Part1 Extracellular signal converting into intracellular signal

Part2 Internal signal regulating recombinase production

Part3 Recombinase turns on downstream effect expression

Part4 Significance of the cell culture environment

Part5 Recombinase filtering and de-shocking

Part6 Robustness of the model

Part1 Extracellular signal converting into intracellular signal

Tips: Skip part1 if the gene circuit is directly regulated by intracellular signal.

SynNotch can convert an extracellular signal into an intracellular signal . Here, we take synNotch-TEV as an example. We use the number of TEV as the intracellular signal after conversion. TEV is an enzyme which is able to release the repressed promoter.

In this section, 4 reactions are considered:

$$G_{synNotch-TEV}\xrightarrow[\quad\quad]{C11}{synNotch-TEV}\ \ \ \ \ \ \ \ \ \ \ (1.1)$$

$$synNotch-TEV+Ts\xrightarrow[\quad\quad]{C12}U+TEV\ \ \ \ \ \ \ \ \ \ \ (1.2)$$

$$synNotch-TEV\xrightarrow[\quad\quad]{C13}\emptyset\ \ \ \ \ \ \ \ \ \ \ (1.3)$$

$$TEV\xrightarrow[\quad\quad]{C14}\emptyset\ \ \ \ \ \ \ \ \ \ \ (1.4)$$

The specific contents are as follows:

$$G_{synNotch-TEV}\xrightarrow[\quad\quad]{C11}{synNotch-TEV}\ \ \ \ \ \ \ \ \ \ \ (1.1)$$

(1.1) represents the expression of synNotch which contains the TEV. Although the actual expression progress of the protein involves many chemical reactions from transcription to modification, these intermediate reactions satisfy the following conditions:

a)The factors affecting these reactions in the system are stable, such as the amount of amino acids, nucleotides, and helicases;

b)The gene is regulated by a constitutive promoter and its expression is not regulated by other products of the system;

c)The product of an intermediate reaction has no effects on the system functions.

We call these three conditions the post-integration conditions [1], and call these reactions the post-integration reaction. In the same way, the expression and degradation of other proteins involved in our model all suit the conditions of the refinement, so they can be considered as one single chemical reaction.

Obviously (1.1) is a post-integration reaction, and the reaction rate is a constant, for the controlling conditions are unchanged.

$$synNotch-TEV+Ts\xrightarrow[\quad\quad]{C12}U+TEV\ \ \ \ \ \ \ \ \ \ \ (1.2)$$

(1.2) represents that the TEV protein is released by the intracellular domain of the sporadic synNotch when the synNotch-TEV protein was stimulated by an external Target signal. The rate of the reaction is calculated by the law of mass action which is in proportion to the number of receptors on the cell membrane and the number of external target signals.

$$synNotch-TEV\xrightarrow[\quad\quad]{C13}\emptyset\ \ \ \ \ \ \ \ \ \ \ (1.3)$$

(1.3) represents the degradation of synNotch membrane proteins linked to TEV proteins The degradation rate is directly proportional to its protein content.

$$TEV\xrightarrow[\quad\quad]{C14}\emptyset\ \ \ \ \ \ \ \ \ \ \ (1.4)$$

(1.4) represents the degradation of TEV protein.

The degradation rate is directly proportional to its protein content.

The figures below show the result of simulation using the Gillespie algorithm.

It is concluded that the initial value of synNotch-TEV is stable at about 20,000 before the Target signal is received, which is used as the initial condition for subsequent translation.

Conclusion: When a stabilized cell contacts different levels of extracellular signals, synNotch membrane protein will release different amounts of TEV. The amount of TEV protein is in proportion of the strength of target signals.

Part2 Internal signal regulating recombinase production

In this part, intracellular proteins can degrade or destroy the repressors, reducing their number. Different levels of protein have different effects on the relieving repression. Then, using Hill equation, we can calculate the expression rate of the recombinase based on the repressor protein number. At the same time, we use the error-reducing reaction to solve the problem of the extra recombinase. Here, we choose TEV as intracellular protein, and TetR as repressor protein.

In this section, 12 reactions are considered:

$$G_{TetR}\xrightarrow[\quad\quad]{C21}TetR\ \ \ \ \ \ \ \ \ \ \ (2.1)$$

$$TetR\xrightarrow[\quad\quad]{C22}\emptyset\ \ \ \ \ \ \ \ \ \ \ (2.2)$$

$$2TetR+tetO\xrightarrow[\quad\quad]{C23}TetOR\ \ \ \ \ \ \ \ \ \ \ (2.3)$$

$$TEV+TetOR\xrightarrow[\quad\quad]{C24}tetO+TEV+U\ \ \ \ \ \ \ \ \ \ \ (2.4)$$

$$TetOR\xrightarrow[\quad\quad]{C25}tetO+U\ \ \ \ \ \ \ \ \ \ \ (2.5)$$

$$G_{rec}\xrightarrow[\quad\quad\quad\quad\quad\quad]{C31-TetOR}rec\ \ \ \ \ \ \ \ \ \ \ (2.6)$$

$$rec\xrightarrow[\quad\quad]{C33}\emptyset\ \ \ \ \ \ \ \ \ \ \ (2.7)$$

$$G_{RDF-inhibitor}\xrightarrow[\quad\quad\quad\quad\quad\quad]{C32-TetOR}RDF-inhibitor\ \ \ \ \ \ \ \ \ \ \ (2.8)$$

$$RDF-inhibitor\xrightarrow[\quad\quad]{C34}\emptyset\ \ \ \ \ \ \ \ \ \ \ (2.9)$$

$$G_{rec-inhibitor}\xrightarrow[\quad\quad]{C35}rec-inhibitor\ \ \ \ \ \ \ \ \ \ \ (2.10)$$

$$rec-inhibitor\xrightarrow[\quad\quad]{C36}\emptyset\ \ \ \ \ \ \ \ \ \ \ (2.11)$$

$$rec+rec-inhibitor\xrightarrow[\quad\quad]{C37}\emptyset\ \ \ \ \ \ \ \ \ \ \ (2.12)$$

The specific content is as follows. (As for each chart in this part, we choose the quantity of substance as the vertical axis, and 0.01 hour as the unit of the horizontal axis. Different subplots correspond to different levels of intracellular protein number).

$$G_{TetR}\xrightarrow[\quad\quad]{C21}TetR\ \ \ \ \ \ \ \ \ \ \ (2.1)$$

(2.1) indicates the expression of the TetR protein. Through the analysis of (1.1) in part1, it is easy to conclude that (2.1) is a post-integration reaction, and the rate of the reaction is constant.

$$TetR\xrightarrow[\quad\quad]{C22}\emptyset\ \ \ \ \ \ \ \ \ \ \ (2.2)$$

(2.2) indicates the degradation of TetR protein. The degradation rate is directly proportional to its protein content.

$$2TetR+tetO\xrightarrow[\quad\quad]{C23}TetOR\ \ \ \ \ \ \ \ \ \ \ (2.3)$$

(2.3) represents the generation of the binary complex TetOR, and the reaction rate is calculated according to the law of mass action in the model.

$$TEV+TetOR\xrightarrow[\quad\quad]{C24}tetO+TEV+U\ \ \ \ \ \ \ \ \ \ \ (2.4)$$

(2.4) indicates that the TEV is a separate TetOR binary complex. This reaction rate is calculated according to the law of mass action.

$$TetOR\xrightarrow[\quad\quad]{C25}tetO+U\ \ \ \ \ \ \ \ \ \ \ (2.5)$$

(2.5) indicates that a small amount of TetOR is self-separate. The reaction rate is calculated according to the law of mass action.

The figure below shows the calculated results.

Using the above calculated amount of repressor, the Hill equation is used to determine the expression rate of the recombinase.

$$G_{rec}\xrightarrow[\quad\quad\quad\quad\quad\quad]{C31-TetOR}rec\ \ \ \ \ \ \ \ \ \ \ (2.6)$$

(2.6) indicates the generation of rec (rec is the abbreviation of recombinase) , which is also a post-integration reaction, but the expression rate is regulated by the repressor TetR. The reaction rate in the model is calculated using the Hill equation, and the rate is ((N/k) ^n+1) ^-1. Accordingly, N is the number of TetR, and K is the half-maximum of the suppression effect.

$$rec\xrightarrow[\quad\quad]{C33}\emptyset\ \ \ \ \ \ \ \ \ \ \ (2.7)$$

(2.7) indicates degradation of rec. The degradation rate is directly proportional to rec number.

$$G_{RDF-inhibitor}\xrightarrow[\quad\quad\quad\quad\quad\quad]{C32-TetOR}RDF-inhibitor\ \ \ \ \ \ \ \ \ \ \ (2.8)$$

(2.8) indicates the generation of RDF. Like rec, it is also a post-integration reaction, whose rate is regulated by the repressor TetR. The reaction rate in the model is calculated by the Hill equation, and the rate is ((N/k) ^n+1) ^-1. Similarly, N is the number of TetR, and K is the half-maximum of suppression effect.

$$RDF-inhibitor\xrightarrow[\quad\quad]{C34}\emptyset\ \ \ \ \ \ \ \ \ \ \ (2.9)$$

Degradation of RDF-inhibitor is represented by (2.9). The degradation rate is directly proportional to the number of RDF-inhibitor. The roles and meanings of this reaction will be discussed in the next part.

$$G_{rec-inhibitor}\xrightarrow[\quad\quad]{C35}rec-inhibitor\ \ \ \ \ \ \ \ \ \ \ (2.10)$$

(2.10) indicates the generation of the rec-inhibitor, which is also a post-integration reaction, and the expression rate is constant.

$$rec-inhibitor\xrightarrow[\quad\quad]{C36}\emptyset\ \ \ \ \ \ \ \ \ \ \ (2.11)$$

(2.11) indicates the degradation of the rec-inhibitor. The degradation rate is directly proportional to rec-inhibitor number.

$$rec+rec-inhibitor\xrightarrow[\quad\quad]{C37}\emptyset\ \ \ \ \ \ \ \ \ \ \ (2.12)$$

(2.12) indicates that the rec-inhibitor combined with the leaked rec. It is used to filter rec. The reaction rate was calculated according to the law of mass action and enables the low expression of rec to be combined to ensure the stability of the system.

It is concluded that internal signal TEV regulates recombinase production. According to (2.4), different concentrations of TEV correspond to different concentrations of TetR. By (2.6), the concentrations of TetR affect rec concentrations. Through the simulation of (2.12), a small amount of rec-inhibitor can almost completely eliminate the rec leakage from the TetR system. This pathway can significantly improves the system stability.

Part3 Recombinase turns on downstream effect expression

In this part, we use the above calculated amount of the rec expression at different signal levels. Using rec and RDF, we can switch the state and calculate the expression of the downstream gene. In this section, 5 reactions are considered:

$$rec+\overline{G}_{downstream}\xrightarrow[\quad\quad]{C41}rec+G_{downstream}\ \ \ \ \ \ \ \ \ \ \ (3.1)$$

$$RDF+G_{dowmstream}\xrightarrow[\quad\quad]{C61}RDF+\overline{G}_{downstream}\ \ \ \ \ \ \ \ \ \ \ (3.2)$$

$$G_{RDF}\xrightarrow[\quad\quad]{C}RDF\ \ \ \ \ \ \ \ \ \ \ (3.3)$$

$$RDF-inhibitor+RDF\xrightarrow[\quad\quad]{C34}\emptyset\ \ \ \ \ \ \ \ \ \ \ \ (3.4)$$

$$G_{gfp}\xrightarrow[\quad\quad]{C}GFP\ \ \ \ \ \ \ \ \ \ \ (3.5)$$

$$rec+\overline{G}_{downstream}\xrightarrow[\quad\quad]{C41}rec+G_{downstream}\ \ \ \ \ \ \ \ \ \ \ (3.1)$$

(3.1) represents that recombinase specifically recognizes the DNA sequence between the two sites and reverses it, so that the downstream gene expression is reversed to an open state. (Downstream genes are GFP and RDF Gene)

$$RDF+G_{dowmstream}\xrightarrow[\quad\quad]{C61}RDF+\overline{G}_{downstream}\ \ \ \ \ \ \ \ \ \ \ (3.2)$$

(3.2) indicates that the reverse recombination factor specifically recognizes the two sites after the recombinase worked on the DNA, and makes the reversal DNA sequence between the sites restored to the shape.

$$G_{RDF}\xrightarrow[\quad\quad]{c}RDF\ \ \ \ \ \ \ \ \ \ \ (3.3)$$

(3.3) indicates the expression of the reverse recombination factor, which exists only when the downstream gene is turned into an open state. The expression of the RDF process is not regulated, and also is a post-integration reaction, so the expression rate is constant.

$$RDF-inhibitor+RDF\xrightarrow[\quad\quad]{C34}\emptyset\ \ \ \ \ \ \ \ \ \ \ \ (3.4)$$

(3.4) indicates that the RDF-inhibitor is combined with the intracellular RDF when the system requests downstream gene opening. The reaction rate is calculated according to the law of mass action. It makes our system stable in the open state. When it is not requested to open, the increase of RDF is not limited, so that it can function to shut down the system and realize real-time feedback.

In this simulation, the off state is the initial state. When the TEV signal number level is less than 5, there is not enough rec to turn on the downstream gene,and RDF is not expressed; when the TEV signal number level is higher than 5, rec and RDF-inhibitor begin to express simultaneously after de-repression. Then，RDF and GFP are simultaneously expressed after the rec turns on the downstream gene. The combination of RDF-inhibitor and RDF ensures that there is not enough RDF in the cell to shut down the downstream gene, which makes the system stable.

$$G_{gfp}\xrightarrow[\quad\quad]{c}GFP\ \ \ \ \ \ \ \ \ \ \ (3.5)$$

The formula (3.5) indicates the production of the effectors. The downstream gene can be expressed when it is reversed to open state. Here, GFP is taken as an example, and the expression is constant after the rectification reaction.

The figure below shows the opening of the downstream gene. 0 means closed; 1 means open.

(a):

s

(b):

The results meet our expectation: The system doesn’t express GFP in low TEV number level. It begins to express GFP when the concentration level reaches the threshold.

Part4 Significance of the cell culture environment

If the normal number of the signal molecules can be added into the culture medium of cells, it will improve the system stability by 10%.

Part5 Recombinase filtering and de-shocking

Part6: Robustness of the model

After adding a lot of disturbances to the previously determined parameters, the calculation results are as follows:

The stability of the algorithm and our path is verified by adding a lot of perturbations on the parameters.