Modelling
Introduction
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.Earlier attempts of reproducing the whole model using the Simbiology Toolbox in MATLAB R2015b were unsuccessful in reproducing the expected results. Possibly because of the complexity of the model and a huge number of reactions and their parameters. 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 Python 3.6 in \href{}{} Spyder 2.3.2 environment, a range of parameter values was scanned to obtain oscillations. On an average, the numerical integration took about 0.15 seconds to run an additional 0.30 seconds to output a graph. 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. Four more differential equations, that of mRNA and unfolded proteins, were added, to introduce delay that is important for oscillations in proteins according to \cite{Stricker}.
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 $r_4$ represents the tetrameric version of the repressor. Dimerisation and Tetramerisation- \[a \ + \ a \xrightleftharpoons[k_{da}]{k_{da}} \ a_2 \] \[r \ + \ r \xrightleftharpoons[k_{dr}]{k_{dr}} \ r_2 \] \[r_2 \ + \ r_2 \xrightleftharpoons[k_{t}]{k_{t}} \ r_4 \] We assume that the dimerisation and promoter binding reactions are fast reactions and attain equilibrium almost instantaneously. 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- \[k_1 = \frac{[ara]^2}{[ara]^2 + (2.5^2)} .\frac{1}{1 + (\frac{[iptg]}{1.8})^2}\] \[k_2 = 2*(0.19 . \frac{1}{1 + (\frac{[iptg]}{0.035})^2} + 0.01)\] $$ \begin{array}{|c|c|c|c|c|} \hline \text{Sr.} & \text{Parameter} & \text{Description} & \text{value} & \text{units} \\ \hline 1 & n_a & \text{copy number of Plasmid having AraC} & 50 & molecules \\\hline 2 & n_r & \text{copy number of Plasmid having LacI} & 25 & molecules \\ \hline 3 & \gamma_{ma}, \gamma_{mr} & \text{Degradation rate of AraC/LacI mRNA} & 0.54 & min^{-1} \\ \hline 4 & b_a,b_r & \text{Transcription rate of AraC, LacI genes} & 0.36 & min^{-1} \\ \hline 5 & \alpha & \text{Increased transcription rate Due to AraC binding} & 20 & \\ \hline 6 & t_{a} & \text{rate of transcription of AraC} & 90 & min^{-1} \\ \hline 7 & t_{r} & \text{rate of transcription of LacI} & 90 & min^{-1} \\ \hline 8 & k_{a} & \text{Rate of degradation of AraC protein} & 0.455 & min^{-1} \\ \hline 9 & k_{r} & \text{Rate of degradation of LacI protein} & 0.182 & min^{-1} \\ \hline 10 & k_{da} & \text{AraC dimerisation constant} & 100 & molecules^{-1} \\ \hline 11 & k_{dr1} & \text{LacI dimerisation constant} & 100 & molecules^{-1} \\ \hline 12 & k_{dr2} & \text{LacI tetramerisation constant} & 100 & molecules^{-1} \\ \hline 13 & k_{fa}, k_{fr} & \text{Rate of folding of proteins} & 0.9 & min^{-1} \\ \hline \end{array} $$ Results
