Team:Hong Kong HKUST/Modelling
Modelling
The tristable switch functions when a selected promoter is induced, two downstream targets are automatically repressed. Through this alternating activation and repression of promoters, the state of the system can be flipped independent of the inducers applied. Therefore, focusing on the regulation and the stability of the system, we aim to establish a model to describe, analyze, and predict the behavior of the system.
Model Equations and Derivation
In order to study the system dynamics and to achieve a better simulation result, we consider every part of the switch by setting the strength of every part as a parameter, which constituted a system of nonlinear ordinary differential equations (ODEs).
Based on mass action kinetics (Phillips, 2013) , a model of genetic circuits, which considered transcription and translation, is constructed. By so doing can we derive the basic equations that describe the change of mRNA and proteins over time:
However, our case, in which three circuits are involved, is much more complicated, as it requires a system with 6 ODEs. Therefore, we sought to reduce the complexity of the system. Grounded on mathematical model of bistable toggle switch in (Gardner, Cantor, & Collins, 2000), we developed a series of three equations to describe the switch.
(I) Quasi Steady –State Approximation:
The modelling begins with the application of Quasi Steady-State Approximation on mRNA dynamic. Assumed that the time scale of mRNA dynamic is much smaller than that of protein dynamic, we then consider the concentration of mRNAs remains constant over time (i.e.$\frac{dr_x}{dt}=\frac{dr_y}{dt}=\frac{dr_z}{dt}=0$, the system of ODEs then reduces into the following form:
Stability Analysis
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