Metabolic Pathways Analysis: A Linear Algebraic Approach - Mathematical Concepts and Methods in Modern Biology - page 247

8.2.2 More on the Nullspace of the Stoichiometric Matrix:
Spanning with Biochemical Pathways and Base Changing

Exercise 8.10. The vectors (total flux vectors v ) you found in Exercise 8.8 , spanning
the nullspace N ( S ) of the stoichiometry matrix S , include two which have at least
one negative entry in a variable associated to an internal reaction. List these two
vectors.

Since any reversible reaction was already broken down graphically into a pair of
double edges (oppositely oriented), and otherwise the arrows of internal reactions
correspond to directions of the associated chemical equations, a negative value on
an internal flux gives a somewhat nonsensical interpretation. Flux vectors with this
problem (one or more negative flux distributions for internal reactions) represent
biochemically impossible outcomes, while flux vectors without this problem represent
chemically feasible pathways through the metabolic system [ 6 ]. This corresponds to
what you have seen pictorially in Exercise 8.8 (c).
Consequently, one goal of [ 6 ] is to find a biologically legitimate spanning set
of flux vectors v for the nullspace N ( S ) of a stoichiometric matrix S , so that each v
corresponds to a biochemically valid pathway through the metabolic systemdescribed
by S . The mechanism [ 6 ] employ is base-changing, as we shall now discuss. First:
Exercise 8.11.

a. Recall the definition of a basis of a vector space here.
b. List a basis for the vector space N ( S ), for the stoichiometric matrix S you found
in Exercise 8.5 .

In seeking a “good basis” for the nullspace N ( S ) of a stoichiometric matrix S [ 6 ],
look for a set of total flux vectors that simultaneously:

1. all represent biochemically valid pathways,
2. span N ( S ), that is, each possible flux vector in N ( S ) is a linear combination of the
basis vectors, and

3. form a linearly independent set: no one flux vector in the basis set (or its graph-

ical interpretation) can be expressed nontrivially as a linear combination of the
remaining ones.

Exercise 8.12. A quick check on your understanding: Is the basis you found for
Exercise 8.11 a “good basis”, in the sense above? Is a “good basis” a basis?

To create a “good basis” as in [6], the authors further require the following condi-
tions be satisfied by a set of total flux vectors for the nullspace N ( S ) of any stoichio-
metric matrix S :

4. All coordinates corresponding to internal fluxes will be positive, that is,

v i 0 ( all internal fluxes ).