Differential equations modeling the interactions between AppA and PpsR.!

The proteins in our regulatory system constitute a network of intermingling elements whose behavior can be narrow into seven reactions. Primarily in this part of the modeling we focused on the PpsR and AppA anti-repressor/repressor system and chose to model it using a system of ordinary differential equations.The system’s repressive activity varies with the redox (O2) state of the cell and Light intensity and thus we wondered how exactly these two variables affect the overall repressive level of PpsR over the expression of PS genes.

Model Construction.

To do this we based ourselves on a previous work that had attempted to study this same regulatory system. [Pandey et al, 2011]. Therefore the mathematical model we developed is based on the ODEs and kinetic parameters outlined there.

The following are its assumptions and basis: AppA inhibits the DNA-binding activity of oxidized PpsR by two mechanisms:

1. By reducing a disulfide bond in PpsR.
2. By a blue-light-dependent sequestration of PpsR proteins into transcriptionally inactive complexes.
At first stage, the reduced form of AppA (A-) reduces a disulfide bond in oxidized PpsR (P4+), which occurs independently of the light conditions.The molecular mechanism of this two-electron transfer is not yet clear.

Some experiments have shown that both PpsR and AppA have two redox-active Thiol groups that can form intramolecular disulfide bonds with a similar midpoint potential, according to this equation:

Then at the second level of regulation, the reduced form of AppA can form a complex with reduced PpsR. During the complex formation, one AppA molecule is associated with two PpsR monomers corresponding to half of a PpsR molecule, which exists as a stable tetramer in solution.

However there is a point were complex formation is inhibited by high intensities of blue- light irradiation (LI ¼ 900mmol/m2s). A subsequent study found that AppA responds to blue light over several orders of magnitude down to 0.2 mmol/m2s.

Other experiments indicate that light absorption induces a structural change in the BLUF domain of AppA, which results in interactions with its C-terminal part, thereby causing the dissociation of PpsR.

To implement the redox-sensing capabilities of AppA, we use the model proposed by Han et al, 2007 according to which AppA utilizes heme as a cofactor, bound to its C-terminal domain, to sense the cytosolic redox conditions, according to this equation:

If the electron transfer from AppA to PpsR in Eq. 1 was indeed effectively irreversible: (kpr ^- << kpr ⁺), as suggested by the experiments of Masuda and Bauer, 2002.PpsR would have to be reoxidized through an AppA-independent mechanism. To account for this possibility, the assumption is that PpsR is reoxidized proportional to the oxygen concentration as:

Finally we assumed mass-action kinetics for the reactions above and the following are the set of ordinary differential equations established:

Were the total amounts of PpsR and AppA molecules are conserved according to:

Later on we reduced this set of ODEs into seven basic reactions, to simplify the code written in Mathematica.

This is the code written in Mathematica.

In addition to contacting the responsible authors and discussing this model, we chose to improve it based on recent publications and preliminary experimental results i.e. we introduced a more subtle way light and oxygen affects the protein concentration dynamic, which is exactly what we were interested in understanding better.

Differential equations Results:

If we assumed that the reduction of PpsR by AppA in Equation 1.

Is effectively irreversible when Keq>>1

We obtain a fifth order polynomial equation that admits at most five real roots, corresponding to five possible stationary states. (Graphics above).

For them to be biologically meaningful we require that they fall within the interval [0,1]. Using our differential equations system we calculated protein concentrations at stable steady states and assumed this to be a natural “resting”, or homeostatic, state given a set of parameters, oxygen and light intensities. As such, this gave us a steady state total repressive force, in essence, as a function of oxygen tension and light intensity. This is because PpsR/AppA’s total repressive strength is a function of the concentrations of both its oxidative and reduced state.

Since this was all done in Mathematica, we were also able to generate an interactive graphical representation of the repressive strength over a range of conditions.

Download the code:Interactive Graphical AP

With this model, we can make predictions in sillico as to how the organism will react to a given environmental surrounding and then go back to the lab and validate that prediction. Also, since we have are able to manipulate the values of the parameters in an easy and intuitive way, we can explore the parameter space and visually observe the change in the concentration curves, which can be of interest to, not only us, but other people that might not be interested in all the math behind interphase.

This parameter tweaking allows us to compare experimental data against sillico predictions; the only difference is that if we focus on the parameters, this can actually give us hints respect to the still-blurred nature of Protein-Protein and Protein-Environment mechanisms that exist in this extremely versatile cell.