Page 183 - Thermodynamics of Biochemical Reactions

11.4 Semigrand Partition Function for a Biochemical Reaction System 183

McQuarrie (2000) gives a nice table like this for several types of partition
functions. The further transformed enthalpy H' of the system at specified pH can
be calculated by use of the Gibbs-Helmholtz equation:

Note that G' = H' - TS'. This shows that the semigrand partition function r'
contains all the thermodynamic information about the system.

11.3 SEMIGRAND PARTITION FUNCTION FOR A
SYSTEM CONTAINING TWO PSEUDOISOMER
GROUPS AT A SPECIFIED pH

For a syslem containing two pseudoisomer groups A and B at a specified pH, the
semigrand partition function is given by

r'(T P, pH, NIsoA, N;,,,) = {exp( -/j~LIs,A)}N;'"h(exp( -ppis,u)}NiSoB (11.3-1)

Substituting this in G' = - kTln r' yields
G' = - kTln[{exp( - flpisoA)}Ni50A{exp( /jp~s,u)}1v~5uB]
-
(1 1.3-2)
=N! isoA p! isoA + NisoBp;soB
If pseudoisomer groups A and B are involved in a reaction
A = 2B (11.3-3)

and there are initially (NisoJO molecules of A, the transformed Gibbs energy of
the system at any time in the reaction is given by

(1 1.3-4)
G' = (NisoA)OdsoA + (2pisoB - k3oA)t'
where 4' is the extent of reaction. The transformed Gibbs energy is minimized at
equilibrium, and so
AG' (11.3-5)
~ dj" = 2dsoB - PisoA =
which is the equilibrium condition.

H 11.4 SEMIGRAND PARTITION FUNCTION FOR A
BIOCHEMICAL REACTION SYSTEM AT
SPECIFIED CONCENTRATIONS OF COENZYMES

At level 3 the semigrand partition function I?' for a sum of reactants that are
pseudoisomers at specified concentrations of coenzymes is given by a sum of
exponential terms raised to a power equal to the number of molecules in the
pseudoisomer group (Alberty, 2001g):
(1 1.4-1)
r" = {exp( -[jpy) + exp(-[jpi) + ...}N15<i
Substituting this in G" = - kTln r" yields
G" = -N:'sokTIn{exp(-flp;) + exp(-[jp';) + ...} = N:50p:'50 (11.4-2)

The semigrand partition function can be written for a system of biochemical
reactions at specified concentrations of coenzymes, and G" = - kTln r". The
fundamental equation for G" can be used to calculate the other thermodynamic