LECTURE 6
Topic 1: Metabolic network and stoichiometric matrix
Topic 2: Hierarchical clustering of multivariate data
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 2 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 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.
Hierarchical clustering
Hierarchical Clustering
Data is not always
available as binary
relations as in the case of
protein-protein
interactions where we
can directly apply
network clustering
algorithms.
AtpB
AtpG
AtpA
AtpB
AtpG
AtpE
AtpA
AtpE
AtpH
AtpH
AtpH
AtpH
In many cases for
example in case of
microarray gene
expression analysis
the data is
multivariate type.
An Introduction to Bioinformatics Algorithms by Jones & Pevzner
Hierarchical Clustering
We can convert multivariate data into networks and can apply
network clustering algorithm about which we will discuss in
some later class.
If dimension of multivariate data is 3 or less we can cluster
them by plotting directly.
An Introduction to Bioinformatics Algorithms by Jones & Pevzner
Hierarchical Clustering
Some data reveal good cluster structure when plotted but some
data do not.
Data plotted in 2
dimensions
However, when dimension is more than 3, we can apply
hierarchical clustering to multivariate data.
In hierarchical clustering the data are not partitioned into a
particular cluster in a single step. Instead, a series of partitions
takes place.
Hierarchical Clustering
Hierarchical clustering is a technique that organizes
elements into a tree.
A tree is a graph that has no cycle.
A tree with n nodes can have maximum n-1 edges.
A Graph
A tree
Hierarchical Clustering
Hierarchical Clustering is subdivided into 2 types
1.
agglomerative methods, which proceed by series of fusions of the n objects
into groups,
2.
and divisive methods, which separate n objects successively into finer
groupings.
Agglomerative techniques are more commonly used
Data can be viewed as a single
cluster containing all objects
to n clusters each containing a
single object .
Hierarchical Clustering
Distance measurements
The Euclidean distance between points
and
, in Euclidean n-space, is defined as:
Euclidean distance between
g1 and g2
(10  10) 2  (8  0) 2  (10  9) 2
 0  64  1  8.0622
Hierarchical Clustering
An Introduction to Bioinformatics Algorithms by Jones & Pevzner
In stead of Euclidean distance correlation can also be used as
a distance measurement.
For biological analysis involving genes and proteins, nucleotide
and or amino acid sequence similarity can also be used as
distance between objects
Hierarchical Clustering
•An agglomerative hierarchical clustering procedure produces
a series of partitions of the data, Pn, Pn-1, ....... , P1. The first Pn
consists of n single object 'clusters', the last P1, consists of
single group containing all n cases.
•At each particular stage the method joins together the two
clusters which are closest together (most similar). (At the first
stage, of course, this amounts to joining together the two
objects that are closest together, since at the initial stage each
cluster has one object.)
Hierarchical Clustering
An Introduction to Bioinformatics Algorithms by Jones & Pevzner
Differences between methods arise because of the
different ways of defining distance (or similarity)
between clusters.
Hierarchical Clustering
How can we measure distances between clusters?
Single linkage clustering
Distance between two clusters A and B, D(A,B) is computed as
D(A,B) = Min { d(i,j) : Where object i is in cluster A and
object j is cluster B}
Hierarchical Clustering
Complete linkage clustering
Distance between two clusters A and B, D(A,B) is computed as
D(A,B) = Max { d(i,j) : Where object i is in cluster A and
object j is cluster B}
Hierarchical Clustering
Average linkage clustering
Distance between two clusters A and B, D(A,B) is computed as
D(A,B) = TAB / ( NA * NB)
Where TAB is the sum of all pair wise distances between objects
of cluster A and cluster B. NA and NB are the sizes of the clusters
A and B respectively.
Total NA * NB edges
Hierarchical Clustering
Average group linkage clustering
Distance between two clusters A and B, D(A,B) is computed as
D(A,B) = = Average { d(i,j) : Where observations i and j are in
cluster t, the cluster formed by merging clusters A and B }
Total n(n-1)/2 edges
Hierarchical Clustering
Alizadeh et al.
Nature 403: 503-511
(2000).
Classifying bacteria
based on 16s rRNA
sequences.