Modeling

We use the following assumptions to make the modeling easier:

• We assume a closed system, which is homogeneous, with no other transport processes to not having any gradient in the system, what leads us to usage of an ODE system and no PDEs.
• As the initial value problem we choose the values measured at the beginning of the experiment.
• The system is well posed, as the ODE-system is Lipschitz continuous, which guarantees existence and uniqueness of solutions.
• We assume a Quasi-Steady-State (QSS) to reduce parameters.

The goodness of fit is tested using statistical parameters such as AIC, BIC and log-Likelihood. We investigate the distribution of residuals to test whether an effect is missed.

A simple ODE for oxalate synthesis could look like this:

$$ \text{With OX} \equiv \text{Oxalate}, \text{OA} \equiv \text{Oxaloacetate}, \alpha \,\, \text{the oxalate synthesis rate}, $$

$$ \emptyset \,\, \text{the oxalate degradation rate.} $$

We included an oscillating function to compensate for translational and stochastic variability and possible leakiness of oahA, as we use little data. We implement data for TCA metabolites from literature for model fitting [Shirai et al, 2007], [Graf et al, 2020], [Reimer et al, 2014]. With this we end up with a non-autonomous ODE, as the rightern side is time dependent. By plotting this ODE (Fig. 1) we see the fit isn't bad, however it can still be improved. Looking into the residuals (Fig. 2), one can say that they are approximately normal.