ODEs used in modeling

The following equations shows the break down of the different equations that will be used in this modeling exercise. By understanding this section, it would make the understanding of the system of ODEs used

Constant synthesis & Linear Synthesis

• Simple ode to describe constant synthesis
• Gives an explicit analytical solution
• Unique solution once a IC is posed

Linear Degradation

• Rate of degradation is proportional to how much of the molecule is present
• Gives an explicit analytical solution
• Constant half life

Simple Forward Reaction

[C] : Complex
kc : Rate constant of complex formation

This equation ignores the fact that dissociation of the complex occurs. We can do so if the dissociation is much slower than the formation.

• Single solvable equation for the unknown C
• Simple, unique solution available with I.C

Phosphorylation and Dephosphorylation

Assumptions:

• Linear kintic rate laws apply only if XT is much less than the Michaelis constants of both kinase and phosphotase

XT : total cost of X protein in phosphorylated and unphosphorylated form
S : protein kinase concentration
k2 : accounts for protein phosphotase

• Modeled after simple linear kinetics
• Gives a hyperbolic signal response curve when X plotted vs S

Regulated Transcription

[P]: Protein Formed
µ: Repression, µ=0;
Activation, µ=1
K: Hill Constant Value of input that gives 50% response
n: Hill coefficient Slope of signal-response curve at this input signal
d: degradation of protein
k1: basal gene expression
k: signal-dependent gene expression
a: correlation between k1 and k, 0<a<1

This ODE attempts to capture characteristics of the mRNA dynamics
For our modeling, all our detection systems activates some form of transcription. Therefore µ=1 in all cases for our modeling exercise.