Steady-state kinetic behaviour of functioning-dependent
structures
Michel Thellier
1,3
, Guillaume Legent
1
, Patrick Amar
2,3
, Vic Norris
1,3
and Camille Ripoll
1,3
1 Laboratoire ‘Assemblages mole
´
culaires: mode
´
lisation et imagerie SIMS’, Faculte
´
des Sciences de l’Universite
´
de Rouen,
Mont-Saint-Aignan Cedex, France
2 Laboratoire de recherche en informatique, Universite
´
de Paris Sud, Orsay Cedex, France
3 Epigenomics Project, GenopoleÒ, Evry, France
Numerous studies have shown that proteins involved
in metabolic or signalling pathways are often distri-
buted nonrandomly, as multimolecular assemblies
[1–15]. Such assemblies range from quasi-static, multi-
enzyme complexes (such as the fatty acid synthase or

transferred from the first enzyme to the second [39,40].
Keywords
enzyme kinetics; metabolic or signalling
pathways; mathematical modelling; protein
associations
Correspondence
M. Thellier, Laboratoire Assemblages
mole
´
culaires: mode
´
lisation et imagerie SIMS
FRE CNRS 2829, Faculte
´
des Sciences de
l’Universite
´
de Rouen, F-76821 Mont-Saint-
Aignan Cedex, France
Fax: +33 2 35 14 70 20
Tel: +33 2 35 14 66 82
E-mail:
(Received 12 January 2006, revised 26 June
2006, accepted 20 July 2006)
doi:10.1111/j.1742-4658.2006.05425.x
A fundamental problem in biochemistry is that of the nature of the
coordination between and within metabolic and signalling pathways. It is
conceivable that this coordination might be assured by what we term func-
tioning-dependent structures (FDSs), namely those assemblies of proteins
that associate with one another when performing tasks and that disassoci-

have allowed this FDS to form. It can therefore be
viewed as a self-organized structure.
Published examples of transient, dynamic multi-
molecular assemblies, that only form in an activity-
dependent manner include: the role of the bifunctional
protein complex cysteine synthetase in the synthesis of
cysteine in Salmonella typhimurium [42]; the metabo-
lite-modulated formation of complexes (especially
binary complexes) of sequential glycolytic enzymes
[4,43,44]; the functional coupling of pyruvate kinase
and creatine kinase via an enzyme–product–enzyme
complex in muscle [45]; the interaction between serine
acetyl-transferase and O-acetylserine(thiol)-lyase in
higher plants [46,47]; the ATP- and pH-dependent
association ⁄ dissociation of the V1 and V0 domains of
the yeast vacuolar H
+
- ATPases [48–50]; the promo-
tion by substrate binding of the assembly of the three
components of protein-mediated exporters involved in
protein secretion in Gram-negative bacteria [51]; the
first step of glycogenolysis in vertebrate muscle tissues
by the sequential formation of a phosphorylase–glyco-
gen complex followed by the binding of phosphorylase
kinase to this previously formed complex [18]; the clus-
tering of the anchoring protein gephyrin with glycine
receptors following glycine receptor activation in
postsynaptic regions of spinal neurons [52–55]; the
clustering of antigen receptors followed by binding of
intracellular proteins, such as protein tyrosine kinases,

when it is not [64–67].
It is striking that these cellular systems that have
very different structures and functions nevertheless
exhibit the common behaviour of assembling into tran-
sient complexes or FDSs when functioning. Why? A
fundamental problem in biochemistry is that of coordi-
nation. The functioning of a protein in a metabolic or
signalling pathway in vivo is coordinated with that of
the other proteins in the same pathway, and the func-
tioning of the pathway itself is coordinated with that
of the other pathways within the cell. In metabolic
pathways, the regulation needed for such coordination
comes in part from the sigmoidal kinetics provided by
allosteric enzymes, due to the fact that subunit–subunit
interactions are added to the classical enzyme–sub-
strate interactions [68]. It is therefore tempting to spe-
culate that FDSs are involved in the coordination
within and between metabolism and signalling.
If FDSs are to have a central role in coordination,
they should be predicted to generate regulatory kinet-
ics via the enzyme–enzyme interactions that constitute
them. In the following, we have endeavoured to test
this prediction by numerically studying the steady-state
kinetics of a model system of two sequential monomer-
ic enzymes, E
1
and E
2
, which, when free, are of the
Michaelis–Menten type (i.e., with a single substrate-

on its activity, the reaction E
1
+E
2
¼ E
1
E
2
does not
exist in this scheme. Note that the symbols used in
Fig. 2 to describe the complexes are such that E
1
S
2
E
2
and E
1
E
2
S
2
mean that S
2
is bound to the catalytic site
of E
1
or of E
2
, respectively, within the FDS, etc. To

1
, the steady-state rate of transformation of S
1
into S
3
is calculated as corresponding to both the rate
of consumption of S
1
, v(s
1
), and the rate of production
of S
3
, v(s
3
), and the shape of the curves {s
1
, v(s
1
)} is
examined in cases involving either free enzymes alone
or an FDS with free enzymes.
It is worth noting that it would only be necessary to
add a few more reactions to Fig. 2 to describe the
interaction of these enzymes with other proteins or
molecules and hence study systems in which, for exam-
ple, small proteins contribute to the formation of the
enzyme–enzyme complexes [15]; the theoretical treat-
ment would be longer but otherwise essentially the
same as that followed here.

switch from low or null current to high current when the potential difference exceeds a threshold; curve (b): Inverse step response: this
behaviour corresponds to a switch from high current to low or null current when the potential difference exceeds a threshold.
M. Thellier et al. Functioning-dependent structures
FEBS Journal 273 (2006) 4287–4299 ª 2006 The Authors Journal compilation ª 2006 FEBS 4289
steady-state conditions [69]. The results are summarized
in Fig. 3A. Briefly, curves monotonically increasing up
to a plateau and exhibiting no inflexion points were
obtained for all parameter values tested. Occasionally,
the shape of these curves was close to that of a hyper-
bola. Cases existed (with the smallest K
2
values in
Fig. 3A) in which the overall rate of reaction became a
quasi-linear function of the concentration of initial sub-
strate, s
1
, almost up to the plateau (which never occurs
when a single enzyme is involved). Hence, under certain
conditions, free enzymes can generate signals or other
behaviours corresponding to a linear relationship
between input (concentration of first substrate) and
output (rate of production of final product) (Fig. 1A).
The system with an FDS
At some parameter values, in the case of an FDS, the
{s
1
, v(s
1
)} curves were similar to those obtained with
the free enzymes, i.e., they increased monotonically

some parameter values, with increasing s
1
, the rate of
consumption of S
1
decreased to almost zero (Fig. 4A).
This means that this FDS system exhibited a sort of
inversed behaviour in which it was active at low s
1
val-
ues (except at the very lowest s
1
values) and inactive at
the high s
1
values. This corresponds to the scenario in
Fig. 1C in which an increasing input leads to an
output in the form of a spike or impulse. Another case
in which an increasing input leads to an output in the
form of an impulse (i.e., corresponding to the scenario
in Fig. 1C) is depicted in Fig. 4B.
At other values of the parameters, with increasing
s
1
, the rate of consumption of S
1
again increased,
reached a maximal value, then decreased, whilst at sat-
urating values of s
1

drops rapidly to a constant and low rate of con-
sumption. This resembles the switch shown in Fig. 1D
curve (b).
Curves with a sigmoid shape, i.e., resembling the
switch shown in Fig. 1D curve (a), were sometimes
obtained (Fig. 5A). At the parameter values tested,
however, the adjustment of the curve to a Hill function
v(s
1
) ¼ v
max
Æ(s
1
)
n
⁄ [(k)
n
+(s
1
)
n
] (in which n is the Hill
Fig. 2. The scheme of the reactions
involved in the functioning of our model of a
two-enzyme FDS. The system comprises 17
different chemical species (free enzymes,
free substrates or products, and binary,
ternary or quaternary complexes) indicated
in the green circles. These species are
linked to one another by 29 chemical reac-

))]}
(Fig. 5B); moreover, the sigmoidicity was rather weak
(Hill coefficient equal to only 1.47).
There were cases in which even more complicated
responses occurred. For example, in Fig. 6 in which
K
10
was varied from 1 to 10
3
and in which all the
other parameters have the values given in the figure
caption, a {s
1
, v(s
1
)} curve similar to those in Fig. 4C
and with a low plateau value was observed with the
smallest K
10
values (Fig. 6, curve a) while the sub-
strate-inhibition effect was less and the plateau was
higher with increasing K
10
values (Fig. 6, curve b).
Finally, with the highest values of K
10
(Fig. 6, curves c
and d), the {s
1
, v(s

these functions include the linear function obtained
when a source of potential difference is connected to
a resistor (Fig. 1A), the constant function obtained
when a current source is connected to a resistor
(Fig. 1B), the impulse function (Fig. 1C) and the
increasing (Fig. 1D, curve a) or decreasing (Fig. 1D,
curve b) step function. We have shown here that the
assembly of only two enzymes can result in a variety
of input ⁄ output relationships including, importantly,
those with characteristics similar to these basic func-
tions. Hence, the assembly of just two enzymes could
provide a macromolecular mechanism for control
processes. This is illustrated by the following exam-
ples. The substrate concentration could be encoded
in a linear response (Fig. 1A). (Note that we occa-
sionally obtained linear responses from a system of
0
0.1
0.2
20.010.00
0
0.06
0.12
0.18
1.050.00
v(s
1
) v(s
1
)

¼ k
4r
¼ k
9r
¼ k
10r
¼ 1, k
4f
calculated according to Eqn (A25), K
1
¼ 10, K
3
¼ 100, K
9
¼ K
10
¼ 1 and
K
2
¼ 0.10 (curve a), 0.05 (curve b), 0.01 (curve c), 0.001 (curve d) and 0.0001 (curve e). Modified from [69]. (B) Case of a two-enzyme FDS:
the parameter values are e
1t
¼ e
2t
¼ 0.5, K ¼ 100, k
1r
¼ 1 (Eqn A6), k
2r
¼ 100, k
3r

18r
¼ k
19r
¼ k
20r
¼ k
21r
¼ k
22r
¼ k
23r
¼ k
24r
¼ k
25r
¼ k
26r
¼ k
27r
¼ k
28r
¼ k
29r
¼ 1, K
1
¼ 10, K
2
¼
0.01, K
5

M. Thellier et al. Functioning-dependent structures
FEBS Journal 273 (2006) 4287–4299 ª 2006 The Authors Journal compilation ª 2006 FEBS 4291
two enzymes that diffused freely, i.e., without FDS.)
Homeostasis results when, despite the concentration
of the initial substrate, s
1
, varying, the rate of pro-
duction of the final product is constant (Fig. 1B).
An impulse that could constitute a signal, results
when, at a narrow range of low concentrations of
substrate s
1
, the rate of production of the final prod-
uct takes the form represented in Fig. 1C (Fig. 4A,B
show a more realistic representation). A switch as
represented in Fig. 1D (curve a) could be based on
the sigmoid curve in the production rate. A switch
from a high rate to a low rate of production occurs
when s
1
exceeds the threshold s
0
at the inflection
point (Fig. 4D) and this could correspond to a sub-
strate-inhibition behaviour. Hence the assembly of
two enzymes into an FDS could allow a switch
behaviour. Alternatively, it could allow this enzyme
system to be efficient at a low substrate concentra-
tion but not at a high concentration where the sub-
strate would become available for enzymes in a

0.0004
0.0008
1.050.00
A
s
1
v(s
1
) v(s
1
)
v(s
1
) v(s
1
)
Fig. 4. Various types of substrate-inhibition {s
1
, v(s
1
)} curves computed in the case of a two-enzyme FDS. (A) Example of an almost total inhi-
bition at high s
1
values (impulse behaviour): the parameter values are e
1t
¼ e
2t
¼ 0.5, K ¼ 100, k
1r
¼ 1 (Eqn A6), k

¼ k
16r
¼ k
17r
¼ k
18r
¼ k
19r
¼ k
20r
¼ k
21r
¼ k
22r
¼ k
23r
¼ k
24r
¼ k
25r
¼ k
26r
¼
k
27r
¼ k
28r
¼ k
29r
¼ 1, K

calculated as indicated in Eqns (A25) to (A27) and Table A2. (B) Another example of an impulse behaviour: the parameter
values are e
1t
¼ e
2t
¼ 0.5, K ¼ 1000, k
1r
¼ 1 (Eqn A6), k
2r
¼ 10
4
,k
3r
¼ k
4r
¼ k
5r
¼ k
6r
¼ k
7r
¼ k
8r
¼ k
9r
¼ k
10r
¼ k
11r
¼ 1, k

¼ k
26r
¼ k
27r
¼ k
28r
¼ 1, k
29r
¼ 10
4
,K
1
¼ 10, K
2
¼
0.0001, K
3
¼ 1000, K
5
¼ 10
6
,K
9
¼ K
10
¼ K
11
¼ 1, K
12
¼ 0.001, K

6r
¼ k
7r
¼ k
8r
¼ k
9r
¼ k
10r
¼ k
11r
¼ k
12r
¼ k
13r
¼ k
14r
¼
k
15r
¼ k
16r
¼ k
17r
¼ k
18r
¼ k
19r
¼ k
20r

¼ 10
3
,K
9
¼ 10, K
3
¼ K
10
¼ K
11
¼ K
12
¼ K
13
¼ K
15
¼ K
17
¼ K
29
¼ 1, K
27
¼ 100 and all the other
K
j
calculated as indicated in Eqns (A25) to (A27) and Table A2. (D) Example of an inversed step response: the parameter values are e
1t
¼
e
2t

¼ k
14r
¼ k
15r
¼
k
16r
¼ k
17r
¼ k
18r
¼ k
19r
¼ k
20r
¼ k
21r
¼ k
22r
¼ k
23r
¼ k
24r
¼ k
25r
¼ k
26r
¼ k
27r
¼ k

¼ 1, K
27
¼ 100, K
29
¼ 10000 and all the other K
j
calculated as indicated
in Eqns (A25) to (A27) and Table A2.
Functioning-dependent structures M. Thellier et al.
4292 FEBS Journal 273 (2006) 4287–4299 ª 2006 The Authors Journal compilation ª 2006 FEBS
of production were sometimes observed with Hill coef-
ficients of less than 2 (Fig. 5) but these could not con-
stitute switches. Compared with the sigmoidicity of
allosteric enzymes [68], that of a two-enzyme FDS –
the only type tested here – is poor. Experimental
results are consistent with this because the formation
of a protein–protein complex of serine acetyl trans-
ferase with O-acetylserine(thiol)-lyase strongly modifies
the kinetic properties of the first enzyme and results in
a transition from a typical Michaelis–Menten beha-
viour to a behaviour displaying positive cooperativity
with respect to serine and acetyl-CoA with a Hill coef-
ficient in the range of 1.3–2.0 [47].
It is probable that many more types of FDSs exist
than those found so far experimentally (see above).
Indeed, many FDSs may have escaped detection pre-
cisely because they tend to dissociate as the substrate
concentration decreases, as generally occurs during
in vitro studies. It may even turn out that most
enzymes and other proteins such as those involved in

1
2
-3 -2 -1 0
B
log s
1
s
1
v(s
1
)
Fig. 5. Example of a sigmoid {s
1
, v(s
1
)} curve computed in the case of a two-enzyme FDS. The parameter values are e
1t
¼ e
2t
¼ 0.5,
K ¼ 100, k
1r
¼ 1 (Eqn A6), k
2r
¼ 10, k
3r
¼ k
4r
¼ k
5r

20r
¼ k
21r
¼ k
22r
¼ k
23r
¼ k
24r
¼ k
25r
¼ k
26r
¼ k
27r
¼ k
28r
¼ k
29r
¼ 1, K
1
¼ K
2
¼ 0.1, K
3
¼ 10, K
5
¼ 1000, K
9
¼ K

¼ {v(s
1
) ⁄ [v
max
–v(s
1
)]}; from the slope of the dashed regression line fitted to the curve, the Hill coefficient was estimated to be of the order
of 1.47.
0
0.2
0.4
2.01.00
a
b
c
d
v(s
1
)
s
1
Fig. 6. Examples of dual-phasic {s
1
, v(s
1
)} curves computed in the
case of a two-enzyme FDS. The parameter values are e
1t
¼ e
2t

¼ k
15r
¼ k
16r
¼ k
17r
¼
k
18r
¼ k
19r
¼ k
20r
¼ k
21r
¼ k
22r
¼ k
23r
¼ k
24r
¼ k
25r
¼ k
26r
¼ k
27r
¼
k
28r

(curve b), 10
2
(curve c) and 10
3
(curve d) and all the other K
j
calcu-
lated as indicated in Eqns (A25) to (A27) and Table A2.
M. Thellier et al. Functioning-dependent structures
FEBS Journal 273 (2006) 4287–4299 ª 2006 The Authors Journal compilation ª 2006 FEBS 4293
different assemblies at different times depending on the
task to be carried out [80].
Biochemists are familiar with the Structure fi Func-
tion relationship with respect to proteins or other
active molecules or cell substructures. They are less
familiar with the idea that the very functioning of
these cellular components may result in their assem-
bling into a dynamic structure from which a better or
even a new functioning emerges. In this case, the rela-
tionship above must be changed into
This leads to the intuition that the very existence of
such a self-organizing relationship in a system is an
indication that this system is a living one. To try to
express this quantitatively, consider the density of
entropy production in a process involving an FDS.
According to the second law of thermodynamics, the
functioning of any system entails a positive production
of entropy that can be written as a bilinear form of the
flux densities of the processes and their conjugated
driving forces [81]. Whichever reaction pathway in our

tion of living systems.
Acknowledgements
We thank Jacques Ricard and Derek Raine for helpful
comments and criticisms.
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Appendix
The basis of the model of a two-enzyme FDS
For computing purpose, it is convenient to list in a
table all the different reactions, R
j
, appearing in Fig. 2
(Table A1). The rate constants, k¢
jf
and k¢
jr

)1
. In the following, when
any reaction, R
j
, in the table proceeds left to right or
right to left, it is written j
f
or j
r
, respectively. With
these conventions, the chain of reactions ‘1
f
-9
f
-2
r
-3
f
-
10
f
-4
r
’ corresponds to the classical case in which the
free enzymes, E
1
and E
2
, transform S
1

) implicates an FDS.
Definition of dimensionless quantities
For easier analysis, we treat our problem using di-
mensionless variables and parameters. If the concen-
tration of any substance, X, is written [X], a
dimensionless concentration, x, may be obtained by
normalizing [X] to the total concentration of enzymes
([E
1
]
t
+[E
2
]
t
),
x ¼½X=ð½E
1

t
þ½E
2

t
ÞðA1Þ
e.g.,
e
1t
¼½E
1

t
þ½E
2

t
Þ;
e
1
s
1
e
2
s
3
¼½E
1
S
1
E
2
S
3
=ð½E
1

t
þ½E
2

t

2
]
t
) for those
that are expressed in mol
)1
Æs
)1
Æm
3
, e.g.,
k
9f
¼ k
0
9f
=k
0
1r
; k
9r
¼ k
0
9r
=k
0
1r
; k
5r
¼ k

þ½E
2

t
ÞÁk
0
5f
=k
0
1r
; etc:
ðA5Þ
With these conventions, it should be noted that k
1r
is always expressed as
Table A1. The various reactions possibly taking place in the system
under study. Reactions R
1
to R
4
correspond to the formation of
enzyme–substrate complexes, reactions R
5
to R
8
and R
22
to R
25
correspond to the formation of the FDS, reactions R

2
within the FDS and reactions
R
14
to R
21
correspond to the fixation of a second substrate by the
FDS. For any of these reactions, j, k’
jf
is the rate constant of the
reaction written left to right and k’
jr
is the rate constant of the reac-
tion written right to left.
Reference number Reaction Rate constants
R
1
E
1
+S
1
¼ E
1
S
1
k¢
1f
,k¢
1r
R

2
+S
3
¼ E
2
S
3
k¢
4f
,k¢
4r
R
5
E
1
S
1
+E
2
¼ E
1
S
1
E
2
k¢
5f
,k¢
5r
R

2
k¢
7f
,k¢
7r
R
8
E
2
S
3
+E
1
¼ E
1
E
2
S
3
k¢
8f
,k¢
8r
R
9
E
1
S
1
¼ E

1
S
2
E
2
k¢
11f
,k¢
11r
R
12
E
1
S
2
E
2
¼ E
1
E
2
S
2
k¢
12f
,k¢
12r
R
13
E

2
S
2
k¢
14f
,k¢
14r
R
15
E
1
S
2
E
2
+S
2
¼ E
1
S
2
E
2
S
2
k¢
15f
,k¢
15r
R

1
¼ E
1
S
1
E
2
S
2
k¢
17f
,k¢
17r
R
18
E
1
E
2
S
2
+S
2
¼ E
1
S
2
E
2
S

1
E
2
S
3
+S
1
¼ E
1
S
1
E
2
S
3
k¢
20f
,k¢
20r
R
21
E
1
E
2
S
3
+S
2
¼ E

22f
,k¢
22r
R
23
E
1
S
1
+E
2
S
3
¼ E
1
S
1
E
2
S
3
k¢
23f
,k¢
23r
R
24
E
1
S

2
E
2
S
3
k¢
25f
,k¢
25r
R
26
E
1
S
1
E
2
S
2
¼ E
1
S
2
E
2
S
2
k¢
26f
,k¢

2
S
2
¼ E
1
S
2
E
2
S
3
k¢
28f
,k¢
28r
R
29
E
1
S
1
E
2
S
3
¼ E
1
S
2
E

1
,S
2
,S
3
,E
1
S
1
,E
1
S
2
,E
2
S
2
,E
2
S
3
,
E
1
S
1
E
2
,E
1

S
2
,
E
1
S
1
E
2
S
3
and E
1
S
2
E
2
S
3
). Assuming that external
mechanisms supply S
1
and remove S
3
as and when
they are consumed and produced, respectively, such
that S
1
is maintained at a constant concentration and
S

Á s
2
þ k
7r
Á e
1
e
2
s
2
À k
7f
Á e
1
Á e
2
s
2
þ k
8r
Á e
1
e
2
s
3
À k
8f
Á e
1

3
þ k
5r
Á e
1
s
1
e
2
À k
5f
Á e
2
Á e
1
s
1
þ k
6r
Á e
1
s
2
e
2
À k
6f
Á e
2
Á e

À k
14f
Á s
2
Á e
1
s
1
e
2
þ k
14r
Á e
1
s
1
e
2
s
2
À k
15f
Á s
2
Á e
1
s
2
e
2

À k
21f
Á s
2
Á e
1
e
2
s
3
þ k
21r
Á e
1
s
2
e
2
s
3
¼ 0 ðA9Þ
de
1
s
1
=ds ¼Àk
1r
Á e
1
s

9r
Á e
1
s
2
À k
22f
Á e
1
s
1
Á e
2
s
2
þ k
22r
Á e
1
s
1
e
2
s
2
À k
23f
Á e
1
s

s
2
À k
6f
Á e
2
Á e
1
s
2
þ k
6r
Á e
1
s
2
e
2
þ k
9f
Á e
1
s
1
À k
9r
Á e
1
s
2

þ k
25r
Á e
1
s
2
e
2
s
3
¼ 0 ðA11Þ
de
2
s
2
=ds ¼Àk
3r
Á e
2
s
2
þ k
3f
Á e
2
Á s
2
þ k
7r
Á e

1
Á e
2
s
2
þ k
22r
Á e
1
s
1
e
2
s
2
À k
24f
Á e
1
s
2
Á e
2
s
2
þ k
24r
Á e
1
s

þ k
8r
Á e
1
e
2
s
3
þ k
10f
Á e
2
s
2
k
10r
Á e
2
s
3
À k
23f
Á e
1
s
1
Á e
2
s
3

¼ 0 ðA13Þ
de
1
s
1
e
2
=ds ¼ k
5f
Á e
2
Á e
1
s
1
À k
5r
Á e
1
s
1
e
2
À k
11f
Á e
1
s
1
e

19f
Á s
3
Á e
1
s
1
e
2
þ k
19r
Á e
1
s
1
e
2
s
3
¼ 0 ðA14Þ
de
1
s
2
e
2
=ds ¼ k
6f
Á e
2

Á s
2
Á e
1
s
2
e
2
þ k
15r
Á e
1
s
2
e
2
s
2
À k
16f
Á s
3
Á e
1
s
2
e
2
þ k
16r

2
À k
13f
Á e
1
e
2
s
2
þ k
13r
Á e
1
e
2
s
3
À k
17f
Á s
1
Á e
1
e
2
s
2
þ k
17r
Á e

e
2
s
3
=ds ¼ k
8f
Á e
1
Á e
2
s
3
À k
8r
Á e
1
e
2
s
3
þ k
13f
Á e
1
e
2
s
2
À k
13r

Á e
1
e
2
s
3
þ k
21r
Á e
1
s
2
e
2
s
3
¼ 0 ðA17Þ
de
1
s
1
e
2
s
2
=ds ¼ k
14f
Á s
2
Á e

1
e
2
s
2
þ k
22f
Á e
1
s
1
Á e
2
s
2
À k
22r
Á e
1
s
1
e
2
s
2
À k
26f
Á e
1
s

1
e
2
s
3
¼ 0 ðA18Þ
de
1
s
1
e
2
s
3
=ds ¼ k
19f
Á s
3
Á e
1
s
1
e
2
À k
19r
Á e
1
s
1

Á e
2
s
3
À k
23r
Á e
1
s
1
e
2
s
3
þ k
27f
Á e
1
s
1
e
2
s
2
À k
27r
Á e
1
s
1

2
s
2
=ds ¼ k
15f
Á s
2
Á e
1
s
2
e
2
À k
15r
Á e
1
s
2
e
2
s
2
þ k
18f
Á s
2
Á e
1
e

2
s
2
þ k
26f
Á e
1
s
1
e
2
s
2
À k
26r
Á e
1
s
2
e
2
s
2
À k
28f
Á e
1
s
2
e

e
2
À k
16r
Á e
1
s
2
e
2
s
3
þ k
21f
Á s
2
Á e
1
e
2
s
3
À k
21r
Á e
1
s
2
e
2

s
2
À k
28r
Á e
1
s
2
e
2
s
3
þ k
29f
Á e
1
s
1
e
2
s
3
À k
29r
Á e
1
s
2
e
2

K
1
; K
2
; K
3
; K
5
; K
9
; K
10
; K
11
; K
12
; K
13
; K
15
; K
17
;
K
27
; K
29
and K ðA23Þ
as our base of independent equilibrium constants. In
this base, K is the equilibrium constant of the overall

Á k
4r
Á k
9f
Á k
10f
Þ=
ðk
1r
Á k
2f
Á k
3r
Á k
4f
Á k
9r
Á k
10r
Þ
¼ðK
1
Á K
3
Á K
9
Á K
10
Þ=ðK
2

,K
16
,K
18
,K
19
,K
20
,K
21
,K
22
,K
23
,K
24
,K
25
,K
26
and K
28
) can be calculated along independent reaction
circuits with a zero balance. For example, the reaction
circuit {5
f
-6
r
-9
r

9
ðA27Þ
and similarly with the circuits L
1
to L
15
, as indicated
in Table A2.
Again, using the maple software, the set of
Eqns (A7) to (A21) is solved, depending on the values
of the parameters of the problem (the 14 fixed equilib-
rium constants, one of the two rate constants, k
jf
or
k
jr
, of each reaction, R
j
, present in Table A1 and the
relative concentrations of the enzymes E
1
and E
2
), and
Eqn (A6) which must always be satisfied. For any
given value of the concentration, s
1
,ofS
1
, the absolute

À k
17r
Á e
1
s
1
e
2
s
2
þ k
17f
Á s
1
Á e
1
e
2
s
2
À k
20r
Á e
1
s
1
e
2
s
3

1
s
2
e
2
þ k
16r
Á e
1
s
2
e
2
s
3
À k
19f
Á s
3
Á e
1
s
1
e
2
þ k
19r
Á e
1
s

number Expression
L
1
5
f
-6
r
-9
r
-11
f
K
6
¼ (K
5
ÆK
11
) ⁄ K
9
L
2
3
f
-6
r
-15
r
-24
f
K

-7
r
-18
r
-24
f
K
7
¼ (K
2
ÆK
5
ÆK
11
ÆK
12
) ⁄ (K
3
ÆK
9
)
L
5
1
f
-7
r
-17
r
-22

K
26
¼ (K
1
ÆK
9
ÆK
15
) ⁄ (K
2
ÆK
12
ÆK
17
)
L
7
10
r
-22
f
-23
r
-27
f
K
23
¼ (K
2
ÆK

19
¼ (K
11
ÆK
12
ÆK
17
ÆK
27
) ⁄ K
L
9
11
r
-14
f
-15
r
-26
f
K
14
¼ (K
2
ÆK
11
ÆK
12
ÆK
17

29
) ⁄
(K
1
ÆK
3
Æ(K
9
)
2
ÆK
10
)
L
11
11
f
-16
f
-19
r
-29
r
K
16
¼ (K
12
ÆK
17
ÆK

ÆK
15
)
L
13
7
f
-8
r
-10
r
-13
f
K
8
¼ (K
2
ÆK
5
ÆK
11
ÆK
12
ÆK
13
) ⁄ (K
3
ÆK
9
ÆK

K
21
¼ (K
2
ÆK
17
ÆK
27
ÆK
29
) ⁄ (K
1
ÆK
9
ÆK
13
)
M. Thellier et al. Functioning-dependent structures
FEBS Journal 273 (2006) 4287–4299 ª 2006 The Authors Journal compilation ª 2006 FEBS 4299