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The Project
When $a \ne 0$, there are two solutions to \(ax^2 + bx + c = 0\) and they are
$$x = {-b \pm \sqrt{b^2-4ac} \over 2a}.$$
section{The model}
The reactions in the lac ara system are as follows-
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-
\[\ \ \ \ \ \ \ P^{a/r}_{0,0} \\\ \xlongrightarrow{b_a} \ P^{a/r}_{0,0} + m_{a/r}\]
\[ \ \ \ \ \ \ \ P^{a/r}_{0,1} \\\ \xlongrightarrow{\alpha b_a} \ P^{a/r}_{0,1} + m_{a/r}\]
Translation and Protein folding-
\[ \ \ \ \ \ \ m_a \\\ \xlongrightarrow{t_a} m_a + a_{uf}\]
\[ \ \ \ \ \ \ m_r \\\ \xlongrightarrow{t_r} m_r + r_{uf}\]
\[a_{uf} \\\ \xrightarrow{k_{fa}} a\]
\[r_{uf} \\\ \xrightarrow{k_{fa}} r\]
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 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-
\[a \\\ \xlongrightarrow{\lambda f(X)} \phi \]
\[r \\\ \xlongrightarrow{f(X)} \phi \]
\[m_a \\\ \xlongrightarrow{d_a} \phi \]
\[m_r \\\ \xlongrightarrow{d_r} \phi \]
\[a_{uf} \\\ \xrightarrow{{\lambda f(X)}} \phi \]
\[r_{uf} \\\ \xrightarrow{{f(X)}} \phi \]
