# Nick Rohacz: Week 6

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## 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).

## State Variables

- The state variables needed to model the process are nitrogen, glutamate, glutamine, a-ketoglutarate

## Differential Equations

- d[glutamate]/dt =-V
_{3}*[glutamate]/(k3+[glutamate])+V_{4}*(([a-keto]*[NH_{4}^{+}])/(k4+[a-keto]*[NH_{4}^{+}]))- V_{2}*(([glutamate]*[NH_{4}^{+}])/(k2+[glutamate]*[NH_{4}^{+}]))+V_{1}*[glutamine]/(k1+[glutamine])+V_{5}*(([a-keto]*[glutamine])/(k_{5}+[a-keto]*[glutamine])) - d[glutamine]/dt =-V
_{1}*[glutamine]/(k1+[glutamine])+V_{2}*(([glutamate]*[NH_{4}^{+}])/(k2+[glutamate]*[NH_{4}^{+}])) - d[a-ketoglutarate]/dt =-V
_{4}*[a-keto](/k4+[a-keto])+V_{3}*[glutamate]/(k3+[glutamate]) - d[NH
_{4}^{+}]/dt = Du+V_{1}*[glutamine]/(k1+[glutamine])+V_{3}*[glutamate]/(k3+[glutamate])

## Discussion

- Glutamate is synthesized from both α-ketoglutarate and glutamine, at the same time, it is broken down to produce both of these substrates. Some pathways, such as

α-ketoglutarate→glutamate and glutamate→glutamine require the consumption of an ammonium ion, while glutamate→α-ketoglutarate gives off an ammonium ion. Due to the presence of two amide groups on the glutamate and zero on α-ketoglutarate, they can be transformed into glutamate through the reduction of NAD^{+}.

- α-ketoglutarate is synthesized through the oxidation of an NADH and gives of an ammonium ion, while at the same time some of the concentration is transformed into glutamate through the reduction of NADPH. Possibility that some concentration is returned to mitochondria to be used in TCA cycle.
- Glutamine is synthesized from glutamate through the addition of an ammonium ion and dephosphorylation of an ATP, at the same time, some of the glutamine is transformed back into glutamate.
- Ammonium is present in the feed therefore it is represented by the term D*u, as well as more ammonium being produced through other reactions, such as the reduction of glutamate, which produces α-ketoglutarate and an ammonium ion. This ammonium is being used as a substrate to transform α-ketoglutarate into glutamate, and glutamate into glutamine.

## Parameters

- Parameters are D, which is dilution rate which varied, V
_{max}which is equal to k*e_{0}, e_{0}being the initial enzyme concentration, however this is split up into 5 different V's due to 5 different enzymes being responsible for each transformation, u which is the feed of nitrogen and carbon, these are constant. - k
_{1}represents GDH, which is responsible for the creation of glutamate from glutamine. - k
_{2}represents GS, which is responsible for the creation of glutamine from glutamate. - k
_{3}represents NADH, which is responsible for the creation of α-keto from glutamate. - k
_{4}represents NAD^{+}, which is responsible for the creation of glutamate from a-keto. - k
_{5}represents GOGAT, which is responsible for the creation of glutamate from a-keto and glutamine

## Relationship

- The data in the second paper focus on different parameters than in the first paper. The dilution rate is one term that was constant in the first paper but changing in the second, while the concentration of nitrogen in the feed was changing in the first and constant in the second. With the nitrogen feed constant in the second paper, the biomass did not increase as much, as in the first paper when more nitrogen resulted in more biomass up to a degree.
- After enough nitrogen was put into the feed, residual concentration was detected, however residual concentration was absent in the second paper.

Nicholas A. Rohacz 02:54, 22 February 2011 (EST)