Scribe: N. Allen

Today's Handouts and Announcements

• Homework Assignment #5 has been posted.
• Dr. Alscher's talk moved to June 23rd.
• Subhag's talk moved to June 26th.

Today's Topics

• Solution cycle for cell models:
  1. Construct molecular network
  2. Create differential equations to model network
  3. Solution cycle to solve DEs:
     1. Estimate initial parameter values
     2. Apply techniques for solving systems of differential equations
     • Euler
     • Backwards-differencing Euler
     • Euler with Heun
     • Euler with n-term Taylor
     • nth-order Runge-Kutta
     • Adams-Bashforth
     • Adams-Moulton
     • Any good super-stable stiff system solver (probably your best choice)
     3. Compute the error of the approximation and refine parameter values
     • Method of steepest descent (Cauchy, 1845) - very slow
     • Multistart steepest descent - extremely slow
     • Newton's nonlinear minimization - needs positive-definite inverse Hessian
     • Newton's nonlinear minimization with Gill-Murray modified Hessian - requires many gradient evaluations
     • Quasi-Newton methods with secant approximations to the Hessian (probably your best choice)
     4. Repeat
  4. Compare model to known cell physiology and adjust objective function
  5. Repeat

• Some more information about yesterday's model and differential equations:
  • Cyclin degradation is caused by APC with adaptor cdc20
  • Active MPF induces production of cdc20 (time delay suggests intermediate mechanics)
  • Empirical observations preclude a linear dependence on amount of Active MPF (predicted to be approximately quadratic)
  • A positive feedback loop exists in the conversion of Inactive MPF to Active MPF using cdc25
  • Inactive->Active threshold at approximately 25 nM
  • Model predicts that the threshold for conversion of Active MPF to Inactive MPF will be at a lower concentration
  • Unverified observations that the Active->Inactive threshold is approximately 8 nM

Today's Sources

• Sources listed on the CS6104 Student Talks page under Jason Zwolak.
• Suggested additional introductory references for numerical approximations of DEs and multidimensional minimization:
  • Kahaner, Moler, and Nash. "Numerical Methods and Software". Prentice Hall, 1989. (also published as Forsythe, Malcolm, and Moler. "Computer Methods for Mathematical Computations". Prentice Hall, 1977.)
  • Boyce and DiPrima. "Elementary Differential Equations and Boundary Value Problems". John Wiley & Sons, 1997.
  • JE Dennis and J More. "Quasi-Newton methods, motivation, and theory". SIAM Review 19 (1974) pp. 46-89