# Sarah Carratt: 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).

## Online Sources

## Student Response

### Variables Needed for a Model

- ammonium → nitrogen
- α-ketogluterate
- glutamate
- glutamine

These four variables are the things that we will need to watch/model as they change over time. In the image, these variables can be seen in context of nitrogen metabolism.

### Differential Equations and Discussion of Terms

[] = concentration of enclosed

D = dilution rate

u = feed concentration

k_{1}, k_{2}, k_{3}, k_{4} = rate constants

V_{max} = enzyme concentrations (constant)

L_{1}, L_{2}, L_{3}, L_{4} = loss of state variable to outside factors/processes in cell and also because of the backwards conversions/cycle

- d
_{[glutamine]}/dt = D*u - V_{max}([glutamine]/k_{1}[glutamine])+ V_{max}([glutamate]/k_{2}[glutamate])- L_{1} - d
_{[glutamate]}/dt = D*u -V_{max}([α-ketogluterate]/k_{3}[α-ketogluterate]) + V_{max}([α-ketogluterate]/k_{4}[α-ketogluterate])- V_{max}([glutamate]/k_{2}[glutamate])+ V_{max}([glutamine]/k_{1}[glutamine])- L_{2} - d
_{[α-ketogluterate]}/dt = D*u-V_{max}([α-ketogluterate]/k_{4}[α-ketogluterate]) + V_{max}([gluterate]/k_{3}[gluterate]) - L_{3} - d
_{[nitrogen]}/dt = D*u + [ammonium] - L_{4}

### Parameters for Model

- V
_{max}(k*[enzymes]_{0}: GDA, GS, NAD-GDH, NADPH-GDH) - D (dilution rate) CONSTANT
- u (includes glucose/ammonium aka carbon/nitrogen)
- ammonium changes
- glucose is constant

### Relationship between ter Schure *et al* and Parameters

All variables are connected to ter Schure. Originally, I was confused with how to include carbon/glucose, but I believe that it is accounted for in the feed concentration and dilution. I shouldn't need a fifth equation for glucose. The major difference between my parameters and ter Schure is that I have not focused on individual enzymes. I tried to factor them into my equation but I'm not sure they can be accounted for in the same ways.

## Correct Answers

STATE VARIABLES:

- α-ketogluterate
- Glutamate
- Glutamine
- Ammonium → Nitrogen

WHAT IS THE SYSTEM?

**Cell**- Chemostat Reactor

UNITS:

- moles/volume
- moles/(volume*time)

EQUATIONS:

- d[α-ketogluterate]/dt = -V
_{4}([α-ketogluterate]/k_{4}+[α-ketogluterate]) + V_{3}([glutamate]/k_{3}+[glutamate]) - d[glutamine]/dt = -V
_{1}([glutamine]/k_{1}+[glutamine]) + V_{2}([glutamate]/k_{2}+[glutamate]) - d[glutamate]/dt = V
_{1}([glutamine]/k_{1}+[glutamine])- V_{2}([glutamate][ammonium]/k_{2}+[glutamate][ammonium]) + V_{3}([α-ketogluterate][ammonium]/k_{3}+[α-ketogluterate][ammonium]) - V_{4}([glutamate]/k_{4}+[glutamate]) + V_{5}([α-ketogluterate][glutamine]/k_{5}+[α-ketogluterate][glutamine]) - d[ammonium]/dt = D*u + V
^{a}_{1}([glutamine]/k^{a}_{1}+[glutamine])+ V^{a}_{4}([glutamate]/k^{a}_{4}+[glutamate])

EQUATIONS WITH SIMPLE VARIABLES:

- d[A]/dt = -V
_{4}([A]/k_{4}+[A]) + V_{3}([B]/k_{3}+[B]) - d[B]/dt = V
_{1}([C]/k_{1}+[C])- V_{2}([B][D]/k_{2}+[B][D]) + V_{3}([A][D]/k_{3}+[A][D]) - V_{4}([B]/k_{4}+[B]) + V_{5}([A][C]/k_{5}+[A][C]) - d[C]/dt = -V
_{1}([C]/k_{1}+[C]) + V_{2}([B]/k_{2}+[B]) - d[D]/dt = D*u + V
^{a}_{1}([C]/k^{a}_{1}+[C])+ V^{a}_{4}([B]/k^{a}_{4}+[B])

A=first substrate (α-ketogluterate), B=second substrate (glutamate), C=third substrate (glutamine), D=fourth substrate (ammonium)

NOTES:

- D*u = source, inflow (dilution rate*feed concentration)
- V
_{#}= enzyme level, accounts for loss, "hides amount of enzyme" (k*e_{0}: GDA, GS, NAD-GDH, NADPH-GDH) - the "L" constant is troubling in terms of units
- strategy: fit to orignial equations, E+S↔ES→E+P and E+P↔EP→E+S
- α-ketogluterate has no nitrogen, glutamate has one, glutamine has two
- food for thought: conserved? 2 substrate model="right"? what if you set d/dt=0 to look at equilibrium? use steady state to find constants?

## Navigation Guide

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