An annotated English translation of Kinetics of stationary reactions [M. I. Temkin, Dolk. Akad. Nauk SSSR. 152, 156 (1963)]

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1 An annotated Englih tanlation of Kinetic of tationay eaction [M. I. Temkin, Dolk. Akad. Nauk R. 52, 56 (963)] Vladilav V. Levchenko *, Ronan Fleming, Hong Qian #, and Daniel A. Bead * * Depatment of hyiology, Medical College of Wiconin, Milwaukee, WI cience Intitute & Cente fo ytem Biology, Univeity of Iceland, Reykjavik, Iceland # Depatment of Applied Mathematic, Univeity of Wahington, eattle, WA Adde coepondence to: D. A. Bead, dbead@mcw.edu Abtact Temkin 963 aticle on one-way fluxe and flux atio in teady-tate eaction ytem bea diectly on cuent eeach in phyical and biological chemity, uch a in the intepetation of metabolic exchange fluxe detemined fom iotopome labeling expeiment. Yet, oiginally publihed in Ruian [Dolk. Akad. Nauk R 52, (963)], thi aticle ha emained inacceible to much of the cientific community. Hee we povide an Englih tanlation of the oiginal aticle with eveal additional claification and coection. Temkin [Dolk. Akad. Nauk R 52, (963)] deived a elationhip between themodynamic diving foce and one-way fluxe in chemical eaction that bea diectly on cuent eeach in phyical and biological chemity. Indeed, mateial in a widely cited Englih-language eview [M.I. Temkin, Advance in Catalyi 28, (979)] elie on pimay liteatue, by the ame autho, publihed in Ruian. Thi tanlation i intended to bing into focu the deivation of the pimay eult efeenced in ubequent Englih-language liteatue. A complete tanlation of the oiginal aticle follow, with additional claification and coection incopoated a footnote. Kinetic of tationay eaction M. I. Temkin, Dolk. Akad. Nauk R. 52, 56 (963) Complex eaction ae combination of imple one, called hee tage o elementay eaction. Hee we ue the tem intemediate to efe to chemical pecie that appea in the chemical equation of elementay eaction, but not in the oveall chemical equation of a given complex eaction. The eaction i called tationay if the concentation of intemediate component emain contant when the concentation of ubtate and poduct that appea in the oveall eaction ae held at fixed concentation. uch eaction egime may be ealized in flow ytem o in eacto without gadient []. Reaction in tatic ytem can often be conideed quai-tationay (Bodenhtein method). uch Thu Temkin i pecifically efeing hee to non-ocillatoy teady-tate ytem. In geneal eaction ytem with oveall eactant concentation held contant can diplay non-ocillatoy o ocillatoy teady tate, o chaotic behavio. The tudy of ocillatoy eaction dynamic began with the pioneeing wok of Belouov and Zabotinky. Othe example include the eveible chnakenbeg eaction cheme in Vellela and Qian [oc. Royal oc. A, to appea (200)] and the eveible Lotka-Voltea eaction in Li et al. [J. Chem. hy. 29, (2008)]. In both of thee cae the oveall eaction involve the teady-tate poduction of poduct B fom ubtate A.

2 condition ae outlined in Fank-Kamenetkii [2] and emenov [3]. Futhe eview i elated to tationay (o quai-tationay) eaction. In the cae of heteogeneou eaction catalyzed on uface, we will aume the eaction uface to be homogeneou. Intemediate component fo uch eaction ae hemoobic 2 atom o molecule and fee uface aea. To obtain an oveall complex equation by addition of elementay eaction 3, the elementay eaction equation mut be multiplied by the appopiate toichiometic numbe [4]. In geneal thee may be ubtantially diffeent et of thee toichiometic numbe coeponding to diffeent oute 4 fo a given oveall complex eaction [4]. By ubtantially diffeent it i meant that a et of toichiometic numbe aociated with a given mechanim cannot be obtained fom anothe et by multiplying by a common facto (which would be equivalent to multiplication of the oveall complex eaction by the ame facto). toichiometic numbe can be factional, negative o be equal to zeo, and by ode of magnitude they ae compaable to. Let u denote x j a the toichiometic coefficient of the intemediate ubtate X j in a chemical equation of elementay eaction : x j > 0, if X j i fomed, and x j < 0, if X j i depleted. The toichiometic numbe aociated with elementay eaction fo the oute p i denoted v ; thee numbe, by definition, atify v xj = 0, () = whee i the total numbe of elementay eaction. Thu thee exit I linealy independent condition of the fom of Equation (), whee I i the numbe of intemediate ubtate. We denote the numbe of linealy independent of et of v olution of ytem of the fom of Equation () a = - I. I.e., thee ae independent eaction oute [4]. ometime thee i moe than one oute aociated with a given oveall chemical equation. Fo example, let u aign numbe and 2 to two oute that yield the ame oveall eaction. Then chemical equation (3) (2) () of the oute 3, defined with the toichiomety v = v v, will have the oveall toichiomety 0 = 0. We will call uch oute empty. (Chitianen call them a cyclical equence [5].) Moe often thee i only one independent oute aociated with a given oveall eaction. In tem of kinetic uch ytem undego one complex eaction; yet ome elementay eaction can be common to diffeent oute. The chemical equation of an elementay eaction i witten in accodance with the flow of it elementay opeation. We define the flux of elementay eaction in the fowad diection and in the evee diection. Fowad and evee fluxe of an elementay eaction coepond to the numbe of tunove of the eaction in each diection pe unit volume pe unit time, o pe unit uface aea pe unit time. The chemical equation of an elementay eaction may be multiplied by an abitay facto 2 I.e., the eaction uface i unifom; thee i no patial heteogeneity in the uface aboption o eaction kinetic. 3 We apply the tem elementay eaction whee Temkin ue the tem стадия, which may be moe popely tanlated a tage. We ue the moe conventional contempoay notation elementay eaction thoughout. 4 The tem oute appea pelled in Latin chaacte in the oiginal pape. oible altenative tanlation ae pathway o mechanim. We ue the tem oute thoughout, which efe to a et of elementay eaction and aociated toichiometic numbe fo each elementay eaction. 2

3 (multiplie) that mut be pecified 5. Given thee definition, the tunove of a eaction i aociated with the diappeaance of molecule of initial ubtance (ubtate) and appeaance of molecule of poduct in a numbe that i indicated in a coefficient of the chemical equation. Denoting the ate of a (coeponding to ate of tunove of a given eaction oute in a given time given eaction oute p a peiod pe unit volume o pe unit uface aea), the total flux though a given elementay eaction i the um of contibution fom the oute, the following citeia of tage tationaity ae atified: v =. (2) And o indeed, in a given volume o on a given uface, ( ) unit time. Combining Equation () and (2), we have x j molecule of X j ae fomed pe = xj ( ) = 0 ; (3) = i.e. numbe of each intemediate ubtance i contant. Fo a given eaction ytem, a total of equation fom Equation (2) may be ued to define + I unknown. That i, value of and I value of activity of independent intemediate ubtance. Equation (3) may be obtained a a diect conequence of tationay-tate ma conevation. A tanfomation fom one et of independent oute { v (), defined by equation of the fom v, } to anothe { v ( ), (2) (2 ) v, } may be v = C v + C v + ( ) () (2) 2 (2 ) () (2) v = C v + C v +, (4) whee C, C 2, etc. ae coefficient. Theefoe the chemical equation of the oute ae tanfomed. If { () (2),, } ae the oute eaction ate fo the oiginal et of oute and { ( ) (2 ),, } ae the ate fo the new et of oute, then = C + C + () ( ) (2 ) 2 (2) ( ) (2 ) 2 22 = C + C + (5) Coefficient along a ow in the ytem of Equation (4) appea in the coeponding column in the ytem 5 Multiplication of a chemical eaction by an abitay facto doe not change the chemity, but doe change the law of ma action. Theefoe the facto ha to be defined and i not abitay. 3

4 of Equation (5) 6. Indeed, ubtituting in the eaction oute ate fom Equation (5) into Equation (2) and taking into account Equation (4), we find that the oute ate { ( ) (2 ),, } atify the condition of tationaity fo all. It i woth noting that the left-hand ide of Equation (2) ha a fom of a cala poduct of two vecto. The teady-tate opeation of chemical tanfomation may be examined uing linea algeba when oute ae decompoed into elementay eaction. Let u aume the oute and 2 coepond to the ame oveall chemical equation. Defining the ( ) () (2 ) (2) () (2 ) tanfomation v = v, v = v v (whee v define an empty oute), it follow fom Equation ( ) () (2) (2 ) (2) (5) that = + and =. Theefoe the ate of eaction on an empty oute i not in geneal equal to zeo. (But it i alway equal to zeo if the empty oute i the only oute). Although oute i left ( ) () unchanged by the tanfomation,. Theefoe it i not poible to unambiguouly aign a defined ate to thi oute, unle the et of all othe independent oute and thei coeponding ate ae defined. Uing thee elationhip it i poible to define a unique ummay oute fo which the eaction ate of all othe oute i equal to zeo. The toichiometic coefficient fo elementay eaction in the ummay ( ) oute i defined by the flux-weighted um of coefficient v p fom an independent et of oute: v v = = v. (6) The eaction ate aociated with the oute defined by thi et of toichiometic coefficient { ν } (the oveall eaction flux) i equal to =. The flux though each othe eaction oute in the new tanfomed et of oute i zeo. The chemical equation of the ummay oute decibe the total (um of all) chemical tanfomation in the ytem. In ome cae the elucidation of kinetic equation elating oute eaction fluxe { () (2),, } to concentation (o activitie) of ubtate and poduct can be ignificantly implified thi way. tating with the equality ( ) ( ) ( ) = (7) take ( ), ( ), and o on accoding to (2). Dividing both ide of equality by 2 3 we get ( p ) ( p ) The elation between Equation (4) and (5) can be demontated a follow. Expeing Equation (4): v = C v, if the total flux though each elementay eaction i maintained by thi tanfomation, then implie ( p ) ( p = C ). p = pp ( p ) ( p ) v = p = p p, which 4

5 v v v v v v = (8) () () () (2) (2) (2) () (2) Equation (8) may be called the equation of tationay eaction. Along with Equation (2) thi equation i independent of the ode in which the elementay eaction ae numbeed. Thi i obviou becaue all elementay eaction in a tationay eaction un imultaneouly. In ome cae implification ae poible. Fo example, ometime the catalytic mechanim fo an oveall eaction contain a elementay eaction in which v = 0 fo all p. uch elementay eaction ae in equilibium ( = ) and the facto / fall out of Equation (8). Futhemoe, when fo one of the elementay eaction i much geate than, thi elementay eaction i efeed to a apid o quai-equilibium. Elementay eaction whee i of imila ode to ae called low. Accoding to Equation (2) fo fat elementay eaction v and, leading to aociated implification of Equation (8). If ome elementay eaction i teated a ieveible, become zeo. It i ometime poible to eaily eliminate ome unknown activitie (o concentation) of intemediate ubtance baed on Equation (8). To do o, it i neceay if poible to aange the equation of elementay eaction in uch an ode that intemediate ubtance which ae fomed in any given elementay eaction ae conumed in the following elementay eaction. In ome cae a paticula odeing allow one to obtain the neceay quantity of equation. Diagam that how the tuctue of the eaction mechanim can often be helpful [6, 3, 5]. Concentation of intemediate ubtance, if need be, may be detemined with the help of the coeponding elementay eaction equence. If equilibium o quai-equilibium elementay eaction ae peent then the activity of an intemediate ubtance which i paticipating in uch a elementay eaction can be elucidated fom the equilibium ma-action atio. Applying Equation (8) in the context of the ummay oute whee all othe oute have zeo flux, we have v v2 2v = (9) An equation of thi fom can be applied diectly to any oveall eaction with only one oute. Let u next conide a ummay oute of the complex eaction (o equivalently the oute aociated with a ingle-oute eaction) in which one of the elementay eaction i ieveible. We futhe aume, without lo of geneality, that the elementay eaction ae odeed o that the ieveible elementay eaction i elementay eaction. Theefoe Equation (9) yield v v2 2v3 2 v = (0) whee + i the fowad flux aociated with elementay eaction. If only one elementay eaction i low (othe ae quai-equilibium o equilibium), thi elementay eaction i called limiting. Fo uch a cae we get equality + = l vl fom (0), whee l i the index of 5

6 the limiting elementay eaction 7. By changing the diection of all elementay eaction and aligning them in the evee ode and equaling l = 0, we deive the equation fo the ate in the evee diection,. Then it i imple to ee that = ; () 23 = + (2) Accoding to Equation () and (2) the value of + and ae independent of the ode in which the elementay eaction ae aligned (becaue + and + do not change when the ode of the elementay eaction change). When we apply Equation (8)-(2) uggeted by Chitianen (5,7) to equence of elementay eaction of pecial type, we aive to the eult uggeted by thi autho. When we have only one way of iotopic exchange then the ate of the iotopic exchange i decibed by an equation of the ame fom a Equation (0), but include only tage that ae paticipating in the exchange and not all elementay eaction of the oveall eaction. uch an equation wa deived by Matuda and Hoiuti [8]. Let u take the eaction with one oute, with the chemical equation of aa + a'a' + = bb + b'b' + We define the aveage (mean) toichiometic value v : v Δ + Δ + G G. (3) v G v2 G2 = Δ +Δ + 2 Hee ΔG i the change of Gibb fee enegy aociated with elementay eaction. Fom the theoy of abolute ate of eaction it i evident that delta Δ G = RTln, a well a 7 Thee i ome confuion hee, pehap in pat due to tanlation. In the paagaph befoe Equation (0) the limiting eaction i defined to be the lat elementay eaction. We have taken the libety of claifying that the index of the lat elementay eaction i defined to be above. Thu the equality = v i tue only if l =. + l l 8 eveal iue elated to Equation () and (2) equie claification. Fit, note that the + and obtained thi way do not coepond to the oveall fowad and evee ate fo the oveall eaction. Indeed, thi incoect aumption wa made in a imila poof decibed by T.L. Hill [Fee Enegy Tanduction in Biology, Academic e Inc., U.. (977) p. 22] and wa late demontated to be incoect by Hill [cf. T.L. Hill, Fee Enegy Tanduction and Biochemical Cycle Kinetic, Dove, (2004) p. 53]. econd, while the analyi hee i limited to ytem in which all but one elementay eaction ae in quaiequilibium, the elationhip of Equation () and (2) do hold tue fo ytem of thi ot fo the moe geneal cae in which any numbe of the eaction ae limiting, a wa demontated coectly by Hill [Fee Enegy Tanduction and Biochemical Cycle Kinetic, Dove (2004) and efeence theein], and Qian et al. [cientia inica 24, (98); 25, 3-40 (982); 27, (984)]. In a moe ecent wok, Bead and Qian [Lo ONE 2: e44, 2007] povide a imple poof of a moe geneal theoem that applie to a boade et of eaction and tanpot pocee than ae conideed hee. 6

7 a ( A) ( A ) b ( ) ( ) a vδ G+ v2δ G2 + =Δ G = RTln K b, whee K i equilibium contant, (A) i the activity B B of ubtance (ubtate) A and o on. It follow that Equation () yield: a ( A) ( A ) b ( ) ( ) /v a + = K b B B. (4) In the Equation (3) contibution elated to equilibium o quai-equilibium elementay eaction dop out becaue delta ΔG = 0 fo them (exact o appoximate). If v of all non-equilibium elementay eaction ae the ame, then v i equal to that oveall value 9. In paticula, if we have limiting elementay eaction, then v = v and we aive to the eult of Boekov [9] and Hoiuti [4]. l. M. I. Temkin, Kinet. Katal., 3, 509 (962). Liteatue Cited 2. D. A. Fank-Kamenetkii, Dolk. Akad. Nauk R., 25, 672 (939). 3. N. N. emenov, Zh. Fiz. Khim. [J. hy. Chem. (UR)], 7, 87 (943). 4. J. Hoiuti, oblem of hyical Chemity, 2, 39 (959). 5. J. A. Chitianen, Advance in Catalyi., 5, 3 (953). 6. A. A. Balandin, Up. Khim. [Ru. Chem. Rev. (UR)] 9, 390, 940; Zh. Fiz. Khim. [J. hy. Chem. (UR)], 5, 65, 629 (94). 7. J. A. Chitianen, Z. phy. Chem., B, 28, 303 (935), 33, 45 (936). 8. A. Mauda, J. Hoiuti. J. Re. Int. fo Catalyi. Hokkaido Univ., 0, 4 (962). 9. G. K. Boekov, Zh. Fiz. Khim. [J. hy. Chem. (UR)] 9, 92 (945). 9 Recall that, hee, Equation () wa poved (albeit incoectly) tictly fo the cae whee thee i only one non-equilibium elementay eaction. Theefoe thee i only one non-equilibium elementay eaction. 7