Part 1. QS-Lysis

1-0. Introduction

Avoideer uses a QS-Lysis gene circuit to restrict the population density of E. coli, therefore allowing the culture to continuously emit odorants long-term. To understand the behavior of this gene circuit and apply this knowledge to our wet experiments, our modeling team focused on modeling of the QS-Lysis circuit.

QS (Quorum sensing) is the ability of bacteria to control their own gene expression based on the population density. In this project, E. coli detects their population density from the membrane-permeable compound AHL's concentration . In our gene circuit, when the culture's population density increases, the production of lysis-inducing proteins also increases, thus connecting quorum sensing and lysis.

Fig.1 The QS-Lysis circuit

Design of the gene circuit with both quorum sensing and lysis circuits, inspired by iGEM Ecuador 2021’s design. The goal of this circuit is to keep the E. coli population below a certain value, thus allowing for continuous, long-term expression.

In our model, we reproduced the change of concentrations of the compounds we observed in our actual gene circuit using ODE models. Since our gene circuit contains an arabinose promoter, we can change the transcription rate of the gene by changing the arabinose concentration in the culture. Thus, we also predicted what the effect of changing the transcription rate was. With this, we predicted the ideal conditions to maintain E. coli cultures' population density.

1-1. ODE Model

ODE models have been utilized previously, both in academic papers [1] and in iGEM [2]. Particularly, the iGEM Ecuador 2021 team reported succeeding in reproducing the lysis time scale observed in their wet lab experiments with their model. We applied a similar model to our gene circuit and built an ODE model of our QS-Lysis circuit.

1-1-A. Intracellular Chemical Reaction

\begin{align} &\text{DNA}_\text{R} \xrightarrow{\text{CR}\cdot \text{kR}} \text{DNA}_\text{R} + \text{mRNA}_\text{R} \\ &\text{mRNA}_\text{R} \xrightarrow{\text{pR}} \text{mRNA}_\text{R} + \text{R} \\ &\text{mRNA}_\text{R} \xrightarrow{\text{dmR}\cdot \mu } \phi \\ &\text{R} \xrightarrow{\text{dR} + \mu} \phi \\ \end{align}

$\text{DNA}_\text{R}$ - LuxR gene
$\text{mRNA}_\text{R}$ - LuxR gene mRNA
$\text{R}$ - LuxR
$\text{CR}$ - The plasmid number of LuxR plasmid
$\text{kR}$ - LuxR transcription rate
$\text{pR}$ - LuxR translation rate
$\text{dmR}$ - LuxR mRNA decomposition rate
$\mu$ - growth rate

\begin{align} &\text{DNA}_\text{I} \xrightarrow{\text{CI}\cdot \text{kI}} \text{DNA}_\text{I} + \text{mRNA}_\text{I} \\ &\text{mRNA}_\text{I} \xrightarrow{\text{pI}} \text{mRNA}_\text{I} + \text{I} \\ &\text{mRNA}_\text{I} \xrightarrow{\text{dmI}\cdot \mu } \phi \\ &\text{I} \xrightarrow{\text{dI} + \mu} \phi \\ &\text{I} \xrightarrow{\text{kH}} \text{H} + \text{I} \\ \end{align}

$\text{DNA}_\text{I}$ - LuxI gene
$\text{mRNA}_\text{I}$ - LuxI gene mRNA
$\text{I}$ - LuxI
$\text{H}$ - AHL
$\text{CI}$ - The plasmid number of LuxI plasmid
$\text{kI}$ - LuxI transcription rate
$\text{pI}$ - LuxI translation rate
$\text{dmI}$ - LuxI mRNA decomposition rate
$\text{kH}$ - AHL synthesis rate

\begin{align} &\text{DNA}_\text{L} \xrightarrow{\alpha \cdot \text{CL}\cdot \text{kL}} \text{DNA}_\text{L} + \text{mRNA}_\text{L} \\ &\text{DNA}_\text{L} + 4\text{R} + 4\text{H} \xrightarrow{\beta \cdot \text{CL}\cdot \text{kL}} \text{DNA}_\text{L}\left( \text{R} \cdot \text{A} \right)_4 \\ &\text{mRNA}_\text{L} \xrightarrow{\text{pL}} \text{mRNA}_\text{L} + \text{L} \\ &\text{mRNA}_\text{L} \xrightarrow{\text{dmL} + \mu} \phi \\ &\text{I} \xrightarrow{\text{dL} + \mu} \phi \\ \end{align}

$\text{DNA}_\text{L}$ - Lysis Protein gene
$\text{mRNA}_\text{L}$ - Lysis Protein gene mRNA
$\text{L}$ - Lysis protein
$\alpha$ - Base lysis Protein expression
$\beta$ - Induced lysis protein expression
$\text{CL}$ - The plasmid number of Lysis protein plasmid
$\text{kL}$ - Lysis Protein transcription rate
$\text{pL}$ - Lysis Protein translation rate
$\text{dmL}$ - Lysis Protein mRNA decomposition rate
$\text{dL}$ - Lysis Protein decomposition rate

1-1-B. Ordinary Differential Equations

\begin{align} & \frac{d N}{d t}=\mu * N *(1-N / N_0)-\frac{k * L^n}{{L_0}^n+L^n} N \\ & \frac{d[H]}{d t}=N * k H *[I]-d H *[H] \\ & \frac{d[R]}{d t}=\frac{\text{CR} * k R * p R}{d m R+\mu *(1-N / N_0)}[D R]-d R *[R]-\mu *[R] *\left(1-\frac{N}{N_0}\right) \\ & \frac{d[I]}{d t}=\frac{\text{CI} * k I * p I}{d m I+\mu(1-N / N_0)}[D I]-d I *[I]-\mu *[I] *\left(1-\frac{N}{N_0}\right) \\ & \frac{d[L]}{d t}=\frac{C L * k L * p L}{d m L+\mu *(1-N / N_0)}[D L] *\left(\frac{\beta *\left(\frac{[R] *[H]}{A 0}\right)^4}{1+\left(\frac{[R] *[H]}{A 0}\right)^4}\right)-d L *[L]-\mu *[L] *\left(1-\frac{N}{N_0}\right)\end{align}

$N$ - Relative number of cells
$N_0$ - Maximum number of cells
$n$ - Hill coefficient
$k$ - Maximum rate of cell lysis
$\text{[H]}$ - AHL concentration
$\text{[R]}$ - LuxR concentration
$\text{[I]}$ ―LuxI concentration
$\text{[L]}$ ―Lysis protein concentration
$\text{[DR]}$ ―LuxRDNA concentration
$\text{[DI]}$ ―LuxIDNA concentration
$\text{[DL]}$ ―Lysis protein DNA concentration

Here are the assumptions we made in our model;

The number of plasmids in any one cell is constant.
All concentrations of chemical compounds' unit are $\text{nM}$ . Cell population density 's unit is arbitrary unit.
The growth rate is the ideal value of cell growth speed minus 1.
The culture follows a logistic growth curve when not undergoing lysis.
When considering dilution of compounds because of growth, dilution rate is considered as $μ(1 - N / N_0)$ .
The culture is homogeneous, and $\text{[X]}$ is the intracellular concentration of compound $\text{X}$ .
The volume of one E. coli cell is $1.1\times 10^{-9}\mu \text{L}$ . Thus, the intracellular concentration of DNA was taken as 1.5 $\text{nM}$ per one plasmid.
kH is not the standard rate constant, but the rate constant of AHL production when the E. coli concentration is $N$ .
The AHL promoter does not leak.
Only the LuxR・AHL tetramer promotes the Plux promoter's expression.
The mRNA concentration is in a steady state and does not factor in the ODEs.

1-1-C. Computer Simulations

We executed our ODE simulations in the Julia program using the parameters below.

Parameter	Value	Description	Reference
μ	2 hr^-1	growth rate	From Wet Data
k	10 hr^-1	Maximum rate of cell lysis	Estimated from [1]
L₀	5 nM	Lysis protein for half maximum expression	Estimated from [1]
N₀	20	Carrying capacity	Estimated
n	2	Cooperative Hill coefficient	Estimated from [2]
CL, CR, CI	15, 10, 15	Plasmid number of lysis gene, LuxR, LuxI	Designed
kR	46.8 hr^-1	LuxR transcription rate	[4]
pR	2.3 hr^-1	LuxR translation rate	Estimated from [2]
dmR, dmI, dmL	14.82 hr^-1	mRNA degradation rate	[3]
dR	12 hr^-1	LuxR degradation rate	[3]
kH	0.01575 hr^-1	AHL synthesis rate (effective)	Estimated from [3]
dH	3.42 hr^-1	AHL degradation rate	[3]
kI	46.8 hr^-1	LuxI transcription rate	From arabinose concentration
pI	2.3 hr^-1	LuxR translation rate	Estimated from [2]
pR	2.3 hr^-1	LuxI translation rate	Estimated
dI	3 hr^-1	luxI degradation rate	Estimated from [3]
kL	15 hr^-1	Lysis protein transcription rate	Estimated
dI	3 hr^-1	luxI degradation rate	Estimated from [3]
pL	2.3 hr^-1	Lysis protein translation rate	Estimated from [2]
b	35 nM	Plux promoter AHL induced constant	[1]
A₀	15	AHL binding affinity constant to Lux promoter	Estimated from [2]
dL	2 hr^-1	Lysis protein degradation rate	Estimated from [2]
DR, DI, DL	1.5 nM/plasmid number	DNA concentration	Estimated

With these parameters, we obtained the following graph.

Fig2. The dynamics of cell population and lysis protein concentration.

Cell population and lysis protein concentration oscillates over time.

From this graph, we can conclude that the cell population oscillates, reaching a maximum of ¼ of the carrying capacity.

In the next part, we will check how we could change the parameters to further control the maximum cell population.

1-2. Arabinose-Mediated Cell Density Control

Because our system's LuxI promoter uses an arabinose promotor, our gene circuit’s expression can be controlled by changing the arabinose concentration. With this, we can further control the cell population, thus increasing our gene circuit’s applicability. Therefore, we examined the behavior of our model in 1-1 by changing parameters responsive to the arabinose concentration.

1-2-A. Correspondence of Arabinose Concentration and Arabinose Promoter Expression

The 2021 iGEM Leiden team [5] conducted measurements relating the concentration of arabinose and arabinose promoter, and reported the corresponding Anderson promoter types. We extracted nine of these Anderson promoters from their data and conducted simulations.

Anderson Promoter	Relative Fluorescence	Arabinose Concentration (%, w/v)
J23112	1	0.001508
J23113	21	0.001562
J23117	162	0.002008
J23115	387	0.002997
J23105	623	0.004562
J23106	1185	0.012405
J23111	1487	0.021234
J23102	2179	0.072769
J23100	25470	140091

Fig3. [Arabinose] = 0.001508

This graph shows a normal logistic curve and almost no lysis protein expression. Thus, we can conclude that the lysis protein expression is too weak to induce lysis and that maintaining the balance within the circuit via changing the arabinose concentration is important in long-term culturing.

Fig4. [Arabinose] = 0.001562

We obtained similar results as above.

Fig5. [Arabinose] = 0.002008

From here, we began to observe some oscillation. The oscillation was centered around a relatively high cell population.

Fig6. [Arabinose] = 0.002997

The population was found to oscillate around a relative value of around 7.5.

Fig7. [Arabinose] = 0.004562

The population was found to oscillate around a relative value of around 6.

Fig8. [Arabinose] = 0.012405

The population was found to oscillate around a relative value of around 3.5.

Fig9. [Arabinose] = 0.021234

The population was found to oscillate around a relative value of around 3.

Fig10. [Arabinose] = 0.072769

The population was found to oscillate around a relative value of around 2.

Fig11. [Arabinose] = 0.140091

Finally, at an arabinose concentration of 0.140091, the cell population oscillated at about a relative value of 0.3

Fig12. Hypothetical simulation

We conducted simulation under the assumption that the arabinose concentration induces 9 times the expression levels of promoter J23016. The cell population is found to oscillate about a constant value.

We also ran a hypothetical scenario where the arabinose concentration induces 9 times the expression levels of promoter J23016. Here, the cell population was also found to oscillate about a constant value.

1-2-B. Analysis

From the results above, we can see that once the promoter's expression exceeds a threshold intensity, cell population oscillation and cell number control are possible and that there is no limit for that threshold intensity in silico.

By changing plasmid copy number or Promoter strength instead of changing arabinose concentration, we can gain similar result.
However, strategy of changing arabinose concentration is superior to other strategy because of some reasons. First, increasing the plasmid number is a burden on the E. coli cell. E. coli has to reproduce their plasmid with propagation process, so high plasmid copy number becomes a disadvantage. Second, replacing a plasmid that has been inserted in a cell takes time, so relying on promoter which we cannot change its strength after inserting can result in much-wasted experiment time. Our model has proven that by changing the arabinose concentration to the appropriate amount for the promoter, the ideal number of cells can still be reached, thus providing more flexibility in working with E. coli.

1-2-C. Conclusion

In the first step of our modeling, we modeled our QS-Lysis circuit using ODEs, and via computer simulations, confirmed the oscillatory nature of the E. coli cell population. From that, we then induced in arabinose-dependent transcription speeds and ran simulations of our system's behavior under various arabinose concentrations, and presented a potential strategy to control cell populations.

In the next step, we will further refine our ODE model by introducing pre- and post-differentiation E. coli cells, and examine their behavior via computer simulations.

1-3. References

[1] Din, M.O., Danino, T., Prindle, A., Skalak, M., Selimkhanov, J., Allen, K.,..., Hasty, J. (2016). Synchronized cycles of bacterial lysis for in vivo delivery. Nature, 536, 81-85. https://doi.org/10.1038/nature18930

[2] iGEM Ecuador 2021 Model

[3] Boada, Y., Vignoni, A., Picó, J. (2020). Multiobjective Identification of a Feedback Synthetic Gene Circuit. IEEE Transactions on Control Systems Technology, 28(1), 208-223. https://doi.org/10.1109/TCST.2018.2885694

[4] iGEM UNILausanne 2020

[5] iGEM Leiden 2021 Measurement

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