Biology Reference
In-Depth Information
c.
Recall:
What is the biological meaning of a vector in
N
(
S
), and of dim
(
N
(
S
))
,
the number of (linearly independent) vectors spanning
N
(
S
)
? What about the
columns of
S
?
d.
For the metabolic system captured by
S
, under what circumstances is the num-
ber of total flux vectors needed to describe the metabolic pathways (via linear
combinations) in the system the same as the difference between the number of
metabolites in the system and the number of chemical reactions in the system?
[Hint: Use your previous answers and the Rank + Nullity Theorem.]
e.
Were the circumstances you identified in part (d) of this exercise met in
Exercise
8.8
? What about in Exercise
8.9
?
f.
By definition of the rank of amatrix, rank
(
S
)
satisfies the inequality rank
(
S
)
m
.
m
whenever conservation relationships hold in the
system, for example, that ATP + ADP equals some constant value for the whole
system. Using this, give a restatement (in English) of the biological meaning of
the Rank + Nullity Theorem for a stoichiometric matrix, tying in your results
from part (d).
g.
Schilling et al. [
7
, pp. 298-299] presents an analysis of the reaction scheme
of a metabolic system consisting of the glyoxylate cycle and related reactions,
as pictured therein. The outcome of solving for the nullspace
N
(
S
)ofastoi-
chiometric matrix
S
for this metabolic system
11
results in a nullspace
N
(
S
) with
dim
As per [
7
, p. 298], rank
(
S
)<
3. Abasis for
N
(
S
) is also pictured below, organized as the columns
of the matrix
B
. For reasons of space, the actual matrix shown is
B
T
; the columns
correspond to the following metabolite abbreviations: Eno, Acn, Sdh, Fum, Mdh,
AspC, Gdh, Pyk, AceEF, GltA, Icd, Icl, Mas, AspCon, Ppc, GluCon:
(
N
(
S
))
=
⎡
⎤
2111211221011100
11112002210110
B
T
⎣
⎦
.
=
10
3211201332111001
−
Using the data from matrix
B
above, that is, from
N
(
S
) (and without attempting to
compute
S
itself
12
), answer the following questions:
i.
How many free variables must have appeared in
E
S
?
ii.
If there were no conservation relationships holding among metabolites in the
system, how many reactions were there?
8.2.3
Conclusion
Biologically “good” bases for
N
lead to the notion of “extreme paths”. For exam-
ple, for
S
as in Exercise
8.18
(g), it is possible to find a “good basis” for
N
(
S
): check that
by setting
b
i
to be the
i
th row of
B
T
, and taking
p
1
=
(
S
)
b
1
−
b
2
,
p
2
=
b
2
,
p
3
=
b
3
−
b
2
,
and
p
4
=
p
3
, one obtains a “biologically good basis.” In [
7
] one can see a graph of
11
As explored in earlier exercises in this module.
12
This could be made into an additional project.
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