J. Phys. Chem. B 2010, 114, 1985–1993
1985
Chemical and Biochemical Thermodynamics: From ATP Hydrolysis to a General Reassessment Stefano Iotti,*,†,‡ Antonio Sabatini,§ and Alberto Vacca§ Dipartimento di Medicina Interna dell’InVecchiamento e Malattie Nefrologiche, UniVersita` di Bologna, I-40138 Bologna, Italy, National Institute of Biostructures and Biosystems, Italy, and Dipartimento di Chimica, UniVersita` di Firenze, I-50019 Sesto Fiorentino, Italy ReceiVed: April 30, 2009; ReVised Manuscript ReceiVed: December 16, 2009
The Legendre-transformed Gibbs energy change for a biochemical reaction, ∆rG′, is shown to be equal to the nontransformed Gibbs energy change, ∆rG, of any single reaction involving selected chemical species of the biochemical system. These two Gibbs energies of reaction have hitherto been thought to have different values. The equality of the quantities means that a substantial part of biochemical and chemical thermodynamics, previously treated separately, can be treated within a unified thermodynamic framework. An important consequence of the equality of ∆rG and ∆rG′ is that the Gibbs energy change of many enzyme reactions can be quantified without specifying which chemical species is the active substrate of the enzyme. Another consequence is that the transformed standard Gibbs energy change of a reaction, ∆rG′0, can be calculated by a simple analytical expression, rather than the complex computational methods of the past. The equality of the quantities is restricted to Gibbs energy changes and does not apply to enthalpy or entropy changes. Introduction The polyanionic nature of the molecules involved in biological reactions implies that such reagents consist of various adducts in equilibrium with Lewis acids such as H+, Mg2+, etc. As a consequence, any biochemical reaction defines a system of multiple chemical reactions occurring at constant pH, pMg, etc. The introduction of a Legendre transform of Gibbs energy, G′, has been necessary to find the criterion for spontaneous chemical change, the use of the Gibbs energy G being inadequate for such complex systems.1,2 The consequence of this thermodynamic reconceptualization3-5 is the existence of two distinct thermodynamic quantities: ∆rG and ∆rG′. The Gibbs energy of reaction ∆rG refers to chemical equations and is expressed in terms of concentration of the individual reagent species, according to the well-known equation
∆rG ) ∆rG0 + RT ln Q
(1)
where the standard Gibbs energy of reaction ∆rG0 is a function of T, P, and ionic strength I. The terms in reaction quotient Q are the actual concentrations of reactants and products. The transformed Gibbs energy of reaction ∆rG′ refers to biochemical equations and is expressed in terms of total (analytical) concentration of reagents according to the equation
∆rG' ) ∆rG' 0 + RT ln Q'
(2)
where the transformed standard Gibbs energy of reaction ∆rG′0 is a function of T, P, I, pH, pMg, etc. and the apparent reaction * To whom correspondence should be addressed. E-mail: stefano.iotti@ unibo.it. Phone: +39 051 305 993. Fax: +39 051 303 962. † Universita` di Bologna. ‡ National Institute of Biostructures and Biosystems. § Universita` di Firenze.
quotient Q′ contains the total concentration of the reagents, that is, the sum of the concentrations of all the species formed by each reagent. At specified pH, pMg, etc., all the species formed by one reagent constitute a pseudoisomer group.6 From the definition of these different thermodynamic quantities, it directly emerges that ∆rG0 * ∆rG′0 as they refer to different reactions, chemical and biochemical, respectively. Nevertheless, the difference between ∆rG and ∆rG′ has never been analyzed in detail, and so far it has been assumed by analogy that they also differ.5,7-11 By analogy with G′, the Legendre transform of enthalpy H′ and entropy S′ were also introduced for biochemical reactions,2,4,5 thus establishing an alternative formalism based on transformed thermodynamic functions applicable to biochemical reactions and to reactions involving polyfunctional macromolecules.12 The apparent equilibrium constant K′, from which the standard transformed Gibbs energy change ∆rG′0 is derived, is used to calculate the reagent total concentrations at equilibrium in biochemical reactions. In addition, the ∆rG′ quantifies the maximum work obtainable from a biochemical reaction just by the use of the total concentration of reagents (sum of species), allowing its assessment by analytical assays. However, the biochemical meaning of ∆rG′ is controversial as all the biochemical reactions are catalyzed by specific enzymes. In turn, the enzymes have specific substrates, and therefore only the ∆rG of the specific reaction involving the enzyme substrate should be meaningful, as claimed by a consistent body of the biochemical literature which assumes that ∆rG′ * ∆rG.7,8,11,13-17 In the course of our research on the in vivo thermodynamics of the coupled system of reactions of ATP hydrolysis and creatine kinase equilibrium in human brain and skeletal muscle,11 we tackled the problem of understanding the relationship between ∆rG and ∆rG′. This study reports both the experimental results and the mathematical demonstration clarifying the relationship between these two thermodynamic quantities.
10.1021/jp903990j 2010 American Chemical Society Published on Web 01/19/2010
1986
J. Phys. Chem. B, Vol. 114, No. 5, 2010
Iotti et al.
Methods and Results The values of the apparent equilibrium constant, K′, and the transformed standard Gibbs energy change, ∆rG′0, for the biochemical reaction of ATP hydrolysis, eq 3
ATP + H2O ) ADP + Pi
(3)
are calculated from the total concentration of ATP, ADP, and Pi, in the mixture at equilibrium eq
K' )
[ADP]eq[Pi] eq
[ATP]
∆rG' 0 ) -RT ln K'
(4)
whereas the values of equilibrium constant K and standard Gibbs energy of reaction ∆rG0, for the chemical reaction 5
ATP4- + H2O ) ADP3- + PO43- + 2H+
(5)
are calculated from the equilibrium concentrations of the species ATP4-, ADP3-, PO43-, and [H+]. eq
K)
eq + 2 [ADP3-]eq[PO34 ] [H ] eq
[ATP4-]
∆rG0 ) -RT ln K
(6) At T ) 310.15 K and I ) 0.25 M, log K ) -12.4511 and ∆rG0 ) 73.93 kJ mol-1. The main reagents in the cytosolic solution are ATP4-, ADP3-, PO43-, Mg2+, K+, Na+, and H+. The chemical model of the main equilibria occurring among these species is shown in Figure 1. Log β values for the formation of complexes of ATP4-, ADP3-, and PO43- with Mg2+, K+, Na+, and H+ are given in Table 1. We begin the empirical demonstration of the equality of ∆rG′ and ∆rG by considering a system with ATP hydrolysis at equilibrium for a solution containing eq[ATP] ) 10-9 M, eq[Pi] ) 0.01 M, pH ) 7, pMg ) 3, [Na+] ) 0.050 M, and [K+] ) 0.150 M. Equation 7 defines the mass balance for ATP.
[ATP] ) [ATP4-] + [HATP3-] + [H2ATP2-] + [NaATP3-] + [KATP3-] + [MgATP2-] + [Mg2ATP] + [MgHATP-] + [MgH2ATP] (7) When the concentration of each complex is replaced by the expression which defines it in terms of the cumulative formation constant β and the concentrations of the free species contained in the complex, eq 8 is obtained.
[ATP] ) [ATP4-](1+βHATP3-[H+] + βH2ATP2-[H+]2 + βNaATP3-[Na+] + βKATP3-[K+]+βMgATP2-[Mg2+] + +
βMg2ATP[Mg ] + βMgHATP-[Mg ][H ] + 2+ 2
2+
βMgH2ATP[Mg2+][H+]2) (8)
Figure 1. Scheme of the chemical equilibria between the three ligands, ATP4-, ADP3-, and PO43-, and the four Lewis acids, H+, Mg2+, K+, and Na+, present in cell cytosol.
TABLE 1: Cumulative Formation Constants (log β) at T ) 310 K and I ) 0.25 mol dm-3 Taken From Iotti et al.11 and References Therein reaction +
log β
ATP + H f HATP ATP4- + 2H+ f H2ATP2ATP4- + Na+ f NaATP3ATP4- + K+ f KATP3ATP4- + Mg2+ f MgATP2ATP4- + 2Mg2+ f Mg2ATP ATP4- + H+ + Mg2+ f MgHATPATP4- + 2H+ + Mg2+ f MgH2ATP ADP3- + H+ f HADP2ADP3- + 2H+ f H2ADPADP3- + Na+ f NaADP2ADP3- + K+ f KADP2ADP3- + Mg2+ f MgADPADP3- + H+ + Mg2+ f MgHADP PO43- + H+ f HPO4 2PO43- + 2H+ f H2PO4PO43- + 3H+ f H3PO4 PO43- + Na+ f NaPO4 2PO43- + H+ + Na+ f NaHPO4PO43- + K+ f KPO4 2PO43- + H+ + K+ f KHPO4PO43- + H+ + Mg2+ f MgHPO4 4-
3-
6.79 10.63 1.11 0.98 4.6 6.21 8.93 11.93 6.54 10.21 0.97 0.85 3.22 7.95 11.69 18.43 20.64 0.67 12.29 0.52 12.12 13.76
It must be emphasized that eq 8 is correct when the free species ATP4- is in equilibrium with the complex species, irrespective of whether the ATP hydrolysis reaction is at equilibrium or not. Substituting the formation constant values from Table 1 and the concentrations of H+, Na+, K+, and Mg2+, eq 9 is obtained.
Chemical and Biochemical Thermodynamics
J. Phys. Chem. B, Vol. 114, No. 5, 2010 1987
[ATP] ) 45.211 × [ATP4-]
(9)
In the same way, the equations of mass balance for ADP and Pi provide the numerical expressions 10 and 11.
[ADP] ) 4.5439 × [ADP3-] [Pi] ) 1.1117 × 10 × [PO4 ] 3-
5
[ADP3-] ) 2.564 × 10-5 /4.5439 ) 5.6427 × 10-6 M (19) [PO43-] ) 5 × 10-3 /1.1117 × 105 ) 4.4976 × 10-8 mM (20)
(10) (11)
Equations 12 and 13 are obtained from eqs 9 and 11 by inserting the given concentrations.
Q ) 1.4342 × 10-23
The ratios Q/K and Q′/K′ are equal at 3.153 × 10-11. The equality implies that ∆rG ) ∆rG′. The value is obtained from eqs 2 and 3.
∆rG ) ∆rG' ) -62.35 kJ mol-1
-9
[ATP ] ) [ATP]/45.211 ) 10 /45.211 )
eq
4-
eq
2.2119 × 10-14 M (12) eq
[PO43-] ) eq[Pi]/1.1117 × 105 ) 0.01/1.1117 × 105 ) 8.9952 × 10-8 M (13)
Putting these values into the equilibrium constant expression 6 gives the value of eq[ADP3-], eq 14 eq
(21)
(22)
To confirm the equality of ∆rG and ∆rG′, we calculated the ∆rG of other chemical reactions among the pseudoisomers of ATP, ADP, and Pi and always obtained the same value for ∆rG. This is to be expected as the value of ∆rG is zero for any reaction at equilibrium. For instance, consider the following reactions. + ATP4- + H2O f ADP3- + PO34 + 2H
∆rG ) w (23)
[ADP3-] ) 10-12.45 × 2.2119 × 10-14 / (8.9952 × 10-8 × 10-14) ) 8.7248 × 10-3 M (14)
MgATP2- a Mg2+ + ATP4-
∆rG ) 0
(24)
and finally the total concentration of ADP is obtained from eq 10.
Mg2+ + ADP3- a MgADP-
∆rG ) 0
(25)
eq
+ PO34 + 2H a H2PO4
[ADP] ) 4.5439 × eq[ADP3-] ) 4.5439 × 8.7248 × 10-3 ) 3.9645 × 10-2 M (15)
Knowing the total concentrations of ATP, ADP, and Pi, the apparent equilibrium constant and corresponding transformed standard Gibbs energy of reaction can be calculated.
K' ) 3.9645 × 10-2 × 0.01/10-9 ) 3.9645 × 105 ∆rG'0 ) -33.24 kJ mol-1
(16)
The reaction quotients Q and Q′ can now be calculated.
Q)
+ 2 [ADP3-][PO34 ][H ] 4-
[ATP ]
Q' )
[ADP][Pi] [ATP]
(17) Consider a solution in which the ATP hydrolysis reaction is not at equilibrium. Let pH ) 7, pMg ) 3, [Na+] ) 0.050 M, and [K+] ) 0.150 M. With total concentrations for [ATP] of 8 mM, for [ADP] of 2 × 10-2 mM, and for [Pi] of 5 mM, Q′ is calculated as 1.250 × 10-5. The reaction quotient Q is calculated from eqs 9, 10, 11, and 17.
[ATP4-] ) 8 × 10-3 /45.211 ) 1.7695 × 10-4 M
(26)
The value of ∆rG for the reaction
MgATP2- + H2O f MgADP- + H2PO4
(27)
is obtained by adding together the values for the four reactions 23, 24, 25, and 26. The ∆rG of reaction 27 is equal to w, the same value as for ∆rG of reaction 23 and also for the biochemical reaction 3. We also calculated the values of ∆rG′ and ∆rG′0 for the biochemical reaction 3 of ATP hydrolysis in the human brain and in skeletal muscle using the same approach. The total concentrations of ATP, ADP, and Pi were taken from values obtained previously by means of 31P NMR measurements.11 The results are shown in Table 2 and are compared with the ∆rG and ∆rG0 of the chemical reaction 11 published previously.11 The free concentrations of Na+ and K+ employed in these TABLE 2: ATP Hydrolysis in Human Skeletal Muscle and Brain: ∆rG0 and ∆rG Values (kJ mol-1) of the Chemical Reaction 27 and ∆rG′0 and ∆rG′ Values (kJ mol-1) of the Biochemical Reaction 3 and the Corresponding Values of pH and pMga resting muscle working muscle brain
pH
pMg
∆rG0
∆ rG
∆rG′0
∆rG′
7.0 6.5 7.0
3.5 3.2 3.8
-27.3 -27.3 -27.3
-63.0 -50.1 -62.1
-35.0 -32.4 -36.0
-63.0 -50.1 -62.1
The concentrations of Na+ and K+ employed in the calculation are 50 and 150 mM, respectively. a
(18)
∆rG ) 0
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Iotti et al.
calculations were 50 and 150 mM, respectively. It should be noted that the error resulting from the use of total rather than free concentrations of Na+ and K+ can be ignored, as it is 0.01 kJ mol-1 for resting muscle, 0.00 kJ mol-1 for working muscle, and 0.01 kJ mol-1 for brain tissue.
is equal to eqγi. ∆rG can then be written as a function of activities, eq 33.
∆rG ) RT
c
∑ νi ln eqci i
) RT
i
i
Mathematical Demonstrations
γc
∑ νi ln eqγieqic i
RT
Demonstration Using Binding Polynomials. Consider the general eq 28 of a reaction in solution
aA + bB ) cC + dD
(28)
This is a generalization of eq 5. Then suppose that the reagents X (X ) A, B, C, or D) form complex species with subsidiary reactants R, S, ... The subsidiary reactants are ions such as H+, Na+, K+, and Mg2+ whose concentrations are known and constant. Substitution of ∆rG0 ) -RT ln K in eq 1 gives an expression for ∆rG, eq 29.
∆rG ) -RT ln K + RT ln Q ) RT ln
Q K
a
[D] c0 [B] c0
eq
d
[C]
c eq
[D]
0
c K ) eq [A]
b
a
c0
( )( ) ( ( )( ) (
) RT ln
c
a
[D] c0 [B] c0
d
b
eq
[A]
c0 eq [C] c0
c
[D] eq [D]
d
b
∑ βr s ...[X][R]r [S]s ... ) i
)( ) )( ) a eq
[B]
c0 eq [D]
eq
[A] [A]
a eq
c
i
i i
∑ βr s ...[R]r [S]s ...) i
i
i
i i
(35)
where [X]t is the analytical concentration of the reagent X and [X] is its free concentration. The factor in round brackets is known as a binding polynomial.18 When the free concentrations of the subsidiary reactants [R], [S], etc. are constant, the binding polynomial PX has a constant value. The above expression can be rewritten as eq 36
[X]t ) PX[X]
(36)
b
So, for individual reagents, eq 37 applies.
d
[A]t ) PA[A]
[B]t ) PB[B]
c0
∑ νi ln eqci i
[X]t ) [X] +
(30)
[B] [B]
b
(31)
which can be written in the general form as eq 32
∆rG ) RT
[X][R]ri[S]si... [XRriSsi...] ) βrisi...[X][R]ri[S]si... (34)
[X] × (1 +
(( ) ( ) ( ) ( ) ) [C] eq [C]
[XRriSsi...]
d
c0
c
βrisi... )
0
c eq [B]
(33) i
In the following treatment, it is assumed that all pseudoisomers contain a single atom of one of the biochemical reagents, X; that is, the formula of a pseudoisomer can be written as XRriSsi... The formation constant of a pseudoisomer and its concentration are given in eq 34
i
[A], [B], etc. are the free concentrations of A, B, etc. in a given solution; eq[A], eq[B], etc. are the free concentration of reactants and products in a mixture at equilibrium; and c0 is the standard concentration. Substitution of expression 30 in eq 29 gives eq 31
[C] 0 ∆rG ) RT ln c [A] c0
i
The equation of mass balance of each reagent is given by eq 35.
( )( ) ( )( )
( )( ) ( )( ) c
a
∑ νi ln eqai
(29)
The reaction quotient Q and the equilibrium constant K are given by eq 30.
[C] c0 Q) [A] c0
)
i
(32)
i
νi are the stoichiometric coefficients, positive for products and negative for reactants. It is evident from eq 32 that ∆rG does not depend on the standard concentration c0. When temperature, pressure, and ionic strength are constant and the concentrations ci and eqci are both small in relation to ionic strength, it can be assumed that activity coefficient γi
[C]t ) PC[C] [D]t ) PD[D] (37)
It is important to note that the proportionality of total to free concentration is valid only when the pseudoisomers are monomeric in reagents A, B, C, or D. When the concentrations [R], [S], etc. are constant, the binding polynomials PA, PB, etc. are the same whether or not the reagents A, B, C, or D are at equilibrium. Therefore: eq[X]t ) PXeq[X]. The apparent equilibrium constant K′ and the apparent reaction quotient Q′ are given by eq 38
[C]tc eq[D]td
eq
K' )
eq
[A]ta eq[B]tb
Q' )
[C]tc[D]td [A]ta[B]tb
and the transformed free energy of reaction by eq 39.
(38)
Chemical and Biochemical Thermodynamics
∆rG' ) RT ln
J. Phys. Chem. B, Vol. 114, No. 5, 2010 1989
Q' K'
(39)
Substitution of expressions 38 in eq 39 gives eq 40.
(( ) ( ) ( ) ( ) ) [C]t
∆rG' ) RT ln
eq
eq
[C]t
d eq[A] a eq[B] b t t
[D]t
c
[A]t
[D]t
[B]t
(40)
Introducing the proportionality relationships 37 gives eq 41.
∆rG' ) RT ln
(( ) ( (( ) ( ) ( ) ( ) ) PC[C]
c
eq
eq
PC [C] [C] eq [C]
) RT ln
c
PD[D]
PD [D]
[D] eq [D]
d
eq
)( d
[A] [A]
PAeq[A] PA[A] a eq
)( ))
[B] [B]
a
PBeq[B] PB[B]
b
b
) ∆rG
TABLE 3: Binding Polynomials of Equation 52 and ∆rG′0 Values (kJ mol-1) of the Chemical Reaction 3 Calculated Using Different Values of Na+ and K+ Concentrations (Millimolar) and pH and pMg [Na+]
[K+]
pH pMg
10.0 50.0 50.0 50.0 50.0
150.0 150.0 150.0 150.0 150.0
7.0 7.0 7.0 7.0 6.5
3.0 3.0 3.5 3.8 3.2
PATP
PADP
PPi
∆rG′0
44.696 45.211 16.472 10.057 30.965
4.1706 4.5439 3.4031 3.1399 4.6916
1.0337 × 105 1.1117 × 105 1.0724 × 105 1.0633 × 105 5.2889 × 105
-32.86 -33.24 -35.01 -36.05 -32.38
The Methods and Results section showed that any chemical reaction involving the pseudoisomers of ATP, ADP, and Pi occurs with the same change in Gibbs energy. Assuming that the pseudoisomers XRriSsi... are in equilibrium with the biochemical reagents and that the concentrations of R, S, etc. are the same whether or not the biochemical reaction 28 is at equilibrium, eq 46 applies
(41) [X]t
which proves the equality of transformed and nontransformed Gibbs energy of reaction. It follows from this equality that the Gibbs energy change of the chemical reaction can be calculated from eq 2 that requires the determination of ∆rG′0 and Q′. Q′ depends on the total reagent concentrations which can be measured much more easily than the concentrations of each chemical species that are needed for the evaluation of Q in eq 1. ∆rG′0 can be calculated by means of the binding polynomials in a much simpler manner than the complex procedure devised by Alberty19 who wrote a specific computer program to overcome this difficulty. Rather than give a general expression, the process will be illustrated using the example of the reaction of ATP hydrolysis. + ATP4- + H2O ) ADP3- + PO34 + 2H
Q)
[ADP
+ 2 ][PO34 ][H ] 4-
[ATP ]
)
[ADP]t [Pi]t PATP [H+]2 ) PADP PPi [ATP]t PATP[H+]2 Q' (43) PADPPPi
This expression for Q is inserted into eq 1 to give eq 44.
PATP[H+]2 ∆rG ) ∆rG + RT ln + RT ln Q' PADPPPi 0
(44)
Since ∆rG ) ∆rG′, it follows that
PATP10-2pH ∆rG' ) ∆rG + RT ln PADPPPi 0
0
(45)
Values of the binding polynomials and ∆rG′0 are given in Table 3. The calculations were performed on an Excel worksheet. Values of [Na+], [K+], pH, and pMg were chosen to be physiologically realistic.
)
[X]t
[X] [X]
(46)
eq
Equation 34 is rewritten as eq 47.
[X] )
eq
[XRriSsi...]
eq
βrisi...[R]ri[S]si...
[XRriSsi...]
[X] )
βrisi...[R]ri[S]si...
(47) Substitution of expressions 47 by eq 46 gives eq 48.
[X]t
(42)
Q is expressed as a function of total concentrations. 3-
eq
eq
[X]
t
)
[XRriSsi...] eq
[XRriSsi...]
(48)
reaching the result that the Gibbs energy of reaction 28 can be calculated by eq 33 indifferently by the use of total or free concentrations of the reagents A, B, C, and D or the concentration of any of the complex species or the activity of the reagents. Demonstration Using Chemical Potentials. The following approach also provides a proof of the equality of ∆rG and ∆rG′. When the subsidiary reactants [R], [S], etc. are in equilibrium with reagents X, all the pseudoisomers of X have the same transformed chemical potential µ′X. For this reason, Alberty18 suggested that the transformed chemical potential, eq 49
µ′i ) µi - riµR - siµS - ...
(49)
be used for biochemical reactions. The value of ∆rG of the reaction 28, at constant T, P, and I, is given by the Gibbs energy required to bring each reactant from its actual concentration to its equilibrium concentration and each product from its equilibrium concentration to its actual concentration, eq 50.
∆rG ) a(eqµA - µA) + b(eqµB - µB) + c(µC - eqµC) + d(µD - eqµD) (50)
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Iotti et al.
The transformed Gibbs energy of reaction is given by a similar expression, eq 51, involving transformed chemical potentials.
∆rG′ ) a(eqµ′A - µ′A) + b(eqµ′B - µ′B) + c(µ′C - µ′C) + d(µ′D - µ′D) (51) eq
eq
MgATP2- + H2O f MgADP- + H2PO4-
Reaction 56 is given by the sum of reaction 55 and the following two reactions20 MgATP2- f Mg2+ + ATP4-
Substituting the transformed chemical potentials given by eq 49 by chemical potentials in any of the terms in brackets of eq 51 gives eq 52.
ADP3- + Mg2+ f MgADP-
µ′i - µ′i ) eqµi - rµR - sµS - ... - µi +
(52)
Equation 51, considering the equality 52, can be rewritten as eq 53
∆rG' ) a(eqµA - µA) + b(eqµB - µB) + c(µC - eqµC) + d(µD - eqµD) ) ∆rG (53) which proves the equality of nontransformed and Legendretransformed free energies of reaction
∆rG ) ∆rG'
(54)
This provides an alternative proof of the equality of ∆rG′ and ∆rG under the assumption that the pseudoisomers formed are all monomeric in the biochemical reagents. It follows from this equality that the Gibbs energy change of any chemical reaction involving pseudoisomers of the biochemical reagents can be calculated from eq 2. Transformed Enthalpy and Entropy The equality ∆rG ) ∆rG′ holds at any temperature since the arguments for its derivation do not depend on any reference to a specific temperature. As a consequence, the temperature derivatives of both ∆rG and ∆rG′ should be equal, and the general relationships of the form
( ∂(∆G/T) ∂(1/T) )
P
) ∆H
( ∂(∆G) ∂T )
) -∆S P
would imply that ∆rH ) ∆rH′ and ∆rS ) ∆rS′. Nevertheless, this is true only if the total amount of substance is constant, i.e., for a close system, which is not the case for a biochemical reaction taking place in living systems which are by definition open systems. In fact, when pH, pMg, etc. are specified in a biochemical reaction, this means that their values are held fixed, and this necessarily implies that the total amount of substance (H+, Mg2+, etc.) is not constant. A simple example is reported showing that ∆rH * ∆rH′. This is illustrated by considering the reactions
ATP4- + H2O f ADP3- + H2PO4-
∆rH1
(55)
∆rH3 ) -18.0 kJ mol-1
(57)
eq
riµRsiµS - ... ) eqµi - µi
∆rH2 (56)
∆rH4 ) 13.0 kJ mol-1 (58)
Hence, ∆rH2 ) ∆rH1 + ∆rH3 + ∆rH4 ) ∆rH1 - 5.0 kJ mol-1. These results show that ∆rH depends on the specific chemical reaction. Each pseudoisomer has its own enthalpy of formation ∆fH, and any of the possible reactions among pseudoisomers holds its own ∆rH. The ∆rH′, which is a linear combination of ∆fH of the pseudoisomers involved, is a fortiori different from any possible ∆rH: ∆rH′ * ∆rH. Moreover, since ∆rH′ - T∆rS′ ) ∆rH - T∆rS (as we showed that ∆rG ) ∆rG′), it follows that also ∆rS′ * ∆rS. Discussion Chemical and biochemical reactions can be described by two types of equations: chemical equations expressed in terms of individual species and biochemical equations which are expressed in terms of the sum of species in equilibrium with H+ and metal ions. Therefore, any biochemical reaction defines a multireaction system formed by the ligands, the polyanionic molecules involved in biological metabolism, and the Lewis acids, H+, and the metal ions present in the cell at millimolar concentrations such as Mg2+, Na+, and K+ (not Ca2+, as it is almost completely bound to proteins like calmodulin and calbinding D-proteins).11 The complexity of the reaction entails a conceptual and experimental problem when, knowing the active species in enzyme binding, we want to determine experimentally the energy released by the specific reaction involving that species. In chemical thermodynamics, Gibbs energy, G, provides the criterion for equilibrium at consant T and P since (dG)T,P ) 0. Alberty1 showed that in biochemical systems the Legendre transformed Gibbs energy, G′, provides the criterion for equilibrium at constant T, P and concentrations of free metal ions since (dG′)T,P,pH,pMg,... ) 0. When several reactions occur simultaneously in a system, as in biochemical reactions, the standard transformed Gibbs energy of reaction, ∆rG′0, is calculated from the apparent equilibrium constant K′,6 which is expressed in terms of total concentration of reagents (sum of species) and can be used to calculate the equilibrium composition of the system. Therefore, for biochemical reactions, ∆rG′0 ) -RT ln K′ and ∆rG′ ) ∆rG′0 + RT ln Q′, or combining the two equations, ∆rG′ ) RT ln(Q′/K′) where Q′ is the quotient of the biochemical reaction examined. This is in complete analogy with chemical reactions for which ∆rG0 ) -RT ln K, ∆rG ) ∆rG0 + RT ln Q, and ∆rG ) RT ln(Q/K) where K is the equilibrium constant and Q is the quotient of the chemical reaction examined. ∆rG′ is the transformed Gibbs energy change of the biochemical reaction, while any of the possible chemical reactions involved should have its own ∆rG value. Although ∆rG0 * ∆rG′0 and Q * Q′, and the thermodynamic potentials G and G′ are different functions as shown by Alberty,2 we found that
Chemical and Biochemical Thermodynamics
∆rG ) ∆rG' This means that all the individual chemical reactions involved in the system described by the biochemical reaction have the same ∆rG value which in turn is equivalent to the transformed Gibbs energy change ∆rG′. We believe it is helpful to list all the assumptions, which are all reasonable in a biochemistry context, upon which is based the above result: 1. Constituents can be divided into two classes: (a) reactants and products, A, B, C, D, etc., which are dilute enough that their production, consumption, and complexation with other constituents does not change the concentrations of those other constituents; (b) other constituents, such as H+, K+, Na+, Mg2+, etc., that form complexes with the reactants and products which are assumed to be buffered so that their concentrations do not change throughout the reaction. 2. All complexes are assumed to be mononuclear in the reactants and products. The reaction can involve arbitrary numbers of A, B, C, and D, and the complexes formed by them can involve arbitrary numbers of other constituents; however, the various reactants and products must not form complexes (“pseudoisomers”) in which more than one of A, B, C, D, etc., are included. 3. All the complexation reactions are at equilibrium throughout the course of the biochemical reaction. 4. All activity coefficients are assumed constant. It must be emphasized that a biochemical reaction is not the fraction-weighted average of the chemical or pseudoisomer reactions as it could be misconstrued if the distinction between biochemical and chemical reactions is not understood. 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 reagents that often consist of species in equilibrium with each other and do not balance elements that are assumed fixed, such as hydrogen ions at constant pH.5 Taking as an example of a biochemical reaction the hydrolysis of 1 mol of ATP occurring at pH ) 7, pMg ) 3, [Na+] ) 0.050 mM, [K+] ) 0.150 mM, T ) 310.15 K, and I ) 0.25 M we have
J. Phys. Chem. B, Vol. 114, No. 5, 2010 1991
0.022 ATP4- + 0.014 HATP3- + 0.014 NaATP3- + 0.032 KATP3- + 0.881 MgATP2- + 0.036 Mg2ATP + 0.002 MgHATPf 0.220 ADP + 0.076 HADP2- + 0.103 NaADP2- + 0.234 KADP2- + 0.365 MgADP- + 0.002 MgHADP + 0.457 HPO42- + 0.251 H2PO4- + 0.091 NaHPO4- + 3-
0.184 KHPO4- + 0.017 MgHPO4 The total electric charge is -2.032 for reactants and -3.241 for products. The change in H, Mg, Na, and K elements is 1.305, -0.536, 0.177, and 0.380, respectively. Figure 2 reports the main species of ATP, ADP, and Pi present in the cell cytosol of a resting human calf muscle,11 showing that the values of ∆rG of the chemical reactions defined by the biochemical reaction of ATP hydrolysis are all the same and in turn equal to ∆rG′ at constant pH and concentrations of the free metal ions. The result that all the individual chemical reactions involved in the system described by the biochemical reaction have the same value of ∆rG is because the Gibbs energy of interconverting between all the different pseudoisomers of any one of the reaction participants is zero. As a consequence, all the reactions involving different pseudoisomers occur with the same value of ∆rG. However, our study demonstrates that ∆rG′ of a biochemical reaction is equal to ∆rG of any chemical reaction between pseudoisomers. This outcome is not trivial since, as shown above, a biochemical reaction is not a fraction-weighted average of the pseudoisomer reactions. The physical reason for the property found can be illustrated by the following example. We can hypothesize that the biochemical reaction of ATP hydrolysis takes place at constant pH, pMg, etc. as a result of three steps: 1. All the pseudoisomers of ATP (in equilibrium with each other) are converted to a given pseudoisomer, for instance MgATP2-. This step has ∆rG ) 0. 2. Then, chemical reaction 27 occurs, and its Gibbs energy change is ∆rG.
Figure 2. Gibbs energy of ATP hydrolysis. The transformed Gibbs energy (∆rG′) of biochemical reaction ATP f ADP + Pi is equivalent to the Gibbs energy ∆rG of any of the chemical reactions involving the pseudoisomers of ATP, ADP, and Pi. Rectangles contain the main pseudoisomers of ATP, ADP, and Pi present in the cell cytosol of human skeletal muscle at pH ) 7, pMg ) 3, [Na+] ) 50 mM, and [K+] ) 150 mM,11 and the size of the formula is roughly indicative of their relative abundance.
1992
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Iotti et al.
3. The products of the above reaction (MgADP- and H2PO4-) are converted to all pseudoisomers of ADP and Pi. This step has ∆rG ) 0. Since Gibbs energy is a state function, the ∆rG′ of the biochemical reaction is equal to the ∆rG of the chemical reaction involving any single pseudoisomer of the reagents. A physical interpretation of such a property can be given observing that the equality ∆rG ) ∆rG′ occurs when dealing with an open system (i.e., not at equilibrium) which is at a steady state. In our case, the evidence of a steady state is given by the constant pH, pMg, etc., assumed when the biochemical reaction takes place. The steady state is the typical condition of living systems and can be considered the ordered state of an open system, a condition that allows living systems to extract work from chemical reactions (∆rG * 0) at a maximal efficiency or, in other words, to increase entropy at the minimal rate.21 This finding unifies the formalism of a substantial part of biochemical and chemical thermodynamics, which so far have been treated separately. This gives a new and wider significance to the Legendre transform of Gibbs energy G′, which has been conceptualized and introduced to find the criterion for spontaneous chemical change in biochemical reactions1,2 and represents a new paradigm in biochemical thermodynamics. Another interesting property found is that the ∆rG value calculated either in terms of activities or in terms of concentrations has the same value if the concentration of the reagents involved is much lower than the ionic strength of the solution (see Mathematical Demonstrations). We discovered the equivalence of the two quantities ∆rG and ∆rG′ studying the thermodynamics of the coupled system of reactions of ATP hydrolysis and creatine kinase equilibrium in vivo in human brain and skeletal muscle by 31P-MRS. We recently developed a new approach for the in vivo assessment of the ∆rG of ATP hydrolysis in human brain and skeletal muscle.11 The method aims to provide a procedure for MRS measurement of the energy released by the specific hydrolysis reaction of MgATP2-, the active species in enzyme binding. In the case of ATP hydrolysis (adenosinetriphosphatase reaction), one of the possible chemical reactions is
ATP4- + H2O f ADP3- + HPO42- + H+
(59)
while the biochemical reaction for ATP hydrolysis is
ATP + H2O f ADP + Pi
(60)
where ATP, ADP, and Pi each represent the sum of all the corresponding protonated and metal complexed species present in the cell, named pseudoisomers.4 In all tissues, the Gibbs energy change for cellular synthesis, mechanical work, and active transport is usually calculated as the ∆rG′ of the adenosinetriphosphatase reaction by the equation
( [ADP][Pi] [ATP] )
∆rG'ATP-hyd ) ∆rG'0ATP-hyd + RT ln
inferred that only the Gibbs energy of hydrolysis of this ionic species is relevant in describing the intracellular energetic status of a tissue7-9 according to the chemical equation
MgATP2- + H2O f MgADP- + H2PO4-
(62)
The corresponding Gibbs energy of reaction is given by
∆rGMgATP2- )
0 ∆rGMgATP 2-
+ RT ln
(
[MgADP-][H2PO4-] [MgATP2-]
)
(63)
According to this view, the thermodynamic status of a tissue should be assessed only by measuring the concentrations of the species involved in reaction 62, which is experimentally difficult. However, in light of property 54 of Gibbs energy found here, this dispute is now meaningless since ∆rGMgATP2- ) ∆rG′ATP-hyd as reported in Table 2. Recently, Nath22 found for the muscle that when added to the standard free energy change upon ATP hydrolysis of -36 kJ mol-1 the part of the free energy of binding Mg-nucleotide to the enzyme, which helps break the actin-myosin/F1 β-ε interactions, yields the total ∆rG′ change of -55 to -60 kJ mol-1 for the entire cycle. It is very interesting to note the correspondence of the above values with those we reported in Table 2 for resting and working muscle. It must also be highlighted that the value of -27 kJ mol-1 for ∆rG′0 of ATP hydrolysis often used in textbooks now appears to be an underestimate. The correct value appears to be 35-36 kJ mol-1 depending on the system (Table 2), and is also in agreement with Nath et al.23 Since ∆rG′ is defined at specified pH and concentration of free metal ions (Mg2+, K+, and Na+, the cations relevant in binding the organic molecules present in cellular milieu), these concentrations should be determined. The total concentrations of K+ and Na+ are known for many cell types, and the bias resulting in using their total instead of free concentrations can be ignored (see Methods and Results), as the difference between total and free concentration is minimal for these cations in the cell cytosol. Therefore, knowing ∆rG′0, whose value can be easily calculated using the binding polynomials, the ∆rG′ of a biochemical reaction can be assessed measuring the total concentrations of reagents and the pH and pMg of the environment where the reaction takes place. At first, we thought that this could have been a specific feature of the ATP system when it is coupled with the creatine kinase equilibrium; however, then we discovered that this is not the case, and we demonstrated mathematically that what we found for ATP hydrolysis can be generalized to any biochemical reaction involving mononuclear species (see Mathematical Demonstration). Taking the biochemical reaction of ATP hydrolysis as an example, this means that the adducts must contain only one molecule of ATP, ADP, and Pi. In addition, our treatment considers that all reactions forming complexes of the reactants and products are at equilibrium.
(61) Conclusions In fact, it is experimentally possible to measure the total concentration of the phosphorylated metabolites involved. However, MgATP2- is the active species in enzyme binding13 and the energy-producing form in active transport14,15 and muscular contraction.16,17 As a consequence, several authors
The value of the transformed Gibbs energy of reaction, ∆rG′, for a biochemical reaction is the same as the value of ∆rG for any chemical reaction between pseudoisomers of the reagents of the biochemical reaction. This means that the maximum work that can be obtained from a biochemical reaction can be
Chemical and Biochemical Thermodynamics calculated from measurements of the analytical concentrations of the reagents, irrespective of which species is the enzyme’s substrate. Even knowing the active species in enzyme binding, it would be difficult or even impossible to measure its concentration, thereby preventing calculation of the reaction Q quotient and hence the corresponding ∆rG. Another consequence of the property found is that it greatly simplifies calculation of the transformed standard Gibbs energy of reaction, ∆rG′0, which can now be obtained in a straightforward manner, avoiding complex computational methods requiring specific computer programs. Last but not least, ∆rG can be calculated either in terms of activities or concentrations giving the same value, thereby facilitating the experimental assessment of the free energy of biochemical reactions occurring in living cells. Acknowledgment. We would like to warmly thank Emeritus Professor John C. Wheeler for his thorough and honest review of the manuscript accompanied by fruitful suggestions. This work was supported by RFO grants from the University of Bologna and PRIN 2007ZT39FN from MIUR to Stefano Iotti. References and Notes (1) Alberty, R. A. Biophys. Chem. 1992, 42, 117–131. (2) Alberty, R. A. Biophys. Chem. 1992, 43, 239–254. (3) Waldram, J. R. The Theory of Thermodynamics; Cambridge University Press: Cambridge, 1985. (4) Alberty, R. A. Biochim. Biophys. Acta 1994, 1207, 1–11.
J. Phys. Chem. B, Vol. 114, No. 5, 2010 1993 (5) Moss, G. P. IUBMB-IUPAC Joint Commission on Biochemical Nomenclature (JCBN), Recommendations for nomenclature and tables in biochemical thermodynamics, 1994; http://www.chem.qmul.ac.uk/iubmb/ thermod/. (6) Alberty, R. A. J. Theor. Biol. 2002, 215, 491–501. (7) Lawson, R. W. J.; Veech, R. L. J. Biol. Chem. 1979, 254, 6528– 6537. (8) Veech, R. L.; Lawson, R. W. J.; Cornell, N. W.; Krebs, H. A. J. Biol. Chem. 1979, 254, 6538–6547. (9) Roth, K.; Weiner, M. W. Magn. Reson. Med. 1991, 22, 505–511. (10) Alberty, R. A. Thermodynamics of biochemical reactions; John Wiley and Sons: New York, 2003. (11) Iotti, S.; Frassineti, C.; Sabatini, A.; Vacca, A.; Barbiroli, B. Biochim. Biophys. Acta 2005, 1708, 164–177. (12) Wyman, J. Proc. Natl. Acad. Sci. U.S.A. 1975, 72, 1464–1468. (13) Kuby, S. A.; Noltman, E. A. ATP-Creatine Transphosphorylase. In The Enzymes, 2nd ed.; Boyer, P. D., Ed.; Academic Press: New York, 1959; pp 515-603. (14) Repke, K. R. H. Ann. N.Y. Acad. Sci. 1982, 402, 272–286. (15) Skou, J. C. Ann. N.Y. Acad. Sci. 1982, 402, 169–184. (16) Ramirez, F.; Marecek, J. F. Biochim. Biophys. Acta 1980, 589, 21– 29. (17) Wells, J. A.; Knoeber, M. C.; Sheldon, M. C.; Werber, M. M.; Yount, R. G. J. Biol. Chem. 1980, 255, 11135–11140. (18) Alberty, R. A. Biophys. Chem. 1998, 70, 109–119. (19) Alberty, R. A. J. Phys. Chem. B 2003, 107, 12324–12330. (20) Smith, R. M.; Martell, A. E. NIST Critically Selected Stability Constants of Metal Complexes Database, Version 4.0; U.S. Department of Commerce, National Institute of Standards and Technology: Gaithersburg, MD 20899 (U.S.A.), 1997. (21) Katchalsky, A.; Curran, P. F. Nonequilibrium Thermodynamics in Biophysics; Harvard University Press: Cambridge, MA, 1965. (22) Nath, S. Int. J. Mol. Sci 2008, 9, 1784–1840. (23) Nath, S. S.; Nath, S. J. Phys. Chem. B 2009, 113, 1533–1537.
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