Open Access

Statistical inference of chromosomal homology based on gene colinearity and applications to Arabidopsis and rice

  • Xiyin Wang1, 2, 4,
  • Xiaoli Shi1, 3, 4,
  • Zhe Li1,
  • Qihui Zhu1, 3,
  • Lei Kong1,
  • Wen Tang1,
  • Song Ge3 and
  • Jingchu Luo1Email author
Contributed equally
BMC Bioinformatics20067:447

DOI: 10.1186/1471-2105-7-447

Received: 16 April 2006

Accepted: 12 October 2006

Published: 12 October 2006

Abstract

Background

The identification of chromosomal homology will shed light on such mysteries of genome evolution as DNA duplication, rearrangement and loss. Several approaches have been developed to detect chromosomal homology based on gene synteny or colinearity. However, the previously reported implementations lack statistical inferences which are essential to reveal actual homologies.

Results

In this study, we present a statistical approach to detect homologous chromosomal segments based on gene colinearity. We implement this approach in a software package ColinearScan to detect putative colinear regions using a dynamic programming algorithm. Statistical models are proposed to estimate proper parameter values and evaluate the significance of putative homologous regions. Statistical inference, high computational efficiency and flexibility of input data type are three key features of our approach.

Conclusion

We apply ColinearScan to the Arabidopsis and rice genomes to detect duplicated regions within each species and homologous fragments between these two species. We find many more homologous chromosomal segments in the rice genome than previously reported. We also find many small colinear segments between rice and Arabidopsis genomes.

Background

Exploration of homology between chromosomes facilitates the understanding of the structure, function and evolution of genomes. Extensive synteny and colinearity have been detected between chromosomes in different species of cereals [1], mammals [2] and yeasts [3] providing a deep insight into the evolution of ancient chromosomes. Between chromosomes of the same species, large-scale homologous segments exist caused by whole genome or segmental duplication [49]. It has been reported that nearly 80% of the Arabidopsis thaliana genome and 45–60% of the rice genome are in large duplicated regions [1012].

Special care should be taken to reveal chromosomal homology due to numerous genomic changes such as chromosomal rearrangements, gene inversions and gene loss [1315]. Many approaches have been developed to identify chromosomal homologues [16] based on genetic maps [17], sequence alignment [18, 19], gene synteny [10] and gene colinearity [2023]. By detecting the density and order of homologous gene pairs between chromosomes, colinearity approach can reveal reliable homologous regions and requires less computational resources. This approach also enables us to develop reasonable statistical tests to evaluate the significance of predicted homologous regions.

The typical implementations of the colinearity strategy are ADHoRe [20], FISH [24] and DiagHunter [25]. The implementations of these approaches have limitations in some aspects, though they have been widely adopted. Firstly, the gap size between neighboring homologous genes which is essential to define and detect true colinearity needs further evaluation [12, 2023, 26]. Secondly, statistical tests to assess predicted homologous regions are mainly based on a prerequisite of balanced gene loss rates between homologous regions. Finally, compositional and structural differences, especially gene density and repetition in genome-wide and local chromosomal regions, have not been fully addressed.

Here we describe a new colinearity approach characterized by improved statistical inference, flexibility and computational efficiency. Firstly, the selection of parameter values is theoretically explored, especially that of the gap length between neighboring genes. Secondly, the statistical test has been substantially strengthened with a mathematical deduction to evaluate the significance of the predicted homologous regions. Finally, the compositional and structural heterogeneity of chromosomes has been considered.

Using a dynamic programming algorithm, we developed ColinearScan and scanned the Arabidopsis and rice genomes to detect duplicated regions in each species and homologous chromosomal regions between these two species. We found 75.0% of Arabidopsis genes and 76.2% of rice genes were in duplicated regions. Moreover, we identified homologous fragments between these two species, in 32.9% of Arabidopsis and 16.8% of rice. Nearly all homologous segments were shorter than 0.6 Mb, indicating massive chromosomal rearrangements after the monocot-dicot divergence [27].

Results

Algorithm

Gene homology matrix

The first step in our colinearity approach is the construction of the gene homology matrix. To find homologous gene pairs between two chromosomes denoted as A and B, protein sequences encoded by genetically or physically positioned genes are used to perform an all-against-all BLAST search [28]. A gene homology matrix (GHM, denoted as H) is then constructed using the homology information from BLAST results. Chromosome A and B, represented by the positioned genes are arranged along H horizontally and vertically (Fig. 1A). A cell of H is filled with "1" if the corresponding genes on chromosome A and chromosome B are homologous, otherwise with "0". Tandem and other repetitive genes are widely distributed in chromosomes showing many "1"s in horizontal or vertical straight lines in the dot matrix map (Fig. 2) and causing problems in revealing true homology. Therefore, we used a general approach, masking the genes appearing more than 10 times in both chromosomes.
Figure 1

A modified Smith-Waterman algorithm to locate colinearity. (A) A simplified gene homology matrix (GHM, denoted as H). Genes A1, A2, ..., A18 on chromosome A are arranged horizontally, and genes B1, B2, ..., B14 on chromosome B are arranged vertically. Each cell of the matrix is filled with "1" or "0" based on the homology information from BLASTP search, e.g., gene A1 and gene B1 are homologous, and gene A2 and B2 are non-homologous. (B) A modified dynamic programming procedure. A scoring matrix S is constructed recursively based on H, with mg set to 2 genes apart. The distance criterion demands that neighboring genes in colinearity are no more than 2 genes apart. Pointers are shown by dark or grey arrow lines. Two collinear paths containing 9 and 5 genes are shown by dark arrow lines reflecting the same colinear relationship between the corresponding chromosomal regions.

Figure 2

Examples of dot maps. (A) A dot map between rice chromosomes 2 and 4. Each dot in the map reflects a homologous gene pair with BLASTP score > 100. The dots are not distributed uniformly in the map. The map is also featured by many horizontal and vertical lines formed by repetitive genes. (B) A dot map between the same chromosomes as (A) with repetitive genes filtered. (C) A dot map of rice chromosome 1 against itself. Self-matching dots form a solid diagonal line. (D) A dot map with self-matching and repetitive genes filtered. A diagonal line reflecting the neighboring homologues can still be seen.

Dynamic programming algorithm

To reveal the homologous genes in colinearity between two chromosomes, we implemented a dynamic programming approach based on the well-known Smith-Waterman algorithm [29]. Using this approach, we can discover the longest putative sister regions represented by several proximal points of colinear homologous gene pairs in nearly diagonal orientations. The points may not be in close proximity due to large-scale gene loss, insertion and translocation. The extent of the proximity of the points is essential to reveal and evaluate the colinear sister segments. Lines forming by the points corresponding to the true colinear segments are either nearly parallel to the main diagonal line or the anti-diagonal line due to DNA segmental inversion. Homologous genes in colinear segments should all have the same or inverse transcriptional directions if no single gene inversions occur. We scan H in two directions, starting from the upper-left and upper-right. Here, we describe the procedure starting from the upper-left, which also applies in the other direction. Transcriptional orientations of the genes are recorded but not used when performing the colinearity search.

To reveal the colinearity represented by the proximal points in H, we introduce a parameter mg (the maximum gap length) between two neighboring points. Then we define another matrix S (the scoring matrix) with the same size as H (Fig. 1B). A cell in matrix S represents the extension of a colinearity path, i.e., the value of each cell is the number of collinear gene pairs in the path accumulated from its starting point. The path extends and the value of the cell increases by 1 if there is a "1" in lower-right neighborhood, and both vertical and horizontal distances are less than mg.

Initially, S is identical to H. We rebuild the matrix S recursively using a dynamic programming procedure:

S(i,j) = max { S(i-1,j-1) + H(i,j), if dist(p(i,j), p(i-1,j-1)) < mg and S(i-1,j-1) > S(pre(i-1,j-1)) S(pre(i-1,j-1)) + H(i,j), if dist(p(i,j), pre(i-1,j-1))  < mg and  S(i-1,j-1) = S(pre(i-1,j-1)) S(i-1,j), if dist(p(i,j), p(i-1,j)) < mg and  S(i-1,j) > S(pre(i-1,j)) S(pre(i-1,j)), if dist(p(i,j), pre(i-1,j)) < mg and  S(i-1,j) = S(pre(i-1,j)) S(i,j-1), if dist(p(i,j), p(i,j-1)) < mg and S(i,j-1) > S(pre(i,j-1)) S(pre(i,j-1)), if dist(p(i,j), pre(i,j-1)) < mg and S(i,j-1) = S(pre(i,j-1)) H(i,j), or else MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@21C9@

where S(i, j) is the score computed, H(i, j) is the homology information, pre(i, j) is the cell leading to the maximum score at the cell p(i, j), and a pointer (denoted by dark or gray arrow lines in Fig. 1B) is created from the cell p(i, j) to pre(i, j), dist(p(i, j), p(a, b)) is the distance between the cells p(i, j) and p(a, b). Eventually, the maximum score in S corresponds to the longest putative collinear segments. The longest colinearity path formed by dots of the homologous gene pairs is revealed by a trace-back procedure according to the pointers created. After the homologous genes in putative colinearity are recorded, we mask these putative colinear segments by setting H(i, j) to 0, rebuild the matrix S, and scan for other putative colinear segments till no sister regions containing more colinear genes than a threshold r could be found.

Maximum gap length

In colinearity methods, mg is the most important parameter which determines the length, quality and extensiveness of the predicated colinearity. The frequency of gene deletion in duplicated chromosomal segments is high and only a small fraction of homologous genes remains in colinearity. A small value of mg will result in finding many small colinear segments, and increase the difficulty of interpreting possible evolutionary events. On the other hand, a large mg value will surely result in high false positives. In fact, mg is dependent on the density of homologous gene pairs between the chromosomes. When the two chromosomes A and B are from the same species, homologous genes between them are mainly distributed at a similar density. On the other hand, when we compare two chromosomes from different species, the density of homologous genes may more divergent. Therefore we adopt two parameters to define the maximum distance between the neighboring dots, the maximum gap between genes in chromosome A (mgA) and the maximum gap between genes in chromosome B (mgB). When the chromosomes are from the same species, we set mgA = mgB.

Under the assumption that homologous genes are uniformly distributed in chromosomes, we explore the possibility of finding sister segments containing equal to or more than r genes by chance. However, this uniform distribution assumption is not very strict since we only need reasonable rather than optimal values of mg. Suppose the length of chromosome A and B are lenA and lenB, and the number of homologous gene pairs between two chromosomes is pnum. The location of a gene is a random variable with a probability density 1 l e n A MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabigdaXaqaaiabdYgaSjabdwgaLjabd6gaUjabdgeabbaaaaa@32D0@ , the joint probability density of the locations of r genes on chromosome A is 1 [ l e n A ] r MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabigdaXaqaamaadmaabaGaemiBaWMaemyzauMaemOBa4MaemyqaeeacaGLBbGaayzxaaWaaWbaaSqabeaacaaMcSUaemOCaihaaaaaaaa@37E3@ , and the joint probability density of the locations on chromosome A and B of r homologous pairs is 1 [ l e n A l e n B ] r MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabigdaXaqaamaadmaabaGaemiBaWMaemyzauMaemOBa4MaemyqaeKaeyyXICTaemiBaWMaemyzauMaemOBa4MaemOqaieacaGLBbGaayzxaaWaaWbaaSqabeaacaaMcSUaemOCaihaaaaaaaa@3F53@ . Therefore, the probability p that r homologous gene pairs are in colinearity by chance can be evaluated by

P p n u m r [ D 1 [ l e n A l e n B ] r d x 1 d x 2 d x r d y 1 d y 2 d y r ] , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@6B8A@

where the multiple integral field D is

[ i = 2 r { x i | 0 < x i x i 1 < m g A } ] { x i | 0 < x i < l e n A } [ i = 2 r { y i | 0 < y i y i 1 < m g B } ] { y i | 0 < y i < l e n B } MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@98FB@

and x1,x2,...,x r ;y1,y2,...,y r are the positions of the genes on the chromosomes, P p n u m r MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGqbaudaqhaaWcbaGaemiCaaNaemOBa4MaemyDauNaemyBa0gabaGaemOCaihaaaaa@3513@ is the number of permutations of homologous genes: p n u m ! ( p n u m r ) ! MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabdchaWjabd6gaUjabdwha1jabd2gaTjabcgcaHaqaaiabcIcaOiabdchaWjabd6gaUjabdwha1jabd2gaTjabgkHiTiabdkhaYjabcMcaPiabcgcaHaaaaaa@3DA4@ . When mgA <<lenA and mgB <<lenB, the integral in the above formula can be approximated by

1 [ l e n A l e n B ] r ( 0 l e n A m g A d x 1 x 1 x 1 + m g A d x 2 x r 1 x r 1 + m g A d x r ) ( 0 l e n B m g B d y 1 y 1 y 1 + m g B d y 2 y r 1 y r 1 + m g B d y r ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@B2A4@

Then we can estimate p by

P p n u m r ( m g A m g B l e n A l e n B ) r 1 . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGqbaudaqhaaWcbaGaemiCaaNaemOBa4MaemyDauNaemyBa0gabaGaemOCaihaaOGaeiikaGYaaSaaaeaacqWGTbqBcqWGNbWzcqWGbbqqcqGHflY1cqWGTbqBcqWGNbWzcqWGcbGqaeaacqWGSbaBcqWGLbqzcqWGUbGBcqWGbbqqcqGHflY1cqWGSbaBcqWGLbqzcqWGUbGBcqWGcbGqaaGaeiykaKYaaWbaaSqabeaacqWGYbGCcqGHsislcqaIXaqmaaGccqGGUaGlaaa@51AE@

If we set the significance level of colinearity to α, then we have

P p n u m r ( m g A m g B l e n A l e n B ) r 1 < α . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGqbaudaqhaaWcbaGaemiCaaNaemOBa4MaemyDauNaemyBa0gabaGaemOCaihaaOGaeiikaGYaaSaaaeaacqWGTbqBcqWGNbWzcqWGbbqqcqGHflY1cqWGTbqBcqWGNbWzcqWGcbGqaeaacqWGSbaBcqWGLbqzcqWGUbGBcqWGbbqqcqGHflY1cqWGSbaBcqWGLbqzcqWGUbGBcqWGcbGqaaGaeiykaKYaaWbaaSqabeaacqWGYbGCcqGHsislcqaIXaqmaaGccqGH8aapiiGacqWFXoqycqGGUaGlaaa@5458@

When the two chromosomes are from the same species, we assume mgA = mgB and denote them as mg, and the value of mg can be evaluated by

m g < [ ( α P p n u m r ) 1 r 1 ( l e n A l e n B ) ] 1 2 . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGTbqBcqWGNbWzcqGH8aapdaWadaqaaiabcIcaOmaalaaabaacciGae8xSdegabaGaemiuaa1aa0baaSqaaiabdchaWjabd6gaUjabdwha1jabd2gaTbqaaiabdkhaYbaaaaGccqGGPaqkdaahaaWcbeqaamaalaaabaGaeGymaedabaGaemOCaiNaeyOeI0IaeGymaedaaaaakiabcIcaOiabdYgaSjabdwgaLjabd6gaUjabdgeabjabgwSixlabdYgaSjabdwgaLjabd6gaUjabdkeacjabcMcaPaGaay5waiaaw2faamaaCaaaleqabaWaaSaaaeaacqaIXaqmaeaacqaIYaGmaaaaaOGaeiOla4caaa@5409@

When the two chromosomes are from different species, the gene densities are often different and thus we adopt different values for the parameters mgA and mgB. The length of gaps between neighboring homologous genes is inversely proportional to the density of homologous genes, let

m g A m g B = p n u m l e n B l e n A p n u m = l e n A l e n B . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabd2gaTjabdEgaNjabdgeabbqaaiabd2gaTjabdEgaNjabdkeacbaacqGH9aqpdaWcaaqaaiabdchaWjabd6gaUjabdwha1jabd2gaTbqaaiabdYgaSjabdwgaLjabd6gaUjabdkeacbaacqGHflY1daWcaaqaaiabdYgaSjabdwgaLjabd6gaUjabdgeabbqaaiabdchaWjabd6gaUjabdwha1jabd2gaTbaacqGH9aqpdaWcaaqaaiabdYgaSjabdwgaLjabd6gaUjabdgeabbqaaiabdYgaSjabdwgaLjabd6gaUjabdkeacbaacqGGUaGlaaa@598E@

Thus, we can estimate the value of mgA and mgB

m g A < ( α P p n u m r ) 1 2 ( r 1 ) l e n A , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGTbqBcqWGNbWzcqWGbbqqcqGH8aapcqGGOaakdaWcaaqaaGGaciab=f7aHbqaaiabdcfaqnaaDaaaleaacqWGWbaCcqWGUbGBcqWG1bqDcqWGTbqBaeaacqWGYbGCaaaaaOGaeiykaKYaaWbaaSqabeaadaWcaaqaaiabigdaXaqaaiabikdaYiabcIcaOiabdkhaYjabgkHiTiabigdaXiabcMcaPaaaaaGccqWGSbaBcqWGLbqzcqWGUbGBcqWGbbqqcqGGSaalaaa@4A77@
m g B < ( α P p n u m r ) 1 2 ( r 1 ) l e n B . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGTbqBcqWGNbWzcqWGcbGqcqGH8aapcqGGOaakdaWcaaqaaGGaciab=f7aHbqaaiabdcfaqnaaDaaaleaacqWGWbaCcqWGUbGBcqWG1bqDcqWGTbqBaeaacqWGYbGCaaaaaOGaeiykaKYaaWbaaSqabeaadaWcaaqaaiabigdaXaqaaiabikdaYiabcIcaOiabdkhaYjabgkHiTiabigdaXiabcMcaPaaaaaGccqWGSbaBcqWGLbqzcqWGUbGBcqWGcbGqcqGGUaGlaaa@4A7F@

Colinearity shadow

Many genes have homologues in their neighborhood as indicated by the distance distribution of homologous genes (Fig. 3). Neighboring homologues may result in many colinear segments parallel to each other in sister regions within the same chromosome, or between the different chromosomes (Fig. 2A, 2C). If for convenience we call the longer ones the 'true' colinearity, then the shorter ones are the 'shadows'. The colinearity shadows may also reflect colinearity, but they do not provide further information and greatly increase the computational time. Thus after a putative colinearity in certain sister regions is found, we mask the neighboring homologous pairs within the corresponding the entire rectangular regions to avoid possible colinearity shadows.
Figure 3

Distance distribution of homologous genes. (A) The distance distribution of rice homologous genes. (B) The distance distribution of Arabidopsis homologous genes.

When candidate colinearity is found in a specific region of GHM from upper-left to lower-right, colinear shadows perpendicular to this path might also be found in the same region scanning from upper-right to lower-left. These shadows may also reflect actual homology between two sister chromosomal segments, and they may occur when gene rearrangements are frequent in specific regions. A large amount of perpendicular shadows may affect efficiency of the algorithm. However, they do not occur very often so we do not mask these shadows.

Statistical test

Many genes are in multi-gene families and repetitive genes are extensively found in plant genomes. Putative colinear regions might be a reflection of extensive occurrences of single gene duplication or translocation. Therefore, it is critical to correctly detect the colinearity pattern by evaluating the significance of the putative colinear regions generated by the above approach, and to develop an effective statistical method to assess whether the putative colinearity represents true homology or is produced by chance. Moreover, the distribution of homologous gene pairs is far from uniform. We use a statistical test considering divergent distribution of homologous gene pairs in different regions, rather than assuming a uniform distribution throughout the genome.

Given that a DNA segment resides in chromosome B, a corresponding colinear segment in chromosome A is generated due to many independent single-gene duplication events. Under the assumption of a uniform distribution of collinearly paired genes in the local chromosomal regions, we obtain the probability epvA to display the significance of the homology,

e p v A = R A 1 i = 2 m ω A i R A i p s s A , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGLbqzcqWGWbaCcqWG2bGDcqWGbbqqcqGH9aqpcqWGsbGudaqhaaWcbaGaemyqaeeabaGaeGymaedaaOGaeyyXIC9aaebCaeaadaWcaaqaaGGaciab=L8a3naaDaaaleaacqWGbbqqaeaacqWGPbqAaaGccqGHflY1cqWGsbGudaqhaaWcbaGaemyqaeeabaGaemyAaKgaaaGcbaGaemiCaaNaem4CamNaem4CamNaemyqaeeaaaWcbaGaemyAaKMaeyypa0JaeGOmaidabaGaemyBa0ganiabg+GivdGccqGGSaalaaa@504F@

where m is the number of collinear gene pairs, ω A i MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFjpWDdaqhaaWcbaGaemyqaeeabaGaemyAaKgaaaaa@3113@ is the gap length between paired gene i and gene i-1, R A i MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGsbGudaqhaaWcbaGaemyqaeeabaGaemyAaKgaaaaa@306C@ is the number of occurrences of the i-th paired gene and its homologues in the putative sister segment in chromosome A, pssA is the length of the sister segment in chromosome A. The above formula can be deduced by extending the colinearity point by point taking repetitive homologues in consideration. The possibility of finding such colinearity under this assumption will be increased by R A i MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGsbGudaqhaaWcbaGaemyqaeeabaGaemyAaKgaaaaa@306C@ times if R A i MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGsbGudaqhaaWcbaGaemyqaeeabaGaemyAaKgaaaaa@306C@ homologous genes of the i-th colinear gene exist in this segment. Similarly, we can define the probability epvB

e p v B = R B 1 i = 2 m ω B i R B i p s s B . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGLbqzcqWGWbaCcqWG2bGDcqWGcbGqcqGH9aqpcqWGsbGudaqhaaWcbaGaemOqaieabaGaeGymaedaaOGaeyyXIC9aaebCaeaadaWcaaqaaGGaciab=L8a3naaDaaaleaacqWGcbGqaeaacqWGPbqAaaGccqGHflY1cqWGsbGudaqhaaWcbaGaemOqaieabaGaemyAaKgaaaGcbaGaemiCaaNaem4CamNaem4CamNaemOqaieaaaWcbaGaemyAaKMaeyypa0JaeGOmaidabaGaemyBa0ganiabg+GivdGccqGGUaGlaaa@505D@

Then we define

epv = max(epvA,epvB)

to measure the possibility of the colinearity appearing by chance in the sister regions. If it is below a threshold, we take the putative colinearity as significant.

However, if we apply the above test directly to the putative different homologous regions, we cannot distinguish between patterns with similar numbers of homologous gene pairs in segments of different size. For example, the putative colinear regions found in small sister segments (Fig. 4B) more likely indicate true segmental homology than that in large sister segments (Fig. 4A). Dense colinearity possibly generated by recent duplications should be evaluated as significant.
Figure 4

Different distribution pattern of homologous genes in sister regions. Same numbers of homologous genes located in two pairs of sister regions with different size. Homologue pairs are more densely located in (B) than in (A). Dark horizontal lines represent chromosomes, round dots denote genes on chromosomes, lines linking the dots indicate gene homology.

Rather than using pssA and pssB, we try to estimate epv on expected scales of the sister regions (essA and essB) determined by the numbers of colinear genes and all homologous genes. Assuming that paired genes are uniformly distributed throughout the chromosomes, we estimate the expected sizes for each pair of sister segments as follows

e s s A = λ c n u m p n u m l e n A , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGLbqzcqWGZbWCcqWGZbWCcqWGbbqqcqGH9aqpiiGacqWF7oaBcqGHflY1daWcaaqaaiabdogaJjabd6gaUjabdwha1jabd2gaTbqaaiabdchaWjabd6gaUjabdwha1jabd2gaTbaacqGHflY1cqWGSbaBcqWGLbqzcqWGUbGBcqWGbbqqcqGGSaalaaa@4A7F@
e s s B = λ c n u m p n u m l e n B , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGLbqzcqWGZbWCcqWGZbWCcqWGcbGqcqGH9aqpiiGacqWF7oaBcqGHflY1daWcaaqaaiabdogaJjabd6gaUjabdwha1jabd2gaTbqaaiabdchaWjabd6gaUjabdwha1jabd2gaTbaacqGHflY1cqWGSbaBcqWGLbqzcqWGUbGBcqWGcbGqcqGGSaalaaa@4A83@

where cnum is the number of colinearly paired genes in the putative sister segments, pnum is the number of all homologous gene pairs between chromosome A and chromosome B, λ is a coefficient to normalize the size of different putative colinear segments. For each pair of putative sister regions we calculate a preliminary coefficient number λ i by ( p s s A p n u m l e n A c n u m + p s s B p n u m l e n B c n u m ) 1 2 . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqGGOaakdaWcaaqaaiabdchaWjabdohaZjabdohaZjabdgeabjabgwSixlabdchaWjabd6gaUjabdwha1jabd2gaTbqaaiabdYgaSjabdwgaLjabd6gaUjabdgeabjabgwSixlabdogaJjabd6gaUjabdwha1jabd2gaTbaacqGHRaWkdaWcaaqaaiabdchaWjabdohaZjabdohaZjabdkeacjabgwSixlabdchaWjabd6gaUjabdwha1jabd2gaTbqaaiabdYgaSjabdwgaLjabd6gaUjabdkeacjabgwSixlabdogaJjabd6gaUjabdwha1jabd2gaTbaacqGGPaqkcqGHflY1daWcaaqaaiabigdaXaqaaiabikdaYaaacqGGUaGlaaa@68F4@ .

where pssA and pssB are the original sizes of the putative i-th sister segments. These preliminary coefficients are averaged throughout the putative homologous sister pairs to obtain the estimate of λ. Finally, we redefine e p v A = R A 1 i = 2 m ω A i R A i e s s A MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGLbqzcqWGWbaCcqWG2bGDcqWGbbqqcqGH9aqpcqWGsbGudaqhaaWcbaGaemyqaeeabaGaeGymaedaaOGaeyyXIC9aaebCaeaadaWcaaqaaGGaciab=L8a3naaDaaaleaacqWGbbqqaeaacqWGPbqAaaGccqGHflY1cqWGsbGudaqhaaWcbaGaemyqaeeabaGaemyAaKgaaaGcbaGaemyzauMaem4CamNaem4CamNaemyqaeeaaaWcbaGaemyAaKMaeyypa0JaeGOmaidabaGaemyBa0ganiabg+Givdaaaa@4F4F@ and e p v B = R B 1 i = 2 m ω B i R B i e s s B MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGLbqzcqWGWbaCcqWG2bGDcqWGcbGqcqGH9aqpcqWGsbGudaqhaaWcbaGaemOqaieabaGaeGymaedaaOGaeyyXIC9aaebCaeaadaWcaaqaaGGaciab=L8a3naaDaaaleaacqWGcbGqaeaacqWGPbqAaaGccqGHflY1cqWGsbGudaqhaaWcbaGaemOqaieabaGaemyAaKgaaaGcbaGaemyzauMaem4CamNaem4CamNaemOqaieaaaWcbaGaemyAaKMaeyypa0JaeGOmaidabaGaemyBa0ganiabg+Givdaaaa@4F59@

by substituting the original size pssA and pssB with the expected size essA and essB. Thus, we assign different significance to the two patterns in Fig. 4, the sister regions with denser collinear genes have a smaller epv value.

Assessment of mg estimation

To test the applicability of the criteria defining mg, we performed a computational simulation test on rice chromosome 1 and Arabidopsis chromosome 1.

First, using the integral formula (r = 4, α = 0.01, length of rice chromosome 1: lenA = 48.2 Mb, length of Arabidopsis chromosome 1: lenB = 30.5 Mb, pnum = 1737), we calculated the maximum gap length in both chromosomes: mgA = 155 Kb, and mgB = 98 Kb. The parameters r and α mean that using such gap length to scan the homologous segments between the chromosomes, the probability of finding sister segments containing 4 or more collinear genes should be <= 0.01 under the assumption of uniform distribution.

Second, we shuffled the positions of the homologous gene pairs on both chromosomes and scanned the longest homologous segments occurring by chance, then checked whether it had 4 or more collinear genes. We repeated this process 1000 times and found collinear segment with 4 or more collinear genes 9 times.

We calculated mgA and mgB between every pair of chromosomes of rice and Arabidopsis, and applied the largest values (mgA = 160 Kb and mgB = 154 Kb) to all chromosome pairs. The candidate homologous segments can be verified by a statistical test. To check how it affects the scanning process between rice chromosome 1 and Arabidopsis chromosome 1 when using larger mgA and mgB, we found colinear segments with 4 or more collinear gene pairs in 33, out of 1000, simulations.

Application to rice and Arabidopsis

We explored the colinear segments within and between the chromosomes in Arabidopsis and rice. The genomic sequences of Arabidopsis thaliana were from GenBank (Accession NC_003070, NC_003071, NC_003074, NC_003075, NC_003076) [30]. Oryza sativa L. ssp. indica genomic sequences were downloaded from RiceGD [31] and the rice genes were predicted using the software BGF [32] from the Beijing Genome Institute. By performing all-against-all BLASTP, we revealed homologous gene pairs within and between Arabidopsis and rice (BLASTP score > 100).

For Arabidopsis, we searched for the duplication regions with the parameter value r = 4, mg = 116 Kb (~25 intervening genes) and λ = 2.6. We used the maximal mg values estimated between each pairs of Arabidopsis chromosomes. At the significant level of epv <= 0.01, we discovered 203 duplicated sister segments out of 350 candidates, among them 3 were possible perpendicular shadows (Supplementary table 1 [see Additional file 1]). About 75.0% of the genome is in duplicated segments and 22.4%, 1.8% of the genes are in segments with a multiplication level > 2 and > 4, respectively (Table 1). The detected coverage of duplicated regions is a little more than that (71%) found by Blanc et al. [18], less than that (89%) reported by Bowers et al. [33]. The longest sister segments contain 106 colinear genes and extend more than 1.8 Mb in chromosome 2 and 1.55 Mb in chromosome 3 (Table 2). In the longest 20 duplicated segments, 88–100% of the colinear genes are of the same relative transcriptional orientation, indicating a low inversion rate.
Table 1

The number and percentage of genes in duplicated blocks in rice and Arabidopsis genomes

Multiplication level 1

2

3

4

5

6

7

8

Arabidopsis

Gene No

13452

3934

1345

269

115

86

0

 

Percentage 2

0.750

0.224

0.071

0.018

0.008

0.003

0.000

Rice

Gene No

17947

11114

6020

2930

1371

846

550

 

Percentage

0.762

0.429

0.223

0.111

0.057

0.031

0.016

1. Multiplication level of a gene displays that it is in a duplicated segment that appears for how many times in a genome.

2. The percentage is the ratio of genes in multiplication levels >= a specific number.

Table 2

The 20 longest duplicated segments in the Arabidopsis genome

Colinear gene number

Gene orientation identity

Epv

Segment A in Arabidopsis

Segment B in Arabidopsis

   

Starting gene

Ending gene

Gene number

Length (Mb)

Starting gene

Ending gene

Gene number

Length (Mb)

106

0.96

1.92E-226

2_3518

2_4078

560

1.88

3_4725

3_5165

440

1.55

83

0.96

3.99E-200

1_1726

1_2130

404

1.54

1_6013

1_6467

454

1.65

82

0.98

3.64E-199

1_6041

1_6467

426

1.56

1_1752

1_2130

378

1.45

78

0.97

1.83E-155

3_840

3_1141

301

0.97

5_151

5_609

458

1.54

70

0.90

1.38E-138

2_3108

2_3445

337

1.26

3_4285

3_4619

334

1.19

69

0.97

1.68E-129

1_3629

1_3991

362

1.43

3_1669

3_2096

427

1.56

64

0.94

1.23E-116

1_639

1_343

296

1.02

2_2172

2_2582

410

1.47

64

0.95

5.65E-127

3_118

3_353

235

0.75

5_1391

5_1691

300

1.05

62

1.00

2.82E-123

1_4258

1_4017

241

0.90

3_1328

3_1603

275

0.97

60

0.97

1.67E-114

4_1800

4_1590

210

0.78

5_3789

5_4037

248

0.90

59

0.97

1.24E-110

2_1607

2_1840

233

0.97

4_2963

4_3236

273

1.02

54

0.91

9.58E-101

4_2685

4_2484

201

0.68

5_4682

5_5062

380

1.35

53

0.98

6.29E-098

2_1326

2_1096

230

0.86

4_2727

4_2960

233

0.85

53

0.96

3.97E-091

3_2507

3_2114

393

1.57

4_1114

4_1412

298

1.27

51

0.94

6.88E-100

2_935

2_1093

158

0.62

4_3464

4_3625

161

0.57

41

0.98

1.07E-063

1_1423

1_1285

138

0.47

2_6

2_245

239

0.98

37

0.92

3.83E-062

2_1308

2_1472

164

0.56

4_3786

4_3958

172

0.57

37

0.89

1.33E-056

4_2327

4_2478

151

0.50

5_4245

5_4549

304

1.10

33

0.88

1.91E-044

1_134

1_271

137

0.44

4_191

4_396

205

0.83

32

1.00

6.68E-055

3_111

3_1

110

0.32

5_1250

5_1390

140

0.47

* The gene names (also in Table 4 and 5) reflect the chromosome and the gene order, e.g. '2_3518' represents the 3518th gene on chromosome 2.

As for rice, we used another set of parameter values r = 4, mg = 334 Kb (~46 intervening genes) and λ = 3.97. At the same significance level as in Arabidopsis, we revealed 309 duplicated segments out of 841 candidates in rice, among them 13 are possible perpendicular shadows (Supplementary table 2 [see Additional file 2]). In our study we found that 76.2% of the genes were in duplicated regions, significantly higher than 20.59% reported by Simillion et al. [20], also higher than those reported by Paterson et al. [10] (61.9%) and Guyot and Keller [5] (52%). The longest colinear sister regions contain 194 homologous genes and extend more than 4.11 and 3.73 Mb in chromosomes 11 and 12 (Table 3), corresponding a segmental duplication event duplicated ~5–7 Mya [15, 34]. About 42.9%, 11.1% of the genome sequences are in a multiplication level of >2 and >4, respectively (Table 1). The transcriptional orientation of colinear genes is in high consistency, similar to that in Arabidopsis. The colinear sister segments at different levels of multiplication are distributed throughout the rice genome.
Table 3

The 20 longest duplicated segments in the rice genome

Colinear gene number

Gene orientation identity

Epv

Segment A in rice

Segment B in rice

   

Starting gene

Ending gene

Gene number

Length (Mb)

Starting gene

Ending gene

Gene number

Length (Mb)

194

0

0.95

11_5

11_700

695

4.11

12_73

12_691

618

3.73

191

0

0.95

2_3549

2_4505

956

6.18

4_3002

4_4139

1137

7.03

157

0

0.94

1_5264

1_3976

1288

8.42

5_3767

5_4423

656

3.85

139

0

0.97

1_6225

1_5317

908

5.68

5_2981

5_3719

738

4.50

126

3.65E-296

0.90

8_3168

8_4097

929

5.92

9_1944

9_2863

919

5.73

122

8.23E-286

0.89

3_2952

3_1610

1342

9.01

7_3177

7_4010

833

5.10

110

3.85E-257

0.94

2_5335

2_4716

619

3.94

6_682

6_1503

821

5.59

88

5.33E-191

0.95

2_2758

2_3516

758

5.07

4_2121

4_2941

820

5.56

59

2.57E-112

0.83

3_5355

3_5625

270

1.61

7_335

7_910

575

3.87

41

1.81E-067

0.98

1_743

1_1160

417

2.97

5_701

5_1067

366

2.75

40

4.49E-072

0.95

2_896

2_661

235

1.54

6_3717

6_4079

362

2.57

34

4.97E-054

0.97

3_3955

3_4295

340

2.28

12_2686

12_2997

311

2.08

33

1.83E-059

0.94

2_643

2_310

333

2.22

6_4127

6_4386

259

1.64

32

1.30E-051

0.94

2_1059

2_907

152

1.05

6_3325

6_3648

323

2.39

30

8.41E-049

0.90

2_1361

2_1112

249

1.76

6_2904

6_3209

305

2.18

23

1.49E-036

0.96

3_661

3_561

100

0.59

10_1721

10_1879

158

1.16

22

5.24E-030

1.00

1_6600

1_6816

216

1.30

5_2616

5_2780

164

1.15

22

9.83E-031

0.95

8_2936

8_3126

190

1.40

9_1646

9_1894

248

1.80

21

3.50E-038

0.95

1_581

1_724

143

0.74

5_559

5_644

85

0.56

21

3.23E-030

0.81

3_5157

3_5293

136

0.92

7_71

7_279

208

1.32

Using parameter values r = 4, mgA = 154 Kb in Arabidopsis, mgB = 160 Kb in rice and λ = 1.47, we found 177 colinear sister segments out of 432 candidates between the chromosomes of two species (Supplementary table 3 [see Additional file 3]), accounting for 32.9% and 16.9% of the Arabidopsis and rice genes, respectively. The longest sister segments are ~0.6 Mb in length and contain ~14 colinear genes, but most segments are much shorter, indicating extensive independent chromosomal rearrangements and gene loss or gain in each genome. The sister copy in rice is always 1–4 times longer than that in Arabidopsis, implying a possible chromosomal expansion in the rice genome (Table 4).
Table 4

The 20 longest collinear regions between the Arabidopsis and rice genome

Colinear gene number

Gene orientation identity

epv

Segment in Arabidopsis

Segments in rice

   

Starting gene

Ending gene

Gene number

Length (Mb)

Starting gene

Ending gene

Gene number

Length (Mb)

14

0.93

5.53E-16

4_3648

4_3741

93

0.61

2_3736

2_3815

79

0.32

11

0.64

1.45E-09

2_5222

2_5287

65

0.38

4_2954

4_3026

72

0.26

11

0.91

2.69E-10

4_2502

4_2430

72

0.46

2_2835

2_2883

48

0.19

10

0.60

1.37E-07

7_562

7_658

96

0.64

3_2215

3_2280

65

0.24

10

0.70

9.51E-08

10_3103

10_3021

82

0.54

2_3733

2_3781

48

0.19

10

0.70

3.61E-09

1_5674

1_5595

79

0.51

2_3286

2_3339

53

0.18

10

0.90

4.37E-10

8_3510

8_3590

80

0.42

5_4866

5_4931

65

0.23

9

0.78

7.81E-07

3_1660

3_1736

76

0.48

2_2221

2_2248

27

0.12

9

0.89

3.63E-07

2_4713

2_4635

78

0.41

5_4870

5_4948

78

0.29

9

0.56

4.60E-08

9_2767

9_2822

55

0.31

4_3435

4_3468

33

0.12

9

0.89

1.44E-08

7_3347

7_3305

42

0.28

1_5880

1_5908

28

0.10

9

1.00

4.51E-10

1_5843

1_5889

46

0.31

3_4249

3_4279

30

0.10

8

0.75

1.32E-05

5_3294

5_3386

92

0.56

5_104

5_198

94

0.29

8

0.75

6.45E-05

6_1100

6_1031

69

0.54

5_5811

5_5863

52

0.21

8

0.75

6.08E-05

7_3242

7_3189

53

0.32

5_5408

5_5441

33

0.12

8

0.75

4.70E-05

4_2502

4_2430

72

0.46

3_4109

3_4154

45

0.16

8

0.75

3.60E-05

4_3665

4_3741

76

0.50

3_4887

3_4952

65

0.23

8

0.63

2.24E-05

6_772

6_682

90

0.57

4_3081

4_3144

63

0.21

8

0.63

1.25E-05

4_2498

4_2425

73

0.49

2_1946

2_2009

63

0.19

8

0.63

1.12E-05

1_5796

1_5720

76

0.43

5_128

5_207

79

0.25

Discussion

Identification of the duplicated segments, especially their distribution pattern in a genome, is essential for further inference on when and how the duplication or species divergence occurred, and whether or not recurrent duplication events happened. The selection of parameter values, in particular the maximum gap length between the neighboring genes, is critical to detect chromosomal homology. However, the selection of maximum gap length in previous reported studies was mainly empirical, which might fail to detect authentic duplicated segments [20, 22, 23]. Many fewer and shorter duplicated segments are discovered when a smaller gap length is adopted, such as in the case of rice [20], whereas more and longer duplicated regions can be found if a larger gap length is adopted. Moreover, different gap length should be used in different genomes such as Arabidopsis and rice, since the density of colinear genes varies due to DNA loss and insertion. By considering the difference in gene density, especially the density of homologous genes in different genomes, we determined the maximum gap length based on statistical analysis. For example, when the duplicated regions in Arabidopsis and rice are detected, the maximum gap lengths were estimated to be 116 Kb and 334 Kb, respectively.

The input data of our approach can be any type of genetic markers such as sequences, genetic markers. Various measurements can be used to represent the distance between markers, such as physical or genetic distances, or gene numbers. In most previous studies, the significance of the predicted colinear regions was evaluated by a permutation test, which is rather time-consuming [20, 23]. We estimate the significance of the predicted colinear segments through statistical inference. The statistical inference has the advantage over the permutation test in terms of computational efficiency. It takes only 2 minutes to calculate the epv to evaluate their significance on a personal computer (AMD AthlonXP 2000+, 512 MB RAM) while running a permutation test takes several hours on the same machine. The massive gene duplications and translocations in its proximal regions will lead to many colinearity shadows, decreasing the computational efficiency. We include a neighborhood masking procedure in ColinearScan to remove colinearity shadows in our algorithm, which dramatically improves the efficiency of detecting duplicated segments in the rice genome.

ADHoRe adopts linear regression analysis to infer duplicated chromosomal segments [20, 21]. The underlying assumption is that gene loss rates have been balanced between sister segments, resulting in a straight line in the dot map. The colinear homologues in a chromosomal segment might be interspersed by individual genes that have no homologues at the corresponding position in its sister segment. At the very beginning of divergence of the sister segments, there should be one-to-one gene homology. Thereafter, massive gene deletions, translocations and chromosomal rearrangements occur and the initial pattern eventually becomes obscured [25]. The homologues with the conservative orders would appear in a straight line in the dot map if gene deletion or insertion had been balanced in different regions of the sister segments, otherwise in a curvy line. Wang et al. [15] explore the gene loss rates in the sister segments in rice and find that nearly straight lines are obtained for some sister segments, e.g., in chromosomes 11 and 12, and in chromosomes 2 and 4. However, curvy lines are also found for some sister segments, e.g., in chromosomes 1 and 5, and in chromosomes 8 and 9. A linearity assumption might fail to detect true duplicated segments. In FISH, Calabrese et al. [24] also adopt a colinearity strategy and develop a different statistical approach to evaluate the extension of collinear points, referred as clump in GHM. However, the value of key parameter p, reflecting the probability that a point occurs in the neighborhood of the former point, is artificially defined, and the maximal gap is deduced from p in their approach. DiagHunter [25] adopts a colinearity method similar to our approach, and the maximal length of the path is predefined. The program stops extending the current path until it reaches the maximal length threshold, or other neighboring points cannot be found.

Polyploidy has been supposed to be prevalent in plants. Recently, genome-wide studies further suggest the ubiquity of polyploidy, even in genomes which have not been considered to undergo genomic duplication [35]. The small genome of Arabidopsis has been reported to have undergone at least one round of duplication by different groups [12, 18]. Here using a different method, we discover that 75.0% of the Arabidopsis genome sequences are in duplicated regions and a significant portion of sequences have multiple copies. The previous studies in the rice genome have been focused on the large obvious duplicated segments, produced by the relatively recent duplication events [15, 36]. Here, we detect 76.2% of rice sequences in duplicated regions, and 42.9% have multiple copies.

The possibility of constructing the monocot-dicot comparative genetic map has been discussed [37] based on the comparison of Arabidopsis and rice sequences [38, 39]. However, a comprehensive detection of homologous regions between these two genomes has not been available. Based on gene colinearity, we detected homologous regions between Arabidopsis and rice, accounting for 32.9% and 16.9% of each genome. All homologous segments were shorter than 0.6 Mb in length, indicating the massive genome rearrangements in both genomes after the monocot-dicot divergence. Though the short homologous segments make it difficult to construct the comparative genetic map between monocot and dicot, the homologues in colinearity found in this study may provide clues for further work in comparative genomics.

Conclusion

We develop an algorithm to detect homologous chromosomal segments with conserved gene order, and we propose a statistical approach to estimate parameters and evaluate the significance of potential homology. We apply this approach to rice and Arabidopsis with high efficiency to detect potential colinear regions and evaluate their significance. We find many more homologous chromosomal segments in rice genomes than previously reported, which consolidated the inference that a polyploidy had occurred in the common ancestor of grasses. We also find many small colinear segments between rice and Arabidopsis genomes, providing clues to the evolutionary history of monocots and dicots.

Availability and requirements

  • Project name: ColinearScan

  • Project home page: http://colinear.cbi.pku.edu.cn

  • Operating systems: Linux, Unix

  • Programming languages: C++, PERL

  • Other requirements: Standard C++ Library, BioPerl and other PERL modules including Getopt::Long and Pod::Usage

  • License: GPL

Notes

Declarations

Acknowledgements

We thank Manyuan Long, Wentian Li, Liping Wei, Min Zhao, Ge Gao, Di Liu for helpful discussions. This study was supported by the National Key Basic Research Program of China (973 No 2003CB715900), Natural Science Foundation of China (NSFC, No 90408015 and 30170232) and the China High-tech Program, the China High Technology Platform and the Program of Introducing Talents of Discipline to Universities, no. B06001.

Authors’ Affiliations

(1)
College of Life Sciences, National Laboratory of Plant Genetic Engineering and Protein Engineering, Center of Bioinformatics, Peking University
(2)
College of Mathematics, Hebei Polytechnic University
(3)
Laboratory of Systematic and Evolutionary Botany, Institute of Botany, Chinese Academy of Sciences
(4)
Beijing Genomics Institute, Chinese Academy of Sciences

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© Wang et al; licensee BioMed Central Ltd. 2006

This article is published under license to 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.