We based on the Michealis-Menten Equation and Hill Equation to describe the kinetics of gene regulatory networks. The M-M Equation and Hill Equation describe the relationship between the production rate of products and the concentration of substrate.

M-M Equations of activation (monotone-increasing) and inhibition (monotone-decreasing) are defined as:

〖f(X)〗_activition=CX/(K+X) 〖f(X)〗_inhibition=CK/(K+X)

Hill Equations of activation (monotone-increasing) and inhibition (monotone-decreasing) are defined as:

〖f(X)〗_activition=(CX^n)/(K^n+X^n ) 〖f(X)〗_inhibition=(CK^n)/(K^n+X^n )

(C is a constant, K is called half maximal effective concentration and n is called hill coefficient.)

In our project, we aimed at all three-node network structures, and each contains 4 nodes (1 input node, 2 regulatory nodes and 1 output node) and vast possible regulatory edges among them.



So we formed ODE equation sets to describe the mutual relationships and form networks. The ODE equation sets involve two 4*4 matrices to respectively bring in the activating effect and inhibitive effect, and a column vector for self-decomposition.



In our wet lab part, we synthetized network and constructed plasmids transferred into Hela cells, and we made some adjustment in our design for a better realization in experiment:
 We assumed that input (I) is not regulated by the network and remained constant.
 We fixed two certain edges (I inhibits A, A inhibits O), because when constructing the system, the components to function as inducers are quite few. So we tend to use 2 repressors to substitute it.
 We limited the maximum number of regulatory nodes to 2 restrict the problem to an acceptable scale.
 We chose relative parameters close to the actual experiments in our simulation. According to previous research, we guaranteed that most parameters are within the already confirmed range, such as reaction rate.