# Computational Biology - Bioinformatik by e2f57F39

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```									          Metabolic Pathway Analysis: Elementary Modes
The technique of Elementary Flux Modes (EFM) was developed prior to extreme
pathways (EP) by Stephan Schuster, Thomas Dandekar and co-workers:
Pfeiffer et al. Bioinformatics, 15, 251 (1999)
Schuster et al. Nature Biotech. 18, 326 (2000)

The method is very similar to the „extreme pathway“ method to construct a basis
for metabolic flux states based on methods from convex algebra.
Extreme pathways are a subset of elementary modes, and for many systems, both
methods coincide.

Are the subtle differences important?

21. Lecture WS 2003/04                      Bioinformatics III        1
Review: Metabolite Balancing
For analyzing a biochemical network, its structure is expressed by the stochiometric
matrix S consisting of m rows corresponding to the substances (metabolites) and n
rows corresponding to the stochiometric coefficients of the metabolites in each
reaction.
A vector v denotes the reaction rates (mmol/g dry weight * hour) and a vector c
describes the metabolite concentrations.

Due to the high turnover of metabolite pools one often assumes pseudo-steady state
(c(t) = constant) leading to the fundamental Metabolic Balancing Equation:

dct 
 0  Sv                           (1)
dt
Flux distributions v satisfying this relationship lie in the null space of S and are able
to balance all metabolites.

Klamt et al. Bioinformatics 19, 261 (2003)

21. Lecture WS 2003/04                    Bioinformatics III                  2
Review: Metabolic flux analysis
Metabolic flux analysis (MFA): determine preferably all components of the flux
distribution v in a metabolic network during a certain stationary growth experiment.

Typically some measured or known rates must be provided to calculate unknown
rates. Accordingly, v and S are partioned into the known (vb, Sb) and unknown part
(va, Sa).

(1) leads to the central equation for MFA describing a flux scenario:

0 = S  v = Sa  va + Sb  vb.

The rank of Sa determines whether this scenario is redundant and/or
underdetermined. Redundant systems can be checked on inconsistencies. In
underdetermined scenarios, only some element of va are uniquely calculable.

Klamt et al. Bioinformatics 19, 261 (2003)

21. Lecture WS 2003/04                 Bioinformatics III                 3
Software: FluxAnalyzer
A network project constructed by FluxAnalyzer. Here, vb consists of R1, R2, 
and va of R3 - R7, whereof R3, R4, R7 can be computed.

Biomass component 1:
BC1[g] = 2[mmol]A + 1 [mmol]C
Biomass component 2:
BC2[g] = 1[mmol]C + 3[mmol]D

1    -1   0    -1      0       0         0   0.8
0    1    -1   0       0       0         0    0
S=                                                       Klamt et al. Bioinformatics
0    0    0    1      -1       1         0    1
0    0    1    0       1      -1         1   1.8        19, 261 (2003)

R1 R2     R3 R4       R5      R6 R7 biomass
synthesis
21. Lecture WS 2003/04                       Bioinformatics III                                   4
Review: structural network analysis (SNA)
Whereas MFA focuses on a single flux distribution, techniques of Structural
(Stochiometric, Topological) Network Analysis (SNA) address general topological
properties, overall capabilities, and the inherent pathway structure of a metabolic
network.

Basic topological properties are, e.g., conserved moieties.

Flux Balance Analysis (FBA9 searches for single optimal flux distributions (mostly
with respect to the synthesis of biomass) fulfilling S  v = 0 and additionally
reversibility and capacity restrictions for each reaction (i  vi  i).

Klamt et al. Bioinformatics 19, 261 (2003)

21. Lecture WS 2003/04                 Bioinformatics III              5
Review: Metabolic Pathway Analysis (MPA)
Metabolic Pathway Analysis searches for meaningful structural and functional units
in metabolic networks. The most promising, very similar approaches are based on
convex analysis and use the sets of elementary flux modes (Schuster et al. 1999,
2000) and extreme pathways (Schilling et al. 2000).

Both sets span the space of feasible steady-state flux distributions by non-
decomposable routes, i.e. no subset of reactions involved in an EFM or EP can hold
the network balanced using non-trivial fluxes.

MPA can be used to study e.g.
- routing + flexibility/redundancy of networks
- functionality of networks
- idenfication of futile cycles
- gives all (sub)optimal pathways with respect to product/biomass yield
- can be useful for calculability studies in MFA

Klamt et al. Bioinformatics 19, 261 (2003)
21. Lecture WS 2003/04                 Bioinformatics III               6
Elementary Flux Modes
Start from list of reaction equations and a declaration of reversible and irreversible
reactions and of internal and external metabolites.

E.g. reaction scheme of monosaccharide                               Fig.1
metabolism. It includes 15 internal
metabolites, and 19 reactions.
 S has dimension 15  19.

It is convenient to reduce this matrix
by lumping those reactions that
necessarily operate together.
 {Gap,Pgk,Gpm,Eno,Pyk},
 {Zwf,Pgl,Gnd}

Such groups of enzymes can be detected automatically.
This reveals another two sequences {Fba,TpiA} and {2 Rpe,TktI,Tal,TktII}.

Schuster et al. Nature Biotech 18, 326 (2000)
21. Lecture WS 2003/04                     Bioinformatics III                7
Elementary Flux Modes
Lumping the reactions in any one sequence gives the following reduced system:

Construct initial tableau by combining
S with identity matrix:

Ru5P

GAP
R5P
F6P
FP2

1      0     ...     0     0     0      1     0       0          Pgi
0      1     ...     0     0     -1     0     2       0          {Fba,TpiA}
0      0     ...     0    -1     0      0     0       1          Rpi                     reversible
0      0     ...     0    -2     0      2     1      -1          {2Rpe,TktI,Tal,TktII}
T(0)=       0      0     ...     0     0     0      0    -1       0          {Gap,Pgk,Gpm,Eno,Pyk}
0      0     ...     0     1     0      0     0       0          {Zwf,Pgl,Gnd}
0      0     ...     0     0     1     -1     0       0          Pfk                     irreversible
0      0     ...     0     0     -1     1     0       0          Fbp
0      0     ...     1     0     0      0     0      -1          Prs_DeoB
Schuster et al. Nature Biotech 18, 326 (2000)
21. Lecture WS 2003/04                                Bioinformatics III                           8
Elementary Flux Modes
Aim again: bring all entries
1                                            0    0    1    0    0
of right part of matrix to 0.
1                                        0    -1   0    2    0
E.g. 2*row3 - row4 gives
1                                   -1   0    0    0    1
„reversible“ row with 0 in column 10
1                             -2   0    2    1    -1
T(0)=                         1                      0    0    0    -1   0
New „irreversible“ rows with 0 entry
in column 10 by row3 + row6 and                                          1               1    0    0    0    0

by row4 + row7.                                                              1           0    1    -1   0    0
1       0    -1   1    0    0
In general, linear combinations                                                      1   0    0    0    0    -1
of 2 rows corresponding
to the same type of directio-
1                                            0    0    1    0    0
nality go into the part of
the respective type in the                      1                                        0    -1   0    2    0

tableau. Combinations by                             2    -1                             0    0    -2   -1   3

different types go into the                                       1                      0    0    0    -1   0
„irreversible“ tableau
T(1)=                                     1           0    1    -1   0    0
because at least 1 reaction is                                                   1       0    -1   1    0    0
irreversible. Irreversible reactions                                                 1   0    0    0    0    -1
can only combined using positive                     1                   1               0    0    0    0    1
coefficients.                                              1             2               0    0    2    1    -1
Schuster et al. Nature Biotech 18, 326 (2000)
21. Lecture WS 2003/04                     Bioinformatics III                                            9
Elementary Flux Modes
Aim: zero column 11.                    1                                             0   0    1    0    0
Include all possible (direction-wise        1                                         0   -1   0    2    0
allowed) linear combinations of                   2    -1                             0   0    -2   -1   3
rows.                                                          1                      0   0    0    -1   0
1           0   1    -1   0    0

T(1)=                                         1       0   -1   1    0    0
1   0   0    0    0    -1
1                   1               0   0    0    0    1
1             2               0   0    2    1    -1

1                                             0   0    1    0    0
2    -1                             0   0    -2   -1   3
1                      0   0    0    -1   0
1   0   0    0    0    -1
T(2)=             1                   1               0   0    0    0    1
1             2               0   0    2    1    -1
1                             1           0   0    -1   2    0
-1                                1       0   0    1    -2   0
continue with columns 12-                                                 1   1       0   0    0    0    0
14.                                                                Schuster et al. Nature Biotech 18, 326 (2000)
21. Lecture WS 2003/04                     Bioinformatics III                                           10
Elementary Flux Modes
In the course of the algorithm, one must avoid
- calculation of nonelementary modes (rows that contain fewer zeros than the row
- duplicate modes (a pair of rows is only combined if it fulfills the condition
S(mi(j))  S(mk(j))  S(ml(j+1)) where S(ml(j+1)) is the set of positions of 0 in this row.

- flux modes violating the sign restriction for the irreversible reactions.

1    1    0   0    2    0     1         0   0    0    ...   ...    0
Final tableau
-2   0    1   1    1    3     0         0   0   ...               ...
0    2    1   1    5    3     2         0   0
T(5) =     0    0    1   0    0    1     0         0   1
5    1    4   -2   0    0     1         0   6
-5   -1   2   2    0    6     0         1   0   ...               ...
0    0    0   0    0    0     1         1   0    0    ...   ...    0

This shows that the number of rows may decrease or increase in the course of the
algorithm. All constructed elementary modes are irreversible.

Schuster et al. Nature Biotech 18, 326 (2000)
21. Lecture WS 2003/04                      Bioinformatics III                                       11
Elementary Flux Modes
Graphical representation of the elementary flux modes of the monosaccharide
metabolism. The numbers indicate the relative flux carried by the enzymes.

Fig. 2

Schuster et al. Nature Biotech 18, 326 (2000)
21. Lecture WS 2003/04          Bioinformatics III                                       12
Two approaches for Metabolic Pathway Analysis?
The pathway P(v) is an elementary flux mode if it fulfills conditions C1 – C3.

(C1) Pseudo steady-state. S  e = 0. This ensures that none of the metabolites is
consumed or produced in the overall stoichiometry.

(C2) Feasibility: rate ei  0 if reaction is irreversible. This demands that only
thermodynamically realizable fluxes are contained in e.

(C3) Non-decomposability: there is no vector v (unequal to the zero vector and to
e) fulfilling C1 and C2 and that P(v) is a proper subset of P(e). This is the core
characteristics for EFMs and EPs and supplies the decomposition of the network
into smallest units (able to hold the network in steady state).
C3 is often called „genetic independence“ because it implies that the enzymes in
one EFM or EP are not a subset of the enzymes from another EFM or EP.

Klamt & Stelling Trends Biotech 21, 64 (2003)

21. Lecture WS 2003/04                     Bioinformatics III                  13
Two approaches for Metabolic Pathway Analysis?
The pathway P(e) is an extreme pathway if it fulfills conditions C1 – C3 AND
conditions C4 – C5.

(C4) Network reconfiguration: Each reaction must be classified either as exchange
flux or as internal reaction. All reversible internal reactions must be split up into
two separate, irreversible reactions (forward and backward reaction).

(C5) Systemic independence: the set of EPs in a network is the minimal set of
EFMs that can describe all feasible steady-state flux distributions.

Klamt & Stelling Trends Biotech 21, 64 (2003)

21. Lecture WS 2003/04                     Bioinformatics III              14
Two approaches for Metabolic Pathway Analysis?

A(ext) B(ext) C(ext)
R1        R2               R3

R4    B      R8

R7
R5
A         C             P
R9

R6       D

Klamt & Stelling Trends Biotech 21, 64 (2003)

21. Lecture WS 2003/04                      Bioinformatics III   15
Reconfigured Network
A(ext) B(ext) C(ext)
R1         R2             R3

R4      B     R8

R7f       R7b
A           C          P
R5         R9

R6         D

3 EFMs are not systemically independent:
EFM1 = EP4 + EP5
EFM2 = EP3 + EP5
EFM4 = EP2 + EP3

Klamt & Stelling Trends Biotech 21, 64 (2003)

21. Lecture WS 2003/04                       Bioinformatics III   16
Property 1 of EFMs
The only difference in the set of EFMs emerging upon reconfiguration consists in
the two-cycles that result from splitting up reversible reactions. However, two-cycles
are not considered as meaningful pathways.

Valid for any network: Property 1
Reconfiguring a network by splitting up reversible reactions leads to the same set of
meaningful EFMs.

Klamt & Stelling Trends Biotech 21, 64 (2003)

21. Lecture WS 2003/04                      Bioinformatics III              17
Software: FluxAnalyzer
What is the consequence of when all exchange fluxes (and hence all
reactions in the network) are irreversible?

EFMs and EPs always co-incide!
Klamt & Stelling Trends Biotech 21, 64 (2003)

21. Lecture WS 2003/04                      Bioinformatics III                  18
Property 2 of EFMs
Property 2
If all exchange reactions in a network are irreversible then the sets of meaningful
EFMs (both in the original and in the reconfigured network) and EPs coincide.

Klamt & Stelling Trends Biotech 21, 64 (2003)

21. Lecture WS 2003/04                      Bioinformatics III              19
Reconfigured Network
A(ext) B(ext) C(ext)
R1         R2             R3

R4      B     R8

R7f       R7b
A           C          P
R5         R9

R6         D

3 EFMs are not systemically independent:
EFM1 = EP4 + EP5
EFM2 = EP3 + EP5
EFM4 = EP2 + EP3

Klamt & Stelling Trends Biotech 21, 64 (2003)

21. Lecture WS 2003/04                       Bioinformatics III   20
Comparison of EFMs and EPs
Problem                          EFM (network N1)                    EP (network N2)

Recognition of                   4 genetically indepen-       Set of EPs does not contain
operational modes:               dent routes                  all genetically independent
routes for converting            (EFM1-EFM4)                  routes. Searching for EPs
exclusively A to P.                                 leading from A to P via B,
no pathway would be found.

Klamt & Stelling Trends Biotech 21, 64 (2003)

21. Lecture WS 2003/04                      Bioinformatics III                     21
Comparison of EFMs and EPs
Problem                          EFM (network N1)                    EP (network N2)

Finding all the                  EFM1 and EFM2 are                   One would only find the
optimal routes:                  optimal because they                suboptimal EP1, not the
optimal pathways for             yield one mole P per                optimal routes EFM1 and
synthesizing P during            mole substrate A                    EFM2.
growth on A alone.               (i.e. R3/R1 = 1),
whereas EFM3 and
EFM4 are only sub-
optimal (R3/R1 = 0.5).

Klamt & Stelling Trends Biotech 21, 64 (2003)

21. Lecture WS 2003/04                      Bioinformatics III                        22
Comparison of EFMs and EPs
Problem                         EFM (network N1)                     EFM (network N1)

Analysis of network             4 pathways convert A                 Only 1 EP exists for
flexibility (structural         to P (EFM1-EFM4),                    producing P by substrate A
robustness,                     whereas for B only one               alone, and 1 EP for
redundancy):                    route (EFM8) exists.                 synthesizing P by (only)
relative robustness of          When one of the                      substrate B. One might
exclusive growth on             internal reactions (R4-              suggest that both
A or B.                         R9) fails, for production            substrates possess the
of P from A 2 pathways               same redundancy of
will always „survive“.               pathways, but as shown by
By contrast, removing                EFM analysis, growth on
reaction R8 already                  substrate A is much more
stops the production of              flexible than on B.
P from B alone.

Klamt & Stelling Trends Biotech 21, 64 (2003)

21. Lecture WS 2003/04                      Bioinformatics III                        23
Comparison of EFMs and EPs
Problem                        EFM (network N1)                       EFM (network N1)

Relative importance            R8 is essential for                    Consider again biosynthesis
of single reactions:           producing P by substrate               of P from substrate A (EP1
relative importance of         B, whereas for A there is              only). Because R8 is not
reaction R8.                   no structurally „favored“              involved in EP1 one might
reaction (R4-R9 all occur              think that this reaction is not
twice in EFM1-EFM4).                   important for synthesizing P
However, considering the               from A. However, without this
optimal modes EFM1,                    reaction, it is impossible to
EFM2, one recognizes the               obtain optimal yields (1 P per
importance of R8 also for              A; EFM1 and EFM2).
growth on A.

Klamt & Stelling Trends Biotech 21, 64 (2003)

21. Lecture WS 2003/04                      Bioinformatics III                             24
Comparison of EFMs and EPs
Problem                        EFM (network N1)                       EFM (network N1)

Enzyme subsets                 R6 and R9 are an enzyme                The EPs pretend R4 and R8
and excluding                  subset. By contrast, R6                to be an excluding reaction
reaction pairs:                and R9 never occur                     pair – but they are not
suggest regulatory             together with R8 in an                 (EFM2). The enzyme
structures or rules.           EFM. Thus (R6,R8) and                  subsets would be correctly
(R8,R9) are excluding                  identified.
reaction pairs.                        However, one can construct simple
(In an arbitrary composable            examples where the EPs would also
steady-state flux distribution they    pretend wrong enzyme subsets (not
might occur together.)                 shown).

Klamt & Stelling Trends Biotech 21, 64 (2003)

21. Lecture WS 2003/04                      Bioinformatics III                               25
Comparison of EFMs and EPs
Problem                        EFM (network N1)                       EFM (network N1)

Pathway length:                The shortest pathway                   Both the shortest (EFM2)
shortest/longest               from A to P needs 2                    and the longest (EFM4)
pathway for                    internal reactions (EFM2),             pathway from A to P are not
production of P from           the longest 4 (EFM4).                  contained in the set of EPs.
A.

Klamt & Stelling Trends Biotech 21, 64 (2003)

21. Lecture WS 2003/04                      Bioinformatics III                           26
Comparison of EFMs and EPs
Problem                        EFM (network N1)                       EFM (network N1)

Removing a                     All EFMs not involving the             Analyzing a subnetwork
reaction and                   specific reactions build up            implies that the EPs must be
mutation studies:              the complete set of EFMs               newly computed. E.g. when
effect of deleting R7.         in the new (smaller) sub-              deleting R2, EFM2 would
network. If R7 is deleted,             become an EP. For this
EFMs 2,3,6,8 „survive“.                reason, mutation studies
Hence the mutant is                    cannot be performed easily.
viable.

Klamt & Stelling Trends Biotech 21, 64 (2003)

21. Lecture WS 2003/04                      Bioinformatics III                           27
Comparison of EFMs and EPs
Problem                        EFM (network N1)                       EFM (network N1)

Constraining                   For the case of R7, all                In general, the set of EPs
reaction                       EFMs but EFM1 and                      must be recalculated:
reversibility:                 EFM7 „survive“ because                 compare the EPs in network
effect of R7 limited to        the latter ones utilize R7             N2 (R2 reversible) and N4
B  C.                         with negative rate.                    (R2 irreversible).

Klamt & Stelling Trends Biotech 21, 64 (2003)

21. Lecture WS 2003/04                      Bioinformatics III                         28
Software: FluxAnalyzer
FluxAnalyzer has
both EPs and EFMs
implemented.

Allows convenient
studies of metabolic
systems.

Klamt et al. Bioinformatics 19, 261 (2003)

21. Lecture WS 2003/04                 Bioinformatics III   29
Software: FluxAnalyzer
Representation of stochiometric
matrix.

Klamt et al. Bioinformatics 19, 261 (2003)

21. Lecture WS 2003/04                 Bioinformatics III   30
Application of elementary modes
Metabolic network structure of E.coli determines
key aspects of functionality and regulation
Compute EFMs for central
metabolism of E.coli.

Catabolic part: substrate uptake
reactions, glycolysis, pentose
phosphate pathway, TCA cycle,
excretion of by-products (acetate,
formate, lactate, ethanol)

Anabolic part: conversions of
precursors into building blocks like
amino acids, to macromolecules,
and to biomass.

Stelling et al. Nature 420, 190 (2002)
21. Lecture WS 2003/04              Bioinformatics III   31
Metabolic network topology and phenotype
The total number of EFMs for given
conditions is used as quantitative
measure of metabolic flexibility.

a, Relative number of EFMs N enabling
deletion mutants in gene i ( i) of E. coli
to grow (abbreviated by µ) for 90 different
combinations of mutation and carbon
source. The solid line separates
experimentally determined mutant
phenotypes, namely inviability (1–40)
from viability (41–90).                                  The # of EFMs for mutant strain
allows correct prediction of
growth phenotype in more than 90%
of the cases.

Stelling et al. Nature 420, 190 (2002)

21. Lecture WS 2003/04             Bioinformatics III                           32
Robustness analysis
The # of EFMs qualitatively indicates whether a mutant is viable or not, but does
not describe quantitatively how well a mutant grows.

Define maximal biomass yield Ymass as the optimum of:
ei
Yi , X / Si    Sk
ei
ei is the single reaction rate (growth and substrate uptake) in EFM i selected for
utilization of substrate Sk.

Stelling et al. Nature 420, 190 (2002)

21. Lecture WS 2003/04                        Bioinformatics III           33
Software: FluxAnalyzer
Dependency of the mutants' maximal
growth yield Ymax( i) (open circles) and
the network diameter D( i) (open
squares) on the share of elementary
modes operational in the mutants. Data
were binned to reduce noise.
Stelling et al. Nature 420, 190 (2002)

Central metabolism of E.coli behaves in a highly robust manner because
mutants with significantly reduced metabolic flexibility show a growth yield
similar to wild type.

21. Lecture WS 2003/04            Bioinformatics III                                34
Growth-supporting elementar modes

Distribution of growth-supporting
elementary modes in wild type
(rather than in the mutants), that is,
share of modes having a specific
biomass yield (the dotted line
indicates equal distribution).

Stelling et al. Nature 420, 190 (2002)

Multiple, alternative pathways exist
with identical biomass yield.

21. Lecture WS 2003/04              Bioinformatics III   35
Can regulation be predicted by EFM analysis?
Assume that optimization during biological evolution can be characterized by the
two objectives of flexibility (associated with robustness) and of efficiency.

Flexibility means the ability to adapt to a wide range of environmental conditions,
that is, to realize a maximal bandwidth of thermodynamically feasible flux
distributions (maximizing # of EFMs).

Efficiency could be defined as fulfilment of cellular demands with an optimal
outcome such as maximal cell growth using a minimum of constitutive elements
(genes and proteins, thus minimizing # EFMs).

These 2 criteria pose contradictory challenges.
Optimal cellular regulation needs to find a trade-off.

Stelling et al. Nature 420, 190 (2002)

21. Lecture WS 2003/04             Bioinformatics III                                       36
Can regulation be predicted by EFM analysis?
Compute control-effective fluxes for each reaction l by determining the efficiency of any EFM
ei by relating the system‘s output  to the substrate uptake and to the sum of all absolute
fluxes.
With flux modes normalized to the total substrate uptake, efficiencies i(Sk, ) for
the targets for optimization -growth and ATP generation, are defined as:

ei                         eiATP
 i S k ,           and  i S k , ATP 
 eil
l
 eil                         l

Control-effective fluxes vl(Sk) are obtained by averaged weighting of the product of reaction-
specific fluxes and mode-specific efficiencies over all EFMs using the substrate under
consideration:
  i S k ,   eil                                i S k , ATP eil
vl S k  
1                                               1
       i
                   i
Ymax
X / Sk             S ,  
l
i     k               Y  max
A / Sk          S , ATP
l
i   k

YmaxX/Si and YmaxA/Si are optimal yields of biomass production and of ATP synthesis.

Control-effective fluxes represent the importance of each reaction for efficient and flexible
operation of the entire network.
Stelling et al. Nature 420, 190 (2002)
21. Lecture WS 2003/04                                        Bioinformatics III                                               37
Prediction of gene expression patterns
As cellular control on longer timescales
is predominantly achieved by genetic
regulation, the control-effective fluxes
should correlate with messenger RNA
levels.

Compute theoretical transcript ratios
(S1,S2) for growth on two alternative
substrates S1 and S2 as ratios of
Calculated ratios between gene expression levels
control-effective fluxes.
during exponential growth on acetate and
exponential growth on glucose (filled circles
Compare to exp. DNA-microarray data               indicate outliers) based on all elementary modes
for E.coli growin on glucose, glycerol,           versus experimentally determined transcript
and acetate.                                      ratios19. Lines indicate 95% confidence intervals
for experimental data (horizontal lines), linear
Excellent correlation!                            regression (solid line), perfect match (dashed
Stelling et al. Nature 420, 190 (2002)            line) and two-fold deviation (dotted line).
21. Lecture WS 2003/04              Bioinformatics III                              38
Prediction of transcript ratios

Predicted transcript ratios for acetate
versus glucose for which, in contrast to
a, only the two elementary modes with
highest biomass and ATP yield
(optimal modes) were considered.

This plot shows only weak correlation.
This corresponds to the approach
followed by Flux Balance Analysis.

Stelling et al. Nature 420, 190 (2002)

21. Lecture WS 2003/04               Bioinformatics III    39
Summary
EFM are a robust method that offers great opportunities for studying functional and
structural properties in metabolic networks.

Klamt & Stelling suggest that the term „elementary flux modes“ should be used
whenever the sets of EFMs and EPs are identical.

In cases where they don‘t, EPs are a subset of EFMs.
It remains to be understood more thoroughly how much valuable information about
the pathway structure is lost by using EPs.

Ongoing Challenges:
- study really large metabolic systems by subdividing them
- combine metabolic model with model of cellular regulation.

Klamt & Stelling Trends Biotech 21, 64 (2003)

21. Lecture WS 2003/04                      Bioinformatics III            40

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