Team:UFMG Brazil/Modeling

$M$ is TMAO. $T$ is TorT. $TM$ is the complex of T and M. $S$ is TorS. $STM$ is the complex of S and TM. $R$ is TorR. $R^{ *}$ is TorR Phosophorilated. Reactions: $M+T{{k_{1} \atop\displaystyle\longrightarrow} \atop {\displaystyle\longleftarrow \atop k_{-1}}}TM$ $2TM+S{{k_{2} \atop\displaystyle\longrightarrow} \atop {\displaystyle\longleftarrow \atop k_{-2}}}STM$ $STM+R{{k_{3} \atop\displaystyle\longrightarrow} \atop {\displaystyle\longleftarrow \atop k_{-3}}} X {{k_{cat}\atop\longrightarrow} \atop {\quad \atop \quad }}R^{ *} + STM$ $R^{ *} + P {{\displaystyle\longrightarrow} \atop {\displaystyle\longleftarrow}} PR^{ *}$ \begin{eqnarray} \begin{cases} {d M(t)\over d t}&=& -k_1 \cdot M\cdot T + k_{-1}\cdot TM\\\\ \\\\ {d T(t)\over d t}&=& -k_1 \cdot M\cdot T + k_{-1}\cdot TM\\\\ \\\\ {d TM(t)\over d t}&=& k_1 \cdot M\cdot T - k_{-1}\cdot TM - k_2\cdot TM^{2} \cdot S + k_{-2} \cdot STM\\\\ \\\\ {d S(t)\over d t}&=& -k_2 \cdot S\cdot TM^{2} - k_{-1}\cdot TM - k_2\cdot TM^{2} \cdot S + k_{-2} \cdot STM\\\\ \\\\ {d STM(t)\over d t}&=& k_2 \cdot TM^{2} \cdot S - k_{-2}\cdot STM \\\\ \\\\ {d R^{ *}(t)\over d t}&=& k_{cat}\cdot STM \over{k_m + STM}\\\\ \\\\ {d mRNA(t)\over d t}&=& {\alpha \cdot R^{ *^{\gamma}} \over{\beta + R^{ *^{\gamma}}}} - \mu \cdot mRNA\\\\ \\ \end{cases} \end{eqnarray}