V18 – extreme pathways
Computational metabolomics: modelling constraints
Surviving (expressed) phenotypes must satisfy constraints imposed on the molecular
functions of a cell, e.g. conservation of mass and energy.
Fundamental approach to understand biological systems: identify and formulate
constraints.
Important constraints of cellular function:
(1) physico-chemical constraints
(2) Topological constraints
(3) Environmental constraints
(4) Regulatory constraints
Price et al. Nature Rev Microbiol 2, 886 (2004)
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Physico-chemical constraints
These are „hard“ constraints: Conservation of mass, energy and momentum.
Contents of a cell are densely packed  viscosity can be 100 – 1000 times higher
than that of water
Therefore, diffusion rates of macromolecules in cells are slower than in water.
Many molecules are confined inside the semi-permeable membrane  high
osmolarity. Need to deal with osmotic pressure (e.g. Na+K+ pumps)
Reaction rates are determined by local concentrations inside cells
Enzyme-turnover numbers are generally less than 104 s-1. Maximal rates are equal to
the turnover-number multiplied by the enzyme concentration.
Biochemical reactions are driven by negative free-energy change in forward
direction.
Price et al. Nature Rev Microbiol 2, 886 (2004)
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Topological constraints
The crowding of molecules inside cells leads to topological (3D)-constraints that affect
both the form and the function of biological systems.
E.g. the ratio between the number of tRNAs and the number of ribosomes in an E.coli
cell is about 10. Because there are 43 different types of tRNA, there is less than
one full set of tRNAs per ribosome  it may be necessary to configure the
genome so that rare codons are located close together.
E.g. at a pH of 7.6 E.coli typically contains only about 16 H+ ions.
Remember that H+ is involved in many metabolic reactions.
Therefore, during each such reaction, the pH of the cell changes!
Price et al. Nature Rev Microbiol 2, 886 (2004)
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Environmental constraints
Environmental constraints on cells are time and condition dependent:
Nutrient availability, pH, temperature, osmolarity, availability of electron acceptors.
E.g. Heliobacter pylori lives in the human stomach at pH = 1
 needs to produce NH3 at a rate that will maintain ist immediate surrounding at a pH
that is sufficiently high to allow survival.
Ammonia is made from elementary nitrogen  H. pylori has adapted by using amino
acids instead of carbohydrates as its primary carbon source.
Price et al. Nature Rev Microbiol 2, 886 (2004)
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Regulatory constraints
Regulatory constraints are self-imposed by the organism and are subject to
evolutionary change  they are no „hard“ constraints.
Regulatory constraints allow the cell to eliminate suboptimal phenotypic states and to
confine itself to behaviors of increased fitness.
Price et al. Nature Rev Microbiol 2, 886 (2004)
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Mathematical formation of constraints
There are two fundamental types of constraints: balances and bounds.
Balances are constraints that are associated
with conserved quantities as energy, mass, redox potential, momentum
or with phenomena such as solvent capacity, electroneutrality and osmotic pressure.
Bounds are constraints that limit numerical ranges of individual variables and
parameters such as concentrations, fluxes or kinetic constants.
Both bound and balance constraints limit the allowable functional states of
reconstructed cellular metabolic networks.
Price et al. Nature Rev Microbiol 2, 886 (2004)
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Genome-scale networks
Price et al. Nature Rev Microbiol 2, 886 (2004)
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Tools for analyzing network states
The two steps that are used
to form a solution space —
reconstruction and the
imposition of governing
constraints — are illustrated
in the centre of the figure.
Several methods are being
developed at various
laboratories to analyse the
solution space.
Ci and Cj concentrations of
compounds i and j;
EP, extreme pathway;
vi and vj fluxes through
reactions i and j;
v1 –v3 flux through reactions
1-3;
vnet, net flux through loop.
Price et al. Nature Rev Microbiol
2, 886 (2004)
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Determining optimal states
Price et al. Nature Rev Microbiol 2, 886 (2004)
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Flux dependencies
Price et al. Nature Rev Microbiol 2, 886 (2004)
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Characterizing the whole solution space
Price et al. Nature Rev Microbiol 2, 886 (2004)
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Altered solution spaces
Price et al. Nature Rev Microbiol 2, 886 (2004)
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Extreme Pathways
introduced into metabolic analysis by the lab of Bernard Palsson
(Dept. of Bioengineering, UC San Diego). The publications of this lab
are available at http://gcrg.ucsd.edu/publications/index.html
The extreme pathway
technique is based
on the stoichiometric
matrix representation
of metabolic networks.
All external fluxes are
defined as pointing outwards.
Schilling, Letscher, Palsson,
J. theor. Biol. 203, 229 (2000)
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Feasible solution set for a metabolic reaction network
(A) The steady-state operation of the metabolic network is restricted to the region
within a cone, defined as the feasible set. The feasible set contains all flux vectors
that satisfy the physicochemical constrains. Thus, the feasible set defines the
capabilities of the metabolic network. All feasible metabolic flux distributions lie
within the feasible set, and
(B) in the limiting case, where all constraints on the metabolic network are known,
such as the enzyme kinetics and gene regulation, the feasible set may be reduced
to a single point. This single point must lie within the feasible set.
Edwards & Palsson PNAS 97, 5528 (2000)
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Extreme Pathways – theorem
Theorem. A convex flux cone has a set of systemically independent generating
vectors. Furthermore, these generating vectors (extremal rays) are unique up to
a multiplication by a positive scalar. These generating vectors will be called
„extreme pathways“.
(1) The existence of a systemically independent generating set for a cone is
provided by an algorithm to construct extreme pathways (see below).
(2) uniqueness?
Let {p1, ..., pk} be a systemically independent generating set for a cone.
Then follows that if pj = c´+ c´´ both c´and c´´ are positive multiples of pj.
Schilling, Letscher, Palsson,
J. theor. Biol. 203, 229 (2000)
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Extreme Pathways – uniqueness
To show that this is true, write the two pathways c´and c´´ as non-negative linear
combinations of the extreme pathways:
k
c  i p i
i 1
k
c   i p i
i , i  0
i 1
k
p j  c  c   i  i p i
i 1
Since the pi are systemically independent,
Therefore both c´and c´´ are multiples of pj.
i  j
i  0, i  0
If {c1, ..., ck} was another set of extreme pathways, this argument would show that
each of the ci must be a positive multiple of one of the pi.
Schilling, Letscher, Palsson,
J. theor. Biol. 203, 229 (2000)
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Extreme Pathways – algorithm - setup
The algorithm to determine the set of extreme pathways for a reaction
network follows the pinciples of algorithms for finding the extremal rays/
generating vectors of convex polyhedral cones.
Combine n  n identity matrix (I) with the transpose of the stoichiometric
matrix ST. I serves for bookkeeping.
S
Schilling, Letscher, Palsson,
J. theor. Biol. 203, 229 (2000)
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I
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ST
17
separate internal and external fluxes
Examine constraints on each of the exchange fluxes as given by
j  bj  j
If the exchange flux is constrained to be positive  do nothing.
If the exchange flux is constrained to be negative  multiply the
corresponding row of the initial matrix by -1.
If the exchange flux is unconstrained  move the entire row to a temporary
matrix T(E). This completes the first tableau T(0).
T(0) and T(E) for the example reaction system are shown on the previous slide.
Each element of this matrices will be designated Tij.
Starting with x = 1 and T(0) = T(x-1) the next tableau is generated in the
following way:
Schilling, Letscher, Palsson,
J. theor. Biol. 203, 229 (2000)
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idea of algorithm
(1) Identify all metabolites that do not have an unconstrained exchange flux
associated with them.
The total number of such metabolites is denoted by .
For the example, this is only the case for metabolite C ( = 1).
What is the main idea?
- We want to find balanced extreme pathways
that don‘t change the concentrations of
metabolites when flux flows through
(input fluxes are channelled to products not to
accumulation of intermediates).
- The stochiometrix matrix describes the coupling of each reaction to the
concentration of metabolites X.
- Now we need to balance combinations of reactions that leave concentrations
unchanged. Pathways applied to metabolites should not change their
concentrations  the matrix entries
Schilling, Letscher, Palsson,
need to be brought to 0.
J. theor. Biol. 203, 229 (2000)
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keep pathways that do not change
concentrations of internal metabolites
(2) Begin forming the new matrix T(x) by copying
all rows from T(x – 1) which contain a zero in the
column of ST that corresponds to the first
metabolite identified in step 1, denoted by index c.
(Here 3rd column of ST.)
1
1
T(0)
1
=
1
1
1
T(1) =
1
-1
1
0
0
0
0
-1
1
0
0
0
1
-1
0
0
0
0
-1
1
0
0
0
1
-1
0
0
0
-1
0
1
-1
1
0
0
0
+
Schilling, Letscher, Palsson, J. theor. Biol. 203, 229 (2000)
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balance combinations of other pathways
(3) Of the remaining rows in T(x-1) add together
all possible combinations of rows which contain
values of the opposite sign in column c, such that
the addition produces a zero in this column.
1
-1
1
0
0
0
0
-1
1
0
0
0
1
-1
0
0
0
0
-1
1
0
0
0
1
-1
0
1
0
0
-1
0
1
1
1
T(0) =
1
1
T(1)
=
Schilling, et al.
JTB 203, 229
18. Lecture WS 2005/06
1
0
0
0
0
0
-1
1
0
0
0
0
1
1
0
0
0
0
0
0
0
0
0
1
0
1
0
0
0
-1
0
1
0
0
1
0
0
0
1
0
-1
0
0
1
0
0
1
0
1
0
0
1
0
-1
0
0
0
0
1
1
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
-1
1
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remove “non-orthogonal” pathways
(4) For all of the rows added to T(x) in steps 2 and 3 check to make sure that no
row exists that is a non-negative combination of any other sets of rows in T(x) .
One method used is as follows:
let A(i) = set of column indices j for with the elements of row i = 0.
For the example above
Then check to determine if there exists
A(1) = {2,3,4,5,6,9,10,11}
another row (h) for which A(i) is a
A(2) = {1,4,5,6,7,8,9,10,11}subset of A(h).
A(3) = {1,3,5,6,7,9,11}
A(4) = {1,3,4,5,7,9,10}
If A(i)  A(h), i  h
A(5) = {1,2,3,6,7,8,9,10,11}where
A(6) = {1,2,3,4,7,8,9}
A(i) = { j : Ti,j = 0, 1  j  (n+m) }
then row i must be eliminated from T(x)
Schilling et al.
JTB 203, 229
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repeat steps for all internal metabolites
(5) With the formation of T(x) complete steps 2 – 4 for all of the metabolites that do
not have an unconstrained exchange flux operating on the metabolite,
incrementing x by one up to . The final tableau will be T().
Note that the number of rows in T () will be equal to k, the number of extreme
pathways.
Schilling et al.
JTB 203, 229
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balance external fluxes
(6) Next we append T(E) to the bottom of T(). (In the example here  = 1.)
This results in the following tableau:
1
1
-1
1
0
0
0
0
0
0
0
0
0
-1
0
1
0
0
-1
0
1
0
1
0
1
0
-1
0
1
0
0
0
0
0
0
0
0
-1
1
-1
0
0
0
0
0
-1
0
0
0
0
0
0
-1
0
0
0
0
0
-1
1
1
1
1
1
1
T(1/E) =
1
1
1
1
1
Schilling et al.
JTB 203, 229
1
1
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balance external fluxes
(7) Starting in the n+1 column (or the first non-zero column on the right side),
if Ti,(n+1)  0 then add the corresponding non-zero row from T(E) to row i so as to
produce 0 in the n+1-th column.
This is done by simply multiplying the corresponding row in T(E) by Ti,(n+1) and
adding this row to row i .
Repeat this procedure for each of the rows in the upper portion of the tableau so
as to create zeros in the entire upper portion of the (n+1) column.
When finished, remove the row in T(E) corresponding to the exchange flux for the
metabolite just balanced.
Schilling et al.
JTB 203, 229
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balance external fluxes
(8) Follow the same procedure as in step (7) for each of the columns on the right
side of the tableau containing non-zero entries.
(In this example we need to perform step (7) for every column except the middle
column of the right side which correponds to metabolite C.)
The final tableau T(final) will contain the transpose of the matrix P containing the
extreme pathways in place of the original identity matrix.
Schilling et al.
JTB 203, 229
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pathway matrix
1
-1
1
1
1
T(final) =
1
-1
1
1
1
PT
=
Schilling et al.
JTB 203, 229
18. Lecture WS 2005/06
1
1
-1
1
1
v2 v3 v4
1
-1
1
1
v1
1
1
v5 v6
-1
b 1 b2
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
b3 b4
1
0
0
0
0
0
-1
1
0
0
0
1
1
0
0
0
0
0
0
0
0
1
0
1
0
0
0
-1
1
0
0
1
0
0
0
1
0
-1
0
1
0
0
1
0
1
0
0
1
-1
0
0
0
0
1
1
0
0
0
0
0
0
0
0
0
1
1
0
0
-1
1
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p1
p7
p3
p2
p4
p6
p5
27
Extreme Pathways for model system
2 pathways p6 and p7 are not shown (right below)
because all exchange fluxes with the exterior are 0.
Such pathways have no net overall effect on the
functional capabilities of the network.
They belong to the cycling of reactions v4/v5 and v2/v3.
v1
v2 v3 v4
v5 v6
b1 b2
b3 b4
1
0
0
0
0
0
-1
1
0
0
0
1
1
0
0
0
0
0
0
0
0
1
0
1
0
0
0
-1
1
0
0
1
0
0
0
1
0
-1
0
1
0
0
1
0
1
0
0
1
-1
0
0
0
0
1
1
0
0
0
0
0
0
0
0
0
1
1
0
0
-1
1
p1
p7
p3
p2
p4
p6
p5
Schilling et al.
JTB 203, 229
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How reactions appear in pathway matrix
In the matrix P of extreme pathways, each column is an EP and each row
corresponds to a reaction in the network.
The numerical value of the i,j-th element corresponds to the relative flux level
through the i-th reaction in the j-th EP.
PLM  PT  P
Papin, Price, Palsson,
Genome Res. 12, 1889 (2002)
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Properties of pathway matrix
A symmetric Pathway Length Matrix PLM can be calculated:
PLM  PT  P
where the values along the diagonal correspond to the length of the EPs.
The off-diagonal terms of PLM are the number of reactions that a pair of extreme
pathways have in common.
Papin, Price, Palsson, Genome Res. 12, 1889 (2002)
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Properties of pathway matrix
One can also compute a reaction participation matrix PPM from P:
PPM  P  PT
where the diagonal correspond to the number of pathways in which the given
reaction participates.
Papin, Price, Palsson, Genome Res. 12, 1889 (2002)
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EP Analysis of H. pylori and H. influenza
Amino acid synthesis in Heliobacter pylori vs.
Heliobacter influenza studied by EP analysis.
Papin, Price, Palsson, Genome Res. 12, 1889 (2002)
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Extreme Pathway Analysis
Calculation of EPs for increasingly large networks is computationally intensive and
results in the generation of large data sets.
Even for integrated genome-scale models for microbes under simple conditions,
EP analysis can generate thousands of vectors!
Interpretation:
- the metabolic network of H. influenza has an order of magnitude larger degree of
pathway redundancy than the metabolic network of H. pylori
Found elsewhere: the number of reactions that participate in EPs that produce a
particular product is poorly correlated to the product yield and the molecular
complexity of the product.
Possible way out?
Papin, Price, Palsson, Genome Res. 12, 1889 (2002)
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Linear Matrices
Transpose of a linear matrix A:
U is a unitary n  n matrix  U
with the n  n identity matrix In.
Ai, j T
 A j , i 
U  UU  I n
*
*
In linear algebra singular value decomposition (SVD) is an important
factorization of a rectangular real or complex matrix, with several applications in
signal processing and statistics.
This matrix decomposition is analogous to the diagonalization of symmetric or
Hermitian square matrices using a basis of eigenvectors given by the spectral
theorem.
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Diagonalisation of pathway matrix?
Suppose M is an m  n matrix with real or complex entries.
Then there exists a factorization of the form
M = U  V* where
U : m  m unitary matrix,
Σ : is an m  n matrix with nonnegative numbers on the diagonal and
zeros off the diagonal,
V* : the transpose of V, an n  n unitary matrix of real or complex
numbers.
Such a factorization is called a singular-value decomposition of M.
U describes the rows of M with respect to the base vectors associated with
the singular values.
V describes the columns of M with respect to the base vectors associated
with the singular values. Σ contains the singular values
One commonly insists that the values Σi,i be ordered in non-increasing
fashion. In this case, the diagonal matrix Σ is uniquely determined by M
(though the matrices U and V are not).
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Single Value Decomposition of EP matrices
For a given EP matrix P  np, SVD decomposes P into 3 matrices
 Σ 0
 V T
P  U
 0 0  n p
where U  nn : orthonormal matrix of the left singular vectors,
V pp : an analogous orthonormal matrix of the right singular vectors,
 rr :a diagonal matrix containing the singular values i=1..r arranged in
descending order where r is the rank of P.
The first r columns of U and V, referred to as the left and right singular vectors, or
modes, are unique and form the orthonormal basis for the column space and row
space of P.
The singular values are the square roots of the eigenvalues of PTP. The magnitude
of the singular values in  indicate the relative contribution of the singular vectors in
U and V in reconstructing P.
E.g. the second singular value contributes less to the construction of P than the first
singular value etc.
Price et al. Biophys J 84, 794 (2003)
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Single Value Decomposition of EP: Interpretation
The first mode (as the other modes) corresponds to a valid biochemical pathway
through the network.
The first mode will point
into the portions of the
cone with highest density
of EPs.
Price et al. Biophys J 84, 794 (2003)
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SVD applied for Heliobacter systems
Cumulative fractional contributions for the singular value decomposition of the EP
matrices of H. influenza and H. pylori.
This plot represents the
contribution of the first
n modes to the overall
description of the system.
Price et al. Biophys J 84, 794 (2003)
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Application of elementary modes
Metabolic network structure of E.coli determines
key aspects of functionality and regulation
Elementary modes will be covered in V19.
The concept is closely related to extreme
pathways. In this example , we will simply
ignore the small difference.
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)
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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)
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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:
Yi , X / Si
ei
 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)
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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.
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Growth-supporting elementary 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.
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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)
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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 
l
 ei
 eil
l
l
Control-effective fluxes vl(Sk) are obtained by averaged weighting of the product of reactionspecific fluxes and mode-specific efficiencies over all EFMs using the substrate under
consideration:
vl S k  
1
max
X / Sk
Y

l



S
,

e
i k i
i
  S ,  
i
k

1
max
A / Sk
Y
l

l



S
,
ATP
e
i k
i
i
  S , ATP
i
k
l
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)
18. Lecture WS 2005/06
Bioinformatics III
45
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
control-effective fluxes.
Compare to exp. DNA-microarray data
for E.coli growin on glucose, glycerol,
and acetate.
Excellent correlation!
Stelling et al. Nature 420, 190 (2002)
18. Lecture WS 2005/06
Calculated ratios between gene expression levels
during exponential growth on acetate and
exponential growth on glucose (filled circles
indicate outliers) based on all elementary modes
versus experimentally determined transcript
ratios19. Lines indicate 95% confidence intervals
for experimental data (horizontal lines), linear
regression (solid line), perfect match (dashed
line) and two-fold deviation (dotted line).
Bioinformatics III
46
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)
18. Lecture WS 2005/06
Bioinformatics III
47
Summary (extreme pathways)
Extreme pathway analysis provides a mathematically rigorous way to dissect
complex biochemical networks.
The matrix products PT  P and PT  P are useful ways to interpret pathway lengths
and reaction participation.
However, the number of computed vectors may range in the 1000sands.
Therefore, meta-methods (e.g. singular value decomposition) are required that
reduce the dimensionality to a useful number that can be inspected by humans.
Single value decomposition may be one useful method ... and there are more to
come.
Price et al. Biophys J 84, 794 (2003)
18. Lecture WS 2005/06
Bioinformatics III
48