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

Variables in Context
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

  1. Vmax (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([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?


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
Sarah Carratt: Week 5 Sarah Carratt: Week 9 Sarah Carratt: Week 14

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

Sarah Carratt_1.18.11 Sarah Carratt_2.3.11 Sarah Carratt_2.22.11
Sarah Carratt_1.20.11 Sarah Carratt_2.8.11 Sarah Carratt_2.24.11
Sarah Carratt_1.25.11 Sarah Carratt_2.10.11 Sarah Carratt_3.1.11
Sarah Carratt_1.27.11 Sarah Carratt_2.15.11 Sarah Carratt_3.3.11
Sarah Carratt_2.1.11 Sarah Carratt_2.17.11 Sarah Carratt_3.8.11

Internal Links

BIOL398-01/S11:Assignments BIOL398-01/S11:People BIOL398-01/S11:Sarah Carratt
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