Modelling
Introduction
Mathematical modelling often gives useful insights into the behaviour of complex systems. Here, we model the synthetic gene oscillator circuit made by Stricker et al.(2008), 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.The oscillator described by Stricker et al. (2008) comprises of two genes AraC and LacI, placed downstream of a $p_{lac/ara-1} $hybrid promoter, which is activated by the binding of AraC protein in the presence of arabinose and repressed by LacI protein in the absence of IPTG. This forms a dual feedback loop that causes the levels of these proteins to oscillate periodically.
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 number of 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 $r_4$ represents 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- \[a \xrightarrow{\lambda f(X)} \phi \] \[r \xrightarrow{f(X)} \phi \] \[m_a \xrightarrow{d_a} \phi \] \[m_r \xrightarrow{d_r} \phi \] \[a_{uf} \xrightarrow{{\lambda f(X)}} \phi \] \[r_{uf} \xrightarrow{{f(X)}} \phi \] These are all the reactions that we have considered. We ignored protein looping, which is considered by Stricer et al. (2008) in their model. When we solved the differential equations corresponding to each reaction using Simbiology in MATLAB, however, we were unable to reproduce the results reported. As a solution, a bottom-up approach of modelling was chosen, in which a simple model is built first and equations, parameters are added as required. The simple model, in this case, was chosen to be the cI-Lac oscillator described by \cite{Hasty}.The two first order coupled differential equations in the model were solved using < a href= "https://www.python.org/"> Python 3.6 in Spyder 2.3.2environment, a range of parameter values was scanned to obtain oscillations. The main difference between the two oscillators is that the promoter in cI-Lac oscillator has two binding sites for cI protein and one for LacI tetramer, whereas the promoter in Ara-Lac oscillator has one binding site for AraC protein and two sites for LacI binding. The model was then modified to fit the ara-lac system and IPTG, and Arabinose inducibility was incorporated into the model. This model did not include the intermediate reactions of transcription and translation and also assumed that the reactions are dimerisation and tetramerisation are fast, and equilibrium is attained instantaneously. These assumptions were relaxed by adding four more differential equations, that of mRNA production and protein unfolding for both AraC and LacI; and also by adding a delay in the differential equation of mRNA. \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- \[c_a = {k_1 k_{da}}\] \[c_r = {k_2 {k_{dr_1}^2} k_{dr_2}}\] $k_1,k_2$ are dependent on IPTG and Arabinose concentration, given by- \[k_1 = \frac{[ara]^2}{[ara]^2 + (2.5^2)} .\frac{1}{1 + (\frac{[iptg]}{1.8})^2}\] \[k_2 = 2*(0.19 . \frac{1}{1 + (\frac{[iptg]}{0.035})^2} + 0.01)\] Results
$$
\begin{array}{|c|c|c|c|c|}
\hline
\text{Sr.} & \text{Parameter} & \text{Description} & \text{value} & \text{units} \\ \hline
1 & n_a & \text{copy number of Plasmid having AraC} & 50 & molecules \\\hline
2 & n_r & \text{copy number of Plasmid having LacI} & 25 & molecules \\
3 & \gamma_{ma}, \gamma_{mr} & \text{Degradation rate of AraC/LacI mRNA} & 0.54 & min^{-1} \\ \hline
4 & b_a,b_r & \text{Transcription rate of AraC, LacI genes} & 0.36 & min^{-1} \\ \hline
5 & \alpha & \text{Increased transcription rate Due to AraC binding} & 20 & \\ \hline
6 & t_{a} & \text{rate of transcription of AraC} & 90 & min^{-1} \\ \hline
7 & t_{r} & \text{rate of transcription of LacI} & 90 & min^{-1} \\ \hline
8 & k_{a} & \text{Rate of degradation of AraC protein} & 0.455 & min^{-1} \\ \hline
9 & k_{r} & \text{Rate of degradation of LacI protein} & 0.182 & min^{-1} \\ \hline
10 & k_{da} & \text{AraC dimerisation constant} & 100 & molecules^{-1} \\ \hline
11 & k_{dr1} & \text{LacI dimerisation constant} & 100 & molecules^{-1} \\ \hline
12 & k_{dr2} & \text{LacI tetramerisation constant} & 100 & molecules^{-1} \\ \hline
13 & k_{fa}, k_{fr} & \text{Rate of folding of proteins} & 0.13 & min^{-1} \\ \hline
\end{array}
$$
