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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 thresholds, 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

Fluorescence 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 effector 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) 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 transcription factor (it means the NorR will be activated to bind 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 genes that bind to a transcriptional activator, and the more genes that bind to a transcriptional activator, the more the rate of expression of the gene increases. The expression of conversion to mathematics is: In fact, the increased rate of transcription of a regulated gene depends on the proportion of genes that bind to a transcriptional activator, and the more transcriptional activators that bind to the gene, the more the rate of expression of the gene increases. The mathematical expression 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 is 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:

Symbol Description

Table 1. The name and symbol of variable