1. NIMBIOS Workshop June 20, 2014 Terrell L. Hodge
Linear Algebraic Approaches to Metabolic Systems Analysis: Adventures for Undergrads… and Up
NIMBIOS Workshop
June 20, 2014
Terrell L. Hodge

2. 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).

3. Outline of Workshop Topics
Metabolic Pathways
Stoichiometric Matrices
Null Spaces: Extreme Pathways
Left Null Spaces: Extreme Pool Maps
Going Further: SVDs and More

4. Metabolic Pathways Context Examples One Mathematical Approach
[diagram from Wikipedia]

5. 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.]

6. 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.
Comments: But what are “chains of consecutive enzymatic reactions that produce specific products”? What IS a metabolic pathway?
Footnote to last point: Some models we deal with may record enzymes as metabolites. Enzymes are…

7. 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+
Comments on glycolysis pathway:

8. Stoichiometric Matrix S
Metabolic system with m metabolites, n reactions
Dynamic mass balance equation dC/dt = Sv
S = (sij) is integer-valued m by n matrix
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)
metabolites
s11
r … e
a … c
t … i
o … n
s …
s1n
s21
s2n :
sm1
smn
Perspectives and items of value for a linear algebra class:
Use of real-life biological context for matrix equations
Organizing multiple linear equations in matrix form, in a (usually, non-square) matrix
Can treat topics without derivatives, by going ahead to Sv = 0, motivated by idea of mass-balancing (input-output balancing) as in other similar homogeneous linear system problems (e.g., see Appendix)

9. Example: Glycolysis [diagram from Voet & Voet]

10. Example: Glycolysis [diagram from Voet & Voet]
Comment on conventions for ‘external’ exchanges.

11. 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 + si2v sinvn, where
vj= rate of reaction j := flux of reaction j
Results from conservation of mass law for the metabolic system: in essence, says at each node (metabolite)
the difference between the rate of production and rate of consumption must be equivalent to the change in concentration of the metabolite over time. In a single reversible reaction, this comes formally (viewing directions in the reaction diagrams) as
dCi/dt = (rate of forward reaction) – (rate of backwards reaction)
or more generally as
dCi/dt = (sum of rates of reactions going out/forward from metabolite Xi) – (sum of rates of reactions going into/backwards from metabolite Xi)
Recall that fluxes are constant in a steady-state system, so it makes sense to represent the list of reaction rates of the metabolites by a vector v that will take constant entries.

12. Example: Glycolysis [diagram from Voet & Voet] Dynamic Mass- Balance
First two
reactions, again.
Exercise:
Repeat for
extended
two-reaction
system, and/or
first three
reactions.

13. 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]

14. 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]

15. 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)
Some perspectives and items of value for a linear algebra class:
Nice example of naturally arising homogeneous system.
Parallels with familiar examples of chemical balancing (elementary and secondary school), and circuit analysis.
Use of real-life biological context for null space of a matrix and calculations of null spaces (by hand or by technology).
Visual (graphical) interpretation of vectors in the null space via pathways (where appropriate).
Biological and graphical interpretation of linear dependence, in the form of ‘column dependencies’. (Many students can replicate the algorithms for computing null spaces, but find it more difficult to grasp their significance; concrete contexts can help.)

16. Example: Glycolysis [diagram from Voet & Voet]
Yikes! Cut and paste error: Basis of null space should be (0,1,1)^T.

17. Example: Glycolysis [diagram from Voet & Voet]

18. Metabolic Pathways Exercise, Step 1: Build Your Own Stoichiometric Matrix
A ‘toy’ example from [Schilling and Palsson].

19. Metabolic Pathways Exercise, Step 2: Read off the “Pathways”
Some perspectives and items of value for a linear algebra class:
Use of real-life biological context for null space of a matrix and bases of (null) spaces and calculating them.
Visual (graphical) interpretation of vectors in the null space via pathways (where appropriate).
Motivation for taking linear combinations/base-changing, due to constraints of physical context.
Biological and graphical interpretation of linear dependence.
[As an aside, although it is a direct question, asking students to recover the free variables and the equations in v1 -v11 that yielded matrix $\mathcal{B}$ turned out to be a revealing exercise – they so frequently go ‘the other way’!]
Figures from [Schilling & Palsson]

20. Metabolic Pathways Exercise: Drawing “Paths”
Some perspectives and items of value for a linear algebra class:
Attempts here to find biological/graphical interpretations for the basis vectors of N(S) lead fairly naturally to discussions of what would make a “good’’ basis.
When using similar material in linear algebra classes, one can ask students to find linear combinations
of bases vectors for N(S) that do provide meaningful paths, as a precursor to attempting to change to
a new basis of N(S) (see next slides).
Figure from [Schilling & Palsson]

21. Metabolic Pathways Exercise, Step 3: Change Basis, Read Pathways
Comments:
What makes the new basis for N(S) appealing, from a biological perspective?
Are there any guarantees one can find such a basis for N(S)?
What are the biological interpretations of such bases elements?
Some perspectives and items of value for a linear algebra class:
Use of real-life biological context for null space of a matrix and calculations of null spaces (by hand or by technology).
Visual (graphical) interpretation of vectors in the null space via pathways (where appropriate).
Motivation for taking linear combinations/base-changing, due to constraints of physical context.
Opportunity to explore base-changing (even direct questions such as “Justify the claim that $\mathcal{P}$ is still a basis”, for example, can be revealing ).
Biological and graphical interpretation of spanning, linear (in)dependence.
Figures from [Schilling & Palsson]

22. Metabolic Pathways Exercise: Drawing “Paths” Again
Some perspectives and items of value for a linear algebra class:
Students really find this exercise fun! It can be adapted quickly into an accessible problem by starting with the “good” basis $\mathcal{P}$ and the perspective of the system $S\mathbf{v} = \mathbf{0}$ as a homogeneous sytem of “node-balancing” equations.
Figure from [Schilling & Palsson]

23. 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.
Some perspectives and items of value for a linear algebra class:
Significance of properties of bases of subspaces (here, null space), in a real-life context
Utility of notion of linear combinations in a constrained setting (e.g., convex hulls), even though subspace notion itself not applicable (good exercise in drawing parallels).
Modifying/adapting techniques from linear algebra for new tasks (finding expas)

24. 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.
Comments on biological interpretations:
Image from
[Schilling, Letscher, Palsson]

25. 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
Comments on biological interpretations:
Expas: systematically independent basis P
(transposed) for convex flux cone: f1 f2 f3 f4
Figures and tables from [Schilling,
Schuster, Palsson & Heinrich]

26. 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 pri-mary 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).
Add data on classification of expas (Type I, II, III) and expool maps (Types A, B, C) to Appendix.

27. HRBC: Some of the Type I Expas [Wiback & Palsson]

28. 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
Further details on outcomes of [Wiback and Palsson]:
Further comments on linear algebra vs. ‘traditional’ ad-hoc(?) biological approach:
Another not-so-subtle point of the last few slides: consider advantages of good mathe-matical framework, such as linear algebra.

29. 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
Do the ‘metabolites’ here include ATP, ADP, etc.?
Add comments re: deficiencies such as anemia, and details on information given by the analysis here.

31. 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]

32. 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.

33. Conservation and Pool Maps Exercise, Step 1: Build Your Own Dynamic Mass-Balance Equations
A ‘toy’ example from [Nikolaev, Burgard, & Maranas]
Reaction v_1 includes a component `arrow’ outward from E towards B.

34. Conservation and Pool Maps Exercise, Step 1: Build Your Own Dynamic Mass-Balance Equations (Solution)
Reaction v_1 includes a component `arrow’ outward from E towards B.
Note difference
from previous sign
conventions; can
use –S in place
of prior S.
Example from [Nikolaev, Burgard, & Maranas]

35. Conservation and Pool Maps Exercise, Step 2: Find Conserved Cycles
Reaction v_1 includes a component `arrow’ outward from E towards B.
Note difference
from previous sign
conventions; can
use –S in place
of prior S.
Diagram from [Nikolaev, Burgard, & Maranas]

36. Conservation and Pool Maps Exercise, Step 2: Find Conserved Cycles
Reaction v_1 includes a component `arrow’ outward from E towards B.
Note difference
from previous sign
conventions; can
use –S in place
of prior S.
Diagram from [Nikolaev, Burgard, & Maranas]

37. 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

38. 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)

39. 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]

40. 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.
Add comments re: pool map details in a few cases here.

41. 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.

42. 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)
Add comments re: perspectives for linear algebra students.
Focus
here.
Picture modified from
[Famili and Palsson]

43. Singular Value Decomposition (SVD): Learning More From S [Price, et. al.]
Diagram from [Price, et. al.]
‘p’ above = ‘t’ below.
Add comments re: last point above, plus experience in linear algebra classes with SVD context.
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).

44. 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.
Comments re: biological significance:
Diagrams from [Price, et. al.]

45. 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.
Some comments on relations between expa and/or extreme pool maps approach and other methods:
[ADD TO THIS LIST STILL!]
EFMs: See [Schilling, Letscher, Palsson], [Cakir, Tacer, Ulgen], [Suthers, Burgard, et. al], Schuster refs…
MCCA and MCPI: See [Nikolaev, Burgard, and Maranas]
SNA: See [Schilling, Letscher, Palsson]
*Still other approaches/enhancements exist, making use of aspects of probability and statistics, etc…

46. 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 post-genomic era.
Çakir, Tacer and Ülgen, Metabolic pathway analysis of enzyme-deficient 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 Genome-Scale Metabolic Reconstructions.

47. 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,
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.