LECTURE 6
Topic 1: Metabolic network and stoichiometric
matrix
Topic 2: Centrality measures of nodes
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.
Centrality measures of nodes
Centrality measures
Within graph theory and network analysis, there are
various measures of the centrality of a vertex within a
graph that determine the relative importance of a
vertex within the graph.
We will discuss on the following centrality measures:
•Degree centrality
•Betweenness centrality
•Closeness centrality
•Eigenvector centrality
•Subgraph centrality
Degree centrality
Degree centrality is defined as the number of links incident
upon a node i.e. the number of degree of the node
Degree centrality is often interpreted in terms of the
immediate risk of the node for catching whatever is flowing
through the network (such as a virus, or some information).
Degree centrality of the
blue nodes are higher
Betweenness centrality
The vertex betweenness centrality BC(v) of a vertex v is
defined as follows:
Here σuw is the total number of shortest paths between
node u and w and σuw(v) is number of shortest paths
between node u and w that pass node v
Vertices that occur on many shortest paths between other
vertices have higher betweenness than those that do not.
Betweenness centrality
σuw
a
c
d
b
f
e
Betweenness centrality of
node c=6
Betweenness centrality of
node a=0
σuw(v)
σuw/σuw(v)
(a,b) 1
0
0
(a,d) 1
1
1
(a,e) 1
1
1
(a,f)
1
1
1
(b,d)
1
1
1
(b,e) 1
1
1
(b,f) 1
1
1
(d,e) 1
0
0
(d,f)
1
0
0
(e,f)
1
0
0
Calculation for node c
Betweenness centrality
•Nodes of high
betweenness centrality
are important for
transport.
•If they are blocked,
transport becomes less
efficient and on the
other hand if their
capacity is improved
transport becomes
more efficient.
•Using a similar
concept edge
betweenness is
calculated.
Hue (from red=0 to blue=max)
shows the node betweenness.
http://en.wikipedia.org/wiki/Between
ness_centrality#betweenness
Closeness centrality
The farness of a vortex is the sum of the shortest-path
distance from the vertex to any other vertex in the graph.
The reciprocal of farness is the closeness centrality (CC).
CC ( v ) 
1
 d (v, t )
t V \ v
Here, d(v,t) is the shortest distance between vertex v and
vertex t
Closeness centrality can be viewed as the efficiency of a
vertex in spreading information to all other vertices
Eigenvector centrality
Let A is the adjacency matrix of a graph and λ is the largest
eigenvalue of A and x is the corresponding eigenvector then
-----(1)
N×N N×1
|A-λI|=0, where I is an
identity matrix
N×1
The ith component of the eigenvector x then gives the eigenvector
centrality score of the ith node in the network.
From (1)
xi 
1
N


Ai , j x j
j 1
•Therefore, for any node, the eigenvector centrality score be
proportional to the sum of the scores of all nodes which are
connected to it.
•Consequently, a node has high value of EC either if it is
connected to many other nodes or if it is connected to others that
themselves have high EC
Subgraph centrality
the number of closed
walks of length k starting
and ending on vertex i in
the network is given by
the local spectral
moments μ k (i), which
are simply defined as the
ith diagonal entry of the
kth power of the
adjacency matrix, A:
Subgraph Centrality in Complex
Networks, Physical Review E 71,
056103(2005)
Closed walks can be trivial or
nontrivial and are directly related to
the subgraphs of the network.
Subgraph centrality
01000000000000
10110100000000
01011100000000
01101101000000
00110100000000
01111010000000
M=
00000100001000
00010000100000
Muv = 1 if there is an edge between
nodes u and v and 0 otherwise.
00000001010011
00000000101011
00000010010000
00000000000010
00000000110101
00000000110010
Adjacency matrix
Subgraph centrality
10110100000000
04223211000000
12432311000000
12352310100000
03223211000000
12332501001000
M2 =
01111020010000
01101102010011
(M2)uv for uv represents the
number of common neighbor of the
nodes u and v.
00010000421122
local spectral moment
00000000110101
00000011240122
00000100102011
00000001221042
00000001221123
Subgraph centrality
The subgraph centrality of the node i is given by
Let λ be the main eigenvalue of the adjacency matrix A. It can be
shown that
Thus, the subgraph centrality of any vertex i is bounded above
by
Table 2.
Summary of
results of eight
real-world
complex
networks.