Metabolic Pathways Analysis: A Linear Algebraic Approach - Mathematical Concepts and Methods in Modern Biology - page 258

Table 8.1
(Internal Fluxes)
v 1 I − 1 A 1 B

v 2 I − 1 A 1 C

(Exchange Fluxes)
A Input

F Output

The I is for “irreversible.”
Exercise 8.19. The vertical dots in the input pictured in Table 8.1 should be replaced
with the data for v 6 . Create an input file 13 for ExPA , based on Figure 8.7

that will model our abstract system.

v 3 through

Exercise 8.20. Use ExPA to find the extreme paths in the representation from

Exercise 8.19.

Exercise 8.21. Now consider the system
⎧
⎨

A −→ B ,

A −→ C ,

B −→ D ,

(8.20)

⎩ C −→ E ,

D −→ F ,

E −→ F .

Represent ( 8.20 ) as a directed graph and compare your answer to the graph
for ( 8.19 ).

Exercise 8.22. Write down at least one more biochemical system which will give
you the directed graph from Exercise 8.21 .
Your answer to the previous exercise highlights a possible weakness of our decision
to model processes using directed graphs. In particular, a directed graph may not
perfectly capture the stoichiometry of a biochemical process. For this reason, it can
be interesting to use hypergraphs . Whereas in a directed graph, each edge must start
at a unique vertex and end at a unique vertex, in a directed hypergraph an edge (really
“hyperedge”) is allowed to have both multiple starting points and multiple endpoints.
For example, a hypergraph representation of ( 8.19 ) would look like Figure 8.8 , with
r 1 and r 4 hyperedges that are not just edges.
Exercise 8.23. When we represent a biochemical process using a hypergraph, what
do the “hyperedges” represent? In what ways is this the same as or different from the
edges in a directed graph?

13 Note: If you're using a PC and plan on running the program without opening the command prompt,
this input file has to be named “source.txt” and has to be placed in the same directory as “ expa.exe ”. If
you intend to use a command prompt or terminal window, the input file can be given any name.