Page 66 - Thermodynamics of Biochemical Reactions

4.1 Fundamental Equation for a Biochemical Reaction System at Specified pH 61

Thus the definition of the transformed Gibbs energy for the system by G' =
G - n,(H)p(H+) automatically brings in the transformed enthalpy H' =
H - n,(H)H,(H+) and the transformed entropy S' = S - n,(H)S,(H+) so that
there is a complete set of transformed thermodynamic properties.
The summation in equation 4.1-13 can be written with fewer than N, - 1
terms because when the pH is specified, groups of terms now have the same value
of pi. These are the terms for the different protonated species of a reactant. When
a group of species differ only in the number of hydrogen atoms that they contain,
these species have the same transformed chemical potential pi at a specified pH,
and this makes them pseudoisomers. Isomers have the same chemical potential at
chemical equilibrium, and pseudoisomers have the same transformed chemical
potential ,ul at equilibrium at a specified pH. For example, the various protonated
species (ATP4-, HATP3-, H,ATP2'-) of ATP have the same transformed
chemical potential ,us at a specified pH. This can be proved by minimizing G' at
specified 7; P, and pH for a system containing the three species of ATP. Since
pseudoisomers have the same transformed chemical potential pi, we can collect
terms for pseudoisomers and use n: = Xnj for the amount of a pseudoisomer group.
Thus equation 4.1-13 can be rewritten as

"
dG' = -S'dT+ VdP + c pidni + RTln(lO)n,(H)dpH (4.1-18)
i=l

where N' is the number of pseudoisomer groups in the system. A pseudoisomer
group may contain a single species. This is the form of the fundamental equation
for G' that is used to treat biochemical reaction systems in a single phase. Note
that this fundamental equation has a new type of term, the last one, that is
proportional to dpH. The number D' of natural variables of G' is N' + 3, which
may be considerably less than the D = N, + 2 for the system described in terms
of species. In writing equation 4.1-18, it is assumed that the binding of H+ by
species is at equilibrium. Acid dissociations are equilibrated much more rapidly
than enzyme-catalyzed reactions.
A very important step has been taken in aggregating species in equation
4.1-18 so that the number of terms proportional to differentials in amounts is
reduced from N, - 1 (in equation 4.1-6) to N' (in equation 4.1-18). Aggregating
groups of species makes it possible to deal with ATP as a reactant at a specified
pH. This more global view makes it easier to think about systems of metabolic
reactions. Within a pseudoisomer group, the transformed chemical potentials of
species at equilibrium are equal, the amounts add, and the standard ther-
modynamic properties of the group are given by the isomer group equations
discussed earlier (3.5-11 to 3.5-18). This matter will be discussed in greater detail
in Section 4.3.
Equation 4.1-18 can be integrated at constant values of the intensive proper-
ties to obtain

(4.1- 19)

Thus the transformed Gibbs energy is additive in the transformed chemical
potentials of pseudoisomer groups just like the Gibbs energy G is additive in the
chemical potentials of species (equation 2.5-12).
Equation 4.1-18 shows that G' is a function of 7; P, {nil, and pH, and so
calculus requires that