Modeling - Math 056 - Homework 6
1. Read: Perelson, A., Neumann, A., Markowitz, M., Leonard, J., Ho, D. HIV-1 Dynamics in Vivo: Virion Clearance Rate, Infected Cell Life-Span, and Viral Generation Time. Science. 271(5255). pp. 1582-1586.
Write a one-page typed summary of the paper including the following:
- What is the biological question being addressed?
- Why is a mathematical model being used?
- Describe the model(s), including terms and assumptions.
- Describe how the model is used (are steady states found? are differential equations solved?)
- What other mathematical techniques are used?
- What is the biological impact of using the model?
2. In a growth chamber, the microorganisms (density N(t)) and their food supply are kept in a chamber separated by a semipermeable membrane from a reservoir containing the stock nutrient whose concentration (C0 > C(t)) is assumed to be fixed. Nutrient can pass across the membrane by a process of diffusion at a rate proportional to the concentration difference. The microorganisms have mortality µ.
(a) Explain the following equations:
dN/dt = N(KmaxC/Kn + C) - µN
dC/dt = D(C0 - C) - αN (KmaxC/Kn + C)
(b) Determine the dimensions of all the quantities in (a).
(c) Bring the equations to a dimensionless form.