LECTURE 5 Topic 1: Metabolic network and stoichiometric matrix Topic 2: Hierarchical clustering of multivariate data

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LECTURE 5Topic 1 Metabolic network and
stoichiometric matrixTopic 2 Hierarchical
clustering of multivariate data
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Typical network of metabolic pathways
Reactions are catalyzed by enzymes. One enzyme
molecule usually catalyzes thousands reactions
per second (102-107) The pathway map may be
considered as a static model of metabolism
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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.

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Example reaction system and corresponding
stoichiometric matrix
There are 6 metabolites and 8 reactions in this
example system
stoichiometric matrix
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Binary form of N
To determine the elementary topological
properties, Stiochiometric matrix is also
represented as a binary form using the following
transformation nij0 if nij 0 nij1 if nij ?0
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Stiochiometric matrix is a sparse matrix
Source Systems biology by Bernhard O. Palsson
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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
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Information contained in the stiochiometric matrix
There are relatively few metabolites (24 or so)
that are highly connected while most of the
metabolites participates in only 2 reactions
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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 lt r(the
number of reactions) K is called kernel matrix
if it satisfies NK0 The kernel matrix K is not
unique
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Information contained in the stiochiometric matrix
The kernel matrix K of the stoichiometric matrix
N that satisfies NK0, contains (r- Rank N) basis
vectors as columns Every possible set of steady
state fluxes can be expressed as a linear
combination of the columns of K
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Information contained in the stiochiometric matrix
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And for steady state flux it holds that J a1
.k1 a2.k2
With a1 1 and a2 1, , i.e. at
steady state v1 2, v2 -1 and v3 -1
That is v2 and v3 must be in opposite direction
for the steady state corresponding to this kernel
matrix which can be easily realized.
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Information contained in the stiochiometric matrix
Reaction System
Stoicheometric Matrix
The stoicheomatric matrix comprises r8 reactions
and Rank 5 and thus the kernel matrix has 3
linearly independent columns. A possible solution
is as follows
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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.
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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
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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.
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Elementary flux modes and extreme pathways
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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
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Elementary flux modes and extreme pathways
Source Systems biology by Bernhard O Palsson
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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.
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Hierarchical clustering
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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.
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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
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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.
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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
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Hierarchical Clustering

• Hierarchical Clustering is subdivided into 2
  types
• agglomerative methods, which proceed by series of
  fusions of the n objects into groups,
• 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 .
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Hierarchical Clustering
Distance measurements
Euclidean distance between g1 and g2
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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
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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.
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• 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.)
•    

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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.
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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
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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
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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
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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
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Hierarchical Clustering
Alizadeh et al. Nature 403 503-511 (2000).
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Classifying bacteria based on 16s rRNA sequences.