Page 113 - Thermodynamics of Biochemical Reactions

6.4 Calculations of Equilibrium Compositions for Systems of Biochemical Reactions 109

Krambeck (1978, 1991) in APL and in Muthemutica and by Smith and Missen
(1982) in Fortran. Krambeck wrote the program equcalc for use on gaseous
mixtures involved in petroleum processing and also adapted it to solution
reactions as equcalcc and equcalcrx. These latter programs, which are given in
BasicBiochemData2, have the advantage that they operate with conservation
matrices and stoichiometric number matrices, respectively, so that they can be
used with systems of any size.
Equcalcc was written to calculate compositions of reaction systems in terms
of species, given the conservation matrix A, the vector of standard Gibbs energies
of formation, and initial amounts of species, but since the fundamental equation
for G at specified pH has the same form as the fundamental equation for G,
equcalcc can be used with the apparent conservation matrix A‘, the vector of
standard transformed Gibbs energies of formation of pseudoisomer groups, and
initial amounts of pseudoisomer groups. However, there is a problem when H,O
is a reactant. as was discussed in the preceding section. When equcalcc is used for
reactions involving water as a reactant, further transformed Gibbs energies of
formation (based on equation 6.3-2) and conservation matrices omitting oxygen
have to be used.
Equcalcrx was written to calculate the equilibrium composition in terms of
species, given the stoichiometric number matrix, but it can be used with the
apparent stoichiometric number matrix at a specified pH. Apparent equilibrium
constants have to be known for a set of independent biochemical reactions for the
system. This program has the advantage over equcalcc that further transformed
Gibbs energies of formation (based on equation 6.3-2) do not have to be
calculated when water is involved as a reactant. Actually equcalcrx obtains A’ by
calculating the null space of (v’)~ and then using equcalcc. Although equcalcc and
equcalcrx appear to require the vector of initial amounts, it is really only the
vector of initial amounts of components that is used.
The inputs for these programs are designated by the following terminology:

1. When equcalcc[as,lnk,no] is applied to a system of R independent chemi-
cal reactions, it requires a C x N conservation matrix as, a list Ink of standard
Gibbs energies of formation of species multiplied by (- l/RT), and a list no of
the initial concentrations of species. It can be used at specified pH by using a
C‘ x N‘ conservation matrix as, a list of standard transformed Gibbs energies of
pseudoisomer groups multiplied by (- l/RT), and a list no of initial concentra-
tions of pseudoisomer groups.
2. When equcalcrx[nt,lnkr,no] is applied to a system of R independent
chemical reactions, it requires a R x N transposed stoichiometric number matrix
nt, a vector of natural logarithms of the equilibrium constants of independent
reactions, and a vector no of the initial concentrations. It can be used at a
specified pH by using a R’ x N’ transposed stoichiometric number matrix nt, a
vector lnkr of natural logarithms of the apparent equilibrium constants of
independent biochemical reactions, and a vector no of the initial concentrations.

The use of these programs is illustrated in Problems 6.4 to 6.8.
The calculation of the equilibrium composition of a system of chemical
reactions with equcalcc is based on minimizing the Gibbs energy subject to the
conservation condition An = n,. This is accomplished by using a Lagrangian L
defined by
L = G - h(An - n,) = (p - hA)n + An, (6.4-1)

where 3, is the 1 x C vector of Lagrange multipliers. At equilibrium the rates of
change of L with respect to the amounts of each of the species must be equal to
zero. Thus at equilibrium,

p = LA (6.4-2)