Team:BEAS China/Model
Summary of Model
In this part, we use model(steady-state) to describe which parameters are affecting the performance of the heavy metal sensors. We also fitted the sensors’ dose-response curves to a Hill function-based biochemical model to describe their input-output relationships.
Equations to describe the mercury sensor
Equations
Table 1: Meaning of modelling terms
Equations
Equation(1): Equation describing the rate of transcription of merR mRNA (MmerR). Note that at steady state:
Equation(2): Equation describing the rate of translation of merR Protein (p1). Note that at steady state:
In the absence of Hg, the equations become: （PmerR would appear at a constant level）
When we add in Hg, the story is more complicated, so theoretically the equations become:
[Hg] concentrations change over time, so this is like adding a "disturbance" in the system, that will effectively remove some merR protein from the system.
Equation(3): Equation describing rate of change of Hg when added in. Hg undergoes a secondary set of reactions with merR when introduced, where the forward reaction destroys it and the backward reaction re-forms it:
Equation(4): Equation describing rate of change of merR-Hg temporary complex formedwhen Hg is added in. Same reasoning as Equation (3).
Equation(5): Equation describing the rate of transcription of [MGFP]. The difference with Equation (1) is that before the promoter was constitutive, and now it is inducible. Now, at steady state:
Note that the term below is a repressive hill function of merR protein，Therefore, when merR is at a high concentration, we are repressing transcription of MGFP.
However, the action of removing merR from the system, if it goes below threshold, we "de-repress" MGFP transcription, so more protein is produced during that period of time.
Equation(6): The system is originally at a steady state, given by:
So originally, the system will tend to a particular steady state. When we add in Hg, this will change, so more PGFP will be produced.
Finally, by combining all the equations together, you can get equation below (steady state):
Equation 7
From equationb 7, we can see that many parameters affecting the output [PGFP]. Considering the challenges in circuits design and cloning, we decided to foucus on adjusting the parameters related to merR to improve the performance of mercury sensor.
Assuming that the mercury concentration is excessive, we can assume that parameter KPmerR & KMmerR is inversely related to the final GFP value. So by simply tuning Promoter Pc to a weaker level, we can get a better GFP output at the same Hg concentration.
Based on this strategy, we then selected constitutive promoters of varying strengths from iGEM promoter library (Fig. 1A) (that is, J23101, J115 and J109) to replace J101 in the mercury sensor. The sensors were then compared under various HgNO3 induction conditions (Fig. 1B). The results showed that the weaker the promoter (that is, the lower the MerR receptor concentration), the more sensitive and higher the dynamic range of the sensor., which is the same as the math model indicated.
Figure 1: A Different constitutively J23 family promoter measured strength (Data source: iGEM) B Tuning mercury receptor meRR’s intracellular density by varying the strength of J23 prmoter
Hill Equation Fitting
We have applied several strategies to improve the performance of the mercury sensor. See our Design.
Basic sensor(different J23 promoter) and sensor with three different amplifiers.
To compare the performance of different designs, we fitted the sensors’ dose–response curves to a Hill function-based biochemical model to describe their input-output relationships.
The equation used to fit the sensors’ dose–response curves to a Hill function based biochemical model to describe their input-GFPput relationships
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The Hill constant EC50 is the inducer concentration that provokes half-maximal activation of a sensor; EC50 is negatively correlated with sensitivity.
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KTop is the sensor’s maximum output expression level; KTop is positively correlated with output amplitude.
Basic sensors
Best fits for the characterized response of the various sensors circuits in this study
Here, for basic sensors, EC50 showed a 2.7-fold decrease and KTop showed a 3.5-fold increase from high to low MerR levels (Fig. 3a & 3b ), confirming that the mercury sensor’s sensitivity and output amplitude were both increased while the MerR intracellular concentration was decreased.
The maximum output (KTop) and EC50 of the sensor’s dose response against the relevant intracellular MerR concentrations
Sensor with amplifiers
Best fits for the characterized response of the various sensors circuits in this study
The maximum output (KTop) and EC50 of the sensor’s dose response with different amplifiers
The characterization of Amplifier 3 shows that this design fully protected the GFP reporter from degradation at high mercury induction levels while achieving significantly lower basal expression through continuous degradation of the reporter GFP at low mercury induction levels.
In summary, Amplifier 3 is sufficient to reduce the sensor’s basal background while also being able to maintain both the sensor’s output amplitude and sensitivity, leading to expanded output dynamic range. What’s more, this strategy can also be applied to other heavy metal sensor circuits, such as As3+ (arsR)，Pb2+ (pbrR), etc.