Math 4600
Midterm practice, Part I
1. There are three chemicals involved in the reaction: enzyme (E), a protein
in the inactivated state (I) and the same protein in the activated state (A).
a) Write a system of dierential equations to describe the following situation.
- When inactivated protein reacts (meets) with enzyme, it becomes activated,
this happens at rate k. The amount of enzyme does not change as a result of
this reaction, it only acts as a catalyst
- Activated protein can inactivate at rate .
- Activated protein results in production of more enzyme at rate .
- Inactivated protein is produced at a constant rate s.
b) Let us say that an enzyme inhibitor is applied. What parameter would
be aected?
2.
T_ = T (T 2 1)
a) Find steady states graphically, on the phase line. Also include direction
arrows and state stability of the steady states
b) Conrm your stability results from a) analytically
3. Consider a system consisting of concentration of calcium inside the cell
(C), and the number of open channels (O). Their interaction is described in
words below, and also by the following system of equations:
O_ = k1 C + k2 O
C_ = k3 C + k4 O
Explain why the constants k1 ; k2 ; k3 ; k4 should be positive or negative, based on
the following:
- when a channel is open it lets calcium into the cell
- Calcium promotes opening of the channels
- In the absence of calcium the channels close
- Calcium is removed from (degraded) inside the cell
4.
x_ = 2x y
y_ = x + y
a) Find and plot nullclines, steady state and direction arrows in the phase plane
b) Find eigenvalues for the steady state. What kind of steady state is it?
5. Look at the following bifurcation diagram.
a) What kind of bifurcation is this?
b) If the parameter s is large, and the initial condition is V = 0, where will this
solution converge? Mark on the diagram.
c) As a continuation of b): if now s is slowly decaying to zero, what will be the
1
corresponding time course of V ? Sketch.
V
10
5
0
s
6. Sketch the qualitative time courses of x and y for the two solutions marked
with a square and a triangle.
y
x
2