Page 222 - Thermodynamics of Biochemical Reactions

222 Mathematica Solutions to Problems

calcdGHT[speciesma-1 := Module[{dGzero, dGzeroT, dHzero,
zi, nH, gibbscoeff, pHterm, isterm, gpfnsp, dGfn, dHfn),(*This program produces
the function of T (in Kelvin), pH, and ionic strength (is) that gives the stndard
transformed Gibbs energy of formation of a reactant (sum of species) and the standard
transformed enthalpy. The input speciesmat is a matrix that gives the standard Gibbs
energy of formation at 198.15 K, the standard enthalpy of formation at 298.15 K, the
electric charge, and the number of hydrogen atoms in each species. There is a row in
the matrix for each species of the reactant. gpfnsp is a list of the functions for
the transformed Gibbs energies of the species. The output is in the form {dGfn,dHfn),
and the energies are expressed in kJ molA-l. The values of the standard transformed
Gibbs energy of formation and the standard transformed enthalpy of formation can be
calculated at any temperature in the range 273.15 K to 313.15 K, any pH in the range 5
to 9, and any ionic strength in the range 0 to 0.35 M by use of the assignment
operator (/.).*)
{dGzero, dHzero, zi, nH) = Transpose[speciesmat1;
gibbscoeff = (9.20483*t)/10A3 - (1.284668*tA2)/10^5 +
(4.95199*tA3)/1OA8; dGzeroT = (dGzero*t)/298.15 +
dHzero*(l - t/298.15); pHterm =
(nH*8.31451*t*Log[10A(-pH)I)/1000;
istermG = (gibbscoeff*(ziA2 - nH)*isA0.5)/
(1 + 1.6*isA0.5); gpfnsp = dGzeroT - pHterm - istermG;
dGfn = -((8.31451*t*Log[Plus @@
(EA(-(gpfnsp/((8.31451*t)/1000))))l)/1000~;
dHfn = -(tA2*D[dGfn/t, tl); {dGfn, dHfn)]
Calculate the functions that yield the standard transformed Gibbs energies of the reactants in the reaction atp + h20 = adp + pi
at 313.15 K.

atp313 = calcdGHT [atpsp] El] / . t + 313.15;

h20313 = calcdGHT [h2osp] El] /. t + 313.15;

adp313 = calcdGHT[adpsp] 111 /. t + 313.15;
pi313 = calcdGHT [pisp] El] /. t -+ 313.15;

Calculate the standard transformed reaction Gibbs energies for the hydrolysis of ATP at 313.15 K, pHs 5, 6, 7, 8, and 9 and
ionic strengths of 0, 0.10, and 0.25 M.

dGerx313 =calctrGerx[atp313+h20313+de== adp313+pi313, {5, 6, 7, 8, 9), {0, 0.1, 0.25}];
TableForm[Transpose[dGerx313], TableHeadings -+
{{"I = 0 MIi, "I = 0.10 Mi*, "1 = 0.25 Mi'}, {"pH 5", l1pH 6", "RH 7", "pH 8", "RH 9")}1
PH 5 PH 6 PH 1 PH 8 PH 9
I=OM -35.9046 -36.4816 -38.212 -43.5265 -49.6803

I = 0.10 M -33.6656 -34.2845 -37.1598 -42.482 -48.4002
I = 0.25 M -32.8611 -33.5895 -36.6852 -42.0518 -41.9136

9.6 Calculation of the standard transformed Gibbs energies of formation of the species of a
reactant from the apparent equilibrium constant of a reaction

To do this all the reactants have to be in the database, except for one or two, The following three programs, which make this
calculation in one step, are based on the concept of the inverse Legendre transform. The program to be used depends on the
number of species in the reactant. The programs produce entries for the database on species.