Sarah Carratt: Week 6

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

• Microbiology Paper
• Bacteriology Paper

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

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

Parameters for Model

1. V_max (k*[enzymes]₀: GDA, GS, NAD-GDH, NADPH-GDH)
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.

Correct Answers

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([gluterate]/k₃+[gluterate])
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 + V^a₁([glutamine]/k^a₁+[glutamine])+ V^a₄([glutamate]/k^a₄+[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 + V^a₁([C]/k^a₁+[C])+ V^a₄([B]/k^a₄+[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?

Navigation Guide

Individual Assignments

Sarah Carratt: Week 2	Sarah Carratt: Week 6	Sarah Carratt: Week 11
Sarah Carratt: Week 3	Sarah Carratt: Week 7	Sarah Carratt: Week 12
Sarah Carratt: Week 4	Sarah Carratt: Week 8	Sarah Carratt: Week 13
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Class Assignments

Shared Journal: Week 1	Shared Journal: Week 6	Shared Journal: Week 11
Shared Journal: Week 2	Shared Journal: Week 7	Shared Journal: Week 12
Shared Journal: Week 3	Shared Journal: Week 8	Shared Journal: Week 13
Shared Journal: Week 4	Shared Journal: Week 9	Shared Journal: Week 14
Shared Journal: Week 5	Shared Journal: Week 10

Class Notes

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Sarah Carratt_1.25.11	Sarah Carratt_2.10.11	Sarah Carratt_3.1.11
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