Page 115 - Thermodynamics of Biochemical Reactions

6.5 Three Levels of Calculation of Compositions for Systems of Biochemical Reactions 11 1

component. As mentioned in the discussion of matrices in Section 5.1, the
reactants listed first in the conservation matrix will be taken as components if they
are sufficiently different. Since G’ = C pin;, G” = X pyny, nb(ATP) = C NATp(i)ni,
and nb(ADP) = C NADp(i)ni, equation 6.5-1 shows that the further transformed
chemical potential of reactant i (sum of species) is given by

p!’ = p! - N ATp(i)d(ATP) - NADp(j)p’(ADP) (6.5-2)
where NATp(i) and NADp(i) are the numbers of ATP and ADP molecules required
to make up the ith reactant; note that these numbers may be positive or negative.
The values of NATp(i) and NADp(i) can be obtained from the apparent conserva-
tion matrix (see equation 6.5-21).
The differential of the Legendre transform in equation 6.5-1 is
dG” = dG‘ - ni(ATP)dp‘(ATP)) - p’(ATP)dnb(ATP) - nL(ADP)dp’(ADP)
p’(ADP)dn’,(ADP) (6.5-3)
The general equation for dG’ at a given pH is equation 4.1-18. This equation can
be written in terms of the amounts of components nh(ATP) and nk(ADP) by use
of equation 6.5-2. This yields
N’ - 2
dG‘ = -S’dT+ VdP + c pi’ dni + p’(ATP)dnh(ATP) + p’(ADP)dnb(ADP)
i= 1
+ n,(H)RTIn(lO)dpH (6.5-4)
where dn’,(ATP) = C NATp(i)diii and dnL(ADP) = C NADp(i)dni. Substituting
equation 6.5-4 into equation 6.5-3 yields the following fundamental equation
for G”:
N‘ 2
~
dG’ = -S’dT+ VdP + 1 py dni - nL(ATP)dp’(ATP) + nk(ADP)dp‘(ADP)
i= 1
+ n,(H)RTln(lO)dpH (6.5-5)
However, p‘(ATP) and p’(ADP) are not convenient independent variables
because they depend on temperature as well as concentration. To eliminate
dp’(ATP) and dp’(ADP) from equation 6.5-5, the following equations are used:

dp‘(ATP) = [““~‘p’]~,p.,[~T~~dT+ [:LATpl ]T,P,pH d[ATP] (6.5-6)
ap’( ADP)
d[ADP] (6.5-7)
dp’(ADP) = ~~’(;.:p)]P,p”,[A~P~dT+ IT,P,pH
The derivatives in the first terms of these equations are -SL(ATP) and
-SL(ADP), and the derivatives in the second terms are calculated using ,LL‘
(ATP) = p”(ATP) + RT In[ATP]. Thus

dp’(ATP) = -SL(ATP)dT+ RTdln[ATP] (6.5-8)
dp’(ADP) = -Sk(ADP)dT+ RTdln[ADP] (6.5-9)
Substituting these equations in equation 6.5-5 yields
N”
dG” = -S”dT+ VdP + c p:dnj‘ + RTln(lO)n,(H)dpH
1=1
- nb(ATP)RTdln[ATP] - n’,(ADP)RTdln[ADP] (6.5- 10)

It can be shown that when the reactions of ATP and ADP with reactants in the
system are at equilibrium, the further transformed chemical potentials of some of
the reactants are equal; these reactants form a pseudoisomer group with amount
n:. Thus holding [ATP] and [ADP] constant makes it possible to reconceptualize
the system into a smaller set of pseudoisomer groups; specifically, the number of