Stoichiometricnetworktheoryfornonequilibrium biochemical
systems
Hong Qian
1,2
, Daniel A. Beard
2
and Shou-dan Liang
3
Departments of
1
Applied Mathematics and
2
Bioengineering, University of Washington, Seattle, USA;
3
NASA Ames Research Center, Moffett Field, CA, USA
We introduce the basic concepts and develop a theory for
nonequilibrium steady-state biochemicalsystems applicable
to analyzing large-scale complex isothermal reaction
networks. In terms of the stoichiometric matrix, we dem-
onstrate both Kirchhoff’s flux law R
‘
J
‘
¼ 0 over a bio-
chemical species, and potential law R
‘
l
‘
¼ 0overa
reaction loop. They reflect mass and energy conservation,
respectively. For each reaction, its steady-state flux J can
be decomposed into forward and backward one-way fluxes
J ¼ J
+
–J
–
, with chemical potential difference Dl ¼ RT
ln(J
–
/J
+
). The product –JDl gives the isothermal heat
dissipation rate, which is necessarily non-negative accord-
ing to the second law of thermodynamics. The stoichio-
metric networktheory (SNT) embodies all of the relevant
fundamental physics. Knowing J and Dl of a biochemical
reaction, a conductance can be computed which directly
reflects the level of gene expression for the particular
enzyme. For sufficiently small flux a linear relationship
between J and Dl can be established as the linear flux–
force relation in irreversible thermodynamics, analogous to
Ohm’s law in electrical circuits.
Keywords: biochemical network; chemical potential; flux;
nonequilibrium thermodynamics; steady-state.
With the completion of the Human Genome Project,
understanding of complex biochemicalsystems is entering a
new era that emphasizes engineering approaches and
quantitative analysis [1]. In order to develop a comprehen-
sive theoryforbiochemical networks, Palsson and col-
leagues have utilized flux balance analysis (FBA) which is
based on the fundamental law of mass conservation [2–4].
In terms of general network theory, flux balance is
Kirchhoff’s current law [5,6]. We recently augmented the
FBA with Kirchhoff’s loop law [7] for energy conservation
[8,9], as well as the second law of thermodynamics. An
analysis combining FBA and energy balance analysis of
Escherichia coli central metabolism has provided significant
insights into the regulation and control mechanism of the
biological system and improved the computational predic-
tions from FBA alone [7].
The objective of the present work is to provide the energy
balance analysis a sound basis in terms of biophysical
chemistry. The earlier work provides one simple and one
realistic example for the general theory developed in the
present paper. Establishing a rigorous thermodynamic
theory for ÔlivingÕ metabolic networks [10] challenges the
current theories of nonequilibrium steady-state systems. The
classic nonequilibrium physics, culminated in the work of
R. Kubo and L. Onsager, deals mainly with transient
processes and transport properties [11,12]. A living meta-
bolic network is sustained under a nonequilibrium steady
state [13,14] via active chemical pumping and heat dissipa-
tion; it is far from equilibrium [15]. The approach of
I. Prigogine and the Brussels group is formal and its
application to chemical reactions has been difficult and
controversial. Balancing chemical energy pumping and heat
dissipation in terms of the stoichiometry of a general
nonlinear reaction network is the focus of the present work.
In rigorous physical chemistry, one of us recently has
developed a nonequilibrium statistical theory which addres-
ses particularly the stochastics as well as thermodynamics of
isothermal nonequilibrium steady states [16–19]. In the
work of Katchalsky et al. [9], the thermodynamics of
biochemical networks have been explored. In particular,
Oster and Perelson [20] have developed an algebraic
topological theory of Kirchhoff’s laws in terms of chains,
cochains, boundary and coboundary operators. But in
terms of the stoichiometric matrix, no explicit, practically
useful formulae have been obtained. The objective of the
current work is to present the mathematical basis of energy
balance analysis for complex reaction networks with
nonlinear graphs. (A simple linear graph can be represented
sufficiently and necessarily by an incidence matrix,
nodes · edges, which has 0 and 1 as elements and exactly
two 1s per column. Nonlinear reaction networks cannot be
represented by a simple linear graph; rather they have to be
represented by nonlinear graphs, also known as hyper-
graphs.) The essential idea is to introduce the metabolite
chemical potentials into the biochemicalnetwork analysis.
While the theory of basic isothermal nonequilibrium
processes can be found in Hill 1974, and Qian 2001, 2002
[13,16,18], its application to metabolic networks, combining
with FBA, has led to a simple and powerful optimization
approach [7]. In addition, we also establish the connection
between the laws of physical chemistry and Kirchhoff’s laws
Correspondence to H. Qian, University of Washington, Seattle,
98195-2420, WA, USA.
E-mail: qian@amath.washington.edu
Abbreviations: SNT, stoichiometricnetwork theory; FBA, flux balance
analysis; cmf, chemical motive force; hdr, heat dissipation rate.
(Received 1 July 2002, revised 4 September 2002,
accepted 7 November 2002)
Eur. J. Biochem. 270, 415–421 (2003) Ó FEBS 2003 doi:10.1046/j.1432-1033.2003.03357.x
for circuits. Finally, the practical aspects of such analysis in
biochemistry is discussed.
The paper is organized as follows. Basic concepts pertinent
to the steady-state stoichiometricnetworktheory (SNT) such
as biochemical reaction flux, chemical potential, conduct-
ance, one-way fluxes and heat dissipation are introduced
using simple examples. In particular, we discuss nonlinear
relationships between reaction flux and chemical potential
and its linear approximation. The latter is analogous to the
Ohm’s law in electrical circuits and is widely known as the
linear flux-force relationship in irreversible thermodynamics.
The materials in this section are closely related to the work of
T. L. Hill [13,14] and Westerhoff and van Dam [10]. In the
following section the loop law is introduced via an example of
a simple nonlinear reaction. Next we present the general
proof of the loop law in terms of an arbitrary stoichiometric
matrix. Combining the flux and loop laws, as well as an
inequality for heat dissipation, the thermodynamically
feasible null space of a stoichiometric matrix can be
significantly restricted. In the case of no external flux
injection and concentration clamping, a zero null space
consistent with chemical equilibrium is uniquely determined
from the three constraints alone. The last section gives a
brief discussion of SNT and its future direction.
Basic concepts
Nonlinear flux-potential relationship
Most of the basic concepts used in the SNT can be found in
classic treaties [10,13,14]. For readers unfamiliar with the
work on nonequilibrium steady-state biochemical reactions,
we introduce some of the necessary essentials using simple
examples. We begin our analysis with the simplest possible
uni-molecular biochemical reaction with balanced input and
output steady-state fluxes J:
!
J
A
*
)
k
1
k
1
B
!
J
ð1Þ
Conceptually, the SNT is the generalization of this simple
example to complex nonlinear reaction networks in terms of
their stoichiometric matrices. The steady-state solution to
reaction (1) is:
c
A
¼
k
1
c
T
þ J
k
1
þ k
1
; c
B
¼
k
1
c
T
J
k
1
þ k
1
ð2Þ
which can be obtained from the kinetic equation in terms of
the law of mass action:
dc
A
dt
¼k
1
c
A
þ k
1
c
B
þ J;
dc
B
dt
¼ k
1
c
A
k
1
c
B
J
ð3Þ
c
A
+c
B
¼ c
T
is the total concentration for molecules in A
and B states. In steady state the total number of A and B
molecules is conserved and the fluxes are balanced.
Without flux J, the reaction in Eqn (1) approaches
chemical equilibrium with c
B
/c
A
¼ k
1
/k
)1
. We focus on the
energetics of reaction (1) in the nonequilibrium steady state
with a nonzero external flux J (also known as boundary
flux). In this case, the chemical potential difference between
states A and B is:
Dl ¼ Dl
o
AB
þ RT ln
c
B
c
A
¼RT ln
J
þ
J
ð4Þ
where T is the temperature, R is the gas constant, and Dl
o
AB
is
the reaction chemical potential in standard state. The flux is
decomposed into forward and backward components
J ¼ J
+
) J
–
where J
+
¼ k
1
c
A
, J
–
¼ k
)1
c
B
. Hence
J
J
þ
¼
k
1
c
B
k
1
c
A
which is known as the mass-action ratio.
RT ¼ 2.48 kJÆmol
)1
at room temperature of 298.16 K
and Dl have units of kJÆmol
)1
. Units for concentrations c,
flux J, and first-order rate constant k are
M
,
M
Æs
)1
,ands
)1
,
respectively. Substituting Eqn (3) into Eqn (4) gives:
Dl
AB
ðJÞ¼RT
ðk
1
c
T
þ JÞk
1
ðk
1
c
T
JÞk
1
;
J ¼
k
1
k
1
c
T
e
Dl
AB
=RT
1
k
1
þ k
1
e
Dl
AB
=RT
ð5Þ
We see that the Dl
AB
is a nonlinear function of J. However,
when J ¼ 0, Dl ¼ 0 and vice versa. More importantly,
–JDl is always non-negative and equals zero if and only if
J ¼ Dl ¼ 0. –JDl is the rate of heat dissipation of the
reaction in nonequilibrium steady state [13,18]. When
J ¼ 0, the reaction is at its thermodynamic chemical
equilibrium.
For small J we can take first-order Taylor expansion for
Eqn (5) at J ¼ 0 and obtain a linear relation
Dl
AB
ðJÞ¼
RT
c
T
1
k
1
þ
1
k
1
J; ð6Þ
which is analogous to Ohm’s law. The linearity is only valid
when J/c
T
k
1
,k
)1
. According to Onsager–Hill’s theory
on uni-molecular cycle kinetics [13,21], the linear ÔOhmÕs
resistance’ r
AB
¼ –Dl
AB
/J, is directly related to the equilib-
rium one-way flux in the absence of finite J.Thatis,inthe
absence of J, the equilibrium probabilities p
eq
A
and p
eq
B
of
a single molecule are in detailed balance, the one-way flux
J
þ
¼ k
1
p
eq
A
¼ J
¼ k
1
p
eq
B
(Onsager’s reciprocal relation),
and
r
AB
¼ RT
1
k
1
þ
1
k
1
¼
RT
p
eq
A
k
1
: ð7Þ
For biochemicalnetwork analysis, it is interesting and
important to note that the conductance, i.e. 1/r
AB
,is
linearly proportional to the concentration of the particular
enzyme which catalyzes the A « B reaction, i.e. k
1
and k
)1
are, at first-order approximation, proportional to the
expression level of the enzyme [E]. A gene regulation or
enzyme activation changes a particular network resistance
but does not directly effect the chemical potential difference
of the reaction per se!
It is also important to keep in mind that beyond the linear
regime, the biochemical resistance is not symmetric. In Eqn
(5) Dl
AB
(–J) „ –Dl
AB
(J). Such nonlinear behavior is
similar to that of diodes in electric circuits, which have
played a pivotal role in electronic circuit technology.
We now use a second example to demonstrate how the
linear result can be useful in analyzing networks with
balanced influx and efflux. We consider the more complex
situation of two reactions in series:
416 H. Qian et al. (Eur. J. Biochem. 270) Ó FEBS 2003
!
J
A
*
)
k
1
k
1
B
*
)
k
2
k
2
C
!
J
ð8Þ
We have in the steady-state:
c
A
¼
k
1
k
2
c
T
þðk
1
þ k
2
þ k
2
ÞJ
k
1
k
2
þ k
1
k
2
þ k
1
k
2
c
B
¼
k
1
k
2
c
T
þðk
1
k
2
ÞJ
k
1
k
2
þ k
1
k
2
þ k
1
k
2
c
C
¼
k
1
k
2
c
T
ðk
1
þ k
1
þ k
2
ÞJ
k
1
k
2
þ k
1
k
2
þ k
1
k
2
and in the linear regime:
Dl
AB
¼
RT
c
T
k
1
k
2
þ k
1
k
2
þ k
1
k
2
k
1
k
1
k
2
J
¼
RT
p
eq
A
k
1
J ¼r
AB
J ð9Þ
and
Dl
BC
¼
RT
c
T
k
1
k
2
þ k
1
k
2
þ k
1
k
2
k
1
k
2
k
2
J
¼
RT
p
eq
B
k
2
J ¼r
BC
J ð10Þ
in which the r
AB
and r
BC
are the linear resistances
introduced in Eqn (7), as expected. Therefore, in the linear
regime, the biochemical flux in a particular reaction and the
chemical potential difference across the reaction are linearly
related via the biochemical resistance.
Flux decomposition and biochemical heat dissipation
In FBA, the net flux in each reaction is determined based on
the Kirchhoff’s flux law as well as certain optimization
criterion [2]. While the net flux is extremely important for
mass conservation, it does not provide sufficient informa-
tion on biochemical energetics. For each reaction in a
biochemical network:
A
*
)
k
1
k
1
B
with nonequilibrium steady-state concentration c
A
and c
B
,
the heat dissipation rate (hdr) of this reaction is [13,17]
JDl ¼ RTðk
1
c
A
k
1
c
B
Þ ln
k
1
c
A
k
1
c
B
ð11Þ
in which the net flux J ¼ k
1
c
A
– k
)1
c
B
is the turnover per
unit time, and the Dl ¼ RT ln(k
)1
c
B
/k
1
c
A
)thechemical
potential change of turnover per mole. One can decompose
J into J
+
and J
–
,whichcanprovideinformationonits
energetics in biochemicalnetwork analysis [13,17]:
J ¼ J
þ
J
; Dl ¼RT ln
J
þ
J
;
hdr ¼ RTðJ
þ
J
Þ ln
J
þ
J
0
ð12Þ
The last inequality is the second law of thermodynamics:
One cannot derive useful work entirely from a single
temperature bath. By summing over all the reactions in a
biochemical network, this energy formula can be used to
compute the total heat dissipation rate [18].
In equilibrium thermodynamics, the chemical potential
Dl of a system is related to its partial molar enthalpy
h and entropy s: l ¼ h ) Ts. While the enthalpy
observes the law of conservation of energy, the l does
not [22]. In the equilibrium Dl is zero for each and every
reaction in the system. In a nonequilibrium steady-state,
the l, h,ands for each species do not change. However,
Dl for each reaction is no longer zero. These chemical
potential differences are maintained by an external
chemical pumping (in terms of the J in Eqn 1). The
work done by an external agent (e.g. a battery), which
equals precisely the chemical potential difference of the
reaction in a steady state, is the amount of heat
dissipated in the steady state. Therefore, the energy
conservation is between the chemical work done to a
system and the heat dissipated by the system [22]. The
amount of energy dissipated can be computed in terms
of the chemical potential differences in the system. We
emphasize the difference here between the equilibrium
Gibbs free energy of a reaction and the chemical
potential of its species in an nonequilibrium isothermal
steady-state.
External flux injection and internal flux distribution
When a throughput flux is injected into a biochemical
system, how is it distributed throughout the entire
network? What is the most probable pathway? These
problems are well understood in terms of linear network
theory, but require further analysis for nonlinear biochemi-
cal networks. In this section we discuss these basic
questions using simple examples and suggest that some
of the results from linear analysis can be applicable to
nonlinear systems. A comprehensive study of this subject
will be published elsewhere.
We start with the simple three-state kinetic cycle with
detailed balance:
(13)
in which a nonzero throughput flux J is introduced. In
steady-state:
c
A
¼
C
T
1 þ
k
1
k
1
þ
k
1
k
2
k
1
k
2
þ
k
2
þ k
2
þ k
3
D
J
c
B
¼
k
1
k
1
C
T
1 þ
k
1
k
1
þ
k
1
k
2
k
1
k
2
k
2
þ k
3
þ k
3
D
J
c
C
¼
k
1
k
2
k
1
k
2
C
T
1 þ
k
1
k
1
þ
k
1
k
2
k
1
k
2
k
2
k
3
D
J;
where
D ¼ k
1
k
3
þ k
3
k
1
þ k
1
k
3
þðk
1
þ k
3
þ k
3
Þk
2
þðk
1
þ k
1
þ k
3
Þk
2
:
Ó FEBS 2003 Nonequilibriumstoichiometricbiochemicalnetworktheory (Eur. J. Biochem. 270) 417
The fluxes are given by
J
AB
¼
k
1
ðk
2
þ k
2
þ k
3
Þþk
1
ðk
2
þ k
3
þ k
3
Þ
D
J
J
AC
¼ J
CB
¼
k
2
k
3
þ k
2
k
3
þ k
2
k
3
D
J:
When J ¼ 0, all fluxes are zero. The ratio J
AB
/J
AC
J
AB
J
AC
¼
k
1
ðk
2
þ k
2
þ k
3
Þþk
1
ðk
2
þ k
3
þ k
3
Þ
k
2
k
3
þ k
2
k
3
þ k
2
k
3
¼
k
1
k
2
þ k
1
k
3
k
2
k
3
¼
r
AC
þ r
CB
r
AB
again as expected from the Ohm’s law. This result is
surprising, since this equality is true even for large J.
Kirchhoff’s loop law for chemical potentials:
an example
We now introduce the loop law for chemical potentials.
This is parallel to the Kirchhoff’s voltage law for electri-
cal circuits. Note that the Kirchhoff’s voltage and current
laws are independent from the Ohm’s law which assumes
a linear relationship between the current and voltage.
Kirchhoff’s laws are much more fundamental than that
of Ohm’s.
The Kirchhoff’s voltage law in electrical circuit is due to
the fact that electrical energy has a potential function and
no curl. This is also the case for chemical reaction
network: for each species in the network, it has a uniquely
defined chemical potential. This is the origin of our loop
law. The loop in a network (a graph) is formally
equivalent to a closed curve in Euclid space. This
equivalence is best seen between a continuous physical
model and a lattice model. A graph is a generalization of
a high-dimensional irregular lattice. In fact, the boundary
flux and clamped concentration could be considered as
analogies to inhomogeneous Dirichlet and Newmann
boundary conditions, respectively.
To demonstrate the essential idea, we first use simple
cyclic chemical reactions as examples. Both unimolecular
and more importantly nonunimolecular reactions will be
considered. A general proof for arbitrary topology (stoichi-
ometric matrix) will be given later.
We first consider the simplest cyclic, unimolecular
reaction with three states:
(14)
When the system is closed and there is no external
flux injection or concentration clamping, the microscopic
reversibility dictates that:
k
1
k
2
k
3
k
1
k
2
k
3
¼ 1
known as the thermodynamic box in chemistry or detailed
balance in physics. The steady state is then a chemical
equilibrium with zero flux: k
1
c
A
) k
)1
c
B
¼ k
2
c
B
)
k
)2
c
C
¼ k
3
c
C
) k
)3
c
A
¼ 0. Therefore, Dl
AB
¼ Dl
BC
¼
Dl
CA
¼ 0, as expected for three resistors in a loop with
no battery (neither current source or voltage source).
Now let us assume that the system is open and the
reaction from B to C is really a second order with ÔchargingÕ:
B
*
)
k
0
2
½D
k
0
2
½E
C
in which the cofactors D and E have fixed concentrations,
[D]and[E]. (One can treat the k
o
2
[D]andk
o
2
[E]ask
2
and
k
)2
if the bimolecular reaction is rate limiting.) An example
of such a situation is a reaction accompanied by ATP
hydrolysis with D and E representing ATP and ADP,
respectively [23,24]. In this case k
o
2
and k
o
2
are second-order
rate constants. The equilibrium constant for the reaction
D $ E is
K
eq
¼
½E
eq
½D
eq
¼
k
o
2
½B
eq
k
o
2
½C
eq
¼
k
o
2
½B
eq
½A
eq
½A
eq
k
o
2
½C
eq
¼
k
1
k
o
2
k
3
k
1
k
o
2
k
3
When [D]and[E] are not at their equilibrium, the amount of
energy in this reaction with fixed concentrations, or
equivalently the amount of work needed to maintain the
concentrations, is
Dl
DE
¼RT ln
K
eq
½D
½E
Now again consider the cyclic reaction in Eqn (14), with
pseudo-first-order rate constants k
2
and k
)2
, we have:
Dl
AB
þ Dl
BC
þ Dl
CA
Dl
DE
¼ 0 ð15Þ
in which the first three terms are chemical potential
differences due to a nonzero flux J,andthelasttermis
the energy, or more precisely Ôchemical motive forceÕ (cmf),
of a biochemical battery. If the flux J is running from
A ! B ! C ! A then the first three Dl are negative. We
call Eqn (15) the law of energy balance. It is formally
analogous to Kirchhoff’s loop law. Multiplying J through-
out Eqn (15), we have energy conservation: chemical
work ¼ dissipated heat.
As a function of the driving force Dl
DE
, the steady-state
cycle flux in the cyclic reaction is [23,24]
J ¼
k
1
k
o
2
k
3
½E e
Dl
DE
=RT
1
k
1
k
3
þ k
3
k
1
þ k
1
k
3
þðk
1
þ k
3
þ k
3
Þk
o
2
½Dþðk
1
þ k
1
þ k
3
Þk
o
2
½E
: ð16Þ
418 H. Qian et al. (Eur. J. Biochem. 270) Ó FEBS 2003
This relationship is highly nonlinear. However for small
in which the r
AB
, r
BC
and r
CA
are the linear resistances, as in
Eqns (9) and (10). Eqn (17) observes the law of serial
resistors.
Energy balance analysis in terms of
stoichiometric matrix for nonlinear
reaction networks
Generalization of the above results on energy balance to
networks with arbitrary topology is not trivial. While it is
straightforward to identify reaction cycles in a system of
unimolecular reactions [13,14], it is not clear how to define
loops for networks of multispecies biochemical reactions.
Here we discuss the methodology for imposing energy
balance in complex biochemical networks [7].
Consider a system of N+N¢ metabolites X
i
(i ¼ 1,2, ,N are dynamic concentrations and
i ¼ N +1,N +2, ,N + N¢ are clamped concentra-
tions) with M+M¢ biochemical reactions (j ¼ 1,2, ,M
for internal reactions and j ¼ M +1,M +2, ,M + M¢
for external boundary fluxes). The jth internal reaction is
characterized by a set of stoichiometric coefficients m
j
i
and j
j
i
in the form
j
j
1
X
1
þ j
j
2
X
2
þj
j
NþN
0
X
NþN
0
Ð
k
j
þ
k
j
m
j
1
X
1
þ m
j
2
X
2
þm
j
NþN
0
X
NþN
0
ð18Þ
in which some of the integers m and j can be zero. If X
n
is an
enzyme for the reaction m,thenm
m
n
¼ j
m
n
¼ 1. More
complex Michaelis–Menten kinetics can also be expressed
in terms of Eqn (18).
The jth boundary flux is characterized by the same
Eqn (18) but all m and j are zero except one. A nonzero
m(j) corresponds to an influx (efflux).
The stoichiometry of this set of reactions can be
mathematically represented by the (N+N¢) · (M+M¢)
incidence matrix S ¼fj
j
i
m
j
i
g [5,6,25–27]:
S ¼
.
.
.
.
.
.
.
.
.
~
SS
NM
.
.
.
.
.
.
.
.
.
.
.
.
SS
NM
0
.
.
.
^
SS
N
0
M
0
j
0
B
B
B
B
B
B
@
1
C
C
C
C
C
C
A
The lower-right 0 block indicates that there should be no
boundary flux to or from a clamped species. Matrix S is the
starting point of the FBA (which also assumes no clamped
metabolites,
^
SS ¼ 0 [2,28]) as well as other modeling
approaches such as metabolic control analysis (MCA).
The matrix
~
SS contains only the dynamic species and internal
fluxes. The null space of the Nð
~
SSÞ, consists of all the
possible internal flux distributions which satisfy flux
balance. All internal fluxes are necessarily cycles [16,29],
although these cycles may not be intuitively obvious for
nonlinear reactions involving many species.
We denote the expression of jth reaction in Eqn (18) by
(R
j
). For each vector belonging to the cycle space
v ¼ (v
1
,v
2
, ,v
M
) in the null space Nð
~
SSÞ, the expression:
X
M
j¼1
m
j
ðR
j
Þð19Þ
is an overall reaction with the left and right sides identical:
X
M
j¼1
m
j
X
N
i¼1
j
j
i
X
i
X
M
j¼1
v
j
X
N
i¼1
m
j
i
X
i
¼
X
N
i¼1
X
i
X
M
j¼1
j
j
i
m
j
i
v
j
¼ 0: ð20Þ
The chemical potential difference of the jth reaction is
expressed as:
Dl
j
¼ Dl
o
j
þ RT ln
Q
NþN
0
i¼1
½X
i
j
j
i
Q
NþN
0
i¼1
½X
i
m
j
i
!
¼
X
N
i¼1
~
SS
ij
l
i
þ Dl
ext
j
;
ð21Þ
where l
i
¼ l
o
i
þ RT ln[X
i
] is the chemical potential of ith
species and
Dl
ext
j
¼
X
NþN
0
i¼Nþ1
^
SS
ij
l
i
is the cmf for the jth reaction. Combining Eqns (20) and (21)
and recalling
~
SS v ¼ 0, we reach the conclusion that:
X
j
v
j
ðDl
j
Dl
o
j
Þ¼0
if Dl
ext
¼ 0, i.e. when there is no externally clamped
concentration. Furthermore in equilibrium, according to the
thermodynamic box expression, for equilibrium constants,
it can be shown that
P
j¼1
m
j
Dl
o
j
¼ 0 (since for any steady-
state
P
j
m
j
ðDl
j
Dl
o
j
Þ¼0; it has to be valid for equilib-
rium in which all Dl
j
¼ 0)Hencewehave:
X
M
v
j
Dl
j
¼ 0
ð22Þ
for any v in the null space Nð
~
SSÞ.
In general, in addition to the external boundary fluxes
being held constant, a steady-state network can also have
chemical energy, i.e. cmf, supplied through clamped
concentrations of certain species. In this case, Dl
ext
„ 0
and Eqn (22) becomes vÆ(Dl
int
– Dl
ext
) ¼ 0overthe
reaction loop v (e.g. Dl
DE
in Eqn (15) is external).
The law of energy balance restricts the Nð
~
SSÞ to a smaller,
thermodynamically feasible subspace [7]. Now let
J ¼
1
k
1
þ
1
k
1
þ
k
2
k
1
k
2
þ
1
k
2
þ
1
k
2
þ
k
1
k
1
k
2
þ
1
k
3
þ
1
k
3
þ
k
2
k
2
k
3
1
Dl
DE
RT
¼ðr
AB
þ r
BC
þ r
CA
Þ
1
Dl
DE
ð17Þ
Ó FEBS 2003 Nonequilibriumstoichiometricbiochemicalnetworktheory (Eur. J. Biochem. 270) 419
v
1
, v
2
, , v
m
be the m linearly independent vectors of the
thermodynamically feasible null space, and define the
matrix K ¼ðv
T
1
; v
T
2
; ; v
T
m
Þ where v
T
denotes the transpose
of the row vector in a column form. Then we obtain an
novel algebraic structure for the biochemical network
theory:
~
SSK ¼ 0;
~
SSJ
int
¼ b
ext
; Dl
int
K ¼ p
ext
; ð23Þ
in which matrices
~
SS and K are known as incidence and loop
matrices in graph theory [5,30]. Vectors J
int
and Dl
int
are
internal fluxes and chemical potentials, both M dimensional.
b
ext
¼
SSJ
ext
is a N-dimensional external flux (typically with
many zero components) and J
ext
is M¢-dimensional.
p
ext
¼ l
ext
^
SSK is the external cmf on reaction loops, and
N¢-dimensional l
ext
is determined by the externally
clamped species. The second and third equations in (23) are
Kirchhoff’s flux law and potential law, respectively. Finally,
in steady-state, heat dissipation rate ¼J
int
j
Dl
int
j
0for
individual jth reaction and hdr ¼J
int
Dl
int
0foran
entire network, where the equals sign holds true if, and only
if, the reactions are in chemical equilibrium. This is the
second law of thermodynamics [18].
Eqn (23) indicates that, in a biochemicalnetwork analysis
that avoids detailed reaction rate constants, the steady-state
flux and potential are on an equal footing. In optimizing
fluxes, an idea originated by Palsson and his colleagues, one
should proceed both analyses in parallel and enforce the
second law. The classical FBA does not determine J
+
and
J
–
separately. Interesting biological questions concerning
the nature of biochemical control follow from the above
analysis: are metabolic networks sustained in nonequilibri-
um steady state by constant boundary flux, or by clamped
concentrations? We believe that providing answers to such
engineering questions will further deepen our understanding
of the regulation and control of metabolism and other
complex biochemical processes.
The practical value of the energy balance relation and non-
negative hdr is to further provide thermodynamic constraints
in the FBA with optimization. The introduction of chemical
potential significantly restricts the null space of
~
SS, Nð
~
SSÞ.In
the case when a network without any external flux and
clamped species:
~
SS J
int
¼ 0 ) l
~
SS J
int
¼ DlJ
int
¼ 0. Since
every term Dl
j
J
j
£ 0, one has Dl
j
J
j
¼ 0forallj. Hence
Dl ¼ J
int
¼ 0. Therefore, the only null space vector J that
satisfies flux balance, energy balance, and non-negative hdr is
zero, as expected for a chemical equilibrium.
Discussion
We have presented the concepts of SNT which serves the
foundation for analyzing nonequilibrium steady-state fluxes
and energetics in biochemical systems. Cornerstone con-
cepts of this theory are flux balance and energy balance, or
equivalently mass and energy conservation. While flux
balance can provide useful predictions of biochemical fluxes
[4] it alone cannot sufficiently restrict the solution space to
guarantee thermodynamically feasible fluxes [7]. Energy
balance introduces proper thermodynamics into network
analysis while simultaneously providing quantitative infor-
mation on control and regulation [7]. Theoretical tools such
as these, which are firmly rooted in rigorous biophysical
chemistry, are essential to the development of computa-
tional and bioinformatic protocols for analysis, simulation,
and design of complex biochemical systems.
The SNT developed in this paper provides a unique
conceptual framework fornetwork analysis of large-scale
metabolic reaction systems. We expect tools from electrical
circuit analysis and nonlinear graph theory will soon
significantly enhance the practical usefulness of this
approach. On the theoretical side, an integration of SNT
with existing theories on metabolic system analysis might
also be possible. We give several possible directions for the
future development of SNT.
Modular analysis of interactions between passive
and active subnetworks
Engineering analysis of large-scale complex systems requires
that one understands such systems in modular terms [31].
Toward this end, we define the basic concepts of passive and
active biochemical subnetworks and examine the conse-
quences of interactions between such subnetworks.
By a passive subnetwork, we mean the collection of the
reactions in the network that contain neither boundary fluxes
nor clamped concentrations. All the species in the passive
subnetwork are dynamic, and all the internal fluxes are
balanced. In this case, by a simple analogy with a subnetwork
of resistors, there should be no current loops in this
subnetwork. When all the connectivity between this subnet-
work and the remaining network is severed, the subnetwork
approaches an equilibrium with zero fluxes. Such a subnet-
work is a passive component in a nonequilibrium steady
state. In contrast, an active subnetwork has to involve either
fixed concentration or influx and efflux. Such a subnetwork is
an open system with energy utilization and heat dissipation.
When a passive subnetwork is coupled to an active one,
the fluxes pass through the former. A fundamental result
from Hill’s theory on cycle kinetics states that that there
should be no cycle flux in the passive subnetwork. Hence, an
active subnetwork cannot induce cycle flux in a passive one
[13,16]. A passive subnetwork can support only a transit flux
distribution.
SNT with Michaelis–Menten kinetics
In the present analysis, we have assumed that the metabolic
kinetics follow the law of mass action. In cells, metabolic
reactions that involve enzymes can all be represented by a
stoichiometric matrix. (Michaelis–Menten kinetics is an
approximated solution to the general enzyme kinetic models
basedonthelawofmassaction.)
SNT analysis which takes the enzymatic reaction into
specific consideration is currently in progress. However, it is
important to firmly establish the physiochemical foundation
of the SNT first based on the complete chemical kinetics.
Relation to MCA
MCA [32–36] is a systematic approach to measure, both
theoretically and experimentally, the control imposed by a
metabolic network upon a particular flux. This evaluation is
done in terms of the concept of flux control coefficients,
which are defined as the fractional change in a flux induced
420 H. Qian et al. (Eur. J. Biochem. 270) Ó FEBS 2003
by a fractional change in an enzyme activity. A second type
of quantities central to MCA is the elasticity coefficient, the
fractional response of the rate of a reaction to a fractional
change in concentration of a metabolite in steady state.
We are currently developing the relations between these
important quantities and SNT.
In closing, we shall comment on the two alternative but
complementary approaches to metabolic network mode-
ling. The traditional approach is based on rate constants
and kinetic equations. FBA with optimization, recently
introduced by Palsson and his colleagues and is now
integrated with energy balance in SNT, articulates an
optimization approach based on a (or several) biological
objective functions. The two approaches could be viewed
from a historical perspective as ÔNewtonianÕ and ÔLagran-
gianÕ, respectively [37]. The challenge for the reductionistic
former is to obtain the detailed information on rate
constants and kinetic equations, while for the integrative
latter it is to discover cellular principles in terms of
optimalities, if they indeed exist.
Acknowledgements
H. Q. thanks J. S. Oliveira for an enlightening conversation and V. Hsu
for many helpful discussions. This work is supported in part by
National Institutes of Health grants NCRR-1243 and NCRR-12609,
and National Aeronautics and Space Administration grant NCC2-
5463.
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Ó FEBS 2003 Nonequilibriumstoichiometricbiochemicalnetworktheory (Eur. J. Biochem. 270) 421
. Stoichiometric network theory for nonequilibrium biochemical
systems
Hong Qian
1,2
, Daniel A. Beard
2
and. a theory for
nonequilibrium steady-state biochemical systems applicable
to analyzing large-scale complex isothermal reaction
networks. In terms of the stoichiometric