Team:NTHU Taiwan/Model
Overview
To prove our design can work, we set up an enzyme kinetics model to simulate how the changing temperature together with different fatty acid sensing promoters allows us to control the amount of final output. The kinetics model consists of 2 parts. The first part is the temperature sensing model, which produces different amounts of fatty acid as the temperature changes. The second part is the fatty acid sensing model. In this model, the sensitivity of various promoters to fatty acid is shown by simulating how much the downstream gene is expressed as different concentrations of fatty acid is initially added. Finally, the two models are analyzed so that we can design different temperature sensing systems by choosing the promoter that best suits our needs.
Temperature Sensing Model
Inferring from our circuit design (shown in the simplified schematic diagram above), and ignoring the transcription and translation time of LipaseA, our temperature sensing design can be modeled using the following reaction equation:
which can then be used to derive the following ODEs.
- $\;\;\;{\dfrac{d[LipA]}{dt}} = -k_{on}(T) [LipA] [lipid] + k_{off}(T) [LipA:lipid] + k_{cat}(T) [LipA:lipid]$
- $\;\;\;{\dfrac{d[LipA:lipid]}{dt}} = k_{on}(T) [LipA] [lipid] - k_{off}(T) [LipA:lipid] - k_{cat}(T) [LipA:lipid]$
- $\;\;\;{\dfrac{d[lipid]}{dt}} = -k_{on}(T) [LipA] [lipid] + k_{off}(T) [LipA:lipid]$
- $\;\;\;{\dfrac{d[fatty\,acid]}{dt}} = k_{cat}(T) [LipA:lipid]$
Using the steady-state approximation, we hypothesize that the LipA:lipid complex will rapidly approach a constant concentration, together with the free ligand approximation we can simplify our equations and arrive at the familiar form of the Michaelis-Menten equations and express our fatty acid production rate as:
- $\;\;\;v = {\dfrac{V_{max} [lipid]}{K_{M} + [lipid]}}$
- $\;\;\;V_{max} = k_{cat}(T) [LipA:lipid]$
- $\;\;\;K_{M}(T) = {\dfrac{k_{off}(T) + k_{cat}(T)}{k_{on}(T)}}$
However, there is a slight difference between our model and the usual Michaelis-Menten kinetics, all our reaction rate constants have to be temperature-dependent instead of being a constant. We tried looking for these reaction rate parameters with temperature variants from existing literature, but there are almost no data on our specific LipaseA. So instead we used known KM and Vmax values of another similar structured Lipase at different temperatures from [1]. This might not be a perfect match with our experimental results, but our model can still provide a framework for other people trying to build a similar system.
| Parameter | Value | Reference | |||||
|---|---|---|---|---|---|---|---|
| 4°C | 15°C | 25°C | 37°C | 50°C | 60°C | ||
| KM (mM) | 20.81 | 18.89 | 17.25 | 16.58 | 17.96 | 18.07 | [1] |
| Vmax (μM min-1) | 1.27 $\times$ 105 | 2.70 $\times$ 105 | 4.98 $\times$ 105 | 5.24 $\times$ 105 | 3.82 $\times$ 105 | 3.56 $\times$ 105 | |
| Kcat (s-1) | 0.25 $\times$ 104 | 0.53 $\times$ 104 | 0.98 $\times$ 104 | 1.03 $\times$ 104 | 0.75 $\times$ 104 | 0.70 $\times$ 104 | |
| [Lipid]initial | 75 mM | assumed | |||||
Using the parameters obtained from [1] and fitting them to different temperatures, we simulated the relationship between temperature and the amount of fatty acid produced after the same amount of time.
Looking at the result we see that the concentration of fatty acid changes noticeably as temperature changes and is highest around 32°C. Furthermore, according to our model, we can adjust the output concentration of fatty acid by changing the initial amount of lipase (which can be achieved by growing different concentrations of E.coli) to fit the sensing range of our fatty acid sensitive promoter.
Fatty Acid Sensing Model
After completing the tunable-temperature-sensing-system, we need a way to detect different concentrations of fatty acid so that different amounts of the downstream gene, such as glutamine or glutamate, can be expressed. In our experiments, however, the RFP protein is used as our downstream gene for easy quantization of gene expression. After researching fatty acid promoters, we discovered that Team NTU-Taida 2012 and Team UPF-CRG-Barcelona 2018 have already done similar designs, so we decided to improve their works and design our own fatty acid sensing promoters and compare their performance.
According to the works of Team UPF-CRG-Barcelona 2018 and Team NTU-Taida 2012, the following differential equations can describe how fatty acid induces the downward gene expression.
- $\;\;\;X_{FadR} = [FadR] \times \dfrac{\beta_{FadR}^{n_1}}{[FA]^{n_2}+\beta_{FA}^{n_1}}$
- $\;\;\;\dfrac{d[RFP]}{dt} = \alpha_{RFP}(\dfrac{\beta^{n_2}}{\beta^{n_2}+X_{FadR}^{n_2}})-\gamma_{RFP}[RFP]$
We decided to refit most of the equation parameters using least square methods based on our own experimental data (download here) and use those parameters to simulate the performance when different amounts of fatty acid are added after the same amount of time. Below are the simulated results of RFP fluorescence of different promoters after different amounts of fatty acid are added.
| Parameter | pFadBA | pFadD+Lac | pFadBA_NTHU | Reference |
|---|---|---|---|---|
| Value | ||||
| βFadR (µM) | 997.31 | 780.63 | 992.955 | data fitted |
| βFA (µM) | 0.127 | 0.129 | 0.127 | |
| αREP (µM min-1) | 2.0297 | 6.54 $\times$ 10-3 | 3.94 $\times$ 10-3 | |
| γREP (min-1) | 4.245 $\times$ 10-3 | 3.067 $\times$ 10-9 | 1.583 $\times$ 10-9 | |
| β (µM) | 0.818 | 0.150 | 0.903 | |
| n2 | 6.8 | 2.2 | 6.4 | |
| n1 | 2 | NTU-Taida 2012 | ||
The final values of RFP at different fatty acid concentrations were simulated using the parameters fitted from our own data for 10 minutes. We chose to run for 10 minutes because we wanted to see how quickly each promoter can react to the change in fatty acid concentration since we want to be as close to real-time as possible for nutrient production. We also decided to include the leakage effect seen from the experiments into our model because we want these models to be as true to reality as possible, hence the non-zero values even when no fatty acid is added.
Model Analysis
There are 3 factors our team considers to be the most important when designing our temperature sensing system. One is how fast the system can react, the other is the range of fatty acid that can still induce a change in the product (linear region), and the last is how much difference in product different fatty acid concentrations can induce. For real-time fertilizer production, we want our system to be able to react quickly, be able to sense a wide range of fatty acid, and have a noticeable difference in output at different fatty acid concentrations.
From the modeled results, we see that each promoter has its strengths and weaknesses. The promoter pfadBA from NTU-Taida 2012 produced the most RFP in ten minutes (partially due to high leakage) and can create at most 2 fold difference. However, the range of fatty acid it can detect is quite narrow, so it acts more like an on-off switch instead of continuous control. The promoter pfadD+lac has a lower leakage, but still produces a moderate amount of product after 10 minutes. It has a longer sensing range of fatty acid, however, the change in fluorescence is practically unnoticeable. A larger difference in a product can be achieved over the same range of fatty acid using our model but requires a long reaction time, which isn’t suitable for our project but could find uses in other systems. The last promoter, pfadBA_NTHU is the least ideal for our project. It has the lowest production rate and a narrow detection range. It’s the only advantage is that it has the lowest leakage, so this system is less susceptible to noise and can work more precisely under the right conditions.
Overall our team thinks despite the large leakage, pfadBA might be the most suitable promoter to use for our project after all, due to its fast response rate and high production. However, we wouldn’t consider the other promoters failures, instead our team believes that their properties may make them more useful in other systems and hope that future iGEMers can think of more creative ways to utilize these designs.
References
- Xiuling Ji et al.,"Purification and characterization of an extracellular cold-adapted alkaline lipase produced by psyc", J.Basic Microbiol, 2015
- Zhang F, et al. (2012) Design of a dynamic sensor-regulator system for production of chemicals and fuels derived from fatty acids. Nature Biotechnology 30(4):354-9.
- iGEM NTU-Taida 2012
- iGEM UPF CRG Barcelona 2018
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