<!DOCTYPE html> Suicide Model

The Suicide Model

Background

To guarantee biosafety, many teams give special consideration to preventing engineered bacteria from escaping into the environment. In our project, we want the engineered bacteria¹ to stay in the desert and not get out to other kinds of ecosystems, that's why we need a mechanism to induce suicide at the proper time.

We choose a stress-induced suicide module named mzaEF to achieve this goal. This module consists of two adjacent genes, mazE and mazF. MazF is a toxic protein that can cleave a certain sequence of mRNA, ACA for E.coli MazF ^[1] and UACAU for B. subtilis MazF^[2], and then trigger programmed cell death. Here are some properties of this suicide system^[3]:

When MazF reaches a thershold, the cell death will be irreversible.
MazE act as an antitoxin which can bind to MazF and inhibit its function.
MazE is a labile protein whereas MazF can stably exist for a long time.

The process of suicide can be divided into 3 steps: Firstly, bacteria will meet arabinose, which neither can be produced^[4] by themselves nor exist in the desert, when they escape from the desert. Then arabinose in the environment will be absorbed by bacteria through the arabinose transport system (AraE, AraNPQ)^[5]^[6], arabinose in bacteria cells can promote transcription of mazE/F as an inducer. After that, once the concentration of toxin protein MazF reaches a threshold, apoptosis will be induced and bacteria will die.^[7]^[8]

Mathematical model

This model is constructed to simulate the physiological process after the engineered bacteria meet arabinose, including arabinose transportation, activation of promoter, transcription& translation, and binding of MazE and MazF (if MazE exists). Then we will get a relationship between concentration (of arabinose, mRNAs , and proteins) and time. With this concentration-time relation, we can know whether the suicide system works well.

For simplicity, we make the following principal assumptions:

We assume that
- regulation at the translation level can be ignored
- Arabinose transportation system in E. coli(AraE, AraFGH) can be a replacement of in B. subtilis, for they share plenty of similarities.^[9]
- The interaction between promoter of araE and its repressor AraR in B. subtilis^[5] can be described by using lacO and repressor LacI^[10]^[11], for they are both negative regulation induction system.
- AraE only transports arabinose, although it can function as a transporter for more than one kind of sugars.^[12]

Equations and Variables

We use 7 ODEs to describe the whole process^[7]^[9] and solve them in Python.

Table 1. ODEs used in model

No.	Equation	Descriptions
1	$\frac{d[A_{in}]}{dt}=\frac{V_E[A_{ex}]}{K_E+[A_{ex}]}[M_{araE}]+\frac{V_{FGH}[A_{ex}]}{K_{FGH}+[A_{ex}]}[M_{araFGH}]-(\mu+\gamma_{Ain})[A_{in}]$ ³	The transport of arabinose from extracellular to intracellular
2	$\frac{d[M_{araE}]}{dt}=\alpha_E+\frac{V_{mE}[A_{in}]^n}{K_{mE}^n+[A_{in}]^n}-(\mu+\gamma_E)[M_{araE}]$	Transcription of araE
3	$\frac{d[M_{araFGH}]}{dt}=\alpha_{FGH}+\frac{V_{mFGH}[A_{in}]^n}{K_{mFGH}^n+[A_{in}]^n}-(\mu+\gamma_{FGH})[M_{araFGH}]$	Transcription of araFGH
4	$\frac{d[M]}{dt}=Dk_M\frac{1+k_1[A_{in}]}{K+k_1[A_{in}]}-\frac{d_{large}(\beta[MazF])^2}{(\beta[MazF])^2+K_t^2}[M]-d_m[M]$	Transcription of mazE/mazF gene
5	$\frac{d[MazF]}{dt}=b_2[M]-a_T[MazF][MazE]-d_c[MazF]+d_{a2}[MazEF]$	Translation of MazF
6	$\frac{d[MazE]}{dt}=b_1[M]-a_T[MazF][MazE]-d_a[MazE]$	Translation of MazE
7	$\frac{d[MazEF]}{dt}=a_T[MazE][MazF]-d_c[MazEF]-d_{a2}[MazEF]$	Binding of MazE and MazF to form MazEF

We comprehensively considered possible reasons that can cause the production and decay of substances and added them to the ODEs, aiming to simulate the phycological processes as close to real situation as possible. The reasons include the intrinsic transcription/translation and decay of mRNAs and proteins, the cleavage of mRNA caused by MazF, and concentration change due to protein binding and disassembling.

For the 3 equations describing arabinose transportation (Eq.1 to 3), all of them contain 1 or 2 Michaelis Menten constant(s): $K_E$ and $K_{FGH}$ , $K_{mE}$ , $K_{mFGH}$ , respectively. These constants are used to decide how the rate changes as $[A_{in}]$ and $[A_{ex}]$ changing. Moreover, as for the two parameters describing decay of concentration, $\mu$ and $\gamma$ , they stand for the decay caused by bacteria growth and degradation of mRNA (or consumption of arabinose in Eq.1), respectively.

The latter 4 equations describe the transcription, translation, and protein binding process which occurs after the promoter was activated. Equation 4 is the key point of our model, it represents how arabinose regulates the araE promoter and the effect of MazF cleaving mRNA. In this equation, we use the term $\frac{1+k_1[A_{in}]^2}{K+k_1[A_{in}]^2}$ ^[11] to indicate the effect on arabinose of activating the promoter upstream mazEF. As $[A_{in}]$ increases, the transcription rate will increase.

The second term of Eq.4 is about the effect of MazF. One parameter that needs to be explained is $\beta$ , the concentration factor of MazF toxic protein. As MazF cleaving mRNA, it functions to all mRNA in bacteria simultaneously. Thus, we need to introduce a concentration factor to get an 'effective concentration' of MazF when it cleaves its own mRNA or some other specific kind of mRNA. In this way, we can roughly use the MazF mRNA to represent other mRNAs that can be cleaved by MazF. And the decrease of mRNA concentration due to cleavage would trigger apoptosis. The meaning and value of parameters can be found below.

Parameters

Table 2. Parameters and their value

Parameter	Descriptions	Value
$V_E$	Maximal transportation rates of the extracellular arabinose by AraE	$48\mathrm{s}^{-1}$
$V_{FGH}$	Maximal transportation rates of the extracellular arabinose by AraFGH	$4.8\mathrm{s}^{-1}$
$K_E$	Michaelis Menten constant for araE facilitated transport of A~ex~	$1.50\times10^{-4}\mathrm{M}$
$K_{FGH}$	Michaelis Menten constant for araFGH facilitated transport of A~ex~	$6\times10^{-6}\mathrm{M}$
$\mu$	loss of internal arabinose/mRNA due to the cellular growth	$5.78\times10^{-4}\mathrm{s}^{-1}$
$\gamma_{Ain}$	Consumption of arabinose through activation of araBAD genes and its degradation	$0.01155\mathrm{s}^{-1}$
$n$	Hill factor	$3$
$\alpha_E$	Constitutive transcriptional rate of araE mRNA	$2.83\times10^{-11}\mathrm{M\cdot s}^{-1}$
$V_{mE}$	maximal induction rate	$4.28\times10^{-9}\mathrm{M\cdot s}^{-1}$
$K_{mE}$	the amount of internal arabinose to get half of the maximal induction rate	$1.06\times10^{-4}\mathrm{M}$
$\gamma_E$	degradation rate of araE mRNA	$0.011552s^{-1}$
$\alpha_{FGH}$	Constitutive transcriptional rate of araFGH mRNA	$7.55\times10^{-11}\mathrm{M\cdot s}^{-1}$
$V_{mFGH}$	maximal induction rate	$1.1266\times10^{-8}\mathrm{M\cdot s}^{-1}$
$K_{mFGH}$	the amount of internal arabinose to get half of the maximal induction rate	$1\times10^{-9}\mathrm{M}$
$\gamma_{FGH}$	degradation rate of araE mRNA	$0.0115\mathrm{s}^{-1}$
$K$	Total amount of repressor	$7200$
$k_1$	Equilibrium constant for the repressor-inducer reaction	$2.52\times10^{10}\mathrm{M}^{-2}$
$D$	Concentration of promoter	$2$
$K_M$	The maximum initiation rate of the promoter	$0.03\mathrm{s}^{-1}$
$d_{large}$	Increased mRNA degradation caused by MazF	$0.198\mathrm{s}^{-1}$
$\beta$	Concentration factor of MazF	$0.1$
$K_t$	Threshold for cleavage of mRNA	$15$
$d_m$	Decay rate of mRNA	$0.002\mathrm{s}^{-1}$
$b_1$	MazE translation rate	$0.122\mathrm{s}^{-1}$
$d_a$	Decay rate of MazE	$0.00231\mathrm{s}^{-1}$
$b_2$	MazF translation rate	$0.009\mathrm{s}^{-1}$
$a_T$	Binding rate of MazE and MazF	$0.000365\mathrm{s}^{-1}$
$d_c$	Decay rate due to cell division	$0.00028\mathrm{s}^{-1}$
$d_{a2}$	Decay rate of MazE within complex MazEF	$0.000231\mathrm{s}^{-1}$

Results

As the inducement of cell death is that MazF cleaves mRNA, we use the level of mRNA to verdict whether the cell is alive, assuming that after a dramatic decrease of mRNA, the apoptosis process would initiate.

Simulation

To be more close to reality, we designed two kinds of limit simulation:

The first one(Simulation Ⅰ) is a two-step simulation: Run the simulation process without any arabinose as the engineered bacteria in the desert. Then we add arabinose into the system ( $[A_{ex}]=10^{-7}\mathrm{M}$), just like when they escaped the desert.
The second kind of simulation(Simulation Ⅱ) considered those bacteria that born out of the desert, so they start with arabinose outside. We set the arabinose concentration at $10^{-7}\mathrm{M}$ in both two simulations.

Our purpose is to use these two limit simulations to cover all situations that the engineered bacteria may encounter. The first kind of simulation refers to bacteria that are born in the desert and get out after a quite long time. Whereas the second kind of simulation represents bacteria that are born out of the desert. If the suicide process can be initiated in these two situations, we can prove that it can work well in any kind of situation. Solving the ODEs, we draw a concentration-time graph for each substance in our system, including internal arabinose, different kinds of mRNA, and proteins (MazE, MazF, MazEF).

We add arabinose 200 minutes after the simulation started. There is no significant difference shown between these two simulations. The reason is that stable level of mRNA/MazF is fairly low, which equals to $4.17\times10^{-3}$ and $0.134$ , respectively. With the external arabinose concentration $[A_{ex}] = 10^{-7}\mathrm{M}$ , which is relatively low for the content of arabinose in soil is considered as approximately 0.03 g/kg^[4], the apoptosis would initiate 30~60 minutes later after arabinose was added. It is proved that our engineered bacteria can commit suicide when they moved out of the desert, thus guarantee the biosafety.

Analysis

Simulation of leak expression

Basal transcription of mazF exists in our system. To know if this leakage of expression can influence the survival of bacteria, we run a simulation when arabinose does not exist.

The level of MazF cannot reach the threshold of 15 and mRNA does not show decrease in this case, so the leakage would not interrupt survival.

This result helped us to design the gene circuit. We were concerned that if the leakage harms bacteria's survival, mazE gene should be added to the gene circuit and have a very high level of expression in order to inhibit MazF. However, putting a large amount of energy on MazE production would bring pressure to the growth of bacteria. Fortunately, the result guided us to cancel mazE in gene circuit.

Determining the reason of apoptosis

Is MazF the only reason causing cell death? To know whether the bacteria can stay alive without MazF, we remove the effect of MazF in two ways: add mazE to the gene circuit, or delete mazF from it.

So the answer to the question above is “YES”. The figures demonstrate that bacteria with mazE or without mazEF would survive well for they do not show a decrease of mRNA. Then we know the intrinsic decay cannot cause a decrease of mRNA. The level of MazF in bacteria with mazE gene shows a huge drop in a very short time, certifying the function of MazE protein to bind to MazF.

Arabinose Concentration

$[A_{ex}]$ ) and run the simulation to a stable state.

The arabinose transportation system can work normally at these five different concentration, gathering arabinose in the cells and making $[A_{in}]$ approximately 10 times of $[A_{ex}]$ . As for mRNA, we could observe a dramatic decrease at each concentration of $A_{ex}$ , which is more than 75%, except for the bacteria at the concentration of $[A_{ex}]=10^{-9}\mathrm{M}$ . And MazF can also be produced normally in these five situations. As a result, we find that the suicide system, including arabinose transportation system and mazEF module(without mazE gene in circuit however), can work normally and initiate suicide successfully at a very small concentration and a large range of extracellular arabinose.

When analyzing the tendency, it is clear that as $[A_{ex}]$ getting higher, the amount/concentration of all the three substance increase as well. However, although the increase at lower concentration interval is obvious, curves of $[A_{ex}]=10^{-7}\mathrm{M}$ and higher concentration( $10^{-6}\mathrm{M}$ and $10^{-5}\mathrm{M}$ ) are very similar to each other, even the highest two curves are almost coincident. All of the three curves of mRNA grow to a peak in 10 minutes and then drop fast to a stationary phase 20 or 30 minutes later. The possible reason is that repression caused by AraR is completely canceled by higher concentration of arabinose. The process of apoptosis would initiate then.

Conclusion and Prospects

Biosafety is a vital issue especially for the synthetic biology project using engineered bacteria like us. The result proved that our bacteria would commit suicide in about 1 hour at a relatively low concentration of arabinose, and can survive well without arabinose.

However, this model is far from a perfect one. Because of the lack of experimental or literature support, we had to introduce two substitutions to our model: E. coli arabinose transport system and LacI repressor. Although we believe that they are reasonable, there would be a gap between the simulation result and reality. The accuracy of this model may not be that satisfying. Time to initiate apoptosis and minimum inducing concentration of arabinose which is around 30 minutes and $10^{-8}\mathrm{M}$ can be slightly erroneous. Luckily, qualitative properties would not change greatly, bacteria could commit suicide without doubt.

Besides, it is the first time for most of us to know and build a mathematical model for a complex biosystem. We realized that dry and wet lab should make good use of each strength. Dry lab can use previous data to simulate experiment and give some advice to the wet lab and even design of project. On the other hand, the data used in modeling are from experiments by the wet lab, and results of simulation should be confirmed by experiment. Only if we do well in both 'lab', we can successfully finish a project.

References

[1] Zhang, Y., Zhang, J., Hoeflich, K. P., Ikura, M., Qing, G., & Inouye, M. (2003, oct). MazF cleaves cellular mRNAs specifically at ACA to block protein synthesis in Escherichia coli. Molecular cell, 12(4), 913–923. doi: 10.1016/s10972765(03)004027
[2] Park, J.H., Yamaguchi, Y., & Inouye, M. (2011, aug). Bacillus subtilis MazF-bs (EndoA) is a UACAU specific mRNA interferase. FEBS letters, 585(15), 2526–2532. Retrieved from https://pubmed.ncbi.nlm.nih.gov/21763692https://www.ncbi.nlm.nih.gov/ pmc/articles/PMC3167231/ doi: 10.1016/j.febslet.2011.07.008
[3] Amitai, S., Yassin, Y., & Engelberg-Kulka, H. (2004, dec). MazF mediated cell death in Escherichia coli: a point of no return. Journal of bacteriology, 186(24), 8295–8300. Retrieved from https://pubmed.ncbi.nlm.nih.gov/15576778https://www.ncbi.nlm.nih.gov/ pmc/articles/PMC532418/ doi: 10.1128/JB.186.24.82958300.2004
[4] Gunina, A., & Kuzyakov, Y. (2015). Sugars in soil and sweets for microorganisms: Review of origin, content, composition and fate. Soil Biology and Biochemistry, 90, 87–100. Retrieved from http://www.sciencedirect.com/science/article/pii/S0038071715002631 doi: https://doi.org/10.1016/j.soilbio.2015.07.021
[5] Mota, L. J., Sarmento, L. M., & de Sá-Nogueira, I. (2001, jul). Control of the arabinose regulon in Bacillus subtilis by AraR in vivo: crucial roles of operators, cooperativity, and DNA looping. Journal of bacteriology, 183(14), 4190–4201. Retrieved from https://pubmed.ncbi.nlm.nih.gov/11418559https://www.ncbi.nlm.nih.gov/pmc/articles/PMC95308/ doi: 10.1128/JB.183.14.41904201.2001
[6] Sá-Nogueira, I., Nogueira, T. V., Soares, S., & de Lencastre, H. (1997, mar). The Bacillus subtilis L- arabinose (ara) operon: nucleotide sequence, genetic organization and expression. Microbiology (Reading, England), 143 ( Pt 3, 957–969. doi: 10.1099/002212871433957
[7] Vet S, Vandervelde A, Gelens L (2019, feb). Excitable dynamics through toxin-induced mRNA cleavage in bacteria. PLOS ONE 14(2): e0212288. https://doi.org/10.1371/journal.pone.0212288
[8] Engelberg-Kulka, H., Hazan, R., & Amitai, S. (2005, oct). mazEF: a chromosomal toxin-antitoxin module that triggers programmed cell death in bacteria. Journal of cell science, 118(Pt 19), 4327– 4332. doi: 10.1242/jcs.02619
[9] Yildirim, N. (2012). Mathematical modeling of the low and high affinity arabinose transport systems in Escherichia coli. Molecular BioSystems, 8(4), 1319–1324. Retrieved from http://dx.doi.org/10.1039/C2MB05352G doi: 10.1039/C2MB05352G
[10] Mackey, M. C., Santillán, M., Tyran-Kamińska, M., & Zeron, E. S. (2012). The utility of simple mathematical models in understanding gene regulatory dynamics. In Silico Biology, 12, 23–53. doi: 10.3233/ISB-140463
[11] Yildirim, N., & Mackey, M. C. (2003, may). Feedback regulation in the lactose operon: a mathematical modeling study and comparison with experimental data. Biophysical journal, 84(5), 2841–2851. doi: 10.1016/S00063495(03)700137
[12] Krispin, O., & Allmansberger, R. (1998, jun). The Bacillus subtilis AraE Protein Displays a Broad Substrate Specificity for Several Different Sugars. Journal of Bacteri ology, 180(12), 3250 LP – 3252. Retrieved from http://jb.asm.org/content/180/12/ 3250.abstract doi: 10.1128/JB.180.12.32503252.1998

1 'Engineered bacteria' and 'bacteria' in this website refers to B. subtilis. ↩

2 The image of polymerase is used as a sketch, got from https://3dprint.nih.gov/discover/3DPX-011523 by Kassube SA, Fang J, Grob P, et al. "Structural insights into transcriptional repression by noncoding RNAs that bind to human Pol II." ↩

3

$A_{ex}$ : external arabinose;

$A_{in}$ : internal arabinose. ↩

Team:XJTU-China/Suicide-Model

XJTU-China