Simulate the Bacteria Quorum Sensing Process and Kill Switch
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.
1. 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.
Gene expression: the expression of AHL and LuxR:
Protein degradation: The degradation of AHL and LuxR:
Changes about AHL: The production of AHL and its diffusion outside the cell:
Changes about RA: The combination and dissociationof AHL and LuxR (noted as RA):
Gillespie Algorithm
The Gillespie algorithm is a mathematical method to simulate biochemical reactions with random processes. The main ideas of this algorithm are as follows:
(a) First, we divide the biochemical reaction process into several elemental reactions. The substances involved and their stoichiometric number are recorded in the form of a matrix.
(b) Due to the rate constant and the concentration of reactants, the probability of different reactions is also different. We calculate the probability that each reaction will occur the next time.
(c) The knowledge of probability theory and stochastic processes tells us that the next reaction's time follows an exponential distribution, with the total reaction rate as its parameter. At that time, according to the possibility introduced in (b), a reaction occurs, causing a change in the number of molecules.
By repeating the above process, we can simulate the random biochemical reactions within the system.
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 simulated 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 (LuxR-AHL), 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 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.
2. 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 RNA thermometer, the toxin/antitoxin system express better in our design.Consider the expression partition knew and kold.The equation of our system will be:
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
So we only use earlier experiment data in fitting. Wevisualize the results of former and improved kill-switch here.
The result shows that our Kill Switch reacts stronger in a smaller temperature range. It's clear that our improvement did a good job, thus improve the safety.
We conducted a further sensitivity analysis, finding that RNA thermometer expression level has a continuous and stable impact on the killing rate.
Modeling and Our Project
Our models do an excellent job of filling in the gaps when were challenging to implement. Firstly, since web-lab experiments cannot be carried out until recently, our model simulates the change of the number of molecules in the engineering bacteria and the process of quorum sensing. Furthermore, we prove that it is feasible to realize the dynamic balance of engineering bacteria quantity through quorum sensing in project design. Our design excellence in the Kill-Switch model is demonstrated by visualizing a better kill rate caused by RNA Thermometer parameters change. As a result, our model fits in well with the project.
Parameters Used for Simulation and Regression
| Signal | Meaning | Value. |
|---|---|---|
| αcon | Gene production rate | 0.0167 nM/s |
| μ | Degradation rate | 0.0038 /s |
| αAHL | AHL production rate | 0.167 nM/s |
| kon | AHL and LuxR combination rate | 0.03 nM-1/s |
| koff | RA disposition rate | 0.167 nM/s |
| kdif | AHL transfer rate between invitro and invivio | 1.83 /s |
| r | Bacteria growth rate | 0.079 /s |
| K | Bacteria competition rate | 1.663e-04 /s |
| α | Additional competition rate | 1.013 /s |
| kA-N | Bacteria density decrease caused by AHL | 1 /s |
| knew | New RNA thermometer expression rate | 0.102360391 |
| kold | Old RNA thermometer expression rate | 0.394639 |
Reference
[1] Melke, P., et al., A cell-based model for quorum sensing in heterogeneous bacterial colonies. PLoS Comput Biol, 2010. 6(6): p. e1000819.
[2] A.B.Goryachev,D.J.TohT.Lee, Systems analysis of a quorum sensing network: Design constraints imposed by the functional requirements, network topology and kinetic constants. Biosystems,Volume 83, Issues 2–3, February–March 2006, Pages 178-187
[3] https://2019.igem.org/Team:Fudan/Model
[4] Darren J. Wilkinson, Stochastic Modelling for Systems Biology.
[5] Martin Overgaard, Jonas Borch, and Kenn Gerdes, RelB and RelE of Escherichia coli Form a Tight Complex That Represses Transcription via the Ribbon–Helix–Helix Motif in RelB. J Mol Biol. 2009 Nov 27; 394(2): 183–196.
See our codes at https://github.com/Isabel-Jiang/2020-model!
Signature: Xiaohui