Page 322 - Thermodynamics of Biochemical Reactions

322 Mathematica Solutions to Problems

This is the expected stoichiometric number matrix.

5.4 The glutamate-ammonia ligase reaction is
glutamate + ATP + ammonia = glutamine + ADP + Pi
It can be considered to be the sum of two reactions. (a) Write the stoichiometric number matrix for this enzyme-catalyzed
reaction and use Nullspace to obtain a basis for the conservation matrix.. (b) Write a conservation matrix that includes a
constraint to couple the two subreactions, and row reduce it to show that it is equivalent to the stoichiometric number matrix
obtained in (a).

(a) Calculate a basis for the conservation matrix from the stoichiometric number matrix and row reduce it.

TableForm[NullSgace[~~-l,-l,-l,l,1,l~~ll
1 0 0 0 0 1
1 0 0 0 1 0
1 0 0 1 0 0
- 1 0 1 0 0 0
- 1 1 0 0 0 0

1 0 0 0 0 1
0 1 0 0 0 1
0 0 1 0 0 1
0 0 0 1 0 - 1
0 0 0 0 1 - 1

This shows that there are five components, in spite of the fact that there are just four elements (C, 0, N, P).

(b) Write the apparent conservation matrix for the glutamate-ammonia ligase reaction and row reduce it. One way to couple
the two subreactions is to require that every time an ATP molecule disappears, a glutamine molecule appears; this leads to
the conservation equation
n(ATP) + n(g1utamine) = const

0 4 13 0 10 4 3

N 1 5 1 5 0 2
P 0 3 0 2 1 0
con1 0 1 0 0 0 1

TableForm[RowReduce [a] , TableHeadings-
}
]
> { { "Glamatel*, "ATP" , llAwn", "ADP", mlPiml}, "Glutmatell, llATP1lr llAmmll, "ADP" , "Pi", llGlutmine"}
{