Beard2and Shou-dan Liang3 Departments of1Applied Mathematics and2Bioengineering, University of Washington, Seattle, USA; 3 NASA Ames Research Center, Moffett Field, CA, USA We introduce t
Trang 1Stoichiometric network theory for nonequilibrium biochemical
systems
Hong Qian1,2, Daniel A Beard2and Shou-dan Liang3
Departments of1Applied Mathematics and2Bioengineering, 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 biochemical systems 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 0 over a
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
J¼ J+– J–, with chemical potential difference Dl¼ 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 network theory (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 biochemical systems is entering a
new era that emphasizes engineering approaches and
quantitative analysis [1] In order to develop a
comprehen-sive theory for biochemical 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 colicentral 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 living 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 biochemical network 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, stoichiometric network 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)
Trang 2for 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 stoichiometric network theory (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 *)k1
k 1
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:
cA¼k1cTþ J
k1þ k1
; cB¼k1cT J
k1þ k1
ð2Þ which can be obtained from the kinetic equation in terms of
the law of mass action:
dcA
dt ¼ k1cAþ k1cBþ J; dcB
dt ¼ k1cA k1cB J
ð3Þ
cA+ cB¼ cTis 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 cB/cA¼ k1/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:
Dl¼ Dlo
ABþ RT ln cB
cA
¼ RT ln Jþ
J
ð4Þ where T is the temperature, R is the gas constant, and Dlo
ABis the reaction chemical potential in standard state The flux is decomposed into forward and backward components
J¼ J+) J– where J+¼ k1cA, J–¼ k)1cB Hence
J
J þ¼k1 c B
k1cA which is known as the mass-action ratio
RT¼ 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 areM,MÆs)1, and s)1, respectively Substituting Eqn (3) into Eqn (4) gives:
DlABðJÞ ¼ RT ðk1cTþ JÞk1
ðk1cT JÞk1
;
J¼k1k1cT e
Dl AB =RT 1
k1þ k1eDl AB =RT
ð5Þ
We see that the DlABis a nonlinear function of J However, when J¼ 0, Dl ¼ 0 and vice versa More importantly, –JDl is always non-negative and equals zero if and only if
J¼ Dl ¼ 0 –JDl is the rate of heat dissipation of the reaction in nonequilibrium steady state [13,18] When
J¼ 0, the reaction is at its thermodynamic chemical equilibrium
For small J we can take first-order Taylor expansion for Eqn (5) at J¼ 0 and obtain a linear relation
DlABðJÞ ¼ RT
cT
1
k1þ 1
k1
which is analogous to Ohm’s law The linearity is only valid when J/cT k1, k)1 According to Onsager–Hill’s theory
on uni-molecular cycle kinetics [13,21], the linear Ohms resistance’ rAB¼ –DlAB/J, is directly related to the equilib-rium one-way flux in the absence of finite J That is, in the absence of J, the equilibrium probabilities peqAand peqB of
a single molecule are in detailed balance, the one-way flux
Jþ¼ k1peqA ¼ J¼ k1peqB (Onsager’s reciprocal relation), and
rAB¼ RT 1
k1
k1
peqAk1
For biochemical network analysis, it is interesting and important to note that the conductance, i.e 1/rAB, is linearly proportional to the concentration of the particular enzyme which catalyzes the A« B reaction, i.e k1and 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) DlAB (–J) „ –DlAB (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:
Trang 3!J A *)k1
k 1
B *)k2
k 2
We have in the steady-state:
cA ¼ k1k2cTþ ðk1þ k2þ k2ÞJ
k1k2þ k1k2þ k1k2
cB ¼ k1k2cTþ ðk1 k2ÞJ
k1k2þ k1k2þ k1k2
cC ¼ k1k2cT ðk1þ k1þ k2ÞJ
k1k2þ k1k2þ k1k2 and in the linear regime:
DlAB¼ RT
cT
k1k2þ k1k2þ k1k2
k1k1k2
J
peqAk1
and
DlBC¼ RT
cT
k1k2þ k1k2þ k1k2
k1k2k2
J
peqBk2
in which the rAB and rBC 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 reacinforma-tion in a
biochemical network:
A *)k1
k 1
B with nonequilibrium steady-state concentration cAand cB,
the heat dissipation rate (hdr) of this reaction is [13,17]
JDl ¼ RTðk1cA k1cBÞ ln k1cA
k1cB
ð11Þ
in which the net flux J¼ k1cA– k)1cBis the turnover per
unit time, and the Dl¼ RT ln(k)1cB/k1cA) the chemical
potential change of turnover per mole One can decompose
J into J+ and J–, which can provide information on its
energetics in biochemical network analysis [13,17]:
J¼ Jþ J; Dl¼ RT ln Jþ
J
;
hdr¼ RTðJþ JÞ ln Jþ
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: l¼ 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, and s 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:
1þ k 1
k1þ k 1 k 2
k1k2
þk2þ k2þ k3
cB¼
k1
k 1CT
1þ k1
k 1þ k1 k2
k 1 k 2
k2þ k3þ k3
cC¼
k 1 k 2
k 1 k 2CT
1þ k 1
k 1þ k 1 k 2
k 1 k 2
k2 k3
where
D¼ k1k3þ k3k1þ k1k3þ ðk1þ k3þ k3Þk2
þ ðk1þ k1þ k3Þk2:
Trang 4The fluxes are given by
JAB¼k1ðk2þ k2þ k3Þ þ k1ðk2þ k3þ k3Þ
JAC¼ JCB¼k2k3þ k2k3þ k2k3
When J¼ 0, all fluxes are zero The ratio JAB/JAC
JAB
JAC
¼k1ðk2þ k2þ k3Þ þ k1ðk2þ k3þ k3Þ
k2k3þ k2k3þ k2k3
¼k1k2þ k1k3
k2k3 ¼rACþ rCB
rAB 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:
k1k2k3
k1k2k3¼ 1 known as the thermodynamic box in chemistry or detailed balance in physics The steady state is then a chemical equilibrium with zero flux: k1cA) k)1cB¼ k2cB)
k)2cC¼ k3cC) k)3cA¼ 0 Therefore, DlAB¼ DlBC¼
DlCA¼ 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 charging:
B *)
k 0
k 0
2
C
in which the cofactors D and E have fixed concentrations, [D] and [E] (One can treat the ko
2 [D] and ko
2[E] as k2and
k)2if 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 koand ko
2are second-order rate constants The equilibrium constant for the reaction
D$ E is
Keq¼ eq
eq
o eq
ko
2 eq
o
eq eq
eqko
2 eq
¼ k1k
ok3
k1ko
2k3 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
DlDE¼ RT ln Keq
Now again consider the cyclic reaction in Eqn (14), with pseudo-first-order rate constants k2and k)2, we have:
DlABþ DlBCþ DlCA DlDE¼ 0 ð15Þ
in which the first three terms are chemical potential differences due to a nonzero flux J, and the last term is the energy, or more precisely chemical motive force (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 work¼ dissipated heat
As a function of the driving force DlDE, the steady-state cycle flux in the cyclic reaction is [23,24]
o
2k3 DlDE =RT 1
k1k3þ k3k1þ k1k3þ ðk1þ k3þ k3Þko
1þ k1þ k3Þko
2
Trang 5This relationship is highly nonlinear However for small
in which the rAB, rBCand rCAare 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 Xi
(i¼ 1,2, .,N are dynamic concentrations and
i¼ N + 1,N + 2, .,N + N¢ are clamped
concentra-tions) with M + M¢ biochemical reactions (j ¼ 1,2, .,M
for internal reactions and j¼ M + 1,M + 2, .,M + M¢
for external boundary fluxes) The jth internal reaction is
characterized by a set of stoichiometric coefficients mjiand jji
in the form
jj1X1þ jj2X2þ jjNþN0XNþN 0Ð
kjþ
k j
mj1X1þ mj2X2þ mjNþN0XNþN0 ð18Þ
in which some of the integers m and j can be zero If Xnis an
enzyme for the reaction m, then mm
n ¼ jm
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 S¼ fjji mjig [5,6,25–27]:
S¼
S~NM
. .
SNM 0
S^N 0 M j 0
0
B
B
B
@
1 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, ^S¼ 0 [2,28]) as well as other modeling
approaches such as metabolic control analysis (MCA)
The matrix ~Scontains only the dynamic species and internal
fluxes The null space of the Nð~SÞ, 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 (Rj) For each vector belonging to the cycle space
v¼ (v1,v2, .,vM) in the null space Nð~SÞ, the expression:
XM j¼1
is an overall reaction with the left and right sides identical:
XM j¼1
mj
XN i¼1
jjiXiXM j¼1
vj
XN i¼1
mjiXi
¼XN i¼1
Xi
XM j¼1
jji mji
The chemical potential difference of the jth reaction is expressed as:
Dlj¼ Dlo
j þ RT ln
QNþN 0
i¼1 ½Xij
j
QNþN 0
i¼1 ½Xim
j
!
¼XN i¼1
~
Sijliþ Dlext
j ; ð21Þ where li¼ lo
i þ RT ln[Xi] is the chemical potential of ith species and
Dlextj ¼ NþNX0
i¼Nþ1
^
Sijli
is the cmf for the jth reaction Combining Eqns (20) and (21) and recalling ~SS v¼ 0, we reach the conclusion that:
X j
vjðDlj DlojÞ ¼ 0
if Dlext¼ 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 thatP
j¼1mjDlo
j ¼ 0 (since for any steady-stateP
jmjðDlj Dlo
jÞ ¼ 0; it has to be valid for equilib-rium in which all Dlj¼ 0) Hence we have:
XM
for any v in the null space Nð~SÞ
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, Dlext „ 0 and Eqn (22) becomes vÆ(Dlint– Dlext)¼ 0 over the reaction loop v (e.g DlDEin Eqn (15) is external) The law of energy balance restricts the Nð~SÞ to a smaller, thermodynamically feasible subspace [7] Now let
J¼ 1
k1
k1þ k2
k1k2þ 1
k2
k2þ k1
k1k2
þ 1
k3
k3þ k2
k2k3
RT
Trang 6v1, v2, ., vmbe the m linearly independent vectors of the
thermodynamically feasible null space, and define the
matrix K¼ ðvT
1; vT
2; ; vT
mÞ where vTdenotes the transpose
of the row vector in a column form Then we obtain an
novel algebraic structure for the biochemical network
theory:
~
S
SK¼ 0; SJ~ int¼ bext; DlintK¼ pext; ð23Þ
in which matrices ~Sand K are known as incidence and loop
matrices in graph theory [5,30] Vectors Jintand Dlint are
internal fluxes and chemical potentials, both M dimensional
bext¼ SSJextis a N-dimensional external flux (typically with
many zero components) and Jext is M¢-dimensional
pext¼ lext^K is the external cmf on reaction loops, and
N¢-dimensional lext 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 rate¼ Jint
j Dlint
j 0 for individual jth reaction and hdr¼ Jint Dlint 0 for an
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 biochemical network 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 ~S, Nð~SÞ I n
the case when a network without any external flux and
clamped species: ~SS Jint¼ 0 ) l~SS Jint¼ DlJint¼ 0 Since
every term DljJj £ 0, one has DljJj¼ 0 for all j Hence
Dl¼ Jint¼ 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 for network 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 netsubnet-work is severed, the subnetsubnet-work 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 based on the law of mass action.)
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
Trang 7by 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 Newtonian and
Lagran-gian, 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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... computa-tional and bioinformatic protocols for analysis, simulation, and design of complex biochemical systemsThe SNT developed in this paper provides a unique conceptual framework for network analysis... 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... the transpose
of the row vector in a column form Then we obtain an
novel algebraic structure for the biochemical network
theory:
~
S
SKẳ 0; SJ~