Car activation

Problem

Most of the reactions in our pathway consider just the propane-intermediate with other cofactors and the enzyme in their reactions. However, there is one reaction that does not fit in this mold: the activation of CAR.

As mentioned before, Car is the enzyme converting butyric acid to butyraldehyde. However, before this enzyme can function in our pathway, we need to activate it using a different enzyme, Sfp. This reaction differs from most in the way the substrate is produced and degraded: Car is produced from DNA with transcription and translation and degraded by protein-degrading enzymes, as opposed to the substrate being created and degraded from enzyme activity only. This gives rise to some questions: What aspects of the cell change the amount of active Car, as well as what is the amount of active Car.

Model

The reaction transforming Car from its inactive (called Car_apo) form into its active (called Car_holo) form is as follows:

--Formulas of the reactions here--

This reaction abides by the irreversible bi-bi ping pong reaction, and is therefore quite nice to model with Michaelis-Menten kinetics. In addition to this equation, we consider two more phenomena that affect the amount of active Car we have: The creation of Car in its apo form, and the degradation of enzymes. These reactions can be written as

--Some more formulas--

For this model, we assume that the amount of enzymes in a cell is constant, i.e. that the speed of enzyme’s creation is equal to its speed of degradation. We also assume that this reaction has no big effect on either the amounts of CoA and 3’,5’-ADP. We assume that the amount of Car, that is the cumulative amount of both its apo- and holo-forms is constant. Since these substrates or products do not affect the rest of the simulation, we have good grounds to assume that we can separate it from the main simulation.

From these assumptions we can build an ODE system:

--The complete ODEs here--

$$ \begin{array}{ccl} \frac {\mathrm{d} {{\mathrm{[Car\_apo]}} } } {\mathrm{d}{t} } \; &=& \; { - \frac { {{\mathrm{Kcat}}_{\mathrm{Car-activation}} \, \cdot \, {\mathrm{[Sfp]}} } \, \cdot \, {{\mathrm{[Car\_apo]}} \, \cdot \, {\mathrm{[CoA]}} } } { { {{\mathrm{[Car\_apo]}} \, \cdot \, {\mathrm{[CoA]}} } \, + \, {{\mathrm{Kmb}}_{\mathrm{Car-activation}} \, \cdot \, {\mathrm{[Car\_apo]}} } } \, + \, {{\mathrm{Kma}}_{\mathrm{Car-activation}} \cdot {\mathrm{[CoA]}} } } } \\ && \\ \; && \; { \, + \,{\mathrm{Car\_creationrate}} } \\ && \\ \; && \; { \, - {{\mathrm{protein\_degradationrate}} \, \cdot \, {\mathrm{[Car\_apo]}} } } \\ && \\ \frac {\mathrm{d} {{\mathrm{[Car\_holo]}} } } {\mathrm{d}{t} } \; &=& \; { \, + \frac { {{\mathrm{Kcat}}_{\mathrm{("Car-activation")}} \, \cdot \, {\mathrm{[Sfp]}} } \, \cdot \, {{\mathrm{[Car\_apo]}} \, \cdot \, {\mathrm{[CoA]}} } } { { {{\mathrm{[Car\_apo]}} \, \cdot \, {\mathrm{[CoA]}} } \, + \, {{\mathrm{Kmb}}_{\mathrm{Car-activation}} \, \cdot \, {\mathrm{[Car\_apo]}} } } \, + \, {{\mathrm{Kma}}_{\mathrm{Car-activation}} \, \cdot \, {\mathrm{[CoA]}} } } } \\ && \\ \; && \; { \, - {{\mathrm{protein\_degradationrate}} \, \cdot \, {\mathrm{[Car\_holo]}} } } \\ && \\ {\mathrm{protein\_degradationrate}} \; &=& \; { { {\mathrm{Car\_creationrate}} } } \\ && \\ \end{array} $$

As you can see, we modeled the creation of Car_apo as a constant flux reaction, and the degradation of different proteins as a reaction abiding the laws of mass action. This is because we assume we aren’t affecting the DNA transcription and translation in our model, and since protein degradation is an enzymatic reaction that is hard to model we simplify it as a mass-action reaction.

Results

We tested our model with two scenarios: One where the Car creation and degradation is disabled, and one where those reactions are active. Since we do not know the rates at which protein creation and degradation happen, we tested our model with values between 0 and 5 µM/s.

Here is a time course of the model with no protein degradation or creation:

--pic of concentration results--

As you can see, since the reaction is irreversible, all of the Car in the simulation is swiftly transformed into its active form.

With the protein creation and degradation, we tested different rates for the protein creation and degradation. Here is a time course with rates varied from 0 to 5 µM/s:

--pic of concentration results, this time with degradation--

As you can see, the amount of active Car in the system varies with the enzyme degradation and creation rates. Within this variation, the values that the relation Car_apo/Car got are between 90% and 100%.

There are some weak points to this model: Since we could not get data about DNA transcription and translation speed, this model is mainly used to find lower limits for the amount of active Car we have in our cells. Also, it might be wrong to assume that a cell reaches equilibrium in its enzymes’ amounts. However, this model shows that unless enzymes are created and degraded with awe-inspiring speeds, our system will have most of its Car enzyme in the active form.