logistic Equation
Let's discuss logistic equation first.In this equation,the population change is caused by two factors: reproduction and competition. The reproductive rate (r) is proportional to the number of bacteria，and the competition rate (K) is proportional to the number of bacteria in the quadratic term.The equation and its solution are as follows.
2.2 Control Group
We can fairly consider that density in control group fits logistic equation.We fit the experimental data as follows.
2.3 Experiment Group and Improvement
Now let's consider the case when kill-switch is turned on.Competitive pressure sharply increases,casing a change in the equation above, in which α indicates additional pressure.
Furthermore, with an improvement in promoter ,the toxin-antitoxin system express better in our design.Consider the expression partition knew and kold.The equation of our system will be:
We visualize it here.
However, when looking at experiment data,we see some abnormal increase throughout time. Here are some possible reasons.
- Toxin-antitoxin plasmid lost nts
- The cellular repair mechanism
- Cell detoxification
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