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 dependent 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:
(II) Non-dimensionalisation
In this model, every part is regarded as unique. Having an independent parameter in each part, though gives a better description of the system; apparently increases the complexity of the ODE system. Therefore, additional assumptions were made to simplify this model by reducing the parameters.
Firstly, it is assumed that each protein has the same degradation rate because of the negligible differences between them. Under this condition, we could apply non-dimensionalization to integrate a series of parameters that model the same circuit into one main parameter represented by "k". In spite of this assumption, six main parameters in the system are remained. In order to further simplify it, these main parameters are generalised to one changeable core parameter on the ground that three circuits of the same strength are the basic requirements for a tristable switch. Furthermore, the Hill’s coefficient of each repressor-protein interaction are close to each other in our experimental design; hence, we assume that $n_1=n_2=n_3=n$. Accordingly, we derived a new version of ODEs of the system:
The main variables in these final ODEs are core parameters (a) and a Hill’s Coefficient (n). Through this model, we can subsequently analyze the system's stability and predict the system's behaviours.
(III) Overall Assumptions:
- Degradation is a unimolecular reaction, depending on the concentration only.
- The copy number of the plasmid (the amount of DNA) is constant.
- mRNA reaches equilibrium much faster than protein, such that mRNA is at steady state when considering protein dynamic.
- Compared to the cell division rate, degradation rates of individual repressor proteins are negligible. As a result, protein dilution rate due to cell division is used to replace the repressors’ degradation rates.
