AMATH/BIOL 382 Problem Set 1
Due January 22, 2010
1.
Notes, chapter 1: Exercise 1.1.11.
2.
Consider the chemical reaction network indicated in Figure 1 with rate constants as
shown.
A
B
C
D
E
F
k
k
k
1
3
2
Figure 1: Closed Network
a) Write a set of six differential equations describing the evolution of the concentrations of
the species from an arbitrary initial concentration profile.
b) Use mass conservations to reduce the system description to three differential equations
and three algebraic equations.
c) Determine the system steady state as a function of the initial concentrations. (Note, the
answer is rather trivial since the system is closed).
d) Verify your result in (c) by simulating the system behaviour from initial condition ([A],
[B], [C], [D], [E], [F]) = (1
,
1
,
1
2
,
0
,
0
,
0).
For the simulation take
k
1
= 3,
k
2
= 1,
k
3
= 4.
Submit a printout.
e) Repeat (ad) for the open system in Figure 2. There will be fewer structural conservations.
The input rate v is constant. For the simulation, take
k
1
= 3,
k
2
= 1,
k
3
= 4,
k
4
= 1,
k
5
= 5,
v = 0
.
5, and the same initial condition.
k
2
A
B
C
D
E
F
k
k
1
3
k
5
k
4
v
Figure 2: Open Network
f) Why is there no (physical) steady state if we choose v = 5?
1
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3.
Notes, chapter 1: Exercise 1.2.1
4.
Consider the system described in Figure 3 below. Take S and P as boundary species with
fixed concentrations and suppose the reaction constants are given (in units of
s

1
) as
k
1
= 1
, k
2
= 11
, k

2
= 8
, k
3
= 0
.
2
and take
S
= 1 mM.
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