J. Phys. Chem. 1992, 96, 9614-9621
9614
FEATURE ARTICLE Degrees of Freedom in Biochemical Reaction Systems at Specified pH and pMg Robert A. Alberty Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 (Received: June 11, 1992; In Final Form: August 31, 1992) When pH and pMg are specified for a biochemical reaction system, thermodynamic properties are described in terms of transformed thermodynamic properties. Under these conditions, chemical reactions are written in terms of sums of species, so that the stoichiometric number matrix Y for the reaction system is replaced by an apparent stoichiometric number matrix v’ and the conservation matrix A is replaced by an apparent conservation matrix A’. The number N of species in a biochemical reaction system may be very large because a reactant like ATP exists as an equilibrium mixture of species. Under certain conditions, ATP exists as ATP‘-, HATP3-, H2ATPZ-,MgATP2-, MgHATP, and Mg2ATP. Thus ATP contributes 1 to the number N’ of reactants and 6 to the number N of species. Similarly, the number C’ of apparent components is smaller than the number C of components. The equilibrium constants of biochemical reactions are referred to as apparent equilibrium constants K’because they arc written in terms of reactants (sums of species) rather than particular species. Apparent equilibrium constants depend on T,P,pH, pMg, and ionic strength. The number F’ of apparent degrees of freedom is given by C’ p + 2, where C’ is the number of apparent components and p is the number of phases. The criterion of equilibrium at T, P,pH, and pMg is the transformed Gibbs energy G’. The fundamental equation for G’ provides the means for calculating the transformed entropy S’ and transformed enthalpy H’.There is a Gibbs-Duhem equation written in terms of the transformed entropy S’, the transformed chemical potentials pciof components,and the amounts nc; of apparent components in the system. Matrix notation is emphasized because of its utility in treating systems involving many reactions.
Introduction The number of independent variables required to describe the equilibrium state of a reaction system is specified by the phase rule, but various choices may be made within this number. Equilibrium compositions of reaction systems are usually calculated at specified temperature, pressure, and amounts of C - 1 components divided by the amount of the Cth component, where C is the number of components. However, in biochemistry, pH and pMg are usually chosen as independent variables in addition to T and P,and this opens up a whole new world of thermodynamics. Biochemists are not unique in their choice of reactant concentrations as independent variables to describe the state of a reaction system. In a re-former for gasoline, the partial pressure of molecular hydrogen may be known or controlled, and equilibrium calculations can be made by essentially changing the standard-state pressure for molecular hydrogen from the usual standard-state pressure to the desired pressure.’ Equilibrium distributions of the alkylbenzenes have been calculated at various partial pressures of ethylene.* In 1987, Cheluget, Missen, and Smith3 showed how composition constraints can be applied by developing a compatible stoichiometric number matrix and adjusting the standard Gibbs energies of formation for the fmed value of the concentration of the species held constant. In 1988, Alberty and Oppenheim4 considered the alkylation of benzene with ethylene and pointed out that when a substance is available to a system through a semipermeable membrane, it need not be represented by a term in the fundamental equation of thermodynamics. This was accomplished by defining a new thermodynamic potential by using a Legendre transform.s Wyman6 and Wyman and Gill’ have used Legendre transforms to define thermodynamic potentials especially suited to the treatment of the binding of ligands by a polyfunctional macromolecule. In 1989, Alberty and Oppenheim9 showed how the transformed entropy of a system can be calculated, and they used a semigrand isothermal isobaric ensemble to show how the standard transformed thermodynamic properties of a system can be calculated by use of the molecular partition functions of the various species. In 1992? they showed how to calculate the full set of thermodynamic properties when the equilibrium partial pressure of a
reactant is specified. Legendre transformed Gibbs energies have also been used to treat equilibria in the polymerizationof polycyclic aromatic hydrocarbons at specified partial pressures of acetylene and molecular hydrogen.I0 Norval, Phillips,Missen, and Smithi1 have discussed constrained equilibria and have made equilibrium calculations for a specified partial pressure of SO2in stack gas. Biochemistry provides an excellent example of the specification of independent concentration variables sin= biochemical equilibria generally depend on the pH and may depend on pMg or the concentrations of other metal ions. Reactants in biochemical reactions are often weak acids, and a number of them have pk”s in the neutral pH region. These species may also bind Mgz+or other metal ions so that the apparent equilibrium constant K’is a function of the free concentrations of these. metal ions. When a biochemical reaction is at equilibrium, the composition is almost always determined experimentally in terms of reactants like ATP, rather than in terms of species. The apparent equilibrium constant K’ is written in terms of sums of concentrations of species of reactants,I2 rather than in terms of species, and [H+] and [Mg*+] are not included in the expression for K’. At specified pH and pMg, standard transformed Gibbs energies of reaction 4 G ’ O and standard transformed enthalpies of reaction A J ’ O can be calculated from K’ and its temperature dependence, and standard transformed Gibbs energies of formation A @ ’ O ( i ) and standard transformed enthalpies of formation 4H’O can be calculated for reactant i, which is a sum of s p a ~ i e s . ~When ~ . ~ pH ~ and pMg are specified as independent variables, the thermodynamictreatment of a reaction system involves a number of new physical quantities and new terminology. To provide a basis for describing these new physical quantities, the fundamental equation of thermodynamics is used and matrix quantities are emphasized because this facilitates the discussion of the thermodynamicsof complex systems that are encountered in biochemistry. To simplify the discussion, the species in the buffer, in addition to H+ and MgZ+,are not included in the fundamental equation, but K’ and the standard thermodynamic properties calculated from it do depend on the ionic strength of the buffer, which must always be specified. The next section describes the fundamental equation of thermodynamics for a simple reaction system in terms of the amounts n, of species in the system and the amounts nc, of components in
0022-365419212096-9614$03.00/0 0 1992 American Chemical Society
The Journal of Physical Chemistry, Vol. 96,No. 24, I992 9615
Feature Article the system. The section after that describes the fundamental equation for a more complicated system in terms of the apparent amounts n,‘ of reactants (sums of species) and the amounts nc; of apparent components.
Fuudamental Equation for the Gibbs Energy C In .Matrix Form The most basic constraints on the composition of a chemical system at equilibrium in a closed system are the element conservation equations and constraints that arise implicitly in the mechanism for chemical change in the system. The we of matrices to represent these conservation equations and the stoichiometry of the chemical reactions involved in a reaction system are described by Smith and Missen.I5 The basic matrix eqs l , 6,8,l l , 14, 18, and 19 used here are discussed in their book. The conservation matrix A is made up of the coefficients of the amounts of species in the conservation equations for the elements present, electric charges, and additional constraints in the reactions. Since the conservation equations for a system may not all be independent, it is sufficient to use only an independent subset of conservation equations. The number of components C is equal to the number of independent rows in A, so that C = rank A (1) If the only constraints are element balances, the components can be taken as an independent set of elements. The number of columns in the A matrix is equal to the number N of species in the system, so that the A matrix is C X N . In certain ranges of pH and pMg, the chemical changes in an enzyme catalyzed reaction can be represented by a single chemical reaction between species. An example is the hydrolysis of adenosine triphosphate at pH > 8 in the absence of Mg2+. Under these conditions, the reactants exist predominently as single species, and the reaction can be written A T P + H 2 0 = ADP3- HP042- H+ (2)
+
+
The conservation matrix A for the system is A+
cr
10 A = H 12 0 13 P L 3
I
2 1
0
10 12
0
0 1
10
4
0
2
where the columns are for A W , HzO, ADP>, HP02-, and H+; and the rows are the numbers of atoms of C,H, 0, and P, as indicated. Rows for nitrogen and charge are not included since they are not independent of the four given. Thus there are four components. Although the number of components is fixed, there is a choice of components. When large molecules are involved, counting atoms is tiresome and groups of atoms can be used instead. For example, the following components can be used: the adenosine moiety, 0 in addition to the 0 in adenosine and PO3, H in addition to the adenosine moiety, and P. This leads to the following conservation matrix A. 1 0 1 0 0
1 1 1 1 0 2 0 1 1 1 3 0 2 1 0
10
o o
1 -iJ
and are, therefore, equivalent. (It may be necessary to reorder the columns to obtain this row-reduced echelon form for another system.) Actually stoichiometricnumbers for the reaction or reactions in a system can be derived from the A matrix since the conservation matrix A and the stoichiometricnumber matrix v are related. The stoichiometric number matrix gives the stoichiometric numbers (positive for products and negative for reactants) for the species in the same order as the columns of A. In Y, there is a column for each reaction, and so Y is N X R, where R is the number of independent reactions in the system. These matrices are related by
AV=Q
(6)
Thus the stoichiometricnumber matrix v is the null space of the conservation matrix. The stoichiometric numbers in reaction 2 can be calculated from the conservation matrices in eqs 3,4, or 5 by using Nullspace in Mathematica.I7
v =
(7)
The rows in this matrix give the stoichiometric numbers of A T P , H20, ADP3-, HP042-,and H+. The rank of the v matrix, 1 in this case, gives the number R of independent reactions required to represent all possible changes in composition that can occur in the system given the conservation equations. R = rank v (8) The form of the expression for the equilibrium constant K(T 9 J ) comes from the fundamental equation of thermodynamics,to be discussed below. The following form is based on the assumption that the reactants form ideal solutions on the molar concentration scale in the supporting electrolyte of ionic strength I.
H,O ADP% H P O ~H+~
o
identity matrix on the left side. Matrices 3 and 4 have the same row-reduced echelon form given by
(4)
The rows are for adenosine, 0 in addition to the 0 in adenosine and PO3, H in addition to adenosine, and P. Matrices 3 and 4 look different, but the solutions of the underlying set of conservation equations are not altered by (1) multiplying a row by a constant different from zero or by (2) adding a multiple of any row to another row. Two matrices can be compared16 by performing a Gaussian reduction to put them in the row-reduced echelon form, which can be done by hand for a small matrix or by use of a computer for a larger matrix. A number of the matrix operations described in this paper have been made using the mat he ma tic^^^ computer program-in this instance RowReduce. This program adds multiples of rows together, producing zero elements when possible. The final matrix has an
K(T’P’T) =
[ADP3-][HP042-][H+] [ATp4-](cO)2
(9)
The concentrations are in moles per liter, and co = 1 M is the standard-state concentration. The co term arises in the derivation and is required to make K dimensionless. The convention is that water is omitted from the equilibrium constant expression when dilute aqueous solutions are considered. Since the equilibrium constants used here are functions of the ionic strength, it is useful to be able to adjust K and the standard thermodynamic properties from one ionic strength to another by use of the extended Debye-Hiickel theory.l*~~~ For a system containing a large number of species, it is usually found that more than one chemical reaction is required to represent all possible changes in composition than can occur within the constraints of the conservation equations. For example, for the hydrolysis of adenosine 5’-triphosphate to adenosine 5’4iphosphate and inorganic phosphate in the neutral pH range in the presence of Mg2+at low concentrations, it is necessary to include all of the following species: ATP“-, HATP3-, MgATP2-, H 2 0 , ADP3-, HADP2-,MgADP2-, HP042-,H2P04-, MgHm4, H+, and Mg2+. The A matrix for this system is given by rlo I12 A = 13 3 0
L
10 13 13 3 0
10 12 13 3 1
o
2 1 0 0
10 12 10 2 0
10 13 10 2 0
10 12 10 2 1
0 0 0 1 2 1 4 4 4 1 1 1 0 0 1
o 01 1 0 0 0
0 0 0 1
I
(10)
where the rows are for C, H, 0, P, and Mg. When the null space
9616 The Journal of Physical Chemistry, Vol. 96, No. 24, 1992
of this matrix is calculated, it is found to be equivalent to the Y matrix obtained from eq 2 (that is, eq 7) with added columns for the acid dissociation and magnesium complex dissociation reactions for the system. The A and Y matrices for a system are not unique, but their row-reduced forms are unique for a given order of the species in the system. So far we have considered systems by writing their A matrix and deriving their Y matrix using eq 6. However, it is often more appropriate to start with a set of independent reactions, that is Y, and derive the conservation matrix A. This can be done by use of the equation vTAT = 0
(11)
which is an alternative form of eq 6. Thus AT is the null space of uT. Using matrix 7, the following A matrix is obtained by use of Nullspace in Mathematica.
rl
11
o o o
Alberty inating from eq 15 or 16 the chemical potential of one species for each of the R independent reactions in the system. c
dG = -S d T
+ V d P + i=Efii dn, I
= -S d T + V d P
+ pcdnc (21)
C is the number of components and ncl is the amount of component i. pc is a vector (1 X C) of the first C species, which are taken
as components, and nc is the amount of component matrix (C X 1). It may be necessary to rearrange the columns in the A matrix so that the first C species can be taken as components. Equation 21 can be written in terms of the A matrix as dG = -S d T + V d P + pcA dn
(22)
since the conservation equations have the form An = ne The chemical terms in eq 22 for reaction 2 are given by
1 0 0 1 0 A = l 1 0 1 0 0 1-1 1 0 0 o ]
This appears to be different from eqs 3 and 4, but all three of these yield matrix 5 by row reduction and are therefore equivalent. The number of components C that are required to describe the composition of a system at chemical equilibrium is given by (13)
C=N-R
where N is the number of reacting species in the system and R is the number of independent reactions between these species. This relation arises because one species is eliminated from the list of unknowns by each independent reaction. For reaction 2, C = 5 - 1 = 4, which can be taken to be C, H, 0, and P. Equation 13 can be written as N = rank Y rank A (14)
+
The fundamental equation of thermodynamics for the Gibbs energy G for an open one-phase system where only pressurevolume work is done is dG = -S d T + V d P +
Thus there are four components. This is an example of the statement by Beattie and Oppenheim5 that (1) the chemical potential of a component of a phase is independent of the choice of components and (2) the chemical potential of a constituent (species) of a phase when considered to be a species is equal to its chemical potential when considered to be a component. The Gibbs-Duhem equation can be written in terms of species, and a t chemical equilibrium it can be written in terms of components. Here it is given only in terms of components. Equation 21 is integrated at constant T, P, and composition to obtain
N
C p i dni i=l
C
(15)
G=
Ewe, = P@C i= 1
(24)
where pi is the chemical potential of species i, ni is its amount, and N is the number of species in the system. As a simplification, the species of the buffer, other than H+ and Mg2+,are omitted from the fundamental equation, but they have an effect on the thermodynamic properties of the reacting species through their contribution to the ionic strength. This equation can be written in terms of the chemical potential matrix p , which is 1 X N, and the amount of species matrix n, whch is N X 1: dG = -S d T + V d P p dn (16)
The differential of the Gibbs energy G at constant T and P is
If the system is closed, the fundamental equation can be written in terms of the extent of reaction matrix 4, which is R X 1 , and the stoichiometric number matrix Y, which is N X R . The extent of reaction matrix is defined by
There are C 2 intensive variables for the system, but only C 1 are independent because of this relation. If there are two phases at equilibrium, there are only C independent intensive variables because there is an additional Gibbs-Duhem equation to be satisfied. Therefore, if there are p phases, the number F of intensive degrees of freedom is given by
C
dG =
Subtracting eq 25 from eq 21 yields the Gibbs-Duhem equation. 0 = -S d T
+
n=n,-vt where n, is the initial amount matrix ( N X 1). Thus dG = -S d T + V d P + p~ d t
(17) (18)
As shown by Smith and Missen,I5 this form of the fundamental equation leads to the criterion for equilibrium in a multireaction system at constant T and P, which is d G / d t = 0 = WY (19) or N
c v l j p i= 0
j = 1 , 2,..., R
(20)
l=l
When a closed system is at chemical equilibrium, the fundamental equation can be written in terms of components by elim-
C
C p i d n , + Enc,dri = pC dnc + (dac)nc (25) i= 1 i= I
+ VdP-
C
Enc,dpl = -S d T +
V d P - (dpc)nc
i= I
(26)
+
+
F=C-p+2
(27)
For the system represented by ATP hydrolysis at pH > 8, C = 4, and so the number of degrees of freedom is F = 4 - 1 + 2 = 5. If the components are taken to be C, H, 0, and P, these degrees of freedom can be taken to be T, P, n(C)/n(O),n(H)/ n(O),and n(P)/n(O). If the components are taken to be adenosine, H, in addition to H in adenosine, P, and 0 in addition to the 0 in adenosine and PO,, these degrees of freedom can be taken to be T, P, n(aden)/n(H20), n(H)/n(H20), and n(P)/n(H20). If the system a t pH > 8 contains in addition glucose 6-phosphate and the enzyme glucose 6-phosphatase, there are two more species (glucose and G6P2-) and one more component. Now, there are 6 degrees of freedom, which can be taken to be T, P, n(aden)/
The Journal of Physical Chemistry, Vol. 96, No. 24, 1992 9617
Feature Article n(H201, n(gluc)/n(H20), n(H)/n(H2O), and n(P)/n(H20). At pH < 8 the description of the equilibrium in the hydrolysis of ATP is more complicated because the reactants exist in several forms at equilibrium and the acid dissociation and magnesium complex dissociation equilibrium expressions have to be satisfied at equilibrium as well. Since the acid and magnesium complex dissociation constants are known for ATP, ADP, and inorganic phosphate, it is possible to calculate20the equilibrium composition for a mixture (including pH and pMg) given the initial concentrations of A T P - and total Mg2+ before addition of ATPase. When pH and pMg are specified, these equations are all changed.
The apparent stoichiometric matrix v’ is obtained from A’ by use of A‘v‘ = 0
(32)
In this case the null space of A’ is
r-11 v’ =
I I 11
(33)
J
which corresponds with ATP + H20 = ADP
+
Pi (34) Fundamental Equation for the Transformed Cibbe Energy C ’ where ATP represents the sum of the species A T P , HATP3-, When a biochemical reaction is carried out in the laboratory MgATPz-, etc. This type of reaction is referred to as a biochemical it is carried out at constant temperature and pressure, but the reaction, rather than a chemical reaction, to indicate that it is Gibbs energy G is not the criterion for equilibrium because the carried out at specified pH and pMg. Equation 34 balances pH is measured and pMg may be calculated and the equilibrium adenosine groups and inorganic phosphate but does not balance composition is dependent on the equilibrium pH and pMg. Thus H or Mg. The expression for the apparent equilibrium constant for the purpose of analyzing the equilibrium data, the independent K’(T,P,pH,pMg,Z) can be written down from eq 34. variables are T,P, pH, and pMg. It is as if the reaction was carried out in a reaction vessel with a semipermeablemembrane connecting it to a reservoir at a specified pH and pMg. The semipermeable membrane allows H+ and Mgz+to be equilibrated This expression can be derived from the fundamental equation between the reservoir and the reacting system but retains all other for the transformed Gibbs energy, which is given later as eq 42. species in the system. In this thought experiment, the reaction In general, the calculation of pMg at equilibrium requires inis carried out under conditions such that H+ and Mg2+are not formation on the magnesium complex ion dissociation constants conserved in the reaction vessel. Thus the conservation equations for all species present, but if the reactants are at low concentrations for H+ and Mg2+are lost, and the new conservation matrix A’ relative to a buffer that binds Mg2+,a satisfactory value of pMg is referred to as the apparent conservation matrix. This leads to can be calculated from the concentration of the buffer, the a reconceptualization of the system and the reactions in it. In equilibrium pH, and the total amount of Mg2+ in the system. order to develop a criterion for equilibrium at specified T, P, pH, stoichiometric number matrix J is N’ X R’, where and pMg, it is necessary to make a Legendre t r a n s f ~ r m . ~ ~ ~ ~ ~ *The ’ ~ Japparent ~ R ’ is the number of independent biochemical reactions. When we consider the hydrolysis of ATP to ADP in the neutral pH range in the presence of Mg2+at low concentrations, the H R’ = rank v‘ (36) and Mg rows and the H+ and Mg2+columns are deleted in the The advantage in dealing with the number N’ of biochemical conservation matrix2’of eq 10. The three columns for ATP species reactions is that it is much smaller than the total number N of are now identical, and so two of them are deleted. Similarly, two reactions, which may include an unknown number of acid disredundant columns are deleted for ADP and two for inorganic sociations and metal complex dissociations. The number c’of phosphate. The apparent conservation matrix for the system at apparent components (pseudocomponents) is given by specified pH and pMg is therefore ATP HZO ADP P, C‘ = N ’ - R’ (37)
F[:”
A’ = 0
13
: :” : 1
10
4 1
(28)
where the rows are for C, 0,and P, and the columns are for ATP, H20, ADP, and Pi (inorganic phosphate), where the symbols represent sums of species. Alternatively, the conservation matrix can be written for adenosine, 0 not in adenosine, and P. A‘=
[: :I [a :
10 1 7 4
(29)
These two matrices have the same row-reduced form and are therefore equivalent. A‘= 0 1 0 1
(30)
:ll
In the conservation matrix A, the number N of columns is equal to the number of species, but in the apparent conservation matrix A’, the number N’ of columns is equal to the number of reactants (sums of species). The rank of A’ gives the number C’ of apparent components (pseudocomponents). The number of apparent components is the number of reactants (sums of species) that have to be specified to describe the equilibrium composition of the system. C‘ = rank A’ (31) Thus, matrix A’ is C’ X N’. The A’ matrix in eq 30 for the hydrolysis of ATP shows that N’ = 4 and C‘ = 3.
The number N’ of reactants (sum of species) in the system is given by N ‘ = rank v’ rank A‘ (38)
+
The number N’ of reactants in the system is smaller than the number N of species because specifying the equilibrium concentration of one species makes groups of species pseudoisomers. That is, their equilibrium distribution is a function only of temperature. In a thermodynamic calculation, pseudoisomers can be considered to be a single species (a pseudospecies), and standard thermodynamic properties can be calculated for the isomer group. In other words, when pH and pMg are specified, reactions and equilibrium constant expressions can be written in terms of reactants (sums of species) rather than species, as described later in connection with eq 45. In general, a biochemical reaction system is defined by specifying a set of independent biochemical reactions, so that it is of interest to use the apparent stoichiometric number matrix v’ to calculate the apparent conservation matrix for the reaction system. The null space of ( v ’ ) ~is (A’)T because
(v’)~(A’)~ =0
(39) In the case of glycoly~is,’~ the null space of (uqT shows that there are seven conservation equations, and so there are seven apparent components. In the case of glycolysis there are only four independent element balances for C, 0, N, and P because H and Mg are not conserved, so that three additional conservation equations arise implicitly from the mechanism; that is, the series of independent reactions in this system of reactions provides
9618 The Journal of Physical Chemistry, Vol. 96, No. 24, 1992
constraints in addition to the element balances. Earlier13 it was thought that there were two additional constraints, but that is an error because H 2 0 was omitted as a reactant. When pH and pMg are specified for a biochemical reaction system, the criterion of equilibrium is no longer the Gibbs energy G. The criterion of equilibrium is the transformed Gibbs energy that is defined by the Legendre transformI3J4 G’ = G - n’(H+) p(H+) - n’(Mg2+) p(Mg2+)
(40)
where n’(H+) is the total amount of H+ (free and bound) in the system and n’(Mg2+) is the total amount of Mg2+(free and bound) in the system. The Legendre transform is a change in variable, and in this case it eliminates the Gibbs energy G from the fundamental equation in favor of the transformed Gibbs energy G and replaces the terms in p(H+) and p(Mg2+) with terms in dp(H+) and dp(Mg2+). However, since A H + ) and p(Mgz+) depend on both temperature and concentration, they are not convenient independent variables, and so a shift is made to pH and pMg. In doing that it is assumed that pi = pio
+ R T In ([i]/co)
(41)
Under these conditions various species of a reactant like ATP become pseudoisomers; and it is convenient to aggregate them and refer to the equilibrium mixture of pseudoisomers as a reactant. The amount of the reactant i is represented by n,‘. The fundamental equation of thermodynamics for the transformed Gibbs energy G’ can be written in terms of the amounts of reactants (sums of species) asI4 dG’ = -S‘ d T + V dP + 2.303nr(H+)RT dpH +
+ i-lCp,’ dn;
(42)
where S’ is the transformed entropy of the system defined by
S’ = S - n’(H+)(SO(H+)- R In ([H+]/co)J n’(Mg2+)(So(Mg2+)- R In ([Mg2+]/co)J (43) and p,‘ is the transformed chemical potential of reactant i. The summation is carried out over the number N’ of reactants in the system that are defined by the columns in the A’ matrix, rather than the number N of species in the system that are defined by columns in the A matrix, and is larger. The fundamentalequation for G’ in eq 42 at specified pH and pMg looks very much like the fundamental equation for G in eq 15. Thus the reactants (like ATP) with amounts n; can be referred to as pseudospecies. Equation 42 can be used to derive the expression (eq 35) for the apparent equilibrium constant K’ for a biochemical reaction like reaction 34. Equation 42 can be used to derive the equilibrium condition for a biochemical reaction. If a single biochemical reaction occurs in a system at specified T, P, pH, and pMg, the apparent extent of reaction can be used in eq 42 to derive
=0 (44) where the i refers to reactants (sums of species), rather than species. Since, for ideal solutions, eq 41 applies to the transformed chemical potential as well as to the chemical potential, the expression for the apparent equilibrium constant can be written cu;p,‘
N’
K’(T,P,pH,pMg,l) = C([il/co)”r
(47) it can be shown that
If A,.Hr0 is independent of temperature in the range TI-T2, then (49) In order to treat multireaction systems, it is useful to write the fundamental equation for the transformed Gibbs energy at specified pH and pMg in matrix form. (dG’)pH,pMg = -S’d T V d P p’ dn’ (50)
+
(45)
(51)
jd is the 1 X N’ transformed chemical potential matrix, n’ is the
N’ X 1 amount of reactant matrix, v’ is the N’ X R’apparent stoichiometric number matrix,and €‘ is the R’ X 1 apparent extent of reaction matrix. Thus the fundamental equation for G’ at specified pH and pMg yields the following equilibrium relationships. pfv‘
=0
(52)
or
..., R’ (53) This equilibrium condition plus p,‘ = p,’O + R T In ([i]/co) yields j = 1, 2,
the general equilibrium expression given in eq 35 for the hydrolysis of ATP. When the system is at chemical equilibrium, the fundamental equation for G’ can also be written in terms of apparent components by eliminating the transformed chemical potential of one of the reactants for each of the R’independent biochemical reactions in terms of reactants (sums of species) by using eq 52. C’
(dG’)pH,pMg = -S’ d T
+ V d P + c p , ’ dn, = -S’ d T + V dP + pc dnc 1-1
(54)
where nc. is the amount of apparent component matrix, which is C’ X 1. Equation 54 can also be written as (dG’)pH,pMK= -s’ d T VdP PctA’ dn’ (55)
+
+
since A’n’ = Q,, where A’ is the apparent conservation matrix. As an illustration, the last term in eq 55 is calculated for the hydrolysis of ATP at specified pH and pMg. Using the row reduced form (eq 30) of eq 28 for the A’ matrix in eq 55 yields (~G’)T,P.~H= .~M~ &’(ATP)p’(H,O)p’(ADP)]
p’(ATP)[dn’(ATP)
+
[: ; !,I[ 1 0 0 1
dn‘(ATP)
dn’(P,)] + F’(H,O)[dn’(H,O)
=
+ dn’(P,)] +
@’(ADP)[dn’(ADP) - dn’(P,)] (56)
1=I
This equilibrium constant expression does not have terms in [H+] and [Mg2+]. The convention is that water is omitted from the equilibrium constant expression when dilute aqueous solutions are considered. The standard transformed Gibbs energy of reaction is given by A,Gr0 = -RT In K ‘ = A,H’O - TA,Sr0 (46) where A$’’ is the standard transformed enthalpy of reaction and A J ‘ O is the standard transformed entropy of reaction. These thermodynamic properties are functions of T, P, pH, pMg, and I. Since
+
(dG’)pH,pMg= -S’d T + V d P + p’d d.$‘
c lJ1;p,’ = 0
N‘
2.303n’(Mg2+)RTdpMg
Alberty
This shows a possible set of three components. Equation 54 (written in terms of pseudocomponents) can be integrated at constant T, P, pH, pMg, and mole fractions of reactants to C‘
G‘ = Ep[nc,‘ = pc,nc. i- I
(57)
The differential of G’ at constant T, P, pH, and pMg is C‘
dG’ =
Cp,’ 1-1
C’
dn,‘
+ Enc,,dp,‘ = pc, dnc, - ( d p c t ) q t l=l
(58)
The Journal of Physical Chemistry, Vol. 96, No. 24, 1992 9619
Feature Article
-
__
TABLE I: CdculPHon of Demees of Freedom F and Apmrent Degrees - of Freedom F ~~
case
N
R
1 2 3
9 17
5 12
15 21
9 15
4 5a 5b
C 4 5 6 6
C‘
-S’ d T
+ V d P - (dpc,)nct (59)
This tells us how many degrees of freedom there are. The transformed Gibbs-Duhem equation (eq 59) for a phase indicates that there are C’ 2 intensive variables for a biochemical reaction system at specified pH and pMg. However, only C’ 1 of these are independent because of eq 59. If there are two phases in equilibrium, there are only C’ independent intensive variables because there is an additional transformed Gibbs-Duhem equation, Thus, if there are p phases, the number F’ of apparent degrees of freedom is given by
+
+
F’ C ‘ - p
+2
(60)
The number of apparent degrees of freedom F’ is less than the number of degrees of freedom F because the pH and pMg have been specified.
Calculation of Degrees of Freedom and Apparent Degrees of
Freedom To calculate the equilibrium composition of a system, it is important to choose the necessary number of degrees of freedom and to specify the values of these independent intensive variables. For any system, there is a certain number of degrees of freedom, but various choices can be made. In this analysis the buffer is ignored, except for H+ and MgZ+,and the ionic strength enters in the values of the equilibrium constants used and not in the degrees of freedom. If the system is described in terms of N species, C = N - R and F = C - p 2. Alternatively, if pH and pMg are specified, the system is discussed in terms of N’ reactants (some of which may be pseudoisomer groups) and R’biochemical reactions. Thus at specified pH and pMg, C‘ = N’ - R’and F’ = C’ - p 2, where F’ is the number of apparent degrees of freedom after pH and pMg are specified. The following five cases are considered from both points of view. They are considered in the order of increasing complexity. Case 1: An Aqueous Solution of ATP in the Neutral pH Range conoaining I m The system involves nine species (including HzO, but excluding OH- since this brings in one more species and one more equilibrium relation, and [H+] will later be treated as an independent variable) and five dissociation reactions. Thus C = 9 - 5 = 4. Since C, N, 0, and P are in the same ratio in all of the ATP species, this combination can be taken as a pseudoelement and be called ATP. Thus the four components can be taken to be ATP, HzO, H, and Mg. The 5 intensive degrees of freedom (F = 4 - 1 2 = 5) can be taken to be T, P, n(ATP)/n(HzO), n(H)/n(HzO), and n(Mg)/n(HzO). When pH and pMg are specified there is a single pseudospecies in water so that N’ = 2, R’ = 0, C’ = 2, and we can take the two components to be ATP and HzO. Since F’ = 3, we can take T, P,and n(ATP)/n(HzO) to be independent variables in addition to pH and pMg. Note that the total number of degrees of freedom is the same when pH and pMg are added to F’,but that it is easier to calculate F‘ than F, and it is the number of independent variables specified in a laboratory experiment in addition to pH and pMg. Case 2 Hydrolysis of ATP with ATPase To Produce ADP and PIin tbe Neutral pH Range in the Presence of Mg2+. This adds 5 species of ADP and 3 species of inorganic phosphate, so there are now 17 species and there are 12 reactions. The set of inde-
+
+
w+
+
R‘
2
0
6
4 5 6 17 8
1 1 2 10 1
7
0 = - S ’ d T + VdP-CncidpLi)= i= 1
N‘
5 7
Subtracting this equation from eq 54 yields the transformed Gibbs-Duhem equation at specified pH and pMg.
’
F
C‘ 2
F’
3
4 5
4
3
4
5
7 7
8 8
pendent reactions includes reaction 1. Thus the number of components is 5 and the number of degrees of freedom is 6. The discussion after eq 23 shows that when pH > 8 and M g + is absent, C = 4 and F = 5 . If pH and pMg are specified, the number N’ of reactants (pseudospecies) is reduced to 4 (ATP, HzO, ADP, and Pi), and the apparent number R’of reactions is reduced to 1 so that the number of pseudocomponents is reduced to 3. Thus the number F’ of apparent degrees of freedom is 4, and these can be taken as T, P, n(aden)/n(HzO), and n(P)/n(HzO). When pH and pMg are added to the apparent number of degrees of freedom, the total is 6, as expected from F. Case 3: Creatine Kinase Reaction in the Neutral pH Range in the Presence of Magnesium Ion. There are 15 species (including HzO) and 9 reactions, including the following: CrP”
+ ADP3- + H+ = A T P - + Cr
(61)
The number of components is equal to the number of elements involved (C, H, 0, N, P, Mg). The 7 degrees of freedom can be taken to be T, P, and the mole fractions of five of the elements. However, since the atoms in adenosine stay together and the atoms of creatine stay together, it is more convenient to take T, P, and the ratios of H, Mg, Cr, adenosine, and P to HzO. When pH and pMg are specified, there are 5 reactants, including HzO, and the biochemical reaction is CrP
+ ADP = ATP + Cr
(62)
where Cr represents the reactant creatine. The four apparent components can be taken to be creatine, adenosine, P, and H20. The five apparent degrees of freedom can be taken to be T, P, n( creatine) / n( HzO), n( aden) / n ( HzO), n( P) /n( HzO), where amounts per liter of water can be used as well. Case 4 ATPase and a Creatine Phosphatase Addition to a Mixture of CrP and ATP in the Neutral pH Raage in the Preseace of Magnesium Ion. There are 21 species and 15 reactions, including reaction 1 and the following reaction:
+
CrPZ- HzO = Cr
+ HPOdZ-
(63)
Inorganic phosphate is present at equilibrium. The number of components is equal to the number of elements, as in the preceding case. When pH and pMg are specified, there are five reactants, including H20. The two reactions are reaction 34 and CrP
+ HzO = Cr + Pi
(64)
The four apparent components can be taken to be aden, Cr, P, and HzO. The five apparent degrees of freedom can be taken to be T, P, n(aden)/n(HzO), n(Cr)/n(H,O), and n(P)/n(H,O), where amounts per liter of water can be used as well. Case i k GIycolysis in the Neutral pH Range. At specified pH and pMg, glycolysis involves 17 reactants (including H20) and 10 biochemical reactions, so that there are 7 apparent components. However, since only 4 elements are conserved (C, 0,N, and P), this indicates that there are 3 additional constraints. There are 8 apparent degrees of freedom. Six of these involve ratios of amounts of elements or peudoelements. There are obviously many choices of independent variables. The choice of independent variables is perhaps best considered in terms of the net reaction, which is discussed in the next section. The number F of degrees of freedom can be calculated by counting species and chemical reactions, but this is not necessary for defining the conditions for an equilibrium system.
9620 The Journal of Physical Chemistry, Vol. 96, No. 24, I992
Case sb: Net Reaction for Glycolysis in the Neutral pH Range. When pH and pMg are specified, the sum of the reactions in glycolysis with the last five reactions multiplied by 2 is glucose 2Pi + 2ADP + 2NAD = 2pyruvate + 2ATP + 2NADH + 2 H 2 0 (65)
+
If this was the only reaction in a system, where would be seven components since there are eight reactants. This system has eight apparent degrees of freedom, the same as glycolysis. The apparent degrees of freedom can be taken to be T,P, and the ratios of six of the components to the seventh component. It is convenient to take water as the seventh component. The point of this section is that it is not necessary to count all the species and reactions in order to make equilibrium calculations on a system of biochemical reactions. If the apparent equilibrium constants K' are known for the reactions involved at the desired pH and pMg, the equilibrium composition can be calculated by specifying in addition T, P,and the necessary number of independent ratios of components.
Deterohtion of the Change in Binding of Protons and Magnesium Ions in a Reaction at Specified pH and pMg When a biochemical reaction occurs at specified pH and pMg, there is a change in the binding of H+ and Mg2+to reactants and products that is represented by AJV(H+) and AJV(Mg2+). The reactants bind a certain amount of H+, for example, and the products bind a different amount under the specified conditions. The difference is the change in binding and is the amount of H+ produced or consumed per mole of biochemical reaction as written. These are dimensionless quantities because they are in moles per mole of reaction. The negatives of these quantities were referred to in earlier article^^^-^^ as n H and n M the amounts of H+ and Mg2+produced per mole of reaction. 'fhe fundamental equation of thermodynamics provides a general way to determine these quantitia from measurements of the apparent equilibrium constant K'. When eq 42 is applied to a biochemical reaction, the changes in binding are given by
Alberty reservoir and pMg reservoir through semipermeable membranes. When the reaction is carried out in this way, H+ and Mg2+are not conserved in the system, but other elements are. Thus the conservation matrix is changed to A' that has columns only for reactants (sums of species), and the apparent stoichiometric number matrix Y' calculated from it deals with reactions written in terms of reactants (sums of species). The apparent equilibrium constant K' written in terms of reactants is the key to all of the thermodynamic properties of a biochemical reaction. The standard transformed Gibbs energy of reaction A,G" is calculated from K', and the standard transformed entropy of reaction A$"" and standard transformed enthalpy of reaction A a ' " are calculated from derivatives of A r c f " with respect to T. The availability of these reaction properties for a series of reactions makes it possible to calculate A,G"(i), AfH'"(i), and AP'"(i) for the various r e a ~ t a n t s . ~ ~ J ~ * * ~ The changes in binding AJV(H+) and AJV(Mg2+)in the reaction are calculated from derivatives of 4G'O with respect to pH and pMg. All of these quantities depend on the ionic strength, and so it is important to specify I as well as pH and pMg. The changes in standard transformed thermodynamic properties in a biochemical reaction can be calculated if there is enough information about the thermodynamics of the species of the reactants, but this is seldom the case. The important point is that measurements of K' for a biochemical reaction make it possible to calculate all of the transformed thermodynamic properties without thermodynamic information on the species. Some properties can be determined directly: can be determined calorimetrically, if a correction is made for the heat effects due to the production (consumption) of H+ and Mg2+by the enzyme catalyzed reaction, and A,.N(H+) can be determined by use of a pH stat. Since there is greater interest in the thermodynamic properties of biochemical reactants at pH 7 and pMg 3 than at other pH and pMg, tables of transformed thermodynamic properties are needed for these conditions.
w'"
Acknowledgment. Acknowledgement is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for support of this research. I am indebted to Robert N. Goldberg and Irwin Oppenheim for helpful discussions.
Nomenclature (UsualDimensions Are Shown in Parentheses)
These equations make it possible to determine AJV(H+) and AJV(Mg2+)without having knowledge of the acid dissociation constants and magnesium complex dissociation constants of the reactants. Taking the cross derivatives yields
(
aA,N( H+) *Mg )T,P,pH =
(
aArN(Mg2+) apH )T,P,pMg
(68)
which expresses the linkage6,22between the change in binding of H+and the change in binding of Mg2+in a biochemical reaction.
Discussion This article has described the thermodynamics of biological reactions on the basis of the fundamental equation of thermodynamics, but the emphasis has been on the interpretation of the equilibrium experiment. In the laboratory the experiment is carried out at constant temperature and pressure and the pH and pMg change a little during the approach to equilibrium. When apparent equilibrium constants are calculated, it is the final pH and pMg that are important. From the point of view of the interpretation of the apparent equilibrium constant, the determination of the equilibrium composition has been made at this pH and pMg. Thus pH and pMg are independent variables in the same sense as T and P. However, specifying pH and pMg is different from specifying T and P in that this requires a reconceptualization of the experiment. It is as if the experiment had been carried out in a reaction chamber connected with a pH
A A' C C'
conservation matrix (C X N) (dimensionless) apparent conservation matrix (C' X N ') (dimensionless) number of components (dimensionless) number of apparent components (pseudocomponents) (dimensionless) CO standard-state concentration (1 M) F number of degrees of freedom (dimensionless) F' number of apparent degrees of freedom (dimensionless) G Gibbs energy of a system (kJ) transformed Gibbs energy of a system (kJ) G' ArG 'O standard transformed Gibbs energy of reaction (kJ mol-') A,GJo(i) standard transformed Gibbs energy of formation of i (kJ mol-I) H enthalpy of a system (kJ) H' transformed enthalpy of a system (kJ) A,H" standard transformed enthalpy of reaction (kJ mol-I) AfH'O(i) standard transformed enthalpy of formation of i (kJ mol-') I ionic strength (mol L-I) K equilibrium constant at specified T,P,and I (dimensionless) K' apparent equilibrium constant at specified T,P,pH, pMg, and I (dimensionless) amount of species i (mol) n, amount of component i (mol) "C, amount of apparent component i (mol) "C; amount of reactant i (sum of species) (mol) ni n amount of species matrix (N X 1) (mol) n' amount of reactant matrix (N'X 1) (mol) amount of component matrix (C X 1) (dimensionless) nC amount of apparent component matrix (C' X 1) (dimennc, sionless) A.N(H+) change in binding of H+ in the reaction (dimensionlessl AiNiM&+) change in bindingof Mg2+ in the reaction (dimensionless) N number of species (dimensionless)
Feature Article N'
P PH PMg
P
R R'
S Si
S'
AJ A$ Pi
Pi' P
B' PC
PC' Y
'V
€
e
'O 'O
(i)
number of reactants (pseudospecies) (dimensionless) number of phases (dimensionless) -[log ( [ H + ] / c o ) ](dimensionless) -[log ( [ M g 2 ' ] / c o ) ] (dimensionless) pressure (bar) number of independent reactions in a system (dimensionless) number of independent biochemical reactions in a system (dimensionless) entropy of a system (J K-I) (partial) molar entropy of species i (J K-I mol-') transformed entropy of a system (J K-I) standard transformed entropy of reaction (J K-I mol-I) standard transformed entropy of formation of i (J K-' mol-I) chemical potential of species i (kJ mol-') transformed chemical potential of reactant i (kJ mol-') chemical potential matrix for species (1 X N) (kJ mol-I) transformed chemical potential matrix of reactants (sums of species) (N' X 1) (kJ mol-') chemical potential matrix for components (1 X C) (kJ mol-') transformed chemical potential matrix for apparent components (1 X C') (kJ mol-I) stoichiometric number matrix ( N X R ) (dimensionless) apparent stoichiometric number matrix (N' X R ' ) (dimensionless) extent of reaction matrix ( R X 1) (dimensionless) apparent extent of reaction matrix (R'X 1) (dimensionless)
References and Notes (1) Ramage, M. P.; Graziani, K. R.; Krambeck, F. J. J . Chem. Eng. Sci. 1980, 35, 41.
The Journal of Physical Chemistry, Vol. 96, No. 24, 1992 9621 (2) Alberty, R. A. J. Phys. Chem. 1985,89, 880. (3) Cheluget, E. L.; Missen, R. W.; Smith, W. R.J . Phys. Chem. 1987, 91, 2428. (4) Alberty, R. A.; Oppenheim, I. J . Chem. Phys. 1988,89, 3689. ( 5 ) Beattie, J. A.; Oppenheim, I. Principles of Thermodynamics; Elsevier: New York, 1979. (6) Wyman, J. Biophys. Chem. 1981, 14, 135. (7) Wyman, J.; Gill, S.J. Binding and Linkage; University Science Books: Mill Valley, CA, 1990. (8) Alberty, R. A.; Oppenheim, I. J. Chem. Phys. 1989, 91, 1824. (9) Alberty, R. A.; Oppenheim, I. J . Chem. Phys. 1992,96,9050-9054. (10) Alberty, R. A. In Chemical Reactions in Complex Systems; Krambeck, F. J., Sapre, A. M., Eds.; Van Nostrand Reinhold: New York, 1991. (1 1) Norval, G. W.; Phillips, M. J.; Missen, R. W.; Smith, W. R. Can. J . Chem. Eng. 1991, 69, 1154-1 192. (12) Wads& I.; Gutfreund, H.; Privalov, J. T.; W a l l , J. T.; Jencks, W. P.;Strong, G. T.; Biltonen, R. L. Interunion Commission on Biothermodynamics (1976). J. Biol. Chem. 1976,251,6879-6885. (13) Alberty, R. A. Biophys. Chem. 1992,42, 117-131. (14) Alberty, R. A. Biophys. Chem. 1992, 43, 239-254. (15) Smith, W. R.; Missen, R. W. Chemical Reaction Equilibrium Analysis: Theory and Algorithms; Wiley-Interscience: New York, 1982. (16) Strang, G. Linear Algebra and Its Applications; Harcourt Brace Jovanovich: Kent, U.K., 1988. (1 7) Wolfram Research, Inc., Champaign, IL. (18) Goldberg, R. N.; Tewari, Y. B. Biophys. Chem. 1991,40,241-261. (19) Clarke, E. C. W.;Glew, D. N.J. Chem.Soc. 1980,176, 1911-1916. (20) Alberty, R. A. Proc. Natl. Acad. Sci. 1991,88, 3268-3271. (21) Alberty, R. A. J. Chem. Educ. 1992,69, 493. (22) Alberty, R. A. J . Biol. Chem. 1968,243, 1337-1343. (23) Alberty, R. A. J. Biol. Chem. 1969, 244, 3290-3302. (24) Alberty, R. A. J . Chem. Educ. 1969,46, 713-719. (25) Alberty, R. A.;Goldberg, R. N. Biochemistry 1992,31,10610-10615.
