Chapter 3.7, Problem 8E

Interpretation Introduction

Interpretation:

1. To non-dimensionalize the system by writing the equation in terms of dimensionless quantities, $\text{x=}\frac{{\text{A}}_{\text{p}}}{\text{K}}{\text{, τ=k}}_{\text{d}}\text{t,}\text{ s=}\frac{{\text{k}}_{\text{p}}\text{S}}{{\text{k}}_{\text{d}}}\text{ and b=}\frac{\text{β}}{\left({\text{k}}_{\text{d}}\text{K}\right)}$

2. To draw the phase portrait of the system assuming there’s no stimuli, i.e., s=0.

3. To draw all qualitatively different phase portraits that can occur for s > 0.

4. To draw the bifurcation diagram for increasing stimuli s, considering the behavior of fixed points.

5. To show that the system is irreversible, i.e., proteins get activated for sufficiently large s, but they don’t get deactivated by decreasing s to zero later.

6. To analyze the reversibility in case ${\text{β<<k}}_{\text{d}}{\text{A}}_{\text{t}}$.

Concept Introduction:

• ➢ To illustrate that the cells’ fate can be determined irreversibly by transient stimuli by using the model of phosphorylation/dephosphorylation cycle, in which phosphorylation can be induced either by using stimuli S or by phosphorylated proteins. Dynamics of phosphorylated proteins are governed by the differential equation

  ${\dot{\text{A}}}_{\text{p}}{\text{=k}}_{\text{p}}\text{SA+β}\frac{{\text{A}}_{\text{p}}^{\text{n}}}{{\text{K}}^{\text{n}}{\text{+A}}_{\text{p}}^{\text{n}}}{\text{-k}}_{\text{d}}{\text{A}}_{\text{p}}$

• ➢ A is the concentration of the unphosphorylated protein and A_p is the concentration of the phosphorylated protein. Total protein concentration A_T=A+ A_pis constant. k_p is the activation rate, and K_d is the inactivation rate.