Volume 12 Supplement 9
Proceedings of the Ninth Annual Research in Computational Molecular Biology (RECOMB) Satellite Workshop on Comparative Genomics
Genomic distance under gene substitutions
 Marília D V Braga^{1}Email author,
 Raphael Machado^{1},
 Leonardo C Ribeiro^{1} and
 Jens Stoye^{2}
DOI: 10.1186/1471210512S9S8
© Braga et al; licensee BioMed Central Ltd. 2011
Published: 5 October 2011
Abstract
Background
The distance between two genomes is often computed by comparing only the common markers between them. Some approaches are also able to deal with noncommon markers, allowing the insertion or the deletion of such markers. In these models, a deletion and a subsequent insertion that occur at the same position of the genome count for two sorting steps.
Results
Here we propose a new model that sorts noncommon markers with substitutions, which are more powerful operations that comprehend insertions and deletions. A deletion and an insertion that occur at the same position of the genome can be modeled as a substitution, counting for a single sorting step.
Conclusions
Comparing genomes with unequal content, but without duplicated markers, we give a linear time algorithm to compute the genomic distance considering substitutions and doublecutandjoin (DCJ) operations. This model provides a parsimonious genomic distance to handle genomes free of duplicated markers, that is in practice a lower bound to the real genomic distances. The method could also be used to refine orthology assignments, since in some cases a substitution could actually correspond to an unannotated orthology.
Background
The genomic distance is often computed taking into consideration only the common markers, that occur in both genomes [1–3]. Approaches to deal with unique markers (that occur in only one genome) also exist, but usually allowing only insertions or deletions of these markers. Insertions and deletions can be shortly called indels. In [4], the operations allowed are inversions and indels, while the models given in [5] and [6] consider indels and the double cut and join (DCJ) operation [7], that is able to represent most large scale mutation events in genomes, such as inversions, translocations, fusions and fissions. The mentioned approaches assign the same weight to all rearrangement operations, including indels, regardless of the size of the affected regions and the particular types of the operations. A drawback in these models is that, if a deletion and a subsequent insertion occur at the same position of the genome, the cost is the same as a deletion and an insertion in different positions.
We analyze genomes with unequal content, but without duplicated markers and extend the results given in [6] to develop a linear time algorithm that exactly computes the genomic distance with substitutions and DCJ operations. The objective of this model is to provide a parsimonious genomic distance to handle genomes free of duplicated markers, that in practice is a lower bound to the real genomic distances. In the present work, we do not study algorithms to generate parsimonious sorting scenarios. Nevertheless, in the analysis of the evolution of human chromosomes X and Y, we manually obtain a parsimonious evolutionary scenario under our model, that is coherent with the results given in [10].
In the remainder of this section we introduce some concepts given in [1] and [6] and define the operation that substitutes markers in a genome  these are the basis of the method that we will present here.
Preliminaries
In the present study duplicated markers are not allowed. Given two genomes A and B, possibly with unequal content, we denote by the “reduced” genome [4], that is the set of markers that occur once in A and once in B. Moreover, the set contains the markers that occur only in A and the set contains the markers that occur only in B. The markers in sets and are also called unique markers. Observe that the sets , and are disjoint.
In the following we adopt definitions which we have given in [6] (some of them are generalizations of concepts introduced by Bergeron et al. [1]).
adjacencies
Each one of the two ends of a linear chromosome is called a telomere and is represented by the symbol ○. For each marker , denote its two extremities by g^{ t } (tail) and g^{ h } (head). A adjacency in genome A (respectively in genome B) is in general a linear string v = γ_{1}ℓγ_{2}, such that γ_{1} and γ_{2} are telomeres or extremities of markers of and ℓ, the string composed of the markers that are between γ_{1} and γ_{2} in A (respectively in B), contains no marker that also belongs to . The string ℓ is said to be the label of v, and the extremities γ_{1} and γ_{2} are said to be adjacent. If ℓ is a nonempty string, v is said to be labeled, otherwise v is said to be clean.
A adjacency γ_{1}ℓγ_{2} can also be represented by . Furthermore, ◦ℓ◦ represents a linear chromosome composed only of markers that are not in . In the same way, a adjacency given by a label ℓ corresponds to a whole circular chromosome composed only of markers that are not in . This is the only case of a adjacency in which we have a circular instead of a linear string.
Two genomes A and B can then be represented by the sets and , containing their adjacencies. For the two genomes in Figure 2, we have , and .
The DCJ operation
A cut performed on a genome A separates two adjacent markers of A. A cut affects a adjacency v of as follows: if v is linear, the cut is done between two symbols of v, creating two open ends in two separate linear strings; if v is circular, the cut creates two open ends in one linear string. A doublecut and join or DCJ applied on a genome A is the operation that generally performs two cuts in , creating four open ends, and joins these open ends in a different way. A DCJ operation can correspond to several rearrangement events, such as an inversion, a translocation, a fusion, or a fission [7].
We represent by ({γ_{1}ℓ_{1}ℓ_{4}γ_{4} , γ_{3} ℓ_{3}  ℓ_{2}γ_{2} } → {γ_{1}ℓ_{1} ℓ_{2} γ_{2}, γ_{3} ℓ_{3}ℓ_{4} γ_{4} }) a DCJ applied on γ_{1}ℓ_{1}ℓ_{4}γ_{4} and γ_{3}ℓ_{3}ℓ_{2}γ_{2} , that creates γ_{1}ℓ_{1}ℓ_{2}γ_{2} and γ_{3}ℓ_{3}ℓ_{4}γ_{4}. Observe that one or more extremities among γ_{1}, γ_{2}, γ_{3} and γ_{4} can be equal to ○ (a telomere), as well as one or more labels among ℓ_{1}, ℓ_{2}, ℓ_{3} and ℓ_{4} can be equal to ε (the empty string). Particular cases include circular adjacencies and are described in [6].
Adjacency graph and the DCJ distance
The adjacency graph AG(A, B) [1] is the bipartite graph that has a vertex for each adjacency in and a vertex for each adjacency in . Then, for each , we have one edge connecting the vertex in and the vertex in that contain g^{ h } and one edge connecting the vertex in and the vertex in that contain g^{ t }.
The number of ABpaths in AG(A, B) is always even and a DCJ operation can be of three types [1, 6]: optimal when it either increases the number of cycles by one, or the number of ABpaths by two; neutral when it does not affect the number of cycles and ABpaths; or counteroptimal when it either decreases the number of cycles by one, or the number of ABpaths by two.
Singletons, ABpaths composed of one single edge, and cycles composed of two edges are said to be DCJsorted. Longer paths and cycles are said to be DCJunsorted. The procedure of using DCJ operations to turn AG(A, B) into DCJsorted components is called DCJsorting of A into B. The DCJ distance of A and B, denoted by d_{ DCJ }(A, B), corresponds to the minimum number of steps required to do a DCJsorting of A into B and can be easily obtained:
Theorem 1 ( [1]) Given two genomes A and B without duplicated markers, we have , where is the set of common markers between A and B, and c and b are the number of cycles and of ABpaths in AG(A, B).
Runs of unique markers
Substitutions
The unique markers in and are represented in AG (A, B) as labels and singletons and, in order to sort A into B, they also have to be considered. Here we propose a model in which only the following operation can be applied to unique markers. A substitution is an operation that affects the label of one single adjacency, by substituting contiguous markers in this label.
Consider the labels ℓ_{1} and ℓ_{2}, where ℓ_{1} = m and ℓ_{2} = n. The substitution of ℓ_{1} by ℓ_{2} in a adjacency is represented by (γ_{1}ℓ_{3}ℓ_{1}ℓ_{4}γ_{2} → γ_{1}ℓ_{3}ℓ_{2}ℓ_{4}γ_{2}) (for better reading in our notation we omit the curly set brackets for singleton sets). One or both extremities among γ_{1} and γ_{2} can be equal to ○ (a telomere), as well as one or both labels among ℓ_{3} and ℓ_{4} can be equal to ε (the empty string). The substitution of ℓ_{1} by ℓ_{ 2 } in a circular singleton is represented by (ℓ_{1}ℓ_{3} → ℓ_{2}ℓ_{3}). Observe that at most one chromosome can be entirely substituted at once (but we do not allow the substitution of a linear by a circular chromosome and viceversa). Moreover, if m = 0, we have an insertion of n contiguous markers. On the other hand, if n = 0, we have a deletion of m contiguous markers. Thus, insertions and deletions, also called indels, are special cases of substitutions.
The DCJsubstitution distance of A and B, denoted by , is the minimum number of DCJs and substitutions required to transform A into B. Since substitutions include indels, is upper bounded by the DCJindel distance, the minimum number of DCJ and indel operations required to transform A into B, that can be computed in linear time [6]. In the present work we give an approach to exactly compute also in linear time.
Results and discussion
The main result of the present study is an exact formula to compute the DCJsubstitution distance in linear time. We achieve this formula by developing the substitutionpotential of two genomes, a property that allows us to obtain a good upper bound to the genomic distance with DCJ operations and substitutions. Then we show how some special DCJ operations reduce the overall number of substitutions and obtain the exact formula. Although the objective of this model is to provide a parsimonious genomic distance, that in practice is a lower bound to real distances, we run some experiments on data from human X and Y chromosomes and obtained a parsimonious sorting scenario that is coherent with the results available in the literature. We also observe that the DCJsubstitution method could be used to refine orthology assignments.
The substitutionpotential
Observe that a adjacency with a nonempty label ℓ can be cut in at least two different positions, either before or after ℓ. Since the position of the cut does not change the effect of the DCJ on d_{DCJ}(A, B), we can choose to cut at positions that allow the concatenation of the labels of the original adjacencies. As a consequence, a set of labels of one genome can be accumulated with DCJ operations. In particular, when we apply optimal DCJs on only one component of the adjacency graph, we can accumulate an entire run in a single adjacency:
Proposition 1 ( [6]) A run can be entirely accumulated in the label of one single adjacency with optimal DCJ operations.
Given a DCJ operation ρ, let Λ_{0} and Λ_{1} be, respectively, the number of runs in AG (A, B) before and after ρ. We define ∆Λ(ρ) = Λ _{1}– Λ _{0}.
Proposition 2 ( [6]) Given any DCJ operation ρ, we have ∆Λ(ρ) ≥ – 2.
In order to obtain the exact formula for the DCJsubstitution distance, we will first analyze the components of the adjacency graph separately. Given two genomes A and B and a component C ∈ AG (A, B), we denote by d_{ DCJ }(C) the minimum number of DCJ operations required to do a separate DCJsorting in C, applying DCJs on vertices of C (or vertices that result from DCJs applied on vertices that were in C). It is possible to do a separate DCJsorting using only optimal DCJs in any component of AG (A, B), thus, in other words, d_{ DCJ }(A, B) = ∑_{ C }_{∈}_{ AG }_{(}_{ A }_{,}_{ B }_{)}d_{ DCJ }(C) [2]. In [6] we have already defined the indelpotential of a component, denoted by λ(C), that is the minimum number of runs that we can obtain by DCJsorting C with optimal DCJ operations only, and can be computed with the formula given in the next proposition.
Proposition 3 ( [6]) Given a component C in AG(A, B), we have , if Λ(C) ≥ 1. Otherwise λ(C) = 0.
Similarly, here we denote by σ(C) the substitutionpotential of a component C, that is the minimum number of substitutions that we can obtain by DCJsorting C with optimal DCJ operations only. In order to find a formula to compute σ(C), we first obtain a stronger version of Proposition 1 where not only the labels of a run are accumulated into a single adjacency, but pairs of consecutive runs are accumulated into adjacent adjacencies (that are adjacencies connected by a single edge in the adjacency graph).
Proposition 4 ( [6]) If γ_{1}γ_{2} is a clean adjacency in a DCJunsorted component C of AG(A, B), such that neither γ_{1} nor γ_{2} are telomeres, then it is always possible to extract a clean cycle from C with an optimal DCJ operation.
Proposition 5 Two consecutive runs in a component C can be entirely accumulated into the labels of two adjacent adjacencies of C with optimal DCJs.
Proof: By Proposition 1 we assume that two consecutive runs of C are accumulated into adjacencies v_{ A } and v_{ B }. If v_{ A } and v_{ B } are not adjacent, there are only clean adjacencies between v_{ A } and v_{ B } in C. By Proposition 4, we can apply optimal DCJs to extract clean cycles until v_{ A } and v_{ B } are adjacent.
Pairs of consecutive runs that are accumulated into adjacent adjacencies can be extracted into a labeled DCJsorted component, that can be sorted with one substitution. Observe that minimizing the number of pairs of consecutive runs is equivalent to minimizing the total number of runs. Hence, we can determine the substitutionpotential from the indelpotential.
Proposition 6 Given a component C in AG (A, B), we have , if Λ(C) ≥ 1. Otherwise σ(C) = 0.
The formulas to compute λ(C) and σ(C), given in Propositions 3 and 6 above, are indeed very similar. Consequently, many of the results obtained in [6] can be adapted to the new substitutionpotential. Let σ_{0} and σ_{1} be, respectively, the sums of the number σ for the components of the adjacency graph before and after a DCJ operation ρ. We then define ∆σ(ρ) = σ_{1}– σ_{0}. Furthermore, let ∆_{ dcj }(ρ) be respectively 0, +1 and +2 depending whether ρ is optimal, neutral or counteroptimal. We also define ∆d(ρ) = ∆_{ dcj }(ρ) + ∆σ(ρ).
Proposition 7 Given a DCJ operation ρ acting on a single component, we have ∆d(ρ) ≥ + 2 if ρ is counteroptimal, or ∆d(ρ) ≥ 0 if ρ is neutral.
We denote by the minimum number of DCJs and substitutions required to sort separately a component C of AG (A, B). The definition of σ and Proposition 7 guarantee that .
Observe that, if C is a singleton in the adjacency graph, , corresponding to the insertion or the deletion of the whole chromosome. We do not allow the substitution of a linear by a circular singleton and viceversa. However, each pair composed by a singleton in genome A and a singleton in genome B (such that both are linear or both are circular) can be sorted with one single substitution, which saves one sorting step per pair. Let P_{ L } and P_{ C } be, respectively, the maximum number of disjoint pairs of linear and circular singletons in the adjacency graph. Together with the DCJsubstitution distance per component, these numbers give a good upper bound for :
The formula given by Lemma 1 above corresponds to the exact distance for a particular set of genomes. Given a adjacency γℓ○ of a genome A such that γ≠○, then γ is said to be a tail of a linear chromosome in A. Two genomes are cotailed if their sets of tails are equal (this includes two genomes composed only of circular chromosomes).
However, for non cotailed genomes the use of DCJs applied to two components of the adjacency graph can lead to a shorter sequence of operations sorting one genome into another, as we will see in the next section.
The DCJsubstitution distance
Recall that ∆σ(ρ) = σ_{1}– σ_{0}, where σ_{0} and σ_{1} are the sums of the number σ for the components of the adjacency graph before and after ρ. A DCJ operation ρ that acts on two components of the adjacency graph is called recombination.
Proposition 8 Given any recombination ρ, we have ∆σ(ρ) ≥ –2.
Proof: Only the recombinations that decrease or do not change the number of runs (∆Λ ≤ 0) have to be analyzed (we can not have ∆σ ≤ –1 if the number of runs increases). Consider the recombination of two paths with i and j runs, that result in two new paths with i′ and j′ runs. The best we can have is when i and j are multiples of 4, i′ and j′ are multiples of 4 minus 1 and ∆Λ = –2, that gives: . The analysis of recombinations involving cycles is analogous.
All recombinations involving at least one cycle are counteroptimal and any counteroptimal recombination has ∆d ≥ 0, thus only path recombinations can have ∆d ≤ –1. The following definitions are similar to those given in [6], except that here we have a larger number of labeled path types.
Consider an integer i ≥ 0. For a second integer k ∈ {1, 3}, let (respectively ) be a sequence with odd 4i + k runs, starting and ending with an run (respectively run). Similarly for k ∈ {2, 4}, let (respectively ), be a sequence with even 4i + k runs, starting with an run (respectively run) and ending with a run (respectively run). An empty sequence (with no run) is represented by ε. Then each one of the notations AA_{ ε }, , , , , , , BB_{ ε }, , , , , , , AB_{ ε }, , , , , , , and represents a particular type of path (AA, BB or AB) with a particular structure of runs (ε, , , , , , , , or ).
Path recombinations that have ∆d ≤ –1 and allow the best reuse of the resultants.
sources  resultants  ∆σ  ∆_{ dcj }  ∆d 

 • + •  –2  0  –2 

 –2  +1  –1 

 –2  +1  –1 

 –2  +1  –1 

 –2  +1  –1 

 –2  +1  –1 

 –2  +1  –1 

 –1  0  –1 

 –1  0  –1 

 –1  0  –1 

 –1  0  –1 
 • + •  –1  0  –1 
 • + •  –1  0  –1 
 • + •  –1  0  –1 
 • + •  –1  0  –1 
 • + •  –1  0  –1 
 • + •  –1  0  –1 
 • + •  –1  0  –1 
 • + •  –1  0  –1 
 • + •  –1  0  –1 
 • + •  –1  0  –1 
 • + •  –1  0  –1 
 • + •  –1  0  –1 
 • + •  –1  0  –1 
 • + •  –1  0  –1 
 • + •  –1  0  –1 
 • + •  –1  0  –1 
 • + •  –1  0  –1 
 • + •  –2  +1  –1 
Recombinations that have ∆d = 0 and create resultants that can be used in recombinations with ∆d ≤ –1 (listed in Table 1).
sources  resultants  ∆σ  ∆_{ dcj }  ∆d 


 –2  +2  0 

 –1  +1  0 

 –1  +1  0 

 –1  +1  0 

 –1  +1  0 

 –2  +2  0 

 –1  +1  0 

 –1  +1  0 

 –1  +1  0 

 –1  +1  0 
Proposition 9 The recombinations with ∆d = 0 involving cycles or circular singletons cannot create new components that can be used as sources of recombinations listed in Tables 1 and 2.
sources  resultants  ∆d  scr  

U 
 2•  –2  –1 
V 
 4•  –3  –3/4 
 4•  –3  –3/4  
W 
 3•  –2  –2/3 
 3•  –2  –2/3  
 3•  –2  –2/3  
 3•  –2  –2/3  

 –2  –2/3  

 –2  –2/3  

 –2  –2/3  

 –2  –2/3  
X  Recombinations from Table 1 with ∆d = –1  –1  –1/2  
Y 
 4•  –2  –1/2 
 4•  –2  –1/2  
Z 
 3•  –1  –1/3 
 3•  –1  –1/3  
 3•  –1  –1/3  
 3•  –1  –1/3  
 3•  –1  –1/3  
 3•  –1  –1/3  
 3•  –1  –1/3  
 3•  –1  –1/3  

 –1  –1/3  

 –1  –1/3  

 –1  –1/3  

 –1  –1/3  

 –1  –1/3  

 –1  –1/3 
The deductions shown in Table 3 can be computed with an approach that greedily maximizes the number of recombinations in U, V, W, X, Y and Z in this order. The U part contains only one operation and the two groups in V are mutually exclusive after applying U. The part W is then the application of all possible remaining groups of two operations with ∆d = –2. Similarly, the part X is only the application of all possible remaining operations with ∆d = –1. After X, the two groups in Y are mutually exclusive and then the same happens to the groups in Z. Although some groups in W, X and Z have some reusable resultants, those are actually never reused (if operations that are lower in the table use as sources resultants from higher operations, the sources of all referred operations would be previously consumed in operations that occupy even higher positions in the table). Due to this fact, the number of operations in U, V , W, X, Y and Z depends only on the initial number of each type of component.
With the results presented in this section we have an exact formula to compute the DCJsubstitution distance:
where P_{ L } and P_{ C } are the numbers of disjoint pairs of linear and circular singletons and U, V, W, X, Y and Z are computed as described above.
The formula given in Theorem 3 is analogous to the one which we have obtained in [6] to compute the DCJindel distance. Both formulas depend on factors that can be computed in linear time [6].
Triangular inequality
Note that, since only unique markers can be substituted in this model, we avoid the “free lunch problem”, mentioned in [5], that is the possibility of transforming any genome A into any genome B by simply substituting the whole content of A by the whole content of B. However, the triangular inequality can be disrupted in the DCJsubstitution distance. In other words, given any three genomes A, B and C without duplicated markers, there is no guarantee that the triangular inequality holds. In a companion paper [11] we provide an efficient way to establish the triangular inequality a posteriori in both the DCJindel [6] and the DCJsubstitution distances.
Experiments
The objective of this model is to provide a parsimonious genomic distance, that in practice is a lower bound to real distances. Nevertheless, we could run some experiments on data from human X and Y chromosomes and obtained a parsimonious sorting scenario that is coherent with the results available in the literature. During evolution, a portion of the human Y chromosome has become increasingly subjected to local mutations, while the X chromosome remained relatively conserved, as we will see in the following. Human X and Y chromosomes are very different and, while X is 155 Mbp long, the Y chromosome is 58 Mbp long. However, they still share pseudoautosomal regions at both extremities and are believed to have evolved from an identical autosomal pair [12] (the autosomes are all nonsex chromosomes). Current theories suggest that the pseudoautosomal region, which originally covered the whole chromosomes, was successively pruned by a few big inversions on the Y chromosome [13] (we call these inversions pruning). After each pruning inversion, several mutations seem to have occurred on the affected part of the Y chromosome, while X remained “closer” to the common ancestor.
Discussion
Our method was designed to find gene mutations, but it could also help to improve orthology assignments, that are the computational prediction of orthologous pairs of genes from different species. No orthology predictor is able to find all assignments correctly. In particular, when comparing two different species, some pairs of orthologous genes that are below the predictor threshold remain unassigned. Since our substitutions establish a relation between different genes in the two compared genomes, they correspond to candidates to be assigned as orthologous genes.
Conclusions and future work
In this work we presented a new model to compare two genomes with unequal content, but without duplicated markers, using substitutions and DCJ operations, and developed a linear time algorithm to exactly compute the DCJsubstitution distance.
Although the objective of this model is to provide a parsimonious genomic distance, that in practice is a lower bound to real distances, based on our method we have manually reconstructed a parsimonious evolutionary scenario of human chromosomes X and Y. We considered biological constraints that are specific to this case and obtained a scenario that is coherent with the results given in the literature.
By reconstructing a parsimonious scenario that minimizes substitutions, we may identify genomic regions that were subject to continuous mutations during evolution and could have a common evolutionary origin. Currently our method is only able to compute the genomic distance, but in a future work we intend to study the space of all parsimonious sorting scenarios and develop methods to systematically identify such regions.
The DCJsubstitution model could also be used to refine orthology assignments, since in some cases a substitution could actually correspond to an unannotated orthology. We also plan on exploring the use of our method in refining orthology in a future work.
Declarations
Acknowledgements
This research was partially supported by the Brazilian research agencies CNPq (grant PROMETRO 563087/20102) and FAPERJ (grant INST E26/111.837/2010).
This article has been published as part of BMC Bioinformatics Volume 12 Supplement 9, 2011: Proceedings of the Ninth Annual Research in Computational Molecular Biology (RECOMB) Satellite Workshop on Comparative Genomics. The full contents of the supplement are available online at http://www.biomedcentral.com/14712105/12?issue=S9.
Authors’ Affiliations
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