8. Lecture SS 20005Cell Simulations1 V9: Characterizing the Fluxome of Biological Cells Qualitative methods for fluxome analysis: - Graph theoretical methods

8. Lecture SS 20005

Cell Simulations 1

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.

8. Lecture SS 20005

Cell Simulations 2

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 log–log 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)

8. Lecture SS 20005

Cell Simulations 3

Different opinion

Arita, PNAS 101, 1543 (2004)

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.

8. Lecture SS 20005

Cell Simulations 4

Different opinion

Arita, PNAS 101, 1543 (2004)

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.

8. Lecture SS 20005

Cell Simulations 5

Different opinion

Arita, PNAS 101, 1543 (2004)

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)

8. Lecture SS 20005

Cell Simulations 6

Basis for analysis

Arita, PNAS 101, 1543 (2004)

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.

8. Lecture SS 20005

Cell Simulations 7

Basis for analysis

Arita, PNAS 101, 1543 (2004)

8. Lecture SS 20005

Cell Simulations 8

Results of Arita

Arita, PNAS 101, 1543 (2004)

In Arita‘s 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.

8. Lecture SS 20005

Cell Simulations 9

Review from bioinformatics III: V19 Extreme Pathwaysintroduced 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)

8. Lecture SS 20005

Cell Simulations 10

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.


J. theor. Biol. 203, 229 (2000)

S

I ST

8. Lecture SS 20005

Cell Simulations 11

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:


J. theor. Biol. 203, 229 (2000)

8. Lecture SS 20005

Cell Simulations 12

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

need to be brought to 0.Schilling, Letscher, Palsson,

J. theor. Biol. 203, 229 (2000)

8. Lecture SS 20005

Cell Simulations 13

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)

1 -1 1 0 0 0

1 0 -1 1 0 0

1 0 1 -1 0 0

1 0 0 -1 1 0

1 0 0 1 -1 0

1 0 0 -1 0 1

1 -1 1 0 0 0

T(0) =

T(1) =

+

8. Lecture SS 20005

Cell Simulations 14

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

1 -1 1 0 0 0

1 0 -1 1 0 0

1 0 1 -1 0 0

1 0 0 -1 1 0

1 0 0 1 -1 0

1 0 0 -1 0 1

T(0) =

T(1) =

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

8. Lecture SS 20005

Cell Simulations 15

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

8. Lecture SS 20005

Cell Simulations 16

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

8. Lecture SS 20005

Cell Simulations 17

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) =

1 -1 1 0 0 0

1 1 0 0 0 0 0

1 1 0 -1 0 1 0

1 1 0 -1 0 1 0

1 1 0 1 0 -1 0

1 1 0 0 0 0 0

1 1 0 0 0 -1 1

1 -1 0 0 0 0

1 0 -1 0 0 0

1 0 0 0 -1 0

1 0 0 0 0 -1

8. Lecture SS 20005

Cell Simulations 18


(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

8. Lecture SS 20005

Cell Simulations 19


(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

8. Lecture SS 20005

Cell Simulations 20

pathway matrix

T(final) =

PT =

Schilling et al.

JTB 203, 229

1 -1 1 0 0 0 0 0 0

1 1 0 0 0 0 0 0

1 1 -1 1 0 0 0 0 0 0

1 1 -1 1 0 0 0 0 0 0

1 1 1 -1 0 0 0 0 0 0

1 1 0 0 0 0 0 0

1 1 -1 1 0 0 0 0 0 0

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

v1 v2 v3 v4 v5 v6 b1 b2 b3 b4

p1 p7 p3 p2 p4 p6 p5

8. Lecture SS 20005

Cell Simulations 21

Extreme Pathways for model system

Schilling et al.

JTB 203, 229

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

v1 v2 v3 v4 v5 v6 b1 b2 b3 b4

p1 p7 p3 p2 p4 p6 p5

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.

8. Lecture SS 20005

Cell Simulations 22

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)

PPP TLM

8. Lecture SS 20005

Cell Simulations 23

Papin, Price, Palsson, Genome Res. 12, 1889 (2002)

A symmetric Pathway Length Matrix PLM can be calculated:

where the values along the diagonal correspond to the length of the EPs.

PPP TLM

Properties of pathway matrix

The off-diagonal terms of PLM are the number of reactions that a pair of extreme

pathways have in common.

8. Lecture SS 20005

Cell Simulations 24

Papin, Price, Palsson, Genome Res. 12, 1889 (2002)

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.

TPM PPP

Properties of pathway matrix

8. Lecture SS 20005

Cell Simulations 25

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)

8. Lecture SS 20005

Cell Simulations 26

C13 flux analysis

Fischer & Sauer, Eur J Biochem 270, 880 (2003)

8. Lecture SS 20005

Cell Simulations 27

C13 flux analysis


8. Lecture SS 20005

Cell Simulations 28

Calculation of metabolic flux ratios


8. Lecture SS 20005

Cell Simulations 29

Calculation of metabolic flux ratios


8. Lecture SS 20005

Cell Simulations 30

Effects of knockouts on absolute fluxes and optimality

Fischer & Sauer, Nature Genetics 37, 636 (2005)

8. Lecture SS 20005

Cell Simulations 31

Effects of knockouts on relative fluxes

(a) Relative fluxes through glycolysis

and the TCA cycle to the synthesis

of glyceraldehyde-3-phosphate and

oxaloacetate, respectively, as

obtained from flux ratio analysis.

The complementing fractions are

contributed by the pentose

phosphate pathway and the

anaplerotic reaction.


(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.

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.

8. Lecture SS 20005

Cell Simulations 32

Effects of knockouts on relative fluxes

(a) Relative fluxes through

glycolysis and the TCA cycle to

the synthesis of glyceraldehyde-

3-phosphate 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 theperipheral biosynthetic network,pathway disruptions are mostlylethal; there are only a fewbypass reactions.

More redundancy exists withincentral carbon metabolism respond to food changes.

8. Lecture SS 20005

Cell Simulations 33

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).


8. Lecture SS 20005

Cell Simulations 34

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.


Documents

8. Lecture SS 20005Cell Simulations1 V9: Characterizing the Fluxome of Biological Cells Qualitative methods for fluxome analysis: - Graph theoretical methods