Stoichiometric network theory for nonequilibrium 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 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 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 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 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 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) 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 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 * ) 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 ¼k 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  ¼RT 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 ðJÞ¼RT 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 biochemical network 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 ¼r 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 ¼r 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 biochemical network analysis [13,17]: J ¼ J þ  J  ; Dl ¼RT 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 Nonequilibrium stoichiometric biochemical network theory (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 ¼RT 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 ½Dþðk 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 þj j NþN 0 X NþN 0 Ð k j þ k j  m j 1 X 1 þ m j 2 X 2 þm 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  ¼ðr AB þ r BC þ r CA Þ 1 Dl DE ð17Þ Ó FEBS 2003 Nonequilibrium stoichiometric biochemical network theory (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 ¼J int j Dl int j  0for individual jth reaction and hdr ¼J 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 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 ~ 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 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 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. References 1. Stephanopoulos, G. (1994) Metabolic engineering. Curr. Opin. Biotechnol. 5, 196–200. 2. Edwards, J.S. & Palsson, B.O. (1998) How will bioinformatics influence metabolic engineering? Biotechn Bioeng. 58, 162–169. 3. Schilling, C.H. & Palsson, B.O. (1998) The underlying pathway structure of biochemical reaction networks. Proc.NatlAcad.Sci. USA 95, 4193–4198. 4. Edwards, J.S. & Palsson, B.O. (2000) The E. coli MG1655 in silico metabolic genotype: its definition, characteristics, and capacities. Proc.NatlAcad.Sci.USA97, 5528–5533. 5. Balabanian, N. & Bickart, T.A. (1981) Linear Network Theory: Analysis, Properties, Design and Synthesis. Matrix,Beaverton,OR. 6. Strang, G. (1986) Introduction to Applied Mathematics. Wellesley- Cambridge Press, Wellesley, MA. 7. Beard, D.A., Liang, S D. & Qian, H. (2002) Energy balance for analysis of complex metabolic networks. Biophys. J. 83, 79–86. 8. Feynman, R.P., Leighton, R.B. & Sands, M.L. (1963) The Feyn- manLecturesonPhysics. Addison-Wesley, Redwood City, CA. 9. Oster, G.F., Perelson, A.S. & Katchalsky, A. (1973) Network thermodynamics: the analysis of biophysical systems. Quart. Rev. Biophys. 6, 1–134. 10. Westerhoff, H.V. & van Dam, K. (1987) Thermodynamics and Control. of Biological Free-Energy Transduction. Elsevier, Amsterdam. 11. Katzir-Katchalsky, A. & Curran, P.F. (1965) Nonequilibrium Thermodynamics in Biophysics. Harvard University Press, Cambridge. 12. Kubo, R., Toda, M. & Hashitsume, N. (1978) Statistical Physics II: Nonequilibrium Statistical Mechanics. Springer-Verlag, New York. 13. Hill, T.L. (1974) Free Energy Transduction in Biology: the Steady- State Kinetic and Thermodynamic Formalism.AcademicPress, New York. 14. Hill, T.L. (1989) Free Energy Transduction and Biochemical Cycle Kinetics. Spinger-Verlag, New York. 15. Nicolis, G. & Prigogine, I. (1977) Self-Organization in Nonequilibrium Systems. Wiley Intersci., New York. 16. Qian, H. (2001) Mathematical formalism for isothermal linear irreversibility. Proc. R. Soc. Lond. A. 457, 1645–1655. 17. Qian, H. (2001) Nonequilibrium steady-state circulation and heat dissipation functional. Phys.Rev.E.64, 022101. 18. Qian, H. (2002) Entropy production and excess entropy in a nonequilibrium steady-state of single macromolecules. Phys. Rev. E. 65, 021111. 19. Qian, H. (2002) Equations for stochastic macromolecular mechanics of single proteins: equilibrium fluctuations, transient kinetics, and nonequilibrium steady-state. J. Phys. Chem. 106, 2065–2073. 20. Oster, G.F. & Perelson, A.S. (1974) Chemical reaction dynamics. Part II: Reaction networks. Arch. Ration. Mech. Anal. 57, 31–98. 21. Hill, T.L. (1982) The linear Onsager coefficients for biochemical kinetic diagrams as equilibrium one-way cycle fluxes. Nature 299, 84–86. 22. Crabtree, B. & Nicholson, B.A. (1988) Thermodynamics and Metabolism. In Biochemical Thermodynamics (Jones, M.N., ed.), Elsevier Scientific, Amsterdam, pp. 359–373. 23. Qian, H. (1997) A simple theory of motor protein kinetics and energetics. Biophys. Chem. 67, 263–267. 24. Qian, H. (2000) A simple theory of motor protein kinetics and energetics II. Biophys. Chem. 83, 35–43. 25. Peusner, L. (1986) Studies in Network Thermodynamics. Elsevier, New York. 26. Poland, D. (1991) Indicator matrices for potential instabilities in open systems. J. Chem. Phys. 95, 7984–7997. 27. Alberty, R.H. (1991) Equilibrium compositions of solutions of biochemical species and heats of biochemical reactions. Proc. Natl Acad. Sci. USA 88, 3268–3271. 28. Oliveira, J.S., Bailey, C.G., Jones-Oliveira, J.B. & Dixon, D.A. (2001) An algebraic-combinatorial model for the identification and mapping of biochemical pathways. Bullet. Math. Biol. 63, 1163–1196. 29. Schuster, S. & Schuster, R. (1991) Detecting strictly balanced subnetworks in open chemical-reactions networks. J. Math. Chem. 6, 17–40. 30. Kalpazidou, S. (1994) Cycle Representation of Markov Process. Springer-Verlag, New York. 31. Hartwell, L.H., Hopfield, J.J., Leibler, S. & Murray, A.W. (1999) From molecular to modular cell biology. Nature 402, C47–C52. 32. Kacser, H. & Burns, J.A. (1973) The control of flux. Symp. Soc. Exp. Biol. 27, 65–104. 33. Heinrich, R. & Rapoport, T.A. (1974) A linear steady-state treatment of enzymatic chains. General properties, control and effector strength. Eur. J. Biochem. 42, 89–95. 34. Fell, D.A. & Sauro, H.M. (1985) Metabolic control and its ana- lysis. Additional relationships between elasticities and control coefficients. Eur. J. Biochem. 148, 555–561. 35. Giersch, C. (1988) Control analysis of metabolic networks. 1. Homogeneous functions and the summation theorems for control coefficients. Eur. J. Biochem. 147, 509–519. 36. Kholodenko, B.N. & Westerhoff, H.V. (1995) The macroworld versus the microworld of biochemical regulation and control. Trends Biochem. Sci. 20, 52–54. 37. Goldstein, H. (1980) Classical Mechanics. Addison-Wesley, Reading, MA. Ó FEBS 2003 Nonequilibrium stoichiometric biochemical network theory (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