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Introduction to Bioinformatics
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Lecture 16
Intracellular Networks
Graph theory
High-throughput Biological Data

Enormous amounts of biological data are being
generated by high-throughput capabilities; even
more are coming
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
genomic sequences
gene expression data
mass spectrometry data
protein-protein interaction data
protein structures
......
Hidden in these data is information that reflects
– existence, organization, activity, functionality …… of biological
machineries at different levels in living organisms
Bio-Data Analysis and
Data Mining

Existing/emerging bio-data analysis and mining tools for
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
DNA sequence assembly
Genetic map construction
Sequence comparison and database search
Gene finding
….
Gene expression data analysis
Phylogenetic tree analysis to infer horizontally-transferred genes
Mass spec. data analysis for protein complex characterization
……
Current prevailing mode of work
Developing ad hoc tools for each
individual application
Bio-Data Analysis and
Data Mining

As the amount and types of data and the needs to establish
connections across multi-data sources increase rapidly, the number of
analysis tools needed will go up “exponentially”
– blast, blastp, blastx, blastn, … from BLAST family of tools
– gene finding tools for human, mouse, fly, rice, cyanobacteria, …..
– tools for finding various signals in genomic sequences, protein-binding sites,
splice junction sites, translation start sites, …..
Many of these data analysis problems are fundamentally the
same problem(s) and can be solved using the same set of tools
Developing ad hoc tools for each application problem (by each
group of individual researchers) may soon become inadequate
as bio-data production capabilities further ramp up
Data Clustering

Many biological data analysis problems can be
formulated as clustering problems
– microarray gene expression data analysis
– arrayCGH data (chromosomal gains and losses)
– identification of regulatory binding sites (similarly, splice
junction sites, translation start sites, ......)
– (yeast) two-hybrid data analysis (for inference of protein
complexes)
– phylogenetic tree clustering (for inference of horizontally
transferred genes)
– protein domain identification
– identification of structural motifs
– prediction reliability assessment of protein structures
– NMR peak assignments
– ......
Data Clustering: an example

Regulatory binding-sites are short conserved sequence fragments in
promoter regions
........acgtttataatggcg ......
........ggctttatattcgtc ......
........ccgaatataatcta .......

Solving binding-site identification as a clustering problem
– Project all fragments into Euclidean space so that similar fragments are
projected to nearby positions and dissimilar fragments to far positions
– Observation: conserved fragments form “clusters” in a noisy background
Data Clustering Problems

Clustering: partition a data set into clusters so that data points of the
same cluster are “similar” and points of different clusters are “dissimilar”

Cluster identification -- identifying clusters with significantly different
features than the background
Multivariate statistics – Cluster analysis
C1 C2 C3 C4 C5 C6 ..
1
2
3
4
5
Raw table
Any set of numbers per
column
•Multi-dimensional problems
•Objects can be viewed as a cloud of
points in a multidimensional space
•Need ways to group the data
Multivariate statistics – Cluster analysis
1
2
3
4
5
C1 C2 C3 C4 C5 C6 ..
Raw table
Any set of numbers per
column
Similarity criterion
Scores
5×5
Similarity
matrix
Cluster criterion
Dendrogram
Cluster analysis – data normalisation/weighting
1
2
3
4
5
C1 C2 C3 C4 C5 C6 ..
Raw table
Normalisation criterion
1
2
3
4
5
C1 C2 C3 C4 C5 C6 ..
Column normalisation
Normalised
table
x/max
Column range normalisation (x-min)/(max-min)
Cluster analysis – (dis)similarity matrix
1
2
3
4
5
C1 C2 C3 C4 C5 C6 ..
Raw table
Similarity criterion
Scores
5×5
Similarity
matrix
Di,j = (k | xik – xjk|r)1/r Minkowski metrics
r = 2 Euclidean distance
r = 1 City block distance
Cluster analysis – Clustering criteria
Scores
5×5
Similarity
matrix
Cluster criterion
Dendrogram (tree)
Single linkage - Nearest neighbour
Complete linkage – Furthest neighbour
Group averaging – UPGMA (phylogeny)
Ward
Neighbour joining – global measure (phylogeny)
Cluster analysis – Clustering criteria
1. Start with N clusters of 1 object each
2. Apply clustering distance criterion iteratively until
you have 1 cluster of N objects
3. Most interesting clustering somewhere in between
distance
Dendrogram (tree)
1 cluster
N clusters
Single linkage clustering (nearest
neighbour)
Char 2
Char 1
Single linkage clustering (nearest
neighbour)
Char 2
Char 1
Single linkage clustering (nearest
neighbour)
Char 2
Char 1
Single linkage clustering (nearest
neighbour)
Char 2
Char 1
Single linkage clustering (nearest
neighbour)
Char 2
Char 1
Single linkage clustering (nearest
neighbour)
Char 2
Char 1
Distance from point to cluster is defined as the
smallest distance between that point and any point in
the cluster
Single linkage clustering (nearest
neighbour)
Let Ci and Cj be two disjoint clusters:
di,j = Min(dp,q), where p  Ci and q  Cj
Single linkage dendrograms typically show
chaining behaviour (i.e., all the time a
single object is added to existing cluster)
Complete linkage clustering
(furthest neighbour)
Char 2
Char 1
Complete linkage clustering
(furthest neighbour)
Char 2
Char 1
Complete linkage clustering
(furthest neighbour)
Char 2
Char 1
Complete linkage clustering
(furthest neighbour)
Char 2
Char 1
Complete linkage clustering
(furthest neighbour)
Char 2
Char 1
Complete linkage clustering
(furthest neighbour)
Char 2
Char 1
Complete linkage clustering
(furthest neighbour)
Char 2
Char 1
Complete linkage clustering
(furthest neighbour)
Char 2
Char 1
Distance from point to cluster is defined as the
largest distance between that point and any point in
the cluster
Complete linkage clustering
(furthest neighbour)
Let Ci and Cj be two disjoint clusters:
di,j = Max(dp,q), where p  Ci and q  Cj
More ‘structured’ clusters than with single
linkage clustering
Clustering algorithm
1.
2.
3.
4.
Initialise (dis)similarity matrix
Take two points with smallest distance as
first cluster
Merge corresponding rows/columns in
(dis)similarity matrix
Repeat steps 2. and 3.
using appropriate cluster
measure until last two clusters are
merged
Average linkage clustering
(Unweighted Pair Group Mean Averaging -UPGMA)
Char 2
Char 1
Distance from cluster to cluster is defined as the
average distance over all within-cluster distances
UPGMA
Let Ci and Cj be two disjoint clusters:
1
di,j = ———————— pq dp,q, where p  Ci and q  Cj
|Ci| × |Cj|
Ci
Cj
In words: calculate the average over all pairwise
inter-cluster distances
Multivariate statistics – Cluster analysis
1
2
3
4
5
C1 C2 C3 C4 C5 C6 ..
Data table
Similarity criterion
Scores
Similarity
matrix
5×5
Cluster criterion
Phylogenetic tree
Multivariate statistics – Cluster analysis
1
2
3
4
5
C1 C2 C3 C4 C5 C6
Similarity
criterion
Scores
6×6
Cluster criterion
Scores
5×5
Cluster criterion
Make two-way ordered
table using dendrograms
Multivariate statistics – Two-way cluster
analysis
C4 C3 C6 C1 C2 C5
1
4
2
5
3
Make two-way (rows, columns) ordered table using dendrograms;
This shows ‘blocks’ of numbers that are similar
Multivariate statistics – Two-way cluster analysis
Graph theory
The river Pregal in Königsberg – the Königsberg
bridge problem and Euler’s graph
Can you start at some land area (S1, S2, I1, I2) and walk each
bridge exactly once returning to the starting land area?
Graphs - definition

Digraphs: Directed graphs

Complete graphs: have all possible
edges

Planar graphs: can be presented in 2D
and have no crossing edges (e.g. chip
design)
Graphs - definition
0 1 1.5 2 5 6 7 9
1 0 2 1 6.5 6 8 8
1.5 2 0 1 4 4 6 5.5
.
.
.
Graph
Adjacency
matrix
An undirected graph has a symmetric adjacency matrix
A digraph typically has a non-symmetric adjacency matrix
Example application – OBSTRUCT: creating
non-redundant datasets of protein structures
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
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Based on all-against-all global sequence alignment
Create all-against-all sequence similarity matrix
Filter matrix based on desired similarity range
(convert to ‘0’ and ‘1’ values)
Form maximal clique (largest complete subgraph) by
ordering rows and columns
This is an NP-complete problem (NP = nonpolynomial) and thus problem scales exponentially
with number of vertices (proteins)
Example application 1 – OBSTRUCT:
creating non-redundant datasets of protein
structures
•
Statistical research on protein structures typically
requires a database of a maximum number of nonredundant (i.e. non-homologous) structures
•
Often, two structures that have a sequence identity of
less than 25% are taken as non-redundant
•
Given an initial set of N structures (with corresponding
sequences) and all-against-all pair-wise alignments:
•
Find the largest possible subset where each
sequence has <25% sequence identity with any
other sequence
Heringa, J., Sommerfeldt, H., Higgins, D., and Argos, P. (1992). Obstruct: a program to obtain largest cliques from a protein sequence set according to
structural resolution and sequence similarity. Comp. Appl. Biosci. (CABIOS) 8, 599-600.
Example application 1 – OBSTRUCT:
creating non-redundant datasets of protein
structures
(Cnt.)
• The problem now can be formalised as follows:
• Make a graph containing all sequences as
vertices (nodes)
• Connect two nodes with an edge if their
sequence identity < 25%
• Make an adjacency matrix following the above
rules
Heringa, J., Sommerfeldt, H., Higgins, D., and Argos, P. (1992). Obstruct: a program to obtain largest cliques from a protein sequence set according to
structural resolution and sequence similarity. Comp. Appl. Biosci. (CABIOS) 8, 599-600.
Example application 1 – OBSTRUCT:
creating non-redundant datasets of protein
structures
(Cnt.)
The algorithm:
• Now try and reorder the rows (and columns in
the same way) such that we get a square only
consisting of 1’s in the upper left corner
• This corresponds to a complete graph (also
called clique) containing a set of non-redundant
proteins
Heringa, J., Sommerfeldt, H., Higgins, D., and Argos, P. (1992). Obstruct: a program to obtain largest cliques from a protein sequence set according to
structural resolution and sequence similarity. Comp. Appl. Biosci. (CABIOS) 8, 599-600.
Example application 1 – OBSTRUCT:
creating non-redundant datasets of protein
structures
(Cnt.)
1 0 1 1 1 0 0 0 1
5
0 1 0 0 1 1 1 0 0
4
1 0 1 1 1 0 1 1 0
6
1 0 1 1 0 0 0 0 1
4
. . . . . . .
..
Adjacency matrix

Heringa, J., Sommerfeldt, H., Higgins, D., and Argos, P.
(1992). Obstruct: a program to obtain largest cliques from a
protein sequence set according to structural resolution and
sequence similarity. Comp. Appl. Biosci. (CABIOS) 8, 599-600.
1.
Order sum array and
reorder rows and
columns accordingly…
2.
Estimate largest possible
clique and take subset of
adj. matrix containing
only rows with enough 1s
3.
For a clique of size N, a
subset of M rows (and
columns), where M  N,
with at least N 1s is
selected.
4.
Go to step 1.
Some books call graphs containing multiple edges or loops a
multigraph, and those without a graph. Other books allow
multiple edges or loops in a graph, but then talk about a graph
without multiple edges and loops as a simple graph.
Remarks
A multigraph might have no multiple edges or loops.
Every (simple) graph is a multigraph, but not every
multigraph is a (simple) graph.
Every graph is finite
Sometimes even “multigraph” folks talk about a “simple
graph” to emphasize that there are no multiple edges
and loops.
Further definitions
K3,3
Further definitions
bipartite
A graph is bipartite if its vertices can be
partitioned into two disjoint subsets U and V
such that each edge connects a vertex from
U to one from V. A bipartite graph is a
complete bipartite graph if every vertex in U
is connected to every vertex in V. If U has n
elements and V has m, then we denote the
resulting complete bipartite graph by Kn,m.
K3,3
The Stable Marriage Algorithm


In mathematics, the stable marriage problem (SMP)
is the problem of finding a stable matching — a
matching in which no element of the first matched set
prefers an element of the second matched set that also
prefers the first element.
It is commonly stated as:
– Given n men and n women, where each person has ranked all
members of the opposite sex with a unique number between 1
and n in order of preference, marry the men and women off
such that there are no two people of opposite sex who would
both rather have each other than their current partners. If there
are no such people, all the marriages are "stable".

In 1962, David Gale and Lloyd Shapley proved that, for
any equal number of men and women, it is always
possible to solve the SMP and make all marriages
stable.
The Stable Marriage Algorithm

Also called the Gale-Shapley algorithm (see preceding
slide)
 Given two non-overlapping equally sized graphs of
men (A, B, C, ..) and women (a, b, c, …), where each
man and woman has a preference list about persons of
the opposite sex
 A pairing denotes a 1-to-1 correspondence between
men and women (each man marries one woman)
 A pairing is unstable if there are couples X-x and Y-y
such that X prefers y to x and y prefers X to Y
– if this happens, pair X-y is called unsatisfied


A pairing in which there are no unsatisfied couples is
called a stable pairing or stable marriage
The Stable Marriage Algorithm forms a bipartite graph
that is stable
A: abcd denotes the preferences of A (likes a the
most, then b, then c, while d is liked least)
The Stable Marriage Algorithm
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

The Gale-Shapley pairing, in the form presented here,
is male-optimal and female-pessimal (it would be the
reverse, of course, if the roles of "male" and "female"
participants in the algorithm were interchanged).
To see this, consider the definition of a feasible
marriage. We say that the marriage between man A
and woman B is feasible if there exists a stable pairing
in which A and B are married. When we say a pairing
is male-optimal, we mean that every man is paired with
his highest ranked feasible partner. Similarly, a femalepessimal pairing is one in which each female is paired
with her lowest ranked feasible partner.
Thuis means that men pair with women higher in their
preference list than where these men appear in the list
of the women paired to them (on average)
Graphs - definition
0 1 1.5 2 5 6 7 9
1 0 2 1 6.5 6 8 8
1.5 2 0 1 4 4 6 5.5
.
.
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Graph
Adjacency
matrix
An undirected graph has a symmetric adjacency matrix
A digraph typically has a non-symmetric adjacency matrix
A Theoretical Framework

Representation of a set of n-dimensional (n-D) points as a graph
– each data point represented as a node
– each pair of points represented as an edge with a weight defined by the
“distance” between the two points
0 1 1.5 2 5 6 7 9
1 0 2 1 6.5 6 8 8
1.5 2 0 1 4 4 6 5.5
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.
.
n-D data points
graph
representation
distance matrix
A Theoretical Framework

Spanning tree: a sub-graph that has all nodes connected and
has no cycles
(a)

(b)
(c)
Minimum spanning tree: a spanning tree with the minimum total
distance
Spanning tree
Prim’s algorithm (graph, tree)

– step 1: select an arbitrary node as the current tree
– step 2: find an external node that is closest to the tree, and add it with its
corresponding edge into tree
– step 3: continue steps 1 and 2 till all nodes are connected in tree.
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(a)
(b)
(c)
(d)
(e)
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Spanning tree
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Kruskal’s algorithm
– step 1: consider edges in non-decreasing order
– step 2: if edge selected does not form cycle, then add it into tree; otherwise
reject
– step 3: continue steps 1 and 2 till all nodes are connected in tree.
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(a)
(b)
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(e)
(d)
reject
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A Theoretical Framework

A formal definition of a cluster:
– C forms a cluster in D only if for any partition C = C1 U C2, the
closest point, from D-C1, to C1 is from C2, in other words, the
closest connection from the points outside C1 must come from
within C2
c2
c1

Key results
For any data set D, any of its cluster is
represented by a sub-tree of its MST
A Theoretical Framework
We can use the result on the preceding slide for
clustering using PRIM’s algorithm
– The order of nodes as selected by PRIM’s algorithm defines a
linear representation, L(D), of a data set D
Any contiguous block in L(D) represents a cluster if
and only if its elements form a sub-tree of the MST, plus
some minor additional conditions (each cluster forms a valley)
– We can plot the node distances in PRIM’s minimum spanning
tree against the order of the nodes as they were added by
PRIM’s algorithm (see earlier slide)
4
A
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E
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3
D
PRIM’s
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A
B
E
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C
D
B
7
5
C
edge weight (distance)
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A B E D C
A Theoretical Framework
So, if we first calculate all pairwise distances in the plots
below, convert the dots to a (complete) graph, use PRIM’s
algorithm to calculate a MST, and then make the plot as on
the preceding slide, we get for real-size data:
Valleys correspond
to clusters (red
bars)
A nice clustering algorithm.... That can also be used for signal/noise filtering
Take home messages
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Revise agglomerative clustering (e.g.
nearest/furthest neighbour, group averaging)
Learn about graphs (complete, (un)directed,
adjacency matrix, bipartite, etc.)
Understand the relationship between a graph
and its adjacency matrix
Understand the Stable Marriage Algorithm
Understand Prim’s and Kruskal’s algorithm
– Make sure you understand the clustering method
based on the minimal spanning tree made by
Prim’s algorithm (preceding slide)