Biochemical Reaction Networks
123
colum ns 2 to 4 o f T. F inally the stoichiom etric coefficients for the three intracellular m etabolites A, B
and C are listed in the colum ns 5 to 7 o f T. T he row s o f the stoichiom etric m atrix clearly specify the
stoichiom etry for the six reactions considered in the netw ork, w ith the first reaction being the conversion
o f the substrate to the intracellu lar m etabolite A and the second reaction being the conversion o f A to B
etc.
W ith the stoichiom etric m atrix specified the balances are easily derived by transposing the m atrix
(changing colum ns and row s) and m u ltiplying w ith the flux vector v:
f - \
0
0
0
0
0 '
I V
'
-V ,
'
r
n ,
0
0
1
0
0
0
V,
P
-1
v
3
rp.2
0
0
0
0
1
0
y2
V5
V ?
=
0
0
0
0
0
1
V3
=
v 6
v 4
0
1
- 1
0
0
0
0
V5
V, - v 2
0
0
1
- 1
- 1
0
0
v , - V, — V.
0 ,
, 0
0
0
1
- 1
- K
NV6
J
1
1
... ^
(
2
)
The top four row s o f this m atrix equation relate the net flow s in and out o f the cells to the intracellular
flux vector and the bottom three row s specify balances for the intracellular m etabolites. It is seen that the
balances for the intracellular m etabolites im pose a set o f constraints on the six fluxes, and the degrees o f
freedom is three, i.e., only three o f the m easurable rates (r5,
r
p,,,
rp2
and rp -,) needs to be m easured in order
to calculate all the fluxes.
In the m a trix e q u a tio n (5 .1 ) th e re a re
J
u n k n o w n s, n a m e ly th e e le m e n ts o f th e flux v e c to r v, and
there are
K
c o n stra in ts im p o s e d b y th e m a s s b a la n c e s fo r th e in tra c e llu la r m e ta b o lite s. T h u s, the
degrees o f fre e d o m in th e e q u a tio n sy ste m is
F=J-K,
a n d i f
F
o f th e m e a su ra b le ra te s are
available fro m e x p e rim e n ts th e re m a in in g
N+M-F
ra te s c a n b e c a lc u la te d . A sy ste m a tic
pro ced u re to d o th is w ill b e p re s e n te d in S e c tio n 5.3.
The m atrix e q u a tio n (5 .1 ) im p lie s th a t th e fo rm a tio n o f in tra c e llu la r m e ta b o lite s e x a c tly b a la n c e s
the c o n su m p tio n , i.e., th e n e t fo rm a tio n ra te o f all in tra c e llu la r m e ta b o lite s is z ero . D u rin g cell
grow th th e b io m a ss w ill, h o w e v e r, e x p a n d a n d i f th e n e t fo rm a tio n ra te o f a n in tra c e llu la r
m etabolite is z ero th is w ill re su lt in a d e c re a s in g c o n c e n tra tio n as a c o n se q u e n c e o f d ilu tio n . F o r
m ost m e ta b o lite s th is d ilu tio n e ffe c t is n e g lig ib le , b u t in a few c a se s o n e m u st c o n s id e r th is
dilution e ffe c t as d is c u s s e d in N o te 5.1.
Note 5.1 D ilution effect on intracellu lar m etabolites
The correct m aterial balance for an in tracellular m etabolite is given by:
dc
met
dt
(
1
)
where rmeI is the net rate o f form ation o f the m etabolite in all the reactions w ithin the cell. In a steady
state m etabolite co ncentration cme, (unit: m oles (g D W )'1) m ust stay constant, and the balance reduces to:
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