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1 Entropy 2011, 13, ; do: /e OPEN ACCESS entropy ISSN Artcle The Mchaels-Menten-Stueckelberg Theorem Alexander N. Gorban 1, and Muhammad Shahzad 1,2 1 Department of Mathematcs, Unversty of Lecester, Lecester, LE1 7RH, UK 2 Department of Mathematcs, Hazara Unversty, Mansehra, 21300, Pakstan Author to whom correspondence should be addressed; E-Mal: ag153@le.ac.uk. Receved: 25 January 2011; n revsed form: 28 March 2011 / Accepted: 12 May 2011 / Publshed: 20 May 2011 Abstract: We study chemcal reactons wth complex mechansms under two assumptons: () ntermedates are present n small amounts (ths s the quas-steady-state hypothess or QSS) and () they are n equlbrum relatons wth substrates (ths s the quasequlbrum hypothess or QE). Under these assumptons, we prove the generalzed mass acton law together wth the basc relatons between knetc factors, whch are suffcent for the postvty of the entropy producton but hold even wthout mcroreversblty, when the detaled balance s not applcable. Even though QE and QSS produce useful approxmatons by themselves, only the combnaton of these assumptons can render the possblty beyond the rarefed gas lmt or the molecular chaos hypotheses. We do not use any a pror form of the knetc law for the chemcal reactons and descrbe ther equlbra by thermodynamc relatons. The transformatons of the ntermedate compounds can be descrbed by the Markov knetcs because of ther low densty (low densty of elementary events). Ths combnaton of assumptons was ntroduced by Mchaels and Menten n In 1952, Stueckelberg used the same assumptons for the gas knetcs and produced the remarkable sem-detaled balance relatons between collson rates n the Boltzmann equaton that are weaker than the detaled balance condtons but are stll suffcent for the Boltzmann H-theorem to be vald. Our results are obtaned wthn the Mchaels-Menten-Stueckelbeg conceptual framework. Keywords: chemcal knetcs; Lyapunov functon; entropy; quasequlbrum; detaled balance; complex balance PACS Codes: Ln; Db

2 Entropy 2011, Introducton 1.1. Man Asymptotc Ideas n Chemcal Knetcs There are several essentally dfferent approaches to asymptotc and scale separaton n knetcs, and each of them has ts own area of applcablty. In chemcal knetcs varous fundamental deas about asymptotcal analyss were developed [1]: Quasequlbrum asymptotc (QE), quas steady-state asymptotc (QSS), lumpng, and the dea of lmtng step. Most of the works on nonequlbrum thermodynamcs deal wth the QE approxmatons and correctons to them, or wth applcatons of these approxmatons (wth or wthout correctons). There are two basc formulaton of the QE approxmaton: The thermodynamc approach, based on entropy maxmum, or the knetc formulaton, based on selecton of fast reversble reactons. The very frst use of the entropy maxmzaton dates back to the classcal work of Gbbs [2], but t was frst clamed for a prncple of nformatonal statstcal thermodynamcs by Jaynes [3]. A very general dscusson of the maxmum entropy prncple wth applcatons to dsspatve knetcs s gven n the revew [4]. Correctons of QE approxmaton wth applcatons to physcal and chemcal knetcs were developed [5,6]. QSS was proposed by Bodensten n 1913 [7], and the mportant Mchaels and Menten work [8] was publshed smultaneously. It appears that no knetc theory of catalyss s possble wthout QSS. Ths method was elaborated nto an mportant tool for the analyss of chemcal reacton mechansm and knetcs [9 11]. The classcal QSS s based on the relatve smallness of concentratons of some of the actve reagents (radcals, substrate-enzyme complexes or actve components on the catalyst surface) [12 14]. Lumpng analyss ams to combne reagents nto quascomponents for dmenson reducton [15,17 19]. We and Prater [16] demonstrated that for (pseudo)monomolecular systems there exst lnear combnatons of concentratons whch evolve n tme ndependently. These lnear combnatons (quascomponents) correspond to the left egenvectors of the knetc matrx: If lk = λl then d(l, c)/dt = (l, c)λ, where the standard nner product (l, c) s the concentraton of a quascomponent. They also demonstrated how to fnd these quascomponents n a properly organzed experment. Ths observaton gave rse to a queston: How to lump components nto proper quascomponents to guarantee the autonomous dynamcs of the quascomponents wth approprate accuracy? We and Kuo studed condtons for exact [15] and approxmate [17] lumpng n monomolecular and pseudomonomolecular systems. They demonstrated that under certan condtons a large monomolecular system could be well-modelled by a lower-order system. More recently, senstvty analyss and Le group approach were appled to lumpng analyss [18,19], and more general nonlnear forms of lumped concentratons were used (for example, concentraton of quascomponents could be a ratonal functon of c). Lumpng analyss was placed n the lnear systems theory and the relatonshps between lumpablty and the concepts of observablty, controllablty and mnmal realzaton were demonstrated [20]. The lumpng procedures were consdered also as effcent technques leadng to nonstff systems and

3 Entropy 2011, demonstrated the effcency of the developed algorthm on knetc models of atmospherc chemstry [21]. An optmal lumpng problem can be formulated n the framework of a mxed nteger nonlnear programmng (MINLP) and can be effcently solved wth a stochastc optmzaton method [22]. The concept of lmtng step gves the lmt smplfcaton: The whole network behaves as a sngle step. Ths s the most popular approach for model smplfcaton n chemcal knetcs and n many areas beyond knetcs. In the form of a bottleneck approach ths approxmaton s very popular from traffc management to computer programmng and communcaton networks. Recently, the concept of the lmtng step has been extended to the asymptotology of multscale reacton networks [23,24]. In ths paper, we focus on the combnaton of the QE approxmaton wth the QSS approach The Structure of the Paper Almost thrty years ago one of us publshed a book [25] wth Chapter 3 enttled Quasequlbrum and Entropy Maxmum. A research program was formulated there, and now we are n the poston to analyze the achevements of these three decades and formulate the man results, both theoretcal and appled, and the unsolved problems. In ths paper, we start ths work and combne a presentaton of theory and applcaton of the QE approxmaton n physcal and chemcal knetcs wth exposton of some new results. We start from the formal descrpton of the general dea of QE and ts possble extensons. In Secton 2, we brefly ntroduce man notatons and some general formulas for excluson of fast varables by the QE approxmaton. In Secton 3, we present the hstory of the QE and the classcal confuson between the QE and the quas steady state (QSS) approxmaton. Another surprsng confuson s that the famous Mchaels-Menten knetcs was not proposed by Mchaels and Menten n 1913 [8] but by Brggs and Haldane [12] n It s more mportant that Mchaels and Menten proposed another approxmaton that s very useful n general theoretcal constructons. We descrbed ths approxmaton for general knetc systems. Roughly speakng, ths approxmaton states that any reacton goes through transformaton of fast ntermedate complexes (compounds), whch () are n equlbrum wth the nput reagents and () exst n a very small amount. One of the most mportant benefts from ths approach s the excluson of nonlnear knetc laws and reacton rate constants for nonlnear reactons. The nonlnear reactons transform nto the reactons of the compounds producton. They are n a fast equlbrum and the equlbrum s ruled by thermodynamcs. For example, when Mchaels and Menten dscuss the producton of the enzyme-substrate complex ES from enzyme E and substrate S, they do not dscuss reacton rates. These rates may be unknown. They just assume that the reacton E + S ES s n equlbrum. Brggs and Haldane nvolved ths reacton nto the knetc model. Ther approach s more general than the Mchaels-Menten approxmaton but for the Brggs and Haldane model we need more nformaton, not only the equlbrum of the reacton E + S ES but also ts rates and constants. When compounds undergo transformatons n a lnear frst order knetcs, there s no need to nclude nteractons between them because they are present n very small amounts n the same volume, and ther concentratons are also small. (By the way, ths argument s not applcable to the heterogeneous catalytc reactons. Although the ntermedates are n both small amounts and n a small volume,.e., n the

4 Entropy 2011, surface layer, the concentraton of the ntermedates s not small, and ther nteracton does not vansh when ther amount decreases [33]. Therefore, knetcs of ntermedates n heterogeneous catalyss may be nonlnear and demonstrate bfurcatons, oscllatons and other complex behavor.) In 1952, Stueckelberg [26] used smlar approach n hs semnal paper H-theorem and untarty of the S-matrx. He studed elastc collsons of partcles as the quas-chemcal reactons v + w v + w (v, w, v, w are veloctes of partcles) and demonstrated that for the Boltzmann equaton the lnear Markov knetcs of the ntermedate compounds results n the specal relatons for the knetc coeffcents. These relatons are suffcent for the H-theorem, whch was orgnally proved by Boltzmann under the stronger assumpton of reversblty of collsons [27]. Frst, the dea of such relatons was proposed by Boltzmann as an answer to the Lorentz objectons aganst Boltzmann s proof of the H-theorem. Lorentz stated the nonexstence of nverse collsons for polyatomc molecules. Boltzmann dd not object to ths argument but proposed the cyclc balance condton, whch means balancng n cycles of transtons between states S 1 S 2... S n S 1. Almost 100 years later, Cercgnan and Lamps [28] demonstrated that the Lorenz arguments are wrong and the new Boltzmann relatons are not needed for the polyatomc molecules under the mcroreversblty condtons. The detaled balance condtons should hold. Nevertheless, Boltzmann s dea s very semnal. It was studed further by Hetler [29] and Coester [30] and the results are sometmes cted as the Hetler-Coestler theorem of sem-detaled balance. In 1952, Stueckelberg [26] proved these condtons for the Boltzmann equaton. For the mcro-descrpton he used the S-matrx representaton, whch s n ths case equvalent for the Markov mcroknetcs (see also [31]). Later, these relatons for the chemcal mass acton knetcs were redscovered and called the complex balance condtons [51,63]. We generalze the Mchaels-Menten-Stueckelberg approach and study n Secton 5 the general knetcs wth fast ntermedates present n small amount. In Subsecton 5.7 the bg Mchaels-Menten-Stueckelberg theorem s formulated as the overall result of the prevous analyss. Before ths general theory, we ntroduce the formalsm of the QE approxmaton wth all the necessary notatons and examples for chemcal knetcs n Secton 4. The result of the general knetcs of systems wth ntermedate compounds can be used wder than ths specfc model of an elementary reacton: The ntermedate complexes wth fast equlbra and the Markov knetcs can be consdered as the constructon stagng for general knetcs. In Secton 6, we delete the constructon stagng and start from the general forms of the obtaned knetc equatons as from the basc laws. We study the relatons between the general knetc law and the thermodynamc condton of the postvty of the entropy producton. Sometmes the knetcs equatons may not respect thermodynamcs from the begnnng. To repar ths dscrepancy, deformaton of the entropy may help. In Secton 7, we show when s t possble to deform the entropy by addng a lnear functon to provde agreement between gven knetc equatons and the deformed thermodynamcs. As a partcular case, we got the defcency zero theorem. The classcal formulaton of the prncple of detaled balance deals not wth the thermodynamc and global forms we use but just wth equlbra: In equlbrum each process must be equlbrated wth

5 Entropy 2011, ts reverse process. In Secton 7, we demonstrate also that for the general knetc law the exstence of a pont of detaled balance s equvalent to the exstence of such a lnear deformaton of the entropy that the global detaled balance condtons (Equaton (87) below) hold. Analogously, the exstence of a pont of complex balance s equvalent to the global condton of complex balance after some lnear deformaton of the entropy Man Results: One Asymptotc and Two Theorems Let us follow the deas of Mchaels-Menten and Stueckelberg and ntroduce the asymptotc theory of reacton rates. Let the lst of the components A be gven. The mechansm of reacton s the lst of the elementary reactons represented by ther stochometrc equatons: α ρ A β ρ A (1) The lnear combnatons α ρa and β ρa are the complexes. For each complex y ja from the reacton mechansm we ntroduce an ntermedate auxlary state, a compound B j. Each elementary reacton s represented n the form of the 2n-tal scheme (Fgure 1) wth two ntermedate compounds: α ρ A Bρ B ρ + β ρ A (2) Fgure 1. A 2n-tal scheme of an extended elementary reacton. There are two man assumptons n the Mchaels-Menten-Stueckelberg asymptotc: The compounds are n fast equlbrum wth the correspondng nput reagents (QE); They exst n very small concentratons compared to other components (QSS). The smallness of the concentraton of the compounds mples that they () have the perfect thermodynamc functons (entropy, nternal energy and free energy) and () the rates of the reactons B B j are lnear functons of ther concentratons. One of the most mportant benefts from ths approach s the excluson of the nonlnear reacton knetcs: They are n fast equlbrum and equlbrum s ruled by thermodynamcs. Under the gven smallness assumptons, the reacton rates r ρ for the elementary reactons have a specal form of the generalzed mass acton law (see Equaton (74) below): r ρ = ϕ ρ exp(α ρ, ˇμ)

6 Entropy 2011, where ϕ ρ > 0 s the knetc factor and exp(α ρ, ˇμ) s the Boltzmann factor. Here and further n the text (α ρ, ˇμ) = α ρˇμ s the standard nner product, exp(, ) s the exponental of the value of the nner product and ˇμ are chemcal potentals μ dvded on RT. For the prefect chemcal systems, ˇμ = ln(c /c ), where c s the concentraton of A and c > 0 are the postve equlbrum concentratons. For dfferent values of the conservaton laws there are dfferent postve equlbra. The postve equlbrum c s one of them and t s not mportant whch one s t. At ths pont, ˇμ =0, hence, the knetc factor for the perfect systems s just the equlbrum value of the rate of the elementary reacton at the equlbrum pont c : ϕ ρ = r ρ (c ). The lnear knetcs of the compound reactons B B j mples the remarkable dentty for the reacton rates, the complex balance condton (Equaton (89) below) ϕ ρ exp(ˇμ, α ρ )= ϕ ρ exp(ˇμ, β ρ ) ρ ρ for all admssble values of ˇμ and gven ϕ whch may vary ndependently. For other and more convenent forms of ths condton see Equaton (91) n Secton 6. The complex balance condton s suffcent for the postvty of the entropy producton (for decrease of the free energy under sothermal sochorc condtons). The general formula for the reacton rate together wth the complex balance condtons and the postvty of the entropy producton form the Mchaels-Menten-Stueckelberg theorem (Secton 5.7). The detaled balance condtons (Equaton (87) below), ϕ + ρ = ϕ ρ for all ρ, are more restrctve than the complex balance condtons. For the perfect systems, the detaled balance condton takes the standard form: r ρ + (c )=rρ (c ). We study also some other, less restrctve suffcent condtons for accordance between thermodynamcs and knetcs. For example, we demonstrate that the G-nequalty (Equaton (92) below) ϕ ρ exp(ˇμ, α ρ ) ϕ ρ exp(ˇμ, β ρ ) ρ ρ s suffcent for the entropy growth and, at the same tme, weaker than the condton of complex balance. If the reacton rates have the form of the generalzed mass acton law but do not satsfy the suffcent condton of the postvty of the entropy producton, the stuaton may be mproved by the deformaton of the entropy va addton of a lnear functon. Such a deformaton s always possble for the zero defcency systems. Let q be the number of dfferent complexes n the reacton mechansm, d be the number of the connected components n the dgraph of the transtons between compounds (vertces are compounds and edges are reactons). To exclude some degenerated cases a hypothess of weak reversblty s accepted: For any two vertces B and B j, the exstence of an orented path from B to B j mples the exstence of an orented path from B j to B. Defcency of the system s [63] q d rankγ 0 where Γ=(γ j )=(β j α j ) s the stochometrc matrx. If the system has zero defcency then the entropy producton becomes postve after the deformaton of the entropy va addton of a lnear functon. The defcency zero theorem n ths form s proved n Secton 7.3.

7 Entropy 2011, Interrelatons between the Mchaels-Menten-Stueckelberg asymptotc and the transton state theory (whch s also referred to as the actvated-complex theory, absolute-rate theory, and theory of absolute reacton rates ) are very ntrgung. Ths theory was developed n 1935 by Eyrng [35] and by Evans and Polany [36]. Basc deas behnd the transton state theory are [37]: The actvated complexes are n a quas-equlbrum wth the reactant molecules; Rates of the reactons are studed by studyng the actvated complexes at the saddle pont of a potental energy surface. The smlarty s obvous but n the Mchaels-Menten-Stueckelberg asymptotc an elementary reacton s represented by a couple of compounds wth the Markov knetcs of transtons between them versus one transton state, whch moves along the reacton coordnate, n the transton state theory. Ths s not exactly the same approach (for example, the theory of absolute reacton rates uses the detaled balance condtons and does not produce anythng smlar to the complex balance). Important techncal tools for the analyss of the Mchaels-Menten-Stueckelberg asymptotc are the theorem about preservaton of the entropy producton n the QE approxmaton (Secton 2 and Appendx 1) and the Mormoto H-theorem for the Markov chans (Appendx 2). 2. QE and Preservaton of Entropy Producton In ths secton we ntroduce nformally the QE approxmaton and the mportant theorem about the preservaton of entropy producton n ths approxmaton. In Appendx 1, ths approxmaton and the theorem are presented wth more formal detals. Let us consder a system n a doman U of a real vector space E gven by dfferental equatons dx = F (x) (3) dt The QE approxmaton for (3) uses two basc enttes: Entropy and slow varables. Entropy S s an ncreasng concave Lyapunov functon for (3) wth non-degenerated Hessan 2 S/ x x j : ds dt 0 (4) In ths approach, the ncrease of entropy n tme s exploted (the Second Law n the form (4)). The slow varables M are defned as some dfferentable functons of varables x: M = m(x). Here we assume that these functons are lnear. More general nonlnear theory was developed n [38,39] wth applcatons to the Boltzmann equaton and polymer physcs. Selecton of the slow varables mples a hypothess about separaton of fast and slow moton. The slow varables (almost) do not change durng the fast moton. After some ntal tme, the fast varables wth hgh accuracy are functons of the slow varables: We can wrte x x M. The QE approxmaton defnes the functons x M as solutons to the followng MaxEnt optmzaton problem: S(x) max subject to m(x) =M (5)

8 Entropy 2011, The reasonng behnd ths approxmaton s smple: Durng the fast ntal layer moton, entropy ncreases and M almost does not change. Therefore, t s natural to assume that x M s close to the soluton to the MaxEnt optmzaton problem (5). Further x M denotes a soluton to the MaxEnt problem. A soluton to (5), x M,stheQE state, the set of the QE states x M, parameterzed by the values of the slow varables M s the QE manfold, the correspondng value of entropy S (M) =S(x M) (6) s the QE entropy and the equaton for the slow varables dm = m(f (x dt M)) (7) represents the QE dynamcs. The crucal property of the QE dynamcs s the preservaton of entropy producton. Theorem about preservaton of entropy producton. Let us calculate ds (M)/dt at pont M accordng to the QE dynamcs (7) and fnd ds(x)/dt at pont x = x M due to the ntal system (3). The results always concde: ds (M) = ds(x) (8) dt dt The left hand sde n (8) s computed due to the QE approxmaton (7) and the rght hand sde corresponds to the ntal system (3). The sketch of the proof s gven n Appendx 1. The preservaton of the entropy producton leads to the preservaton of the type of dynamcs: If for the ntal system (3) entropy producton s non-negatve, ds/dt 0, then for the QE approxmaton (7) the producton of the QE entropy s also non-negatve, ds /dt 0. In addton, f for the ntal system (ds/dt) x =0f and only f F (x) =0then the same property holds n the QE approxmaton. 3. The Classcs and the Classcal Confuson 3.1. The Asymptotc of Fast Reactons It s dffcult to fnd who ntroduced the QE approxmaton. It was mpossble before the works of Boltzmann and Gbbs, and t became very well known after the works of Jaynes [3]. Chemcal knetcs has been a source for model reducton deas for decades. The deas of QE appear there very naturally: Fast reactons go to ther equlbrum and, after that, reman almost equlbrum all the tme. The general formalzaton of ths dea looks as follows. The knetc equaton has the form dn dt = K sl(n)+ 1 ɛ K fs(n) (9) Here N s the vector of composton wth components N > 0, K sl corresponds to the slow reactons, K fs corresponds to fast reacton and ɛ>0 s a small number. The system of fast reactons has the lnear conservaton laws b l (N) = j b ljn j : b l (K fs (N)) 0. The fast subsystem dn dt = K fs(n)

9 Entropy 2011, tends to a stable postve equlbrum N for any postve ntal state N(0) and ths equlbrum s a functon of the values of the lnear conservaton laws b l (N(0)). In the plane b l (N) =b l (N(0)) the equlbrum s asymptotcally stable and exponentally attractve. Vector b(n) =(b l (N)) s the vector of slow varables and the QE approxmaton s db dt = b(k sl(n (b)) (10) In chemcal knetcs, equlbra can be descrbed by condtonal entropy maxmum (or condtonal extremum of other thermodynamc potentals). Therefore, for these cases we can apply the formalsm of the quasequlbrum approxmaton. The thermodynamc Lyapunov functons serve as tools for stablty analyss and for model reducton [40]. The QE approxmaton, the asymptotc of fast reactons, s well known n chemcal knetcs. Another very mportant approxmaton was nvented n chemcal knetcs as well. It s the Quas Steady State (QSS) approxmaton. QSS was proposed n [7] and was elaborated nto an mportant tool for analyss of chemcal reacton mechansms and knetcs [9 11]. The classcal QSS s based on the relatve smallness of concentratons of some of actve reagents (radcals, substrate-enzyme complexes or actve components on the catalyst surface) [13,14]. In the enzyme knetcs, ts nventon was tradtonally connected to the so-called Mchaels-Menten knetcs QSS and the Brggs-Haldane Asymptotc Perhaps the frst very clear explanaton of the QSS was gven by Brggs and Haldane n 1925 [12]. Brggs and Haldane consder the smplest enzyme reacton S + E SE P + E and menton that the total concentraton of enzyme ([E] +[SE]) s neglgbly small compared wth the concentraton of substrate [S]. After that they conclude that d [SE] s neglgble compared wth d [S] and d [P ] and dt dt dt produce the now famous Mchaels-Menten formula, whch was unknown to Mchaels and Menten: k 1 [E][S] =(k 1 + k 2 )[ES] or [ES]= [E][S] K M +[S] and d dt [P ]=k 2[ES]= k 2[E][S] K M +[S] (11) where the Mchaels-Menten constant s K M = k 1 + k 2 k 1 There s plenty of msleadng comments n later publcatons about QSS. Two most mportant confusons are: Enzymes (or catalysts, or radcals) partcpate n fast reactons and, hence, relax faster than substrates or stable components. Ths s obvously wrong for many QSS systems: For example, n the Mchaels-Menten system all reactons nclude enzyme together wth substrate or product. There are no separate fast reactons for enzyme wthout substrate or product. Concentratons of ntermedates are constant because n QSS we equate ther tme dervatves to zero. In general case, ths s also wrong: We equate the knetc expressons for some tme dervatves to zero, ndeed, but ths just explots the fact that the tme dervatves of ntermedates

10 Entropy 2011, concentratons are small together wth ther values, but not oblgatory zero. If we accept QSS then these dervatves are not zero as well: To prove ths we can just dfferentate the Mchaels-Menten formula (11) and fnd that [ES] n QSS s almost constant when [S] K M, ths s an addtonal condton, dfferent from the Brggs-Haldane condton [E] +[AE] [S] (for more detals see [1,14,33] and a smple detaled case study [41]). After a smple transformaton of varables the QSS smallness of concentraton transforms nto a separaton of tme scales n a standard sngular perturbaton form (see, for example [33,34]). Let us demonstrate ths on the tradtonal Mchaels-Menten system: d[s] = k 1 [S][E]+k 1 [SE] dt d[se] = k 1 [S][E] (k 1 + k 2 )[SE] dt [E]+[SE]=e = const, [S]+[P]=s = const (12) Ths s a homogeneous system wth the sochorc (fxed volume) condtons for whch we wrte the equatons. The Brggs-Haldane condton s e s. Let us use dmensonless varables x =[S]/s, ξ =[SE]/e: s dx e dt = sk 1x(1 ξ)+k 1 ξ dξ dt = sk 1x(1 ξ) (k 1 + k 2 )ξ To obtan the standard sngularly perturbed system wth the small parameter at the dervatve, we need to change the tme scale. Ths means that when e 0 the reacton goes proportonally slower and to study ths lmt properly we have to adjust the tme scale: dτ = e s dt: dx dτ = sk 1x(1 ξ)+k 1 ξ e dξ s dτ = sk 1x(1 ξ) (k 1 + k 2 )ξ For small e/s, the second equaton s a fast subsystem. Accordng to ths fast equaton, for a gven constant x, the varable ξ relaxes to sx ξ QSS = K M + sx exponentally, as exp( (sk 1 x + k 1 + k 2 )t). Therefore, the classcal sngular perturbaton theory based on the Tkhonov theorem [42,43] can be appled to the system n the form (14) and the QSS approxmaton s applcable even on an nfnte tme nterval [44]. Ths transformaton of varables and ntroducton of slow tme s a standard procedure for rgorous proof of QSS valdty n catalyss [33], enzyme knetcs [45] and other areas of knetcs and chemcal engneerng [13]. It s worth to menton that the smallness of parameter e/s can be easly controlled n experments, whereas the tme dervatves, transformaton rates and many other quanttes just appear as a result of knetcs and cannot be controlled drectly. (13) (14)

11 Entropy 2011, The Mchaels and Menten Asymptotc QSS s not QE but the classcal work of Mchaels and Menten [8] was done on the ntersecton of QSS and QE. After the brllantly clear work of Brggs and Haldane, the name Mchaels-Menten was attached to the Brggs and Haldane equaton and the orgnal work of Mchaels and Menten was consdered as an mportant partcular case of ths approach, an approxmaton wth addtonal and not necessary assumptons of QE. From our pont of vew, the Mchaels-Menten work ncludes more and may gve rse to an mportant general class of knetc models. Mchaels and Menten studed the fermentatve splttng of cane sugar. They ntroduced three compounds : The sucrose-ferment combnaton, the fructose-ferment combnaton and the glucose-ferment combnaton. The fundamental assumpton of ther work was that the rate of breakdown at any moment s proportonal to the concentraton of the sucrose-nvertase compound. They started from the assumpton that at any moment accordng to the mass acton law [S ][E] =K [S E] (15) where [S ] s the concentraton of the th sugar (here, =0for sucrose, 1 for fructose and 2 for glucose), [E] s the concentraton of the free nvertase and K s the th equlbrum constant. For smplfcaton, they use the assumpton that the concentraton of any sugar n queston n free state s practcally equal to that of the total sugar n queston. Fnally, they obtan e[s 0 ] [S 0 E]= (16) K 0 (1 + q[p ]) + [S 0 ] where e =[E]+ [S E], [P ]=[S 1 ]=[S 2 ] and q = 1 K K 2. Of course, ths formula may be consdered as a partcular case of the Brggs-Haldane formula (11) f we take k 1 k 2 n (11) (.e., the equlbraton S + E SE s much faster than the reacton SE P + E) and assume that q = 0n (16) (.e., fructose-ferment combnaton and glucose-ferment combnaton are practcally absent). Ths s the truth but may be not the complete truth. The Mchaels-Menten approach wth many compounds whch are present n small amounts and satsfy the QE assumpton (15) s a seed of the general knetc theory for perfect and non-perfect mxtures. 4. Chemcal Knetcs and QE Approxmaton 4.1. Stochometrc Algebra and Knetc Equatons In ths secton, we ntroduce the basc notatons of the chemcal knetcs formalsm. For more detals see, for example, [33]. The lst of components s a fnte set of symbols A 1,...,A n. A reacton mechansm s a fnte set of the stochometrc equatons of elementary reactons: α ρ A β ρ A (17) where ρ = 1,...,m s the reacton number and the stochometrc coeffcents α ρ,β ρ are nonnegatve ntegers.

12 Entropy 2011, A stochometrc vector γ ρ of the reacton (17) san-dmensonal vector wth coordnates γ ρ = β ρ α ρ (18) that s, gan mnus loss n the ρth elementary reacton. A nonnegatve extensve varable N, the amount of A, corresponds to each component. We call the vector N wth coordnates N the composton vector. The concentraton of A s an ntensve varable c = N /V, where V > 0 s the volume. The vector c = N/V wth coordnates c s the vector of concentratons. A non-negatve ntensve quantty, r ρ, the reacton rate, corresponds to each reacton (17). The knetc equatons n the absence of external fluxes are dn dt = V ρ r ρ γ ρ (19) If the volume s not constant then equatons for concentratons nclude V and have dfferent form (ths s typcal for the combuston reactons, for example). For perfect systems and not so fast reactons, the reacton rates are functons of concentratons and temperature gven by the mass acton law for the dependance on concentratons and by the generalzed Arrhenus equaton for the dependance on temperature T. The mass acton law states: c α ρ (20) r ρ (c, T )=k ρ (T ) where k ρ (T ) s the reacton rate constant. The generalzed Arrhenus equaton s: ( ) T Saρ R k ρ (T )=A ρ exp ( E ) aρ T 0 RT (21) J where R = s the unversal, or deal gas constant, E K mol aρ s the actvaton energy, S aρ s the actvaton entropy (.e., E aρ TS aρ s the actvaton free energy), A ρ s the constant pre-exponental factor. Some authors neglect the S aρ term because t may be less mportant than the actvaton energy, but t s necessary to stress that wthout ths term t may be mpossble to reconcle the knetc equatons wth the classcal thermodynamcs. In general, the constants for dfferent reactons are not ndependent. They are connected by varous condtons that follow from thermodynamcs (the second law, the entropy growth for solated systems) or mcroreversblty assumpton (the detaled balance and the Onsager recprocal relatons). In Secton 6.2 we dscuss these condtons n more general settngs. For nondeal systems, more general knetc law s needed. In Secton 5 we produce such a general law followng the deas of the orgnal Mchaels and Menten paper (ths s not the same as the famous Mchaels-Menten knetcs ). For ths work we need a general formalsm of QE approxmaton for chemcal knetcs.

13 Entropy 2011, Formalsm of QE Approxmaton for Chemcal Knetcs QE Manfold In ths secton, we descrbe the general formalsm of the QE for chemcal knetcs followng [34]. The general constructon of the quas-equlbrum manfold gves the followng procedure. Frst, let us consder the chemcal reactons n a constant volume under the sothermal condtons. The free energy F (N,T) =Vf(c, T ) should decrease due to reactons. In the space of concentratons, one defnes a subspace of fast motons L. It should be spanned by the stochometrc vectors of fast reactons. Slow coordnates are lnear functons that annulate L. These functons form a subspace n the space of lnear functons on the concentraton space. Dmenson of ths space s s = n dm L. It s necessary to choose any bass n ths subspace. We can use for ths purpose a bass b j n L, an orthogonal complement to L and defne the basc functonals as b j (N) =(b j,n). The descrpton of the QE manfold s very smple n the Legendre transform. The chemcal potentals are partal dervatves μ = F(N,T) = f(c, T ) (22) N c Let us use μ as new coordnates. In these new coordnates (the conjugated coordnates ), the QE manfold s just an orthogonal complement to L. Ths subspace, L, s defned by equatons μ γ =0 forany γ L (23) It s suffcent to take n (23) not all γ L but only elements from a bass n L. In ths case, we get the system of n dm L lnear equatons of the form (23) and ther soluton does not cause any dffculty. For the actual computatons, one requres the nverson from μ to c. It s worth to menton that the problems of the selecton of the slow varables and of the descrpton of the QE manfold n the conjugated varables can be consdered as the same problem of descrpton of the orthogonal complement, L. To fnalze the constructon of the QE approxmaton, we should fnd for any gven values of slow varables (and of conservaton laws) b the correspondng pont on the QE manfold. Ths means that we have to solve the system of equatons for c: b(n) =b; (μ(c, T ),γ ρ )=0 (24) where b s the vector of slow varables, μ s the vector of chemcal potentals and vectors γ ρ form a bass n L. After that, we have the QE dependence c QE (b) and for any admssble value of b we can fnd all the reacton rates and calculate ḃ. Unfortunately, the system (24) can be solved analytcally only n some specal cases. In general case, we have to solve t numercally. For ths purpose, t may be convenent to keep the optmzaton statement of the problem: F mn subject to gven b. There exsts plenty of methods of convex optmzaton for soluton of ths problem. The standard toy example gves us a fast dssocaton reacton. Let a homogeneous reacton mechansm nclude a fast reacton of the form A + B AB. We can easly fnd the QE approxmaton

14 Entropy 2011, for ths fast reacton. The slow varables are the quanttes b 1 = N A N B and b 2 = N A + N B + N C whch do not change n ths reacton. Let the chemcal potentals be μ A /RT =lnc A + μ A0, μ B /RT = ln c B +μ B0, μ AB /RT =lnc AB +μ AB0. Ths corresponds to the free energy F = VRT c (ln c +μ 0 ), the correspondent free entropy (the Masseu-Planck potental) s F/T. The stochometrc vector s γ =( 1, 1, 1) and the equatons (24) take the form c A c B = b 1 V, c A + c B + c AB = b 2 V, c AB c A c B = K where K s the equlbrum constant K = exp(μ A0 + μ B0 μ AB0 ). From these equatons we get the expressons for the QE concentratons: (1 c A (b 1,b 2 )= 1 b 1 2 V 1 K + b 1 2 V 1 ) 2 + b 1 + b 2 K KV c B (b 1,b 2 )=c A (b 1,b 2 ) b 1 V, c AB(b 1,b 2 )= b 1 + b 2 2c A (b 1,b 2 ) V The QE free entropy s the value of the free entropy at ths pont, c(b 1,b 2 ) QE n Tradtonal MM System Let us return to the smplest homogeneous enzyme reacton E + S ES P + S, the tradtonal Mchaels-Menten System (12) (t s smpler than the system studed by Mchaels and Menten [8]). Let us assume that the reacton E + S ES s fast. Ths means that both k 1 and k 1 nclude large parameters: k 1 = 1κ ɛ 1, k 1 = 1κ ɛ 1. For small ɛ, we wll apply the QE approxmaton. Only three components partcpate n the fast reacton, A 1 = S, A 2 = E, A 3 = ES. For analyss of the QE manfold we do not need to nvolve other components. The stochometrc vector of the fast reacton s γ =( 1, 1, 1). The space L s one-dmensonal and ts bass s ths vector γ. The space L s two-dmensonal and one of the convenent bases s b 1 =(1, 0, 1), b 2 =(0, 1, 1). The correspondng slow varables are b 1 (N) =N 1 +N 3, b 2 (N) =N 2 +N 3. The frst slow varable s the sum of the free substrate and the substrate captured n the enzyme-substrate complex. The second of them s the conserved quantty, the total amount of enzyme. The equaton for the QE manfold s (15): k 1 c 1 c 2 = k 1 c 3 or c 1 c 2 c 1 c 2 = c 3 c 3 because k 1 c 1c 2 = k 1 c 3, where c = c (T ) > 0 are the so-called standard equlbrum values and for perfect systems μ = RT ln(c /c ), F = RT V c (ln(c /c ) 1). Let us fx the slow varables and fnd c 1,2,3. Equatons (24) turn nto c 1 + c 3 = b 1,c 2 + c 3 = b 2,k 1 c 1 c 2 = k 1 c 3 Here we change dynamc varables from N to c because ths s a homogeneous system wth constant volume. If we use c 1 = b 1 c 3 and c 2 = b 2 c 3 then we obtan a quadratc equaton for c 3 : k 1 c 2 3 (k 1 b 1 + k 1 b 2 + k 1 )c 3 + k 1 b 1 b 2 =0 (25)

15 Entropy 2011, Therefore, c 3 (b 1,b 2 )= 1 ( b 1 + b 2 + k ) ( 1 1 b1 + b2 + k ) 2 1 4b 1 b 2 2 k 1 2 k 1 The sgn s selected to provde postvty of all c. Ths choce provdes also the proper asymptotc: c 3 0 f any of b 0. For other c 1,2 we should use c 1 = b 1 c 3 and c 2 = b 2 c 3. The tme dervatves of concentratons are: ċ 1 = k 1 c 1 c 2 + k 1 c 3 + v n c n 1 v out c 1 ċ 2 = k 1 c 1 c 2 +(k 1 + k 2 )c 3 + v n c n 2 v out c 2 ċ 3 = k 1 c 1 c 2 (k 1 + k 2 )c 3 + v n c n 3 v out c 3 ċ 4 = k 2 c 3 + v n c n 4 v out c 4 here we added external flux wth nput and output veloctes (per unte volume) v n and v out and nput concentratons c n. Ths s done to stress that the QE approxmaton holds also for a system wth fluxes f the fast equlbrum subsystem s fast enough. The nput and output veloctes are the same for all components because the system s homogeneous. The slow system s ḃ 1 =ċ 1 +ċ 3 = k 2 c 3 + v n b n 1 v out b 1 (26) ḃ 2 =ċ 2 +ċ 3 = v n b n 2 v out b 2 (27) ċ 4 = k 2 c 3 + v n c n 4 v out c 4 where b n 1 = c n 1 + c n 3, b n 2 = c n 2 + c n 3. Now, we should use the expresson for c 3 (b 1,b 2 ): ( 1 ḃ 1 = k 2 b 1 + b 2 + k ) ( 1 1 b1 + b2 + k 1 2 k 1 2 k 1 ċ 4 =k ( ḃ 2 =v n b n 2 v out b 2 b 1 + b 2 + k 1 k 1 ) 2 4b 1 b 2 ) ( 1 b1 + b2 + k ) 2 1 4b 1 b 2 2 k 1 + v n b n 1 v out b 1 + v n c n 4 v out c 4 It s obvous here that n the reduced system (28) there exsts one reacton from the lumped component wth concentraton b 1 (the total amount of substrate n free state and n the substrate-enzyme complex) nto the component (product) wth concentraton c 4. The rate of ths reacton s k 2 c(b 1 b 2 ). The lumped component wth concentraton b 2 (the total amount of the enzyme n free state and n the substrate-enzyme complex) affects the reacton rate but does not change n the reacton. Let us use for smplfcaton of ths system the assumpton of the substrate excess (we follow the logc of the orgnal Mchaels and Menten paper [8]): (28) [S] [SE],.e., b 1 c 3 (29) Under ths assumpton, the quadratc equaton (25) transforms nto ( 1+ b 2 + k ) 1 c 3 = b 2 + o b 1 k 1 b 1 ( c3 b 1 ) (30)

16 Entropy 2011, and n ths approxmaton b 2 b 1 c 3 = (31) b 1 + b 2 + k 1 k 1 (compare to (16) and (11): Ths equaton ncludes an addtonal term b 2 n denomnator because we dd not assume formally anythng about the smallness of b 2 n (29)). After ths smplfcaton, the QE slow equatons (28) take the form ḃ 1 = k 2b 2 b 1 b 1 + b 2 + k 1 k 1 ḃ 2 = v n b n 2 v out b 2 ċ 4 = k 2 b 2 b 1 b 1 + b 2 + k 1 k 1 + v n b n 1 v out b 1 + v n c n 4 v out c 4 Ths s the typcal form n the reduced equatons for catalytc reactons: Nomnator n the reacton rate corresponds to the brutto reacton S + E P + E [33,49] Heterogeneous Catalytc Reacton For the second example, let us assume equlbrum wth respect to the adsorpton n the CO on Pt oxdaton: CO+Pt PtCO; O 2 +2Pt 2PtO (for detaled dscusson of the modelng of CO on Pt oxdaton, ths Mona Lza of catalyss, we address readers to [33]). The lst of components nvolved n these 2 reactons s: A 1 = CO, A 2 = O 2, A 3 = Pt, A 4 = PtO, A 5 = PtCO (CO 2 does not partcpate n adsorpton and may be excluded at ths pont). Subspace L s two-dmensonal. It s spanned by the stochometrc vectors, γ 1 =( 1, 0, 1, 0, 1), γ 2 =(0, 1, 2, 2, 0). The orthogonal complement to L s a three-dmensonal subspace spanned by vectors (0, 2, 0, 1, 0), (1, 0, 0, 0, 1), (0, 0, 1, 1, 1). Ths bass s not orthonormal but convenent because of nteger coordnates. The correspondng slow varables are b 1 =2N 2 + N 4 =2N O2 + N PtO (32) b 2 = N 1 + N 5 = N CO + N PtCO (33) b 3 = N 3 + N 4 + N 5 = N Pt + N PtO + N PtCO For heterogeneous systems, cauton s needed n transton between N and c varables because there are two volumes and we cannot put n (33) c nstead of N : N gas = V gas c gas but N surf = V surf c surf, where where V gas s the volume of gas, V surf s the area of surface. There s a law of conservaton of the catalyst: N Pt + N PtO + N PtCO = b 3 = const. Therefore, we have two non-trval dynamcal slow varables, b 1 and b 2. They have a very clear sense: b 1 s the amount of atoms of oxygen accumulated n O 2 and PtO and b 2 s the amount of atoms of carbon accumulated n CO and PtCO. The free energy for the perfect heterogeneous system has the form F = V gas RT ( ) ) c c (ln 1 + V c surf RT c (ln A gas A surf ( c c ) ) 1 (34)

17 Entropy 2011, where c are the correspondng concentratons and c = c (T ) > 0 are the so-called standard equlbrum values. (The QE free energy s the value of the free energy at the QE pont.) From the expresson (34) we get the chemcal potentals of the perfect mxture ( ) c μ = RT ln (35) The QE manfold n the conjugated varables s gven by equatons: μ 1 μ 3 + μ 5 =0; μ 2 2μ 3 +2μ 4 =0 It s trval to resolve these equatons wth respect to μ 3,4, for example: c or wth the standard equlbra: μ 4 = 1 2 μ 2 + μ 3 ; μ 5 = μ 1 + μ 3 c 4 c 4 = c 3 c2, c 3 c 2 c 5 c 5 = c 1 c 3 c 1 c 3 or n the knetc form (we assume that the knetc constants are n accordance wth thermodynamcs and all these forms are equvalent): The next task s to solve the system of equatons: k 1 c 1 c 3 = k 1 c 5,k 2 c 2 c 2 3 = k 2 c 2 4 (36) k 1 c 1 c 3 = k 1 c 5,k 2 c 2 c 2 3 = k 2 c 2 4, 2V gas c 2 + V surf c 4 = b 1, V gas c 1 + V surf c 5 = b 2,V surf (c 3 + c 4 + c 5 )=b 3 (37) Ths s a system of fve equatons wth respect to fve unknown varables, c 1,2,3,4,5. We have to solve them and use the soluton for calculaton of reacton rates n the QE equatons for the slow varables. Let us construct these equatons frst, and then return to (37). We assume the adsorpton (the Langmur-Hnshelwood) mechansm of CO oxdaton (the numbers n parentheses are used below for the numeraton of the reacton rate constants): (±1) CO+Pt PtCO (±2) O 2 +2Pt 2PtO (3) PtO+PtCO CO 2 +2Pt (38) The knetc equatons for ths system (ncludng the flux n the gas phase) s CO Ṅ 1 = V surf ( k 1 c 1 c 3 + k 1 c 5 )+V gas (v n c n 1 v out c 1 ) O 2 Ṅ 2 = V surf ( k 2 c 2 c k 2 c 2 4)+V gas (v n c n 2 v out c 2 ) Pt Ṅ 3 = V surf ( k 1 c 1 c 3 + k 1 c 5 2k 2 c 2 c k 2 c k 3 c 4 c 5 ) (39) PtO Ṅ 4 = V surf (2k 2 c 2 c 2 3 2k 2 c 2 4 k 3 c 4 c 5 ) PtCO Ṅ5 = V surf (k 1 c 1 c 3 k 1 c 5 k 3 c 4 c 5 ) CO 2 Ṅ 6 = V surf k 3 c 4 c 5 + V gas (v n c n 6 v out c 6 )

18 Entropy 2011, Here v n and v out are the flux rates (per unt volume). For the slow varables ths equaton gves: ḃ 1 =2Ṅ2 + Ṅ4 = V surf k 3 c 4 c 5 +2V gas (v n c n 2 v out c 2 ) ḃ 2 = Ṅ1 + Ṅ5 = V surf k 3 c 4 c 5 + V gas (v n c n 1 v out c 1 ) ḃ 3 = Ṅ3 + Ṅ4 + Ṅ5 =0 Ṅ 6 = V surf k 3 c 4 c 5 + V gas (v n c n 6 v out c 6 ) (40) Ths system looks qute smple. Only one reacton, PtO+PtCO CO 2 +2Pt (41) s vsble. If we know expressons for c 3,5 (b) then ths reacton rate s also known. In addton, to work wth the rates of fluxes, the expressons for c 1,2 (b) are needed. The system of equatons (37) s explctly solvable but the result s qute cumbersome. Therefore, let us consder ts smplfcaton wthout explct analytc soluton. We assume the followng smallness: Together wth ths smallness assumptons equatons (37) gve: b 1 N 4, b 2 N 5 (42) b 3 c 3 = V surf (1+ k 1 b 2 1 k k 1 V gas + 2 b 1 2 k 2 1 k 2 b 1 b 3 c 4 = 2 k 2 V gas V surf (1+ k 1 V gas ) b 2 k 1 V gas + ) 1 k 2 b 1 2 k 2 V gas (43) c 5 = k 1 b 2 b 3 k 1 V gas V surf (1+ k 1 b 2 k 1 V gas + ) 1 k 2 b 1 2 k 2 V gas In ths approxmaton, we have for the reacton (41) rate k 1 1 k 2 b1 b 2 b 2 3 r = k 3 c 4 c 5 = k 3 k 1 2 k 2 Vgas 3/2 ( Vsurf 2 1+ k 1 b 2 k 1 V gas + Ths expresson gves the closure for the slow QE equatons (40) Dscusson of the QE procedure for Chemcal Knetcs ) 2 1 k 2 b 1 2 k 2 V gas We fnalze here the llustraton of the general QE procedure for chemcal knetcs. As we can see, the smple analytc descrpton of the QE approxmaton s avalable when the fast reactons have no jont reagents. In general case, we need ether a numercal solver for (24) or some addtonal hypotheses about smallness. Mchaels and Menten used, n addton to the QE approach, the hypothess about smallness of the amount of ntermedate complexes. Ths s the typcal QSS hypothess. The QE approxmaton was modfed and further developed by many authors. In partcular, a computatonal optmzaton approach

19 Entropy 2011, for the numercal approxmaton of slow attractng manfolds based on entropy-related and geometrc extremum prncples for reacton trajectores was developed [47]. Of course, valdty of all the smplfcaton hypotheses s a crucal queston. For example, for the CO oxdaton, f we accept the hypothess about the quasequlbrum adsorpton then we get a smple dynamcs whch monotoncally tends to the steady state. The state of the surface s unambguously presented as a contnuous functon of the gas composton. The pure QSS hypothess results for the Langmur-Hnshelwood reacton mechansm (38) wthout quasequlbrum adsorpton n bfurcatons and the multplcty of steady states [33]. The problem of valdty of smplfcatons cannot be solved as a purely theoretcal queston wthout the knowledge of knetc constants or some addtonal expermental data. 5. General Knetcs wth Fast Intermedates Present n Small Amount 5.1. Stochometry of Complexes In ths Secton, we return to the very general reacton network. Let us call all the formal sums that partcpate n the stochometrc equatons (17), the complexes. The set of complexes for a gven reacton mechansm (17) sθ 1,...,Θ q. The number of complexes q 2m (two complexes per elementary reacton, as the maxmum) and t s possble that q<2m because some complexes may concde for dfferent reactons. A complex Θ s a formal sum Θ = n j=1 ν ja j =(ν,a), where ν s a vector wth coordnates ν j. Each elementary reacton (17) may be represented n the form Θ ρ Θ + ρ, where Θ ± ρ are the complexes whch correspond to the rght and the left sdes (17). The whole mechansm s naturally represented as a dgraph of transformaton of complexes: Vertces are complexes and edges are reactons. Ths graph gves a convenent tool for the reacton representaton and s often called the reacton graph. Let us consder a smple example: 18 elementary reactons (9 pars of mutually reverse reactons) from the hydrogen combuston mechansm (see, for example, [48]). H+O 2 O+OH; OH + H 2 H+H 2 O; HO 2 +H H 2 +O 2 ; H+OH+M H 2 O+M; H 2 O 2 +H H 2 +HO 2 There are 16 dfferent complexes here: O+H 2 H+OH; O+H 2 O 2OH; HO 2 +H 2OH; H+O 2 +M HO 2 +M; (44) Θ 1 =H+O 2, Θ 2 =O+OH, Θ 3 =O+H 2, Θ 4 =H+OH, Θ 5 =OH+H 2, Θ 6 =H+H 2 O, Θ 7 =O+H 2 O, Θ 8 = 2OH, Θ 9 =HO 2 +H, Θ 10 =H 2 +O 2, Θ 11 =H+OH+M, Θ 12 =H 2 O+M, Θ 13 =H+O 2 +M, Θ 14 =HO 2 +M, Θ 15 =H 2 O 2 +H, Θ 16 =H 2 +HO 2

20 Entropy 2011, The reacton set (44) can be represented as Θ 1 Θ 2, Θ 3 Θ 4, Θ 5 Θ 6, Θ 7 Θ 8 Θ 9 Θ 10, Θ 11 Θ 12, Θ 13 Θ 14, Θ 15 Θ 16 We can see that ths dgraph of transformaton of complexes has a very smple structure: There are sx solated pars of complexes and one connected group of four complexes Stochometry of Compounds For each complex Θ j we ntroduce an addtonal component B j, an ntermedate compound and B ± ρ are those compounds B j (1 j q), whch correspond to the rght and left sdes of reacton (17). We call these components compounds followng the Englsh translaton of the orgnal Mchaels-Menten paper [8] and keep complexes for the formal lnear combnatons Θ j. An extended reacton mechansm ncludes two types of reactons: Equlbraton between a complex and ts compound (q reactons, one for each complex) Θ j B j (45) and transformaton of compounds Bρ B ρ + (m reactons, one for each elementary reacton from (17). So, nstead of the reacton (17) we can wrte α ρ A Bρ B ρ + β ρ A (46) Of course, f the nput or output complexes concde for two reactons then the correspondng equlbraton reactons also concde. It s useful to vsualze the reacton scheme. In Fgure 1 we represent the 2n-tal scheme of an elementary reacton sequence (46) whch s an extenson of the elementary reacton (17). The reactons between compounds may have several channels (Fgure 2): One complex may transform to several other complexes. The reacton mechansm s a set of multchannel transformatons (Fgure 2) for all nput complexes. In Fgure 2 we grouped together the reactons wth the same nput complex. Another representaton of the reacton mechansm s based on the groupng of reactons wth the same output complex. Below, n the descrpton of the complex balance condton, we use both representatons. The extended lst of components ncludes n + q components: n ntal speces A and q compounds B j. The correspondng composton vector N s a drect sum of two vectors, the composton vector for ntal speces, N, wth coordnates N ( =1,...,n) and the composton vector for compounds, Υ, wth coordnates Υ j (j =1,...,q): N = N Υ. The space of composton vectors E s a drect sum of n-dmensonal E A and q-dmensonal E B : E = E A E B. For concentratons of A we use the notaton c and for concentratons of B j we use ς j. The stochometrc vectors for reactons Θ j B j (45) are drect sums: g j = ν j e j, where e j s the jth standard bass vector of the space R q = E B, the coordnates of e j are e jl = δ jl : g j =( ν j1, ν j2,..., ν jn, 0,...,0, 1, 0,...,0) (47) }{{} l

21 Entropy 2011, The stochometrc vectors of equlbraton reactons (45) are lnearly ndependent because there exsts exactly one vector for each l. The stochometrc vectors γ jl of reactons B j B l belong entrely to E B. They have jth coordnate 1, lth coordnate +1 and other coordnates are zeros. To exclude some degenerated cases a hypothess of weak reversblty s accepted. Let us consder a dgraph wth vertces Θ and edges, whch correspond to reactons from (17). The system s weakly reversble f for any two vertces Θ and Θ j, the exstence of an orented path from Θ to Θ j mples the exstence of an orented path from Θ j to Θ. Of course, ths weak reversblty property s equvalent to weak reversblty of the reacton network between compounds B j. Fgure 2. A multchannel vew on the complex transformaton. The hdden reactons between compounds are ncluded n an oval S Energy, Entropy and Equlbra of Compounds In ths secton, we defne the free energy of the system. The basc hypothess s that the compounds are the small admxtures to the system, that s, the amount of compounds B j s much smaller than amount of ntal components A. Followng ths hypothess, we neglect the energy of nteracton between compounds, whch s quadratc n ther concentratons because n the low densty lmt we can neglect the correlatons between partcles f the potental of ther nteractons decay suffcently fast when the dstance between partcles goes to [50]. We take the energy of ther nteracton wth A n the lnear approxmaton, and use the perfect entropy for B. These standard assumptons for a small admxtures gve for the free energy: F = Vf(c, T )+VRT q j=1 ς j ( uj (c, T ) RT ) +lnς j 1 (48)