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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