Metabolic Pathways Analysis: A Linear Algebraic Approach - Mathematical Concepts and Methods in Modern Biology - page 253

the associated system with nodes indexed by the metabolites listed, and the problem
that b 2 poses biochemically. (But given your experience to date, you should guess
what that will be!) Since any three linearly independent vectors in

n

, n ≥ 3 span a

R

three-dimensional vector space, there is a geometric interpretation p 1 , p 3 , and p 4

as the edges of a convex cone, the “flux cone” of S in the space. (Do see [ 7 ] for repre-
sentative pictures and more discussion.) The edges p i determine all possible total flux
vectors for the system S represents, in that any positive convex linear combination
of them (i.e., point in the flux cone) is a total flux vector. This leads to the notion of
the p i as so-called extreme pathways . Although they are biologically a bit difficult to
describe, their cone yields all total flux vectors for which the system has no changes
in concentrations of the metabolites, so metabolites are conserved, and we arrive back
to the ideas we discussed in Section 8.1 .
The papers we have noted so far represent an early attempt to apply linear alge-
braic ideas to this area of metabolite conservation. In a series of papers over the
years, their authors and their colleagues have employed other concepts in linear alge-
bra [ 8 ] and more significant linear algebraic techniques (e.g., the SVD [ 9 ]). They
have combined linear algebraic techniques with notions from convex analysis and
other areas to try to address questions we have raised here, and to tackle biochemical
reaction systems that are much more sophisticated than the tutorial examples in this
chapter, taken from their early papers, address. The reader is encouraged to check
http://gcrg.ucsd.edu/Researchers for a wealth of developments. In this context, there
has been developed a program expa (free) for computing extreme pathways, and
subsequently much more extensive software, the COBRA toolbox http://opencobra.
sourceforge.net/openCOBRA/Welcome.html (requires Matlab or Python). A signif-
icant problem that arises early on, when working with more sophisticated reaction
systems, is that our naive intuition linking paths in the graph/cartoon of the biochem-
ical reaction system to extreme paths and the stoichiometry quickly breaks down
or can become stretched in a biologically incorrect direction. A final section of this
chapter, which could be broken apart as a project unto itself, includes a compan-
ion tutorial for the free download expa and gives the interested reader a chance to
explore these issues and a potential solution to them by working with hypergraphs
instead of graphs. These additional materials will also allow you to explore large (and
hence, more realistically interesting) biochemical reaction systems, as was proposed
would be useful in the introduction. Papers using related linear algebraic techniques
combined with convex analysis and differential equations have enabled researchers to
isolate new metabolic pathways for E. coli [ 10 ], to identify potential disease mecha-
nisms in red blood cells [ 5 ], and a host of other biomechanical engineering and other
applications, see, e.g., the survey [ 11 ]. Other papers explore the links between varying
versions of biologically “special” sets of vectors (such as the “extreme flux modes”
of [ 13 ], to name just one) e.g., [ 12 ] and modeling of biochemical reaction networks
using many other types of modeling tools, such as graph theoretic and other algebraic
approaches to system dynamics, e.g., [ 14 , 15 ], algebraic geometric methods, e.g., by
[ 16 ], and more. It is even possible to use these ideas, in combination with aspects of

p 2 ,