Mathematical Biology:
Metabolic
Network
Analysis
Lecture 2 (13th February 2017):
Modeling Metabolic Networks II
dr. Sander Hille
[email protected]
http://pub.math.leidenuniv.nl/~hillesc
Snellius, Niels Bohrweg 1, room 402
Mathematisch
Instituut
Spring semester 2017
Summary of last week
Mathematisch
Instituut
Modeling of (bio)chemical reaction networks:
0.) Hypergraph representation
(Usual description in biochemistry)
ATP
Glucose
ADP
G6P
F6P
1.) Weighted bi-partite graph representation
R1:
R1
A + B → 2C
A
B
A
R1
B
(2)
Sander Hille
2
C
C
Stoichiometric matrix
Spring semester 2017
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1
Summary of last week
Mathematisch
Instituut
Modeling of (bio)chemical reaction networks:
1.) Weighted bi-partite graph representation
2.) (Directed) substrate graph
Nodes: the chemical compounds involved
Edges: arrow from A to B for each reaction that uses A as substrate and
produces B
A
C
B
One-way, irreversible
(loss of information)
E
D
(loss of information)
2b.) (Undirected) substrate graph
(3)
Sander Hille
Metabolic Nw. An. L2
Spring semester 2017
Summary of last week
Mathematisch
Instituut
Modeling of (bio)chemical reaction networks:
1.) Weighted bi-partite graph representation
2.) (Directed) substrate graph
Nodes: the chemical compounds involved
Edges: arrow from A to B for each reaction that uses A as substrate and
produces B
A
One-way, irreversible
(loss of information)
C
B
E
D
(loss of information)
2b.) (Undirected) substrate graph
(3)
Sander Hille
Spring semester 2017
Metabolic Nw. An. L2
2
Summary of last week
Mathematisch
Instituut
Modeling of (bio)chemical reaction networks:
1.) Weighted bi-partite graph representation
2.) (Undirected / directed) substrate graph
3.) (Directed) Reaction graph
Nodes: the chemical reactions involved
Edges: arrow from R1 to R2 if reaction R2 uses a product of reaction R1 as
substrate
(loss of information)
3b.) (Undirected) Reaction graph
(4)
Spring semester 2017
Sander Hille
Metabolic Nw. An. L2
An additional network model
Mathematisch
Instituut
(from the general Chemical Reaction Network Theory: CRNT)
4.) CRNT network model
Issue: e.g. catalyzed reactions
G6P + E → F6P + E
E: enzyme – catalyst, here phosphoglucose isomerase
R1
G6P
R1
F6P
G6P
F6P
E
E
Stoichiometric matrix
No net change in amount of E !!
(5)
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An additional network model
Mathematisch
Instituut
(from the general Chemical Reaction Network Theory: CRNT)
4.) CRNT network model
Issue: e.g. catalyzed reactions
G6P + E → F6P + E
E: enzyme – catalyst, here phosphoglucose isomerase
R1
G6P
R1
G6P
F6P
F6P
E
E
Stoichiometric matrix
G6P
R1
F6P
Ambiguity !!
(5)
Sander Hille
Spring semester 2017
Metabolic Nw. An. L2
An additional network model
Mathematisch
Instituut
(from the general Chemical Reaction Network Theory: CRNT)
4.) CRNT network model
A chemical reaction network is a triple
1.
of finite sets:
is the set of basic species / molecules
that undergo chemical transitions
2.
is the set of all complexes, i.e. all combinations of
species (together with their multiplicities) that are either substrate
or product of a reaction.
3.
is the set of all reactions, i.e. all
transitions between complexes that are possible.
(6)
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Spring semester 2017
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An additional network model
Mathematisch
Instituut
(from the general Chemical Reaction Network Theory: CRNT)
4.) CRNT network model
A chemical reaction network is a triple
(6)
of finite sets:
1.
is the set of basic species / molecules
that undergo chemical transitions
2.
is the set of all complexes, i.e. all combinations of
species (together with their multiplicities) that are either substrate
or product of a reaction.
3.
is the set of all reactions, i.e. all
transitions between complexes that are possible.
Sander Hille
Spring semester 2017
Metabolic Nw. An. L2
An additional network model
Mathematisch
Instituut
(from the general Chemical Reaction Network Theory: CRNT)
4.) CRNT network model
Nodes: the complexes in
Arrows: the reactions in
G6P + E → F6P + E
G6P
R1
F6P
E
(7)
Sander Hille
Spring semester 2017
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An additional network model
Mathematisch
Instituut
(from the general Chemical Reaction Network Theory: CRNT)
4.) CRNT network model
-- Provides an unambiguous representation
of any chemical reaction network
-- Characteristics of the graph representation relate to possible
dynamics of the reaction network
(‘Feinberg’s deficiency-one theorem’)
-- The graph representation is hard to interpret in terms of chemical
transitions of species, however…
G6P
R1
F6P
E
(8)
Sander Hille
Spring semester 2017
Metabolic Nw. An. L2
An additional network model
Mathematisch
Instituut
(from the general Chemical Reaction Network Theory: CRNT)
4.) CRNT network model
-- Provides an unambiguous representation
of any chemical reaction network
-- Characteristics of the graph representation relate to possible
dynamics of the reaction network
(‘Feinberg’s deficiency-one theorem’)
-- The graph representation is hard to interpret in terms of chemical
transitions of species
There is no need for a CRNT network model if one does
not include the catalysts in a metabolic network model
… so we do not, and model with bi-partite graphs
(8)
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Mathematisch
Instituut
Elementary biochemical
reactions
(9)
Sander Hille
Metabolic Nw. An. L2
Spring semester 2017
Elementary biochemical reactions
Mathematisch
Instituut
-- 1. Isomerases -Three types of elementary biochemical reactions:
1.) Isomeric reactions, (‘subgroup reorganisation’)
Isomer:
chemical compound with the same molecular formula, but different
structural formula
Example: in glycolysis…
ATP
ADP
1
Glucose
2
G6P
3
F6P
Phosphoglucose
isomerase
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Elementary biochemical reactions
Mathematisch
Instituut
-- 1. Isomerases -Three types of elementary biochemical reactions:
1.) Isomeric reactions, (‘subgroup reorganisation’)
G6P
F6P
α-D-glucose-6-phosphate
β-D-fructose-6-phosphate
Same
molecular formula
C6H13O9P
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C6H13O9P
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Elementary biochemical reactions
Mathematisch
Instituut
-- 1. Isomerases -Three types of elementary biochemical reactions:
1.) Isomeric reactions, (‘subgroup reorganisation’)
G6P
F6P
Phosphoglucose
isomerase
(EC 5.3.1.9)
(Enzyme Commission number;
classifies catalyzed reactions)
α-D-glucose-6-phosphate
β-D-fructose-6-phosphate
Isomerase: Enzyme that changes the structural formula of a compound, not
its molecular formula
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Elementary biochemical reactions
Mathematisch
Instituut
-- 1. Isomerases -Three types of elementary biochemical reactions:
1.) Isomeric reactions, (‘subgroup reorganisation’)
Modelling:
R1+
(a)
M1
M2
R1-
(pair of unidirectional (irreversible)
reactions)
Labelling ‘+’/’-’ indicates that the same enzyme
catalyzes both forward (+) and backward (-) reactions
(b)
M1
M2
R1
(introduce new class of reversible reactions,
catalyzed by the same enzyme)
Arrow indicates the choice of positive direction
(‘forward reaction’)
(‘Tripartite graph’)
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Elementary biochemical reactions
Mathematisch
Instituut
-- 1. Isomerases -Three types of elementary biochemical reactions:
1.) Isomeric reactions, (‘subgroup reorganisation’)
Stoichiometric matrix:
R1+ R1-
R1+
(a)
(b)
M1
M1
R1-
R1
M2
M1
M2
R1
M2
M1
M2
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Elementary biochemical reactions
-- 2. Bimolecular association / dissociation --
Mathematisch
Instituut
Three types of elementary biochemical reactions:
2.) Bimolecular association or dissociation, (‘splitting’)
or
M1 + M2 → M3
M1→ M2 + M3
Example: again in glycolysis…
ATP
ADP
DHAP
1
Glucose
2
4
3
G6P
F6P
F1,6BP
GAP
Andolase
… or starch production (polymerisation)
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Elementary biochemical reactions
-- 2. Bimolecular association / dissociation --
Mathematisch
Instituut
Three types of elementary biochemical reactions:
2.) Bimolecular association or dissociation, (‘splitting’)
or
M1 + M2 → M3
M1→ M2 + M3
Modelling:
M1
M2
R1
M3
M1
R2
M2
M3
Stoichiometric matrix:
R1
(16) Sander Hille
R2
M1
M1
M2
M2
M3
M3
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Elementary biochemical reactions
Mathematisch
Instituut
-- 3. Co-factor coupled reactions -Three types of elementary biochemical reactions:
3.) Co-factor coupled reactions, (‘carrier mediated’)
A moiety P is ‘donated’ by a carrier (co-factor) C to an accepting
compound A
A + CP → AP + C
Example: again in glycolysis… -- phosphorylation -ATP
ADP
1
Glucose
2
G6P
3
F6P
Hexokinase
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Spring semester 2017
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Elementary biochemical reactions
Mathematisch
Instituut
-- 3. Co-factor coupled reactions -Three types of elementary biochemical reactions:
3.) Co-factor coupled reactions, (‘carrier mediated’)
A moiety P is ‘donated’ by a carrier (co-factor) C to an accepting
compound A
ATP
A + CP → AP + C
A:
P:
C:
(18) Sander Hille
1
Glucose
Glucose
Phosphate group
The currency metabolite ADP
D-glucose
ADP
G6P
Hexokinase
α-D-glucose-6-phosphate
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Elementary biochemical reactions
Mathematisch
Instituut
-- 3. Co-factor coupled reactions -Three types of elementary biochemical reactions:
3.) Co-factor coupled reactions, (‘carrier mediated’)
A moiety P is ‘donated’ by a carrier (co-factor) C to an accepting
compound A
A + CP → AP + C
Modelling:
(M1 = A; M2 = AP)
(a)
M1
R1
M2
(Carrier/currency metabolites is not explicitly
modelled, e.g. energy balance is not important)
(b)
M1
R1
M2
(Carrier/currency metabolites are explicitly modelled,
e.g. energy balance is taken into account )
C1
C2
(C1 = C; C2 = CP: currency metabolites)
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Elementary biochemical reactions
Mathematisch
Instituut
-- 3. Co-factor coupled reactions -Three types of elementary biochemical reactions:
3.) Co-factor coupled reactions, (‘carrier mediated’)
A moiety P is ‘donated’ by a carrier (co-factor) C to an accepting
compound A
Stoichiometric matrix:
(b)
R1
M1
M2
R1
C1
C2
M1
M2
C1
C2
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Note: further classification
-- types of metabolites / reactions --
Mathematisch
Instituut
So, two types of metabolites may be identified in model:
M1 M2
-- Specific metabolites
-- Currency metabolites (ADP / ATP, NAD+ / NADH)
C1
C2
And two types of reactions:
R1
-- Irreversible reactions
-- Reversible reactions (catalyzed by the same enzyme)
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R1
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Mathematisch
Instituut
Modularisation
and
virtual and physical
compartmentation
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Modularisation
Mathematisch
Instituut
Often one is interested in modeling only part of
an organism’s metabolic network:
-- ‘Reductionist approach’
-- Full network is not accessible
computationally
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Modularisation
Metabolic Nw. An. L2
Mathematisch
Instituut
Often one is interested in modeling only part of
an organism’s metabolic network:
-- ‘Reductionist approach’
-- Full network is not accessible
computationally
-- Part of the network may be sufficient
for studying research question of interest
So one (subjectively) selects a subnetwork:
a module
(simplified E.coli metabolic model)
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Modularisation
Mathematisch
Instituut
-- example: glycolytic pathway --
Orth, Fleming, and Palsson (2010)
‘Reconstruction and Use of Microbial
Metabolic Networks: the Core
Escherichia coli Metabolic Model as
an Educational Guide’
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Modularisation
Mathematisch
Instituut
-- example: glycolytic pathway -Extracellular glucose (nutrient)
Glu
Intracellular glucose-6-phosphate
G6P
Flucose-6-phosphate
F6P
(Flucose-1,6-biphosphate is omitted)
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F1,6BP
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Modularisation
Mathematisch
Instituut
-- example: glycolytic pathway -Extracellular glucose (nutrient)
Glu
Intracellular glucose-6-phosphate
G6P
Flucose-6-phosphate
F6P
F1,6BP
GAP
DHAP
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Modularisation
Mathematisch
Instituut
-- example: glycolytic pathway -M1
F6P
M1
module
R1
F1,6BP
R1’
M2
GAP
DHAP
R2
(Hypergraph representation)
M3
M4
(bipartite graph representation)
Or
M3
M4
(bipartite graph representation
with non-elementary
- modular - reaction )
M3
M1
MR1
M4
‘Modular reaction’
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Modularisation
-- example: glycolytic pathway --
Mathematisch
Instituut
M3
M1
MR1
M4
The function of modular reaction 1 is:
-- ‘to transform one M1 (F6P) into one M3 (DHAP) and one M4 (GAP)’
Glycolytic pathway / glycolysis:
The function of glycolysis is:
Glu
2
‘Glycolysis’
2
ADP
2
2
2
NAD+
NADH
ATP
Pyr
(25) Sander Hille
-- ‘to transform one glucose into
two pyruvate (central metabolite),
together with a net gain of two
ATP (energy) and two NADH (redox power)
(currency metabolites)
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Modularisation
-- example: glycolytic pathway --
Mathematisch
Instituut
M3
M1
MR1
M4
The function of modular reaction 1 is:
-- ‘to transform one M1 (F6P) into one M3 (DHAP) and one M4 (GAP)’
Glycolytic pathway / glycolysis:
Recall ‘catabolism’:
Glu
2
‘Glycolysis’
2
ADP
2
2
2
NAD+
NADH
ATP
(25) Sander Hille
Pyr
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Modularisation
Mathematisch
Instituut
-- a hierarchy of network models -R1
‘Glycolysis’
Glu
2
ADP
2
2
2
NAD+
NADH
ATP
ADP
Glu
R1
G6P
R2
2
F6P
ATP
Pyr
ADP
DHAP
MR1
R3
ATP
GAP
MR2
‘Exergonic
reactions’
4
ADP
(26) Sander Hille
2
2
2
4
NAD+
NADH
ATP
Pyr
Metabolic Nw. An. L2
Spring semester 2017
Modularisation
Mathematisch
Instituut
-- a hierarchy of network models -R1
‘Glycolysis’
Glu
2
ADP
ADP
Glu
R1
ATP
G6P
R2
F6P
2
2
2
2
NAD+
NADH
ATP
Pyr
ADP
DHAP
MR1
R3
ATP
GAP
MR2
‘Exergonic
reactions’
4
ADP
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2
2
2
4
NAD+
NADH
ATP
Pyr
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Modularisation
Mathematisch
Instituut
-- a hierarchy of network models -R1
‘Glycolysis’
Glu
2
ADP
2
2
2
NAD+
NADH
ATP
ADP
Glu
R1
G6P
R2
2
F6P
ATP
Pyr
ADP
DHAP
MR1
R3
ATP
GAP
MR2
‘Exergonic
reactions’
4
ADP
(26) Sander Hille
2
2
2
4
NAD+
NADH
ATP
Pyr
Metabolic Nw. An. L2
Spring semester 2017
Modularisation
Mathematisch
Instituut
-- a hierarchy of network models -R1
‘Glycolysis’
Glu
2
ADP
ADP
Glu
R1
ATP
G6P
R2
F6P
2
2
2
2
NAD+
NADH
ATP
ADP
DHAP
MR1
R3
ATP
GAP
Note: currency metabolites are often
copied – to avoid graphs with
complex connections…
(27) Sander Hille
Pyr
MR2
‘Exergonic
reactions’
4
ADP
Spring semester 2017
2
2
2
4
NAD+
NADH
ATP
Pyr
Metabolic Nw. An. L2
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Modularisation
Mathematisch
Instituut
-- a hierarchy of network models -R1
‘Glycolysis’
Glu
2
ADP
ADP
Glu
G6P
R1
ATP
R2
F6P
2
2
2
2
NAD+
NADH
ATP
Pyr
ADP
DHAP
MR1
R3
ATP
GAP
MR2
‘Exergonic
reactions’
4
Note: only net inputs and outputs
are presented in higher level module
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2
ADP
Spring semester 2017
2
Pyr
4
2
2
2
NAD+
NADH
ATP
Metabolic Nw. An. L2
Mathematisch
Instituut
Exchange reactions
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Exchange reactions
Mathematisch
Instituut
-- transport over membranes --
Cells are enclosed by the cell membrane and
cell wall.
Metabolites (nutrients) must cross this barrier.
Cells may have many internal compartments
and membranes too…
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Compartmentation
-- Nature’s diversity --
Eukaryotes
Prokaryotes
Compartmentalized
cells with (many)
organelles,
e.g. nucleus
Unicellular;
No internal
compartments in
cells
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Mathematisch
Instituut
Spring semester 2017
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21
Compartmentation
Mathematisch
Instituut
-- Nature’s diversity --
Eukaryotes
Prokaryotes
Compartmentalized
cells with (many)
organelles,
e.g. nucleus
Unicellular;
No internal
compartments in
cells
Source: Introduction to the Archaea (http://www.ucmp.berkeley.edu/archaea/archaea.html)
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Compartmentation
Mathematisch
Instituut
-- Nature’s diversity -E. coli
‘Tree of life’
Source: Introduction to the Archaea (http://www.ucmp.berkeley.edu/archaea/archaea.html)
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A history of life
Mathematisch
Instituut
(and the solar system)
(time measured in years before this day…)
Formation of the Sun
First life forms on Earth
4.57 bio
0.5 bio
3.5 bio
2
3
4.54 bio
End of life
supportability
Cambrian
explosion of life
NOW
Formation of the Earth
Homo sapiens
First dinosaurs
First bacteria
First multicellular
organism
3.5 bio
0.5 bio
1
First fungi
2 bio
1.5 bio
First plants
0.2 mio
230 mio
0.5 bio
1
3
65 mio
(1 bio = 1 billion = 109; 1 mio = 1 million = 106)
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Extinction of dinosaurs
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Exchange reaction
Mathematisch
Instituut
-- example: glycolytic pathway -Glu
Extracellular glucose (nutrient)
Glucose uptake
Intracellular glucose-6-phosphate
G6P
Flucose-6-phosphate
F6P
Nutrients have to cross the
cell membrane !!
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Cell membranes
Mathematisch
Instituut
Consist of a phospholipid bilayer
with various proteins
Fluid-like structure!
Amoeba (slime mould)
(Dictyostelium discoideum)
Membrane lipids
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(Borislav Mitev)
Metabolic Nw. An. L2
Cell membranes
Mathematisch
Instituut
water
‘Head’: hydrophilic
‘Tails’: hydrophobic
Membrane lipid(s) (Borislav Mitev)
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Cell membranes
Mathematisch
Instituut
water
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Diversity in bacteria
Mathematisch
Instituut
-- different membrane structures -Gram-negative and Gram-positive bacteria
Examples:
Bacillus anthraxis
Bacillus subtilis
Listeria …
Cell membrane
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Cell wall
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Diversity in bacteria
Mathematisch
Instituut
-- different membrane structures -Gram-negative and Gram-positive bacteria
Examples:
Bacillus anthraxis
Examples:
Escherichia coli
Bacillus subtilis
Salmonella …
Legionella …
Listeria …
Periplasmic
space
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Plasma
membrane
Cell membrane
Spring semester 2017
Cell wall
Metabolic Nw. An. L2
A real-life example
Mathematisch
Instituut
-- Glucose uptake by E. coli --
Glucose dissolves in plasma membrane and can enter the
cell through diffusion (and leak away…)
There is also an active uptake mechanism, directly
converting glucose into G6P
D-glucose
α-D-glucose-6-phosphate (G6P)
The anion (negative ion) G6P can hardly diffuse through the
charged plasma membrane.
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A real-life example
Mathematisch
Instituut
-- Glucose uptake by E. coli -Cartoon of the active uptake mechanism:
EnzIIb
G6P
Glu
G6P
EnzIIc
G6P
Pyr
Environment
EnzIIb
Transmembrane
transport protein
EnzIIa
HPr
EnzI
P
P
P
P
PEP
G6P
Phosphorelay (signalling pathway)
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A real-life example
Mathematisch
Instituut
-- Glucose uptake by E. coli -Metabolic network model
for glucose uptake in E. coli:
ATP
Ge
Gi
ER1
ADP
R1
Currency metabolite
G6P
EnzIIb
ER2
Environment
EnzIIbP
Cofactor
Intracellular
Enzymes !!
ER1: Diffusive uptake of glucose through cell membrane (‘uptake is positive’)
(ER = ‘Exchange reaction’ – exchange between compartments)
ER2: Active glucose uptake through cell membrane through EnzIIb
R1:
Conversion catalyzed by hexokinase
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A real-life example
Mathematisch
Instituut
-- Glucose uptake by E. coli -Metabolic network model
for glucose uptake in E. coli
(Corrected…)
Ge
ATP
Gi
ER1
ADP
R1
Currency metabolite
G6P
Pyr
ER2
Intracellular
PEP
Environment
Cofactor
Signaling molecules / enzymes have been removed
ER1: Diffusive uptake of glucose through cell membrane (‘uptake is positive’)
(ER = ‘Exchange reaction’ – exchange between compartments)
ER2: Active glucose uptake through cell membrane through EnzIIb
R1:
Conversion catalyzed by hexokinase
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A real-life example
Mathematisch
Instituut
-- Glucose uptake by E. coli -Detailed network graph:
ATP
Ge
Gi
ER1
ADP
R1
G6P
Pyr
ER2
PEP
Substrate graph:
ADP
Ge
ATP
Gi
ADP
PEP
Ge
G6P
Gi
(41) Sander Hille
G6P
Pyr
Pyr
(directed)
PEP
ATP
(undirected)
Spring semester 2017
Metabolic Nw. An. L2
28
A real-life example
Mathematisch
Instituut
-- Glucose uptake by E. coli -Detailed network graph:
ATP
Ge
Gi
ER1
ADP
R1
G6P
Pyr
ER2
PEP
Reaction graph:
R1
ER1
ER2
(R1 uses a product of ER1 (namely Gi); ER2 does not use any products of the other
reactions, nor does ER1 or R1 from ER2)
(42) Sander Hille
Metabolic Nw. An. L2
Spring semester 2017
Summary
Mathematisch
Instituut
-- types of reactions -Types of reactions:
-- Irreversible or reversible reactions
R1
R1
-- Internal or exchange reactions
R1
R1
ER2
ER1
Exchange reactions transport (and possibly
transform) metabolites over membranes that
separate physical compartments
… or are internal reactions that are part of
the virtual boundary of a module
(= virtual compartment)
Glucose uptake
ADP
DHAP
F6P
R3
F1,6BP
R4
GAP
ATP
Virtual exchange reactions
(43) Sander Hille
Spring semester 2017
Metabolic Nw. An. L2
29