(IJCSIS) International Journal of Computer Science and Information Security,
Vol. 7, No. 1, 2010

A Probabilistic Model For Sequence Analysis
B. M. Karan+, G. Sahoo++

Amrita Priyam
+

Dept. of Computer Science and Engineering
Birla Institute of Technology
Ranchi, India.
.
Abstract— This paper presents a probabilistic approach for

II.

DNA sequence analysis. A DNA sequence consists of an
arrangement of the four nucleotides A, C, T and G and
different representation schemes are presented according
to a probability measure associated with them. There are
different ways that probability can be associated with the
DNA sequence: one way is when the probability of an
occurrence of a letter does not depend on the previous one
(termed as unsuccessive probability) and in another
scheme the probability of occurrence of a letter depends on
its previous letter (termed as successive probability).
Further, based on these probability measures graphical
representations of the schemes are also presented. Using
the digram probability measure one can easily calculate an
associated probability measure which can serve as a
parameter to check how close is a new sequence to already
existing ones.
Keywords-Successive Probability; Unsuccessive
Transition Probability; Digram Probability;

I.

Dept. of Electronics and Electrical Engineering
++
Dept. of Information Technology
Birla Institute of Technology
Ranchi, India

DNA SEQUENCE

DNA sequences is a succession of letters representing the
primary structure of a real or hypothetical DNA molecule or
strand, with the capacity to carry information as described by
the central dogma of molecular biology. There are 4
nucleotide bases (A – Adenine, C – Cytosine, G – Guanine, T
– Thymine). DNA sequencing is the process of determining
the exact order of the bases A, T, C and G in a piece of DNA.
In essence, the DNA is used as a template to generate a set of
fragments that differ in length from each other by a single
base. The fragments are then separated by size, and the bases
at the end are identified, recreating the original sequence of
the DNA. The most commonly used method of sequencing
DNA the dideoxy or chain termination method was developed
by Fred Sanger in 1977 (for which he won his second Nobel
Prize). The key to the method is the use of modified bases
called dideoxy bases; when a piece of DNA is being replicated
and a dideoxy base is incorporated into the new chain, it stops
the replication reaction.

Probability;

Most DNA sequencing is carried out using the chain
termination method. This involves the synthesis of new DNA
strands on a single standard template and the random
incorporation of chain-terminating nucleotide analogues. The
chain termination method produces a set of DNA molecules
differing in length by one nucleotide. The last base in each
molecule can be identified by way of a unique label.
Separation of these DNA molecules according to size places
them in correct order to read off the sequence.

INTRODUCTION

A DNA sequence is a succession of the letters A, C, T and G.
The sequences are any combination of these letters. A physical
or mathematical model of a system produces a sequence of
symbols according to a certain probability associated with
them. This is known as a stochastic process, that is, it is a
mathematical model for a biological system which is governed
by a set of probability measure. The occurrence of the letters
can lead us to the further study of genetic disorder. The
stochastic process is also known mathematically as Discrete
Markov Process. There are different ways to use probabilities
for depicting the DNA sequences. One scheme is where each
occurrence of a letter is independent of the occurrence of any
other letter. That is, the probability of occurrence of a letter
does not depend on the occurrence of the previous one. In yet
another scheme, the occurrence of a letter depends on the
occurrence of the previous one. The earlier one is termed in as
unsuccessive probability and the later one is termed as
successive probability. From this study we can further show
the relationship between sequence and genetic variations. It
can also lead to a more powerful test for identifying particular
classes of genes or proteins which has been illustrated by an
example.

III.

DIFFERENT PROBABILISTIC APPROACHES FOR SEQUENCE
REPRESENTATION

A DNA sequence is essentially represented as a string of four
characters A, C, T, G and looks something like
ACCTGACCTTACG. These strings can also be represented in
terms of some probability measures and using these measures
it can depicted graphically as well. This graphical
representation matches the Markov Hidden Model. Some of
these schemes are presented in this paper. A physical or
mathematical model of a system produces a sequence of
symbols according to a certain probability associated with
them. This is known as a stochastic process. There are

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(IJCSIS) International Journal of Computer Science and Information Security,
Vol. 7, No. 1, 2010

different ways to use probabilities for depicting the DNA
sequences

structure is to give the “digram” probabilities p(i,j),
i.e., the relative frequency of the digram i j. The
letter frequencies p(i), (the probability of letter i), the
transition probabilities pi(j) and the digram
probabilities p(i,j) are related by the following
formulas:

A. Unsuccessive Probability
In this representation scheme the probability of the next
occurrence of a letter does not depend on the previous letter.
There can be two different representation schemes in this: one
which uses equal probability for each letter and another which
assumes a fixed probability for each letter.
•

p (i ) = ∑ p (i, j ) = ∑ p ( j , i ) = ∑ p ( j ) p j
j

j

p (i, j ) = p (i ) pi ( j )

∑ p ( j ) = ∑ p(i) = ∑ p(i, j ) = 1

Equal Probability - Suppose we have a 4 letter code
consisting of the 4 letters A, C, T, G which can be
chosen each with equal probability 0.25, successive
choices being independent. This would lead to a
sequence of which the following is a typical example

i

j

•

ACCATGGACTTAGCTACTGG
•

j

Unequal Probability - For a DNA sequence series of
length 20, where each letter has a probability of 0.3,
0.2, 0.3, 0.2 respectively, with successive choices are
independent. A typical message from this source is:
ACTTGAAATTCGGACCTGAT

i

i, j

In the example presented in this paper we have used
the letter probability for representing the sequence.
Word Probability - A process can also be defined
which produces a text consisting of a sequence of
words. Since a DNA sequence consists of typically 4
letters A, C, T, G any word could be made up of these
four letters. For example the typical DNA sequence
may consist of any combination of the following
words:
ACTG TACG ACGT AATC AGTG TCCA CAAG
CCTG
This can be depicted graphically as in figure 2.

IV.

ALGORITHM

Fig 2: Graphical Representation of Word Probability
Sequence

Fig 1: Graphical Representation of Unequal
Probability Sequence

B. Successive Probability
The various probability (transition and digram) are calculated
according to the algorithm given below:

This section presents two schemes for representing a DNA
sequence: one where successive letter depends on the
preceding one and in another scheme the letters are used to
form words and a probability is associated with each word.
•

ALGORITHM
SERIES (DATA, N, LOC): Here DATA is a linear array with
N elements and LOC acts as a pointer that keeps the record of
occurrence of the nucleotide.

Letter Probability - A more complicated structure is
obtained if successive symbols are not chosen
independently but their probabilities depend on
preceding letters. In the simplest case of this type a
choice depends only on the preceding letter and not
on ones before that. The statistical structure can then
be described by a set of transition probabilities pi(j),
the probability that a letter i is followed by letter j.
The indices i and j range over all the possible
symbols. A second equivalent way of specifying the

1. Initialize LOC = 0 and initialize all variables to zero.
2. Repeat while LOC ≠ N,
IF DATA [LOC] = ‘A’
Set A = A+ 1
IF DATA [LOC +1 ] = ‘A’ SET AA = AA+1

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IF DATA [LOC +1 ] = ‘C’ SET AC = AC+1
IF DATA [LOC +1 ] = ‘T’ SET AT = AT+1
IF DATA [LOC +1 ] = ‘ G’ SET AG = AG +1

TRANSITION PROBABILITY
Pi(j)
A
C
T
G

Else IF DATA [LOC] = ‘ C’
Set C = C+1
IF DATA [ LOC + 1] = ‘A’ SET CA = CA+1
IF DATA [LOC +1 ] = ‘ C’ SET CC = CC+1
IF DATA [LOC +1 ] = ‘ T’ SET CT = CT+1
IF DATA [LOC +1 ] = ‘ G’ SET CG = CG +1

T
A’ * AT’
C’ * CT’
T’ * TT’
G’ * GT’

G
A’ * AG’
C’ * CG’
T’ * TG’
G’ * GG’

Type 1:
TABLE I

A
C
T
G

Else IF DATA [LOC ] = ‘ G’
Set G = G+1
IF DATA [LOC +1 ] = ‘A’ SET GA = GA +1
IF DATA [LOC +1] = ‘C’ SET GC = GC +1
IF DATA [LOC +1] = ‘T’ SET GT = GT +1
IF DATA [LOC+1] = ‘G’ SET GG = GG+1

A
0.13
0.09
0.06
0.08

C
0.06
0.04
0.05
0.04

T
0.09
0.05
0.05
0.05

G
0.08
0.02
0.07
0.06

Type 2:
TABLE II

3. SET A’ = A/N , C’ = C/N , G’ = G/N, T’ = T / N
AA’ = AA* A’; AC’ = AC*A’; AT ‘ = AT* A’;
AG ‘ =AG* A’; CA’ = CC* C’ ; CC’ = CC*C’;
CT’ = CT * C’; CG’ = CG * C’; TA’ = TA * T’;
TC’ = TC * T’; TT’ =TT * T’; TG’ = TG* T’;
GA’ = GA *G’; GC’ = GC *G’; GT’ = GT * G’;
GG’ = GG * G’;

A
C
T
G

A
0.13
0.08
0.06
0.08

C
0.06
0.04
0.05
0.04

T
0.09
0.04
0.06
0.04

G
0.08
0.02
0.07
0.06

T
0.08
0.06
0.06
0.04

G
0.08
0.02
0.09
0.07

Type 3:
TABLE

End

A
C
T
G

LETTER FREQUENCY
(i)
A
C
T
G

C
A’ * AC’
C’ * CC’
T’ * TC’
G’ * GC’

On using the algorithm on the different H1N1 viruses we get
the following transition probability tables:

Else IF DATA [LOC] = ‘T’
Set T = T+1
IF [LOC +1 ] = ‘A’
SET TA = TA+ 1
IF [LOC +1 ] = ‘C’
SET TC = TC+1
IF [LOC + 1 ] = ‘ T’ SET TT = TT + 1
IF [LOC + 1 ] = ‘ G’ SET TG = TG +1

4.

A
A’ * AA’
C’ * CA’
T’ * TA’
G’ * GA’

P(i)
No of A’S / Size of string = (A’ Say )
No of C’S / Size of string = ( C’ Say )
No of T’S / Size of string = ( T’ Say )
No of G‘S / Size of string = (G’ say )

A
0.08
0.08
0.05
0.08

C
0.05
0.03
0.06
0.06

III

A graphical representation for the type 1 H1N1 virus can be
constructed based on the digram probability as follows:

DIGRAM PROBABILITY
P(i, j)
A
C
T
G

A
No. of AA *
A’ = AA’
No. of CA *
C’ = CA’
No. of TA *
T’ = TA’
No. of GA *
G’ = GA’

C
No. of AC *
A’ = AC’
No. of CC *
C’ = CC’
No. of TC *
C’ = TC’
No. of GC *
C’ = GC’

T
No. of AT *
A’ = AT’
No. of CT *
C’ = CT’
No. of TT *
T’ = TT’
No. of GT *
T’ = GT’

G
No. of AG *
A’ = AG’
No. of CG *
C’ = CG’
No. of TG * T’
= TG’
No. of GG *
G’ = GG’
Fig 3: Graphical Representation of Type 1 H1N1 Sequence

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Now if any new sequence comes which has to be categorized
as any one of the three types we can use the transition
probability table of the existing types and calculate the
probability measure. For example consider a new sequence
tggtgctggc

REFERENCES
[1]
[2]

C. E. Shanon, “A Mathematical Theory of Communciation”.
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[4] W. J. Ewens, G. R. Grant, “Statistical Methods in Bioinformatics: AN
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[5] L. R. Rabenir, “A Tutorial on Hidden Markov Models and Selected
Applications in Speech Recognition”, Proceedings of the IEEE, Vol. 77,
No. 2, Feb 1989.
[6] J. L. Massey, “Applied Digital Information Theory”, Lecture Notes.
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[8] I. Csiszar, P. C. Shields, “Information Theory Statistics: A Tutorial”,
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[10] Andras Telcs, “Transition Probability Estimates for Reversible Markov
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[11] Y. Ephraim, L. R. Rabiner, “On the Relations Between Modeling
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Probability for Type 1: 0.07 * 0.06 * 0.05 * 0.07 * 0.04 * 0.05
* 0.07 * 0.06 * 0.04 = 4.9 X 10 -12
Probability for Type 2: 0.06 * 0.06 * 0.04 * 0.07 * 0.04 * 0.04
* 0.07 * 0.06 * 0.04 = 2.7 X 10 -11
Probability for Type 3: 0.09 * 0.07 * 0.04 * 0.09 * 0.06 * 0.06
* 0.09 * 0.07 * 0.06 = 3 X 10 -11
Since the probability measure for type 3 comes out to be the
greatest we can conclude that the chance for the new DNA
sample to be the type 3 virus is more.

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