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Open Access Highly Accessed Research article

The number of reduced alignments between two DNA sequences

Helena Andrade1, Iván Area2, Juan J Nieto13* and Ángela Torres4

Author Affiliations

1 Departamento de Análise Matemática, Facultade de Matemáticas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain

2 Departamento de Matemática Aplicada II, E.E. Telecomunicación, Universidade de Vigo, 36310 Vigo, Spain

3 Faculty of Science, King Abdulaziz University, P.O. Box 80203, 21589 Jeddah, Saudi Arabia

4 Departamento de Psiquiatría, Radioloxía e Saúde Pública, Facultade de Medicina, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain

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BMC Bioinformatics 2014, 15:94  doi:10.1186/1471-2105-15-94

The electronic version of this article is the complete one and can be found online at: http://www.biomedcentral.com/1471-2105/15/94


Received:10 January 2014
Accepted:19 March 2014
Published:1 April 2014

© 2014 Andrade et al.; licensee BioMed Central Ltd.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Background

In this study we consider DNA sequences as mathematical strings. Total and reduced alignments between two DNA sequences have been considered in the literature to measure their similarity. Results for explicit representations of some alignments have been already obtained.

Results

We present exact, explicit and computable formulas for the number of different possible alignments between two DNA sequences and a new formula for a class of reduced alignments.

Conclusions

A unified approach for a wide class of alignments between two DNA sequences has been provided. The formula is computable and, if complemented by software development, will provide a deeper insight into the theory of sequence alignment and give rise to new comparison methods.

AMS Subject Classification

Primary 92B05, 33C20, secondary 39A14, 65Q30

Keywords:
DNA sequence; Alignment; Difference equation

Background

Let us consider a DNA sequence as a mathematical string

<a onClick="popup('http://www.biomedcentral.com/1471-2105/15/94/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/15/94/mathml/M1">View MathML</a>

where xi∈{A,G,C,T} is one of the four nucleotides, i=1,2,…,n, i.e. A denotes adenine, C cytosine, G guanine and T thymine. In these conditions, the sequence x is of length n.

Our main goal is to compare the sequence x with another DNA sequence

<a onClick="popup('http://www.biomedcentral.com/1471-2105/15/94/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/15/94/mathml/M2">View MathML</a>

to measure the similarity between both strings and also to determine their residue-residue correspondences.

Sequence comparison and alignment is a central and crucial tool in molecular biology. For example, Pairwise Sequence Alignment is used to identify regions of similarity that may indicate functional, structural and/or evolutionary relationships between two biological sequences (protein or nucleic acid) [1].

For some recent developments and directions we refer the reader to [2-7] and [8] for a general review of different alignments methods.

To align the sequences CGT and ACTT, one can use EMBOSS Needle for nucleotide sequence [9] that creates an optimal global alignment of the two sequences using the Needleman-Wunsch algorithm to get

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Following Lesk [10], in order to compare the amino acids appearing at their corresponding positions in two sequences, theirs correspondences must be assigned and a sequence alignment is the identification of residue-residue correspondence. For some references on sequence alignment we refer the reader to [10-16].

To compare two sequences, there exist mainly three different possibilities leading to three different numbers of total alignments [10,11,13]:

1. The total number of alignments denoted by f(n,m) that was solved in [13].

2. A gap in a sequence is followed by another gap in the other sequence as in Alignments 1 and 2 for the sequences x=CGT and y=ACTT (see Tables 1 and 2 below) Considering the two alignments as equivalents to the Alignment 3 (see Table 3) without gap in those positions, we have the number of reduced alignments denoted by h(n,m), and obviously h(n,m)<f(n,m). This case has been solved in [11], and we give here another representation in terms of hypergeometric series.

3. In the interesting case that the alignments 1 and 2 are equivalent, but different from alignment 3 we have a number or reduced alignments g(n,m) where h(n,m)<g(n,m)<f(n,m). This last case is new and we present an explicit formula for g.

Table 1. Alignment 1

Table 2. Alignment 2

Table 3. Alignment 3

Results and discussion

Number of f(x,y) alignments

The total number of alignments f(x,y) satisfies the following recurrence relation [13]

<a onClick="popup('http://www.biomedcentral.com/1471-2105/15/94/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/15/94/mathml/M4">View MathML</a>

with initial conditions f(n,0)=f(0,m)=1 for n,m=1,2,3,…. The solution of the above partial difference equation is given by

<a onClick="popup('http://www.biomedcentral.com/1471-2105/15/94/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/15/94/mathml/M5">View MathML</a>

(see formula (10) in [13]) and the generating function [17,18] is

<a onClick="popup('http://www.biomedcentral.com/1471-2105/15/94/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/15/94/mathml/M6">View MathML</a>

Therefore the coefficients f(n,m) in the expansion

<a onClick="popup('http://www.biomedcentral.com/1471-2105/15/94/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/15/94/mathml/M7">View MathML</a>

are given in terms of a hypergeometric series by

<a onClick="popup('http://www.biomedcentral.com/1471-2105/15/94/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/15/94/mathml/M8">View MathML</a>

This relation seems to be new in this form. Here, the generalized hypergeometric series is defined as (see e.g. [19, Chapter 16])

<a onClick="popup('http://www.biomedcentral.com/1471-2105/15/94/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/15/94/mathml/M9">View MathML</a>

and (A)k=A(A+1)⋯(A+n−1), with (A)0=1, denotes the Pochhammer’s symbol. It is assumed that bj≠−k in order to avoid singularities in the denominators. If one of the parameters aj equals to a negative integer, then the sum becomes a terminating series.

Number of h(x,y) alignments

In this case, the recurrence relation for the h(n,m) coefficients is [11]

<a onClick="popup('http://www.biomedcentral.com/1471-2105/15/94/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/15/94/mathml/M10">View MathML</a>

with initial conditions h(n,0)=h(0,m)=1. Therefore, the generating function [17,18] is

<a onClick="popup('http://www.biomedcentral.com/1471-2105/15/94/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/15/94/mathml/M11">View MathML</a>

and the coefficients in the expansion

<a onClick="popup('http://www.biomedcentral.com/1471-2105/15/94/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/15/94/mathml/M12">View MathML</a>

are given by

<a onClick="popup('http://www.biomedcentral.com/1471-2105/15/94/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/15/94/mathml/M13">View MathML</a>

where

<a onClick="popup('http://www.biomedcentral.com/1471-2105/15/94/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/15/94/mathml/M14">View MathML</a>

The above coefficients can be written in terms of (terminating) hypergeometric series as

<a onClick="popup('http://www.biomedcentral.com/1471-2105/15/94/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/15/94/mathml/M15">View MathML</a>

Number of g(x,y) alignments

As indicated before, the main aim of this paper is to give an explicit representation in this case. The recurrence relation for the g(n,m) coefficients is [11]

<a onClick="popup('http://www.biomedcentral.com/1471-2105/15/94/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/15/94/mathml/M16">View MathML</a>

with initial conditions g(n,0)=g(m,0)=1. Thus, the generating function [17,18] is

<a onClick="popup('http://www.biomedcentral.com/1471-2105/15/94/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/15/94/mathml/M17">View MathML</a>

(1)

Theorem 1. The coefficientsαn,min the expansion

<a onClick="popup('http://www.biomedcentral.com/1471-2105/15/94/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/15/94/mathml/M18">View MathML</a>

(2)

are explicitly given by

<a onClick="popup('http://www.biomedcentral.com/1471-2105/15/94/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/15/94/mathml/M19">View MathML</a>

(3)

where

<a onClick="popup('http://www.biomedcentral.com/1471-2105/15/94/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/15/94/mathml/M20">View MathML</a>

(4)

<a onClick="popup('http://www.biomedcentral.com/1471-2105/15/94/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/15/94/mathml/M21">View MathML</a>

(5)

<a onClick="popup('http://www.biomedcentral.com/1471-2105/15/94/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/15/94/mathml/M22">View MathML</a>

(6)

<a onClick="popup('http://www.biomedcentral.com/1471-2105/15/94/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/15/94/mathml/M23">View MathML</a>

(7)

<a onClick="popup('http://www.biomedcentral.com/1471-2105/15/94/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/15/94/mathml/M24">View MathML</a>

(8)

<a onClick="popup('http://www.biomedcentral.com/1471-2105/15/94/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/15/94/mathml/M25">View MathML</a>

(9)

<a onClick="popup('http://www.biomedcentral.com/1471-2105/15/94/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/15/94/mathml/M26">View MathML</a>

(10)

and [ x] denotes the integer part of x.

Proof. If we expand,

<a onClick="popup('http://www.biomedcentral.com/1471-2105/15/94/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/15/94/mathml/M27">View MathML</a>

(11)

we have two summands to be computed, namely

<a onClick="popup('http://www.biomedcentral.com/1471-2105/15/94/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/15/94/mathml/M28">View MathML</a>

(12)

<a onClick="popup('http://www.biomedcentral.com/1471-2105/15/94/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/15/94/mathml/M29">View MathML</a>

(13)

In order to compute the first sum (12) let us introduce

<a onClick="popup('http://www.biomedcentral.com/1471-2105/15/94/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/15/94/mathml/M30">View MathML</a>

(14)

Therefore, the summation to be done reads as

<a onClick="popup('http://www.biomedcentral.com/1471-2105/15/94/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/15/94/mathml/M31">View MathML</a>

where U, V, A and B must be computed in terms of the initial indices.

The product of binomials can be simplified to

<a onClick="popup('http://www.biomedcentral.com/1471-2105/15/94/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/15/94/mathml/M32">View MathML</a>

Thus,

<a onClick="popup('http://www.biomedcentral.com/1471-2105/15/94/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/15/94/mathml/M33">View MathML</a>

and then

<a onClick="popup('http://www.biomedcentral.com/1471-2105/15/94/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/15/94/mathml/M34">View MathML</a>

Finally, the summation reads as

<a onClick="popup('http://www.biomedcentral.com/1471-2105/15/94/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/15/94/mathml/M35">View MathML</a>

where

<a onClick="popup('http://www.biomedcentral.com/1471-2105/15/94/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/15/94/mathml/M36">View MathML</a>

A similar work with the second summand (13) leads to the final result.

Some numerical values are g(10,10)=2003204, g(50,50)=2.71972×1034, g(100,100)=7.55997×1069, and we note that g(n,n)>1080 for n≥115. This last inequality is relevant since 1080 is an estimation of the number of protons of our universe [13].

Conclusions

A unified approach for a wide class of alignments between two DNA sequences has been provided. We conclude also that our approach gives an explicit formula filling a gap in the theory of sequence alignment. The formula is computable and, if complemented by software development, will provide a deeper insight into the theory of sequence alignment and give rise to new comparison methods. It may be used also, in the future, to get explicit formulas and compute the number of total, reduced, and effective alignments for multiple sequences.

Methods

We have performed a number of numerical computations to compare our formulae and Mathematica®; [20] command Coefficient for the series expansion of (1), on a MacBook Pro featuring a 45 nm “Penryn” 2.66 GHz Intel “Core 2 Duo” processor (P8800), with two independent processor “cores” on a single silicon chip, 8 GB of 1066 MHz DDR3 SDRAM (PC3-8500). We would like to mention that our approach is amazingly fast, since e.g. g(100,100) is computed by using Mathematica®; in 0.125165 seconds by using the new formulas presented in this paper, while the use of Mathematica®; command Coefficient needs 99.167659 seconds.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Each of the authors HA, IA, JJN and AT, contributed to each part of this study equally and read and approved the final version of the manuscript.

Acknowledgements

The authors are grateful to Prof. Marko Petkovs̆ek for helpful comments. The work of I. Area has been partially supported by the Ministerio de Economía y Competitividad of Spain under grant MTM2012–38794–C02–01, co-financed by the European Community fund FEDER. J.J. Nieto also acknowledges partial financial support by the Ministerio de Economía y Competitividad of Spain under grant MTM2010–15314, co-financed by the European Community fund FEDER.

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