Welcome to the Modeling page of the Hydro Guardian. By combining two specialized sub-models into a unified system, our model simulates real-world conditions to predict how various environmental factors influence bacterial resistance development.
To effectively deploy our SynBio Hydro Guardian biosensor, it is crucial to understand when and where it should be utilized. Resistance mechanisms in bacteria are complex and can be unpredictable, making targeted interventions difficult without proper foresight. By understanding the development of resistance, we can optimize the application of our biosensor, ensuring it is used in the right environments at the right times.
In creating a powerful Hydro Guardian, detection limits are critical. To empower the Wetlab team with the necessary information, we used computational modeling to gather key data on the thresholds at which resistant bacteria begin to proliferate. This approach allowed us to identify the detection limits, providing the Wetlab team with the insights they needed to fine-tune our Hydro Guardian, and contributes to our Sustainable Development Goals.
Our model is designed with sophistication and precision, capturing the complexity of real-world conditions. It is a fusion of two distinct yet interconnected components: one focusing on antibiotic resistance through antibiotics and the other through metal. We build our model on established measurements and methods TextText. This comprehensive approach integrates a range of parameters, including bacterial growth rates, dynamic system behaviors, and varying concentrations of metals and antibiotics. By incorporating these diverse factors, the model offers a detailed and nuanced understanding of how different environmental conditions influence bacterial resistance and the overall performance of the biosensor.
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To better understand the risks of antibiotic contamination in waste waters, we adapted a model that simulates the complex dynamics between sensitive E. coli populations, resistant strains (both mutation-driven and horizontal gene transfer-driven), and antibiotic concentration. By adjusting these parameters, we can predict when multi-resistant strains are likely to emerge through antibiotic contamination and determine the necessary limits for our biosensor to detect and control their spread effectively.
All of our models solve systems of ordinary differential equations (ODEs) that describe the bacterial populations using the Runge-Kutta method. This method allows us to numerically integrate the system over time, providing a high degree of accuracy. The ODEs are based on Sutradhar et al. 2021 and were further developed and refined in the context of the Hydro Guardians project Text.
The change in antibiotic concentration \(C\) over time is modeled through the rate \(E\) at which the antibiotic is introduced into the system and the antibiotic clearance proportional to its current concentration.
$$ \frac{\text{d}C}{\text{d}t} = E - k_e \cdot C $$
The equations for sensitive bacteria \(S\) and resistant bacteria (\(R_m\) for mutation-based resistance and \(R_p\) for plasmid-based resistance) describe their population dynamics over time. All three equations follow logistic growth, limited by the total population capacity \(N_{\text{max}}\), and account for various factors influencing bacterial survival and competition.
The population of sensitive bacteria grows logistically, with a limiting factor determined by the combined populations of resistant and sensitive bacteria. This is further influenced by influx and efflux rates, antibiotic-induced mortality, and loss due to mutation and horizontal gene transfer. The killing effect of antibiotics is modeled as a concentration-dependent term, where the antibiotic concentration \(C\) reduces the population of sensitive bacteria.
$$ \frac{\text{d}S}{\text{d}t} = \alpha_S \cdot \left(1 - \frac{R_m + R_p + S}{N_{\text{max}}}\right) \cdot S + g_S - k_T \cdot S - \delta_{\text{max}} \cdot \left(\frac{C}{C + C_{S_{50}}}\right) \cdot S - m(C) \cdot S - \beta \cdot \frac{S \cdot R_p}{R_m + R_p + S} - \beta \cdot \frac{S \cdot R_m}{R_m + R_p + S} $$
The population of resistant bacteria due to mutations follows a similar logistic growth pattern but also incorporates the fitness cost (\(\alpha_{R_m}\)) associated with resistance. The antibiotic concentration \(C\) reduces the population of resistant mutants in a concentration-dependent manner while increasing the mutation rate \(m(C)\).
$$ \frac{\text{d}R_m}{\text{d}t} = \alpha_{R_m} \cdot \left(1 - \frac{R_m + R_p + S}{N_{\text{max}}}\right) \cdot R_m + g_{R_m} - k_T \cdot R_m - \delta_{\text{max}} \cdot \left(\frac{C}{C + C_R}\right) \cdot R_m + m(C) \cdot S $$
Similar to \(R_m\), this population undergoes logistic growth, factoring in the fitness cost of resistance (\(\alpha_{R_p}\)). However, instead of mutation, the last two terms capture the effects of horizontal gene transfer with the gene transfer rate \(\beta\), which enables the transfer of resistance traits between sensitive bacteria and plasmid-carrying resistant strains. This process contributes to the further expansion of the \(R_p\) population.
$$ \frac{\text{d}R_p}{\text{d}t} = \alpha_{R_p} \cdot \left(1 - \frac{R_m + R_p + S}{N_{\text{max}}}\right) \cdot R_p + g_{R_p} - k_T \cdot R_p - \delta_{\text{max}} \cdot \left(\frac{C}{C + C_R}\right) \cdot R_p + \beta \cdot \frac{S \cdot R_p}{R_m + R_p + S} + \beta \cdot \frac{S \cdot R_m}{R_m + R_p + S} $$
Utilizing those ODEs, we can classify the danger of different antibiotic concentrations in terms of multi-resistant E. coli strains. With the right parameters, it is also possible to run the model for different antibiotics and compare their respective influence on the system's behavior. As described above, there are three different E. coli populations (sensitive \(S\), resistance via mutation \(R_m\), and resistance via horizontal gene transfer \(R_p\)) to consider when analyzing the model results. Fig. 1 displays an exemplary illustration for the bacterial population development over time, in a microcosm with an erythromycin influx rate of 10 mg/L/hour.
The population of sensitive bacteria decreases rapidly in less than 65 hours. In the same time period, the resistant bacteria populations rise to a constant maximum. The resistance through mutation (hiding behind the total resistant line) is around two magnitudes higher than through horizontal gene transfer (HGT) and is therefore here the main cause of resistant bacteria. Fig. 2a displays the system with a two-magnitude lower antibiotic influx rate of 0.1 mg/L/hour.
The resistant bacteria through mutation (still hiding behind the total resistant population) are still considerably higher than those originating from horizontal gene transfer. At a lower antibiotic concentration, an increase in the resistant bacteria populations is observable. If a longer time period is considered, the system seems to be in equilibrium. The sensitive bacteria population is constant after a slight drop in the first few hours the system is introduced to the antibiotic.
| Parameter | Description | Value or Range | Unit | Source |
|---|---|---|---|---|
| \(k_e\) | Antibiotic clearance rate | 1.97 | 1/h | |
| \(\alpha_S\) | Growth rate of susceptile bacteria S | 0.5 | 1/h | Text |
| \(\alpha_{R_m}\), \(\alpha_{R_p}\) | Growth rate of bacteria resistant from mutation/plasmid | \(\alpha_S \cdot(1-\alpha)\) | CFU/L | |
| \(\alpha\) | Fitness cost for resistance carry | varied | - | TextTextText |
| N\(_\text{max}\) | Carrying capacity of the microcosm | \(10^9\) | CFU/mL | |
| \(g_S,\,g_{R_m},\,g_{R_p}\) | Bacterial influx rates | 0 | 1/h | |
| \(k_T\) | Bacterial efflux rate | 0.11 | 1/h | Email exchange with I. Sutradhar |
| \(\delta_\text{max}\) | Bacterial killing rate in response to antibiotic | 1.97 | 1/h | Text |
| \(C_{S_{50}}\) | Antibiotic concentration where the killing action is half of the maximum value for S | 12.5 | µg/mL | Text |
| \(C_{R}\) | Antibiotic concentration where the killing action is half of the maximum value for \(R_{m, p}\) | 200 | µg/mL | Text |
| \(\beta\) | Gene transfer rate | 0.001 | 1/h | Text |
| m(C) | Mutation frequency under antibiotic | varies | 1/h | Text |
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Water contaminated with heavy metals harbors known dangers, such as heavy metal poisoning. Another important, but not immediately obvious, factor is the resulting heavy metal resistance of bacteria. But why is this the case? For example, E. coli strains exposed to a heavy metal contaminated residue develop heavy metal resistance. During the process, co-selection (co-resistance or cross-resistance) can occur, resulting in the development of resistance mechanisms that also lead to resistance to antibiotics, further increasing the spread of multi-resistant pathogens.
We want to analyze the development limits of multi-resistant pathogens in E. coli caused by co-selection in heavy metal residues. Drawing on established methods, from Arya et al. 2021, for modeling resistance, we use a framework based on differential equations that describe bacterial growth, resistance gene transfer, and mortality under different metal concentrations Text. By adjusting these parameters, particularly the death rates, we aim to identify the metal concentrations at which co-selecting resulting in the spread of multi-resistant pathogens occurs in E. coli strains.
All of our models solve systems of ordinary differential equations (ODEs) that describe the bacterial populations using the Runge-Kutta method. This method allows us to numerically integrate the system over time, providing a high degree of accuracy. The model simulates bacterial population dynamics under metal exposure, focusing on the persistence of antibiotic-resistant bacteria.
Sensitive bacteria (S) and resistant bacteria (R) are modeled with logistic growth equations, incorporating the fitness cost (\(\alpha\)) of plasmid carriage for resistant bacteria and the impact of metal toxicity on death rates. The model also accounts for conjugation and plasmid loss during cell division, allowing for a dynamic understanding of how metal concentrations drive bacterial competition and resistance persistence.
For the growth of sensitive bacteria we are considering the growth of sensitive bacteria, the death of sensitive bacteria due to the presence of a heavy metal in the microcosm, as well as the conjugation and the plasmid loss due to segregation.
$$\frac{\text{d} S}{\text{d} t} = r\left(1-\frac{N}{N_\mathrm{Max}}\right) S - \delta_S S - \frac{\beta S R}{N} + r\left(1-\frac{N}{N_\mathrm{Max}}\right)(1-\alpha) \varepsilon R$$
For the growth of resistant bacteria we are considering the growth of resistant bacteria, the death of resistant bacteria, as well as the conjugation and the plasmid loss due to segregation.
$$\frac{\text{d} R}{\text{d} t} = r\left(1-\frac{N}{N_\mathrm{Max}}\right)(1-\alpha)(1-\varepsilon) R - \delta_R R + \frac{\beta S R}{N}$$
The death rate for sensitive bacteria (\(\delta_S\)) is influenced by metal concentrations (M), with resistant bacteria exhibiting lower death rates due to their metal resistance.
$$\delta_S = \delta_R + \frac{E_\mathrm{Max} M^\mathrm{H}}{MIC^\mathrm{H} + M^\mathrm{H}}$$
is used to calculate the death rate \(\delta_S\) of sensitive bacteria in dependence of the base death rate \(\delta_R\) of resistant bacteria and the specific heavy metal and its concentration in the microcosm M. The equation is specified for the respective metals through the metal characteristic parameters \(E_\mathrm{Max}\), the maximal death rate due to the metal, H, the Hill coefficient, and MIC, the Minimum inhibitory concentration for the respective heavy metal. Currently the model can describe the influence of five different heavy metal compounds, (Copper (CuSO4), Zinc (ZnSO4), Mercury (HgCl2), Lead (Pb(NO3)2), and Silver (AgNO3)) Text.
Utilizing those ODEs, we can classify the danger of different heavy metal compounds in terms of multi-resistant E. coli strains. Furthermore, those ODEs can be used to obtain Minimal Co-Selective Concentrations (MCSCs) for the respective heavy metals. The MCSC values can then be used to support and improve a heavy metal detection system like our Hydro Guardian.
At first, the different heavy metal compounds were compared on how they promote multi-resistant E. coli strains. Therefore, the resistant bacteria ratio, $$\text{Res. Bac. Ratio} = \frac{R}{S + R}$$ is used. An exemplary illustration is displayed in Fig. 1. For different copper (CuSO4) concentrations, the growth of resistant bacterial strains is either promoted or inhibited. At a metal concentration higher than 28 mg/L, the resistant bacteria ratio increases rapidly, while for a concentration around 12 mg/L, it increases linearly in a timeline of 100 hours. Opposing observations are made with concentrations around and below 2 mg/L, where the resistant bacteria ratio decreases. At around 6 mg/L, the resistant bacteria ratio seems to stay the same for the time period considered.
To take a more precise look into the sensitive and resistant bacteria populations, the bacteria population density in CFU/mL is useful. An exemplary illustration for copper is displayed in Fig. 2a and b. By looking at the respective population changes, a more precise analysis for a specific concentration is possible. At 42 mg/L copper concentration in the system, the observation from Fig. 1 is more clear. The resistant bacteria dominate the system in less than 50 hours. The sensitive decrease rapidly for 100 hours. Fig. 2b displays the case for 5.5 mg/L, where the resistant bacteria population is decreasing very slightly after 200 hours. Fig. 2b illustrates the case of a Minimal Co-Selective Concentration (MCSC) at which no significant increase of the resistant bacteria population is found. Below the value of 5.5 mg/L copper in an E. coli system, the resistant bacteria will decrease over time. Above this value, the resistant bacteria will take over the system over time. The time needed to take over the system decreases with an increase in metal concentration, see Fig. 1.
We took a deeper look into the MCSCs for the respective heavy metals, using metal-specific values from Arya et al. 2021 Text. Fig. 3 displays the MCSC values that are obtained using literature values. The respective MCSC values decrease from lead to mercury. Therefore, lead starts to promote the growth of resistant bacteria above a concentration of 21.5 mg/L, which is around three magnitudes higher than the value of 0.0156 mg/L for mercury. Resulting in the order from least to most dangerous in terms of resistant bacteria promotion: 1. lead, 2. copper, 3. zinc, 4. silver, and 5. mercury.
| Parameter | Description | Value or Range | Unit | Source |
|---|---|---|---|---|
| r | Specific growth rate for sensitive bacteria S | 0.5 | 1/h | TextTextText |
| \(\alpha\) | Fitness cost for resistance carry | 0.25 | - | TextText |
| \(N_\text{max}\) | Carrying capacity of the microcosm | \(6.71\cdot10^7\) | CFU/L | Assumed |
| \(\delta_R\) | Base death rate of bacteria – applied for resistant bacteria | 0.025 | 1/h | Text |
| \(\delta_S\) | Death rate for sensitive bacteria – varies for different heavy metals | Eq. 3 | 1/h | varied |
| \(\varepsilon\) | Plasmid loss probability | 0.000144 | - | Text |
| \(\beta\) | Gene transfer rate | 0.001 | 1/h | Text |
| Parameters for the heavy metal compounds: Copper (CuSO4) / Zinc (ZnSO4) / Mercury (HgCl2) / Lead (Pb(NO3)2) / Silver (AgNO3) | ||||
| MIC | Minimum inhibitory concentration | 212.79 / 2760.31 / 1.85 / 1728.7 / 0.48 | mg/L | TextText |
| \(E_\text{max}\) | Maximum death rate caused by metal presence | 1.74 / 1.37 / 5.89 / 18.74 / 2.42 | 1/h | TextText |
| H | Hill coefficient | 1.54 / 0.72 / 1.44 / 1.82 / 5.19 | - | TextText |
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Most of the times, our daily waste water is not only contaminated by either antibiotic or heavy metal residues, but rather contaminated with both types of residues. We aim to help people get a better understanding of how the respective residues influence the spread of multi-resistant pathogens, and why the combination is concerning.
All of our models solve systems of ordinary differential equations (ODEs) that describe the bacterial populations using the Runge-Kutta method. This method allows us to numerically integrate the system over time, providing a high degree of accuracy.
To achieve an accurate description of the change in multi-resistant bacteria populations, we combined the two models concerning bacteria growth in antibiotic and heavy metal-contaminated water, respectively. We wanted to support our Wetlab team even better than we did before, providing them with more accurate detection limits that should be achieved by our Hydro Guardian.
In order to combine both sub-models we used a seemingly trivial but effective approach: Merging differential equation systems. Due to the fact that both sub-models describe the growth of sensitive bacteria and the development of one or more kinds of resistant bacteria, both systems of differential equations follow the same structure. Therefore, we combined both sensitive bacteria equations into one, adding and subtracting specific growth terms originating either from antibiotic or heavy metal influence.
The change in antibiotic concentration \(C\) over time is modeled, equivalent to the sub antibiotic model, through the rate \(E\) at which the antibiotic is introduced into the system and the antibiotic clearance proportional to its current concentration.
$$ \frac{\text{d}C}{\text{d}t} = E - k_e \cdot C $$
The population of sensitive bacteria grows logistically, with a limiting factor determined by the combined populations of resistant and sensitive bacteria. This is further influenced by influx and efflux rates, antibiotic-induced mortality, and loss due to mutation and horizontal gene transfer. The killing effect of antibiotics and heavy metals is modeled as a concentration-dependent term, where the antibiotic concentration \(C\) reduces the population of sensitive bacteria, while the concentration-dependent killing due to heavy metals is input through the \(\delta_S\) term.
$$ \frac{\text{d}S}{\text{d}t} = \alpha_S \cdot \left( 1 - \frac{R_m + R_p + R_i + S}{N_{\text{max}}} \right) \cdot S + g_S - k_T \cdot S - \delta_S \cdot S - \delta_{\text{max}} \cdot \left( \frac{C}{C + C_{S_{50}}} \right) \cdot S - \beta \cdot S \cdot \frac{R_p}{R_m + R_p + R_i + S} - \beta \cdot S \cdot \frac{R_m}{R_m + R_p + R_i + S} - \beta \cdot S \cdot \frac{R_i}{R_m + R_p + R_i + S} - m_A(C) \cdot S $$The population of resistant bacteria due to mutations follows a similar logistic growth pattern but also incorporates the fitness cost \(\alpha_{R_m}\) associated with resistance. The antibiotic concentration \(C\) reduces the population of resistant mutants in a concentration-dependent manner while increasing the mutation rate \(m(C)\).
$$ \frac{\text{d}R_m}{\text{d}t} = \alpha_{R_m} \cdot \left( 1 - \frac{R_m + R_p + R_i + S}{N_{\text{max}}} \right) \cdot R_m + g_{R_m} - k_T \cdot R_m - \delta_R \cdot R_m - \delta_{\text{max}} \cdot \left( \frac{C}{C + C_R} \right) \cdot R_m + m_A(C) \cdot S $$Similar to \(R_m\), this population undergoes logistic growth, factoring in the fitness cost of resistance \(\alpha_{R_p}\). However, instead of mutation, the last two terms capture the effects of horizontal gene transfer with the gene transfer rate \(\beta\), which enables the transfer of resistance traits between sensitive bacteria and plasmid-carrying resistant strains. This process contributes to the further expansion of the \(R_p\) population.
$$ \frac{\text{d}R_p}{\text{d}t} = \alpha_{R_p} \cdot \left( 1 - \frac{R_m + R_p + R_i + S}{N_{\text{max}}} \right) \cdot \left( 1 - \epsilon \cdot \left( C \neq 0 \right) \right) \cdot R_p + g_{R_p} - k_T \cdot R_p - \delta_R \cdot R_p - \delta_{\text{max}} \cdot \left( \frac{C}{C + C_R} \right) \cdot R_p + \beta \cdot S \cdot \frac{R_p}{R_m + R_p + R_i + S} + \beta \cdot S \cdot \frac{R_m}{R_m + R_p + R_i + S} $$Like both resistant types \(R_m\) and \(R_p\), this population also undergoes logistic growth factoring in the fitness cost of resistance \(\alpha_{R_i}\). The last term captures the effect of horizontal gene transfer with the gene transfer rate \(\beta\), like in \(R_p\). Different to the sub-model for heavy metals, terms like an antibiotic killing term are added. The plasmid loss due to segregation is inhibited while an antibiotic concentration \(C \neq 0\) is in the system.
$$ \frac{\text{d}R_i}{\text{d}t} = \alpha_{R_i} \cdot \left( 1 - \frac{R_m + R_p + R_i + S}{N_{\text{max}}} \right) \cdot \left( 1 - \epsilon \cdot \left( C \neq 0 \right) \right) \cdot R_i + g_{R_i} - k_T \cdot R_i - \delta_R \cdot R_i - \delta_{\text{max}} \cdot \left( \frac{C}{C + C_R} \right) \cdot R_i + \beta \cdot S \cdot \frac{R_i}{R_m + R_p + R_i + S} $$Utilizing those modified ODEs we can classify the danger of different heavy metal compounds and different antibiotics in one system at the same time. The model can be used to predict the development of sensitive and resistant bacteria populations of E. coli strains and gives a precise indication of the origin of the respective resistance. Fig. 1 displays a system with both an antibiotic and heavy metal contamination.
At a erythormycin influx rate of 2 mg/L/hour and a copper (CuSO4) contamination of 24 mg/L, the sensitive bacteria decrease slightly for the first 70 hours, then they rapidly decrease to about 0 CFU/mL after 200 hours. All resistant bacteria species increase, taking over the space in the restricted microcosm from the sensitive bacteria. The resistance through horizontal gene transfer (HGT) is lower than the resistance through mutation. The resistant bacteria originating of HGT induced by heavy metal presence (HGTM) is the lowest of the three species. The HGT induced by antibiotic presence gives a higher bacteria population density but is, like in the sub-model, still significant lower than the resistant bacteria population by mutation.
The model still functions for only antibiotic or only heavy metal contamination. Fig. 2 displays the system with no heavy metal contamination.
There are no resistant bacteria induced by heavy metal presence, because there is no heavy metal in the system. The resistant bacteria through mutation and HGT induced by antibiotics still persist. Fig. 3a and b display the system with no antibiotic influx for a short time period and a long time period, respectively.
For no antibiotic influx in the system but still an starting population of antibiotic resistant bacteria, the heavy metal contamination contributes by killing the sensitive bacteria faster than the resistant bacteria, and therefore giving space in the microcosm for the different resistant bacteria species. Compared to Fig. 1 the HGT induced by heavy metal presence more dominant than before. The system needs more time with only heavy metal presence to be dominated by resistant bacteria, resulting in the death of all sensitive bacteria.
Fig. 1, 2, and 3 illustrate that the model can simulate a system with both antibiotic and heavy metal contamination as well as the specific cases, already displayed in the sub-models.
| Parameter | Description | Value or Range | Unit | Source |
|---|---|---|---|---|
| \(k_e\) | Antibiotic clearance rate | 1.97 | 1/h | |
| \(\alpha_S\) | Growth rate of susceptile bacteria S | 0.5 | 1/h | TextText |
| \(\alpha_{R_m}\), \(\alpha_{R_p}\) | Growth rate of bacteria resistant from mutation/plasmid | \(\alpha_S \cdot(1-\alpha)\) | CFU/L | |
| \(\alpha\) | Fitness cost for resistance carry | varied | - | TextTextText |
| N\(_\text{max}\) | Carrying capacity of the microcosm | \(10^9\) | CFU/mL | |
| \(g_S,\,g_{R_m},\,g_{R_p}\) | Bacterial influx rates | 0 | 1/h | |
| \(k_T\) | Bacterial efflux rate | 0.11 | 1/h | Email exchange with I. Sutradhar |
| \(\delta_\text{max}\) | Bacterial killing rate in response to antibiotic | 1.97 | 1/h | Text |
| \(C_{S_{50}}\) | Antibiotic concentration where the killing action is half of the maximum value for S | 12.5 | µg/mL | Text |
| \(C_{R}\) | Antibiotic concentration where the killing action is half of the maximum value for \(R_{m, p}\) | 200 | µg/mL | Text |
| \(\beta\) | Gene transfer rate | 0.001 | 1/h | TextText |
| m(C) | Mutation frequency under antibiotic | varies | 1/h | Text |
| \(\delta_R\) | Metal induced base death rate of bacteria – applied for resistant bacteria | 0.025 | 1/h | Text |
| \(\delta_S\) | Metal induced death rate for sensitive bacteria – varies for different heavy metals | Eq. 3 - Metal model | 1/h | varied |
| \(\varepsilon\) | Plasmid loss probability | 0.000144 | - | Text |
| Parameters for the heavy metal compounds: Copper (CuSO4) / Zinc (ZnSO4) / Mercury (HgCl2) / Lead (Pb(NO3)2) / Silver (AgNO3) | ||||
| MIC | Minimum inhibitory concentration | 212.79 / 2760.31 / 1.85 / 1728.7 / 0.48 | mg/L | TextText |
| \(E_\text{max}\) | Maximum death rate caused by metal presence | 1.74 / 1.37 / 5.89 / 18.74 / 2.42 | 1/h | TextText |
| H | Hill coefficient | 1.54 / 0.72 / 1.44 / 1.82 / 5.19 | - | TextText |
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This section provides links to our model files, which are integral to our project. By exploring these models, you can gain valuable insights into the parameters and simulations we are working on. You will also find Python scripts that can be easily modified to meet your specific needs, offering greater flexibility and adaptability in your research. Additionally, a ReadMe file is included, providing simple, step-by-step instructions on how to use the models. We encourage collaboration and hope these resources inspire further advancements in synthetic biology.
Access the model files on our GitHub repository.
