Lecture 4
•Reaction system as ordinary differential
equations
•Reaction system as stochastic process
•Metabolic network and stoichiometric matrix
•Graph spectral analysis/Graph spectral
clustering and its application to metabolic
networks
Introduction
Metabolism is the process through which living cells
acquire energy and building material for cell components
and replenishing enzymes.
Metabolism is the general term for two kinds of
reactions: (1) catabolic reactions –break down of
complex compounds to get energy and building blocks,
(2) anabolic reactions—construction of complex
compounds used in cellular functioning
How can we model metabolic reactions?
What is a Model?
Formal representation of a system using
--Mathematics
--Computer program
Describes mechanisms underlying
outputs
Dynamical models show rate of changes
with time or other variable
Provides explanations and predictions
Typical network of metabolic pathways
Reactions are
catalyzed by
enzymes. One
enzyme molecule
usually catalyzes
thousands reactions
per second (~102107)
The pathway map
may be considered
as a static model of
metabolism
Dynamic modeling of metabolic reactions is the
process of understanding the reaction rates i.e.
how the concentrations of metabolites change
with respect to time
An Anatomy of Dynamical Models
Stochastic
Deterministic
Discrete
Time
--No Space --
-- Space --
--No Space --
-- Space --
Discrete
Variables
Finite State
Machines
Boolean
Networks;
Cellular
Automata
Discrete
Time
Markov
Chains
Continuous
Variables
Iterated
Functions;
Difference
Equations
Iterated
Functions;
Difference
Equations
Discrete Time
Markov
Chains
Stochastic
Boolean
Networks;
Stochastic
Cellular
Automata
Coupled
Discrete Time
Markov
Chains
Boolean
Differential
Equations
Coupled
Boolean
Differential
Equations
Continuous
Time Markov
Chain
Coupled
Continuous
Time Markov
Chains
Ordinary
Differential
Equations
Partial
Differential
Equations
Stochastic
Ordinary
Differential
Equations
Stochastic
Partial
Differential
Equations
Continuous Discrete
Time
Variables
Continuous
Variables
Differential equations
Differential equations are based on the rate of
change of one or more variables with respect to
one or more other variables
An example of a differential equation
Source: Systems biology in practice by E. klipp et al
An example of a differential equation
Source: Systems biology in practice by E. klipp et al
An example of a differential equation
Source: Systems biology in practice by E. klipp et al
Schematic representation of the upper part of the Glycolysis
Source: Systems biology in practice by E. klipp et al.
ODEs representing a reaction system
The ODEs representing
this reaction system
Realize that the
concentration of metabolites
and reaction rates v1, v2, ……
are functions of time
The rate equations can be solved as follows using a number of constant
parameters
The temporal evaluation of the concentrations using the
following parameter values and initial concentrations
Notice that because of bidirectional reactions Gluc-6-P and Fruc-6-P reaches peak
earlier and then decrease slowly and because of unidirectional reaction Fruc1,6-P2
continues to grow for longer time.
The use of differential equations assumes that the
concentration of metabolites can attain continuous value.
But the underlying biological objects , the molecules are
discrete in nature.
When the number of molecules is too high the above
assumption is valid.
But if the number of molecules are of the order of a few
dozens or hundreds then discreteness should be
considered.
Again random fluctuations are not part of differential
equations but it may happen for a system of few
molecules.
The solution to both these limitations is to use a stochastic
simulation approach.
Stochastic Simulation
Stochastic modeling for systems biology
Darren J. Wilkinson
2006
Molecular systems in cell
Molecular systems in cell
[m1(out)]
[ ]: concentration of ith object
[m1(in)]
[m2]
[m3]
[m5]
[p1]
[p4]
[p2]
[r1]
[r2]
[m4]
[p3]
[r3]
[r４]
Molecular systems in cell
[m1(out)]
cj: cj’: efficiency of jth process
c1
c3
[m1(in)]
c2
[m2]
c4
[m3]
[p1]
[p2]
c6
[r1]
c7
c8
[r2]
c9
[m5]
[p4]
c5
[m4]
c12
[p3] c10
[r４]
[r3]
c11
c13
[m1(out)] Molecular systems for small molecules in cell
c1
[m1(in)]
h1=c1 [m1(out)]
h3=c3 [m2]
c3  p2 ,r2
h5=c4 [m3]
c4  p4 ,r4
c3
c2
h2=c2 [m1(in)]
c2  p1 ,r1
[m2]
c4
[m3]
c5
[m4]
h4=c5 [m2]
c5  p3 ,r3
Stochastic selection of reaction based on(h1, h2, h3, h4, h5)
[m5]
[m1(out)]=100 Molecular systems for small molecules in cell
c1
[m1(in)]
h1=c1 [m1(out)]
= 100 c1
h3=c3 [m2]
c5  p2 ,r2
h5=c4 [m3]
c4  p4 ,r4
c3
c2
h2=c2 [m1(in)]
c2  p1 ,r1
[m2]
c4
[m3]
c5
[m4]
h4=c5 [m2]
c5  p3 ,r3
Stochastic selection of reaction based on(100 c1, h2, h3, h4, h5)
Reaction 1
[m5]
[m1(out)]=99
Molecular systems for small molecules in cell
c1
[m1(in)]=1
h1=c1 [m1(out)]
= 99 c1
h3=c3 [m2]
=0
c3
c2
[m2]=0
h2=c2 [m1(in)]
= 1 c2
c4
[m3]=0
c5
h5=c4 [m3]
=0
[m4]=0
h4=c5 [m2]
=0
Stochastic selection of Reaction based on (99 c1, 1 c2, 0, 0, 0)
 Reaction 1
[m5]=0
[m1(out)]=98
Molecular systems for small molecules in cell
c1
[m1(in)]=2
h1=c1 [m1(out)]
= 98 c1
h3=c3 [m2]
=0
c3
c2
[m2]=0
h2=c2 [m1(in)]
= 2 c2
c4
[m3]=0
c5
h5=c4 [m3]
=0
[m4]=0
h4=c5 [m2]
=0
Stochastic selection of Reaction based on (98 c1, 2 c2, 0, 0, 0)
 Reaction 1
[m5]=0
[m1(out)]=97
Molecular systems for small molecules in cell
c1
[m1(in)]=3
h1=c1 [m1(out)]
= 97 c1
h3=c3 [m2]
=0
c3
c2
[m2]=0
h2=c2 [m1(in)]
= 3 c2
c4
[m3]=0
c5
h5=c4 [m3]
=0
[m4]=0
h4=c5 [m2]
=0
Stochastic selection of Reaction based on (97 c1, 3 c2, 0, 0, 0)
 Reaction 2
[m5]=0
[m1(out)]=97
Molecular systems for small molecules in cell
c1
[m1(in)]=2
h1=c1 [m1(out)]
= 97 c1
h3=c3 [m2]
=1 c3
c3
c2
[m2]=1
h2=c2 [m1(in)]
= 2 c2
[m3]=0
c5
h5=c4 [m3]
=0
c4
[m4]=0
h4=c5 [m2]
=1 c5
Stochastic selection of Reaction based on (97 c1, c2, 1 c3, 0, 1 c5)
 Reaction 1
[m5]=0
[m1(out)]=96
Molecular systems for small molecules in cell
c1
h3=c3 [m2]
=1 c3
c3
[m1(in)]=3
h1=c1 [m1(out)]
= 97 c1
c2
[m2]=1
h2=c2 [m1(in)]
= 3 c2
[m3]=0
c5
h5=c4 [m3]
=0
c4
[m4]=0
h4=c5 [m2]
=1 c5
Stochastic selection of Reaction(96 c1, 3 c2, 1 c3, 0, 1 c5)
Reaction 3
[m5]=0
[m1(out)]=96
Molecular systems for small molecules in cell
c1
h3=c3 [m2]
=0
c3
[m1(in)]=3
h1=c1 [m1(out)]
= 97 c1
c2
[m2]=0
h2=c2 [m1(in)]
= 3 c2
[m3]=1
c5
h5=c4 [m3]
=1 c4
c4
[m4]=0
h4=c5 [m2]
=0
Stochastic selection of Reaction based on (96 c1, 3 c2, 0, 1 c4 , 0)
…
[m5]=0
[m1(out)]
Input data
c1
c3
[m1(in)]
c2
[m2]
c4
[m3]
c5
m1(out)
c1
m1(in)
c
m1(in) 2
m2
Reaction parameters and Reactions
Initial concentrations [m1(out)]
[m1(in)]
c3
m2
m2
c5
[m2]
[m5]
[m4]
m3
m3
c4
m5
m5
[m3] [m4]
[m5]
Gillespie Algorithm
Step 0: System Definition
objects (i = 1, 2,…, n) and their initial quantities: Xi(init)
reaction equations (j=1,2,…,m)
Rj: m(Pre)j1 X1 + ...+ m(Pre)jn Xn = m (Post) j1 X1 +...+ m (Post) jnXn
reaction intensities: ci for Rj
Step 1: [Xi]Xi(init)
Step 2: hj: :probability of occurrence of reactions
based on cj (j=1,2,..,m) and [Xi] (i=1,2,..,n)
Step 3: Random selection of reaction
Here a selected reaction is represented by index s.
Step 4: Quantities for individual objects are revised base on selected reaction
equation
[Xi] ← [Xi] – m (Pre)s + m(Post)s
Gillespie Algorithm (minor revision)
Step 0: System Definition
objects (i = 1, 2,…, n) and their initial quantities Xi(init)
reaction equations (j=1,2,…,m)
Rj: m(Pre)j1 X1 + ...+ m(Pre)jn Xn = m (Post) j1 X1 +...+ m (Post) jnXn
reaction intensities: ci for Rj
Step 1: [Xi]Xi(init)
Step 2: hj: :probability of occurrence of reactions
based on cj (j=1,2,..,m) and [Xi] (i=1,2,..,n)
Step 3: Random selection of reaction
Here a selected reaction is represented by index s.
Step 4: Quantities for individual objects are revised base on selected reaction
equation X’i = [Xi] – m (Pre)s + m(Post)s
X’i  0
Yes
Step 5: [Xj] X’i
No
X’i  Ximax
Yes
No
Software: Simple Stochastic Simulator
1.Create stoichiometric data file and initial condition file
Reaction Definition: REQ**.txt
R1
R2
[X1] = [X2]
[X2] = [X1]
Objects used are assigned by [ ] .
Stoichiometetric data and ci: REACTION**.txt
Reaction Parameter
R1
R2
ci
1
1
[X1]
1
0
[X1]
0
1
[X2]
0
1
ci is set by user
Initial condition: INIT**.txt
[X1]
[X2]
100
100
Initial quantitiy
0
0
max number (for ith object, max number is set by 0 for ith , [Xi]0
http://kanaya.naist.jp/Lecture/systemsbiology_2010
[X2]
1
0
Software: Simple Stochastic Simulator
2. Stochastic simulation
Stoichiometetric data and ci: REACTION**.txt
Initial condition: INIT**.txt
Simulation results: SIM**.txt
Reaction Parameter
c:
1.0
1.0
//
time
0.00
0.0015706073545097992
0.015704610011372147 100.0
0.01670413203960951 101.0
….
….
150
[X1]
100.0
101.0
100.0
99.0
[X2]
100.0
99.0
100
[X1]
[X2]
50
0
0
10
20
30
40
50
Example of simulation results
# of type of chemicals =2
[X1][X2] c=1, [X1]=1000, [X2]=0
1000
900
800
700
600
[X1]
[X2]
500
400
300
200
100
0
0
2
4
6
8
[X1][X2] [X2][X1]
c1=c2=1
[X1]=1000
1000
900
800
700
600
[X1]
[X2]
500
400
300
200
100
0
0
1
2
3
4
5
6
7
8
9
10
# of type of chemicals =3
[X1][X2][X3], [X1]=1000, c=1
1000
900
800
700
600
500
400
300
200
100
0
[X1]
[X2]
[X3]
0
2
4
6
8
10
[X1] [X2][X3], [X1]=1000, c=1
1000
900
800
700
600
[X1]
[X2]
[X3]
500
400
300
200
100
0
0
5
10
15
20
[X1][X2][X3], [X1]=1000, c=1
1000
900
800
700
600
500
400
300
[X1]
[X2]
[X3]
200
100
0
0
2
4
6
8
10
[X1][X2][X3],[X1]=1000, c=1
1000
900
800
700
600
[X1]
[X2]
[X3]
500
400
300
200
100
0
0
2
4
6
8
loop reaction [X1][X2][X3][X1], [X1]=1000,
c=1
1000
900
800
700
600
[X1]
[X2]
[X3]
500
400
300
200
100
0
0
2
4
6
8
10
Representation of Reaction
Data Set
Reaction Data
[X1]
[X1] + [X2]
[X2]
Initial Condition
c1
c2
c3
2[X1]
[X1]= X1(init)
[X2]= X2(init)
2[X2]
Φ
Example 2 EMP
glcK
glcK
pgi
pgi
pgi
pgi
pfk
fbp
fbaA
fbaA
tpiA
tpiA
gapA
gapB
pgk
pgk
pgm
pgm
eno
eno
ATP + [D-glucose] -> ADP + [D-glucose-6-phosphate]
ATP + [alpha-D-glucose] -> ADP + [D-glucose-6-phosphate]
[D-glucose-6-phosphate] <-> [D-fructose-6-phosphate]
[D-fructose-6-phosphate] <-> [D-glucose-6-phosphate]
[alpha-D-glucose-6-phosphate] <-> [D-fructose-6-phosphate]
[D-fructose-6-phosphate] <-> [alpha-D-glucose-6-phosphate]
ATP + [D-fructose-6-phosphate] -> ADP + [D-fructose-1,6-bisphosphate]
[D-fructose-1,6-bisphosphate] + H(2)O -> [D-fructose-6-phosphate] + phosphate
[D-fructose-1,6-bisphosphate] <-> [glycerone-phosphate] + [D-glyceraldehyde-3-phosphate]
[glycerone-phosphate] + [D-glyceraldehyde-3-phosphate] <-> [D-fructose-1,6-bisphosphate]
[glycerone-phosphate] <-> [D-glyceraldehyde-3-phosphate]
[D-glyceraldehyde-3-phosphate] <-> [glycerone-phosphate]
[D-glyceraldehyde-3-phosphate] + phosphate + NAD(+) -> [1,3-biphosphoglycerate] + NADH + H(+)
[1,3-biphosphoglycerate] + NADPH + H(+) -> [D-glyceraldehyde-3-phosphate] + NADP(+) + phosphate
ADP + [1,3-biphosphoglycerate] <-> ATP + [3-phospho-D-glycerate]
ATP + [3-phospho-D-glycerate] <-> ADP + [1,3-biphosphoglycerate]
[3-phospho-D-glycerate] <-> [2-phospho-D-glycerate]
[2-phospho-D-glycerate] <-> [3-phospho-D-glycerate]
[2-phospho-D-glycerate] <-> [phosphoenolpyruvate] + H(2)O
[phosphoenolpyruvate] + H(2)O <-> [2-phospho-D-glycerate]
Example 2 EMP
D-glucose
alpha-D-glucose
D-glucose-6-phosphate
alpha-D-glucose-6-phosphate
D-fructose-6-phosphate
[D-fructose-1,6-bisphosphate]
[glycerone-phosphate]
[D-glyceraldehyde-3-phosphate]
[1,3-biphosphoglycerate]
[3-phospho-D-glycerate]
[2-phospho-D-glycerate]
[phosphoenolpyruvate]
Metabolic network and stoichiometric matrix
Typical network of metabolic pathways
Reactions are
catalyzed by
enzymes. One
enzyme molecule
usually catalyzes
thousands reactions
per second (~102107)
The pathway map
may be considered
as a static model of
metabolism
What is a stoichiometric matrix?
For a metabolic network consisting of m substances
and r reactions the system dynamics is described by
systems equations.
The stoichiometric coefficients nij assigned to the
substance Si and the reaction vj can be combined
into the so called stoichiometric matrix.
Example reaction system and corresponding stoichiometric matrix
There are 6 metabolites and 8 reactions in this example system
stoichiometric matrix
Binary form of N
To determine the elementary topological properties,
Stiochiometric matrix is also represented as a binary
form using the following transformation
nij’=0 if nij =0
nij’=1 if nij ≠0
Stiochiometric matrix is a sparse matrix
Source: Systems biology by
Bernhard O. Palsson
Information contained in the stiochiometric matrix
Stiochiometric matrix contains many information e.g.
about the structure of metabolic network , possible set
of steady state fluxes, unbranched reaction pathways
etc.
2 simple information:
•The number of non-zero entries in column i gives the
number of compounds that participate in reaction i.
•The number of non-zero entries in row j gives the
number of reactions in which metabolite j participates.
So from the stoicheometric matrix,
connectivities of all the metabolites can be
computed
Information contained in the stiochiometric matrix
Source: Systems
biology by
Bernhard O.
Palsson
There are relatively few metabolites (24 or so) that are
highly connected while most of the metabolites
participates in only a few reactions
Information contained in the stiochiometric matrix
In steady state we know that
The right equality sign denotes a linear equation system
for determining the rates v
This equation has non trivial solution only for Rank N <
r(the number of reactions)
K is called kernel matrix if it satisfies NK=0
The kernel matrix K is not unique
Information contained in the stiochiometric matrix
The kernel matrix K of the stoichiometric
matrix N that satisfies NK=0, contains (rRank N) basis vectors as columns
Every possible set of steady state fluxes can
be expressed as a linear combination of the
columns of K
Information contained in the stiochiometric matrix
-
And for steady state flux it holds that J = α1 .k1 + α2.k2
With α1= 1 and α2 = 1,
v3 =-1
, i.e. at steady state v1 =2, v2 =-1 and
That is v2 and v3 must be in opposite direction of v1 for the steady
state corresponding to this kernel matrix which can be easily
realized.
Information contained in the stiochiometric matrix
Reaction System
Stoicheometric Matrix
The stoicheomatric matrix comprises r=8 reactions and Rank =5
and thus the kernel matrix has 3 linearly independent columns. A
possible solution is as follows:
Information contained in the stiochiometric matrix
Reaction System
The entries in the last row of the kernel matrix is always zero.
Hence in steady state the rate of reaction v8 must vanish.
Information contained in the stiochiometric matrix
If all basis vectors contain the same entries for a set of
rows, this indicate an unbranched reaction path
Reaction System
The entries for v3 , v4 and v5 are equal for each column of the
kernel matrix, therefore reaction v3 , v4 and v5 constitute an
unbranched pathway . In steady state they must have equal rates
Elementary flux modes and extreme pathways
The definition of the term pathway in a metabolic
network is not straightforward.
A descriptive definition of a pathway is a set of
subsequent reactions that are in each case linked by
common metabolites
Fluxmodes are possible direct routes from one
external metabolite to another external metabolite.
A flux mode is an elementary flux mode if it uses a
minimal set of reactions and cannot be further
decomposed.
Elementary flux modes and extreme pathways
Elementary flux modes and extreme pathways
Extreme pathway is a concept similar to elementary flux mode
The extreme pathways are a subset of elementary flux modes
The difference between the two definitions is the
representation of exchange fluxes. If the exchange fluxes are all
irreversible the extreme pathways and elementary modes are
equivalent
If the exchange fluxes are all reversible there are more
elementary flux modes than extreme pathways
One study reported that in human blood cell there are 55
extreme pathways but 6180 elementary flux modes
Elementary flux modes and extreme pathways
Source:
Systems
biology by
Bernhard O
Palsson
Elementary flux modes and extreme pathways
Elementary flux modes and extreme pathways
can be used to understand the range of
metabolic pathways in a network, to test a set
of enzymes for production of a desired product
and to detect non redundant pathways, to
reconstruct metabolism from annotated
genome sequences and analyze the effect of
enzyme deficiency, to reduce drug effects and to
identify drug targets etc.
Lecture7
Topic1: Graph spectral analysis/Graph spectral
clustering and its application to metabolic networks
Topic 2: Concept of Line Graphs
Topic 3: Introduction to Cytoscape
Graph spectral analysis/
Graph spectral clustering
PROTEIN STRUCTURE: INSIGHTS FROM GRAPH
THEORY
by
SARASWATHI VISHVESHWARA, K. V. BRINDA and N. KANNANy
Molecular Biophysics Unit, Indian Institute of Science
Bangalore 560012, India
Adjacency Matrix
Laplacian matrix L=D-A
Degree Matrix
Eigenvalues and eigenvectors
Eigenvalues of a matrix A are the roots of the following equation
|A-λI|=0, where I is an identity matrix
Let λ is an eigenvalue of A and x is a vector such that
-----(1)
N×N N×1
N×1
then x is an eigenvector of A corresponding to λ .
Node 1 has 3 edges, nodes 2, 3 and 4 have 2 edges each and node
5 has only one edge. The magnitude of the vector components of
the largest eigenvalue of the Adjacency matrix reflects this
observation.
Node 1 has 3 edges, nodes 2, 3 and 4 have 2 edges each and node
5 has only one edge. Also the magnitude of the vector
components of the largest eigenvalue of the Laplacian matrix
reflects this observation.
The largest eigenvalue (lev) depends upon the highest degree in the
graph.
For any k regular graph G (a graph with k degree on all the vertices), the
eigenvalue with the largest absolute value is k.
A corollary to this theorem is that the lev of a clique of n vertices
is n − 1.
In a general connected graph, the lev is always less than or equal to (≤ )
to the largest degree in the graph.
In a graph with n vertices, the absolute value of lev decreases
as the degree of vertices decreases.
 The lev of a clique with 11 vertices is 10 and that of a linear
chain with 11 vertices is 1.932
a linear chain with 11 vertices
In graphs 5(a)-5(e), the highest degree is 6. In graphs 5(f)-5(i), the highest degree is 5, 4, 3
and 2 respectively.
It can be noticed that the lev is generally higher if the graph contains vertices of high degree.
The lev decreases gradually from the graph with highest degree 6 to the one with highest
degree 2. In case of graphs 5(a)-5(e), where there is one common vertex with degree 6
(highest degree) and the degrees of the other vertices are different (less than 6 in all cases),
the lev differs i.e. the lev also depends on the degree of the vertices adjoining the highest
degree vertex.
This paper combines graph 4(a) and graph 4(b) and constructs a Laplacian
matrix with edge weights (1/dij ), where dij is the distance between vertices
i and j. The distances between the vertices of graph 4(a) and graph 4(b) are
considered to be very large (say 100) and thus the matrix elements
corresponding to a vertex from graph 4(a) and the other from graph 4(b) is
considered to have a very small value of 0.01. The Laplacian matrix of 8
vertices thus considered is diagonalized and their eigenvalues and
corresponding vector components are given in Table 3.
The vector components corresponding to the
second smallest eigenvalue contains the desired
information about clustering, where the cluster
forming residues have identical values. In Fig. 4,
nodes 1-5 form a cluster (cluster 1) and 6-8 form
another cluster (cluster 2).
Metabolome Based Reaction Graphs of M. tuberculosis
and M. leprae: A Comparative Network Analysis
by
Ketki D. Verkhedkar1, Karthik Raman2, Nagasuma R. Chandra2, Saraswathi Vishveshwara1*
1 Molecular Biophysics Unit, Indian Institute of Science, Bangalore, India, 2 Bioinformatics
Centre, Supercomputer Education and Research Centre, Indian Institute of Science,
Bangalore, India
PLoS ONE | www.plosone.org
September 2007 | Issue 9 | e881
Construction of network
Stoichrometric matrix
Following this method the networks of
metabolic reactions corresponding to 3
organisms were constructed
R1
R2
R3
R4
Analysis of network parameters
Giant component of the reaction network of e.coli
Giant components of the reaction networks of M. tuberculosis and
M. leprae
Analyses of sub-clusters in the giant component
Graph spectral analysis was performed to detect sub-clusters of
reactions in the giant component.
To obtain the eigenvalue spectra of the graph, the adjacency
matrix of the graph is converted to a Laplacian matrix (L), by the
equation:
L=D-A
where D, the degree matrix of the graph, is a diagonal matrix in
which the ith element on the diagonal is equal to the number of
connections that the ith node makes in the graph.
It is observed that reactions belonging to fatty acid biosynthesis
and the FAS-II cycle of the mycolic acid pathway in M.
tuberculosis form distinct, tightly connected sub-clusters.
Identification of hubs in the reaction networks
In biological networks, the hubs are thought to be
functionally important and phylogenetically oldest.
The largest vector component of the highest eigenvalue
of the Laplacian matrix of the graph corresponds to the
node with high degree as well as low eccentricity. Two
parameters, degree and eccentricity, are involved in the
identification of graph spectral (GS) hubs.
Identification of hubs in the reaction networks
Alternatively, hubs can be ranked based on their connectivity
alone (degree hubs).
It was observed that the top 50 degree hubs in the reaction
networks of the three organisms comprised reactions involving
the metabolite L-glutamate as well as reactions involving
pyruvate. However, the top 50 GS hubs of M. tuberculosis and M.
leprae exclusively comprised reactions involving L-glutamate
while the top GS hubs in E. coli only consisted of reactions
involving pyruvate.
The difference in the degree and GS hubs suggests that the most
highly connected reactions are not necessarily the most central
reactions in the metabolome of the organism