Page 321 - Thermodynamics of Biochemical Reactions

Matrices in Chemical and Biochemical Thermodynamics 321

TableFonu [nutrexpected, TableHeadings-> { { *lrxlln, "rx2 'I, 11rx311, *1rx4" 1, { "ATP4-
'I, "H+", "H20", "HP042-", "ADP3-", "HATP3- " , "HADP2- " , "H2P04-" 1 } 1
ATP4- H+ H20 HP042- ADP3- HATP3- HADP2- H2P04-
rxl -1 1 -1 1 1 0 0 0
rx2 1 1 0 0 0 -1 0 0
rx3 0 1 0 0 1 0 -1 0

rx4 0 1 0 1 0 0 0 -1
We can check whether NullSpace[a] and nuexpected are equivalent by row reducing each of them.

1
TableForm [RowReduce [NUllSgace [a] 1
1 0 0 0 - 1 - 1 1 0
0 1 0 0 1 0 -10
0 0 1 0 0 1-1-1
0 0 0 1 - 1 0 1 - 1
TableForm[RowReduce[nutrexpectedll

1 0 0 0 - 1 - 1 1 0
0 1 0 0 1 0 -10
0 0 1 0 0 1-1-1
0 0 0 1 - 1 0 1 - 1

Since these last two matrices are identical, NullSpace[a] and nuexpected are equivalent.

5.3 (a) Construct the conservation matrix A' for the hydrolysis of ATP to ADP in terms of reactants. (b) Calculate a basis
for the stoichiometric matrix from the conservation matrix and show that it is consistent with ATP + HI 0 = ADP + Pi.

(a) Construct the conservation matrix for the hydrolysis of ATP to ADP in terms of reactants.

TablePorm[aa,TableHeadings->{ {asCmr, "O", "P"), {"ATP", "H20", "ADP", ''Pi''} 11
ATP H20 ADP Pi
c 10 0 10 0
0 13 1 10 4

P 3 0 2 1

TableForm[RowReduce[aal ,TableHeadings->{ {llATP1l, 11H20m1, a'ADP1l), {''ATP", "H20", "ADP" , ''Pi" 111
ATP H20 ADP Pi
ATP 1 0 0 1
H20 0 1 0 1

ADP 0 0 1 -1
(b) Calculate a basis for the stoichiometric number matrix from the conservation matrix.

TableForm[nu=NullSgace[aa] ,TableHeadings->{ {"rx"}, {"ATP", "H20". IIADPml, "Pi1'} 11

ATP H20 ADP Pi
rx -1 -1 1 1