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• Introduction
• Assumption
• Symbol Description
• Fluorescence intensityof GFP (1)
• Fluorescence intensity of GFP (2)
• Model solving

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Model

Introduction

In the affected area of ​​patients with inflammatory bowel disease, the concentration of nitric oxide is significantly increased, so we chose it as the input signal of our anti-inflammatory device. However, the nitric oxide is very unstable, so we have introduced an amplifier which can converts unstable gas signals into stable intracellular signals for sustained high-level output. The amplifier is based on a positive feedback loop. Transcription activator B-A can self-drive in a manner independent of the input signal for a period of time after the signal is input, and the metabolic flow in this cycle can be transferred to the output circuit (Figure 1).

Figure 1. Schematic Design of the Synthetic amplifier After the signal is input, the transcriptional activator B-A is generated, which includes a DNA binding domain and a transcriptional activation domain. On the one hand, B-A can activate the expression of the reporter gene, and on the other hand can activate the expression of B-A itself. The constitutively expressed Binder will compete with B-A to suppress the leakage of the device.

In addition, when using an amplifier based on a positive feedback loop, we need to strictly limit its activation until the input signal is strong enough, which is beneficial to suppress leakage of the device. To this end, we introduce the concept of threshold, which is to achieve competition between B-A and B by constitutively expressing Binder with a certain intensity (Figure 1). In this way, the amplifier can only be effectively activated when the input signal is strong enough. We established a mathematical model to predict the performance of the amplifier under different restrictions.

Assumption

(1) The sequence that B-A and B bind is the same. So, it can be considered that both are combined with the same substrate.

(2) B-A and B have the same promotion or inhibition effect on the amplifier and output circuit.

Symbol Description

Table 1. The name and symbol of variable. Correction: X2 represents for the concentration of Binder-Activator and X3 represents for the concentration of Binder.

Fluorescence intensity of GFP with Constant Inflammatory Signal

1.Available when the bacterial resources are extremely rich

Firstly, the amplifier we described in Figure 1 can be simplified to the Figure 2.

Figure 2. Simple circuit of the amplifier

The concentration of the Binder-Activator expressed by the input circuit is $P_2\left(x_1\right)$ when the inflammatory signal is at a concentration of $x_1$ per unit time. The amount of Binder-Activator or GFP expressed by the amplifier or output circuit is $\text{P}\left(x_2\text{,}{x}_3\right)$ for the Binder-Activator and Binder at a concentration of $x_2$ and $x_3$ , respectively. When the amplifier is used, it can be seen from the Figure 2 that Binder-Activator ( $x_2$ ) has two synthetic pathways, one is that the inflammatory signal ( $x_1$ ) promotes the synthesis of the input circuit, and the other is that $x_2$ facilitates the synthesis of the amplifier by $x_3$ . So, we can get this equation:

......(1)

There is only one synthetic pathway of y , that is, $x_2$ and $x_3$ work together with the output circuit to release y, so it can be obtained by assumption 2:

......(2)

Ⅰ. $P_1\left(x_1\right)$ expression solving

The gene (equivalent to the binding sequence of Binder-Activator) is abbreviated as G, various transcriptional activators are abbreviated as S, and various transcriptional repressors (e.g. Binder) are abbreviated as I. The binding of $x_1$ to the NorR (it means the NorR will bind NO to activate the promoter PnorV) of input circuit is a reversible reaction, so the binding reaction of $x_1$ to the input circuit can be expressed as:

......(3)

k1 and k2 are the reaction rate constants of the forward reaction and the reverse reaction, respectively. Refer to the Michaelis-Menten equation, we do the following analysis: when the reaction reaches equilibrium, the concentration of SG does not change, that is, the rate of SG generation and decomposition is equal, then we can get the following equation:

......(4)

......(5)

......(6)

In fact, the increased rate of transcription of a regulated gene depends on the proportion of transcriptional activators that bind to the gene, and the more transcriptional activators that bind to the gene, the more the rate of expression of the gene increases. The expression of conversion to mathematics is:

......(7)

Suppose that when the substrate concentration is large enough, $P_1\left(x_1\right)$ will take the maximum value, set to $P_{\mathrm{1max}}$ , and [S] will also be much larger than [G], so we can get this:

......(8)

Bring the formula (6), (8) into equation (7) to get the analytical expression of $P_1\left(x_1\right)$ :

......(9)

Although the body's immune system can make timely adjustments to the inflammatory response, $x_1$ is considered to be a fixed value in a sufficiently short period of time, and $P_1\left(x_1\right)$ can also be considered as a constant that varies with $x_1$ , abbreviated as A. The calculations that follow are handled this way.

Ⅱ. $\text{P}\left(x_2\text{,}{x}_3\right)$ expression analysis

Since $x_2$ , $x_3$ binds to the same site in the gene, the gene will be activated when $x_2$ binds it while being inhibited when $x_3$ binds it. This can be regarded as the competition between $x_2$ and $x_3$ . Similar to the analysis we used to solve the expression of , we can get:

......(10)

......(11)

In the reaction, $x_2$ represents the concentration of S, and $x_3$ represents the concentration of I. The increase of the transcription rate of the gene depends on the concentration of SG, that is, the amount of the transcriptional activator that binds to the gene, and the gene that binds to $x_3$ can no longer bind to $x_2$ , resulting in inhibition of transcription. So,

Analogy to the analysis we used to solve the expression of $P_1\left(x_1\right)$ , we list the following relationships: When it reaches equilibrium:

......(12)

......(13)

From this we can get the following equation:

......(14)

......(15)

Sort out the above two formulas to get:

......(16)

The same way we get:

......(17)

When [S] is large enough, $\text{P}\left(x_2\text{,}{x}_3\right)$ will take the maximum value and set it to . $P_{\mathrm{2max}}$ [S] will also be much larger than [G] and another in the denominator, so there are:

......(18)

Bringing Equations (16), (18) into Equation (17) yields:

......(19)

So, we get the expression of $\text{P}\left(x_2\text{,}{x}_3\right)$ . Because the role of $x_3$ in the system is to set a threshold for the amplification of $x_2$ , and is set to a constitutive expression, $x_3$ can be regarded as a fixed value of the regulation, and $x_3$ becomes another variable constant.

III. Calculation of : $x_2$

From the analysis of Equation (1) and the above two steps, the relationship between $x_2$ and time after adding the amplifier can be obtained:

......(20)

The relationship between GFP fluorescence intensity and time:

......(21)

When the amplifier is not used, y is directly synthesized by the inflammatory signal ( $x_1$ ), from which we can get:

......(22)

2. Available when the bacterial resources are restricted

The resources in bacteria are actually limited. In addition to using resources to synthesize the GFP, bacteria also need resources to maintain their normal life activities. At this time, there is competition between the amplifier and the bacteria's own life activities. The eventual result is that after a limited amount of resources are exhausted, the bacteria die and the amplifier stops working. Therefore, the following bacterial resource attenuation model is established. Assuming that the total amount of resources in the bacteria is q, then when the resources decays, the following formula holds:

......(23)

......(24)

Among the above relationship, K is the total amount of initial resources before decay, and is the time constant of the decay process, we can get:

......(25)

Solution:

......(26)

Then, when there is no amplifier, the model is corrected to:

......(27)

When there is an amplifier, the model is corrected to:

......(28)

Fluorescence intensity of GFP with attenuated Inflammatory Signal

The above model consistently believes that the intensity of the inflammatory signal is constant. However, in the actual biological environment, the inflammatory signal will be attenuated to varying degrees with distance, time and drug release. Therefore, the attenuation model of the inflammatory signal intensity is first established as follows.

Figure 3. The effect of attenuated inflammatory signal intensity on GFP output when bacterial resources are limited with an amplifier

It is easy to get the inflammatory signal intensity decay rate as follows:

......(29)

Where $K_{\mathrm{s}}$ is the intensity of the inflammatory signal at the initial moment and $\text{T}$ is the time constant of the decay process of the inflammatory signal. Solution:

......(30)

Then the amount $P_1\left(x_1\right)$ of Binder-Activator expressed in the input circuit is as follows:

......(31)

Considering the attenuation of inflammatory signals and the limitation of bacterial resources, when there is no amplifier, the model is corrected to:

......(32)

Considering the attenuation of inflammatory signals and the limitation of bacterial resources, when adding an amplifier, the model is corrected to:

......(33)

Model solving

1. Parameter assignment table

By referring to some previous work, the parameters are as follows:

Table 2. Parameter assignment

2. Model solving without amplifier

Ⅰ. When the intensity of the inflammatory signal is constant and the bacterial resources are considered to be unlimited, record , then . So, the fluorescence intensity is:

......(34)

That is:

......(35)

Ⅱ. When the intensity of the inflammatory signal is constant but the bacterial resources are limited, record . So, the fluorescence intensity is:

......(36)

That is:

......(37)

Figure 4. The effect of bacterial resources on the GFP output without amplifier

III. When the inflammatory signal is attenuated, but the bacterial resources are considered to be unlimited, the output GFP is solved as:

......(38)

That is:

......(39)

IV. When the inflammatory signal is attenuated and the bacterial resources are considered to be limited, the GFP is solved as:

......(40)

That is:

......(41)

Figure 5. The effect of bacterial resources on the GFP output when inflammatory signal is attenuated without amplifier

3. Model solving with amplifier

Ⅰ. When the intensity of the inflammatory signal is constant and the bacterial resources are considered to be unlimited, record , then:

......(42)

Using MATLAB to solve for , it is:

......(43)

Let $x_2=\text{g}\left(\text{t}\right)$ , then we get:

......(44)

Then we get GFP as:

......(45)

Ⅱ. When the intensity of the inflammatory signal is constant but the bacterial resources are limited, record . So, the fluorescence intensity is:

......(46)

Figure 6. The effect of bacterial resources on the GFP output with amplifier

Figure 7. GFP output curve considering bacterial resources limitations with amplifier

III. When the inflammatory signal is attenuated and the bacterial resources are limited, the Runge-Kutta method is used to solve the differential equation and the GFP curve is obtained as follows. Compared with the GFP curve without amplifier, the addition of the amplifier has obvious effect.

Figure 8. GFP output curve considering inflammatory signal attenuation and bacterial resources limitation

4. Analysis of the regulatory role of Thresholder

When the intensity of the inflammatory signal is attenuated and the bacterial resources are limited, in order to avoid the leakage of the positive feedback loop, we set the threshold regulate component Binder, the concentration can be varied between 10 and 100 mM. The regulation curve is as follows.

Figure 9. Binder regulation curve considering inflammatory signal attenuation and bacterial resources limitation

In conclusion, the introduction of Thresholder does have a regulatory effect on the output of GFP and is very obvious, which has a positive effect on suppressing the leakage of the positive feedback loop.

Our established model is based on some assumptions and approximations and lacks the support of experimental data. But we believe that our work can provide a reference to other work that requires the use of a positive feedback loop and threshold. In addition, we will conduct experiments to test the validity of our model in the future.

[1] Smole,A., Lainsˇcek,D., Bezeljak,U., Horvat,S. and Jerala,R. (2017) A synthetic mammalian therapeutic gene circuit for sensing and suppressing inflammation. Mol. Ther., 25, 102–119.

[2] Michaelis L, Menten M L. Die kinetic der invertinwirkung. Biochem. Z. , 1913, 49: 334-336

[3] Runge, Carl David Tolmé (1895), "Über die numerische Auflösung von Differentialgleichungen", Mathematische Annalen, Springer, 46 (2): 167–178,

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