V9: Characterizing the Fluxome of Biological Cells

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V9 Characterizing the Fluxome of Biological Cells

• Qualitative methods for fluxome analysis
• Graph theoretical methods
• Jeong Barabasi scale-free metabolic networks
• Arita different view resulting from different
  representation
• stochiometric analyses
• Extreme pathway analysis
• Large-scale experimental fluxome measurement on
  B. subtilis wildtype
• and 137 mutants.

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First breakthrough scale-free metabolic networks
(d) The degree distribution, P(k), of the
metabolic network illustrates its scale-free
topology. (e) The scaling of the clustering
coefficient C(k) with the degree k illustrates
the hierarchical architecture of metabolism (The
data shown in d and e represent an average over
43 organisms). (f) The flux distribution in the
central metabolism of Escherichia coli follows a
power law, which indicates that most reactions
have small metabolic flux, whereas a few
reactions, with high fluxes, carry most of the
metabolic activity. It should be noted that on
all three plots the axis is logarithmic and a
straight line on such loglog plots indicates a
power-law scaling. CTP, cytidine triphosphate
GLC, aldo-hexose glucose UDP, uridine
diphosphate UMP, uridine monophosphate UTP,
uridine triphosphate.
Barabasi Oltvai, Nature Reviews Genetics 5, 101
(2004)
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Different opinion

According to the formal definition, in a
small-world network, (i) most nodes (metabolites
in our case) have a low connection degree, and
the degree distribution follows a power law also
referred to as scale-freeness (ii) high-degree
nodes, called hubs, dominate the network, and
most nodes are clustered around hubs and (iii)
the average path length (AL i.e., the average of
the shortest path length over all pairs of nodes
in the network) remains the theoretical minimum,
that of a random graph. Because of its topology
with few hubs, a small-world network may be
resistant to random failures any peripheral node
is likely to have a low connection degree and is
therefore expendable. In biological networks, the
hubs are thought to be functionally important and
phylogenetically oldest.
Arita, PNAS 101, 1543 (2004)
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Different opinion
Although several groups confirmed the small-world
property of small-molecule metabolisms in
multiple data sources, the details of their
results differ depending on the purpose of the
analysis and its data-preparation scheme.
Notable differences are attributable to the
reversibility of enzymatic reactions and to the
treatment of metabolically ubiquitous compounds
referred to as coenzymes or inorganics. Table 1
summarizes differences in the major analyses and
compares the average path length (AL) and hub
metabolites they identified.
Arita, PNAS 101, 1543 (2004)
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Different opinion
All of these studies used the same algorithmic
procedure, and discrepancies are ascribable to
the different aims of their network analyses.
Jeong et al. computed the proximity of
metabolites by regarding all substrates and
products in the same reaction as adjacent (Fig. 1)
Arita, PNAS 101, 1543 (2004)
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Basis for analysis
To reproduce biochemical pathways in the
traditional metabolic map, however, metabolites
to be linked cannot be defined per se by
compounds or reactions. The biochemical link
between metabolites is context-sensitive it
depends on the conserved structural moieties in
the adjacent reactions. To accurately compute
the reaction connectivity as in the traditional
metabolic map, we used digitally compiled atomic
mappings, i.e., atomic position pairs between
substrates and products corresponding to the
substructural moieties conserved in each
reaction. With this information, we reassessed
the global properties of metabolic networks with
special emphasis on the small-world hypothesis.
Arita, PNAS 101, 1543 (2004)
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Basis for analysis
Arita, PNAS 101, 1543 (2004)
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Results of Arita
In Aritas conclusion, metabolic pathway
discussions should not be based on
substrate-level network topology. Because the
superficial connectivity on metabolic maps does
not always correspond to pathways, structural
information of metabolites is indispensable for
computing biochemical pathways.
Arita, PNAS 101, 1543 (2004)
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Review from bioinformatics III V19 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/ind
ex.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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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. Schilling,
Letscher, Palsson, J. theor. Biol. 203, 229 (2000)
S
I
ST
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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
dont 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 need to be
brought to 0.
Schilling, Letscher, Palsson, 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.) Schilling, Letscher, Palsson, J.
theor. Biol. 203, 229 (2000)
T(0)
T(1)

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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. Schilling, et al. JTB 203, 229
T(0)
T(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) . 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 Schilling et
al. JTB 203, 229
T(1/E)
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balance external fluxes
(7) Starting in the n1 column (or the first
non-zero column on the right side), if Ti,(n1)
? 0 then add the corresponding non-zero row from
T(E) to row i so as to produce 0 in the n1-th
column. This is done by simply multiplying the
corresponding row in T(E) by Ti,(n1) 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 (n1) 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. Sc
hilling et al. JTB 203, 229
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pathway matrix
T(final) PT Schilling et al. JTB
203, 229
v1 v2 v3 v4 v5 v6 b1 b2 b3
b4
p1 p7 p3 p2 p4 p6 p5
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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.
Schilling et al. JTB 203, 229
v1 v2 v3 v4 v5 v6 b1 b2 b3
b4
p1 p7 p3 p2 p4 p6 p5
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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.
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 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 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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Exp. distribution of metabolic fluxes in 137 B.
subtilis mutants

Relative (a-e) and absolute (f-h, light gray
arrows) carbon fluxes during exponential growth
on glucose. Relative fluxes in a-c were
analytically quantified from the mass isotope
distribution by metabolic flux ratio analysis.
They specify the contribution of a given pathway
or reaction to the synthesis of a particular
metabolite. Wild-type values are indicated by
asterisks. CoA, coenzyme A PP, pentose
phosphate. Observation Absolute fluxes in and
out of the cell varied by 31 55 around the
wt. Relative fluxes inside the cell varied only
by 3-8!
Fischer Sauer, Nature Genetics 37, 636 (2005)
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C13 flux analysis
Fischer Sauer, Eur J Biochem 270, 880 (2003)
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C13 flux analysis
Fischer Sauer, Eur J Biochem 270, 880 (2003)
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Calculation of metabolic flux ratios
Fischer Sauer, Eur J Biochem 270, 880 (2003)
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Calculation of metabolic flux ratios
Fischer Sauer, Eur J Biochem 270, 880 (2003)
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Effects of knockouts on absolute fluxes and
optimality

Fischer Sauer, Nature Genetics 37, 636 (2005)
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Effects of knockouts on relative fluxes
(a) Relative fluxes through glycolysis and the
TCA cycle to the synthesis of glyceraldehyde-3-pho
sphate and oxaloacetate, respectively, as
obtained from flux ratio analysis. The
complementing fractions are contributed by the
pentose phosphate pathway and the anaplerotic
reaction.

Extreme flux re-partitioning critical reaction
knockouts in TCA cycle (OdHA, SdhC, MdH),
glycolysis (Pgi), or PPP (Zwf, GndA) Only 10
mutants without metabolic functions had altered
intracellular fluxes.
(b) Relative carbon fluxes to acetate and biomass
formation. Black circle wild type. Metabolic
genes the categories of central carbon
metabolism, biosynthetic reactions and catabolic
reactions. Regulatory genes the categories of
transcriptional regulators and signal
transduction. The 10 mutants mentioned also have
different ratios between catabolism and anabolism.
Fischer Sauer, Nature Genetics 37, 636 (2005)
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Effects of knockouts on relative fluxes
(a) Relative fluxes through glycolysis and the
TCA cycle to the synthesis of glyceraldehyde-3-pho
sphate and oxaloacetate, respectively, as
obtained from flux ratio analysis. The
complementing fractions are contributed by the
pentose phosphate pathway and the anaplerotic
reaction.

Extreme repartioning in the peripheral
biosynthetic network, pathway disruptions are
mostly lethal there are only a few bypass
reactions. More redundancy exists within central
carbon metabolism ? respond to food changes.
Fischer Sauer, Nature Genetics 37, 636 (2005)
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Effects of knockouts on absolute fluxes and
optimality

Absolute molecular fluxes at the three key
divergent branch points of glucose catabolism
(a) Glucose-6-phosphate (Glc6P), (b)
acetyl-coenzyme A (acetyl-coA) and (c) the
branching between anabolism and catabolism. A
linear correlation between partitioned fluxes
shows a rigid branch point with a
rate-independent flux splitting. The wild type is
highlighted by a black circle. (d) Growth
optimality in 137 investigated mutants. Lines
indicate equal biomass productivity (g (g glucose
h)-1). The white area indicates improved biomass
productivity in the mutant compared with the wild
type (thick line).
Fischer Sauer, Nature Genetics 37, 636 (2005)
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Conclusions for B. subtilis

• Systematic large-scale flux analysis shows that
  the control architecture of central metabolism is
  designed to provide a rigid flux distribution
  that is largely independent of the rate and yield
  of biomass formation.
• Key factor underlying the evolved robustness of
  metabolic networks to sustain proliferation in
  the face of environmental and genetic
  perturbations.
• Possible design principle
• Maintain B. subtilis in a standby mode that
  allows rapid responses to variations in
  environmental conditions of its natural soil
  habitat.

Fischer Sauer, Nature Genetics 37, 636 (2005)