Chemical and biochemical thermodynamics reunification (IUPAC Technical Report)

1 Introduction

In a chemical reaction, all species involved are explicitly considered and their atoms and charges balanced. At equilibrium, the thermodynamic (or standard) equilibrium constant, K, for the chemical reaction is

where γ i and c _i are the activity coefficient and the concentration (molarity) of the ith chemical species (reactant or product), respectively; c i 0 is the standard concentration of species i; and we assume that all c i 0 are equal to 1 mol dm⁻³ (we assume that this does not influence the results presented here). This is the case for the “chemical” convention. It differs from the biochemical convention, which, for example, uses 10⁻⁷ mol dm⁻³ as the standard concentration for the species H⁺. ν i is the corresponding stoichiometric coefficient (positive for products and negative for reactants) and eq stands for equilibrium. K is therefore of dimension 1 (“dimensionless”) and defined by the relation Δ_r G ⁰ = ‒RT ln K [1], [2], [3], [4], [5], [6], [7].

A biochemical reaction involves many species and simultaneous ancillary reactions. Therefore, the determination of the equilibrium composition requires many equilibrium expressions and many conservation equations. A different approach has thus far proved more suitable.

Networks of biochemical reactions take place under steady-state conditions in living organisms. Steady state is the typical condition of the living systems that allows work to be done by chemical reactions at maximal efficiency, or, in other words, to increase entropy at the minimal rate [8]. Another typical feature of living systems is homeostasis, which has the effect of keeping the conditions at which biochemical reactions occur within a narrow range. For example, in a biochemical reaction, the concentration of certain ancillary chemical species, such as H⁺ and Mg²⁺ ions, remains essentially constant.

The polyanionic nature of the molecules involved in biochemical reactions implies that these reagents may act as Lewis bases B and react with Lewis acids, such as H⁺ and Mg²⁺, to form adducts such as BH⁺, BMg²⁺, etc. Consequently, the biochemical reagents are often a mixture of different chemical species. Therefore, biochemical reactions require the use of an equilibrium constant, usually named the apparent equilibrium constant or conditional equilibrium constant. This is the equivalent of an equilibrium constant for a biochemical reaction, where reagents and products are written in terms of the sum of species instead of specific species [4], [5], [6], [7]. It is valid at a given pH, temperature, and ionic strength.

Given the biochemical reaction a A + b B = c C + d D, where A, B, C, and D are biochemical reagents (“sum of species”), the conditional equilibrium constant is as follows (activity coefficients γ _i are assumed equal to 1 and are therefore omitted for sake of clarity, see below):

where c ⁰ is the standard concentration [4], [5], [6], [7] considered equal for all species (1 mol dm⁻³, according to the “chemical convention”). The value of the conditional equilibrium constant K′ depends, besides T, p, and I, also on pH and pMg (according to IUPAC convention, in this paper pH and pMg are defined as − log 10 a H + and − log 10 a Mg 2 + , respectively [1], [2], [3]). In vivo, the biochemical reactions occur at constant pH and pMg.

The amount of H⁺ and Mg²⁺ bound to the chemical species of the reactants A and B is different from the amount of these ions bound to the chemical species of the products C and D. Ions H⁺ and Mg²⁺ are thus produced or consumed during the course of the reaction. For this reason, chemical and biochemical reactions require a different thermodynamic formulation, since chemical equations are written in terms of specific ionic and elemental species and balance elements and charge, whereas biochemical equations are written in terms of biochemical reactants that consist of species in equilibrium with each other and do not balance elements that are assumed to be fixed, such as H⁺ and Mg²⁺. Therefore, when pH and pMg are specified, the conditional equilibrium constant K′ for a biochemical reaction is written in terms of sums of species and can be used to calculate a standard Gibbs energy of reaction Δ_r G′⁰, where the prime symbol (′) has been used to indicate that H⁺ and Mg²⁺ ions concentrations are constant, but not at the standard value.

For example, in the cytosolic solution, the biochemical reactant ATP (adenosine 5′-triphosphate) is composed of the chemical species ATP⁴⁻, HATP³⁻, MgATP²⁻, MgHATP⁻, H₂ATP²⁻, and Mg₂ATP. The biochemical reaction of the hydrolysis of ATP is formally written as:

where ADP (adenosine 5′-diphosphate) is understood to be composed of the chemical species ADP³⁻, HADP²⁻, H₂ADP⁻, MgADP⁻, and MgHADP; and P_i (inorganic phosphate) is similarly understood to be composed of the chemical species PO 4 3 − , HPO 4 2 − , H 2 PO 4 − , and MgHPO₄.

Alberty defined pseudoisomers as all the chemical species that form a biochemical reagent [4], [5], [6], [7]. Referring to what is written above, the number of pseudoisomers of ATP are six, while those of ADP are five and those of P_i are four. The complex species are the adducts of these Lewis bases with the Lewis acids H⁺ and Mg²⁺. The equilibria between the pseudoisomers of ATP, ADP, and P_i are shown in Fig. 1.

Fig. 1:

Equilibria between the pseudoisomers of ATP (a), ADP (b) and P_i (c) (from. ref. 16).

However, MgATP²⁻ is the active species in enzyme binding in the cellular active transport and the form responsible for the energy production and muscular contraction [9]. As a consequence, the biologically relevant ATP pseudoisomer is MgATP²⁻ and the chemical reaction related to ATP hydrolysis is:

Taking into account the equilibrium H₂PO₄ ⁻ ⇆ H⁺ + HPO₄ ²⁻, the equilibrium constant for the chemical reaction (3) can be written as:

The value of K depends on temperature T, pressure p, and ionic strength I of the solution. The conditional equilibrium constant for the biochemical reaction (2) is

Besides T, p, and I, its value depends also on pH and pMg. The corresponding values of the standard Gibbs energy of reaction can be obtained from the values of the equilibrium constants:

The Δ_r G′⁰, named the standard “conditional” Gibbs energy of reaction, is the reference state Gibbs energy of reaction for the biochemical reaction under the specific experimental condition of pH and pMg.

In the “Recommendations for nomenclature and tables in biochemical thermodynamics”, the IUPAC-IUBMB Joint Commission on Biochemical Nomenclature (JCBN) states [7], “When pH and pMg are specified, a whole new set of transformed thermodynamic properties come into play. These properties are different from the usual Gibbs energy G, enthalpy H, and entropy S and they are referred to as the transformed Gibbs energy G′, transformed enthalpy H′, transformed entropy S’.” As a consequence, two categories of thermodynamics based on different concepts and different formulations were established: (i) chemical thermodynamics, which employ conventional thermodynamic quantities to deal with chemical reactions; and (ii) biochemical thermodynamics, which employ transformed thermodynamic quantities to deal with biochemical reactions.

In his works, Alberty has shown how to obtain the transformed thermodynamic quantities Δ_r G′⁰, Δ_r H′⁰, and Δ_r S′⁰ from Δ_f G ⁰, Δ_f H ⁰, and Δ_f S ⁰ of the specific chemical species [10], [11]. According to Alberty, Δ_r G and Δ_r G′ provide the Gibbs energy of the chemical and biochemical reactions, respectively. However, it has been shown that Δ_r G = Δ_r G′ regardless of the reaction involved [12]. This fact has not been sufficiently divulged, and it is still not generally well known by the scientific community. It is worth underlining that this equality is restricted to Gibbs energy changes and does not apply to enthalpy or entropy changes.

In addition, it has been subsequently demonstrated that the ‘transformed’ thermodynamic quantities can be obtained without carrying out any ‘transformation’, but simply by considering the chemical potential of the H⁺ and Mg²⁺ ions constant in the course of the reaction [13].

This means that the thermodynamic quantities G′, H′, and S′ can also be obtained without any “transformation” and the classification into chemical thermodynamics and biochemical thermodynamics can be misleading. Mathematical clarification of the methods allows the two worlds of chemical and biochemical thermodynamics, which are often treated separately, to be reunified within the same thermodynamic framework. In the next sections, the two different methods will be described in detail.

2 Alberty’s method

According to Alberty, the species formed by a given biochemical reagent constitute a pseudoisomer group [4], [5], [6], [7]. Starting from this consideration, Alberty develops a procedure based on transformed Gibbs energies [4], [7], [10], [11]. Since the chemical species constituting the pseudoisomer group are at equilibrium with H⁺ and Mg²⁺ ions (Δμ _reaction = 0), i.e.

the chemical potential μ _i of the ith complex species ( i.e. [ X H N H + M g N M g 2 + ] ( N H + + 2 N M g 2 + ) + ) is

where μ x is the chemical potential of the free species (ATP⁴⁻, ADP³⁻, and PO₄ ³⁻ if we take, for example, the pseudoisomers of ATP, ADP, and P_i reported in Fig. 1); and N H + ( i ) , N M g 2 + ( i ) are, respectively, the number of H⁺ and Mg²⁺ ions bound to ith complex species (all the species bound to H⁺ and Mg²⁺ if we take, for example, the pseudoisomers of Fig. 1). The chemical potentials μ i are adjusted by subtracting N H + ( i ) μ H + and N M g 2 + ( i ) μ M g 2 + .

The chemical potentials, μ ′ , have been named transformed chemical potentials by Alberty and are equal, as shown by Eq. (9), to the chemical potential, μ _X, of the free species. All the chemical species of a pseudoisomer group have the same transformed chemical potential, which is the chemical potential of the free species. The transformed chemical potential, expressed as a function of the concentration (mol dm⁻³) for an ideal solution [4], [5], [6], [7] is:

At a given pH and pMg, the chemical potentials of H⁺ and Mg²⁺ ions have a defined value that can be subtracted from the standard potential:

Substituting in Eq. (10), one obtains

The standard transformed chemical potential, µ i ′ 0 , can be replaced, in the calculations, by the experimental quantity Δ f   G i ′ 0 = Δ f   G i 0 − N H + ( i ) μ H + − N Mg 2 + ( i ) μ Mg 2 + [14]. The transformed standard Gibbs energy of formation Δ r G ′ 0 of the jth isomer group is calculated using

where N _iso is the number of pseudoisomers in the jth isomer group [4], [5], [6], [7]. As a consequence, Δ r G ′ 0 for a biochemical reaction, understood as a reaction involving biochemical reagents, each constituted by a pseudoisomer group, is given by the difference between the transformed standard Gibbs energy of formation of the products and that of the reactants [4], [5], [6], [7]. For reaction (2), we have

3 Thermodynamics of biochemical reactions

An alternative procedure to the one proposed by Alberty for the study of the thermodynamics of a biochemical reaction has been developed and can be used without having to apply transformations of the Gibbs energy [12], [13], [15], [16].

The fundamental equation for the Gibbs energy is given by

where the differential of the amount of species i is dn _i and N is the number of different kinds of species in the system. When making calculations in chemical thermodynamics, μ _i can be replaced with Δ_f G _i , the Gibbs energy of formation of species i. At constant T and p, substituting μ _i with μ i 0 + R T   ln ( c i / c 0 ) , where c ⁰ is the standard concentration, one obtains

In a biochemical reaction, however, pH and pMg remain constant and, consequently, their chemical potential remains constant. For this general approach, the considered equilibria are for a “dilute ideal” solution, in which the activity coefficients of all reactants and products are considered equal to one [13], [16]. Although these conditions cannot be expected to hold rigorously in a biological system, this approximation simplifies the following formal development and calculation of the thermodynamic quantities without compromising their validity. The problem of calculating and using the activity coefficients has been extensively discussed elsewhere [12], [17], [18]. Equation (16) can, therefore, be rewritten as follows:

where N − 2 are the chemical species that make up the reactants and products of the biochemical reaction (N = 17 for reaction (2)). The prime symbol (′) indicates that the chemical potential of the H⁺ and Mg²⁺ ions has a determined and constant value. By analogy with the terminology used for the equilibrium constant K′ of the biochemical reactions, the Gibbs energy G′ is called “conditional” Gibbs energy.

As reported above, a biochemical reagent is a kind of “sum of chemical species”. Indicating the biochemical reagents with the letter j and the individual chemical species of these biochemical reagents with the letter i, the intrinsic mole fraction f _i,j of the chemical species i related to the biochemical reagent j is given by:

where n _i,j is the amount of the chemical species i of the biochemical reagent j and n _j is the total amount of the biochemical reagent j. For example, referring to the equilibrium (a) in Fig. 1: biochemical reagent: ATP; individual chemical species: ATP⁴⁻, HATP³⁻, H₂ATP²⁻, MgATP²⁻, MgHATP⁻ and Mg₂ATP; n _i,j refers to n _{ATP⁴⁻ ,ATP} , n _{HATP³⁻ ,ATP} , n _{H 2 ATP²⁻ ,ATP} , n _{MgATP²⁻ ,ATP,} n _{MgHATP⁻ ,ATP} , n _Mg2ATP,ATP ; n _j  = n _{ATP⁴⁻ ,ATP} + n _{HATP³⁻ ,ATP} + n _{H2ATP²⁻ ,ATP} + n _{MgATP²⁻ ,ATP} + n _{MgHATP⁻ ,ATP} + n _{Mg 2 ATP,ATP}. The detailed procedure for the calculation of n _i,j is reported in [13]. Placing ν _i,j  = f _i,j or ν _i,j  = ‒f _i,j when the chemical species i is a product or a reagent of the reaction, Δ_r G′ of a biochemical reaction, when the extent of reaction is 1 mol at constant pH and pMg, is given by

where N_B is the number of biochemical reactants and products involved in the biochemical reaction, N(j) is the number of chemical species of the biochemical reagent j, μ i , j 0 is the standard chemical potential of the chemical species i, and c _j is the concentration of the biochemical reagent j. The stoichiometric numbers ν H + and ν Mg 2 + are equal to the numerical value of the amount of ions H⁺ and Mg²⁺ that is consumed (ν < 0) or produced (ν > 0) when the extent of the biochemical reaction is 1 mol.

The standard chemical potential μ i , j 0 can be replaced by the experimental quantity Δ f G i , j 0 . Equation (14) becomes

When the concentrations c _j of the biochemical reagents is equal to c ⁰, Eq. (20) provides the conditional standard Gibbs energy value for the biochemical reaction:

The intrinsic mole fractions f _i,j are calculated with a procedure previously described [15]. In the following, X stands for the free species (ATP⁴⁻, ADP³⁻, etc.) of the biochemical reactant X _j (ATP, ADP, etc.) and [XH _h Mg _m ]^(h+2m)+ for a generic complex species of X (where h and m indicate, respectively, the number of H⁺ and Mg²⁺ ions bound to X species; ionic charges from now on are omitted for simplicity). Remembering Eq. (9), one can write the following equality:

Rearranging this equation, one obtains

Substituting μ X 0 with Δ f G X 0 and μ XH h Mg m ′ 0 with

one obtains

For a 1 M solution of the biochemical reactant X, one has:

Dividing both members by c _X and taking the reciprocal one obtains:

Once c X is obtained, the concentration of the complex species is calculated according to equation (25):

The concentrations of the chemical species have been calculated for a 1 M solution of the biochemical reagent and their numerical value corresponds to their intrinsic mole fraction.

The stoichiometric coefficients ν H + and ν Mg 2 + are numerically equal to the difference between the amount of H⁺ and Mg²⁺ ions bound to the reacting chemical species minus that bound to those produced. If we indicate with N _i,j (H⁺) and N _i,j (Mg²⁺) the number of H⁺ and Mg²⁺ ions bound to the ith adduct of the biochemical reactant/product j, the coefficients of these ions are:

If the value of ν H + is positive, that means that H⁺ ions are released during the biochemical reaction, while if it is negative, they are consumed. The same goes for the Mg²⁺ ions.

Once the stoichiometric coefficients of all the chemical species participating in the reaction are obtained, the standard enthalpy of reaction can be easily calculated:

4 Thermodynamics of ATP hydrolysis

In this paragraph we report, as a numerical example of application of Eq. (21), the calculation of Δ_r G′⁰ for ATP hydrolysis (reaction (2)) at pH = 7 and pMg = 3.

The first column of Table 1 shows the list of the chemical species (reactants and products) involved in the hydrolysis reaction. In the second column the values of Δ_f G ⁰ (T = 298.15 K, I = 0.25 m) of all chemical species implied in the biochemical reaction are reported [7], [11], [13]. At first, we calculate the standard conditional Gibbs energies of formation Δ f G i ′ 0 . As shown in eq. (24), we need to know the Gibbs energy of formation Δ_f G of H⁺ and Mg²⁺ ions [7], [11], [13].

These values, reported in the last two rows of the table’s column 8, must then be subtracted from the standard Gibbs energies of formation Δ f G i 0 to get the standard conditional Gibbs energies of formation Δ f G i ′ 0 . The values of N _i (H⁺) and N _i (Mg²⁺) are reported in column 3 and 4, respectively, and the values of Δ f G i ′ 0 , calculated according to Eq. (24), are reported in column 5. The values of exp [ ( Δ f G X 0 − Δ f G i ′ 0 ) / R T ] (see Eq. (27)) are reported in column 6. The concentrations of each chemical species XH _h Mg _m (indicated as c) were calculated according to equations (27) and (28) and reported in column 7. These concentrations, each divided by the standard concentration c ⁰ = 1 m, are simply the intrinsic mole fraction f _i of each chemical species in respect to the corresponding parent biochemical species (i.e. f _i is the amount of substance of a specific chemical species (pseudoisomer) XH _h Mg _m divided by the sum of all (h, m) species that together form the pseudoisomer group XH _h Mg _m ). The Δ f G i = Δ f G i 0 + R T ln f i of each chemical species is reported in column 8. The stoichiometric numbers ν _i , reported in column 9, are negative for reactants and positive for products. The stoichiometric numbers ν H + and ν Mg 2 + are calculated according to Eq. (29), ν H 2 O = − 1 , while for the other chemical species ν _i is numerically equal to c = f _i c ^o . Then, in column 10 the products ν _i Δ_f G _i are reported; according to Eq. (21), the sum of these values is −32.45 kJ mol⁻¹, i.e. the standard conditional Gibbs energy of hydrolysis of ATP.

The stoichiometric numbers, ν i , j , of Eq. (20), reported in column 9 of the table, refer to the chemical species, i, j, that are all the individual reactants/products involved in the biochemical reaction (col. 1). We can write the corresponding chemical equation for the hydrolysis of ATP in which the reactants/products are the chemical species reported in column 1 and in which the stoichiometric numbers ν are the values reported in column 9 of Table 1:

This chemical equation is written in terms of specific ionic species and balanced elements and charge and it has been named the balanced biochemical reaction [13]. Using the balanced biochemical reaction approach, the calculations needed to obtain Δ_r N (H⁺) = ν H + and Δ_r N (Mg²⁺) = ν Mg 2 + are trivial, while the Alberty’s procedure [14] involves much more complex calculations.

As already shown [11], at constant pH and pMg, when the chemical species of each biochemical reagent are in equilibrium with each other, the Δ_r G of any chemical reaction between various chemical species has the same value of the Δ_r G′ of the biochemical reaction. It would be difficult to measure the concentration of the single chemical species involved in the energy-producing chemical reaction, while it is relatively easy to measure the total concentration of the biochemical reactants.

In the case of ATP hydrolysis, MgATP²⁻ is the active species [9] and, as a consequence, the actual reaction of ATP hydrolysis, in vivo, is the chemical reaction (3). In the muscle cytosol, we observe [9]:

Using the stoichiometric numbers, ν, reported in Table 1 and the procedure reported in detail in the reference publication [12], we obtain:

The Δ_r G of the chemical reaction (3) at T = 298.15 K, p = 10⁵ Pa, I = 0.25 M, pH = 7 and pMg = 3 is:

At the same conditions of T, p, I, pH, and pMg, the Δ_r G′ of the biochemical reaction (2) is

The assertion that the Δ_r G of any chemical reaction between various chemical species has the same value of the Δ_r G′ of the biochemical reaction is verified. It is worth underlining that it has also been demonstrated that Δ_r G can be calculated either in terms of activities or concentrations, giving the same value, under the assumption that all activity coefficients are constant [12].

5 Conclusions

The presence of two categories of thermodynamics based on different concepts and different formalisms has often been presumed in the literature: (i) chemical thermodynamics, which make use of the conventional thermodynamic properties and is only suitable to deal with chemical reactions; and (ii) biochemical thermodynamics, which make use of transformed thermodynamic properties and is only suitable to deal with biochemical reactions.

The complexity of biochemical reactions entails a conceptual and experimental problem to determine experimentally the energy released by a specific enzymatic reaction and to assess the associated thermodynamic properties.

The analysis above demonstrates that biochemical reactions, which occur at constant pH and pMg, can be conveniently described by conventional thermodynamics [16]. The two worlds of chemical and biochemical thermodynamics should be viewed as being within the same consistent thermodynamic framework.

Biochemists and analytical chemists are used to dealing with chemical equilibria occurring in a solution at constant pH and having reagents that are the “sum of species”. The corresponding equilibrium constant, K′, is referred to as conditional [19], [20] whereas Δ_rG′, Δ_rH′, and Δ_rS′ have been so far termed, according to Alberty, transformed thermodynamics properties. We suggest the unification of the terminology using the appellation conditional for both K′ and Δ_rG′, Δ_rH′, and Δ_rS′.

The prime symbol is used to indicate that the equilibrium constant and the thermodynamic properties refer to reactions that occur, not only at constant T, p, and I, but also at certain fixed pH and pMg values. The term transformed thermodynamics should be abolished and conditional thermodynamics used instead.

Membership of the sponsoring body

Membership of the IUPAC Physical and Biophysical Chemistry Division for the period 2018–2019 was as follows:

President: Ronald Weir (Canada); Past President: Angela K. Wilson (USA); Vice President: Tim Wallington (USA); Secretary: Attila Császár (Hungary); Titular Members: Frances Separovic (Australia); Roberto Marquardt (France); Pierangelo Metrangolo (Italy); Hiroko Tokoro (Japan); Bert Weckhuysen (Netherlands); Jeremy G. Frey (UK); Associate Members: Zhigang Shuai (China); Vladislav Tomišic (Croatia); Theo Christian Kurtén (Finland); Modou Fall (Senegal); Michal Korenko (Slovakia); National Representatives: Ioanna Zoi (USA); Chia-Chun Jay Chen (China/Taipei); Joaquim Luís Bernades Martins de Faria (Portugal); Leticia Gonzalez (Austria); Keith C. Gordon (New Zealand); Seung-Joon Jeon (South Korea); Gamini Rajapakse (Sri Lanka); Vessela Tsakova (Bulgaria); Ilya Vorotyntsev (Russia); Malgorzata Witko (Poland).