Page 184 - Thermodynamics of Biochemical Reactions

184 Chapter 11 Use of Semigrand Partition Functions

properties of a system. The fundamental equation for G" is
dG" = -S"dT + VdP + EprdNy + kTln(lO)N,(H)dpH

- X N,(coe,)kTdln[coe,] ( 1 1.4-3)

( 1 1.4-4)

(1 1.4-5)

(1 1.4-6)

NJH) = (1 1.4-7)
"(toe,) = - ( (7G" ) = -( ? In r" )
1
/zT i In[coe,] ? In[coe,] (1 1.4-8)
The subscripts on the partial derivatives have been omitted because they are
complicated, as indicated by the fundamental equation. The change in "binding"
of coenzymes in a reaction can be studied at constant concentrations of coen-
zymes, just as the change in binding of hydrogen ions in a reaction can be studied
at constant pH. The further transformed enthalpy H" of the system can be
calculated by use of the Gibbs-Helmholtz equation or from G" = H" - TS".
For glycolysis at specified 7; P, pH, [ATP], [ADP], [Pi], [NAD,,,], and
[NAD,,,]. the semigrand partition function is given by (see Section 6.6)

r" = (exp( -/jpi~))~i(exp( -/)pi)).'; ( 1 1.4-9)
which leads to
GI1 = N" (1 1.4-10)
,& + ";hi;
Note that these two further transformed chemical potentials each involve summa-
tions of exponential terms over the C, reactants and C, reactants. The standard
further transformed Gibbs energies of formation of C, and C, can be calculated
at the desired pH, ionic strength, and concentrations of coenzymes using equation
11.4-2. The apparent equilibrium contants K" for C, = 2C, can be calculated.
Solving a quadratic equation yields [C,] and [C,] as shown in Section 6.6. The
distribution of reactants within these two pseudoisomer groups can then be
calculated. The concentrations of species within the reactants can also be
calculated. Thus no thermodynamic information is lost in going to the level 3
calculations. This method can be applied to larger systems by specifying the
Concentrations of more coenzymes, but the number of coenzymes that can be
specified is C" - 1 or less because at least one component must remain.

11.5 DISCUSSION

The thermodynamics of biochemical rections at specified pH and specified
concentrations of coenzymes can be represented by semigrand partition functions.
Partition functions are always sums of exponential terms. For a single reactant at
specified pH, r' has a term for each species that is weighted by a factor that is
exponential in N,,(i)pH. For a mixture of reactants, the partition function P' for
the system is a product of partition functions of reactants each raised to the power
of the number of molecules of the reactant. Thus the partition function for a
mixture of reactants is also a summation of exponential terms. For a sum of
reactants at specified concentrations of coenzymes, for example, C,. the partition
function I-'' is a sum of exponential terms, with a term for each reactant (e.g..