Math 5110/6830
Instructor: Alla Borisyuk
Homework 9.2
Due: November 10
1. (EK) Full 2-equation system that we used to derive Michaelis-Menten
kynetics
dc
= k1 rc + (k 1 + k1 c)x1
dt
dx1
dt
= k1 rc (k 1 + k2 + k1 c)x1
can be studied by phase-plane methods.
a) Show that the nullclines have the following form:
-nullcline: x1 = Kc=(a + c);
-nullcline: x1 = Kc=(b + c).
Identify K , a, and b in terms of the original parameters r0 , k1 , k 1 , and k2 .
Which is larger, a or b?
b) Sketch the nullclines in the phase plane. How many points of intersection
are there? Determine the direction of ow along the nullclines and along the
axes.
c) Draw a trajectory with the initial condition in which all receptors are
unoccupied and the initial nutrient concentration is c0 . What is the long-term
behavior of this solution? Sketch the corrsponding x1 (t) time-course. Explain
your result in terms of the original cellular process.
2. Starting with the same equaions as in problem 1 do the non-dimensionalizing
change of variables that works on a short time scale (that is the one that I mentioned briey at the end of the class): t = ~t , c = c0 c , x1 = rx1 with
1
~ =
c
x1
k1 c0
and , K and dened as before, and show that the equations become:
dc
dt
=
dx1
dt
c
+ (K
+ c )x1 ;
= c (K + c )x1 :
Now assume that is approximately zero (why can we do that?) and solve
the resulting equations analytically (exact formula), with c(0) = c0 , x1 (0) = 0.
Plot the solution x1 (t), i.e. vs. the original time. Remember this solution is
only valid for t on the order of ~, so plot accordingly. How does this solution
compare to the one you found in problem 1?
1