First Evening Session Exercises (Smith)

  1. Read Ch 1 and 2 of “Computational Cell Biology” by Smith (in preparation, posted on ftp server under 1Smith).  While reading Ch 2 do whatever exercises look appropriate.
  2. Read Appendix B of “Computational Cell Biology” by Fall et al. 1.  entitled “Solving and analyzing dynamical systems using XPPAUT” and reproduce each figure as it is presented.
  3. Use XPPAUT and the ode files provided with my lectures to explore one dimensional dynamical systems and bifurcations (fold, pitchfork, transcritical).
    1. For a particular value of the bifurcation parameter u,  range over initial conditions and observe solutions that are consistent with the corresponding phase diagram  (drawn by hand) or algebraic expressions for steady-states you have derived.
    2. Use AUTO to numerically calculate the pitchfork and transcritical bifurcation diagrams (fold was done this afternoon).
    3. Work through the biomolecular association reaction example in Smith’s 2nd lecture (PDF on ftp server).
  4. Show that the fold bifurcation is “structurally stable” while the pitchfork and transcritical bifurcations are not.
  5. Do the Chapter 1 exercises from  “Computational Cell Biology” of Fall et al.  (Fall et al. is on server; exercises on p. 20).
  6. Write an ODE file from scratch for a nonlinear ODE that you can solve analytically (e.g., using separation of variables).  Compare the numerical solutions generated by XPPAUT with the analytical solution (e.g., as graphed in Matlab or other software package).   Explore how the global error (difference between numerical and analytical result at a specific time) depends on the time step DT for Euler’s method and/or Runge-Kutta (both of which can be selected in XPP).
  7. Read about Numerical Methods for Ordinary Differential Equations
  8. Read any of the “Perspectives in Biological Modeling” that have been posted.  Maybe start with Tyson 2007.
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