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Taming combinatorial explosion of elementary flux modes by appropriate classification of metabolites

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Taming combinatorial explosion of elementary flux ... Formate. Acetate. CO2. Ornithine. Carbamate. CO2. NH3. ADP. fragm. lipid metab. Mycoplasma pneumoniae ... – PowerPoint PPT presentation

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Title: Taming combinatorial explosion of elementary flux modes by appropriate classification of metabolites


1
Taming combinatorial explosion of elementary flux
modes by appropriate classification of
metabolites
  • Stefan Schuster, Jörn Behre
  • Friedrich Schiller University Jena
  • Dept. of Bioinformatics
  • Thomas Dandekar
  • Würzburg University
  • Dept. of Bioinformatics

2
Introduction
  • Serious problem in Metabolic Pathway Analysis
    Combinatorial explosion of pathways
  • This causes running time and memory problems
  • For example E. coli system (Stelling et al.,
    2002) 110 reactions ? about 500.000 elementary
    modes

3
Problem of combinatorial explosion
In large networks, the number of elementary modes
is usually very large.
The number of modes depends both on the number
of reactions and on the number of metabolites
Worst case exponential increase
P
S
P
S
S
S
2
n
1
1
2
3
n-1
Number of modes 2
4
One option
  • Restriction to convex basis (or related concept
    of extreme pathways). That is the minimum number
    of elementary modes needed to span the flux cone.
  • However The remaining elementary modes are often
    biochemically meaningful as well.

Green convex basis
5
Another option
  • Only one substrate and/or product admitted at a
    time.
  • For example, in E. coli system with 110
    reactions, ca. 500.000 modes occur with glucose,
    glycerol, succinate and acetate as substrates
    simultaneously. When only glucose about 27.000
    modes, when only acetate about
    600 modes.
  • Similar approach in several papers by Palsson
    group (extreme pathways)
  • Similar approach in many experimental studies

6
Still another option
  • Decomposition of the network
  • Note that Metabolic Pathway Analysis as a whole
    is based on decomposition

7
First decomposition procedure
S. Schuster, T. Pfeiffer, F. Moldenhauer, I.
Koch, T. Dandekar Exploring the pathway
structure of metabolism Decomposition into
subnetworks and application to Mycoplasma
pneumoniae, Bioinformatics 18 (2002) 351-361.
  • In addition to the pre-defined external
    metabolites, set all metabolites participating in
    more than 4 reactions to external status
  • Thus, the network disintegrates into subnetworks
  • Determine the elementary flux modes of the
    subnetworks separately

8
A)
S
9
A)
S
B)
S
23 modes
10
A)
S
B)
S
23 modes
C)
23 modes
S
11
Mannitol
Mycoplasma pneumoniae
Subsystem 1 Sugar import
Fructose
Glucose
Green boxes additional external metabolites
Subsystem 3
F6P
Nucleotide metab.
R5P
Subsystem 2
PPP, glycolysis,
dUMP
dTMP
fragm. lipid metab.
Serine
GA3P
ATP
Subsystem 5
C1 pool
ADP
Acetate
Subsystem 4
Glycine
Lower glycolysis
Met
f-Met
Formate
CO2
Subsystem 6
Ornithine
NH3
Lactate
Arginine
Arginine degrad.
Carbamate
CO2
12
Second decomposition procedure
T. Dandekar, F. Moldenhauer, S. Bulik, H.
Bertram, S. Schuster A method for classifying
metabolites in topological pathway analyses
based on minimization of pathway number.
BioSystems 70 (2003) 255-270.
  • Treat the classification of external and internal
    metabolites as a combinatorial optimization
    problem
  • Optimality criterion Minimization of the number
    of elementary modes (simplest description of the
    system)
  • Side constraint Each enzyme should be involved
    in at least one elementary mode
  • Solution by exhaustive search or by stochastic
    optimization (e.g. Metropolis algorithm).

13
What is the simplest description of this set of
reactions?
S1
S2
S3
S4
S5
It is one pathway! For this description, S1 and
S5 should be external metabolites, and all
others, internal.
14
Branched system
S
S
S
S
S
S
S
S
S
S
S
S
Red external Blue internal
S
S
S
S
S
S
S
S
S
S
Minimum number of elementary modes (5)
S
15
S
In this case, same solution as with the first
decomposition procedure. Not always the
case. In contrast to the first decomposition
method, the second takes into account the global
properties of the system.
16
Example Part of monosaccharide metabolism
Pyr
ATP
X5P
S7P
E4P
ADP
Ru5P
CO2
PEP
NADPH
GAP
F6P
R5P
NADP
6PG
2PG
3PG
GO6P
ATP
NADPH
ADP
NADP
G6P
GAP
F6P
FP
1.3BPG
2
NADH
NAD
DHAP
ATP
ADP
Red predefined external metabolites
17
1st solution
Pyr
ATP
X5P
S7P
E4P
ADP
Ru5P
CO2
PEP
NADPH
GAP
F6P
R5P
NADP
6PG
2PG
3PG
GO6P
ATP
NADPH
ADP
NADP
G6P
GAP
F6P
FP
1.3BPG
2
NADH
NAD
DHAP
ATP
ADP
Red predefined ext. metabolites
Green additional ext. metabolites
18
Pyr
ATP
X5P
2
2
-4
4
1
S7P
E4P
2
1
1
5
1
ADP
Ru5P
1
1
1
CO2
PEP
-2
-2
1
NADPH
2
4
2
2
R5P
GAP
F6P
2
3
3
1
6
1
5
NADP
1
1
-2
2
6PG
2PG
1
6
R5Pex
2
1
3
5
3
1
6
3PG
GO6P
ATP
NADPH
2
1
5
3
3
1
-1
6
ADP
1
NADP
2
1
1
1
G6P
GAP
F6P
1.3BPG
FP2
1
1
2
2
1
-1
-2
1
5
5
NADH
NAD
DHAP
-5
1
2
ATP
ADP
1
1
7 elementary modes in glycolysis- pentose-phosphat
e system
S. Schuster, D.A. Fell, T. Dandekar Nature
Biotechnol. 18 (2000) 326-332
19
2nd solution
Pyr
ATP
X5P
S7P
E4P
ADP
Ru5P
CO2
PEP
NADPH
GAP
F6P
R5P
NADP
6PG
2PG
3PG
GO6P
ATP
NADPH
ADP
NADP
G6P
GAP
F6P
FP
1.3BPG
2
NADH
NAD
DHAP
ATP
ADP
Red predefined ext. metabolites
Green additional ext. metabolites
20
R5PI
R5P
Ru5P
RPPK
PPM
ATP
R1P
PRPP
PNPase
HGPRT
PRPP
IMP
HYPX
HYPXext
AMPDA
AdPRT
NUCII
AMP
AD
INO
HXtrans
ADA
NUCI
ADO
AMP
ATP
ADtrans
ApK
AK
ADP
ADP
Scheme of nucleotide metabolism from Schuster,
Hilgetag, Woods, Fell, J. Math. Biol. 45 (2002)
153-181
ADext
21
Nucleotide metabolism
  • In the system considered in J. Math. Biol. 45
    (2002), 11 elementary modes are obtained.
  • Minimizing the number of modes gives a unique
    solution in which AMP and R5P are external in
    addition. It involves 7 elementary modes.

22
R5PI
R5P
Ru5P
RPPK
PPM
ATP
R1P
PRPP
PNPase
HGPRT
PRPP
IMP
HYPX
HYPXext
AMPDA
AdPRT
NUCII
AMP
AD
INO
HXtrans
ADA
NUCI
ADO
AMP
ATP
ADtrans
ApK
AK
ADP
ADP
ADext
23
Glutathione metabolism (Dandekar et al.,
BioSystems 70, 2003, 255-270)
24
Glutathione metabolism
  • 59 reactions, 79 metabolites
  • 12 metabolites participate in exactly 2
    reactions, one producing it and one consuming it.
    They are set internal. CO2 also participates in 2
    reactions. However, both produce it. Therefore,
    CO2 is set external.
  • All metabolites involved in exactly 1 reaction
    each are set external.
  • There remain 19 branch point metabolites.
  • This gives rise to 219 524,288 combinations of
    classifying metabolites.

25
Glutathione metabolism
  • For this system, exhaustive search is feasible.
  • On a usual PC (in 2003), it took 96 h.
  • Minimum number of elementary modes 42.
  • Multiplicity of solution 48.
  • Additional extremum principle Minimum number of
    metabolites set external.
  • For the considered system, this number is 57.
    Realized by 4 solutions.
  • By Metropolis algorithm, one of these solutions
    is found after about 90 iterations (20 sec).

26
Glutathione metabolism Frequency distribution of
mode number of all combinations
(59 reactions, 79 metabolites)
27
Stochastic optimization
  • For larger systems, exhaustive seach no longer
    feasible, due to huge number of modes and huge
    number of combinations.
  • Stochastic optimization (e.g. Metropolis) gives
    good results.
  • Advantage When starting with the situation where
    all metabolites are external, a solution can be
    found even if elem. modes in other situations are
    not computable numerically.
  • In the starting situation, each reaction
    represents an elem. mode on its own. Thus, the
    final number of elem. modes is less than, or
    equal to, the number of reactions.

28
Example E. coli metabolism
J. Stelling, S. Klamt, K. Bettenbrock, S.
Schuster, E.D. Gilles Nature 420 (2002)
190-193.
  • Central metabolism of Escherichia coli
  • 89 substances and 110 reactions, involving
  • Substrate uptake
  • Central carbon metabolism (including glycolysis,
    pentose phosphate pathway, TCA cycle and
    glyoxylate shunt)
  • Monomer and precursor synthesis (including
    synthesis of all proteinogenic amino acids and
    nucleotide triphosphates)
  • Polymer synthesis
  • Biomass production (growth)
  • By-product excretion

29
Example E. coli metabolism
  • Exhaustive search no longer feasible
  • Metropolis algorithm finds a combination
    external/internal giving 47 elem. modes.
  • Note the reduction from about 500.000 modes to 47
    modes!
  • However, most elem. modes are very short.
  • System is so large that many substances have a
    high connectivity.

30
Connectivity of metabolites
Name Connectivity Status ATP 38
external NADPH 28 external Glu 19
internal CO2 19 external NADH 18
external Pyr 15 external alKG 14
internal AcCoA 14 external N 11
internal Asp 9 internal PEP 9
external Gln 8 internal Ac 7
external F6P 7 external G3P 7
external G6P 7 external Succ 7
external Fum 7 external PRPP
6 internal
31
Length distribution of elementary modes
Number of reactions involved Number of elem.
Modes 1 25 2 10 3 5 4
4 5 1 6 1 52 1 (biomass
production)
32
Example of elem. mode of length 3
X5P
S7P
E4P
GAP
F6P
R5P
GAP
F6P
33
Conclusions
  • Presented extremum principle of minimizing the
    number of elem. modes reduces combinatorial
    explosion enormously.
  • The method leads to a disintegration into
    subsystems. Often the same subsystems as with the
    first decomposition method.
  • Is the concept of biochemical pathway uniquely
    determined in an objective way?
  • We think No. Depends on how much detail is
    included and on the biochemical focus of the
    model.
  • Presented extremum principle reduces
    arbitrariness and ambiguity in defining the
    concept of biochemical pathway.

34
Conclusions (2)
  • Considered extremum principle gives good results
    for small and medium-size systems
  • For large systems, combinatorial explosion is
    tamed as well (perhaps too much), but the
    resulting elementary modes are very short. Many
    metabolites are set external.
  • Open question How can the method be refined?
  • Open question What is the relation between the
    elem. modes in the subsystems and in the complete
    system?

35
Cooperations
  • Thomas Dandekar, Helge Bertram (U Würzburg)
  • Steffen Klamt, Jörg Stelling, Ernst Dieter Gilles
  • (MPI Magdeburg)
  • David Fell (Brookes U Oxford)
  • Thomas Pfeiffer, Sebastian Bonhoeffer (ETH
    Zürich)
  • Peer Bork (EMBL Heidelberg)
  • Reinhart Heinrich, Ferdinand Moldenhauer, Sascha
    Bulik (Berlin)
  • Hans Westerhoff (VU Amsterdam)
  • Special thanks to Bas Teusink (Wageningen) for
    discussion on this topic
  • Acknowledgement to DFG and BMBF for financial
    support

36
Current group members
  • Jörn Behre
  • Dr. Axel von Kamp
  • Dr. Mikhail Pachkov
  • Dimitar Kenanov
  • Dr. Ina Weiß
  • Beate Knoke
  • Ralf Bortfeldt
  • Gunter Neumann

Involved in work on pathway analysis
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