Synthetic biology wants to simulate the activity of gene expression by quantitative means, and to assist in wet experiments and provide some data support for them, this model was developed by introducing a pathway expressing AidH and an antimicrobial peptide to E.coli for degradation of AHL and killing of SRBs in the environment.
Fig.1 Gene Layer Flowchart
For biosafety reasons, we designed the following suicide switch. The suicide switch induces the expression of the antitoxin protein CcdA through AHL and, at the same time, constitutively expresses the toxin protein CcdB and the AHL-binding protein LuxR. E. coli can only survive when CcdA binds CcdB to form the complex CcdAB.
Therefore, when the concentration of AHL is lower than the threshold of survival of E. coli, the induced expression of CcdA is blocked, and the CcdB is not entirely bound, ultimately leading to the lysis of E.coli when our engineered bacteria are in natural water because of the concentration of AHL below the threshold. Therefore, when the AHL concentration is lower than the survival threshold of E. coli, the CcdA-induced expression is blocked and cannot fully bind to CcdB, ultimately leading to the cleavage of E. coli . The low induced concentration of AHL leads to a more sophisticated expression system, which poses a challenge for wet experiments. For this reason, we modeled the system kinetically to provide a reference for selecting the gene block for wet experiments.
Fig.2 QS-based Induced Expression System
We need to optimize the promoter and RBS to assist the wet experiment better. First, for the constitutively expressed LuxR, we wanted to enhance its expression because this would increase the sensitivity of LuxR binding to AHL, so we chose to replace B0030 with the stronger B0034 (http://parts.igem.org/), and the promoter J12306 with the stronger J23T00. Second, for the induced expression of CcdA, we chose to replace B0032 with the stronger B0034, which, according to the results of the subsequent wet experiments, does not bring a tremendous metabolic burden to our engineered bacterium E. coli.
We summarize the equations of the quorum sensing model as follows:
\begin{equation} V_{\,AHL-diff}\;\;=\;\;r_{AHL}\,(\,[\,E-AHL_{in}\,]\;-\;[\,AHL_{ex}\,]\,) \tag{1} \end{equation}
\begin{equation}\label{eq2} \delta_{E-AHL_{in}-1}\;=\;K_{AHL_{in}}[\,E-LuxI\,]-d_{AHL_{in}}[\,E-AHL_{in}\,] \tag{2} \end{equation}
\begin{equation}\label{eq3} \delta_{E-AHL_{in}-2}\;=\;k_{A-R}\,[\,E-AHL_{in}\,] - k'_{A-R}\,[\,E-(A-R)\,] \tag{3} \end{equation}
\begin{equation}\label{eq4} V_{Aidh-AHL-c}\;=\;\frac{k_{cat}\,[\,AidH_c\,]\,[\,E-AHL_c\,]}{K_m\;+\;[\,E-AHL_c\,]} \tag{4} \end{equation}
\begin{equation}\label{eq5} V_{Aidh-AHL-e}\;=\;\frac{k_{cat}\,[\,AidH_e\,]\,[\,AHL_e\,]}{K_m\;+\;[\,AHL_e\,]} \tag{5} \end{equation}
\begin{equation}\label{eq6} \frac{d\,[\,E-AHL_{in}\,]}{dt}\;=\;\delta_{E-AHL_{in}-1}-\delta_{E-AHL_{in}-2}-V_{\,AHL-diff}\;-\; V_{Aidh-AHL-c} \tag{6} \end{equation}
\begin{equation}\label{eq7} \frac{d\,[\,AHL_{ex}\,]}{dt}\;\;=\;\;V_{\,AHL-diff}\,⋅\,(\,\frac{V_{ECO}\,[\,ECO\,]}{DW_{ECO}}\,)\,-\,d_{AHL_{ex}}\,[\,AHL_{ex}\,]\;-\; V_{Aidh-AHL-e} \tag{7} \end{equation}
\begin{equation}\label{eq8} \frac{d\,[\,E-mRNA_{LuxI}\,]}{dt}\;=\;K_{mRNA_{LuxI}}-d_{mRNA_{LuxI}}\,[\,E-mRNA_{LuxI}\,] \tag{8} \end{equation}
\begin{equation}\label{eq9} \frac{d\,[\,E-mRNA_{LuxR}\,]}{dt}\;=\;K_{mRNA_{LuxR}}-d_{mRNA_{LuxR}}\,[\,E-mRNA_{LuxR}\,] \tag{9} \end{equation}
\( K_{mRNA_{LuxI}} \) is the reaction rate constant generated by \( mRNA_{LuxI}\).
\( K_{mRNA_{LuxR}} \) is the reaction rate constant generated by \( mRNA_{LuxR} \).
\( d_{mRNA_{LuxI}} \) is \( mRNA_{LuxI} \) degradation reaction rate constant
\( d_{mRNA_{LuxR}} \) is the reaction rate constant for \( mRNA_{LuxR} \) degradation
\begin{equation}\label{eq10} \frac{d\,[\,E-LuxI\,]}{dt}\;=\;K_{LuxI}\,[\,mRNA_{LuxI}\,]\,-\,d_{LuxI}\,[\,E-LuxI\,] \tag{10} \end{equation}
\begin{equation}\label{eq11} \frac{d\,[\,E-LuxR\,]}{dt}\;=\;K_{LuxR}\,[\,mRNA_{LuxR}\,]\,-\,d_{LuxR}\,[\,E-LuxR\,] \tag{11} \end{equation}
\( K_{LuxI} \) is the \(LuxI\) generated reaction rate constant
\( K_{LuxR} \) is \(LuxR\) generated reaction rate constant
\( d_{LuxI} \) is the rate constant for \(LuxI\) degradation
\( d_{LuxR} \) is the reaction rate constant for \(LuxR\) degradation
The rate of change of the AHL - LuxR complex is determined by the following two reversible and degradation reactions
\begin{equation} E-LuxR\;\;⇌\;\;E-(A-R) \end{equation}
\(k_{A-R}\) is the rate constant for the forward reaction
\(k'_{A-R}\) is the rate constant for the reverse reaction
\begin{equation} 2(E-(A-R))\;\;⇌\;\;E-(A-R)_{2} \end{equation}
\(k_{(A-R)_2}\) is the rate constant for the forward reaction
\(k'_{(A-R)_2}\) is the rate constant for the reverse reaction
\begin{equation} E-(A-R) \;\;→\;\; ϕ \end{equation}
\(d_{A-R}\) is the degradation rate constant for the AHL - LuxR compound
Therefore, the differential equation for the AHL - LuxR complex is as follows:
\begin{equation} \delta_{E-(A-R)-1}\;=\;k_{A-R}\,[\,E-AHL_{in}\,][\,E-LuxR\,]\,-\,k'_{A-R}\,[\,E-(A-R)\,] \tag{12} \end{equation}
\begin{equation} \delta_{E-(A-R)-2}\;=\;2⋅k_{(A-R)_{2}}\,[\,E-(A-R)\,]^2\,-\,2⋅k'_{(A-R)_{2}}\,[\,E-(A-R)_{2}\,] \tag{13} \end{equation}
\begin{equation} \frac{d\,[\,E-(A-R)\,]}{dt}\;=\;-d_{A-R}\,[\,E-(A-R)\,]\;+\;\delta_{E-(A-R)-1}-\delta_{E-(A-R)-2} \tag{14} \end{equation}
Changes in the \( (A-R)_2 \) complex are determined by two reversible reactions and degradation
\begin{equation} 2(E-(A-R))\;\;⇌\;\;E-(A-R)_{2} \end{equation}
The differential equation for the AHL-LuxR dimer is as follows :
\begin{equation} \delta_{E-(A-R)_2-1}\;=\;k_{(A-R)_{2}}\,[\,E-(A-R)\,]^2\,-\,k'_{(A-R)_{2}}\,[\,E-(A-R)_{2}\,] \tag{15} \end{equation}
\begin{equation} \delta_{E-(A-R)_2-2}\;=\;k_{Plux-(A-R)_{2}}\,[\,E-(A-R)\,][\,Plux\,]\,-\,k'_{Plux-(A-R)_{2}}\,[\,mRNA_{CcdA}\,] \tag{16} \end{equation}
\begin{equation} \frac{d\,[\,E-(A-R)_{2}\,]}{dt}\;=\;\delta_{E-(A-R)_2-1}\;-\;\delta_{E-(A-R)_2-2}\;-\;d_{(A-R)_{2}}\,[\,E-(A-R)_{2}\,] \tag{17} \end{equation}
For the suicide system of the engineered bacterium E. coli, we used the CcdA/CcdB toxin-antitoxin system, where CcdA is the antitoxin protein and CcdB is the toxin protein. Quantitative simulation of this expression system will help to complete the construction of the subsequent predator-prey model and deepen the understanding of the biosafety design module of this project.
\begin{equation} \delta_{mRNA_{CcdA}}\;\;=\;\;k_{P-(A-R)_{2}}\,[\,E-(A-R)\,][\,Plux\,]\,-\,k'_{P-(A-R)_{2}}\,[\,mRNA_{CcdA}\,] \tag{18} \end{equation}
\begin{equation} \frac{d\,[\,mRNA_{CcdA}\,]}{dt}\;\;=\;\;\delta_{mRNA_{CcdA}}\;-\;d_m\,[\,mRNA_{CcdA}\,] \tag{19} \end{equation}
\begin{equation} \frac{d\,[\,mRNA_{CcdB}\,]}{dt}\;\;=\;\;K_{CcdB}\;-\;d_m\,[\,mRNA_{CcdB}\,] \tag{20} \end{equation}
\( k_{Plux-(A-R)_2}\) is the rate constant for the forward reaction
\( k'_{Plux-(A-R)_2}\) is the rate constant for the reverse reaction
\( K_{CcdB}\) is the rate constant for the reaction generated by \( mRNA_{CcdB}\)
\( d_{m}\) is the reaction rate constant for \( mRNA_{CcdB}\) degradation
\begin{equation} \frac{d\,[\,CcdA\,]}{dt}\;\;=\; k_{TL-A}\,[\,mRNA_{CcdA}\,] \;-\; \delta_{A}\,[\,CcdA\,]\,-\,k_{AB}[\,CcdA\,]\,[\,CcdB\,]\,+\,k'_{AB}\,[\,CcdAB\,] \tag{21} \end{equation}
\begin{equation} \frac{d\,[\,CcdB\,]}{dt}\;\;=\; k_{TL-B}\,[\,mRNA_{CcdB}\,] \;-\; \delta_{B}\,[\,CcdB\,]\,-\,k_{AB}[\,CcdA\,]\,[\,CcdB\,]\,+\,k'_{AB}\,[\,CcdAB\,] \tag{22} \end{equation}
\( k_{TL-A}\) is the reaction rate constant generated by \( CcdA\)
\( k_{TL-B}\) is the $CcdB$ generated reaction rate constant
\( \delta_A\) is the reaction rate constant for \( CcdA\) degradation
\( \delta_B\) is the reaction rate constant for \( CcdB\) degradation
\( k_{AB}\) is the forward reaction rate constant for \( CcdA\) combined with \(CcdB\)
\( k'_{AB}\) is the rate constant for the reverse reaction of \( CcdA\) with \(CcdB\)
\begin{equation} \frac{d\,[\,CcdAB\,]}{dt}\;\;=\; k_{AB}[\,CcdA\,]\,[\,CcdB\,]\,-\,k'_{AB}\,[\,CcdAB\,]\;-\;\delta_{AB}\,[\,CcdAB\,] \tag{23} \end{equation}
\( \delta_{AB}\) is the reaction rate constant for \(CcdAB\) degradation
\begin{equation} \frac{d\,[\,mRNA_{AidH}\,]}{dt}\;\;=\;\;K_{AidH}\;-\;\delta_{mRNA_{AidH}}\,[\,mRNA_{AidH}\,] \tag{24} \end{equation}
\begin{equation} \frac{d\,[\,AidH_c\,]}{dt} \;\;=\;\; k_{AidH}\,[\,mRNA_{AidH}\,] \;-\; \delta_{AidH}\,[\,AidH_c\,] \;-\;r_{AidH}\,(\,[\,AidH_c\,]\;-\;[\,AidH_e\,]\,) \tag{25} \end{equation}
\begin{equation} \frac{d\,[\,AidH_e\,]}{dt} \;\;=\;\; r_{AidH}\,(\,[\,AidH_c\,]\;-\;[\,AidH_e\,]\,) \;-\; \delta_{AidH}\,[\,AidH_e\,] \tag{26} \end{equation}
\( K_{AidH}\) is the rate constant for \( mRNA_{AidH}\) generation
\( k_{AidH}\) is \( AidH\) generated reaction rate constant
\( \delta_{mRNA_{AidH}}\) is \( mRNA_{AidH}\)degradation reaction rate constant
\( \delta_{AidH}\) is the reaction rate constant for \( AidH\) degradation
Protein expression mimicry is a more complex process involving mRNA transcription, peptide chain expression, and post-expression modification of the peptide chain
\begin{equation} \frac{d\,[\,mRNA-antipeptide\,]}{dt} \;\;=\;\; K_{anti} \;-\; \delta_{m-anti}\,[\,mRNA\,] \tag{27} \end{equation}
\begin{equation} \frac{d\,[\,antipeptide\,]}{dt} \;\;=\;\; k_{anti}\,[\,mRNA\,] \;-\; \delta_{anti}\,[\,Protein\,] \tag{28} \end{equation}
\( K_{anti}\) is the reaction rate constant generated by \( mRNA - antipeptide\)
\( k_{anti}\) is the reaction rate constant for \( antipeptide\) generation
\( \delta_{m-anti}\) is the reaction rate constant for \( mRNA - antipeptide\) degradation
\( \delta_{anti}\) is the reaction rate constant for \( antipeptide\) degradation
To find the intracellular AHL concentration threshold at which the E. coli suicide system is switched on, as well as the corresponding levels of the toxin protein CcdB, we set up a series of intracellular AHL concentration gradients, viewed the changes in the concentration of each substance through the means of differential equation modeling, and fitted the AHL concentration thresholds based on the results of the kinetic simulations using the gradient descent approach.
Fig.3 Changes in CcdA, CcdB, and CcdAB concentrations with intracellular AHL concentration
As can be seen from the figure, with the increase of intracellular AHL concentration in E.coli, the concentration of the toxin protein CcdB decreased significantly, indicating that a large amount of CcdB binds to the antitoxin protein CcdA to form CcdAB. In addition, we can also see that when the intracellular concentration of AHL is lower, it will lead to the death of E.coli due to the excessive concentration of CcdB. When the concentration of intracellular AHL is higher, the CcdB concentration will be deficient and lower than the CcdA concentration, indicating that E.coli can survive generally at this time. Attention will be at a secondary level and lower than that of CcdA, and the level of antitoxin protein is higher than that of toxin protein, indicating that E.coli can survive generally at this time.
To design AHL concentration gradients for wet experiments as a reference and provide critical data support for cell-level meta-cellular automata modeling, we performed curve fitting for the relationship between CcdB concentration and intracellular AHL concentration. Finally, we applied the curve equations to cell-level meta-cellular automata modeling.
Fig.4 CcdB - Intracellular AHL Curve Fits
As can be seen from the figure, the suicide threshold for AHL corresponds to 1.4129 nM, and we obtained the CcdB - intracellular AHL curve
\begin{equation} CcdB\;\;=\;\;\frac{1.351}{0.007\;+\;e^{4.667(AHL-1.724)}} \end{equation}
In order to better provide modeling guidance for wet experiments to be carried out, we need to carry out sensitivity analysis on some key parameters so as to analyze the influence of these parameters on the model results.
First, we performed a sensitivity analysis for the degradation constant of CcdAB because the concentration of CcdAB significantly affected the amount of free toxin protein CcdB and antitoxin protein CcdA, i.e., it was closely related to the suicide of engineered bacteria E. coli.
Fig.5 CcdAB - d_AB Sensitivity Analysis
As shown in the figure, it can be seen that when d_AB changes from 0.01 to 0.02, CcdAB changes most significantly; when d_AB changes from 0.02 to 0.1, CcdAB changes gradually and slowly. It can be seen that changes in d_AB in the range below 0.07 have a significant effect on ccdAB concentration, and above 0.07 have a lesser impact on ccdAB concentration.
Second, since the concentration of the toxin protein CcdB is directly related to the death of E. coli, we need to analyze the sensitivity of the degradation constants of CcdB
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Fig.6 ccdA/ccdB/ccdAB - d_B Sensitivity Analysis
The figure shows that CcdB changes most significantly when d_B changes from 0.01 to 0.02; CcdAB changes gradually and slowly when d_B changes from 0.02 to 0.1. It can be seen that changes in d_B in the range below 0.08 have a significant effect on the ccdB concentration, and above 0.08 have a lesser impact on the ccdB concentration.
Due to the need for promoter and RBS optimization in wet experiments, we performed sensitivity analyses of the constitutive promoter strength parameter K_B and RBS strength P_B and induced expression RBS strength P_A, respectively.
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Fig.7 ccdA/ccdB/ccdAB - K_B Sensitivity Analysis
As can be seen from the figure, when K_B is very small, the concentrations of both CcdB and CcdAB are small, and free CcdA is more significant in the case of low AHL concentration. With the increase of the K_B value, the final stabilized concentration of CcdA is getting smaller and smaller in the case of high AHL concentration, and the absolute stabilized attention of CcdB and CcdAB is getting larger and larger. The change in K_B value significantly affects the stabilized concentrations of ccdAB, ccdA, and ccdB at high AHL concentrations. In addition, it can also be analyzed that the lethal AHL threshold of CcdB increases with the increase of K_B.
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Fig.8 ccdA/ccdB/ccdAB - P_B Sensitivity Analysis
As can be seen from the figure, when P_B is very small, the concentrations of both CcdB and CcdAB are small, and free CcdA is prominent in the case of low AHL concentration. With the increase of P_B value, the final stabilized concentration of CcdA is getting smaller and smaller in the case of high AHL concentration, and the absolute stabilized attention of CcdB and CcdAB is getting larger and larger. The change in P_B value significantly affects the stabilized concentrations of ccdAB, ccdA, and ccdB at high AHL concentrations. In addition, it can also be analyzed that the lethal AHL threshold of CcdB increases with the increase of P_B.
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Fig.9 ccdA/ccdB/ccdAB - P_A Sensitivity Analysis
As can be seen from the figure, when P_A is very small, the concentrations of both CcdB and CcdAB are significant, and free CcdA is small in the case of low AHL concentration. With the increase of the P_A value, the final stabilized concentration of CcdA in the case of high AHL concentration becomes larger and larger, and the final stabilized attention of CcdB and CcdAB becomes smaller and smaller. The change in P_A value significantly affects the stabilized concentrations of ccdAB, ccdA, and ccdB at high AHL concentrations. In addition, it can also be analyzed that the lethal AHL threshold of CcdB becomes smaller with the increase of P_A.
Since SRB (Sulfate Reducing Bacteria) regulates biofilm formation in the sewer environment through a community sensing system, when the community sensing molecules reach a specific concentration (threshold), they will induce the expression of biofilm formation-related genes. In the first part of the Simulation of E.coli, we simulated the secretion of AidH by E.coli. This would degrade the AHL produced by SRB and thus inhibit the induced expression of proteins related to biofilm formation in SRB. So, in this part, we simulated the induced expression of SRB biofilm formation-related proteins to lay the foundation for the interactions between the two bacteria in the third part of the Predator-prey Model Simulation.
\begin{equation} \frac{d\,[\,S-mRNA_{LuxI}\,]}{dt}\;=\;K_{mRNA_{LuxI}}-d_{mRNA_{LuxI}}\,[\,S-mRNA_{LuxI}\,] \tag{30} \end{equation}
\begin{equation} \frac{d\,[\,S-mRNA_{LuxR}\,]}{dt}\;=\;K_{mRNA_{LuxR}}-d_{mRNA_{LuxR}}\,[\,S-mRNA_{LuxR}\,] \tag{31} \end{equation}
\(K_{mRNA_{LuxI}}\) is the reaction rate constant generated by \(S-mRNA_{LuxI}\)
\(K_{mRNA_{LuxR}}\) is the reaction rate constant generated by \(S-mRNA_{LuxR}\)
\(d_{mRNA_{LuxI}}\) is the reaction rate constant for \(S-mRNA_{LuxI}\) degradation
\(d_{mRNA_{LuxR}}\) is \(S-mRNA_{LuxR}\) reaction rate constant for degradation
\begin{equation} \frac{d\,[\,S-LuxI\,]}{dt}\;=\;K_{LuxI}\,[\,mRNA_{LuxI}\,]\,-\,d_{LuxI}\,[\,S-LuxI\,] \tag{32} \end{equation}
\begin{equation} \frac{d\,[\,S-LuxR\,]}{dt}\;=\;K_{LuxR}\,[\,mRNA_{LuxR}\,]\,-\,d_{LuxR}\,[\,S-LuxR\,] \tag{33} \end{equation}
\(K_{LuxI}\) is the \(S-LuxI\) generated reaction rate constant
\(K_{LuxR}\) is the rate constant for \(S-LuxR\) generation
\(d_{LuxI}\) is the rate constant for \(S-LuxI\) degradation
\(d_{LuxR}\) is the reaction rate constant for \(S-LuxR\) degradation
The rate of change of the AHL - LuxR complex is determined by the following two reversible and degradation reactions :
\begin{equation} S-AHL\;+\;S-LuxR\;\;⇌\;\;S-(A-R) \end{equation}
\(k_{A-R}\) is the rate constant for the forward reaction \(k'_{A-R}\) is the rate constant for the reverse reaction
\begin{equation} 2(S-(A-R))\;\;⇌\;\;S-(A-R)_{2} \end{equation}
\(k_{(A-R)_2}\) is the rate constant for the forward reaction \(k'_{(A-R)_2}\) is the rate constant for the reverse reaction
\begin{equation} S-(A-R) \;\;→\;\; ϕ \end{equation}
\(d_{A-R}\) is the degradation rate constant for AHL - LuxR complexes
The differential equations for the AHL - LuxR complex are as follows:
\begin{equation} \delta_{S-(A-R)-1} \;=\; k_{A-R}\,[\,S-AHL_{in}\,][\,S-LuxR\,]\,-\,k'_{A-R}\,[\,S-(A-R)\,] \tag{34} \end{equation}
\begin{equation} \delta_{S-(A-R)-2} \;=\; 2⋅k_{(A-R)_{2}}\,[\,S-(A-R)\,]^2\,-\,2⋅k'_{(A-R)_{2}}\,[\,S-(A-R)_2\,] \tag{35} \end{equation}
\begin{equation} \frac{d\,[\,S-(A-R)\,]}{dt}\;=\; \delta_{S-(A-R)-1} - \delta_{S-(A-R)-2}-d_{A-R}\,[\,S-(A-R)\,] \tag{36} \end{equation}
Changes in the \((A-R)_2\) complex are determined by two reversible reactions and degradation
\begin{equation} 2(S-(A-R))\;\;⇌\;\;S-(A-R)_{2} \end{equation}
\(k_{(A-R)_2}\) is the rate constant for the forward reaction \(k'_{(A-R)_2}\) is the rate constant for the reverse reaction
\begin{equation} S-(A-R)_{2}\;+\;Plux\;\;⇌\;\;S-Plux-(A-R)_{2} \end{equation}
\(k_{Plux-(A-R)_2}\) is the rate constant for the forward reaction \(k'_{Plux-(A-R)_2}\) is the rate constant for the reverse reaction
\begin{equation} S-(A-R)_2 \;\;→\;\; ϕ \end{equation}
\(d_{(A-R)_2}\) is the degradation rate constant for AHL - LuxR complexes
The differential equation for the AHL-LuxR dimer is as follows :
\begin{equation} \delta_[\,S-(A-R)_{2}-1\,]\;=\;k_{(A-R)_{2}}\,[\,S-(A-R)\,]^2\,-\,k'_{(A-R)_{2}}\,[\,S-(A-R)_{2}\,] \tag{37} \end{equation}
\begin{equation} \delta_[\,S-(A-R)_{2}-2\,]\;=\;k_{Plux-(A-R)_{2}}\,[\,S-(A-R)\,][\,Plux\,]\,-\,k'_{Plux-(A-R)_{2}}\,[\,mRNA_{biofilm}\,] \tag{38} \end{equation}
\begin{equation} \frac{d\,[\,S-(A-R)_{2}\,]}{dt}\;=\;\delta_[\,S-(A-R)_{2}-1\,]-\delta_[\,S-(A-R)_{2}-2\,]-d_{(A-R)_{2}}\,[\,S-(A-R)_{2}\,] \tag{39} \end{equation}
Fig.10 AHL expression of SRB
As can be seen from Figure 10, 1 g / L of SRB can excrete about 10.9 nM of AHL to the outside.
Combining the two parts of Simulation of SRB and Simulation of E.coli, we further extended the model into a predator-prey model in the hope that the final population of the two bacteria produces an oscillating effect.
Fig.11 Interaction between SRB and E. coli
We hope the engineered bacterium E. coli and the SRB in the sewer form a predator-prey model. The specific mechanism is that E. coli excretes AidH to degrade the AHL exocytosed by the SRB. When the concentration of AHL is too low, it inhibits the formation of the SRB biofilm and induces the suicide of E. coli, which guarantees biosafety.
To provide data and relationship support for automata modeling at the cellular level, we visualized the relationships of E.coli-AidH, AHL_e-AidH_e, and SRB-AMP based on modeling for differential equations and enzyme kinetic equations, respectively.
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Fig.12 Various relationship diagrams
As can be seen from the figure, for E.coli-AidH, the production of AidH by the engineered bacterium E. coli is a linear relationship; for AHL_e-AidH_e, the concentration of extracellular AHL decreases linearly with the increase of extracellular AidH and eventually stabilizes; and for SRB-AMP, the concentration of SRB decreases linearly with the increase of AMP concentration and eventually stabilized.
To better understand our biosafety suicide module, we modeled the growth of SRB-E.coli by means of differential equations.
Meanwhile, the necessity of the SRB inhibitory module was discussed several times within the team, so the SRB-E.coli interaction process with and without the inhibitory module was simulated in the modeling work. Combined with the modeling results, the final decision was made to introduce the antimicrobial module in the engineered bacteria.
Fig.13 Changes in concentration of SRB-E.coli and AHL-AidH without SRB repressing module
Without the SRB repressing module, the number of SRBs and E.coli will keep oscillating in a certain range for a certain period of time, with a pulse width of about 200 hours, during which the epithelium can grow to about 70% of its original size according to the calculation in the Cell layer model, which makes it clear that the SRB repressing module is necessary. The module is necessary.
Fig.14 Changes in concentration of SRB-E.coli and AHL-AidH with SRB repressing module
And in the simulation with the SRB repressing module, there was an oscillatory effect in the concentrations of AHL and AidH with increasing time; similarly, we found an oscillatory effect in SRB and E.coli. Overall, the concentrations of AHL and AidH converged to a stable low value at about 1100 h (about day 46); the concentrations of SRB and E.coli converged to a stable 0 value at about 1150 h (about day 48), which suggests that the SRB in the sewers will be largely removed from the sewers from about day 48 after we put the engineered bacteria, and at the same time, our engineered bacteria will die to avoid leakage.
| Parameters | Value | Explanations |
|---|---|---|
| \(K_{mRNA_{LuxI}}\) | 0.1 | Reaction rate constants generated by \(mRNA_{LuxI}\) |
| \(K_{LuxI}\) | 0.5 | Rate constant for LuxI production |
| \(d_{LuxI}\) | 0.05 | Rate constant for LuxI degradation |
| \(K_{mRNA_{LuxR}}\) | 0.3 | Rate constant for \(mRNA_{LuxR}\) production |
| \(d_{mRNA_{LuxR}}\) | 0.03 | Rate constant for \(mRNA_{LuxR}\) degradation |
| \(K_{LuxR}\) | 0.5 | Rate constant for LuxR production |
| \(d_{LuxR}\) | 0.05 | Rate constant for LuxR degradation |
| \(K_{AHL_{in}}\) | 0.65 | Rate constant for \(AHL_{in}\) production |
| \(d_{AHL_{in}}\) | 0.05 | Rate constant for \(AHL_{in}\) degradation |
| \(r\) | 0.005 | Fick's law diffusion coefficient |
| \(k_{A-R}\) | 0.005 | Forward rate constant for AHL-LuxR complex formation |
| \(k\prime_{A-R}\) | 0.05 | Reverse rate constant for AHL-LuxR complex dissociation |
| \(d_{A-R}\) | 0.2 | Rate constant for AHL-LuxR complex degradation |
| \(k_{\left(A-R\right)_2}\) | 0.003 | Forward rate constant for AHL-LuxR dimer formation |
| \(k\prime_{\left(A-R\right)_2}\) | 0.03 | Reverse rate constant for AHL-LuxR dimer dissociation |
| \(d_{\left(A-R\right)_2}\) | 0.02 | Rate constant for AHL-LuxR dimer degradation |
| \(k_{Plux-\left(A-R\right)_2}\) | 0.05 | Forward rate constant for AHL-LuxR dimer binding to \(P_{lux}\) |
| \(k\prime_{Plux-\left(A-R\right)_2}\) | 0.0062 | Reverse rate constant for AHL-LuxR dimer dissociation from \(P_{lux}\) |
Table 1. All parameters used above, explanations and corresponding references
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