Pathway Modeling
in Metabolic Networks
Stefan Schuster
Friedrich Schiller University Jena
Dept. of Bioinformatics
JENA
Battle of Jena and Auerstedt
October 14, 1806
Famous people at Jena University:
Friedrich Schiller
(1759-1805)
Matthias Schleiden
(1804-1881)
Discoverer of the
living plant cell
Ernst Haeckel
(1834-1919,
Biogenetic rule)
Introduction
• Metabolism is bridge between genotype and
phenotype
• Technological relevance of metabolism:
Synthesis of specific products (antibiotics, amino
acids, ethanol, dyes, odorants)
• Production of edibles: cheese, bread, wine, etc.
• Degradation of xenobiotics
Metabolic networks are complex due to their
size and the presence of bimolecular reactions.
Hypergraphs.
Source:
Introduction (2)
• Structure and (nonlinear) dynamics of
metabolic networks cannot be understood
intuitively
• Theoretical methods needed
• These methods should be systemic
(Systems Biology) rather than too
reductionist
Metabolic Pathway Analysis (or
Metabolic Network Analysis)
• Decomposition of the network into the
smallest functional entities (metabolic
pathways)
• Does not require knowledge of kinetic
parameters!!
• Uses stoichiometric coefficients and
reversibility/irreversibility of reactions
non-elementary flux mode
elementary flux modes
S. Schuster and C. Hilgetag: J. Biol. Syst. 2 (1994) 165-182;
StS, T. Dandekar, D.A. Fell: Trends Biotechnol. 17 (1999) 53-60;
StS, D.A. Fell, T. Dandekar: Nature Biotechnol. 18 (2000) 326-332
An elementary mode is a minimal set of enzymes that
can operate at steady state with all irreversible reactions
used in the appropriate direction
All flux distributions in the living cell are non-negative
linear combinations of elementary modes
Related concept: Extreme pathway (Schilling,
Letscher and Palsson, J. theor. Biol. 203 (2000) 229)
- distinction between internal and exchange reactions,
all internal reversible reactions are split up into forward
and reverse steps
Mathematical background
Steady-state condition NV(S) = 0
Sign restriction for irreversible fluxes: Virr
0
If the kinetic parameters are unknown, one can try
to solve this for V.
The equation/inequality system is linear and
homogeneous in V.
However, usually there is a manifold of solutions,
which then represents a convex region.
All edges correspond to elementary modes.
In addition, there may be elementary modes in the interior.
Geometrical interpretation
Elementary modes correspond to generating vectors
(edges) of a convex polyhedral cone (= pyramid)
in flux space (if all reactions are irreversible)
Rate 3
Rate 2
generating vectors
Rate of enzyme 1
ATP
X5P
CO2 Ru5P
NADPH
NADP
S7P
Pyr
E4P
ADP
R5P
GAP
PEP
F6P
6PG
2PG
GO6P
3PG
ATP
NADPH
NADP G6P
ADP
F6P
FP
2
GAP
DHAP
ATP
NAD
1.3BPG
NADH
ADP
Part of monosaccharide metabolism
Red: external metabolites
ATP
Pyr
ADP
PEP
2PG
ATP
3PG
ADP
G6P
F6P
FP
GAP
2
DHAP
ATP
ADP
1st elementary mode: glycolysis
NAD
1.3BPG
NADH
F6P
ATP
FP2
ADP
2nd elementary mode: fructose-bisphosphate cycle
ATP
X5P
CO2 Ru5P
NADPH
NADP
S7P
E4P
ADP
GAP
R5P
PEP
F6P
6PG
2PG
GO6P
3PG
ATP
NADPH
NADP
Pyr
ADP
G6P
F6P
FP
2
GAP
DHAP
ATP
NAD
1.3BPG
NADH
ADP
4 out of 7 elementary modes
S. Schuster, D.A. Fell, T. Dandekar:
Nature Biotechnol. 18 (2000) 326-332
Algorithm for computing
elementary modes
Related to Gauss-Jordan method
Starts with tableau (NT I)
Pairwise combination of rows so that one column
of NT after the other becomes null vector
Test before each combination whether resulting
row is elementary
S. Schuster et al.:
Nature Biotechnol. 18 (2000) 326-332
J. Math. Biol. 45 (2002) 153-181.
Example:
S2
3
4
P1
1
1 0

 −1 0
(
0)
T =
−1 1

 1 −1

S1
2
P2
M 1 0 0 0

M 0 1 0 0
M 0 0 1 0

M 0 0 0 1 
1 0

 −1 0
(
0)
T =
−1 1

 1 −1

0 0

0 1
(
1)
T =
0 −1

0 0

M 1 0 0 0

M 0 1 0 0
M 0 0 1 0

M 0 0 0 1 
M 1 1 0 0

M 1 0 1 0
M 0 1 0 1

M 0 0 1 1 
These two rows should
not be combined
Final tableau:
T(2 )
 0 0 M 1 1 0 0

= 
0 0 M 0 0 1 1
S2
3
4
P1
1
S1
2
P2
Algorithm is faster, if this column is processed first.
1 0

 −1 0
(
0)
T =
−1 1

 1 −1

M 1 0 0 0

M 0 1 0 0
M 0 0 1 0

M 0 0 0 1 
Alternative algorithm
R. Urbanczik, C. Wagner: An improved algorithm for
stoichiometric network analysis: theory and applications.
Bioinformatics 21 (2005) 1203-1210.
First, compute nullspace matrix, K.
NK = 0. Choice of K such that it contains identity matrix.
Perform pair-wise combinations of columns to obtain
further elementary modes.
Empirically, it shows a higher performance. However,
this may depend on the type of network.
Software involving routines
for computing elementary modes
EMPATH - J. Woods
METATOOL - Th. Pfeiffer, F. Moldenhauer,
A. von Kamp (In versions 5.x, Wagner algorithm)
GEPASI - P. Mendes
JARNAC - H. Sauro
In-Silico-DiscoveryTM - K. Mauch
FluxAnalyzer (in MATLAB) - S. Klamt
ScrumPy - M. Poolman
Alternative algorithm in MATLAB – C. Wagner, R. Urbanczik
PySCeS – B. Olivier et al.
On-line computation:
pHpMetatool - H. Höpfner, M. Lange
History of pathway analysis
• „Direct mechanisms“ in chemistry (Milner 1964,
Happel & Sellers 1982)
• Clarke 1980 „extreme currents“
• Seressiotis & Bailey 1986 „biochemical pathways“
• Leiser & Blum 1987 „fundamental modes“
• Mavrovouniotis et al. 1990 „biochemical pathways“
• Fell (1990) „linearly independent basis vectors“
• Schuster & Hilgetag 1994 „elementary flux modes“
• Liao et al. 1996 „basic reaction modes“
• Schilling, Letscher and Palsson 2000 „extreme
pathways“
Robustness of metabolism
• Number of elementary modes leading from a
given substrate to a given product can be
considered as a measure of redundancy
• This characterizes flexibility - number of
alternatives between which the network can
switch if necessary
System under study:
(Work together with J. Stelling, S. Klamt, K. Bettenbrock
and E.D. Gilles, Max Planck Inst., Magdeburg)
• Central metabolism of Escherichia coli
• 89 substances, 110 reactions
• Four representative substrates: glucose, acetate,
glycerol, and succinate
• 0.64 protein + 0.185 RNA + 0.03 DNA + 0.1 lipids +
0.015 lipopolysaccharides + 0.015 glycogen biomass
• If at least one elementary mode leads to biomass
production, the mutant is predicted to be viable.
Results for the E. coli model
• If all four substrates present simultaneously:
507,632 elementary modes
• To cope with combinatorial explosion, we
allow only one substrate at a time
• For example, glucose: 27,099 elem. modes
• acetate: 598 elem. modes
Computing the elementary modes
for mutants
• One enzyme gene at a time was “knocked out”
in silico.
• This was done for 90 different combinations of
single mutants and substrates.
• Comparison of our theoretical predictions on
viability with experimental data from the
literature.
• In 81 out of the 90 cases, the predictions were
correct
Prediction of viability of
mutants through # of elem. modes
true negatives
false positives
false negatives
true positives
J. Stelling, S. Klamt, K. Bettenbrock, S. Schuster, E.D. Gilles,
Metabolic network structure determines key aspects of
functionality and regulation. Nature 420 (2002) 190-193
To analyse robustness more
quantitatively:
• Plot of #(elem. modes) vs. maximum
biomass yield
• For comparison, plot of #(elem. modes)
vs. network diameter = average
#(reactions between any two
substances)
Maximal growth yield (●), network diameter (○).
J. Stelling, S. Klamt, K. Bettenbrock, S. Schuster, E.D. Gilles,
Metabolic network structure determines key aspects of
functionality and regulation. Nature 420 (2002) 190-193
Difference between redundancy
and robustness
A)
Q1
1
2
P1
3
P2
S1
Knockout of enzyme 1
implies deletion of
2 elem. modes
B)
1
S1
3
S2
Q1
2
4
P1
P2
… implies deletion of
1 elem. mode only
Proposed measure of network
robustness
r
∑
R1 =
i)
(
z
i =1
r⋅z
r: number of reactions
z: number of elem. modes
zi: number of elem. modes remaining after knockout of
enzyme i.
T. Wilhelm, J. Behre and S. Schuster: Analysis of structural
robustness of metabolic networks. IEE Proc. Syst. Biol.
1 (2004) 114 - 120.
Metabolic network/
essential products
Number of
R1
elementary modes (robustness)
R2
R3
Human
erythrocyte
ATP, hypoxanthine, NADPH,
2,3DPG
667
0.3834
0.3401
0.3607
Ala, Arg, Asn, His§
667
0.5084
0.3207
0.4295
Arg, Asn, His, Ile
656
0.5211
0.3451
0.4427
Arg, Asn, Ile, Leu
567
0.5479
Arg, Asn, Leu, Pro
540
0.5360
His, Ile, Leu, Lys
802
0.5112
Ile, Leu, Pro, Val
597
0.5488
E. coli
Recently generalized
0.4964
to0.4763
multiple knockouts:
J. Behre, T. Wilhelm, …
S.0.4586
Schuster: 0.4836
J. 0.3482
theor. Biol. 252
(2008),
0.4437
433–441.
0.4675
0.5058
Another Biochemical Application:
Can sugars be produced from lipids?
(Work with David Fell, Oxford)
• Known in biochemistry for a long time that many
bacteria and plants can produce sugars from lipids
(via C2 units) while animals cannot
?
Glucose
AcCoA is linked with glucose by a chain
of reactions. However, no elementary
mode realizes this conversion along
that chain.
CO2
PEP
Pyr
AcCoA
Cit
Oxac
CO2
IsoCit
Mal
CO2
OG
Fum
Succ
SucCoA
CO2
Glucose
Elementary mode representing
conversion of AcCoA into glucose.
It requires the glyoxylate shunt.
CO2
PEP
AcCoA
Pyr
Cit
Oxac
CO2
Mal
Mas
Gly
Icl
IsoCit
OG
Fum
Succ
SucCoA
CO2
CO2
The glyoxylate shunt is present in green plants, yeast,
many bacteria (e.g. E. coli) and others and – as
the only clade of animals – in nematodes.
This example shows that a description by usual
graphs in the sense of graph theory is insufficient…
S. Schuster, D.A. Fell: Modelling and simulating metabolic networks.
In: Bioinformatics: From Genomes to Therapies (T. Lengauer, ed.)
Wiley-VCH, Weinheim 2007, pp. 755-805.
L. Figuereido, S. Schuster, D.A. Fell: Can sugars be produced from
fatty acids? Bioinformatics, under revision
A successful theoretical prediction
Red elementary mode: Usual TCA cycle
Blue elementary mode: Catabolic pathway
predicted in Liao et al. (1996) and Schuster
et al. (1999) for E. coli.
Glucose
CO2
PEP
Pyr
AcCoA
Cit
Oxac
CO2
Mal
IsoCit
Gly
OG
Fum
Succ
SucCoA
CO2
CO2
Glucose
PEP
Pyr
Oxac
Red elementary mode: Usual TCA cycle
Blue elementary mode: Catabolic pathway
predicted in Liao et al. (1996) and Schuster
et al. (1999). Experimental hints in Wick et al.
(2001). Experimental proof in:
E. Fischer and U. Sauer:
CO2
A novel metabolic cycle catalyzes
AcCoA glucose oxidation and anaplerosis
in hungry Escherichia coli,
J. Biol. Chem. 278 (2003)
Cit
46446–46451
CO2
Mal
IsoCit
Gly
OG
Fum
Succ
SucCoA
CO2
CO2
Optimization: Maximizing molar yields
ATP
X5P
CO2 Ru5P
NADPH
NADP
S7P
E4P
ADP
GAP
R5P
PEP
F6P
6PG
2PG
GO6P
3PG
ATP
NADPH
NADP
Pyr
ADP
G6P
F6P
FP
2
GAP
DHAP
ATP
NAD
1.3BPG
NADH
ADP
ATP:G6P yield = 3 ATP:G6P yield = 2
Maximization of tryptophan:glucose yield
Model of 65 reactions in the central metabolism of E. coli.
26 elementary modes. 2 modes with highest tryptophan:
glucose yield: 0.451.
PEP
Pyr
Schuster, Dandekar, Fell,
Trends Biotechnol. 17 (1999) 53
Glc
233
G6P
Anthr
3PG
PrpP
GAP
105
Trp
Tryptophan
Conclusions
• Elementary modes are an appropriate
concept to describe biochemical pathways
• Information about network structure can be
used to derive far-reaching conclusions about
performance and robustness of metabolism
• Elementary modes reflect specific
characteristics of metabolic networks such as
steady-state mass flow, thermodynamic
constraints and and systemic interactions
(Systems Biology)
Conclusions (2)
• It can be tested whether connected routes can
carry fluxes at steady state
• A complete list of potential pathways can be
generated. Thereafter, experimental search for
realized pathways.
• Elementary modes allow one to compute - for
various substrate-product pairs - the maximal
yields that can potentially be achieved
Cooperations
• David Fell (Brookes U Oxford)
• Thomas Dandekar (U Würzburg)
• Steffen Klamt, Ernst Dieter Gilles (MPI
Magdeburg)
• Jörg Stelling (ETH Zürich)
• Sebastian Bonhoeffer (ETH Zürich)
• Thomas Pfeiffer (Harvard)
• and others
• Acknowledgement to DFG and BMBF (Germany) and
FCT (Portugal) for financial support
Introduction (3)
• Application in Functional genomics.
Considering gene products in their
functional context.
• Medical application: Inherited diseases
caused by enzyme deficiencies
• Biotechnological applications: Increaase in
yield, robustness to knockouts
Theoretical Methods
Dynamic Simulation
Stability and bifurcation analyses
Metabolic Control Analysis (MCA)
Metabolic Pathway Analysis
Metabolic Flux Analysis (MFA)
Optimization, Evolutionary Game
Theory
• and others
•
•
•
•
•
•
Problem:
• Many kinetic parameters unknown. Maximal
velocities depend on enzyme concentrations.
• What conclusions can be drawn from the
information we have? Known for most enzymes:
stoichiometry, reversibility
Theoretical Methods
Dynamic Simulation
Stability and bifurcation analyses
Metabolic Control Analysis (MCA)
Metabolic Pathway Analysis
Metabolic Flux Analysis (MFA)
Optimization, Evolutionary Game
Theory
• and others
•
•
•
•
•
•