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The Model
Mathematical modelling often gives useful insights into the behaviour of complex systems. The purpose of this project was to model the synthetic gene oscillator circuit made by \cite{Stricker}, where the protein product of the first gene (AraC) activates the expression of both the genes and the protein product of the second gene (LacI) inhibits expression of both the genes. It is expected that such a model would help test what modifications can be done in the mentioned oscillator, to couple it with other genes to achieve oscillations in their protein products, which are essential in the cell cycle.
section{The model}
The reactions in the lac ara system are as follows-
\[P^{a/r}_{0,j} \ + \ a_2 \xrightleftharpoons[k_a]{k_a} \ P^{a/r}_{1,j} \]
\[P^{a/r}_{i,0} \ + \ r_4 \xrightleftharpoons[k_r]{2k_r} \ P^{a/r}_{i,1} \]
\[P^{a/r}_{i,1} \ + \ r_4 \xrightleftharpoons[2k_r]{k_r} \ P^{a/r}_{i,2} \]
where $P^{a/r}_{i,j}$ represent the states of promoters on the (a)ctivator/(r)epressor plasmids with $i \ \ \epsilon \ \ (0, 1)$ AraC dimers $(a_2)$ bound and $ j \ \ \epsilon \ (0, 1, 2) $ LacI tetramers $(r_4)$ bound.
Transcription-
\[\ \ \ \ \ \ P^{a/r}_{0,0} \xrightarrow{b_a} \ P^{a/r}_{0,0} + m_{a/r}\]
\[ \ \ \ \ \ \ P^{a/r}_{0,1} \xrightarrow{\alpha b_a} \ P^{a/r}_{0,1} + m_{a/r}\]
Translation and Protein folding-
\[\ \ \ m_a \xrightarrow{t_a} m_a + a_{uf}\]
\[ \ \ \ m_r \xrightarrow{t_r} m_r + r_{uf}\]
\[a_{uf} \xrightarrow{k_{fa}} a\]
\[r_{uf} \xrightarrow{k_{fa}} r\]
where $m_{a/r}$ represents the number of activator/repressor transcripts; $a_{uf}$ and $r_{uf}$ are the unfolded
monomeric versions of the activator and repressor; a and r are the folded monomeric versions of
activator and repressor; $a_2$ and $r_2$ are the dimeric versions of activator and repressor; and $r_4$ is
the tetrameric version of the repressor.
Dimerisation and Tetramerisation-
\[a \ + \ a \xrightleftharpoons[k_{da}]{k_{da}} \ a_2 \]
\[r \ + \ r \xrightleftharpoons[k_{dr}]{k_{dr}} \ r_2 \]
\[r_2 \ + \ r_2 \xrightleftharpoons[k_{t}]{k_{t}} \ r_4 \]
Degradation-
\[a \xrightarrow{\lambda f(X)} \phi \]
\[r \xrightarrow{f(X)} \phi \]
\[m_a \xrightarrow{d_a} \phi \]
\[m_r \xrightarrow{d_r} \phi \]
\[a_{uf} \xrightarrow{{\lambda f(X)}} \phi \]
\[r_{uf} \xrightarrow{{f(X)}} \phi \]
\begin{align}
\frac{dm_a}{dt} &= \frac{n_a b_a (1+\alpha c_a a^2)}{(1 + c_a a^2)(1+ 2c_r r^4 + {c_r}^2 r^8)} - \gamma_{m_a} m_a \\
\frac{dm_r}{dt} &= \frac{n_r b_r(1+\alpha c_a a^2)}{(1 + c_a a^2)(1+2c_rr^4 + c_r^2 r^8)} - \gamma_{m_r} m_r \\
\frac{da_{uf}}{dt} &= t_a m_a - k_{fa} a_{uf} - \gamma_{a_{uf}} a_{uf} \\
\frac{dr_{uf}}{dt} &= t_r m_r - k_{fr} r_{uf} - \gamma_{r_{uf}} r_{uf} \\
\frac{da}{dt} &= k_{fa} a_{uf} - k_a a \\
\frac{dr}{dt} &= k_{fr} r_{uf} - k_r r \\
\end{align}
Where $a=$ Concentration of AraC protein, $r=$ Concentration of LacI protein.
$\tau_a$,$\tau_r$ are parameters dependent on various constants as follows-
%\[\displaystyle{\tau_y = \frac{m_1}{m_2} \sqrt[4]{\frac{k_1^2 k_{da}^2}{k_2 k_{dr_1} k_{dr_2}}}}\]
\[c_a = {k_1 k_{da}}\]
\[c_r = {k_2 {k_{dr_1}^2} k_{dr_2}}\]
$k_1,k_2$ are dependent on IPTG and Arabinose concentration, given by-
\[k_1 = \frac{[ara]^2}{[ara]^2 + (2.5^2)} .\frac{1}{1 + (\frac{[iptg]}{1.8})^2}\]
\[k_2 = 2*(0.19 . \frac{1}{1 + (\frac{[iptg]}{0.035})^2} + 0.01)\]
$$
\begin{array}{|c|c|c|c|c|}
\hline
\text{Sr.} & \text{Parameter} & \text{Description} & \text{value} & \text{units} \\ \hline
1 & \gamma_{ma}, \gamma_{mr} & \text{Degradation rate of AraC/LacI mRNA} & 0.54 & min^{-1} \\ \hline
2 & b_a,b_r & \text{Transcription rate of AraC, LacI genes} & 0.36 & min^{-1} \\ \hline
3 & \alpha & \text{Increased transcription rate Due to AraC binding} & 20 & \\ \hline
4 & t_{a} & \text{rate of transcription of AraC} & 90 & min^{-1} \\ \hline
5 & t_{r} & \text{rate of transcription of LacI} & 90 & min^{-1} \\ \hline
6 & k_{a} & \text{Rate of degradation of AraC protein} & 0.455 & min^{-1} \\ \hline
7 & k_{r} & \text{Rate of degradation of LacI protein} & 0.182 & min^{-1} \\ \hline
8 & k_{da} & \text{AraC dimerisation constant} & 100 & molecules^{-1} \\ \hline
9 & k_{dr1} & \text{LacI dimerisation constant} & 100 & molecules^{-1} \\ \hline
10 & k_{dr2} & \text{LacI tetramerisation constant} & 100 & molecules^{-1} \\ \hline
\end{array}
$$
