Metabolic Pathways Analysis: A Linear Algebraic Approach - Mathematical Concepts and Methods in Modern Biology - page 238

cellular processes in common to all creatures, as studied by clinicians and doctors,
biologists, chemists, engineers, and many others.

8.2 BIOCHEMICAL REACTION NETWORKS, METABOLIC
PATHWAYS, AND THE STOICHIOMETRY MATRIX
8.2.1 Stoichiometric Matrix I: Nullspaces, Linear Dependence,
and Spanning Sets
Suppose one wishes to consider a finite sequence of chemical reactions, involving m
chemical compounds C 1 ,

C m .
Recording a reaction in standard chemistry notation, each reaction would take the
form

C 2 ,...,

a 1 C i 1 + a 2 C i 2 +···+ a k C i k −→ a k + 1 C i k + 1 +···+ a t C i t ,

1 i 1 ,..., i t m , a i 1 ,..., a i t ∈ R . (8.1)

That is, in a given equation in the list, some of the possible compounds will be
used (the C i j s from among the possible choices C 1 ,..., C m ) in some amounts (the

numbers a j , one for each compound C i j involved in the reaction). Let's consider this
concretely in some steps of glycolysis, a cellular metabolic pathway that is part of
the even more complex pathway of cellular respiration. For example, a possible first
step in the metabolic pathway of glycolysis in humans is the glucokinase reaction:

) .
(8.2)
In reaction ( 8.2 ), the biochemical molecules/compounds that are input, i.e., the sub-
strates GLC and ATP, and those that are output, i.e., the products G6P and ADP,
are called metabolites. The addition of ATP yielding ADP is a step repeated again
in glycolysis and is very important—it is essentially a way for the cell to release
and recapture energy in a controlled fashion. Alternatively, it's not uncommon for
biologists to think of the reaction step ( 8.2 )simplyas

GLC + ATP −→ G6P + ADP

( glucose ) ( adenosine triphosphate ) ( glucose 6 − phosphate ) ( adenosine diphosphate

GLC −→ G6P ,

that is, glucose is transformed, in the presence of ATP, into glucose 6-phosphate, with
ADP being tossed off in the process. In this view, the reaction captures glucose for the
cell, transforming it into a product that cannot wander back off outside the cell wall,
whence further reactions in glycolysis continue its transformation. Such shifting of
perspective, e.g., from (8.2) to just GLC

G 6 P , is a common step in modeling
(bio)chemical reaction systems—what constitutes a “metabolite” depends upon what
one's interest in the process is. For this reason, we will use the term metabolite to
mean any compound of interest involved in a biochemical reaction, a use somewhat
more loose than by biologists.

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