A novel, surprisingly simple but efficient approach to simulating microbial metabolism for variable temperatures.
Simulation
Legend
Settings
Cuminic acid (auxotrophic substrate)
Rhamnose (substrate inducing
Cas12A2 mediated apoptosis)
20°C:
72 hours:
This is a live simulation of the PseuPomona kill-switches in a batch experiment. Change the settings (temperature, presence of Cuminic acid and Rhamnose), check which values you want to visualize, click the run button, and wait for the simulation. The resulting plot and legend will then be shown. The source code of the model can be found on GitLab.
What is an Ordinary Differential Equation
An Ordinary Differential Equation (ODE) model is a set of equations that takes current values/concentrations and inserts them into equations resulting in outputs which are then used as inputs for the next cycle in time. This relatively simple system can be used to model surprisingly complex systems. In biochemical models the variables used as inputs often represent resources, as is the case in this model which describes how substrate, mRNA, protein, Ribosomes, and tRNA concentrations change over time in E. coli.
Figure 1. A visualization of a set of Ordinary Differential Equations, inputs and equations produce outputs which are used in the next cycle. The inputs and equations used in this model are shown in [Table 1].
Sechkar et al., (2023) [1] constructed an ODE model simulating the resources and growth-rate of E. coli based on the nutrient concentration. We adapted this model to include the effects of temperature and introduced nutrient consumption. The model was then used to predict the effectiveness of the Rhamnose and Cuminic acid Biocontainment systems.
Figure 2. A visualization of the resources taken into account and their various interactions[1].
Equations & Legend
| Equation | Variable |
|---|---|
| $katex$\dfrac{dm_a}{\phantom{b}dt}=F_ac_a\alpha_a\lambda(\color{red}\epsilon\color{black},B)(\color{blue}Q_{10_{mRNA_{deg}}}\color{black}\beta_a+\lambda(\color{red}\epsilon\color{black},B))\cdot m_a$katex$ | Metabolic RNA |
| $katex$\dfrac{dp_a}{\phantom{.}dt}=\dfrac{\color{red}\epsilon\color{black}(t^c)}{\phantom{b}n_a} \cdot \dfrac{\phantom{hayabus}m_a / k_a}{1+\dfrac{1}{1-\Phi_q}\Sigma_{j\color{red}\epsilon\color{black}(a,r)}m_j/k_j}R-\lambda(\color{red}\epsilon\color{black},B)p_a$katex$ | Metabolic protein |
| $katex$\dfrac{dm_r}{\phantom{b}dt}=F_r(P)\cdot c_r\alpha_r\lambda(\color{red}\epsilon\color{black},B)-(\color{blue}Q_{10_{mRNA_{deg}}}\color{black}\beta_r+\lambda(\color{red}\epsilon\color{black},B))m_r$katex$ | Ribosomal RNA |
| $katex$\dfrac{dR}{dt}=\dfrac{\color{red}\epsilon\color{black}(t^c)}{n_r}\cdot \dfrac{\phantom{hayabus}m_r/k_r}{1+\dfrac{1}{1-\Phi_q}\Sigma_{j\color{red}\epsilon\color{black}(a,r)m_j/k_j}}R-\lambda(\color{red}\epsilon\color{black},B)R$katex$ | Ribosome count |
| $katex$\dfrac{dt^c}{dt}=\color{red}\nu\color{black}(t^u,\sigma)\cdot p_a -\color{red}\epsilon\color{black}(t^c)\cdot B-\lambda(\color{red}\epsilon\color{black},B)\cdot t^c$katex$ | Charged tRNA |
| $katex$\dfrac{dt^u}{dt}=\color{red}\psi\color{black}(P)-\color{red}\nu\color{black}(t^u,\sigma)\cdot p_a + \color{red}\epsilon\color{black}(t^c) \cdot B - \lambda(\color{red}\epsilon\color{black},B) \cdot t^u$katex$ | Uncharged tRNA |
| $katex$\dfrac{dN}{dt}=100 \cdot \lambda \cdot N$katex$ | Cell count (CFU) |
| $katex$\dfrac{dS}{dt}=\phi_S-\lambda \cdot \dfrac{N}{Y_{NS}}$katex$ | Substrate (Glucose equivalents mol/L) |
Table 1. The Ordinary Differential Euquations that the Genetic Circuit Model is built upon.
| Unit |
Description |
Unit | Description |
|---|---|---|---|
| $katex$\alpha$katex$ | Promoter Strength | F | Regulation function |
| R | Ribosomes (nM) | $katex$\lambda$katex$ | Dilution rate ($katex$h^{-1}$katex$) |
| $katex$\beta$katex$ | Degradation rate($katex$h^{-1}$katex$) | $katex$\nu$katex$ | tRNA charging rate ($katex$\dfrac{nM}{\phantom{b}h}$katex$) |
| $katex$\epsilon$katex$ | Translation elongation rate ($katex$\#AA h^{-1}$katex$) | $katex$\psi$katex$ | tRNA synthesis rate ($katex$\dfrac{nM}{\phantom{b}h}$katex$) |
| $katex$t^c$katex$ | Charged tRNA ($katex$nM$katex$) | $katex$t^u$katex$ | Uncharged tRNA |
| m | mRNA ($katex$nM$katex$) | p | Protein ($katex$\#AA$katex$) |
| $katex$\sigma$katex$ | Nutrient quality | ppGpp | Guanosine tetraphosphate ($katex$nM$katex$) |
Table 2. Legend with all the variables inherited from sechkar [1].
Temperature introduced
The $katex$Q_{10}$katex$ temperature coefficient is commonly used to simulate the effect of temperature on biochemical processes [2].A simple formula can be used to calculate the reaction rate
$katex$R(T)=R_{base}\cdot Q_{10}^{\dfrac{T-T_{base}}{\phantom{bb}10}}$katex$
For a given reaction at any given temperature, where $katex$R_{base}$katex$ is $katex$5 Mh^{-1}$katex$, $katex$T_{base}$katex$ is $katex$20°C$katex$, and $katex$Q_{10}$katex$ is $katex$3.57$katex$, the $katex$R(T)$katex$ can be calculated from any given $katex$T$katex$.
In this model the effect of $katex$Q_{10}$katex$ was directly applied to protein elongation-, mRNA synthesis-, tRNA charging-, tRNA-synthesis-, and mRNA degradation-rates. The values used to determine the $katex$Q_{10}$katex$ factors of these rates are shown in [table 3].
| Factor |
Value | Source |
|---|---|---|
| $katex$Q_{10_{default}}$katex$ |
$katex$3.57$katex$ | [3] |
| $katex$Q_{10_{\epsilon}}$katex$ | $katex$7.1$katex$ | [4] |
| $katex$Q_{10_{\nu}}$katex$ | $katex$7.1$katex$ | [4] |
| $katex$Q_{10_{\psi}}$katex$ | $katex$7.1$katex$ | [4] |
| $katex$Q_{10_{mRNAdeg}}$katex$ | $katex$1.75$katex$ | [5] |
Table 3. $katex$Q_{10_{factor}}$katex$ values and their sources.
Temperature dependency of biomass production
Colourectal, the 2022 Wageningen iGEM team, performed E. coli biomass measurements over time in batch conditions for different temperatures [Figure 3]. The model was simulated in conditions that are as close as possible to the Colourectal measurements [Figure 4]. The behaviour is qualitatively similar, but there are some quantitative differences. These small deviations are expected when reducing complex biological systems to simplified computer models.
Figure 3. Biomass (OD600) of E. coli measured over time at different temperatures in batch conditions [6].
Figure 4. The ribosomes per cell multiplied by the cell count simulated by the model and used as a proxy fo biomass over time in batch conditions. The three temperatures match the temperatures used in the Colourectal experiments.
The implementation of the ccdA-ccdB system
The ccdA-ccdB system is a thoroughly used and sufficiently researched kill-switch with moderate complexity. The system produces a toxin and anti-toxin under the control of promoters which are chosen depending on when the toxin and anti-toxin are desired. The resulting concentrations of ccdA and ccdB control whether cells die or proliferate.
ccdA-ccdB validation
Colourectal, the 2022 Wageningen iGEM team, obtained in vivo measurements with a temperature dependent ccdA-ccdB system. They placed ccdA under a constitutive promoter and ccdB under a temperature dependent promoter which resulted in a very efficient temperature-dependent kill-switch [Figure 5]. The system is designed to survive at higher temperatures where ccdB is not produced because the promoter becomes inactive resulting in lower toxin concentrations, the anti-toxin (ccdA) is placed under a constitutive promoter and therefore becomes dominant at higher temperatures [Figure 5].
Figure 5. (Left) The ccdAB toxin-antitoxin system was tested in temperatures ranging from 37°C to 20°C, with and without the toxin on a temperature dependent promoter [6]. Survivability of the pscpA strains is at 0% for 30°C and colder. (Right) At lower temperatures the expression of ccdA and ccdB are very close. As temperature increases however, the ccdA expression increases significantly whereas the ccdB expression drops to near zero values.
PseuPomona introduced
We introduced our PseuPomona kill-switches detailed in biosafety to our model to provide information on the effectiveness of the kill-switches in various temperatures. The absence of cuminic acid is shown to be more effective at killing microbes at lower temperatures, and the presence of rhamnose reduces growth, but the most effective way to kill the bug at any given temperature is the combination of both systems [Figure 6]. The absence of cuminic acid and presence of rhamnose combined is enough to create a strong downward trend in the number of Colony Forming Units in the root system. These results lead to doubts that either of the cell death systems is enough to kill the cells in situ separately from eachother, but the results also indicate that a combination of both systems would efficiently eradicate PseuPomona from the soil.
Figure 6. Pairplot of the 4 different combinations of the presence and absence of cuminic acid and rhamnose in the soil for different temperatures over different timespans. A combination of both killswitches being active (bottom-right) results in a steady decline in colony forming units across all temperatures.
