# Sarah Carratt: 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).

## Student Response

### Variables Needed for a Model

Variables in Context
1. ammonium → nitrogen
2. α-ketogluterate
3. glutamate
4. 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

k1, k2, k3, k4 = rate constants

Vmax = enzyme concentrations (constant)

L1, L2, L3, L4 = loss of state variable to outside factors/processes in cell and also because of the backwards conversions/cycle

1. d[glutamine]/dt = D*u - Vmax([glutamine]/k1[glutamine])+ Vmax([glutamate]/k2[glutamate])- L1
2. d[glutamate]/dt = D*u -Vmax([α-ketogluterate]/k3[α-ketogluterate]) + Vmax([α-ketogluterate]/k4[α-ketogluterate])- Vmax([glutamate]/k2[glutamate])+ Vmax([glutamine]/k1[glutamine])- L2
3. d[α-ketogluterate]/dt = D*u-Vmax([α-ketogluterate]/k4[α-ketogluterate]) + Vmax([gluterate]/k3[gluterate]) - L3
4. d[nitrogen]/dt = D*u + [ammonium] - L4

### Parameters for Model

2. D (dilution rate) CONSTANT
3. u (includes glucose/ammonium aka carbon/nitrogen)
1. ammonium changes
2. 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.

STATE VARIABLES:

1. α-ketogluterate
2. Glutamate
3. Glutamine
4. Ammonium → Nitrogen

WHAT IS THE SYSTEM?

1. Cell
2. Chemostat Reactor

UNITS:

1. moles/volume
2. moles/(volume*time)

EQUATIONS:

1. d[α-ketogluterate]/dt = -V4([α-ketogluterate]/k4+[α-ketogluterate]) + V3([glutamate]/k3+[glutamate])
2. d[glutamine]/dt = -V1([glutamine]/k1+[glutamine]) + V2([glutamate]/k2+[glutamate])
3. d[glutamate]/dt = V1([glutamine]/k1+[glutamine])- V2([glutamate][ammonium]/k2+[glutamate][ammonium]) + V3([α-ketogluterate][ammonium]/k3+[α-ketogluterate][ammonium]) - V4([glutamate]/k4+[glutamate]) + V5([α-ketogluterate][glutamine]/k5+[α-ketogluterate][glutamine])
4. d[ammonium]/dt = D*u + Va1([glutamine]/ka1+[glutamine])+ Va4([glutamate]/ka4+[glutamate])

EQUATIONS WITH SIMPLE VARIABLES:

1. d[A]/dt = -V4([A]/k4+[A]) + V3([B]/k3+[B])
2. d[B]/dt = V1([C]/k1+[C])- V2([B][D]/k2+[B][D]) + V3([A][D]/k3+[A][D]) - V4([B]/k4+[B]) + V5([A][C]/k5+[A][C])
3. d[C]/dt = -V1([C]/k1+[C]) + V2([B]/k2+[B])
4. d[D]/dt = D*u + Va1([C]/ka1+[C])+ Va4([B]/ka4+[B])

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

NOTES:

1. D*u = source, inflow (dilution rate*feed concentration)
2. V# = enzyme level, accounts for loss, "hides amount of enzyme" (k*e0: GDA, GS, NAD-GDH, NADPH-GDH)
3. the "L" constant is troubling in terms of units
4. strategy: fit to orignial equations, E+S↔ES→E+P and E+P↔EP→E+S
5. α-ketogluterate has no nitrogen, glutamate has one, glutamine has two
6. 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?