Balanced biochemical reactions

Let us consider a general biochemical reaction(1)where ν_A, ν_B, ν_C and ν_D are the stoichiometric coefficients. The biochemical reactants A, B, C , D are formed by N_A species A_i, N_B species B_i, N_C species C_i and N_D species D_i respectively.

We take as example for our exposition the hydrolysis reaction of glucose-6-phosphate. The chemical reaction is(2)and the biochemical reaction is(3)G6P represents the sum of the free species G6P²⁻ and of all the complex species formed by G6P²⁻ with the Lewis's acids H⁺ and Mg²⁺; Glu stands for glucose; Pi represents the sum of the free species and of all the complex species formed by with the Lewis acids H⁺ and Mg²⁺. In the pH and pMg ranges present in the cell cytosol matrix (pH≈7 and pMg≈3.5) the complex species of G6P²⁻ and are: HG6P⁻, MgG6P, and MgHPO₄. A scheme of such equilibria is shown in Fig. 1. Here and hereafter pH and pMg values are concentration based, i.e. pH = −log[H⁺] and pMg = −log[Mg²⁺].

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Figure 1. Chemical equilibria between the biochemical reactants of glucose-6-phosphate hydrolysis and H⁺ and Mg²⁺.

https://doi.org/10.1371/journal.pone.0029529.g001

The following assumptions are made:

1. free and complex species of each biochemical reactant are at equilibrium with each other;
2. water activity is constant;
3. temperature T, pressure P, and ionic strength I of the solution are constant;
4. pH and pMg are constant;
5. standard concentration c⁰ is 1 M.

The procedure of assessing the change in standard Gibbs energy, in standard enthalpy, in standard entropy and in binding of hydrogen and magnesium ions of a biochemical reaction, starts from balancing elements and charge. The steps needed to balance the biochemical reaction are the following:

1. the concentrations of all the free and complex species at equilibrium have to be calculated by solving the mass balance equations for standard concentration c⁰ of the biochemical reactants at given pH and pMg;
2. the stoichiometric coefficient of each species is equal to its own concentration multiplied by the stoichiometric coefficient of the corresponding biochemical reactant;
3. the stoichiometric coefficients of Mg²⁺ and H⁺ are obtained by balancing the atoms of Mg and H in both terms of the reaction.

The correctness of the whole procedure may be verified by checking that the total ionic charge is the same in both terms of the reaction.

Step 1 of the procedure for balancing the biochemical reaction is based on the calculation of the concentration of free and complex species in a 1 M solution of the biochemical reactants at specified pH and pMg. Almost all biochemical reactions have complex species that are mononuclear with respect to each reactant. In this case the mass balance equation of each biochemical reactant is easily solved using binding polynomials [13] and the free and complex species concentration are obtained [8]. More generally, in case of poly-nuclear complexation the mass balance equations can be solved using any of the available computer speciation programs, like Hyss [14], therefore the general validity of the procedure is maintained. It is to be pointed out that the sum of the concentrations of the species forming each biochemical reactant is equal to c⁰. The ratio between the concentration c_i of the i-th species and the concentration c⁰ of the biochemical reactant will be referred as fractional population of species i and indicated as f_i∶ f_i = c_i/c⁰.

In step 2, the stoichiometric coefficients of the balanced biochemical reaction are obtained multiplying the concentration of each species by the stoichiometric coefficient of the corresponding biochemical reactant:(4)where are the stoichiometric coefficients of the species A_i, B_i, C_i and D_i, respectively. Referring to the general biochemical equation (1), we have(5)(6)As a consequence, for the balanced biochemical reaction corresponding to the biochemical reaction (3), the following equations hold:(7)

In step 3 the Mg and H atoms have to be balanced. Taking the stoichiometric coefficients negative for reactants and positive for products and indicating with and the number of atoms of Mg and H contained in reactant i, the Mg²⁺ and H⁺ stoichiometric coefficients are(8)Referring to the Aberty's scheme of the reaction chamber connected to semi-permeable membranes [5], the changes in binding Δ_rN(H⁺) and Δ_rN(Mg²⁺) represent, respectively, the amount of ions H⁺ and Mg²⁺ entering or exiting (depending if they are reactants or products) from the reaction chamber through semi-permeable membranes. It must be underlined that the balanced biochemical reaction so obtained represents the actual transformation processes taking place in the cytosolic solution when the reactants, at a given pH and pMg, transform into products.

The calculations of the stoichiometric coefficients are detailed in Supporting Information S1. The balanced biochemical reaction (T = 298.15 K, I = 0.25 M, pH = 7, pMg = 3) is:(9)It must be noted that the stoichiometric coefficients of biochemical reactions are usually fractional numbers, as compared to integer coefficients of chemical reactions. The change in the binding of hydrogen and magnesium ions is a direct result of the balancing of the reaction (see Supporting Information S1 and Eq. (S1–14)). The procedure here proposed is of general validity and can be applied to any biochemical reaction with no restriction (see Supporting Information S1 for more details).

Calculation of Δ_rH^′0, Δ_rG^′0 and Δ_rS^′0

The conditional standard enthalpy change Δ_rH^′0 of the biochemical reaction can now be calculated as a sum of the products of the stoichiometric coefficient of each reactant species and its standard enthalpy of formation:(10)This value matches with that obtained by Alberty [4]. The values of the terms in the summation of Eq. (10) are reported in Table 1 (fifth column). It is worth to underline that this procedure does not require the computation of the transformed standard enthalpy of formation of reactants and products.

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Table 1. Calculation of Δ_rH^′0 and Δ_rG^′0 for the hydrolysis of G6P.

https://doi.org/10.1371/journal.pone.0029529.t001

The conditional standard Δ_rG^′0 value of the generic biochemical reaction (1) is the change in Gibbs energy when the reagents (sum of species) form the products (sum of species) all in the standard state of concentration c⁰. The concentration c_i of each species is equal to its fractional population f_i times the concentration c⁰ of the biochemical reactant: c_i = f_i c⁰. The conditional Gibbs energy of formation of species i at concentration c_i, indicated as , is:(11)The concentrations of H⁺ and Mg²⁺ are respectively 10^−pH and 10^−pMg and therefore(12)(13)The fractional population of H₂O is equal to one and then(14)The Δ_rG^′0 value of the biochemical reaction is(15)The values of Gibbs energy of formation Δ_fG′ of the reactants of the balanced biochemical reaction (9) are calculated by Eqs. (11–14). Substituting them into (15) we obtain Δ_rG^′0 = −11.61 kJ mol⁻¹ which perfectly agrees with the value reported by Alberty [4]. The values of the terms in the summation (15) are reported in the last column of Table 1. Therefore, the complex computations required to obtain the standard transformed Gibbs energy of formation are unnecessary.

The value of Δ_rS^′0 can be calculated: Δ_rS^′0 = (Δ_rH^′0−Δ_rG^′0)/T. However, Δ_rS^′0 may also be obtained directly by Δ_fS⁰of reactants of the balanced biochemical reaction. It must be underlined that reactants are not at standard concentration and therefore the Δ_fS⁰ values have to be corrected for the entropy of dilution:(16)(17)(18)(19)The Δ_rS ^′0 of the biochemical reaction is:(20)

The balanced biochemical equation allows to obtain the values of Δ_rN(H⁺), Δ_rN(Mg²⁺), Δ_rH^′0, Δ_rG^′0 and Δ_rS^′0 directly from conventional thermodynamic properties of species without the need of the complex computations required by the use of the Legendre transformed thermodynamic properties [6]. One of the advantages of this approach is, for instance, that it does not require the use of Δ_fH^′0 and Δ_fG^′0 data tables calculated for different values of pH and pMg [7].

Balanced biochemical reaction holds the same features of chemical equations. Balanced biochemical equations are of general validity and can be used for any biochemical reaction. The only difference between a balanced biochemical reaction and a chemical reaction is that in the former the concentration of H⁺ and Mg²⁺ are kept constant. This is the reason why the stoichiometric coefficients of the balanced biochemical reaction are functions of pH and pMg.

Minimum value of G and G′ at equilibrium

It is a common belief that the Legendre transformed G′, H′ and S′ are the only thermodynamic properties appropriate to deal with biochemical reactions occurring at constant pH and pMg and that the transformed Gibbs energy G′, and not G, is minimized at equilibrium [3]. This opinion originates from the consideration that the biochemical reaction occurs in an open system with a change of the quantity of H⁺ and Mg²⁺ ions. The reaction can be exemplified as occurring in a reaction chamber connected to reservoirs with constant [H⁺] and [Mg²⁺] through semi-permeable membranes. Given that, hydrogen and magnesium are not conserved in the reaction chamber and this explains the reason why biochemical reactions are written in terms of sums of species [3]. As a consequence, according to this view, when the pH and pMg are specified only the transformed Gibbs energy G′ and not G provides the means for determining whether the reactions will go to the right or the left [5].

We recently showed that Δ_rG′ of a biochemical reaction is equal to Δ_rG of any single reaction involving selected chemical species of the biochemical system [8]. On this basis we hypothesize that also Δ_rG′ of biochemical reaction and Δ_rG of the corresponding balanced biochemical reaction, are the same at constant T, P, pH and pMg. In fact, a balanced biochemical reaction is equivalent to a chemical reaction and, as a consequence, it can be treated using the conventional chemical thermodynamics.

We calculated the change of G of the balanced biochemical reaction (9) and G′ of the biochemical reaction (3) as a function of the extent of reaction ξ at constant T, P, pH and pMg. We recall that the values of G and G′ refer to the standard Gibbs energy of formation of elements taken as zero.

The initial state is a solution with a volume V⁰ = 1 dm³, containing 1 mol of G6P at concentration c⁰. The final state is a solution with a volume 2V⁰ = 2 dm³, containing 1 mol of Glu, 1 mol of Pi both at concenntration c⁰/2. In the course of reaction pH and pMg are constant while the volume V of the solution changes: V = (1+ξ)V ⁰. V ⁰, being equal to 1, will be omitted in the following equations. G′ of the biochemical reaction (3) is(21)The plot of G′ as a function of the extent of reaction ξ is reported in Fig. 2 (plot a) and shows a minimum at the value of ξ = 0.995412. The equilibrium constant expressed as a function of ξ is(22)giving Δ_rG^′0 = −11.61 kJ mol⁻¹.

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Figure 2. Plot of G′ (a) and G (b) as function of the extent of reaction ξ.

https://doi.org/10.1371/journal.pone.0029529.g002

G of the balanced biochemical reaction (9) is(23)

The plot of G as function of the extent of reaction ξ is reported in Fig. 2 (plot b) and shows that the minimum also occurs at ξ = 0.995412.

These results show that both G and G′ have the minimum at the same value of ξ. The difference G−G′ has a constant value of −554.83 kJ mol⁻¹ in the whole range of ξ.

In agreement with Alberty [3], this difference is given by(24)where and are the number of hydrogen and magnesium atoms in species i and n_i is the amount of i. According to Alberty the number of species N would not include H⁺ and Mg²⁺ and, as a consequence, the difference G−G′ would not be constant in the course of reaction. On the contrary, using the stoichiometric coefficients (n_i = ν_i) of the balanced biochemical reaction (9), which includes the species H⁺ and Mg²⁺, we exactly obtain the constant value of −554.83 kJ mol⁻¹ in the whole range of ξ. This result contradicts the established opinions that: i) the Legendre transformed G′ only is the appropriate thermodynamic function to deal with biochemical reactions occurring at specified pH and pMg, ii) the transformed Gibbs energy G′ and not G is minimized at equilibrium because the system exchanges H⁺ ed Mg²⁺. We show indeed that employing the balanced biochemical reaction there is no need of the Legendre transformed potential G′ to properly describe the thermodynamic state of a biochemical system, G being fully adequate to this scope.

Conclusions

The balanced biochemical equation allows a novel and simpler approach to the thermodynamics of the biochemical reactions. Indeed, the balanced biochemical equation is conceptually simpler as it just requires the knowledge of classical thermodynamics with no need to be acquainted to Legendre transforms thermodynamics, which in this case represents an un-necessary complication. In fact, as shown in Fig. 3, the balanced biochemical reaction approach entails less calculation steps compared to the Legendre transforms approach. The most elaborate procedure of the balanced biochemical reaction approach is the calculation of equilibrium concentrations of the different species; nevertheless this step is also required in the Legendre transforms approach for the calculation of Δ_fH^′0.

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Figure 3. Flow diagram showing the different approaches to obtain the transformed thermodynamic properties: balanced biochemical reaction (left) and transformed formation functions (right).

https://doi.org/10.1371/journal.pone.0029529.g003

In addition, using the balanced biochemical reaction approach, the calculations needed to obtain Δ_rN(H⁺) and Δ_rN (Mg²⁺) are trivial, while the Legendre transforms procedure [15] entails the following complex calculations:(25)

The quantities of the species entering and exiting the system are specified in the balanced biochemical reaction. This is the reason why Δ_rG^′0, Δ_rH^′0 and Δ_rS^′0 can be directly calculated from the conventional standard formation thermodynamic properties of the species taking part to the biochemical reaction. It must be recalled that the value of Δ_rG^′0 is obtained by the conditional constant K^′ (Δ_rG^′0 = −RT lnK′) also known as apparent constant [2]. Coherently to the results of this study, where Δ_rG^′0, Δ_rH^′0 and Δ_rS^′0 are obtained without using the Legendre transforms, we propose to name these thermodynamics properties “conditional” rather than “transformed”.

Finally, we also showed that the thermodynamic potential G is fully adequate to deal with biochemical reactions. The use of the transformed Gibbs energy G′ is correct, but unnecessary, and G is indeed minimized at equilibrium and therefore can properly give a criterion of spontaneous chemical change for a biochemical reaction. The introduction of the concept of the balanced biochemical reaction greatly simplifies the approach to the thermodynamics of complex systems, as the biochemical reactions are, making avoidable the use of the Legendre transforms. The biochemical balanced equations shows the possibility to deal the thermodynamics of biochemical reaction using the conventional thermodynamic properties.

In conclusion, the simple procedure of balancing the biochemical reactions is of general validity and makes possible to use the same approach when dealing with biochemical and chemical reactions. This allows the re-unification of the two worlds of chemical and biochemical thermodynamics, so far treated separately, within the same thermodynamic framework. Maybe, the balanced biochemical reaction can be regarded as the egg of Columbus for a notably easier approach to the biochemical thermodynamics.