This pathway consists of two signal inputs, three possible steady states: a strong signal at Pulse1 will cause a Gene of interest 1 expression, a strong signal at Pulse 2 will result in a Gene of interest 2 expression, and when both have strong inputs, both do NOT express. After designed this pathway, we want to figure out some properties, such as switching-response time, the stability of the entire system, etc. And these will be able to lead us to improve our circuit.
Modeling
In order to simulate the reaction of this prokaryotic pathway in engineering bacteria, we establish a delay differential equation system based on Michaelis-Menten equation and chemical reaction kinetics to simulate the operation of this pathway, and then translate it into mathematical language and into program.
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Brief Parameter Table of DDEs Model
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Formulary
more details
| Parameter | Description | Value | Source |
|---|---|---|---|
| trc pulse1 | rate of pulse1 transcription | - | estimated |
| trc pulse2 | rate of pulse2 transcription | - | estimated |
| copynum | copy number of plasmid | 50 | estimated |
| trl CⅠ | rate of CⅠ translation | 0.05 | estimated |
| trl CⅡ | rate of CⅡ translation | 0.2 | estimated |
| trl CⅢ | rate of CⅢ translation | 0.3 | estimated |
| trl mCro | rate of Cro translation | 0.6 | estimated |
| trl Gene_of_interset_1 | rate of Gene_of_interset_1 translation | 1 | estimated |
| trl Gene_of_interset_2 | rate of Gene_of_interset_2 translation | 1 | estimated |
| deg mCⅠ | rate of mCⅠ degradation | 0.5 | estimated |
| deg mCⅡ | rate of mCⅡ degradation | 3.7 | estimated |
| deg mCⅢ | rate of mCⅢ degradation | 2.5 | estimated |
| deg mCro | rate of mCro degradation | 3.65 | estimated |
| deg mGene_of_interset_1 | rate of mGene_of_interset_1 degradation | 5.7 | estimated |
| deg mGene_of_interset_2 | rate of mGene_of_interset_2 degradation | 0.5 | estimated |
| deg CⅠ | rate of CⅠ degradation | 0.03 | estimated |
| deg CⅡ | rate of CⅡ degradation | 0.74 | estimated |
| deg CⅢ | rate of CⅢ degradation | 0.5 | estimated |
| deg Cro | rate of Cro degradation | 0.5 | estimated |
| deg Gene_of_interset_1 | rate of Gene_of_interset_1 degradation | 1 | estimated |
| deg Gene_of_interset_1 | rate of Gene_of_interset_2 degradation | 1 | estimated |
| Vmax mCⅡ | maximum transcription rate of promoter pRE | 0.2 | estimated |
| Vmax mCⅢ | maximum transcription rate of promoter pR | 0.15 | estimated |
| Vmax mCro | maximum transcription rate of promoter pRE | 0.2 | estimated |
| Vmax mgene_of_ineterest_1 | maximum transcription rate of promoter pR | 5 | estimated |
| Vmax mgene_of_ineterest_2 | maximum transcription rate of promoter pRM | 5 | estimated |
| KmCⅡ | apparent association constant for CⅠ binding with pRE | 0.3 | estimated |
| KmCⅠ2 | apparent association constant for CⅠ2 binding with pR | 0.5 | estimated |
| KmCro | apparent association constant for Cro binding with pRM | 0.025 | estimated |
| K1 | CI dimerization reaction constant | 0.05 | Burz et al.(1994); Shea and Ackers(1985) |
| K2 | CI dimerization reaction constant | 0.6 | estimated |
| VcⅡmax | maxinum rate of Ftsh break down CⅡ | 2.4 | estimated |
| VcⅢmax | maxinum rate of Ftsh break down CⅢ | 2.5 | estimated |
| KCⅡ | Michaelis-Menten constant of FtsH breaks down CⅡ | 1.5 | estimated |
| KmCⅢ | Michaelis-Menten constant of FtsH breaks down CⅢ | 1 | estimated |
| τ1 | time for CⅠ2 binding with promoter pR | 1 | estimated |
| τ2 | time for CⅡ activating promoter pRE | 1 | estimated |
| τ3 | time for CⅡ activating promoter pRE | 1 | estimated |
| τ4 | time for Cro binding with promoter pRM | 1 | estimated |
| τ5 | time for CⅠ2 binding with promoter pR（Gene） | 1 | estimated |
Analysis
Then we can solve some specific problems with the help of these equations and mathematical tools, and get some basic properties about the pathway and to evaluate its working state:
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Test of switch function
After designing a switch-enabled path, the primary task is to determine whether it can work properly, and secondly, to understand how the signal input conditions will lead to switching.
We adjust the pulse1 and pulse2 input signal strength ratio and statistics of product expression at this time to observe the work of the state of this pathway.
We can see that our switching circuit faithfully fulfills his duty to express only one protein under a specific signal input. While the other one is suppressed to be at a low level of expression. Besides, from the figure we can clearly figure out when the ratio between the two signals is about 10, we can realize the conversion quickly.
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Test of filtering performance
We did the following test to check out our filtering performance:
When we keep ratio between the G1 / G2 signal about 1:10, we input a sinusoidal signal with Gaussian white noise to G2:
The ratio of this noise signal is 1. And if our filter can handle it, it shows that our filter can work as a weakening or even eliminating part against most of the biological noise.
We can see that the output does not contain too much noise in the waveform, which is still a typical protein generation curve. And we have the output signal analysis. We find that the output noise signal ratio increased to 75 (without filtering the ratio is 1), the majority of the noise has been removed.
By tracking the variables in the delay differential equation, we can explain the principle of our filter from another point of view:
We have the following reaction in our design:
When there are still Cl and Cl2 in the cell. The balanced system made of these two components works as the capacitance in the circuit to adjust signal fluctuations. When Cl is increased due to signal fluctuations, Cl2 synthesis rate increases to consume more Cl, vice versa. It also explains why such a system of G2 filtering effect performances better than a simply attenuation as the G1. Since this balanced system can play a role as a capacitor in the general circuit, we can thus imitate the basic circuit in electronic circuits (such as LC shocks, etc.) to design the biological circuit to realize different functions. -
Parameter Sensitivity Analysis / Robustness Evaluation
Next, we analyzed the sensitivity of our pathway parameters and take a look at two important parameters in our pathway, the robustness of the response time and the expression level:
Firstly, we obtained the parameters that are more sensitive to our model by pre-analysis, and then we adjust them one by one and observe the change of the steady-state and the response time of the model.
1. KmCI2(apparent association constant for CI2 binding with pR)
We can see that when the value of KmCI2 increases, the response time of G1 decreases but the expression level rises to a stable value, and G2 shows considerable stability, which gives us a revelation: when we improve the circuit, we can use the promoter of larger inhibition constant to reduce the response time while keeping the expression level unchanged.
2. Kmcro(apparent association constant for Cro binding with pRM)
Similarly, the response time of G1 and G2 showed fairly good robustness, but the expression of G2 was sensitive to the value of Kmcro . With the increase of Kmcro, the expression of G2 also increased. But we can also use its sensitivity to control expression levels.
3.KmcII(apparent association constant for CII binding with pRE)
From the figure above, we can see that the response time and expression level of G1 are relatively stable, while G2 is more sensitive to this parameter. However, considering that KmcII value is very small, G2 is almost non-expressed and the leakage of gene expression cannot be avoided, the fluctuation is due to G2 leakage expression. It can be considered like that G2 response time is stable. Another interesting point is that the sudden increase in G2 expression can be considered as a switch, which provides a way for us to design more multivariable switches later. We can turn G2 “off” by providing another competitive binding substance to "change" the value of Kmcll.
Summary
In general, the designed filter and switch functions are achieved. And in the case of little external disturbance our system can be relatively stable. This prokaryotic pathway also exhibits a high level of plasticity, which provides us with many ways for further application and future improvement.
