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Model

Our model functions supplementally to the Measurement experiments performed with pBAD and the constitutive promoters. Firstly, it shows very clearly which processes influence the production of the toxin/antitoxin. But most importantly, it determines which constitutive promoters would be the best combination to provide the most optimal ratio for both toxin-antitoxin systems. This is based on the determined ~130 factor difference in activity between the antitoxins, which supports the combination of BBa_J23100 with BBa_J23103 or BBa_J23113.

Background

The aim of DOPL LOCK is to enable out-of-lab, semi-contained GMO applications, and to prevent the release of genetically modified material (see Description page). This is done by both preventing the escape of the organism as well as horizontal gene transfer (HGT). For DOPL LOCK to function, there should be homeostasis between both toxins and antitoxins to prevent bacterial cells from being killed in the semi-contained environment [1]. To determine the conditions to reach toxin-antitoxin (TA) homeostasis, we have created a model which builds on the foundation laid by the measurements performed with pBAD and the constitutive promoters (see Measurements page). Our model predicts the concentration of the toxin and antitoxin based on the concentration of the inducer and the promoters used in the system. The toxin gene is placed downstream a constitutive promoter while the antitoxin gene is placed downstream an inducible promoter. This way, biocontainment of the bacteria can be regulated by the presence of the inducer, since the lack of this compound will result in the depletion of the antitoxin and the remaining toxin will kill the cell. Since the antitoxin A and B both are behind the same inducible promoter, the two TA systems should be kept in balance by modifying the constitutive promoters of the Toxin A and B. This is essential since each TA system has different degradation rates and requires different toxin-antitoxin expression levels, especially considering various classes of TA systems [2]. The model predicts the optimal ratio of the inducer for each constitutive promoter , based on the ideal ratio between both toxins and antitoxins and the ideal concentration of the inducer. Thereby, the model determines which constitutive promoters would provide the optimal ratio for both TA systems. Furthermore, the model shows which processes influence the production of the toxin or antitoxin. This brings our DOPL LOCK system closer to the ideal situation of homeostasis and lowers the burden on cells, by only expressing ideal levels of toxin and antitoxin. This is important because having high protein production in the cell can cost a lot of resources.

Theory

In our DOPL LOCK system, antitoxin production is regulated by the inducible promoter pBAD while toxin production is under control of a constitutive promoter from the Anderson family (see iGEM Parts Registry). The production of the antitoxin is visualized in a schematic (Figure 1). The first parameter taken up in our model is the uptake of the inducer (p.kin) (Table 1) [5] of pBAD, L-arabinose, which enters the cell via high-affinity transport [3]. After entering the cell, L-arabinose can either bind to the promoter, forming a complex (p.kon) or be degraded (p.d3) inside the cell [4]. This complex can unbind as well (p.koff) and is therefore part of an equilibrium. This means that both the binding and unbinding reactions are taking place continuously to maintain equilibrium and enable inhibition of production by removing the inducer. The complex of the pBAD promoter with L-arabinose enables the start of antitoxin transcription. The transcription rate (p.k1) is another parameter taken up in the model. In the case of the toxin-antitoxin system HOK/SOK, the RNA is already functional as toxin/antitoxin and therefore the only reaction taking place is the degradation of RNA (p.d1). In the case of TA system MazE/F, the formed mRNA can be either degraded (p.d1) or translated into the protein (p.k2), which is the toxin/antitoxin. The functional protein can be degraded as well (p.d2).

Schematic overview model

Figure 1: Display of the processes involved in the production of the antitoxin. It starts with the uptake of arabinose (p.kin), then the complex forming between arabinose and the promoter (p.kon & p.koff), transcription (p.k1), translation (p.k2) and the degradation of RNA (p.d1), the protein (p.d2) and arabinose (p.d3).

Table 1: Parameters that are used in the model for MazE/F or HOK/SOK production

Name	Value (MazE/MazF)	Value (HOK/SOK)	Source
p.CN (copy number)	1	1	Assumed
p.d1 (mRNA degradation rate)	0.6931	0.6931	[5]
p.d2 (protein degradation rate)	0.02	N/A since RNA TA system	[8]
p.k2 (translation rate)	3.6	N/A since RNA TA system	[6]
p.k1 (transcription rate)	1.7	1.7	[5]
p.kin (transport rate L-arabinose)	288	288	[5]
p.kon (binding rate)	0.0072	0.0072	[7]
p.koff (unbinding rate)	0.0072	0.0072	[7]
p.d3 (degradation inducible compound	0.0693	0.0693	[5]

Model

The model is based on the production of the antitoxin that is shown in Figure 1. In the model, a set of equations is used. These derivatives display the change in concentration of each compound or structure over time. Based on the starting concentrations, the current concentration of each compound/structure is calculated. The model is programmed as follows:

function[dxdt] = model\_TF(t,x,p)
 %L-arabinose extracellular
 dxdt(1,1) = -p.kin\*x(1);
 %Promoter bound to inducible compound
 dxdt(2,1) = +p.kon\*x(5)\*x(6)-p.koff\*x(2);
 %mRNA
 dxdt(3,1) = p.k1\*x(2)-p.d1\*x(3);
 %Protein
 dxdt(4,1) = p.k2\*x(3)-p.d2\*x(4);
 %L-arabinose intracellular
 dxdt(5,1) = -p.kon\*x(5)\*x(6) + p.koff\*x(2) - p.d3\*x(5) + p.kin\*x(1);
 %Transcription Factor
 dxdt(6,1) = -p.kon\*x(5)\*x(6);
 end

For each derivative, the parameters (Table 1), such as the transcription rate and translation rate, influencing the concentration are taken into account. These parameters depend on which promoter is used and which TA system is simulated. The derivatives change over time and are dependent on the variable parameters at that moment in time, for example the L-arabinose concentration. The results can be displayed in plots which show the trends and changes during the first hours after adding in L-arabinose at the start of the simulation.

Assumptions

For the model, several assumptions were made. They are summed up in the following list.

The plasmid copy number of both plasmids is the same, and therefore set to 1. This plasmid copy number determines the number of plasmids with the promoters/transcription factors present, which influences the production of RNA and protein expression. Even though the plasmid copy number fluctuates as described [9]. We have assessed 16 different combinations of plasmids with different Origins of Replications (Oris) during our lab time to find a compatible pair, as described in our engineering cycles. We chose the combination with similar expression levels, as described in our Results page. With this information, we assume the plasmid copy number is almost the same in the final DOPL LOCK system. Within our model, these results will provide the basis to assume the plasmid copy number is the same. By assuming both are 1, one factor that could influence the final concentration is taken out of the equation.
We assume that the L-arabinose is not used as an energy source by the bacteria. Although some bacterial strains like Escherichia coli DH5a may metabolize L-arabinose, other strains do not [10]. This assumption is based on the usage of the E. coli TOP10 strain that is araBADC- and araEFGH+, which means L-arabinose can be transported, but not metabolized [4]. For our experiments, we used E. coli TOP10.
The TA genes in the host chromosome are only transcribed in stressful conditions [11]. When E. coli are maintaining homeostasis the conditions will not be stressful, thus the toxins will not be transcribed at significant levels. We, therefore, assume that these pre-existing TA genes will not influence the expression and function of the TA systems in external plasmids.
All parameters are stationary and only dependent on the concentrations in the cell. The toxin build-up in this model is assumed to have no impact on the transcription/translation functionality of the cell. The inducer uptake is assumed to be static and only dependent on the extracellular L-arabinose concentration.

Results model MazE/F & HOK/SOK

For our model, we chose MazE/MazF and HOK/SOK. We chose to model MazE/MazF because it is a protein TA system and the results in our lab unexpectedly showed no effect of MazE/MazF, and HOK/SOK as an example of RNA TA systems. The model is used to predict the final concentration of antitoxin of the TA systems MazE/F and HOK/SOK, based on the following initial conditions, assuming L-arabinose is added at t=0, with an arbitrary chosen concentration(Table 2):

Table 2: Initial conditions for the model

	Initial condition (mM)
L-arabinose extracellulair	5
Promoter complex	0
mRNA	0
Protein	0
L-arabinose intracellulair	0
Available promoters	100%

TA system MazE/MazF

The model for MazE/F is explained in the theory, only with applicable parameters. The overview of the processes in the cell is illustrated in Figure 2, in which the final protein is the antitoxin. Table 3 shows all the final parameters used in the model for MazE/MazF.

Table 3: Parameters that are used in the model for MazE/F production

Name	Value	Source
p.CN (copy number)	1	Assumed
p.d1 (mRNA degradation rate)	0.6931	[5]
p.d2 (protein degradation rate)	0.02	[8]
p.k2 (translation rate)	3.6	[6]
p.k1 (transcription rate)	1.7	[5]
p.kin (transport rate)	288	[5]
p.kon (binding rate)	0.0072	[7]
p.koff (unbinding rate)	0.0072	[7]
p.d3 (degradation inducible compound	0.0693	[5]

Schematic overview model

Figure 2: Display of the processes involved in the production of the antitoxin. It starts with the uptake of L-arabinose (p.kin), then the complex forming between L-arabinose and the promoter (p.kon & p.koff), transcription (p.k1), translation (p.k2) and the degradation of RNA (p.d1), the protein (p.d2) and L-arabinose (p.d3).

The toxin MazF and antitoxin MazE are proteins and interact in a ratio of 1:1 [12]. The model for the antitoxin MazE results in the following figures (Figures 3 and 4), which shows the concentration in mM of proteins per cell based on the initial conditions.

Production of antitoxin

Figure 3: The production of the antitoxin over time.

Development of concentrations

Figure 4: Development of concentrations over time.

Our experimental results show that up to 1% L-arabinose concentration did not show any significant effect of the toxin MazF on the bacteria, in 16 hours. Based on the molar mass of L-arabinose at 150.13 g/mole, we calculated that 1% L-arabinose in 200 µL solutions results in 1.3 * 10^-5 mM L-arabinose. By using this concentration as initial condition, while keeping the model the same, the following production of the toxin MazF was modelled to understand why no effect was observed (Figure 5).

production toxin low

Figure 5: The production of the toxin over time with initial L-arabinose concentration of 1.3 * 10^-5 mM.

As is visible, the highest concentration of MazF reached is about 3 * 10 ^-4 mM. It is suggested that this concentration is not sufficient for toxicity, based on the results from CCU Taiwan in 2017, who showed that after 3 hours bacterial cells have not survived by an initial concentration of 6 mM (Figure 6).

production toxin high

Figure 6: The production of the toxin over time with initial L-arabinose concentration of 6 mM.

The initial condition of 6 mM L-arabinose results in a maximum concentration of about 110 mM, which is several magnitudes higher than the maximal concentration used in our experiments. To use MazE/MazF in our DOPL LOCK system, a constitutive promoter is needed for MazF, which produces a similar concentration of MazF, to ensure the lethality of the toxins on the cells.

TA system HOK/SOK

The model used for HOK/SOK is the same as explained before, but without translation, since HOK and SOK are both RNAs. The overview of the processes in the cell is visible in Figure 5, in which the mRNA is the antitoxin. Table 4 shows all the parameters used in the model and the references for each parameter.

Table 4. Parameters that are used in the model for HOK/SOK production

Name	Value	Source
p.CN (copy number)	1	Assumed
p.d1 (mRNA degradation rate)	0.6931	[5]
p.k1 (transcription rate)	1.7	[5]
p.kin (diffusion rate)	288	[5]
p.kon (binding rate)	0.0072	[7]
p.koff (unbinding rate)	0.0072	[7]
p.d3 (degradation inducible compound	0.0693	[5]

Schematic overview model2

Figure 7: Display of the processes involved in the production of the antitoxin. It starts with the uptake of L-arabinose (p.kin), then the complex forming between L-arabinose and the promoter (p.kon & p.koff), transcription (p.k1) and the degradation of RNA (p.d1) and L-arabinose (p.d3).

The HOK/SOK TA system is dependent on RNAs, which interact in a ratio of 1:1 [13]. The toxin (HOK) is a mRNA and the antitoxin (SOK) is a non-coding RNA. The model for the antitoxin results in the following figures (Figure 8,9):

production antitoxin SOK

Figure 8: The production of the antitoxin over time.

Concentrations SOK model

Figure 9: Development of concentrations over time.

Constitutive promoters

Looking at the results (Figure 3,8), we see that SOK reached its maximum concentration earlier than MazE. This is reasonable since SOK is a RNA antitoxin and has one step in expression less than MazE, which needs to undergo translation before being active. However, when comparing the different maximal concentrations, the SOK concentration is about 0.74 mM maximum, while MazE is about 95 mM. This is a difference of a factor of ~130 between the antitoxins. This means that there should be a factor of this order between the constitutive promoters. When using the Anderson family, the best combination is BBa_J23100 with BBa_J23103 or BBa_J23113 upstream the toxin genes, combined with pBAD upstream the antitoxin genes, based on the measured activity of the promoters by previous iGEM teams (see iGEM Parts Registry).

The model functions supplementally to the measurement experiments performed with pBAD and the constitutive promoters. In these measurements, an L-arabinose concentration of 0.14% was found to give pBAD the same transcriptional activity as BBa_J23100 (see Measurement page). The model provides an additional insight into the different processes that take place between adding L-arabinose and the fluorescence or toxicity of the proteins and therefore can be used to extrapolate the results from fluorescence measurements to other proteins in a substantiated way. An arabinose concentration of 0.14% is expected to enable a working homeostasis for DOPL LOCK.

All in all, this model supports the project in several ways. At first, it shows very clearly which processes influence the production of the toxin/antitoxin. This enabled us to understand cellular processes and toxin effectivity. But most importantly, it determines which constitutive promoters would be the best combination to provide the most optimal ratio for both TA systems. This brings our system closer to the ideal situation of homeostasis and lowers the burden on cells.

Model in the lab

This model can provide an approximation of the ideal concentration of inducer for the antitoxins to be similar in concentration to the toxin. This provides the possibility to start the experiments with the full system within a smaller range of inducer concentrations. Furthermore, the best complementary systems can be determined and (one of the) constitutive promoters before the toxin could be interchanged to make the ratio between toxin and antitoxin optimal for both TA systems.

Checking model with fluorescent proteins

To confirm the model used for the pBAD promoter, the following experiment could be carried out in the laboratory. This experiment allows coupling between the concentrations of L-arabinose and the protein concentration produced by the cell, which enables us to corroborate the output of the model with reality.

Transform the cells with a plasmid containing the pBAD promoter and the mCherry gene.
Grow the cells without L-arabinose until OD₆₀₀ = 0.5.
Add different concentrations of L-arabinose.
Measure the fluorescence and OD₆₀₀ every 10 minutes for five hours.

Besides this experiment (based on the experiments from the Measurement page), two calibration curves have to be measured to compare the data. The first calibration curve between the fluorescence and the concentration mCherry and the second calibration curve between the OD₆₀₀ and the number of cells should be measured. The first to determine the concentration of the fluorescent protein based on the fluorescence measured, the second to determine the number of cells from the OD₆₀₀. Both the OD₆₀₀ and fluorescence are measured by the plate reader during the experiments. The average plasmid copy number of both plasmids should be determined as well, to make the model fit better to the circumstances. However, for now it is assumed to be 1 for both plasmids since they are assumed to be equal to each other.

With the parameters already found in literature and the plasmid copy number, the model can be run and the graphs and data resulting from the model can be compared with the data from the experiment. The data from the experiment first should be analyzed by calculating the concentration from the fluorescence and the number of cells from the OD₆₀₀ by using the calibration curves. Then, the concentration (c) should be divided by the number of cells (N), after which it is multiplied with the number of Avogadro (NA) and the volume of an E. coli cell (V). This gives the following formula:

After analyzing this data, the model can be adjusted slightly to fit the retrieved experimental data better, by adjusting the value of some of the parameters or changing the functions of the derivatives. The adjusted model can be used for the final calculations of the antitoxin concentrations in DOPL LOCK.

Although we started with the proposed experiments, we were unable to finish them due to lab time constraints.

Conclusion

In our project, we designed a model with the aim to establish a balance between the toxin and antitoxin in our DOPL LOCK system. To this end, our model provides insight into the processes concerning the expression of the TA system under the control of the arabinose-inducible promoter. Furthermore, our model is designed to predict the required inducer concentration and paired constitutive promoter to achieve the balance state between the toxin and antitoxin. This brings our DOPL LOCK system closer to the ideal situation of homeostasis. Moreover, the model is designed to be modular and can easily be adapted to other genetic circuits using inducible promoters for gene regulation. This also contributes to the intended modularity of DOPL LOCK, since a range of different cassettes of the DOPL LOCK systems can be described and assessed by our model.

Model link

Click here to download the model

References

Yamaguchi, Y., Park, J. H., & Inouye, M. (2011). Toxin-Antitoxin Systems in Bacteria and Archaea. Annual Review of Genetics, 45(1), 61–79. https://doi.org/10.1146/annurev-genet-110410-132412
Zhang, S. P., Wang, Q., Quan, S. W., Yu, X. Q., Wang, Y., Guo, D. D., Peng, L., Feng, H. Y., & He, Y. X. (2020). Type II toxin–antitoxin system in bacteria: activation, function, and mode of action. Biophysics Reports, 6(2–3), 68–79. https://doi.org/10.1007/s41048-020-00109-8
Vasicek, E. M., O'Neal, L., Parsek, M. R., Fitch, J., White, P., & Gunn, J. S. (2021). L-L-arabinose Transport and Metabolism in Salmonella Influences Biofilm Formation. Frontiers in Cellular and Infection Microbiology, 11. https://doi.org/10.3389/fcimb.2021.698146
Balzer, S., Kucharova, V., Megerle, J., Lale, R., Brautaset, T., & Valla, S. (2013). A comparative analysis of the properties of regulated promoter systems commonly used for recombinant gene expression in Escherichia coli. Microbial Cell Factories, 12(1). https://doi.org/10.1186/1475-2859-12-26
Yildirim, N. (2012). Mathematical modeling of the low and high affinity L-arabinose transport systems in Escherichia coli. Molecular BioSystems, 8(4), 1319. https://doi.org/10.1039/c2mb05352g
Team UC Davis. (2013). Team:UC Davis/Modeling - 2013.igem.org. IGEM. Consulted on 23-07-2021 from https://2013.igem.org/Team:UC_Davis/Modeling
Team St-Andrews. (2011). Team:St Andrews/modelling - 2011.igem.org. IGEM. Consulted on 23-07-2021 from https://2011.igem.org/Team:St_Andrews/modelling
Dubuc, E. (2017). Development of synthetic regulatory gene networks using a cell-free gene expression system. Doctoral Dissertation, Radboud University.
Jahn, M., Vorpahl, C., Hübschmann, T. et al. Copy number variability of expression plasmids determined by cell sorting and Droplet Digital PCR. Microb Cell Fact 15, 211 (2016). https://doi.org/10.1186/s12934-016-0610-8
Koirala, S., Wang, X., & Rao, C. V. (2016). Reciprocal Regulation of l -Arabinose and d -Xylose Metabolism in Escherichia coli. Journal of Bacteriology, 198(3), 386–393. https://doi.org/10.1128/jb.00709-15
Ghafourian S, Raftari M, Sadeghifard N, Sekawi Z. Toxin-Antitoxin Systems: Classification, Biological Function and Application in Biotechnology. Current Issues in Molecular Biology. 2014; 16(1):9-14. https://doi.org/10.21775/cimb.016.009
Dai, J., Chen, Z., Hou, J., Wang, Y., Guo, M., Cao, J., Wang, L., Xu, H., Tian, B., & Zhao, Y. (2021). MazEF Toxin-Antitoxin System-Mediated DNA Damage Stress Response in Deinococcus radiodurans. Frontiers in Genetics, 12. https://doi.org/10.3389/fgene.2021.632423
Gerdes, K., & Wagner, E. G. H. (2007). RNA antitoxins. Current Opinion in Microbiology, 10(2), 117–124. https://doi.org/10.1016/j.mib.2007.03.003

Team:Leiden/Model

Model

Background

Theory

Model

Assumptions

Results model MazE/F & HOK/SOK

TA system MazE/MazF

TA system HOK/SOK

Constitutive promoters

Model in the lab

Checking model with fluorescent proteins

Conclusion

Model link

References

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