# Team:HZAU-China/Circuit

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## Design principle of rewirable circuit

### Design Principle of Rewirable Circuit

If you want to design your own rewirable circuit, you may need more information about how to design. Our design principle can help you to find the pattern that satisfies the specification. Our method is based on the large-scale search.

##### 5.1 ODE sets with matrices

In order to implement large scale calculation, we introduce the ordinary differential equation sets with matrices. The transcription regulatory network can be abstracted into a matrix $R$. We use 1, -1, 0 to stand for three possible gene relationships: activation, repression, no regulation. The elements $r_{ij}$ in $R$ indicates that gene $i$ is regulated by gene $j$. This matrix is used to index the topological structure. For calculation, we need to decompose this matrix into two adjacent matrices to represent activation relationship $R_1$ and repression relationship $R_2$, respectively. According to the matrices, we can use the following equations to simulate the gene expression dynamics. $$\begin{split} \left( \begin{array}{c} \frac{d{mRNA}_{x_1}}{dt}\\ \vdots\\ \frac{d{mRNA}_{x_N}}{dt}\\ \end{array} \right) &= R_1 \cdot \left( \begin{array}{c} \frac{\beta_1\cdot x_1^n}{K^n+x_1^n}\\ \vdots\\ \frac{\beta_N\cdot x_N^n}{K^n+x_N^n}\\ \end{array} \right) + R_2 \cdot \left( \begin{array}{c} \frac{\beta_{N+1}\cdot K^n}{K^n+x_1^n}\\ \vdots\\ \frac{\beta_{2N}\cdot K^n}{K^n+x_N^n}\\ \end{array} \right) - I \cdot K_{R} \cdot \left( \begin{array}{c} {mRNA}_{x_1}\\ \vdots\\ {mRNA}_{x_N}\\ \end{array} \right)\\ \left( \begin{array}{c} \frac{dx_1}{dt}\\ \vdots\\ \frac{dx_N}{dt}\\ \end{array} \right) &= K_{tl}\cdot \left( \begin{array}{c} {mRNA}_{x_1}\\ \vdots\\ {mRNA}_{x_N}\\ \end{array} \right) -K_{P}\cdot \left( \begin{array}{c} x_1\\ \vdots\\ x_N\\ \end{array} \right) \end{split}$$,

where $I$ is the identity matrix. Ignoring some detailed information, we use this coarse-grained model to perform a complete search of the topological space.

##### 5.2 Function-topology maps

The rewirable circuit can be user-defined. First, we define the functions that are used to solve some problems. Be sure that these functions can be represented by some quantitative features. For each function, you should construct a fitness function based on the quantitative features. Then we enumerate the completed topological space. For each topology, a lot of parameter sets are sampled using the Latin hypercube sampling method. We calculate different fitness functions for each topology and get a function-topology table.

##### 5.3 Topological structure matching

There are two criteria to decide whether two topological structure of regulation can be converted. If the index matrices $R_A$ and $R_B$ only differ in one row, which means only one gene is regulated in a different way in the genetic circuit, we can use two inducible promoters to drive this gene alternately (Fig. 1a). In this case, we conclude that the two topological structures match. Another case is that $R_A$ and $R_B$ can convert mutually by exchanging two rows. In this case, promoter transposition of two genes can swap the ways they are regulated (Fig. 1b).

Figure 1. Sketches of matching topological structure.

So we can use the index matrices to match topological structures that are likely to convert.

##### 5.4 General Steps for designing rewirable circuit

1. List the specifications for the rewirable circuit;

2. Design fitness functions for each function you want to realise based on the quantitative specifications;

3. Scan the topological space to calculate the fitness functions for each topological structure;

4. Match topological structures that can convert mutually;

5. Select a pair of topological structures whose value of fitness function satisfies the specification.

##### References

Ma, W., Trusina, A., El-Samad, H., Lim, W. A., & Tang, C. (2009). Defining network topologies that can achieve biochemical adaptation. Cell, 138(4), 760-773.

##### Contacts
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