detectIR: A Novel Program for Detecting Perfect and Imperfect Inverted Repeats Using Complex Numbers and Vector Calculation

Congting Ye; Guoli Ji; Lei Li; Chun Liang

doi:10.1371/journal.pone.0113349

. 2014 Nov 19;9(11):e113349. doi: 10.1371/journal.pone.0113349

detectIR: A Novel Program for Detecting Perfect and Imperfect Inverted Repeats Using Complex Numbers and Vector Calculation

Congting Ye ^1,², Guoli Ji ^1,^3,^*, Lei Li ^1,², Chun Liang ^2,^4,^*

Editor: Silvio C E Tosatto⁵

PMCID: PMC4237412 PMID: 25409465

Abstract

Inverted repeats are present in abundance in both prokaryotic and eukaryotic genomes and can form DNA secondary structures – hairpins and cruciforms that are involved in many important biological processes. Bioinformatics tools for efficient and accurate detection of inverted repeats are desirable, because existing tools are often less accurate and time consuming, sometimes incapable of dealing with genome-scale input data. Here, we present a MATLAB-based program called detectIR for the perfect and imperfect inverted repeat detection that utilizes complex numbers and vector calculation and allows genome-scale data inputs. A novel algorithm is adopted in detectIR to convert the conventional sequence string comparison in inverted repeat detection into vector calculation of complex numbers, allowing non-complementary pairs (mismatches) in the pairing stem and a non-palindromic spacer (loop or gaps) in the middle of inverted repeats. Compared with existing popular tools, our program performs with significantly higher accuracy and efficiency. Using genome sequence data from HIV-1, Arabidopsis thaliana, Homo sapiens and Zea mays for comparison, detectIR can find lots of inverted repeats missed by existing tools whose outputs often contain many invalid cases. detectIR is open source and its source code is freely available at: https://sourceforge.net/projects/detectir.

Introduction

An inverted repeat is a nucleotide sequence fragment that can form self-complementary pairing between its two halves. The perfect inverted repeats are also known as palindrome where one of these two halves is exactly the reverse complement of the other; in contrast, imperfect inverted repeats contain nucleotide pairs that are not reversely complementary (i.e., mismatched), often with a non-palindromic spacer (loop or gaps) in the middle [1] [2] [3]. Abundant inverted repeats are present in both prokaryotic and eukaryotic genomes with nonrandom distributions, and they are involved in many biological processes including DNA replication [4], DNA transition [5] and DNA methylation [6]. In yeast, long inverted repeats were demonstrated to be mitotic recombination hotspots, and quasipalindromes (imperfect inverted repeats) underwent deletion more frequently [7]. In mouse embryonic stem cells, inverted repeats get involved in the generation of unstable chromosomal rearrangements [8]. Inverted repeats of > = 6 complementary nucleotides, either perfect or imperfect, can form secondary structures – cruciforms in double stranded DNA [9]. Some DNA-binding proteins have their two binding sites arranged as in an inverted repeat [3] [4] [10]. Using atomic force microscopy images, the DNA-binding protein PARP-1 was shown to bind the cruciform structure generated by a 106-nt inverted repeat within an E. coli plasmid [5]. PARP-1 was found to participate in chromatin structure coordination and gene expression regulation [11], and it did show a binding preference to cruciform structures than loops or linear DNAs [9]. In humans, a 14-nt imperfect inverted repeat sequence located in distal promoter region of human HFE gene can form a cruciform structure that binds PARP-1 protein to repress HFE transcription, and increased ion level can trigger PARP-1 breakdown to release such transcriptional repression [12]. Interestingly, using 2D electrophoretic analysis of DNA replication intermediates, single-stranded hairpins formed by imperfect inverted repeats with a central non-palindromic spacer, rather than double-stranded cruciforms, proved to be responsible for replication stalling that induces genome instability [13]. On the other hand, as a part of gene expression production, hairpin is a common and important secondary structural element in RNA transcripts, and its single-stranded template DNA must have a relevant inverted repeat [3], suggesting that inverted repeats in transcribed DNAs might play a potential regulatory role in resultant RNA transcripts. For instance, in yeast and mammalian pre-mRNAs, inverted repeats located in introns were shown to affect alternative splicing [14] [15] [16]. Moreover, inverted repeats have been utilized for gene silencing in fungi and plants for many years [17] [18] [19]. They also delineate transposon element boundaries. For instance, miniature inverted repeat transposable elements (MITEs) are characterized by their terminal inverted repeats [20]. Genes away from MITEs show higher expression than those that contain or are close to MITEs [21]. Clearly, the role of inverted repeats in gene expression regulation is worthy of further wet-lab experimental investigation and validation. The identification and characterization of inverted repeats at genome-wide scale will offer an important glimpse and survey that will facilitate our understanding of inverted repeats and their biological functions.

Lu et al. examined the distribution of perfect inverted repeats (palindromes) in human genome [22]. They found that palindromes show higher abundance in introns than exons while upstream regions (i.e., 2,000 bp upstream from translational start site) also contain rich palindromes that can serve as binding sites for transcription factors. Interestingly, they also scanned the human genome for imperfect inverted repeats (i.e., near-perfect palindromes of < = 4 mismatches between two halves, with a short spacer in the middle) and found a similar distribution pattern as perfect inverted repeats [22]. In yeast Saccharomyces cerevisiae genome, both palindromes and imperfect inverted repeats (i.e., the pairing stem length >6 nt and the spacer length less than 77 nt) were significantly richer than randomized genome [3]. In particular, imperfect inverted repeats with short spacers, which have a greater susceptibility to cruciform extrusion than long spacers, were significantly enriched in intergenic regions near 3′ gene ends than near 5′ gene ends [3]. Using yeast relatives S. paradoxus, S. mikatae and S. bayanus, Humphey-Dixon and coworkers studied the conservation of both perfect and imperfect inverted repeats in yeast S. cerevisiae genome, and they found that both conserved inverted repeats in promoters and inverted repeats in the promoters of highly expressed genes are most frequently located near the transcriptional start sites, indicating their potential function in transcriptional regulation [23]. As more and more genome sequences are available for different species, it is important for us to conduct genome-wide comparative study to determine the distributions, properties, and conservation of inverted repeats among different species, either distantly or closely related, in order to deepen our understanding of inverted repeats and their biological importance.

To better understand the roles of inverted repeats in genome organization and evolution, developing efficient programs to conduct genome-wide detection of inverted repeats is particularly important. Recently, a MATLAB-based tool findIR [24] was created for detecting perfect inverted repeats. In comparison with the existing similar tools that adopt conventional string comparison algorithms, findIR deployed a novel algorithm that uses prime number scoring system and turns sequence search into the calculation, search and comparison of numbers. Consequently, findIR proved to have obviously higher accuracy in detecting perfect inverted repeats than several popular tools, and it was capable of processing genome-scale inputs. Unfortunately, it is difficult to use prime numbers to represent imperfect inverted repeats that contain non-complementary pairs in the pairing stem (i.e., two halves) and a central non-palindromic spacer. The search and validation strategy of findIR is not designed for detecting imperfect inverted repeats. Moreover, findIR is limited to detect perfect inverted repeats of length shorter than 1,000 nt and the robust vector calculation power of MATLAB has not been utilized at all by findIR to enhance the program efficiency.

Here, we developed a novel program detectIR that also turns sequence search into numerical calculation and manipulation using complex numbers, which can represent both perfect and imperfect IRs accurately and efficiently. In comparison with findIR, the novelty of detectIR lies in a novel mapping schema that utilizes complex numbers, a distinctive and effective strategy of search and validation to evaluate candidates of both perfect and imperfect inverted repeats, and the utilization of MATLAB built-in vector calculation power that enables simultaneous detection of inverted repeats of same length to improve the program efficiency significantly.

Design and Implementation

Most existing bioinformatics tools in inverted repeat detection rely on string comparison that is often computational resource demanding and less accurate. Using a novel algorithm that employs prime number scoring system and numerical calculation and search to detect perfect inverted repeats (or palindromes), findIR was demonstrated to have much higher accuracy than BioPHP (http://www.biophp.org/minitools/find_palindromes), MATLAB built-in palindromes function and EMBOSS palindrome tool [25] in detecting perfect inverted repeats. However, both EMBOSS palindrome tool and MATLAB built-in palindromes function can detect imperfect inverted repeats that contain a central non-palindromic spacer (loop) and/or non-complementary pairs in the pairing stem. In the core algorithm of findIR, nucleotide bases are first mapped to a prime number scoring system in which scores of reversely complementary bases can cancel each other out, and a cumulative score is computed for each base along the whole target sequence. Then, findIR searches all the pairs of positions whose cumulative scores are the same to construct candidates of perfect inverted repeats. findIR finally validates the candidate based on the principle that if a subsequence is a valid perfect inverted repeat, the number of nested perfect inverted repeats within the subsequence, all of which share the same center, should be equal to the half-length of the subsequence. This search and validation strategy makes findIR impossible to detect imperfect inverted repeats. Moreover, the prime number scoring system has its innate difficulty to represent imperfect inverted repeats that often contains a central non-palindromic spacer and/or non-complementary pairs in the pairing stem. In addition, findIR is limited to detect perfect inverted repeats of length shorter than 1,000 nt, presenting a size constraint for large genomes where longer inverted repeats may exist. Here, we developed a novel program called detectIR that maps the nucleotide sequence to a complex number vector so that both perfect and imperfect inverted repeats can be searched (see examples in Figure 1). Compared with findIR, detectIR adopted a totally different algorithm in perfect inverted repeat detection, which had been modified and extended to search for imperfect inverted repeats. In particular, for both perfect and imperfect inverted repeat detection, we have taken advantage of MATLAB built-in vector calculation to search and validate inverted repeats candidates of the same size simultaneously. Our program proves to be much faster and more accurate in detecting both perfect and imperfect inverted repeats than the previously mentioned tools. Moreover, our program can accept a large genome input like chromosome 1 of Homo sapiens and Zea mays that can often result in an execution crash in other tools.

Circles highlight the nucleotide bases and numbers represent the start and end genomic coordinates of the imperfect inverted repeats. (A) An imperfect inverted repeat containing only mismatches (un-pairing pair). (B) An imperfect inverted repeat containing both mismatches and a spacer (gaps). (C) An imperfect inverted repeat containing only a spacer (gaps) in the middle. (D) The *detectIR* output for the aforementioned 3 imperfect inverted repeats in the combined format of dot-bracket notation and FASTA. The dots represent the spacer or mismatch nucleotides while brackets indicate the pairing relations. The FASTA description line (*e.g.*, >4483|20|2|0) has the following explanation: >GenomicStartPosition|InvertedRepeatLength|MismatchNumber|GapNumber.

Within the prime number scoring system (also called cumulative scoring system) implemented in findIR [24], a subsequence between two bases whose cumulative scores are identical will be recognized as a perfect inverted repeat candidate. This conclusion is based on the assumption that only the scores between reversely complementary bases can completely eliminate each other in the system. If two bases have identical score, the numbers of reversely complementary bases should be equal in the corresponding subsequence [24]. Unfortunately, this assumption is not correct for some cases, because scores also can be partly eliminated between non-complementary bases. For instance, when using the prime number mapping schema: A → 3, T → −3, G → 7, C → −7, the scores of seven nucleotides A can be eliminated out by that of three nucleotides C (same to T and G). Therefore, if two bases have an identical accumulative score, the numbers of complementary bases in the corresponding subsequence between them are not always equal. In findIR, such a case would be still recognized as a perfect inverted repeat candidate. Although these cases could be filtered out later by the downstream process, it clearly increases the computational workload. On the other hand, these cases can be avoided by using large prime numbers (e.g., 10007, 10009), which was exactly adopted by the findIR [24].

Different from prime numbers, complex numbers can effectively represent both perfect and imperfect inverted repeats that contain non-complementary pairs in the stem and a central non-palindromic spacer. The complex numbers have been utilized in detecting symmetric palindromes (e.g., ACGGCA, the palindromes without reverse complementary) by Gupta et al. [26], using the following mapping schema:

Firstly, they divided an input nucleotide sequence into subsequences with the length of a desired palindrome, and then utilized this mapping method to convert each subsequence into numerical series and rearranged the subsequence. Secondly, they use periodicity transformation to calculate a periodic sequence that is closest to the rearranged subsequence. Lastly, through calculating a coefficient between the rearranged subsequence and the periodic sequence to determine and verify each subsequence to be a valid symmetric palindrome [26]. Therefore, their method was not designed for both perfect and imperfect inverted repeat detection that requires nucleotide reverse complementary. In contrast, our usage of complex numbers is different from their approach by deploying a novel mapping schema:

Using this mapping method, only the scores between reversely complementary bases can really cancel each other out within the numeric scoring system. This method makes sure that the numbers of reversely complementary nucleotides A and T (also C and G) are equal within the sequence candidates of perfect inverted repeat. Furthermore, we can precisely define the upper bound of the score of the imperfect inverted repeats using this complex number mapping schema - the sum of the absolute values of the real part and the imaginary part of C is less than or equal to m, Inline graphic (C denotes the score of the subsequence, m represents the maximal number of mismatches allowed – the sum of all nucleotides within the central non-palindromic spacer and the non-complementary pairs in the pairing stem).

Algorithm for perfect inverted repeat detection

The algorithm for perfect inverted repeat detection is based on the principle that a perfect inverted repeat of length h (h>4) must contain a nested perfect inverted repeat of length h-2, which shares the same center. So if we have obtained all the perfect inverted repeats of length h-2, extension of one base at both ends will derive inverted repeat candidates with length h. Among all these inverted repeat candidates, we then filter out those whose terminal bases are not reversely complementary.

The details of our algorithm are shown in Figures 2 and 3. Here, l is the minimal length, L is the maximal length of perfect inverted repeats to be detected, and N is the length of the input sequence S (l ≤ L ≤ N, l and L should be even number).

(A) Map a nucleotide sequence to a complex number vector and calculate its cumulative score value. (B) Calculate the scores of all the subsequences of length l (here l = 4, N is the length of input sequence). (C) Select out perfect inverted repeat candidates and determine valid perfect inverted repeats. (D) Extend one base at both ends of subsequences to obtain longer perfect inverted repeats.

Arrows in the same color represent the borders of inverted repeats.

(1). Mapping sequence to vector

The input sequence S is mapped to a complex number vector M. For example, if input sequence S is ‘ATCGAACGAATTCGTTAACC’, M = [1,-1, j, -j,1, 1, j, -j, 1, 1, -1, -1, j, -j, -1, -1, 1, 1, j, j]. Then the cumulative value V of M is calculated (Figure 2A).

(2). Calculation of subsequence score

The scores of all the subsequences of length l are calculated, a subsequence score is defined as the sum of the subsequence’s corresponding vector. The scores of all the subsequences of length l can be obtained by the following formula,

The i-th element C(i) represent the score of the subsequence S(i:i+l-1) = s_is_i+ ₁ …s_i+l- ₂ s_i+l- ₁ (Figure 2B).

(3). Identification of candidates

A subsequence whose score is equal to 0 will be recognized as a candidate of perfect inverted repeat. In this step we will identify all the candidates of length l by finding the zero elements of C,

P represents the indices of the zero elements in vector C.

(4). Validation of candidates

For each perfect inverted repeat candidate, perfect pairing needs to be found between the front half and rear half of bases. We will keep valid perfect inverted repeat candidates, while filter out invalid cases (Figure 2C).

(5). Extension of inverted repeats

Base on the aforementioned principle, extend one base at both ends of inverted repeats identified in the previous step, and select out cases whose new terminal bases are reversely complementary (Figure 2D). For instance, if subsequence S(i:i+h-3) is a perfect inverted repeat of length h-2 identified in previous step, the extended subsequence S(i-1:i+h-2) will be a potential inverted repeat of length h. If its terminal bases are either G/C or A/T,

Then, S(i-1:i+h-2) is a perfect inverted repeat of length h.

(6). Repetition

Repeat the step 5 until h = L, then all the perfect inverted repeats of length ranged between l and L have been identified.

Algorithm for imperfect inverted repeat detection

Different from the aforementioned algorithm for perfect inverted repeat detection, to find imperfect inverted repeats of length h, allowing maximal mismatch number m, which is defined as the sum of all nucleotides within the central non-palindromic spacer and the non-complementary pairs in the pairing stem, we will firstly find all subsequences of length h-2 with mismatch number ≤ m, with the assumption that the nested subsequence of an imperfect inverted repeat with mismatch number ≤ m must be a sequence with mismatch number ≤ m which share the same center. The major difference between the nested subsequence and the imperfect inverted repeat is that the terminal bases of imperfect inverted repeat must be reversely complementary (‘AAAAATTTCT’), while the former does not require meeting this condition (‘AAAATTTC’).

Figure 4 shows the core algorithm for imperfect inverted repeat detection. l is the minimal length and L is the maximal length of imperfect inverted repeats to be detected, and N is the length of the input sequence (l ≤ L ≤ N), m is the maximal number of mismatches allowing in each imperfect inverted repeat. The first two steps 1 and 2 are same to the detection of perfect inverted repeats.

(A) Select out imperfect inverted repeat candidates (here, we just show part of candidates), report valid imperfect inverted repeats and pick up valid subsequences for the downstream process described in the next step (here l = 4, m = 2). (B) Extend one base at both ends of subsequences to obtain longer imperfect inverted repeats. The subsequences locating at both ends of a sequence (*i.e.*, ‘ATCGAA’) should not be selected for the next round extension.