# Alondra Vega: Week 6

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## Contents |

## Instructions

- List the state variables needed to model the process of interest.
- Propose at least one system of differential equations you think will model the dynamics.
- Discuss the terms in your equation(s) in order to justify your choices.
- List all parameters your model requires for numerical simulation.
- Discuss the relationship between the data in the papers by ter Schure
*et al*and the state variables (and parameters).

## Model Trial 1

- There are a couple of state variables of interest in our model. State variables are the "things" for a lack of a better term that change over time. For our model we will be interested in modeling what happens with nitrogen, glutamine, glutamate, and α-ketogluterate. This means that we essentially need at least four differential equations that will model these amino acids and nitrogen over time.
- The Set of differential equations. The cell is our system in this set of differential equations.
- d[glutamine]/dt = - V
_{max}([glutamine]/k_{1}+[glutamine])+ V_{max}([glutamate]/k_{2}+[glutamate]).- The conentration of glutamine correlates to all the glutamine that is ound in the cell. This means that whatever amount is lost to other processes in the cell will be zero, since the entire concentration is in the cell. The same will follow for glutamate and α-ketogluterate. This differential equation tries to describe what is happening with glutamate over time.V
_{max}shows the enzyme concentrations, thus it is a constant. Glutamine is "transformed" into glutamate, thus we lose some of glutamate. This is why we subtract the concentration of glutamine and we try to take into account the enzyme production that goes into play by dividing it (k_{1}). There is back procedure where glutamate will go back and help make glutamine again. We add that concentration and a different V_{max}along with a different k, since it represents different enzymes.

- The conentration of glutamine correlates to all the glutamine that is ound in the cell. This means that whatever amount is lost to other processes in the cell will be zero, since the entire concentration is in the cell. The same will follow for glutamate and α-ketogluterate. This differential equation tries to describe what is happening with glutamate over time.V
- d[glutamate]/dt = -V
_{max}([glutamate]/k_{3}+[glutamate]) + V_{max}([α-ketogluterate][NH_{a}^{+}]/k_{4}+[α-ketogluterate][NH_{a}^{+}])- V_{max}([glutamate][NH_{a}^{+}]/k_{2}+[glutamate][NH_{a}^{+}]/)+ V_{max}([glutamine]/k_{1}+[glutamine])+V_{max}([α-ketogluterate][glutamine]/(k_{5}+[α-ketogluterate][glutamine]).- This differential equation tries to show how glutamate is changing over time. It shows the gain and loss from the amino acid α-ketogluterate, with different V
_{max}values since different enzymes are being used in each reaction. It also shows the effect that glutamate has on the system when only looking at glutamine.

- This differential equation tries to show how glutamate is changing over time. It shows the gain and loss from the amino acid α-ketogluterate, with different V
- d[α-ketogluterate]/dt = -V
_{max}([α-ketogluterate]/k_{4}+[α-ketogluterate]) + V_{max}([gluterate]/k_{3}+[gluterate]).- This differential equation tries to model the change over time of the amino acid α-ketogluterate. Again the V
_{max}shown in the formula are different, since it takes different enzymes to carry out the different processes.

- This differential equation tries to model the change over time of the amino acid α-ketogluterate. Again the V
- d[nitrogen]/dt = Du + V
^{a}_{1}[glutamine]/(k_{1}+[glutamine]+V_{4}^{a}[glutamate/(k_{4}^{a}+[glutamate].- For the nitrogen model, I feel that the only item that is affecting the nitrogen production is the gain of ammonium into the cell. The dilution rate and the feed concentration should also be taken into account.

- d[glutamine]/dt = - V
- The parameters are described as follows. D is the dilution rate and it is fixed. The glucose and ammonium concentrations are both included in u since it is the feed concentration. This value will change according to how much ammonium is added, since that is the treatment variable. The glucose concentration is fixed. The enzymes are the ones that carry the reactions out. for the purposes of this model we have GDA, GS, NAD-GDH, and NADPH-GDH. These enzymes are incorporated in the V
_{max}and in the k variables. That is what makes those values unique. We learned in class that V_{max}= ke_{0}(k times the initial amount of enzyme). - I feel that my state variables do correspond to the
*ter Schure et al.*paper to an extent. In my differential equations I do not mention the genes that are involved in these processes and I feel that they play a key role. I am just not sure how to include them in there. I also, do not mention carbon, just ammonium. I feel that this is appropriate since that is the item that we are trying to make conclusions and make a final conclusion on. Other than those things, I feel that I have included everything that is on the paper.