Its equilibrium state could be defined by the

1
More about Markov model. The simple two-state
model from the previous lecture was defined by
four transition probabilities (of which two were
independent). They can be described by the so
called transition matrix
A B
A 1-p p
B q 1-q
P
Its equilibrium state could be defined by the
detailed equilibrium conditionnumber of
customers leaving the state A at the step k, (fk
p). equals the number of customers joining the
state A by leaving state B
f p (1-f) q ? f
q/(pq)
2
In general, with s possible states, we can write
the transition probability matrix as
Any row in this matrix corresponds to the state
from which transition is made, and any column
corresponds to the state to which transition is
made. Thus probabilities in any particular row
should sum to 1, while there is no such
requirement for the columns (Why? Explain in
plain words) . Together with an initial
probability distribution, transition matrix
determines the Markov process.
3
ExampleRandom walks (EvensGrant, p136 Random
walk theory underpins BLAST theory
The practical example involves a gambler ( G)
having i dollars and its adversary (A) with (s-i)
dollars. At each step a coin with probability p
for heads is tossed. If heads come up, A gives
G one dollar, and if tails up, then G gives a
dollar to A. This continues until the random
variable reaches either 0 or s.The random
variable is a current fortune of the gambler
taking values 0,1,2,,s dollars.
G 0 1 2 3
s-1 s
4
ExampleRandom walks (EvensGrant, p136 Random
walk theory underpins BLAST theory
If G has 1, he throws a coin, and will either
gain 1 (probability p) or will lose it (q). He
will never stay with 1, and thus p110
If G is left without money, he stays in this
state with p001, and all other transitions
become impossible
5
The transition matrix are widely used in
bioinformatics, for sequences analysis. Make sure
that you understand why P for the gambler has
exactly this shape. Why, for instance, p3,50?
What is the probability of moving from state i to
the state j in two steps? We have to sum up the
probabilities for all possible trajectories
connecting i and j. The result is defined by the
product PP of two matrixes (not two numbers!)
m
The product of two matrixes, a and b is the
matrix c whose components are
j
i
With this in mind, Eq. 1 can be written as a
product
Similarly, the n-step transition probability is
6
Graphical representation of the Markov Chain
Try finding the error
7
Graphical representation
The corrected diagram
8
Working in groups

• Find the product of matrixes A and B

• open Lect7/Math/Lect7_MatrixExample and use it
  for practice

• Draw a graphical representation for the
  transition matrixes between the states E1, E2 ,
  E3 (a) and E4 (for b)

(a)
(b)
9
One application of Markov Model for sequence
analysis
In the human genome wherever the dinucleotide CG
occurs (frequently written CpG to distinguish it
from the C-G base pair across the two strands)
the C nucleotide (cytosine) is typically
chemically modified. There is a relatively high
chance of this modification that mutates C into a
T, and in general CpG dinucleotides are rarer in
the genome than would be expected from the
independent probabilities of C and G. For
biologically important reasons the mutation
modification process is suppressed in short
stretches of the genome, such as around the
promoters or start regions of many genes. In
these regions we see many more CpG dinucleotides
than elsewhere. Such regions are called CpG
islands.
You can find the tool for searching the CpG
islands at http//bioinformatics.org/sms/cpg_isla
nd.html
The CpG dinucleotide is present at approximately
20 of its expected frequency in vertebrate
genomes, a deficiency thought due to a high
mutation rate from the methylated form of CpG to
TpG and CpA.
10
What sort of probabilistic model might we use for
CpG island regions? It should be a model that
generates sequences in which the probability of a
symbol depends on the previous symbol. The
simplest such model is a classical Markov chain.
A Markov chain for DNA can be drawn like this
11
where there is a state for each of the four
letters A, C, G, and T in the DNA alphabet. The
transition probabilities pst are associated with
each arrow in the figure and determine the
probabilities of a certain residue following
another residue (or one state following another
state). Markov models for the CpG island are
illustrated below. From a set of human DNA
sequences a total of 48 putative CpG islands have
been extracted. Two Markov chain models have
been developed, one for the regions labeled as
CpG islands (the model) and the other from
the remainder of the sequence (the - model).
The transition probabilities for each model were
set using the equation
12
where cts is the number of times letter t
followed letter s in the island regions. These
are the maximum likelihood estimators for the
transition probabilities. The resulting tables are
13
(No Transcript)
14
The equation (2) therefore becomes
Eq. (3) is used to calculate the likelihood of
the sequence. For instance, we use (3) to obtain
the likelihood values of a sequence under
model and - model respectively. Comparing them,
we can conclude if the sequence belongs to the
CpG island region or not. More specifically, we
can calculate the log-odds ratio
15
Exercise 1
16
A brief introduction to the HMM
         Very efficient programs for searching a
text for a combination of words are widely
available The same methods can be used for
searching for patterns in biological sequences,
but often they fail. Why?          Biological
spelling is much more sloppy than English
spelling proteins with the same function from
two different organisms are most likely spelled
differently, that is, the two amino acid
sequences differ. It is not rare that two such
homologous sequences have less than 30 identical
amino acids. Similarly in DNA many interesting
signals vary greatly even within the same genome.
Some well-known examples are ribosome binding
sites and splice sites, but the list is
long.          Fortunately there are usually
still some subtle similarities between two such
sequences, and the question is how to detect
these similarities.
17

• The variation in a family of sequences can be
  described statistically, and this is the basis
  for most methods used in biological sequence
  analysis

• A hidden Markov model (HMM) is a statistical
  model, which is very well suited for many tasks
  in molecular biology, although they have been
  mostly developed for speech recognition since the
  early 1970s.
• The most popular use of the HMM in molecular
  biology is as a probabilistic profile of a
  protein family, which is called a profile HMM.
  From a family of proteins (or DNA) a profile HMM
  can be made for searching a database for other
  members of the family.
• Probably the main contribution of HMM comparative
  to some other methods is that the profile HMM
  treats gaps in a systematic way.

18
From Regular expressions to HMM
In programs like grep, JavaScript and Perl,
regular expressions can be used for searching
text files for a pattern.
Using regular expressions is a very elegant and
efficient way to search for some protein
families, but difficult for other. The
difficulties arise because protein spelling is
much more free than English spelling. Therefore
the regular expressions sometimes need to be very
broad and complex. Imagine a DNA motif like this
A regular expression for this is AT CG AC
ACGT A TG GC , meaning that the first
position is A or T, the second C or G, and so
forth. The term ACGT means that any of the
four letters can occur any number of times.
19
The problem with the above regular expression is
that it does not in any way distinguish between
the highly implausible sequence TGCT- - AGG
and the consensus sequence ACAC- -ATC. It is
possible to make the regular expression more
discriminative by splitting it into several
different ones, but it easily becomes messy. The
alternative is to score sequences by how well
they fit the alignment.
20
Such a scoring is shown in the diagram below.
This is the HMM derived from the same alignment
Transition Probabilities
States
insertion state is represented by the state
above the other states.
21
It is now easy to score the consensus sequence
ACACATC. The probabilities of different states
(bases) must be multiplied by the transition
probabilities
Making the same calculation for the exceptional
sequence yields only 0.0023 10-2
22
Probabilities and log-odds scores for the 5
sequences in the alignment and for the consensus
sequence and the exceptional sequence
23
What is the Log-odds score?
The probability itself is not the most convenient
number to use as a score, and the log-odds score
shown in the last column of the table is usually
better. It is the logarithm of the probability of
the sequence divided by the probability according
to a null model. The null model is one that
treats the sequences as random strings of
nucleotides, so the probability of a sequence of
length L is
24
The probabilities of the model in the previous
picture have been turned into log-odds by taking
the logarithm of each nucleotide probability and
subtracting log 0.25. The transition
probabilities have been converted to simple
logs. When a sequence fits the motif very well
the log-odds is high. When its very unlikely,
the log-odds score becomes negative. Here is an
example for the consensus sequence
25
I will stop right here. This discussion was based
on a very well-written article (see the
resources and a link on my site),
Reading assignment pp.8-13 of that article,
Profile HMM section. This reading will prepare
you for better understanding of the project
presentation devoted to HMM