Page 114 - Thermodynamics of Biochemical Reactions

110 Chapter 6 Systems of Biochemical Reactions

This is equivalent to the equilibrium condition pv = 0, and so the objective is to
calculate the Lagrange multipliers. The number of Lagrange multipliers is equal
to the number of components. Once the Lagrange multipliers have been obtained,
the chemical potentials can be calculated using equation 5.3-2 and the equilibrium
mole fractions can be calculated using
RT ]
LA - p*
x=exp[ (6.4-3)

where the elements of p*
pf = p; + RTln[Bj] (6.4-4)
are the chemical potentials of the species.

6.5 THREE LEVELS OF CALCULATIONS OF
COMPOSITIONS FOR SYSTEMS OF
BIOCHEMICAL REACTIONS

In Chapter 4 we saw how specifying the pH and using the transformed Gibbs
energy G‘ provides a more global view of a biochemical reaction. This process of
making Legendre transforms can be continued by specifying the concentrations
of coenzymes like ATP, ADP, NAD,,, and NAD,,, (Alberty,l993c, 2000b, c.
2002a). These reactants are produced and consumed by many reactions, and so,
in a living cell, their concentrations are in steady states. Thus the thermodynamics
of a system of enzyme-catalyzed reactions can be discussed at three levels.
Description of a reaction system in terms of species is referred to as level 1,
discussion in terms of reactants (sums of species) at specified pH (e.g., or specified
pH and pMg) is referred to as level 2, and discussion in terms of reactants at
specified steady state concentrations of coenzymes is referred to as level 3. For the
purpose of derivations we can imagine that the system at level 3 is connected with
reservoirs of coenzymes at their steady state concentrations by means of semiper-
meable membranes. The maximum number of components that can be specified
in this way is one less than the number of components in the system. Table 6.1
shows the criteria for spontaneous change and equilibrium for various specifica-
tion of independent variables.
When the concentrations of ATP and ADP are in a steady state, these
concentrations can be made natural variables by use of a Legendre transform that
defines a further transformed Gibbs energy G” as follows.

G” = G‘ - nL(ATP)p’(ATP) - nL(ADP)p’(ADP) (6.5- 1 )
In this equation nLATP) is the amount of the ATP component in the system, that
is, the total amount of ATP free and bound. Thus nL(ATP) and p’(ATP) are
conjugate variables, and nL(ADP) and p’(ADP) are conjugate variables. It may
seem remarkable that ATP and ADP at a specified pH can be taken as
components, but any group or combination of atoms can be taken as a

Table 6.1 Levels of Thermodynamic Treatment

Criterion for spontaneous change and
equilibrium at constant independent
Level Independent Variables variables