Linear Algebraic Approaches to
Metabolic Systems Analysis:
Adventures for Undergrads… and Up
NIMBIOS Workshop
June 20, 2014
Terrell L. Hodge
NOTE:


The ideas discussed in these slides provide but one in-road into
the use of algebraic (particularly linear algebraic, here) methods
for the representation and analysis of metabolic and other
biochemical reaction networks. This is a fast-moving field and
there is far more research that has been done on this topic that
is much newer.
These slides complement undergraduate curriculum materials
in Chapter 8 of Modern Concepts and Methods in Modern
Biology, but we will also soon start with and use other materials
for today’s presentation and exercises (posted in the NIMBIOS
blog).
Outline of Workshop Topics





Metabolic Pathways
Stoichiometric Matrices
Null Spaces: Extreme Pathways
Left Null Spaces: Extreme Pool Maps
Going Further: SVDs and More
Metabolic Pathways



Context
Examples
One
Mathematical
Approach
[diagram from Wikipedia]
The Meaning of “Life”?




Living organisms/systems are, thermodynamically,
open systems that tend to maintain a steady state.
Steady state: “All rates of flows in the system are
constant, so the system does not change with time.”
[A steady state is a stable state!]
Eventual state of closed thermodynamic system is
equilibrium.
Equilibrium: Death for a living system. (However,
individual reactions in a system may be close to
equilibrium.) [Death is also a very stable state.]
Metabolic pathways
[Voet & Voet]

“Metabolism is the overall process through which living
systems acquire and utilize free energy to carry out their
various functions.”

Metabolism is enacted through metabolic pathways: chains of
“consecutive enzymatic reactions that produce specific
products for use by an organism”.

The metabolites in a metabolic pathway are usually taken to
be the substrates, intermediates, and reactants in this chain of
reactions.
Example: Glycolysis
[diagram from Voet & Voet]
‘Glykos’ = ‘sweet’,
‘lysis’ = ‘loosening’ (Greek)

Widely shared mechanism
of life forms for energy
extraction
Overall reaction:
GLU + 2NAD+ + 2ADP + 2Pi
2NADH + 2PYR + 2ATP
+ 2H2O + 4H+

Stoichiometric Matrix S
Metabolic system with m metabolites, n
reactions
 Dynamic mass balance equation
dC/dt = Sv
 S = (s ) is integer-valued m by n
ij
matrix
metabolites
s11
s21
–
sij = 0 if metabolite i not involved in
reaction j
:
–
sij < 0 if metabolite i is a substrate in
reaction j (|sij| moles (units)
consumed in reaction j)
:
–
sij > 0 if metabolite i is a product of
reaction j (|sij| moles formed in
reaction j)
sm1
r…
e
a…
c
t …
i
o…
n
s…
s1n
s2n
:
:
smn
Example: Glycolysis
[diagram from Voet & Voet]
Example: Glycolysis
[diagram from Voet & Voet]
Dynamic Mass-Balance Equation(s)
dC/dt = Sv
Metabolic system with m metabolites, n reactions
 Ci = [Xi] := concentration in the metabolic system of
Xi := metabolite i, for i= 1,..,m
 dCi/dt = si1v1 + si2v2 + ... + sinvn, where
vj= rate of reaction j := flux of reaction j
Example: Glycolysis
Dynamic MassBalance
[diagram from Voet & Voet]
First two
reactions, again.

Exercise:
Repeat for
extended
two-reaction
system, and/or
first three
reactions.

A Linear Algebraic Perspective
Sv = 0
Null Space of S
N(S)
xTS = 0
Left Null Space of S
N(ST)
Picture modified from
[Famili and Palsson]
A Linear Algebraic Perspective
Focus
here.
Sv = 0
Null Space of S
N(S)
xTS = 0
Left Null Space of S
N(ST)
Picture modified from
[Famili and Palsson]
Null Space of S



Sv = 0: steady-state solutions to dynamic
mass balance equation dC/dt = Sv
N(S):={y in Rn | Sy = 0}; a vector v in N(S) is
a flux vector for the metabolic system (steady
state)
Vectors in N(S) give dependencies among
columns of S (i.e., reactions)
Example: Glycolysis
[diagram from Voet & Voet]
Example: Glycolysis
[diagram from Voet & Voet]
Metabolic Pathways Exercise, Step 1:
Build Your Own Stoichiometric Matrix

A ‘toy’ example from [Schilling and Palsson].
Metabolic Pathways
Exercise, Step 2:
Read off the
“Pathways”
Figures from [Schilling & Palsson]
Metabolic Pathways Exercise: Drawing
“Paths”
Figure from [Schilling & Palsson]
Metabolic Pathways
Exercise, Step 3:
Change Basis,
Read Pathways
Figures from [Schilling & Palsson]
Metabolic Pathways Exercise: Drawing
“Paths” Again
Figure from [Schilling & Palsson]
From the Null Space of S to Extreme
Pathways



Null space of S: Standard methods yield
mathematically valid basis of N(S), but
resulting vectors may not be biologically valid
total flux vectors.
Base-changing: Aim for “biologically valid”
basis of N(S); does such necessarily exist?
Even if so, what about uniqueness?
Next: Convex hulls, extreme pathways, and
examples.
Biologically “Good”
Flux Vectors

In the convex hull flux cone(S),
there is an analogue of a basis for
N(S), only better: a generating set
of ‘independent’ flux vectors P =
{p1,…, pt}, unique up to taking
scalar multiples, and for which
every w in flux cone(S) is a unique
non-negative linear combination of
vectors in P.
Image from
[Schilling, Letscher, Palsson]
Example: Extreme
Pathways (Expas)
[Schilling, Schuster, Palsson & Heinrich]
Look here!
Basis (transposed) for N(S):
b1
b2
b3
p1 = f1 := b1 – b2,, p2 = f2 := b1,
p3 = f3 := b3 – b2 , p4 = f4 := b3
Expas: systematically independent basis P
(transposed) for convex flux cone:
f1
f2
Figures and tables from [Schilling,
Schuster, Palsson & Heinrich]
f3
f4
Expas Example: Human Red Blood
Cell
(HRBC) [Wiback & Palsson]


Model accounts for 39
metabolites and 32 internal
metabolic reactions, as well
as 19 external ones (12 primary exchange and 7
currency exchange fluxes).
Resulting flux cone(S) has
|P| = 54; further partitioning
into ‘types’ yields 39 expas
of interest (36 Type I, 39
Type II).
HRBC: Some of the Type I Expas
[Wiback & Palsson]
HRBC: More Type I
and II Expas
[Wiback & Palsson]
Outcomes include:
 Unique and mathematically
precise description of
pathways, including key
‘historical pathways’, but
extending to many ‘less
intuitive’ paths that reflect
network properties
 Opportunity to predict
system effects of
enzyomapathy and other
‘load’ capacities on
individual reactions
Another not-so-subtle point of the last few
slides: consider advantages of good mathematical framework, such as linear algebra.
Enzyomapathies in HRBC
[Çakir, Tacer & Ülgen]




Human red blood cell model with 44 metabolites and
39 reactions
Investigates 5 (of about 20 known) enzyomapathies
(in this case, enzyme deficiencies) using metabolic
pathway analysis
Follows work including [Wiback & Palsson], but
using EFMs (elementary flux modes)
One aim: identify targets for drug intervention for
diseases caused by enzyme alterations/dysfunction
Recall: A Linear Algebraic Perspective
Sv = 0
Null Space of S
N(S)
xTS = 0
Left Null Space of S
N(ST)
Focus
here.
Picture modified from
[Famili and Palsson]
Left Null Space of S



N(ST) = {x in Rm | STx = 0} = N(-ST)
= {x in Rm | xTS = 0} *
v in N(ST) is a potential “pool map”, defining
a conservation relationship**
Vectors in N(ST) give dependencies among
rows of S (metabolites)
* Rm consists of column vectors.
**Details in the Appendix.
Conservation and Pool Maps Exercise,
Step 1: Build Your Own Dynamic MassBalance Equations

A ‘toy’ example from [Nikolaev, Burgard, & Maranas]
Conservation and Pool Maps Exercise,
Step 1: Build Your Own Dynamic MassBalance Equations (Solution)
Note difference
from previous sign
conventions; can
use –S in place
of prior S.

Example from [Nikolaev, Burgard, & Maranas]
Conservation and Pool Maps
Exercise, Step 2: Find
Conserved Cycles
Note difference
from previous sign
conventions; can
use –S in place
of prior S.

Diagram from [Nikolaev, Burgard, & Maranas]
Conservation and Pool Maps
Exercise, Step 2: Find
Conserved Cycles
Note difference
from previous sign
conventions; can
use –S in place
of prior S.

Diagram from [Nikolaev, Burgard, & Maranas]
Alternate Perspective:
Reaction Maps to Compound Maps
e.g., as in [Famili and Palsson]





Metabolites Nodes
Reactions Arrows
S, N(S)
Substrates Tails of Edges
Products Heads of Edges





Reactions Nodes
Metabolites Arrows
-ST, N(-ST)
Substrates Edges Entering Nodes
Products Edges Exiting Nodes
Left Null Space of S: Compound Maps
and Extreme Pool Maps

As before, basis of N(-ST) lacks uniqueness
and may not be biologically interesting

As before, compute convex basis, call
resulting (unique) vectors extreme pool
maps (extreme pools)
Example: Extreme Pool
Maps in Glycolysis
[Nikolaev, Burgard & Maranas]


Glycolysis represented with 11
metabolites (16 if include ATP,
ADP, NAD+, NADH, H20), and
13 reactions.
Flux cone(-ST) has a
systematically independent
basis with |P| = 8 vectors, so
there are 8 extreme pool
maps.
Diagram
from
[Nikolaev,
Burgard &
Maranas]
Example: Extreme Pool Maps in
Glycolysis
[Nikolaev, Burgard &
Maranas]


Glycolysis offered
as both a rich and
sufficiently small
real-life system for
direct computation
and analysis of
extreme pools.
However, paper
considers
alternative methods
to elucidate and
analyze extreme
pools for larger
systems, paralleling
alternate ‘flux
coupling’ methods
for extreme paths.
Metabolic Pathway Analysis, Extreme
Pathways, and Extreme Pools: Some
Consequences






Yields mathematically precise definition of metabolic pools and
pathways that take a systems/network approach
Yields ‘unique’ generating set, with properties similar to vector
space bases (‘minimality’ and ‘spanning’)
Gives geometrically and graphically appealing interpretations
Algorithms and programs exist for computing extreme paths
and extreme pools
Linear algebra framework provides accessible mathematical
framework that is rich in computational power and is a base for
many other mathematical structures
Extends current biological ‘intuition’, suggests mechanisms for
understanding how living systems maintain steady states and
fight or fall to disease, as well as proper design of medical
interventions.
Linear Algebra Again: The Four
Fundamental Subspaces and S
Sv = 0
Null Space of S
N(S)
xTS = 0
Left Null Space of S
N(ST)
Focus
here.
Picture modified from
[Famili and Palsson]
Singular Value Decomposition (SVD):
Learning More From S [Price, et. al.]
Diagram from [Price, et. al.]
‘p’ above = ‘t’ below.



Recall convex basis of expas P = {p1,.., pt} for flux
cone(S). Set P to be the matrix with columns p1,..pt.
Find SVD(P) = USVT.
Analysis allows for comparison of extreme pathways for
different metabolic systems, and may assist in
identifying key branch points (targets for regulation).
Singular Value Decomposition (SVD):
Learning More From S [Price, et. al.]
The
column vectors of U (‘modes’) give information re: flux
variability within the cone. The singular values measure variance in
directions given by the corresponding U vectors.
Diagrams from
[Price, et. al.]
Some Issues in Use of Extreme
Pathways/Extreme Pool Maps





In small to medium systems, expas/expools can be calculated,
but giving their biological interpretations is not automated!
Scaling to genome-level an issue: Implementation of original
algorithms for computing flux cones (like expa) problematic
for large systems: computational round-off error for large S,
combinatorial explosion and NP completeness issues arise…
Perspective may be enhanced by comparison with other linear
algebra/convex analysis/linear programming methods*, e.g.,
EFMs, FCA, MCCA and MCPI, FluxAnalyzer…
Dynamic and regulatory information are not, in general, treated
in MPA (metabolic pathway analysis) approach.
Biologists are not out of jobs: good biological data and physical
approaches to flux determination (isotope labeling, etc.) still
important.
*Still other approaches/enhancements exist, making use of aspects of
probability and statistics, etc…
References





Schilling and Palsson, The
underlying pathway of biochemical
reaction networks.
Famili and Palsson, The convex
basis of the left null space of the
stochiometric matrix leads to the
definition of metabolically
meaningful pools.
Schilling, Schuster, Palsson and
Heinrich, Metabolic pathway
analysis: Basic concepts and
scientific applications in the postgenomic era.
Çakir, Tacer and Ülgen, Metabolic
pathway analysis of enzymedeficient human red blood cells.
Suthers, Burgard, Dasika,
Nowroozi, Van Dien, Keasling,
Maranas, Metabolic flux elucidation
for large-scale models using 13C
labeled isotopes.




Price, Reed, Papin, Famili and
Palsson, Analysis of metabolic
capabilities using singular value
decomposition of extreme pathway
matrices.
Schilling, Letscher, and Palsson,
Theory for systematic definition of
metabolic pathways and their use in
interpreting metabolic function from
a pathway-oriented perspective.
Wiback and Palsson, Extreme
pathway analysis of human red
blood cell metabolism.
Nikolaev, Burgard, and Maranas,
Elucidation and Structural Analysis
of Conserved Pools for GenomeScale Metabolic Reconstructions.
References





Voet and Voet, Biochemistry (3rd
Ed.)
Bell and Palsson, expa, a program
for calculating extreme pathways in
biochemical reaction networks.
Becker, Feist, Mo, Hannum,
Palsson, and Herrgard, Quantitative
prediction of cellular metabolism
with constraint-based models: the
COBRA Toolbox.
Schuster, Dandekar, and Fell,
Detection of elementary flux modes
in biochemical networks: a
promising tool for pathway analysis
and metabolic engineering.
Klamt, Stelling, Ginkel and Giles,
FluxAnalyzer: exploring structure,
pathways, and flux distributions in
metabolic networks on interactive
flux maps.





Palsson, Representing
Reconstructed Networks
Mathematically: The Stochiometric
Matrix (lecture series).
Systems Biology Research Group,
http://gcrg.ucsd.edu/
Schuster, Fell, and Dandekar, A
general definition of metabolic
pathways useful for systematic
organization and analyis of complex
metabolic networks.
Schuster and Hilgetag, On
elementary flux modes in
biochemical reaction systems at
steady state.
Schuster, Hilgetag, Woods and Fell,
Reaction routes in biochemical
reaction systems: algebraic
properties, validated calculation
procedure and example from
nucleotide metabolism.