Biophysical Chemistry

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1 Bophyscal Chemstry 155 (211) Contents lsts avalable at ScenceDrect Bophyscal Chemstry ournal homepage: Recommendatons for termnology and databases for bochemcal thermodynamcs Robert A. Alberty a, Athel Cornsh-Bowden b, Robert. Goldberg c,d,, Gordon G. Hammes e, Keth Tpton f, Hans V. Westerhoff g a Chemstry Department, Massachusetts Insttute of Technology, 77 Massachusetts Avenue, Cambrdge, MA 2139, USA b Centre atonal de la Recherche Scentfque, 31 Chemn Joseph-Aguer, B.P. 71, 1342 Marselle Cedex 2, France c Bochemcal Scence Dvson, atonal Insttute of Standards and Technology, Gathersburg, MD 2876, USA d Department of Chemstry and Bochemstry, Unversty of Maryland, Baltmore County, Baltmore, MD 2125, USA e Department of Bochemstry, Duke Unversty, Durham, C 2771, USA f Department of Bochemstry, Trnty College Dubln, College Green, Dubln 2, Ireland g Department of Molecular Cell Physology, BoCentrum Amsterdam, Faculty of Bology, Vre Unverstet, De Boelelaan 185, 181 HV Amsterdam, The etherlands artcle nfo abstract Artcle hstory: Receved 9 March 211 Receved n revsed form 17 March 211 Accepted 17 March 211 Avalable onlne 24 March 211 Keywords: Bochemcal equatons Apparent equlbrum constants Gbbs energy of reacton Enzyme knetcs Haldane relatons Legendre transform Standard thermodynamc propertes Enthalpy Entropy Chemcal equatons are normally wrtten n terms of specfc onc and elemental speces and balance atoms of elements and electrc charge. However, n a bochemcal context t s usually better to wrte them wth onc reactants expressed as totals of speces n equlbrum wth each other. Ths mples that atoms of elements assumed to be at fxed concentratons, such as hydrogen at a specfed ph, should not be balanced n a bochemcal equaton used for thermodynamc analyss. However, both knds of equatons are needed n bochemstry. The apparent equlbrum constant K for a bochemcal reacton s wrtten n terms of such sums of speces and can be used to calculate standard transformed Gbbs energes of reacton Δ r G. Ths property for a bochemcal reacton can be calculated from the standard transformed Gbbs energes of formaton Δ f G of reactants, whch can be calculated from the standard Gbbs energes of formaton of speces Δ f G and measured apparent equlbrum constants of enzyme-catalyzed reactons. Tables of Δ r G of reactons and Δ f G of reactants as functons of ph and temperature are avalable on the web, as are functons for calculatng these propertes. Bochemcal thermodynamcs s also mportant n enzyme knetcs because apparent equlbrum constant K can be calculated from expermentally determned knetc parameters when ntal veloctes have been determned for both forward and reverse reactons. Specfc recommendatons are made for reportng expermental results n the lterature. 211 Elsever B.V. All rghts reserved. 1. Preamble Although thermodynamc consderatons affect many aspects of bochemstry, ncludng knetc analyss of enzyme-catalyzed reactons, desgn and use of buffers for controllng the ph or pmg, dentfcaton of reversble steps n metabolc pathways, mcrocalormetry, etc., there are some respects n whch bochemcal practce requres standardzaton wth thermodynamc prncples, the desgn of experments, nomenclature, and the manner n whch results are reported. eglect of these factors can create problems for the analyss and comparson of the results These recommendatons were prepared under the auspces of the Internatonal Unon of Bochemstry and Molecular Bology (IUBMB) by R. A. Alberty (U.S.A.) (Convener), A. Cornsh-Bowden (France), R.. Goldberg (U.S.A.), G. G. Hammes (U.S.A.), K. Tpton (Ireland), and H. V. Westerhoff (The etherlands). Correspondng author at: Bochemcal Scence Dvson, atonal Insttute of Standards and Technology, Gathersburg, MD 2876, USA. Tel.: ; fax: E-mal addresses: alberty@mt.edu (R.A. Alberty), acornsh@bsm.cnrs-mrs.fr (A. Cornsh-Bowden), robert.goldberg@nst.gov (R.. Goldberg), hamme1@mc.duke.edu (G.G. Hammes), ktpton@mal.tcd.e (K. Tpton), hw@bo.vu.nl (H.V. Westerhoff). of bochemcal data. For example, napproprate buffers or napproprate expermental desgns may have been used for controllng the concentratons of bochemcally mportant onc reactants, such as ATP, and especally the on MgATP 2. Addtonally, dfferent conventons may have been used n regards to standard states, concentraton of water as a reactant, etc. In an early attempt to remedy ths stuaton, the Inter-Unon Commsson on Bothermodynamcs set up ontly by IUPAC, IUB, and IUPAB publshed Recommendatons for Measurements and Presentaton of Bochemcal Equlbrum Data [1] n 1976, slghtly revsed n 1985 as Recommendatons for the Presentaton of Thermodynamc and Related Data n Bology [2]. Subsequently, IUPAC n conuncton wth IUBMB publshed Recommendatons for omenclature and Tables n Bochemcal Thermodynamcs [3] n 1994, n whch the maor nnovaton was the ntroducton of transformed thermodynamc propertes at specfed ph and pmg. Further development of transformed thermodynamc propertes snce then has led to new applcatons and ths document, prepared under the auspces of the IUBMB, has two prncpal ams: (1) to provde a concse update to the scentfc formalsm that underles the applcaton of transformed thermodynamc propertes to bochemcal thermodynamcs, and /$ see front matter 211 Elsever B.V. All rghts reserved. do:1.116/.bpc

2 9 R.A. Alberty et al. / Bophyscal Chemstry 155 (211) (2) to update some of the recommendatons gven n 1994, wth partcular attenton to the reportng of expermental results. 2. Introducton Thermodynamcs as normally appled to bochemcal systems dffers from chemcal thermodynamcs n allowng the ph to be specfed n addton to the temperature and pressure. The mportance of ths dfference can be llustrated by the consderaton that the standard state of a solute s defned n chemcal thermodynamcs as a hypothetcal deal soluton at a molalty of 1 mol kg 1 [4]. If ths were done for the hydrogen on n bochemcal reactons, t would make t mpossble to form any mpresson of behavor at neutral ph ust from knowledge of the standard thermodynamc propertes of the speces nvolved; one would also need knowledge of the acd dssocaton constants of all the speces that onzed between ph and ph 7. In practce, therefore, bochemsts have long consdered the standard state of the hydrogen on to be mol L 1, and the standard state of H 2 O s often treated as the concentraton that actually exsts n the system. The practcal effect of the frst pont s that an equlbrum constant K defned at any ph s related n a smple way to the correspondng Gbbs energy of reacton accordng to Eq. (2-7) below. In some bochemcal publcatons the prme s omtted for the equlbrum constant, wrtten smply as K, but ncluded for the standard transformed Gbbs energy Δ r G. However, for the sake of consstency and greater clarty, the prme wll be ncluded for equlbrum constants n ths document, and the standard transformed Gbbs energy Δ r G wll be wrtten wth a subscrpt r. In addton, the qualfcaton transformed wll be used to underlne the fact that an equlbrum constant K s a transformed form of the equlbrum constant K. In most bochemcal contexts the transformaton s assumed, and does not need to be explct, but t s useful to nclude t n any comparson of bochemcal and chemcal conventons. For reactons occurrng n aqueous meda, the practcal effect of defnng the standard state of H 2 O as the pure solvent s that H 2 Ocanbe omtted from expressons for equlbrum constants. However, common practce s nconsstent n ths respect: an expresson for the chemcal reacton catalyzed by an enzyme wll sometmes nclude H 2 Oasareactant, but omt t from the expresson for the Gbbs energy of reacton, and sometmes also from the expresson for the equlbrum constant. Chemcal thermodynamcs treats ons that dffer n ther degree of protonaton as separate speces, but t becomes very complcated f one tres to do ths wth all reactants n an enzyme-catalyzed reacton that have sgnfcant concentratons n the ph range consdered. For example, adenosne trphosphate s a mxture of three speces (ATP 4,HATP 3, and H 2 ATP 2 ) n the mddle ph range, and dssolved carbon doxde s a mxture of four speces (CO 2 (aq), H 2 CO 3,HCO 3,andCO 3 2 ). It becomes much easer to handle these mxtures f the ph s specfed, because the ratos between the equlbrum concentratons of the varous speces are then fxed and thermodynamc propertes of the enttes ATP and CO 2 tot (.e., CO 2 (aq), H 2 CO 3,HCO 3,andCO 3 2 ) can be calculated. A practcal reason for the use of the apparent equlbrum constant K s that most methods of measurement are unable to dstngush between the varous bochemcal speces and, n fact, t s sums of speces that are almost always measured. Ths makes the apparent equlbrum constant a convenent quantty for reportng the results of equlbrum measurements. Before proceedng further, t s necessary to ntroduce some chemcal equlbrum propertes more formally than has been done to ths pont. Chemcal thermodynamcs s based prmarly on the Gbbs energy G, whch s a functon of temperature, pressure, and amounts of speces. Ths s the thermodynamc potental that provdes the crteron for spontaneous change and equlbrum when the ndependent varables are temperature, pressure and amounts of speces. Chemcal thermodynamcs can be sad to be prmarly based on G because the equlbrum constant K for a chemcal reacton s gven by K =exp( Δ r G /RT), where Δ r G s the standard reacton Gbbs energy. If K s determned over a range of temperature, the change n entropy n a chemcal reacton Δ r S can be calculated usng Δ r S = Δ rg : ð2 1Þ T P The change n enthalpy n a chemcal reacton Δ r H can be calculated usng Δ r H = T 2 ðδ r G =TÞ : ð2 2Þ T P If hydrogen ons are requred to balance a chemcal reacton, the concentraton of H + s ncluded n the expresson for the chemcal equlbrum constant. Thus, the hydrolyss of ATP 4 s descrbed by means of a chemcal equaton such as ATP 4 þ H 2 O ¼ ADP 3 þ HPO 2 4 þ H þ ð2 3Þ A chemcal equaton balances atoms of elements and charge. However, f the ph s specfed, the correspondng bochemcal equaton does not balance hydrogens and, therefore, does not balance electrc charge. Chemcal Eq. (2-3) leads to the followng expresson for the chemcal equlbrum constant. h h ADP 3 HPO 2 4 H þ K = : ð2 4Þ ATP 4 ð c Þ 2 Chemcal reactons lke ths are often referred to as reference reactons (see Secton 7). It should be noted that the choce of the reference reacton (2-3) was arbtrary, and that several other reference reactons could have been selected n ts place. Also, the standard concentraton c =1 mol L 1 s ncluded n the above equaton n order to keep the equlbrum constant dmensonless. When bochemcal reactons are consdered at specfed ph they should be wrtten n a form that does not explctly show charges, because charges cannot be balanced f the ph s fxed: ATP þ H 2 O ¼ ADP þ phosphate ð2 5Þ Smlar consderatons apply to fxed pmg: for example, n Eq. (2-5) ATP refers to the equlbrum mxture of ATP 4, HATP 3,H 2 ATP 2, MgATP 2, MgHATP, and Mg 2 ATP at the specfed ph and pmg. The expresson for the apparent equlbrum constant s gven by K = ½ADPŠ½phosphateŠ ½ATPŠc ð2 6Þ Expressons for apparent equlbrum constants K are wrtten n terms of sums of speces, rather than speces, and apparent equlbrum constant K of enzyme-catalyzed reactons usually vary wth the ph. The qualfcaton apparent emphaszes ths dfference from chemcal thermodynamcs. There are several reasons why [H + ] cannot be ncluded n the expresson for an apparent equlbrum constant. The frst s that [H + ]sdefned by the specfed ph, and cannot then be calculated from the value of the apparent equlbrum constant, as n chemcal thermodynamcs; n other words, chemcal thermodynamcs treats [H + ] as a dependent varable, whereas bochemcal thermodynamcs treats t as an ndependent varable, because the nvestgator chooses the ph. A second reason s that f the ph s specfed, t s n prncple held constant durng the reacton, ths beng normally acheved, at least approxmately, by the use of a sutable buffer, or more exactly wth a ph-stat, a devce that holds the ph constant by addng acd or alkal. It follows that hydrogen atoms are not conserved n a bochemcal reacton system lke atoms of other elements. As a buffer wll allow some change n ph as the system

3 R.A. Alberty et al. / Bophyscal Chemstry 155 (211) approaches equlbrum, the ph should be measured at equlbrum, because that s the ph to whch the apparent equlbrum constant refers to. When an apparent equlbrum constant K s used, t s essental to specfy the bochemcal reacton that t refers to, and to nclude all unts, and other relevant condtons, always T, P, ph, and onc strength I, and often pmg or pca as well. The quantty ph s convenently measured wth a ph electrode. However, the quanttes pmg and pca, n the absence of electrodes of suffcent accuracy, are generally calculated from a knowledge of the concentratons of the solutes and the requred metalon bndng constants. The apparent equlbrum constant s related to the standard transformed Gbbs energy of reacton Δ r G by Δ r G = RT lnk : ð2 7Þ The property Δ r G (notδ r G ) has to be used when the ph s specfed because t s a functon of the transformed Gbbs energy of formaton Δ f G that s dscussed later (Secton 5). The ordnary Gbbs energy G does not provde the crteron for spontaneous change and equlbrum when the ndependent varables (varables set by the nvestgator) nclude varables (typcally ph) n addton to T and P. The subscrpt r refers to a reacton and s not necessary, but t s useful n dstngushng the standard transformed Gbbs energy of reacton Δ r G fromthestandard transformed Gbbs energy of formaton Δ f G () of reactant. Although the apparent equlbrum constants and thermodynamc propertes serve many purposes n bochemstry, they cannot replace the chemcal functons entrely, because these reman necessary for studyng changes that occur when the ph (or pmg, etc.) changes: acd dssocatons and dssocatons of complex ons need to be represented by chemcal equatons, and mechansms ncludng ph effects nvolve chemcal equlbrum constants. Effects of onc strength are very mportant n bochemcal thermodynamcs, and so the onc strength should always be specfed n reportng expermental results. In the extended Debye Hückel equaton [5], the onc strength effects are proportonal to the squared electrc charge of an on and so the effect of onc strength on the thermodynamc propertes of ATP 4 are 16 tmes greater than for a chlorde on. The role of onc strength n bochemcal thermodynamcs s very dfferent from the effect of ph because onc strength affects the propertes of the solvent, n the same sense that addng an nert organc lqud changes the propertes of the solvent. Because of these dfferences between chemcal thermodynamcs and bochemcal thermodynamcs, t s necessary to use a Legendre transform (see Eqs. (3-1) and (3-2) and Secton 5) todefne a new thermodynamc potental, the transformed Gbbs energy G, as shown later n Eq. (6-1), that provdes the crteron for spontaneous change and equlbrum when the ph s specfed. The transformed Gbbs energy nvolvng the ph was ntroduced n four papers n 1992 [6 9]. More complete dscussons of the uses of transformed thermodynamc propertes are avalable n two books [1,11]. There s another sgnfcant dfference between chemcal thermodynamcs and bochemcal thermodynamcs. Enzyme-catalyzed reactons conserve atoms of elements other than hydrogen, but some enzyme-catalyzed reactons conserve groups of atoms n addton. For example, the enzyme glutamate ammona lgase (EC ) [12] catalyzes a reacton [13] wth net effect as follows: ATP þ glutamate þ ammona ¼ ADP þ phosphate þ glutamne ð2 8Þ However, although ths process can be regarded as the sum of two smpler reactons: ATP þ H 2 O ¼ ADP þ phosphate glutamate þ ammona ¼ H 2 O þ glutamne ð2 9Þ ð2 1Þ glutamate ammona lgase does not catalyze these reactons separately, though there are other enzymes (EC and EC respectvely) that do. The complete reacton catalyzed by glutamate ammona lgase conserves not only C, O,, and P atoms, but also an addtonal component, components beng the enttes that are conserved n a reacton (n studes of metabolc systems the term moety s sometmes used wth the same meanng). To understand the mportance of components, consder the calculaton of the equlbrum composton for the glutamate ammona lgase reacton: the expresson for the apparent equlbrum constant and conservaton equatons for C, O,, and P provde fve mathematcal relatonshps between the concentratons at equlbrum, but fve equatons cannot contan enough nformaton to allow calculaton of the sx ndependent concentratons of the sx reactants shown n Eq. (2-8). There must, therefore, be an addtonal conservaton relatonshp, and the use of lnear algebra (see Secton 8) allows ths to be dentfed, so that the equlbrum composton can be calculated. Some enzyme-catalyzed reactons nvolve three addtonal components, and, n metabolc systems that nclude numerous reactons, there may be more. For example, a model of glycolyss n the paraste Trypanosoma bruce requred four constrants, three of them obvous (conservaton of adenne nucleotdes, etc.), but the fourth was not at all obvous and nvolved 12 dfferent reactants dstrbuted between two compartments [14]. In such cases, the prncples of the lnear algebra are unchanged from those for sngle reactons, but ther applcaton s only feasble f carred out by computer [15]. 3. Basc thermodynamcs The frst law of thermodynamcs ntroduces the nternal energy U and states that the change n U n a homogeneous system s gven by ΔU=q+w, where q s the heat flow nto the system, and w s the work done on the system. The enthalpy H s defned by H ¼ U þ PV: ð3 1Þ The second law of thermodynamcs ntroduces the entropy S and has two parts. The frst part states that dq/t s ndependent of pathway for a reversble process, where d ndcates a dfferental (an nfntesmal change). Ths makes t possble to defne entropy S by ds=dq rev /T, where the subscrpt rev mples a relatonshp that s vald only for a reversble process. The second part states that when a change of an solated system from one state to another takes place spontaneously, ΔS s greater than zero. Ths provdes a way to predct when a change n state can take place spontaneously, based on calormetrc measurements. However, the second part of the second law cannot be used drectly to predct whether a chemcal reacton wll occur spontaneously at specfed temperature and pressure, because ths depends on addtonal knetc and mechanstc consderatons that are beyond the scope of thermodynamcs. The thrd law of thermodynamcs states that the entropy of each pure element or substance n a perfect crystallne form s zero at absolute zero. Ths provdes a way to determne the entropy of a crystallne substance by makng calormetrc measurements. The use of entropes determned n ths way has allowed for the determnaton of the entropes of a substantal number of crystallne bochemcal substances. These entropes can then be combned wth standard enthalpes of combuston, solubltes, and enthalpes of soluton to calculate the formaton propertes of bochemcal speces n aqueous meda. Ths method has been used [16] to obtan the standard formaton propertes of adenosne and the ATP, ADP, and AMP seres n aqueous meda. Each of the three propertes U, S, andh provdes a crteron for spontaneous change and equlbrum n chemcal thermodynamcs. These crtera are summarzed by (du) S,V, (ds) U,V, and (dh) S,P, where the subscrpts ndcate the propertes that are held constant.

4 92 R.A. Alberty et al. / Bophyscal Chemstry 155 (211) However, none of them apples to changes at constant T and P, andto remedy ths omsson, Gbbs defned the thermodynamc property that we now call the Gbbs energy (often called the Gbbs free energy) G by the use of G = H TS: ð3 2Þ Eqs. (3-1) and (3-2) are referred to as Legendre transforms [17], and each defnes a new thermodynamc property by subtractng a product of ntensve and extensve propertes from a prevously defned thermodynamc property. The pars of propertes are referred to as conugate propertes, and ther products have the dmensons of energy, for example, TS n Eq. (3-2) has the dmensons of energy. ote that the conugate property to V s P, whch accounts for the plus sgn n Eq. (3-1). By subtractng TS from H (see reacton 3-2) one obtans a crteron for spontaneous change and equlbrum at specfed T and P: (dg) T,P. The thermodynamcs of systems of chemcal reactons at specfed T and P s based on the use of G. We are also ndebted to Gbbs for the ntroducton of the chemcal potental μ of speces. The relatons between the varous thermodynamc propertes of a chemcal reacton system are obtaned from the fundamental equaton for G that shows how changes n the ndependent varables T, P, and n (amounts of chemcal speces) determne the dfferental of the Gbbs energy. The fundamental equaton for the Gbbs energy s gven by dg = SdT + VdP + μ dn = SdT + VdP + Δ f G dn ; ð3 3Þ where the dfferental of the amount of speces s dn and s the number of dfferent knds of speces n the system. When makng calculatons n chemcal thermodynamcs, μ can be replaced wth Δ f G, the Gbbs energy of formaton of speces. The ndependent varables for a system wthout reactons are T, P, andn 1, n 2,, n ; these are referred to as the natural varables for the Gbbs energy. Integraton of the fundamental equaton for G at constant values of the ntensve varables yelds G = μ n = Δ f G n : ð3 4Þ The fundamental equaton shows how S, V, and Δ f G can be calculated. For example, Δ f G =( G/ n ) T,P when the amounts of other speces are held constant. 4. Thermodynamcs of chemcal reactons The effects of onc strength on thermodynamc propertes of speces n aqueous soluton are handled n dfferent ways n chemcal thermodynamcs and n bochemcal thermodynamcs, and so onc strength effects n chemcal reactons are dscussed frst. When a chemcal reacton occurs n a closed system, the changes n the amounts n of speces depend on the stochometrc numbers ν n the balanced chemcal equaton (ν s the Greek letter nu): v B =; ð4 1Þ where B represents speces, and the ν values are postve for products and negatve for reactants. The amount n of speces at any stage n a chemcal reacton s gven by n = n + v ξ; ð4 2Þ where n s the ntal amount of speces n the system and ξ s the extent of reacton (ξ s the Greek letter x). It s evdent from ths defnton that ξ s an extensve property wth the unt mole. The dfferental of the amount of speces s gven by dn = v dξ: ð4 3Þ When a sngle chemcal reacton occurs n a closed system, substtuton of Eq. (4-3) n Eq. ( ) yelds dg = SdT + VdP + v Δ f G!dξ = SdT + VdP + Δ r Gdξ; ð4 4Þ where Δ r G s the reacton Gbbs energy gven by Δ r G = G ξ T;P = v Δ f G : ð4 5Þ At equlbrum, the Gbbs energy of the system s at a mnmum wth ( G/ ξ) T,P =. At the mnmum Gbbs energy, the equlbrum condton s v Δ f G eq =; ð4 6Þ where Δ f G eq s the Gbbs energy of formaton of speces at equlbrum. otce that ths relaton has the same form as the chemcal equaton (Eq. (4-1)). The Gbbs energy of formaton Δ f G of a speces n aqueous soluton depends on the concentraton of the speces accordng to Δ f G = Δ f G + RTlnðγ ½B ŠÞ; ð4 7Þ where Δ f G s the standard Gbbs energy of formaton of speces, γ s the actvty coeffcent of speces, and [B ] s ts concentraton. Actvty coeffcents are dmensonless, and so the logarthmc term should be dvded by c, whch s the standard concentraton (1 M), so that the logarthm s taken of a dmensonless quantty. But the c s omtted as a smplfcaton. In dlute aqueous solutons, γ s equal to unty for uncharged speces, but γ of an on depends on ts electrc charge, the onc strength, and temperature. When actvty coeffcents are used n ths way, Δ f G, Δ f H, and K are ndependent of onc strength. In bochemcal thermodynamcs, t s more practcal to take Δ f G, Δ f H, and K to be a functon of onc strength when dealng wth speces. Eq. (4-7) can be wrtten as Δ f G = Δ f G + RTlnγ + RTln½B Š: In bochemcal thermodynamcs, ths equaton s wrtten as Δ f G = Δ f G + RTln½B Š; ð4 8Þ ð4 9Þ where the standard Gbbs energy of formaton Δ f G s a functons of onc strength as well as temperature. Ths s useful because, when chemcal equlbrum constants K are needed n bochemcal thermodynamcs (for example, K a and K ref ), they are expressed n terms of the concentratons of speces wthout actvty coeffcents. When ths s done, K s a functon of the onc strength. Substtutng Eq. (4-9) at equlbrum n Eq. (4-6) yelds v Δ f G = RT v ln½b Š eq = RT ln ½BŠ v eq = RT lnk; ð4 1Þ

5 R.A. Alberty et al. / Bophyscal Chemstry 155 (211) where K s K = ½B Š v eq : ð4 11Þ Ths equaton s often used wthout the subscrpts eq that ndcates that the concentratons are equlbrum values. Eq. (4-1) shows that the standard reacton Gbbs energy Δ r G s gven by Δ r G = v Δ f G = RT lnk: Substtutng Eq. (4-9) n Eq. (4-5) yelds Δ r G = Δ r G + RTlnQ; ð4 12Þ ð4 13Þ where the reacton quotent Q has the form of the expresson for the equlbrum constant, but wth arbtrary concentratons of speces. In calculatng the equlbrum composton for a chemcal reacton system, t s mportant to understand the concept of components (see end of Secton 2). Here, we use the chemcal defnton of components as the substances of fxed composton (.e., pure chemcal compounds) that comprse a mxture. The number of components C n a mxture s the mnmum number of these chemcal substances that are needed to prepare ths mxture n all of the phases n whch t exsts. Of course the atoms of all elements are conserved n chemcal thermodynamcs, but the number of components s often less than the number of elements because only ndependent conservaton equatons can be used to calculate the equlbrum composton. Varous choces can be made of components, but the number of components C s ndependent of the choce. In a mult-reacton system, there are also varous sets of chemcal reactons that can be used to calculate the equlbrum composton, but the number R of ndependent reactons s ndependent of the choce. When a chemcal reacton system contans speces, =C+R. The value of a chemcal equlbrum constant must always be accompaned by a chemcal equaton, the drecton of the reacton to whch the equlbrum constant refers, and, when Eq. (4-9) s used n aqueous solutons (as t s n bochemcal thermodynamcs), the onc strength must be specfed n addton to the temperature. Each speces n a chemcal reacton contrbutes ts Δ f G to the standard reacton Gbbs energy Δ r G and to the equlbrum constant; ths makes t possble to construct tables of standard thermodynamc propertes of speces at zero onc strength. Measurements of K over a range of temperatures make t possble to calculate Δ f G, Δ f H, and Δ f S (or S ) for speces. Calormetrc measurements of enthalpes of reacton make t possble to calculate Δ f H, and calormetrc measurements on crystals down to close to absolute zero make t possble to obtan standard molar entropes of crystallne substances. These propertes of speces are gven n the BS Tables [4] and CODATA Tables [18] at K (25 C) and zero onc strength. IUPAC's Quanttes, Unts and Symbols for Physcal Chemstry, 3rd ed. [19] ncludes symbols for chemcal thermodynamcs. 5. Legendre transform to ntroduce the ph as an ndependent varable n bochemcal thermodynamcs In hs book on thermodynamcs from the vewpont of a physcst, Callen [2] wrtes the choce of varables n terms of whch a gven problem s formulated, whle a seemngly nnocuous step, s often the most crucal step n the soluton. He s referrng to ndependent varables,.e., varables under the control of the nvestgator. Specfyng the ph may seem to be an nnocuous step, but t has maor effects n bochemcal thermodynamcs. The ndependent varables for a thermodynamc system determne the equlbrum state that s fnally reached. Intensve ndependent varables have extensve conugates, and extensve ndependent varables have ntensve conugates. Snce the equlbrum compostons reached by most enzyme-catalyzed reactons are affected by the ph, t s necessary to defne a new thermodynamc property, the transformed Gbbs energy G by use of a Legendre transform (see frst part of Secton 3) n whch a product of conugate varables n c (H)μ(H + )s subtracted from G [6,7]. The amount of the hydrogen component n c (H) s the total amount of hydrogen atoms n the chemcal reacton system that s gven by n c (H)= H ()n,where H () sthenumberofhydrogen atoms n speces. The Legendre transform that defnes the transformed Gbbs energy G s n G = G n c ðhþμðh þ Þ = G n c ðhþ Δ f G o ðh þ Þ RTlnð1ÞpH : ð5 1Þ The second form of ths equaton shows how the chemcal potental of H + (aq) s related to the ph. (ote: hydron s the general name for the caton H + wthout regard to sotopc state.) The standard Gbbs energy of formaton of H + (aq), Δ f G (H + ), s equal to zero at zero onc strength by defnton, but t s not equal to zero at hgher onc strengths. The ph n Eq. (5-1) s defned as log 1 [H + ], rather than as log 1 {γ(h + )[H + ]}. At K, the dfference (ph log 1 {γ(h + )[H + ])=.11 at.1 mol L 1 onc strength and.14 at.25 mol L 1 onc strength [21]. Logarthms can only be calculated for dmensonless quanttes. Usng a Legendre transform s the only way the ph can be made an ndependent varable. The fundamental equaton for the transformed Gbbs energy s 1 dg = S dt + VdP + μ dn + RTlnð1Þn c ðhþdph; ð5 2Þ where S =S n c (H)S(H + ). (Ths s the Legendre transform defnng the transformed entropy at specfed ph. There s a correspondng Legendre transform defnng the transformed enthalpy H at specfed ph.) The transformed chemcal potental of speces s gven by μ = μ H ðþμðh þ Þ: ð5 3Þ ote that the term for H + n the summaton n Eq. (5-2) s equal to zero because μ(h + ) H (H + )μ(h + )=. The Legendre transform has replaced the term μ(h + )dn(h + ) n the summaton wth a new type of term, RTln(1)n c (H)dpH, where n c (H) s the amount of hydrogen atoms n the system. Then, 1 dg = S dt + VdP + Δ f G dn + RTlnð1Þn c ðhþdph: ð5 4Þ Here, μ has been replaced by Δ f G to permt numercal calculatons to be made. The summaton n Eq. (5-4) has a term for each speces, except for H +, but Δ f G has the same value for ATP 4,HATP 3,andH 2 ATP 2,forexample [11]. ThusEq.(5-4) can be wrtten, as follows, n terms of amounts of reactants (sums of speces lke the three speces of ATP) n and the transformed Gbbs energy of formaton of the reactants Δ f G : dg = S dt + VdP + Δ f G dn + RTlnð1Þn c ðhþdph: ð5 5Þ =1 The speces have been represented by, and so the reactants (sums of speces) are represented by. There are dfferent reactants. The natural varables for the transformed Gbbs energy are T, P, ph, and n 1, n 2,, n. Ths reducton n the number of natural varables for G s a result of the specfcaton of the ph. The transformed Gbbs energy provdes the followng crteron for spontaneous change and equlbrum: (dg ), where the temperature, pressure, ph, onc strength, and amounts of reactants are specfed. Fundamental Eq. (5-5) shows how S, V, Δ f G,andRTln(1)n c (H) can be calculated from G. ote that n c (H)=Σn H ðþ. The average number of hydrogen atoms n a reactant

6 94 R.A. Alberty et al. / Bophyscal Chemstry 155 (211) s represented by H (). Eq. (5-5) shows that Δ f G =( G / n )=Δ f G + RTln[B ]atconstantt, P, ph and amounts of other reactants. It s the Maxwell relatons [21,22], whch are mxed partal dervatves, that yeld useful relatons between the varous thermodynamc propertes. If the VdP term s gnored and the transformed enthalpy H s ntroduced usng G =H TS, there are fve Maxwell relatons. They are wrtten here n terms of standard formaton propertes for reactants because the concentraton terms cancel on the two sdes of the equatons: Δ f S = Δ f G T Δ f S T Δ f G ph = RT lnð1 = RT ln ð 1 Þ HðÞ Þ T HðÞ T Δ f H Δ fg = T 2 = T T ð5 6Þ ð5 7Þ ð5 8Þ ð5 9Þ Δ f H ph = RT2 lnð1þ HðÞ : ð5 1Þ T ote that ln(1) s approxmately These equatons are mportant because they show that when Δ f G can be expressed as a functon of T, ph, and onc strength based on expermental data, all the other thermodynamc propertes of a bochemcal reactant can be obtaned by takng partal dervatves. ote that there are alternatve routes to ( Δ f S / ph) and ( Δ f H / ph) because they can be calculated drectly or they can be calculated from the functon for H (). 6. Equatons for the standard transformed formaton propertes of a reactant Eq. (5-3) shows that the speces that make up a reactant have transformed Gbbs energes of formaton Δ f G as well as Gbbs energes of formaton Δ f G. Δ f G = Δ f G H ðþδ f GðH þ Þ ð6 1Þ Substtutng Δ f G =Δ f G +RTln[B ] and Δ f G[H + ]=Δ f G [H + ]+RTln (1 ph ) nto Eq. (6-1) yelds Δ f G = Δ f G n + RTln½B Š H ðþ Δ f G o ½H þ Š RTlnð1ÞpHÞ where z s the charge on speces, and α s the parameter n the Debye-Hückel equaton that can be represented by a power seres n the temperature [23]. The 1.6 kg 1/2 mol 1/2 s an emprcal parameter that s assumed to be ndependent of temperature. When Eq. (6-4) s substtuted for Δ f G and Δ f G [H + ] n Eq. (6-3), the followng equaton yelds the standard transformed Gbbs energy of speces at a specfed onc strength: Δ f G ðþ= I Δ f G ði =Þ H ðþrtln ð1þph α z 2 H ðþ I 1 = 2 = 1+1:6I 1 = 2 : ð6 5Þ Ths shows that Δ f G (I) safunctonofoncstrengthforuncharged speces that contan hydrogen atoms, as well as charged speces. There s an excepton to ths statement when z 2 H ()= for a speces. The standard transformed Gbbs energy of formaton of a speces s ndependent of onc strength and ph when z = and H ()=. Eq. (6-5) shows how the standard transformed Gbbs energy of formaton of a bochemcal reactant consstng of a sngle speces s calculated from the standard Gbbs energy of formaton of the speces at zero onc strength. When there are two or more speces n a pseudosomer group (lke ATP 4, HATP 3, and H 2 ATP 2 at specfed ph), the standard transformed Gbbs energy of formaton Δ f G of the pseudosomer group s calculated usng [24] ( Δ f G so h = RTln exp Δ f G ) = RT ; ð6 6Þ where so s the number of speces n the pseudosomer group. The standard transformed enthalpy of reactant s gven by Δ f H = so r Δ f H ; ð6 7Þ where r s the equlbrum mole fracton of speces that s gven by nh r = exp Δ f G ðreactantþ Δf G o = ðrtþ : ð6 8Þ These calculatons are suffcently complcated that a computer s requred, as well as software wth symbolc and calculus capabltes [11,25]. Mathematcal applcatons for computers can be used to derve the functon of ph, and onc strength that yelds the standard transformed Gbbs energy Δ f G of a reactant at K by combnng Eqs. (6-5) and (6-6). The functon for Δ f G s readly evaluated for desred phs, and onc strengths. The average number of hydrogen atoms H () n the reactant at K at the specfed ph and onc strength can be calculated usng Eq. (6-8). Ths property can also be calculated usng = Δ f G + RTln½B Š; ð6 2Þ H ðþ= r HðÞ ð6 9Þ where the standard transformed Gbbs energy of formaton of speces s gven by Δ f G = Δ f G n HðÞ Δ f G o ½H þ Š RTlnð1ÞpHÞ : ð6 3Þ The standard Gbbs energes of formaton Δ f G andδ f G (H + ) depend on the onc strength I. The standard Gbbs energy of formaton of speces at onc strength I can be estmated usng the extended Debye Hückel equaton [5]. Δ f G ðþ= I Δf G 2 ði =Þ αz I 1 = 2 = 1+1:6I 1 = 2 ; ð6 4Þ where s the number of dfferent speces and r s the equlbrum mole fracton of the th speces n the reactant that can be calculated usng the bndng polynomal [26 28], whch utlzes the pks of speces. When the standard enthalpes of formaton Δ f H at zero onc strength are known at K for the speces of a reactant, the Δ f G (T,I=) can be calculated over a range of temperatures usng [28,29] Δ f G T ðt; I =Þ = 298:15 K Δ f G ð298:15 K;I =Þ + 1 T Δ 298:15 K f H ð298:15 K; I =Þ: ð6 1Þ

7 R.A. Alberty et al. / Bophyscal Chemstry 155 (211) Ths s based on the assumpton that Δ f H s ndependent of temperature over a modest temperature range, e.g K to K. Other propertes at specfed temperature, ph, and onc strength can be calculated by the use of Eqs. (5-6) to (5-1). Tables of transformed thermodynamc propertes of reactants can be produced, but ths does not fully satsfy the needs of bochemsts. It s more useful to have mathematcal functons n a computer that gve these propertes so that they can be calculated at desred temperatures, phs, and onc strengths (see Secton 1). 7. Thermodynamcs of bochemcal reactons When a bochemcal reacton occurs alone, the changes n the amounts n of reactants depend on the stochometrc numbers v n the balanced bochemcal reacton equaton. The balanced bochemcal equaton s represented by v B =; ð7 1Þ =1 where B s a sum of speces and s the number of dfferent reactants. Bochemcal reactons balance the atoms of all elements except for hydrogen when they are carred out at a specfed ph. (If pmg s held constant, the bochemcal equaton does not balance magnesum atoms.) The stochometrc numbers v are postve for products and negatve for reactants. The amount n of reactant at any stage n a bochemcal reacton s gven by n = n + v ξ ; ð7 2Þ where n s the ntal amount of reactant and ξ s the extent of bochemcal reacton. The dfferental of the amount of reactant s gven by dn = v dξ : ð7 3Þ When a sngle bochemcal reacton occurs, the dfferental of the transformed Gbbs energy (see Eq. (5-5)) s gven by dg = S dt + VdP + v Δ f G!dξ + RTlnð1Þn ðhþdph =1 The rate of change of G wth extent of reacton for a system havng a sngle reacton at constant T, ph, and onc strength s gven by G ξ T;P;pH = v Δ f G = Δ r G = Δ r G o + RTlnQ =1 ð7 7Þ where the bochemcal reacton quotent Q has the form of the apparent equlbrum constant, but wth arbtrary concentratons of reactants. It s the sgn of Δ r G that determnes whether a bochemcal reacton wll go to the rght or the left at specfed reactant concentratons: a negatve value ndcates that the reacton can spontaneously go to the rght. The transformed Gbbs energy of the system s at a mnmum at equlbrum, where ( G / ξ ) T,P,pH =. At the mnmum transformed Gbbs energy, the equlbrum condton s v Δ f G eq =: ð7 8Þ =1 otce that ths relaton has the same form as the bochemcal equaton (Eq. (7-1)). To calculate the apparent equlbrum constant for an enzymecatalyzed reacton, the mathematcal functons yeldng Δ f G are added and subtracted accordng to Δ r G = =1 v Δ f G = RTlnK ; ð7 9Þ where the apparent equlbrum constant K s gven by K = ½ Š v : ð7 1Þ B =1 Whenever the value of an apparent equlbrum constant s gven, t s necessary to show the way the bochemcal reacton s wrtten. It s possble to make tables wth values of K at specfed temperatures, phs, and onc strengths, but t s much more useful to have mathematcal functons for K n a computer that can be evaluated at desred temperatures, phs, and onc strengths. Other thermodynamc propertes of the enzyme-catalyzed reacton can be obtaned by takng partal dervatves of Δ r G. Apparent equlbrum constants that depend on ph can be expressed by = S dt + VdP + Δ r G dξ + RTlnð1Þn ðhþdph; ð7 4Þ K = K ref 1 nph f ðphþ; ð7 11Þ where Δ r G s the transformed reacton Gbbs energy. The Maxwell equatons of ths fundamental equaton are lke Eqs. (5-6) to (5-1), except that the changes n propertes apply to a bochemcal reacton rather than a reactant, and H s replaced by Δ r H, the change n bndng of H + n the bochemcal reacton. The thrd and fourth terms n Eq. (7-4) show that the change n bndng of H + n an enzymecatalyzed reacton s gven by [27] 1 Δ Δ r H = r G RT lnð1þ ph : ð7 5Þ Ths property of an enzyme-catalyzed reacton can also be calculated usng [27,28] Δ r H = v HðÞ; =1 where H () s gven by Eq. (6-9). ð7 6Þ where K ref s a chemcal equlbrum constant, n s the number of H + n the reference reacton (a postve nteger f H + s produced and a negatve nteger f H + s consumed). The functon f(ph) brngs n the pks of the substrates. In thermodynamcs, the choce of a reference reacton s arbtrary, and so n does not have a mechanstc sgnfcance, but ths equaton does suggest there are two types of ph effects those nvolvng pks that extend over a couple of unts of ph and effects of 1 nph that extend over the whole range of ph consdered. The value of n can be determned from knetc data. Ths secton has dealt wth the ph as an ndependent varable, but that rases a queston as to what other ntensve varables can be ntroduced nto bochemcal thermodynamcs. When the reacton system contans magnesum ons and magnesum complex ons are formed wth one or more reactants, the followng Legendre transform can be used to ntroduce pmg= log 1 [Mg 2+ ] as an ndependent varable n addton to ph: G = G n c ðhþμðh þ Þ + n c ðmgþμðmg 2+ Þ: ð7 12Þ

8 96 R.A. Alberty et al. / Bophyscal Chemstry 155 (211) where n c (Mg) s the amount of the magnesum component, that s the total amount of magnesum atoms n the system. Snce dssocaton constants of magnesum complex ons are known for the ATP seres, the effects of pmg on standard transformed thermodynamc propertes n the ATP seres have been calculated [3]. Snce H + and Mg 2+ compete, the effect of ph on the bndng of Mg 2+ by a reactant s equal to the effect of pmg on the bndng of H + : H pmg = Mg ph ; ð7 13Þ where the frst partal dervatve s at constant ph and the second partal dervatve s at constant pmg. These are called recprocal effects. When H 2 O s a reactant n an enzyme-catalyzed reacton, there s a problem n calculatng equlbrum compostons because [H 2 O] s omtted n the expresson for the apparent equlbrum constant, but Δ f G (H 2 O) s requred n the calculaton of the apparent equlbrum constant. Ths problem, whch s dscussed n the next secton, can be solved by usng a Legendre transform to defne a further transformed Gbbs energy G usng the followng Legendre transform [31,32]. G ¼ G n c ðoþμ ðh 2 OÞ¼G n c ðoþδ f G ðh 2 OÞ ð7 14Þ Ths s possble because, when H 2 O s nvolved n an enzymecatalyzed reacton, oxygen atoms are not conserved because an essentally nfnte amount of oxygen atoms (n comparson wth the amounts of reactants) s avalable from the solvent. Ths transformaton leads to the further transformed Gbbs energy of formaton Δ f G of a reactant. Snce Δ f G (H 2 O)=, H 2 O can be left out of the calculaton of the standard transformed Gbbs energy of formaton of an enzyme-catalyzed reacton. Δ r G = v Δ f G = RT lnk = ð7 15Þ The ν are stochometrc numbers when H 2 O s omtted, and reactants are defned as pseudosomer groups when oxygen atoms are not conserved. In makng calculatons on systems of bochemcal reactons, t may be of nterest to specfy the concentratons of coenzymes because they are nvolved n many reactons and may be n steady states. Ths provdes a more global vew of a system of bochemcal reactons [33]. When the concentratons of coenzymes are specfed, the crteron for spontaneous change and equlbrum s provded by the further transformed Gbbs energy defned by the followng Legendre transform: G = G n c ðcoenzymeþ Δ r G ðcoenzymeþ + RT ln½coenzymeš ; ð7 16Þ where n c (coenzyme) s the amount of the coenzyme component. The dentfcaton of components s dscussed n the next secton. A reacton system has a fnte number of components, and Legendre transforms can be used to specfy all but one component. Applyng Legendre transforms to all components n a system of reactons yelds the Gbbs Duhem equaton that s a relaton between the ntensve propertes of a system that s equal to zero. Thus, all but one of the ntensve varables for a reacton system at equlbrum can be specfed. 8. Stochometry 8.1. Stochometry of chemcal reactons The stochometry of chemcal reactons s well understood, but t nvolves more than the fact that chemcal reactons balance atoms of elements and electrc charge. A chemcal equaton can be represented by a matrx multplcaton, and ths s of more than ust theoretcal nterest, because t means that technques developed n lnear algebra for matrx manpulaton can be appled drectly to chemcal equatons. Consder the oxdaton of methane to carbon doxde and water n the gas phase, whch can be wrtten as CH 4 2O 2 þ CO 2 þ 2H 2 O ¼ ð8 1Þ Each speces n the reacton can be represented by a column vector gvng the number of carbon atoms, hydrogen atoms, and oxygen atoms: = 4 5 ð8 2Þ Ths equaton can be wrtten as a matrx multplcaton: = 4 5 ð8 3Þ 1 2 The matrx on the left (a 3 4 matrx) s referred to as a conservaton matrx (A), and the 4 1 column vector s referred to as a stochometrc number matrx (ν) because t gves the stochometrc numbers n Eq. (8-1). Ths matrx multplcaton s represented by Aν = ; ð8 4Þ where s the correspondng zero matrx. Ths equaton s useful because t makes t possble to calculate the stochometrc number matrx from the conservaton matrx. Ths operaton s called calculatng the null space of A, and computer applcatons can be used to carry out ths operaton. It s necessary to say that ths produces a bassforthe stochometrc number matrx because the stochometrc number matrx s not unque; for example, multplyng a balanced chemcal equaton by an nteger yelds a balanced chemcal equaton. Eq. (8-4) can also be wrtten n terms of the transposed matrces. ν T A T = ð8 5Þ where T ndcates the transpose. Calculatng the null space of ν T yelds the transposed conservaton matrx, or more precsely a bass for A T. The number of dfferent speces n a chemcal reacton s represented by, the number of dfferent elements (that s, components) s represented by C, and the number of reactons s represented by R. ote that =C+R. More precsely, the number of components C s the number of ndependent elements (that s, elements wth conservaton equatons that do not dffer by an nteger factor), and R s the number of ndependent reactons. For a multreacton system, the conservaton equaton s C and the stochometrc number matrx s R, so that the zero matrx s (C )( R)= C R Stochometry of bochemcal reactons Bochemcal equatons can also be wrtten as matrx multplcatons. The hydrolyss of ATP to ADP at a specfed ph can be wrtten as ATP H 2 O þ ADP þ phosphate ¼ ð8 6Þ Each reactant s represented by a column vector gvng the number of carbon atoms, oxygen atoms, ntrogen atoms, and phosphorus atoms: = ð8 7Þ

9 R.A. Alberty et al. / Bophyscal Chemstry 155 (211) Hydrogen atoms are not conserved because the ph s specfed. Row 3 can be deleted because t s a scalar product of row 1 and therefore provdes no new nformaton. Therefore, the ntrogen row s deleted to obtan = 4 5 ð8 8Þ Ths bochemcal equaton can be wrtten as a matrx multplcaton = ð8 9Þ Ths matrx multplcaton s represented by A ν = ð8 1Þ The prmes are needed to ndcate that the reactons are wrtten n terms of reactants (sums of speces) at specfed ph. Calculatng the null space of A yelds ν. Calculatng the null space of (ν ) T yelds (A ) T. T T ν A = ð8 11Þ ote that the conservaton matrx s C, the stochometrc number matrx s R, and =C +R. The number of components n an enzyme-catalyzed reacton s mportant because the conservaton equatons for components are constrants on the equlbrum that can be reached. As mentoned n Secton 2, couplng by the enzyme mechansm ncreases the number of components beyond the number expected for the conservaton of elements other than hydrogen. Thus, n enzyme-catalyzed reactons, the number of components can be larger than the number of dfferent elements. In Eqs. (8-3) and (8-9), the elements are taken to be components, but the row reducton of A shows that alternatvely reactants can be taken as components. Row reducton of A for ATP hydrolyss (see Eq. (8-9)) yelds the followng conservaton matrx: ATP H 2 O ADP Phosphate ATP 1 1 H 2 O 1 1 ADP 1 1 Ths shows that the components can be taken to be ATP, H 2 O, and ADP, rather than C, O, and P. Couplng ntroduces components beyond the elements, and ths s the way these addtonal components can be assocated wth reactants [33]. Fundamental equatons and Maxwell equatons can be wrtten n matrx notaton, whch s useful n makng calculatons on systems of enzyme-catalyzed reactons [1]. One of the most basc calculatons n bochemcal thermodynamcs s the calculaton of the equlbrum composton. When ths s done for chemcal reactons or enzyme-catalyzed reactons, there s no analytc soluton, and the equlbrum composton has to be calculated by use of an teratve method. Thermodynamc equlbrum calculatons are dscussed n the lterature [1,11,25]. There are specal problems when H 2 O s a reactant because ts concentraton does not change. 9. Standard apparent reducton potentals for half reactons of enzyme-catalyzed reactons Any oxdoreductase reacton can be wrtten as the sum of two half reactons, that are ndependent of each other n the sense that the two electrochemcal half reactons smply exchange formal electrons. In enzyme-catalyzed reactons these electrons are exchanged between half reactons through groups n the catalytc ste. When the speces propertes of reactants n a half reacton have been determned, the standard apparent reducton potental E for the half reacton can be calculated over a range of ph. In wrtng half reactons for enzymecatalyzed reactons, H + are omtted because t s understood that H + are suppled or neutralzed to hold the ph at the specfed value. (Ths s ust a specfc nstance of the general polcy of not ncludng H + n the expressons for bochemcal reactons.) Half reactons do balance atoms, except for hydrogen atoms, but they do not balance electrc charge. If the Δ f H of the speces n all the reactants are known n addton to Δ f G, Δ r G can be calculated over a range of temperature as well as ph by usng E = Δ r G / ν e F. The number of electrons exchanged between the half reactons s ν e, and F s the Faraday constant ( C mol 1 ). Then the E for the half reacton can be calculated over a range of temperature as well as ph [34]. Tables of standard apparent reducton potentals are arranged n such a way that any oxdzed reactant (sum of speces) wll react wth the reduced reactant n a half reacton wth a lower E when the reactant concentratons are all 1 M, except for H 2 O. The reactant n ths sentence can be a sum of reactants, lke CO 2 tot+pyruvate. The electrons n the two half reactons of an oxdoreductase reacton must cancel. The rule n makng such a table s that there should be no fractonal stochometrc numbers, but half reactons can be multpled by an nteger or dvded by an nteger wthout changng the standard apparent reducton potental of the half reacton. Half reactons are wrtten by conventon as reducton reactons,.e. wth the electrons appearng as reactants on the left-hand sde of each half reacton. In aqueous solutons, standard apparent reducton potentals at ph 7 for half reactons of bochemcal nterest are largely restrcted to the range.81 V to.42 V at K, ph 7, and.25 mol L 1 onc strength because half reactons wth potentals hgher than.81 V lead to the spontaneous producton of O 2 gas and half reactons wth potentals lower than.42 V lead to the spontaneous producton of H 2 gas. O 2 ðgþþ4e ¼ 2H 2 O E ¼ :81V ð9 1Þ 2e ¼ H 2 ðgþ E ¼ :42V ð9 2Þ In the hydrogen half reacton, H + s not shown on the left because t s understood that the half reacton occurs at a specfed ph. The standard transformed Gbbs energy of an oxdoreductase reacton s gven by Δ r G = ve FE = ve F E h E 1 Þ = Δr G h Δr G 1 ; ð9 3Þ where E s the standard apparent electrode potental for the oxdoreductase reacton. E h s the standard apparent reducton potental for the half reacton hgher n the table, and E l s the standard apparent reducton potental lower n the table. Δ r G h andδ r G l are the standard transformed Gbbs energes of the two half reactons. The apparent equlbrum constant for an oxdoreductase reacton s gven by K = exp v e FE = RT : ð9 4Þ If E can be calculated or measured over a range of phs, the change n bndng of H + n the oxdoreductase reacton can be calculated from the dfference n E / ph for the two half reactons. The change n bndng of H + n the oxdoreductase reacton s equal to the dfference between the change n bndng n the hgher half reacton Δ r Hh and the change n bndng n the lower half reacton Δ r Hl. Δ r H = Δ r Hh Δ r Hl = ν e F E h RTlnð1Þ ph E l : ð9 5Þ ph