\frac{dx_{1}}{dt} = \left(1 \cdot k_{12} \cdot k_{1} \cdot k_{5} \cdot x_{1} \cdot \left(1 - x_{1} / k_{4}\right) + -1 \cdot k_{12} \cdot k_{2} \cdot \left(1 - k_{5}\right) \cdot x_{1}\right) / k_{12}\\
\frac{dx_{2}}{dt} = \left(1 \cdot k_{12} \cdot k_{2} \cdot \left(1 - k_{5}\right) \cdot x_{1} + -1 \cdot k_{12} \cdot k_{3} \cdot x_{2}\right) / k_{12}\\
\frac{dx_{3}}{dt} = \left(1 \cdot k_{12} \cdot \left(k_{6} \cdot x_{1} + k_{7} \cdot x_{2}\right) + -1 \cdot k_{12} \cdot k_{8} \cdot x_{3}\right) / k_{12}\\
\frac{dx_{4}}{dt} = \left(1 \cdot k_{12} \cdot k_{9} \cdot x_{4} \cdot \left(1 - x_{4} / k_{10}\right) + -1 \cdot k_{12} \cdot k_{11} \cdot x_{3} \cdot x_{4}\right) / k_{12}