Page 116 - Thermodynamics of Biochemical Reactions

112 Chapter 6 Systems of Biochemical Reactions

pseudo isomer groups is reduced from N' + 2 to N". A double prime is used on
the amounts in the summation to indicate that these are amounts of pseudoisomer
groups at specified [ATP] and [ADP] as well as pH. The further transformed
entropy of the system is given by
S" = S' - nL(ATP)SL(ATP) - ni(ADP)Sd(ADP) (6.5-1 1)
where SL(ATP) is the molar transformed entropy of ATP at the specified pH and
ionic strength. There is also a further transformed enthalpy given by
H" = G" + TS",
The fundamental equation for G" given in equation 6.5-10 leads to several
new types of relations between properties. First consider the equation for dG" for
a system containing a single pseudoisomer group; that is, the summation is
replaced with A, Gys,dnj',,. The Maxwell equation between this term and the term
in dpH is

?A,G::,
(~)~.~,n~,",[*~P~,~ADP~ RTln(lO)RH (6.5-12)
=
where n,(H) is replaced by NH, which is a more useful symbol for the average
binding of hydrogen atoms by the pseudoisomer group containing different
reactants. This equation is like equation 4.7-3. It gives the average binding of
hydrogen atoms by the pseudoisomer group as a function of the pH at specified
concentrations of ATP and ADP.
The Maxwell equations between A,G~~, dnib, and the terms in dln[ATP] and
dln[ADP] are

(6.5-1 3)

(6.5-14)

In these equations NArP is equal to the rate of change of n,(ATP) with respect to
the amount of the pseudoisomer group, and RAUp is equal to the rate of change
of n,(ADP) with respect to the amount of the pseudoisomer group. Thus these
equations give ArATp and NADp as functions of the independent variables. The
binding of a component can be negative, as pointed out in Section 3.3.
Taking the derivative of equation 6.5-13 with respect to ln[ADP] yields the
same result as taking the derivative of equation 6.5-14 with respect to InCATP]:
therefore,

We have seen this type of reciprocal relation twice earlier: see equation 1.3-17
and equation 4.8-9. There are also reciprocal relations between the binding of
ATP and hydrogen ions and between ADP and hydrogen ions.
The complete Legendre transform for the system we are discussing yiclds the
Gibbs-Duhem equation for the system:
0 = -S"dT+ VdP + C" n:idpj' + RTln(lO)n,(H)dpH - n;(ATP)RTdln[ATP]

i= 1
- n: (ADP) R T dln[AD P] (6.5- 16)
This relation between the C" + 5 intensive properties of the system shows that
the number of independent intensive degrees of freedom is F" = C" + 4. Since this
is a one-phase system, the total number of degrees of freedom is D" = C" + 5.
The expressions for apparent equilibrium constants K" are written in terms
of concentrations of the N" pseudoisomer groups; thus [ATP] and [ADP] do not
appear explicitly in equilibrium constant expressions for the system. The criterion