Biology Reference
In-Depth Information
FIGURE 8.5
Graph for Exercise
8.7
(b).
Exercise 8.8.
a.
Find the entire space
N
(
S
) for the stoichiometric matrix
S
you found in Exercise
8.5
. Before beginning, make a prediction as to the number of free variables you
will find, and try to complete the rest of the accompanying statement below.
My prediction for the number of free variables is: _____, and the relation to
the minimal number of vectors spanning
N
(
S
) is: _____.
Express your solution as linear combinations of vectors, with coefficients the
free variables, that is, express
N
(
S
) as a span of a set of vectors. (Feel free to use
your calculator here to find the rref
E
S
of
S
.)
b.
Argue (without a lot of computation) that the spanning set of vectors for
N
(
S
)
that you found in part (a) forms a linearly independent set.
c.
As in Exercise 8.7(b), give the graphical interpretation of each element of the
spanning set for
N
(
S
) by using as many copies of Figure
8.5
as needed. Explain
whether these results are consistent with your expectations formed fromExercise
8.7
. (We will explore this outcome further in some later problems.)
Exercise 8.9.
a.
For the stoichiometric matrix
S
as in Exercise
8.2
, find the nullspace
N
(
S
)of
S
.
b.
Then
, explain the biochemical significance of the result of part (a).
c.
Can you hazard a guess also as to the graphical significance of the result of part
(a)? [Hint: Take a look at the glycolysis network represented in Figure 8.3.]
d.
Would you expect the same result as in part (a) if you used instead the stoi-
chiometric matrix
S
for the entire metabolic network given by glycolysis? Be as
specific as you can about the results you would expect, without doing any actual
computations.
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