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

Variables Needed for a Model



 * 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)
 * 4) ammonium changes
 * 5) 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?