Tài liệu Báo cáo khoa học: Steady-state kinetic behaviour of functioning-dependent structures docx

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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 lUniversite

de Rouen,
Mont-Saint-Aignan Cedex, France
2 Laboratoire de recherche en informatique, Universite

de Paris Sud, Orsay Cedex, France
3 Epigenomics Project, Genopoleề, Evry, France
Numerous studies have shown that proteins involved
in metabolic or signalling pathways are often distri-
buted nonrandomly, as multimolecular assemblies
[115]. Such assemblies range from quasi-static, multi-
enzyme complexes (such as the fatty acid synthase or
the a-oxo acid dehydrogenase systems [5]) to transient,
dynamic protein associations [2,3,7,15,16]. Comparison
of yeast and human multiprotein complexes has shown
that conservation across species extends from single
proteins to protein assemblies [11]. Multi-molecular
assemblies may comprise proteins but also nucleic
acids, lipids, small molecules and inorganic ions. Such
assemblies may interact with membranes, skeletal ele-
ments and or cell organelles [3,4,15,17]. They have
been termed metabolons, transducons and repairo-
somes in the case of metabolic pathways [3,10,1823],
signal transduction [24] and DNA repair [12], respect-
ively, or, more generally, hyperstructures [17,2528].
According to Srere [3], metabolons are enzyme
assemblies in which intermediates are channelled from
each enzyme to the next without diffusion of these
intermediates into the surrounding cytoplasm [2
7,9,15,23,2933]. Potential advantages of channelling
[7,9,15,30,31,34,35] are (i) reduction in the size of the
pools of intermediates (a point, however, contested by
some authors [36,37]), (ii) protection of unstable or
scarce intermediates by maintaining them in a protein-
bound state, (iii) avoidance of an underground meta-
bolism in which intermediates become the substrates of
other enzymes [38], and (iv) protection of the cytoplasm
from toxic or very reactive intermediates. The terms sta-
tic and dynamic channelling have been used to describe,
respectively, the channelling in a quasipermanent me-
tabolon and in a transient association between two
enzymes occurring while the intermediate metabolite is
transferred from the rst 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
lUniversite

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: Michel.Thellier@univ-rouen.fr
(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-
ate when no longer performing them. To investigate a role in coordination
for FDSs, we have studied numerically the steady-state kinetics of a model
system of two sequential monomeric enzymes, E
1
and E
2
. Our calculations
show that such FDSs can display kinetic properties that the individual
enzymes cannot. These include the full range of basic input output charac-
teristics found in electronic circuits such as linearity, invariance, pulsing
and switching. Hence, FDSs can generate kinetics that might regulate and
coordinate metabolism and signalling. Finally, we suggest that the occur-
rence of terms representative of the assembly and disassembly of FDSs in
the classical expression of the density of entropy production are character-
istic of living systems.
Abbreviation
FDS, functioning-dependent structure.
FEBS Journal 273 (2006) 42874299 ê 2006 The Authors Journal compilation ê 2006 FEBS 4287
We propose here to generalize the concept of dynamic
channelling or, more precisely, the concept of a struc-
ture that dynamically and transiently forms to carry out
a process, into that of functioning-dependent structure
(FDS) [41]. In other words, an FDS is a dynamic, multi-
molecular structure that assembles when functioning
and that disassembles when no longer functioning, and
thus is created and maintained by the very fact that it is
in the process of accomplishing a task. The lifetime of
such a structure may be short or long, depending only
on the duration of the process that is catalysed by the
FDS. An FDS catalyses efciently the processes that
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 enzymeproductenzyme
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 [4850]; 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
rst step of glycogenolysis in vertebrate muscle tissues
by the sequential formation of a phosphorylaseglyco-
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 [5255]; the
clustering of antigen receptors followed by binding of
intracellular proteins, such as protein tyrosine kinases,
to the cytoplasmic portion of the receptors in the case
of signalling through lymphocyte receptors (reviewed
in [56]); the organization of functional rafts in the
plasma membrane upon T-cell activation [57]; the gly-
cine decarboxylase complex in higher plants [58]; the
assembly of water-soluble, cytosolic proteins with the
membrane-anchored avocytochrome b
558
for the cata-
lysis of the NADPH-dependent reduction of O
2
into
the superoxide anion O
2

in stimulated phagocytic cells
[59]; the dynamic association of HSP90 with the
RPM1 disease resistance protein in the response of
Arabidopsis plants to infection by Pseudomonas syrin-
gae [60]; the association of protein complexes with
assembling actin molecules in the lamellipodium tip of
moving cells [61]; the clustering of glutamate receptors
opposite the largest and most physiologically active
sites of presynaptic release [62]; the differential nucleo-
tide-dependent binding of Bfp proteins in the transduc-
tion of mechanical energy to the biogenesis machine of
Escherichia coli [63]. Even the Golgi apparatus of Sac-
charomyces cerevisiae can be viewed as a dynamic
structure with a size that depends on its functioning
such that it grows when it is secreting and shrinks
when it is not [6467].
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 subunitsubunit
interactions are added to the classical enzymesub-
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 enzymeenzyme 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
MichaelisMenten type (i.e., with a single substrate-
binding site and no regulatory site). Our results show
that the metabolite-induced association of these two
enzymes into an FDS [20] may, under steady-state con-
ditions, confer to the FDS basic regulatory kinetic fea-
tures, that the individual enzymes lack. These include
the full range of input output characteristics found in
electronic circuits such as a linear relationship between
input and output, an output limited to a narrow range
of inputs, a constant output whatever the input, and
even switch-like behaviours (Fig. 1). Hence a metabo-
lite-induced FDS could generate a wide variety of kin-
etics that could serve as signals.
Modelling a two-enzyme FDS
The different substances and reactions that can possibly
take place when an FDS is involved in the overall trans-
Functioning-dependent structures M. Thellier et al.
4288 FEBS Journal 273 (2006) 42874299 ê 2006 The Authors Journal compilation ê 2006 FEBS
formation of an initial substrate, S
1
, into a nal prod-
uct, S
3
, via reactions catalysed by two enzymes, E
1
and
E
2
, are represented in Fig. 2. In total, 29 reactions act
on 17 substances (free substances and complexes) and,
to account for a formation of the FDS solely dependent
on its activity, the reaction E
1
+E
2
ẳ 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
write down the steady-state conditions of functioning of
the system (further details given in Appendix), (i) we
assume 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
3
at a zero concentration, and (ii)
we use the set of algebraic equations obtained by wri-
ting down the mass balance of the 15 other species
involved. For convenience, we have reasoned using di-
mensionless variables (note that capital letters are used
for chemical species and small letters for dimensionless
concentrations). We have also taken into account the
fact that the law of mass action has to be satised what-
ever the pathway from S
1
to S
3
. When all calculations
are carried out for any given value of the concentration,
s
1
,ofS
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
enzymeenzyme complexes [15]; the theoretical treat-
ment would be longer but otherwise essentially the
same as that followed here.
Results
Kinetics of the overall reaction of transformation
of S
1
into S
3
The system with only the free enzymes, E
1
and E
2
The overall rate of functioning of two free sequential
enzymes of the MichaelisMenten type involved in a
metabolic pathway has already been computed as a
function of the concentration of initial substrate under
output
input
output
input
output
input
output
input
AB
DC
(b)
(a)
Fig. 1. Classical input output relationships in electrical circuits. (A) Linear response: this behaviour is obtained when a generator is connec-
ted to a load (resistor). (B) Constant response: this behaviour is obtained when a source of current is connected to a load; whatever the
value of the load, and therefore whatever the value of the potential difference, the current is unchanged. (C) Impulse response: the output
is non-null only for a particular value (or a narrow range of values) of the input. (D) curve (a): Step response: this behaviour corresponds to a
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) 42874299 ê 2006 The Authors Journal compilation ê 2006 FEBS 4289
steady-state conditions [69]. The results are summarized
in Fig. 3A. Briey, curves monotonically increasing up
to a plateau and exhibiting no inexion 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 rst substrate) and
output (rate of production of nal 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
without an inexion point up to a plateau and some-
times exhibited an extended linear response with v(s
1
)
proportional to s
1
over a large range of s
1
values
(Fig. 3B, curves c and d). However, at other parameter
values, the {s
1
, v(s
1
)} curves exhibited a variety of
forms that were not found with the free enzymes. For
instance, in Fig. 3B, the curves (a) and (b) exhibited
substrate-inhibition behaviour, i.e., with increasing s
1
,
the rate of consumption of S
1
initially increased then,
after reaching a maximal value, decreased.
The occurrence of {s
1
, v(s
1
)} curves with a substrate-
inhibition shape was examined further (Fig. 4). At
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
the rate of consumption of S
1
reached a plateau (instead of decreasing to zero)
(Fig. 4C). Moreover, at the largest K
1
values (K
1
ẳ
10
4
), the rate of consumption of S
1
almost immediately
reached the plateau (Fig. 4C, curve d), which means
that the response of the system became effectively
independent of s
1
(except again at the very lowest s
1
values). This corresponds to the scenario in Fig. 1B in
which the output is independent of the input.
A curve is shown (Fig. 4D) that over a wide range
of low values of s
1
has a relatively constant and high
rate of consumption of S
1
but that with higher 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
) ẳ v
max
ặ(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-
tions numbered R
1
to R
29
as indicated in
the rectangles.
Functioning-dependent structures M. Thellier et al.
4290 FEBS Journal 273 (2006) 42874299 ê 2006 The Authors Journal compilation ê 2006 FEBS
coefcient, v
max
is the maximal rate of reaction and k
is the value of s
1
that gives v(s
1
) ẳ 0.5ặv
max
) was not
entirely satisfactory because a perfect straight line was
not obtained (r
2
ẳ 0.985) when using the Hill system
of coordinates, {log s
1
, log [v(s
1
) (v
max
v(s
1
))]}
(Fig. 5B); moreover, the sigmoidicity was rather weak
(Hill coefcient 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 gure
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
1
)} curves increased monotonically
to a plateau but with two inexion points that con-
ferred on them a dual-phasic aspect. Dual-phasic kin-
etic curves are often exhibited by both natural and
articial enzymatic and transport systems [7072];
although the functional advantage of such kinetics is
not clear, it is interesting that this complex behaviour
can be revealed by an FDS with as few as two
enzymes.
Discussion
The consequences of channelling on metabolism have
been extensively explored by modelling. In channelling,
the intermediate metabolites are conned to very small
volumes within a metabolon and have short half-lives.
It may therefore be invalid to assume that the local
statistical distribution of any molecule is Poissonian
and therefore that the classical macroscopic law of
kinetics can be used to describe the reaction rates
[29,7375]. Indeed, certain models based on this invalid
assumption may even lead to an apparent violation of
the second law of thermodynamics [73]. The model
developed here is based on the classical macroscopic
laws of kinetics but, importantly, is self-consistent in
the sense that it uses the same assumptions to deter-
mine and compare the kinetics of two enzymes freely
diffusing or assembled into a FDS.
Numerous command or control devices used in
engineering are made from elements with input out-
put functions as shown in Fig. 1. In electronics,
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
)
A
s
1
B
a
b
c
d
a
b
c
d
e
s
1
Fig. 3. Examples of computed {s
1
, v(s
1
)} curves. (A) Case of a system made of two free enzymes: the parameter values are e
1t
ẳ e
2t
ẳ 0.5,
K ẳ 100, k
1r
ẳ 1 (Eqn A6), k
2r
ẳ 100, k
3r
ẳ k
4r
ẳ k
9r
ẳ k
10r
ẳ 1, k
4f
calculated according to Eqn (A25), K
1
ẳ 10, K
3
ẳ 100, K
9
ẳ K
10
ẳ 1 and
K
2
ẳ 0.10 (curve a), 0.05 (curve b), 0.01 (curve c), 0.001 (curve d) and 0.0001 (curve e). Modied from [69]. (B) Case of a two-enzyme FDS:
the parameter values are e
1t
ẳ e
2t
ẳ 0.5, K ẳ 100, k
1r
ẳ 1 (Eqn A6), k
2r
ẳ 100, k
3r
ẳ k
4r
ẳ k
5r
ẳ k
6r
ẳ k
7r
ẳ k
8r
ẳ k
9r
ẳ k
10r
ẳ k
11r
ẳ
k
12r
ẳ k
13r
ẳ k
14r
ẳ k
15r
ẳ k
16r
ẳ k
17r
ẳ k
18r
ẳ k
19r
ẳ k
20r
ẳ k
21r
ẳ k
22r
ẳ k
23r
ẳ k
24r
ẳ k
25r
ẳ k
26r
ẳ k
27r
ẳ k
28r
ẳ k
29r
ẳ 1, K
1
ẳ 10, K
2
ẳ
0.01, K
5
ẳ 1000, K
3
ẳ K
10
ẳ K
11
ẳ K
12
ẳ K
13
ẳ K
15
ẳ K
17
ẳ K
29
ẳ 1, K
27
ẳ 100, K
9
ẳ 10 (curve a), 10
2
(curve b), 10
3
(curve c), 10
4
(curve
d) and all the other K
j
calculated as indicated in Eqns (A25) to (A27) and Table A2.
M. Thellier et al. Functioning-dependent structures
FEBS Journal 273 (2006) 42874299 ê 2006 The Authors Journal compilation ê 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 nal 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 nal 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 inection
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 efcient at a low substrate concentra-
tion but not at a high concentration where the sub-
strate would become available for enzymes in a
different metabolic pathway.
A strongly sigmoid curve from low to high rates of
production was not revealed by our calculations (see
above). Weakly sigmoid curves from low to high rates
0
0.04
0.08
40.020.00
s
1
C
a
b
c
d
s
1
0
0.02
0.04
1.050.00
B
0
0.01
0.02
1.050.00
s
1
D
0.0000
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
ẳ e
2t
ẳ 0.5, K ẳ 100, k
1r
ẳ 1 (Eqn A6), k
2r
ẳ 10
4
,k
3r
ẳ k
4r
ẳ
k
5r
ẳ k
6r
ẳ k
7r
ẳ k
8r
ẳ k
9r
ẳ k
10r
ẳ k
11r
ẳ k
12r
ẳ k
13r
ẳ k
14r
ẳ k
15r
ẳ k
16r
ẳ k
17r
ẳ k
18r
ẳ k
19r
ẳ k
20r
ẳ k
21r
ẳ k
22r
ẳ k
23r
ẳ k
24r
ẳ k
25r
ẳ k
26r
ẳ
k
27r
ẳ k
28r
ẳ k
29r
ẳ 1, K
1
ẳ 10, K
2
ẳ 0.0001, K
5
ẳ 10
6
,K
3
ẳ K
9
ẳ K
10
ẳ K
11
ẳ K
12
ẳ K
13
ẳ K
17
ẳ 1, K
15
ẳ K
27
ẳ 100, K
29
ẳ 1000
and all the other K
j
calculated as indicated in Eqns (A25) to (A27) and Table A2. (B) Another example of an impulse behaviour: the parameter
values are e
1t
ẳ e
2t
ẳ 0.5, K ẳ 1000, k
1r
ẳ 1 (Eqn A6), k
2r
ẳ 10
4
,k
3r
ẳ k
4r
ẳ k
5r
ẳ k
6r
ẳ k
7r
ẳ k
8r
ẳ k
9r
ẳ k
10r
ẳ k
11r
ẳ 1, k
12r
ẳ 10
3
,
k
13r
ẳ k
14r
ẳ k
15r
ẳ k
16r
ẳ k
17r
ẳ k
18r
ẳ k
19r
ẳ k
20r
ẳ k
21r
ẳ k
22r
ẳ k
23r
ẳ k
24r
ẳ k
25r
ẳ k
26r
ẳ k
27r
ẳ k
28r
ẳ 1, k
29r
ẳ 10
4
,K
1
ẳ 10, K
2
ẳ
0.0001, K
3
ẳ 1000, K
5
ẳ 10
6
,K
9
ẳ K
10
ẳ K
11
ẳ 1, K
12
ẳ 0.001, K
13
ẳ 100, K
15
ẳ 1000, K
17
ẳ 1, K
27
ẳ 100, K
29
ẳ 10000 and all the other
K
j
calculated as indicated in Eqns (A25) to (A27) and Table A2. (C) Examples of an only partial inhibition at high s
1
values: the parameter val-
ues are e
1t
ẳ e
2t
ẳ 0.5, K ẳ 100, k
1r
ẳ 1 (Eqn A6), k
2r
ẳ 100, k
3r
ẳ k
4r
ẳ k
5r
ẳ k
6r
ẳ k
7r
ẳ k
8r
ẳ k
9r
ẳ k
10r
ẳ k
11r
ẳ k
12r
ẳ k
13r
ẳ k
14r
ẳ
k
15r
ẳ k
16r
ẳ k
17r
ẳ k
18r
ẳ k
19r
ẳ k
20r
ẳ k
21r
ẳ k
22r
ẳ k
23r
ẳ k
24r
ẳ k
25r
ẳ k
26r
ẳ k
27r
ẳ k
28r
ẳ k
29r
ẳ 1, K
1
ẳ 10 (curve a), 10
2
(curve b), 10
3
(curve c) and 10
4
(curve d), K
2
ẳ 0.01, K
5
ẳ 10
3
,K
9
ẳ 10, K
3
ẳ K
10
ẳ K
11
ẳ K
12
ẳ K
13
ẳ K
15
ẳ K
17
ẳ K
29
ẳ 1, K
27
ẳ 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
ẳ
e
2t
ẳ 0.5, K ẳ 1000, k
1r
ẳ 1 (Eqn A6), k
2r
ẳ 10
4
,k
3r
ẳ k
4r
ẳ k
5r
ẳ k
6r
ẳ k
7r
ẳ k
8r
ẳ k
9r
ẳ k
10r
ẳ k
11r
ẳ 1, k
12r
ẳ 10
3
,k
13r
ẳ k
14r
ẳ k
15r
ẳ
k
16r
ẳ k
17r
ẳ k
18r
ẳ k
19r
ẳ k
20r
ẳ k
21r
ẳ k
22r
ẳ k
23r
ẳ k
24r
ẳ k
25r
ẳ k
26r
ẳ k
27r
ẳ k
28r
ẳ 1, k
29r
ẳ 10
4
,K
1
ẳ 10, K
2
ẳ 0.0001, K
3
ẳ 75, K
5
ẳ
10
6
,K
9
ẳ K
10
ẳ K
11
ẳ 1, K
12
ẳ 0.001, K
13
ẳ 100, K
15
ẳ 1000, K
17
ẳ 1, K
27
ẳ 100, K
29
ẳ 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) 42874299 ê 2006 The Authors Journal compilation ê 2006 FEBS
of production were sometimes observed with Hill coef-
cients 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 proteinprotein complex of serine acetyl trans-
ferase with O-acetylserine(thiol)-lyase strongly modies
the kinetic properties of the rst enzyme and results in
a transition from a typical MichaelisMenten beha-
viour to a behaviour displaying positive cooperativity
with respect to serine and acetyl-CoA with a Hill coef-
cient in the range of 1.32.0 [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
signalling assemble into FDSs in vivo when function-
ing. These FDSs may be connected to more permanent
structures such as membranes and the cytoskeleton.
They may even be connected to one another to form a
network integrating FDSs responsible for metabolism
and for signal transduction [11,76]. Such a vision of
intracellular organization is supported by many studies
showing the recruitment of proteins into functional
structures (reviewed in [35]) and the coordination of
multiple functions via the formation of networks of
signalling complexes [11,16,7779]. More than 50 dif-
ferent types of protein assemblies, containing up to 35
proteins, have been identied in functions that include
transcription regulation, cell-cycle cell-fate control,
RNA processing, and protein transport [13]. It could
be argued that the concept of FDS should not be lim-
ited to the intracellular level. Indeed, a concept similar
to that of the FDS has been employed at the multi-
cellular level to explain how neurones participate in
0
0.04
0.08
2.01.00
A
log z
1
-3
-2
-1
0
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
ẳ e
2t
ẳ 0.5,
K ẳ 100, k
1r
ẳ 1 (Eqn A6), k
2r
ẳ 10, k
3r
ẳ k
4r
ẳ k
5r
ẳ k
6r
ẳ k
7r
ẳ k
8r
ẳ k
9r
ẳ k
10r
ẳ k
11r
ẳ k
12r
ẳ k
13r
ẳ k
14r
ẳ k
15r
ẳ k
16r
ẳ k
17r
ẳ k
18r
ẳ
k
19r
ẳ k
20r
ẳ k
21r
ẳ k
22r
ẳ k
23r
ẳ k
24r
ẳ k
25r
ẳ k
26r
ẳ k
27r
ẳ k
28r
ẳ k
29r
ẳ 1, K
1
ẳ K
2
ẳ 0.1, K
3
ẳ 10, K
5
ẳ 1000, K
9
ẳ K
10
ẳ K
11
ẳ K
12
ẳ
K
13
ẳ K
15
ẳ K
17
ẳ K
29
ẳ 1, K
27
ẳ 100 and all the other K
j
calculated as indicated in Eqns (A25) to (A27) and Table A2. (A) Curve represented
using the direct system of coordinates, {s
1
, v(s
1
)}. (B) Curve represented using the Hill system of coordinates, {log s
1
, log z
1
} with
z
1
ẳ {v(s
1
) [v
max
v(s
1
)]}; from the slope of the dashed regression line tted to the curve, the Hill coefcient 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
ẳ e
2t
ẳ
0.5, K ẳ 100, k
1r
ẳ 1 (Eqn A6), k
2r
ẳ 100, k
3r
ẳ k
4r
ẳ k
5r
ẳ k
6r
ẳ
k
7r
ẳ k
8r
ẳ k
9r
ẳ k
10r
ẳ k
11r
ẳ k
12r
ẳ k
13r
ẳ k
14r
ẳ k
15r
ẳ k
16r
ẳ k
17r
ẳ
k
18r
ẳ k
19r
ẳ k
20r
ẳ k
21r
ẳ k
22r
ẳ k
23r
ẳ k
24r
ẳ k
25r
ẳ k
26r
ẳ k
27r
ẳ
k
28r
ẳ k
29r
ẳ 1, K
1
ẳ K
9
ẳ 10, K
2
ẳ 0.01, K
3
ẳ K
11
ẳ K
12
ẳ K
13
ẳ
K
15
ẳ K
17
ẳ K
29
ẳ 1, K
5
ẳ 1000, K
27
ẳ 100, K
10
ẳ 1 (curve a), 10
(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) 42874299 ê 2006 The Authors Journal compilation ê 2006 FEBS 4293
different assemblies at different times depending on the
task to be carried out [80].
Biochemists are familiar with the Structure 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
ux densities of the processes and their conjugated
driving forces [81]. Whichever reaction pathway in our
system is chosen to connect S
1
and S
3
(Fig. 2), under
steady-state conditions the only molecules that undergo
transformation are S
1
and S
3
while the other molecules
remain unchanged. Hence, the corresponding density of
entropy production, r, is that of the overall reaction of
transformation of S
1
into S
3
, and r does not depend
on whether the system is catalysed via free enzymes or
an FDS. Out of steady state, however, the situation is
different because the free enzymes, E
1
and E
2
, can act
immediately on their substrates whereas the FDS
enzymes must assemble into an FDS before they can
act. Consequently, if r is expressed in the standard
way, terms representing the entropic cost of FDS
assembly disassembly are present only in the descrip-
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, kÂ
jf
and kÂ
jr
, of the for-
ward and reverse reactions are also indicated in the
table. Note that, depending on the molecularity of the
terms in the left-hand side of the reactions, kÂ
1f
to kÂ
8f
and kÂ
14f
to kÂ
25f
are expressed in mol
)1
ặs
)1
ặm
3
, while
the other rate constants (kÂ
9f
to kÂ
13f
,kÂ
26f
to kÂ
29f
and
all the kÂ
jr
) are expressed in s
)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
into S
3
via the
liberation of S
2
by E
1
, the diffusion of S
2
in the reac-
tion medium and the recapture of S
2
by E
2
. Any other
chain of reactions equivalent to S
1
S
3
(e.g. 12
f
-17
f
-
26
f
-28
f
-16
r
) implicates an FDS.
Denition 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 ẳẵX=ẵE
1

t
ỵẵE
2

t
ịA1ị
e.g.,
e
1t
ẳẵE
1

t
=ẵE
1

t
ỵẵE
2

t
ị;e
1
ẳẵE
1
=ẵE
1

t
ỵẵE
2

t
ị;
e
2
ẳẵE
2
=ẵE
1

t
ỵẵE
2

t
ị;
e
1
s
1
e
2
s
3
ẳẵE
1
S
1
E
2
S
3
=ẵE
1

t
ỵẵE
2

t
ị;etc:
A2ị
Because kÂ
1r
is expressed in s
)1
, a dimensionless
expression, s, of the time, t, may be written as
s ẳ k
0
1r
t A3ị
Similarly, the dimensionless expression, k
j
, of the
rate constants, kÂ
j
, will be obtained by normalization
to kÂ
1r
for the rate constants that are expressed in s
)1
,
and by normalization to kÂ
1r
([E
1
]
t
+[E
2
]
t
) for those
that are expressed in mol
)1
ặs
)1
ặm
3
, e.g.,
k
9f
ẳ k
0
9f
=k
0
1r
; k
9r
ẳ k
0
9r
=k
0
1r
; k
5r
ẳ k
0
5r
=k
0
1r
; etc: A4ị
and
k
1f
ẳẵE
1

t
ỵẵE
2

t
ịk
0
1f
=k
0
lr
;
k
5f
ẳẵE
1

t
ỵẵE
2

t
ịk
0
5f
=k
0
1r
; etc:
A5ị
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
enzymesubstrate complexes, reactions R
5
to R
8
and R
22
to R
25
correspond to the formation of the FDS, reactions R
9
to R
11
,R
13
and R
26
to R
29
correspond to the transformation of S
1
into S
2
by
enzyme E
1
,orS
2
into S
3
by enzyme E
2
, reaction R
12
corresponds
to the channelling of S
2
from E
1
to E
2
within the FDS and reactions
R
14
to R
21
correspond to the xation 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
ẳ E
1
S
1
kÂ
1f
,kÂ
1r
R
2
E
1
+S
2
ẳ E
1
S
2
kÂ
2f
,kÂ
2r
R
3
E
2
+S
2
ẳ E
2
S
2
kÂ
3f
,kÂ
3r
R
4
E
2
+S
3
ẳ E
2
S
3
kÂ
4f
,kÂ
4r
R
5
E
1
S
1
+E
2
ẳ E
1
S
1
E
2
kÂ
5f
,kÂ
5r
R
6
E
1
S
2
+E
2
ẳ E
1
S
2
E
2
kÂ
6f
,kÂ
6r
R
7
E
2
S
2
+E
1
ẳ E
1
E
2
S
2
kÂ
7f
,kÂ
7r
R
8
E
2
S
3
+E
1
ẳ E
1
E
2
S
3
kÂ
8f
,kÂ
8r
R
9
E
1
S
1
ẳ E
1
S
2
kÂ
9f
,kÂ
9r
R
10
E
2
S
2
ẳ E
2
S
3
kÂ
10f
,kÂ
10r
R
11
E
1
S
1
E
2
ẳ E
1
S
2
E
2
kÂ
11f
,kÂ
11r
R
12
E
1
S
2
E
2
ẳ E
1
E
2
S
2
kÂ
12f
,kÂ
12r
R
13
E
1
E
2
S
2
ẳ E
1
E
2
S
3
kÂ
13f
,kÂ
13r
R
14
E
1
S
1
E
2
+S
2
ẳ E
1
S
1
E
2
S
2
kÂ
14f
,kÂ
14r
R
15
E
1
S
2
E
2
+S
2
ẳ E
1
S
2
E
2
S
2
kÂ
15f
,kÂ
15r
R
16
E
1
S
2
E
2
+S
3
ẳ E
1
S
2
E
2
S
3
kÂ
16f
,kÂ
16r
R
17
E
1
E
2
S
2
+S
1
ẳ E
1
S
1
E
2
S
2
kÂ
17f
,kÂ
17r
R
18
E
1
E
2
S
2
+S
2
ẳ E
1
S
2
E
2
S
2
kÂ
18f
,kÂ
18r
R
19
E
1
S
1
E
2
+S
3
ẳ E
1
S
1
E
2
S
3
kÂ
19f
,kÂ
19r
R
20
E
1
E
2
S
3
+S
1
ẳ E
1
S
1
E
2
S
3
kÂ
20f
,kÂ
20r
R
21
E
1
E
2
S
3
+S
2
ẳ E
1
S
2
E
2
S
3
kÂ
21f
,kÂ
21r
R
22
E
1
S
1
+E
2
S
2
ẳ E
1
S
1
E
2
S
2
kÂ
22f
,kÂ
22r
R
23
E
1
S
1
+E
2
S
3
ẳ E
1
S
1
E
2
S
3
kÂ
23f
,kÂ
23r
R
24
E
1
S
2
+E
2
S
2
ẳ E
1
S
2
E
2
S
2
kÂ
24f
,kÂ
24r
R
25
E
1
S
2
+E
2
S
3
ẳ E
1
S
2
E
2
S
3
kÂ
25f
,kÂ
25r
R
26
E
1
S
1
E
2
S
2
ẳ E
1
S
2
E
2
S
2
kÂ
26f
,kÂ
26r
R
27
E
1
S
1
E
2
S
2
ẳ E
1
S
1
E
2
S
3
kÂ
27f
,kÂ
27r
R
28
E
1
S
2
E
2
S
2
ẳ E
1
S
2
E
2
S
3
kÂ
28f
,kÂ
28r
R
29
E
1
S
1
E
2
S
3
ẳ E
1
S
2
E
2
S
3
kÂ
29f
,kÂ
29r
M. Thellier et al. Functioning-dependent structures
FEBS Journal 273 (2006) 42874299 ê 2006 The Authors Journal compilation ê 2006 FEBS 4297
k
1r
ẳ k
0
1r
=k
0
1r
ẳ 1 A6ị
The basic equations of the steady-state problem
It is apparent in Table A1 that the 29 reactions under
consideration (R
1
to R
29
) involve 17 different chem-
ical species (E
1
,E
2
,S
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
2
,E
1
E
2
S
2
,E
1
E
2
S
3
,E
1
S
1
E
2
S
2
,E
1
S
2
E
2
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
3
at a zero concentration, the steady-state condition
of functioning of the system is obtained by writing
down the mass balance of the 15 other species
involved. Using the dimensionless quantities, this is
written
de
1
=ds ẳ k
1r
e
1
s
1
k
1f
e
1
s
1
ỵ k
2r
e
1
s
2
k
2f
e
1
s
2
ỵ k
7r
e
1
e
2
s
2
k
7f
e
1
e
2
s
2
ỵ k
8r
e
1
e
2
s
3
k
8f
e
1
e
2
s
3
ẳ 0 A7ị
de
2
=ds ẳ k
3r
e
2
s
2
k
3f
e
2
s
2
ỵ k
4r
e
2
s
3
k
4f
e
2
s
3
ỵ k
5r
e
1
s
1
e
2
k
5f
e
2
e
1
s
1
ỵ k
6r
e
1
s
2
e
2
k
6f
e
2
e
1
s
2
ẳ 0 A8ị
ds
2
=ds ẳk
2f
e
1
s
2
ỵ k
2r
e
1
s
2
k
3f
e
2
s
2
ỵ k
3r
e
2
s
2
k
14f
s
2
e
1
s
1
e
2
ỵ k
14r
e
1
s
1
e
2
s
2
k
15f
s
2
e
1
s
2
e
2
ỵ k
15r
e
1
s
2
e
2
s
2
k
18f
s
2
e
1
e
2
s
2
ỵ k
18r
e
1
s
2
e
2
s
2
k
21f
s
2
e
1
e
2
s
3
ỵ k
21r
e
1
s
2
e
2
s
3
ẳ 0 A9ị
de
1
s
1
=ds ẳk
1r
e
1
s
1
ỵ k
1f
e
1
s
1
ỵ k
5r
e
1
s
1
e
2
k
5f
e
2
e
1
s
1
k
9f
e
1
s
1
ỵ k
9r
e
1
s
2
k
22f
e
1
s
1
e
2
s
2
ỵ k
22r
e
1
s
1
e
2
s
2
k
23f
e
1
s
1
e
2
s
3
ỵ k
23r
e
1
s
1
e
2
s
3
ẳ 0 A10ị
de
1
s
2
=ds ẳk
2f
e
1
s
2
k
2r
e
1
s
2
k
6f
e
2
e
1
s
2
ỵ k
6r
e
1
s
2
e
2
ỵ k
9f
e
1
s
1
k
9r
e
1
s
2
k
24f
e
1
s
2
e
2
s
2
ỵ k
24r
e
1
s
2
e
2
s
2
k
25f
e
1
s
2
e
2
s
3
ỵ k
25r
e
1
s
2
e
2
s
3
ẳ 0 A11ị
de
2
s
2
=ds ẳk
3r
e
2
s
2
ỵ k
3f
e
2
s
2
ỵ k
7r
e
1
e
2
s
2
k
7f
e
1
e
2
s
2
k
10f
e
2
s
2
ỵ k
10r
e
2
s
3
k
22f
e
1
s
1
e
2
s
2
ỵ k
22r
e
1
s
1
e
2
s
2
k
24f
e
1
s
2
e
2
s
2
ỵ k
24r
e
1
s
2
e
2
s
2
ẳ 0 A12ị
de
2
s
3
=ds ẳ k
4f
e
2
s
3
k
4r
e
2
s
3
k
8f
e
1
e
2
s
3
ỵ k
8r
e
1
e
2
s
3
ỵ k
10f
e
2
s
2
k
10r
e
2
s
3
k
23f
e
1
s
1
e
2
s
3
ỵ k
23r
e
1
s
1
e
2
s
3
k
25f
e
1
s
2
e
2
s
3
ỵ k
25r
e
1
s
2
e
2
s
3
ẳ 0 A13ị
de
1
s
1
e
2
=ds ẳ k
5f
e
2
e
1
s
1
k
5r
e
1
s
1
e
2
k
11f
e
1
s
1
e
2
ỵ k
11r
e
1
s
2
e
2
k
14f
s
2
e
1
s
1
e
2
ỵ k
14r
e
1
s
1
e
2
s
2
k
19f
s
3
e
1
s
1
e
2
ỵ k
19r
e
1
s
1
e
2
s
3
ẳ 0 A14ị
de
1
s
2
e
2
=ds ẳ k
6f
e
2
e
1
s
2
k
6r
e
1
s
2
e
2
k
12f
e
1
s
2
e
2
ỵ k
12r
e
1
e
2
s
2
k
15f
s
2
e
1
s
2
e
2
ỵ k
15r
e
1
s
2
e
2
s
2
k
16f
s
3
e
1
s
2
e
2
ỵ k
16r
e
1
s
2
e
2
s
3
ẳ 0 A15ị
de
1
e
2
s
2
=ds ẳ k
7f
e
1
e
2
s
2
k
7r
e
1
e
2
s
2
k
13f
e
1
e
2
s
2
ỵ k
13r
e
1
e
2
s
3
k
17f
s
1
e
1
e
2
s
2
ỵ k
17r
e
1
s
1
e
2
s
2
k
18f
s
2
e
1
e
2
s
2
ỵ k
18r
e
1
s
2
e
2
s
2
ẳ 0 A16ị
de
1
e
2
s
3
=ds ẳ k
8f
e
1
e
2
s
3
k
8r
e
1
e
2
s
3
ỵ k
13f
e
1
e
2
s
2
k
13r
e
1
e
2
s
3
k
20f
s
1
e
1
e
2
s
3
ỵ k
20r
e
1
s
1
e
2
s
3
k
21f
s
2
e
1
e
2
s
3
ỵ k
21r
e
1
s
2
e
2
s
3
ẳ 0 A17ị
de
1
s
1
e
2
s
2
=ds ẳ k
14f
s
2
e
1
s
1
e
2
k
14r
e
1
s
1
e
2
s
2
ỵ k
17f
s
1
e
1
e
2
s
2
k
17r
e
1
s
1
e
2
s
2
ỵ k
22f
e
1
s
1
e
2
s
2
k
22r
e
1
s
1
e
2
s
2
k
26f
e
1
s
1
e
2
s
2
ỵ k
26r
e
1
s
2
e
2
s
2
k
27f
e
1
s
1
e
2
s
2
ỵ k
27r
e
1
s
1
e
2
s
3
ẳ 0 A18ị
de
1
s
1
e
2
s
3
=ds ẳ k
19f
s
3
e
1
s
1
e
2
k
19r
e
1
s
1
e
2
s
3
ỵ k
20f
s
1
e
1
e
2
s
3
k
20r
e
1
s
1
e
2
s
3
ỵ k
23f
e
1
s
1
e
2
s
3
k
23r
e
1
s
1
e
2
s
3
ỵ k
27f
e
1
s
1
e
2
s
2
k
27r
e
1
s
1
e
2
s
3
k
29f
e
1
s
1
e
2
s
3
ỵ k
29r
e
1
s
2
e
2
s
3
ẳ 0 A19ị
de
1
s
2
e
2
s
2
=ds ẳ k
15f
s
2
e
1
s
2
e
2
k
15r
e
1
s
2
e
2
s
2
ỵ k
18f
s
2
e
1
e
2
s
2
k
18r
e
1
s
2
e
2
s
2
ỵ k
24f
e
1
s
2
e
2
s
2
k
24r
e
1
s
2
e
2
s
2
ỵ k
26f
e
1
s
1
e
2
s
2
k
26r
e
1
s
2
e
2
s
2
k
28f
e
1
s
2
e
2
s
2
ỵ k
28r
e
1
s
2
e
2
s
3
ẳ 0 A20ị
de
1
s
2
e
2
s
3
=ds ẳ k
16f
s
3
e
1
s
2
e
2
k
16r
e
1
s
2
e
2
s
3
ỵ k
21f
s
2
e
1
e
2
s
3
k
21r
e
1
s
2
e
2
s
3
ỵ k
25f
e
1
s
2
e
2
s
3
k
25r
e
1
s
2
e
2
s
3
ỵ k
28f
e
1
s
2
e
2
s
2
k
28r
e
1
s
2
e
2
s
3
ỵ k
29f
e
1
s
1
e
2
s
3
k
29r
e
1
s
2
e
2
s
3
ẳ 0 A21ị
Now, each of the 29 reactions, R
j
, in Table A1 has
an equilibrium constant, K
j
, equal to the ratio of its
forward to its reverse rate constant
K
j
ẳ k
jf
=k
jr
A22ị
Using the maple software (Maplesoft Europe, Zug,
Switzerland), the rank of the 29 ã 17 matrix of the
Functioning-dependent structures M. Thellier et al.
4298 FEBS Journal 273 (2006) 42874299 ê 2006 The Authors Journal compilation ê 2006 FEBS