Nonlinear Dynamics and Chaos
Nonlinear Dynamics and Chaos
2nd Edition
ISBN: 9780813349107
Author: Steven H. Strogatz
Publisher: PERSEUS D
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Chapter 3.7, Problem 8E
Interpretation Introduction

Interpretation:

  1. To non-dimensionalize the system by writing the equation in terms of dimensionless quantities, x=ApK, τ=kdt, s=kpSkd and b=β(kdK)

  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 β<<kdAt.

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

    A˙p=kpSA+βApnKn+Apn-kdAp

  • ➢ A is the concentration of the unphosphorylated protein and Ap is the concentration of the phosphorylated protein. Total protein concentration AT=A+ Apis constant. kp is the activation rate, and Kd is the inactivation rate.

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Please help solve the following whilst showing all working out. Is part of exam revision questions but no solution is given
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