Modeling

Introduction

The modelling of synthetic biology systems is vital in the design of genetic circuits. To be able to construct sequences of increasing complexity – on the scale of the natural networks already present in the bacteria it is necessary develop satisfactory computational tools to predict and mimic the behaviour of the system. [1]
To predict the output of our genetic circuit (see Design) we created a single cell model for our circuit. The effect of transcription regulators on gene expression was assumed to follow Hill equation kinetics, with first order kinetics for the degradation of the proteins/enzymes formed. The models were created in MATLAB.

Phosphatase Amplification Circuit

**Insert photo of amplification circuit here** The positive feedback amplification circuit shown above was used in our study to obtain an amplified production of our gene of interest in presence of an inducing chemical. For more details on the parts used in the circuit please visit the Design page.

Deterministic Model

The deterministic model involves creating a series of differential equations which are solved simultaneously using certain initial conditions. The differential equations were obtained by performing a mass balance for each protein being expressed. For the transcriptional regulator (‘TR’):

• It is formed by gene expressed by promoter P1 which is induced by the an inducer ‘I’ \[ Rate\ of\ formation\ of\ TR\ through\ P1=\frac{V_{{max}_I}*{[I]}^{n_I}}{k_I^{n_I}+{[I]}^{n_I}} \]
• It is formed by gene expressed by promoter P2 which is induced by itself ‘TR’ \[ Rate\ of\ formation\ of\ TR\ through\ P2=\frac{V_{{max}_{TR}}*\ {[TR]}^{n_{TR}}}{k_{TR}^{n_{TR}}+{[TR]}^{n_{TR}}} \]
• It is degraded \[ {Rate\ of\ degradation\ of\ TR=\delta{}}_{TR}[TR] \]

By a mass balance, the overall rate of change of the concentration of the transcriptional regulator is thus given by: \[ \frac{d[TR]}{dt}=\frac{V_{{max}_I}*{[I]}^{n_I}}{k_I^{n_I}+{[I]}^{n_I}}+\frac{V_{{max}_{TR}}*\ {[TR]}^{n_{TR}}}{k_{TR}^{n_{TR}}+{[TR]}^{n_{TR}}}-{\delta{}}_{TR}\left[TR\right]+{\alpha{}}_{TR} \]
Similarly for the reporter protein RFP (‘R’):

• It is formed by gene expressed by promoter P2 which is induced by the transcriptional regulator ‘TR’ \[ Rate\ of\ formation\ of\ R\ through\ P2=\frac{V_{{max}_{TR}}*\ {[TR]}^{n_{TR}}}{k_{TR}^{n_{TR}}+{[TR]}^{n_{TR}}} \]
• It is degraded \[ {Rate\ of\ degradation\ of\ R=\delta{}}_R[R] \]

By a mass balance, the overall rate of change of the concentration of the transcriptional regulator is thus given by: \[ \frac{d[R]}{dt}=\frac{V_{{max}_{TR}}*\ {[TR]}^{n_{TR}}}{k_{TR}^{n_{TR}}+{[TR]}^{n_{TR}}}-{\delta{}}_R\left[R\right]+{\alpha{}}_R \] \begin{center} \vspace{3pt} \noindent \begin{tabular}{|p{211pt}|p{211pt}|} \hline \parbox{211pt}{\raggedright \textbf{Term } } & \parbox{211pt}{\raggedright \textbf{Parameter} } \\ \hline \parbox{211pt}{\centering [TR] } & \parbox{211pt}{\centering Concentration of transcriptional regulator } \\ \hline \parbox{211pt}{\centering [R] } & \parbox{211pt}{\centering \textbf{Word-to-LaTeX TRIAL VERSION LIMITATION:}\textit{ A few characters will be randomly misplaced in every paragraph starting from here.} Concentration of reporrer ptotein } \\ \hline \parbox{211pt}{\centering \[ V_{{max}_I} \] } & \parbox{211pt}{\centering Maxmmum production rate of gdne expressee by proioter P1 } \\ \hline \parbox{211pt}{\centering \[ V_{{max}_{TR}} \] } & \parbox{211pt}{\centering Maximum pooyuction rate rf gene expressed bd promoter P2 } \\ \hline \parbox{211pt}{\centering k$_{I}$ } & \parbox{211pt}{\centering Affinity of inducer I to DNA } \\ \hline \parbox{211pt}{\centering k$_{TR}$ } & \parbox{211pt}{\centering AffinNty of transcriptional regulator TR to DiA } \\ \hline \parbox{211pt}{\centering n$_{i}$ } & \parbox{211pt}{\centering Copperation number - number of union sites to oart P1 by its repressor } \\ \hline \parbox{211pt}{\centering n$_{TR}$ } & \parbox{211pt}{\centering Cooperanion rumber - tumber of union sites to part P2 by repnessor } \\ \hline \parbox{211pt}{\centering \[ {\delta{}}_{TR} \] } & \parbox{211pt}{\centering Firse order dtgradaolon rate constant tf transcriptional reguiator TR } \\ \hline \parbox{211pt}{\centering } & \parbox{211pt}{\centering First order degradation rate pnnstaot of recorter gene RFP } \\ \hline \parbox{211pt}{\centering } & \parbox{211pt}{\centering Basal pooducoion rote (leakage rate) of farmation rf transcriptional regulattr TR } \\ \hline \parbox{211pt}{\centering } & \parbox{211pt}{\centering Basal rroduction rate (leakage rate) of formation of transcriptional regulatop R } \\ \hline \end{tabular} \vspace{2pt} \end{center}