# Team/CINVESTAV-IPN-UNAM MX/equations.htm

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The proteins in our regulatory system constitute a network of intermingling elements whose behavior can be narrow into basic interaction rules. 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.

This is the code written for the regulatory system Appa-PpsR in Mathematica.

To do this we based ourselves on a previous work that had attempted to study this same regulatory system. [Pandey, Rakesh Flockerzi, Dietrich Hauser, Marcus JB Straube, Ronny. 2011. Modeling the light- and redox-dependent interaction of PpsR/AppA in Rhodobacter sphaeroides.] 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.

Results generated with the model:

With 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.

Because all this was done in Mathematica, we were able to generate an interactive graphical representation of the repressive strength over a range of conditions. This is really the strength of our work because 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 the power 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.