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