Biology Reference
In-Depth Information
In the form of Eq. (
8.12
) above, the sequence of balance/node equations corre-
sponds to a homogeneous linear system in the 11 variables
v
1
,...,v
7
,
b
1
,...,
b
4
,
that is, in the flux variables. There is one equation for each node.
Exercise 8.5.
Write down the coefficient matrix for the homogeneous linear system
described in Exercise
8.4
.
Wonderfully, the coefficient matrix you just found is just the stoichiometric matrix
S
for the system! Writing the total flux vector
v
T
=
(v
1
,...,v
11
)
:=
(v
1
,...,v
7
,
T
(so that
b
1
,...,
v
i
sarethe
internal fluxes as before), the homogeneous system described by the node/balance
equations has the form
b
4
)
v
8
:=
b
1
,v
9
:=
b
2
,v
10
:=
b
3
,v
11
:=
b
4
, and the other
S
v
=
0
.
(8.13)
This time, however, we have obtained
S
by focusing on the rows, instead of the
columns.
Exercise 8.6.
How many reaction equations occur in the system described by this
graph/stoichiometric matrix
S
?
Recall that the nullspace
N
(
S
)of
S
is the set of all solutions to the homogeneous
system (
8.13
). Equation (
8.13
) is interpretable as a statement of conservation of mass
[
6
]. Each vector
v
in the nullspace
N
(
S
) describes the relative distribution of “fluxes,”
and the variable entries
v
1
,...,v
n
of each flux vector
v
give values that represent the
activity of the individual reactions, indicated by their flow rates. Thus, the product
Sv
assigns the flux throughout the entire metabolic system represented by
S
[
6
,p.
4194]. Flux vectors
v
satisfying
S
v
=
0 correspond to steady-state solutions to a
“dynamic mass-balance equation”
d
dt
T
=
S
v
, where
X
=
(
X
1
,...,
X
m
)
for
X
i
the
concentration of the
i
th metabolite (compound
C
i
), and
dX
i
=
a
i
,
1
v
1
+···+
a
i
,
n
v
n
dt
is the change in concentration of the
i
th metabolite, and as before,
v
j
the rate (“flux”)
of reaction
j
.
Exercise 8.7.
T
. Show that
v
a.
Let
v
.
b.
Represent the total flux vector
v
from part (a) by shading in Figure
8.5
the
corresponding edges of the graph which have nonzero fluxes. (Recall that a zero
entry means there is no flux, hence no involvement, of a particular reaction.)
What do you notice about the resulting walk in the graph?
c.
Let
v
=
(
1
,
1
,
0
,
0
,
0
,
1
,
0
,
−
1
,
0
,
0
,
1
)
∈
N
(
S
)
T
. Repeat the instructions for parts (a)
=
(
1
,
1
,
0
,
0
,
0
,
0
,
0
,
−
1
,
1
,
0
,
0
)
and (b) above, using Figure
8.5
.
d.
Check that
v
and
v
are linearly independent vectors.
e.
Using your observations from parts (a) to (d), find and sketch the graphical
interpretation of another total flux vector,
v
, which is linearly independent from
v
and
v
(that is, show
v
,
v
}
is a linearly independent set). Use Figure
8.5
.
f.
Now, find a total flux vector
w
for which
{
v
,
v
,
{
v
,
w
}
form a linearly dependent
set. Do you have a graphical interpretation for
w
?
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