Marília D. V. Braga

Footprints of inversions at present and past pseudoautosomal boundaries in human sex chromosomes.

The human sex chromosomes have stopped recombining gradually, which has left five evolutionary strata on the X chromosome. Y inversions are thought to have suppressed X–Y recombination but clear evidence is missing. Here, we looked for such evidence by focusing on a region—the X-added region (XAR)—that includes the pseudoautosomal region and the most recent strata 3 to 5. We estimated and analyzed the whole set of parsimonious scenarios of Y inversions given the gene order in XAR and its Y homolog. Comparing these to scenarios for simulated sequences suggests that the strata 4 and 5 were formed by Y inversions. By comparing the X and Y DNA sequences, we found clear evidence of two Y inversions associated with duplications that coincide with the boundaries of strata 4 and 5. Divergence between duplicates is in agreement with the timing of strata 4 and 5 formation. These duplicates show a complex pattern of gene conversion that resembles the pattern previously found for AMELXY, a stratum 3 locus. This suggests that this locus—despite AMELY being unbroken—was possibly involved in a Y inversion that formed stratum 3. However, no clear evidence supporting the formation of stratum 3 by a Y inversion was found, probably because this stratum is too old for such an inversion to be detectable. Our results strongly support the view that the most recent human strata have arisen by Y inversions and suggest that inversions have played a major role in the differentiation of our sex chromosomes.

The Solution Space of Sorting by DCJ

In genome rearrangements, the double cut and join (DCJ) operation, introduced by Yancopoulos et al. in 2005, allows one to represent most rearrangement events that could happen in multichromosomal genomes, such as inversions, translocations, fusions, and fissions. No restriction on the genome structure considering linear and circular chromosomes is imposed. An advantage of this general model is that it leads to considerable algorithmic simplifications compared to other genome rearrangement models. Recently, several works concerning the DCJ operation have been published, and in particular, an algorithm was proposed to find an optimal DCJ sequence for sorting one genome into another one. Here we study the solution space of this problem and give an easy-to-compute formula that corresponds to the exact number of optimal DCJ sorting sequences for a particular subset of instances of the problem. We also give an algorithm to count the number of optimal sorting sequences for any instance of the problem. Another interesting result is the demonstration of the possibility of obtaining one optimal sorting sequence by properly replacing any pair of consecutive operations in another optimal sequence. As a consequence, any optimal sorting sequence can be obtained from one other by applying such replacements successively, but the problem of finding the shortest number of replacements between two sorting sequences is still open.

Exploring the Solution Space of Sorting by Reversals, with Experiments and an Application to Evolution

In comparative genomics, algorithms that sort permutations by reversals are often used to propose evolutionary scenarios of rearrangements between species. One of the main problems of such methods is that they give one solution while the number of optimal solutions is huge, with no criteria to discriminate among them. Bergeron et al. started to give some structure to the set of optimal solutions, in order to be able to deliver more presentable results than only one solution or a complete list of all solutions. However, no algorithm exists so far to compute this structure except through the enumeration of all solutions, which takes too much time even for small permutations. Bergeron et al. state as an open problem the design of such an algorithm. We propose in this paper an answer to this problem, that is, an algorithm which gives all the classes of solutions and counts the number of solutions in each class, with a better theoretical and practical complexity than the complete enumeration method. We give an example of how to reduce the number of classes obtained, using further constraints. Finally, we apply our algorithm to analyse the possible scenarios of rearrangement between mammalian sex chromosomes.

Double Cut and Join with Insertions and Deletions

Many approaches to compute the genomic distance are still limited to genomes with the same content, without duplicated markers. However, differences in the gene content are frequently observed and can reflect important evolutionary aspects. While duplicated markers can hardly be handled by exact models, when duplicated markers are not allowed, a few polynomial time algorithms that include genome rearrangements, insertions and deletions were already proposed. In an attempt to improve these results, in the present work we give the first linear time algorithm to compute the distance between two multichromosomal genomes with unequal content, but without duplicated markers, considering insertions, deletions and double cut and join (DCJ) operations. We derive from this approach algorithms to sort one genome into another one also using DCJ operations, insertions and deletions. The optimal sorting scenarios can have different compositions and we compare two types of sorting scenarios: one that maximizes and one that minimizes the number of DCJ operations with respect to the number of insertions and deletions. We also show that, although the triangle inequality can be disrupted in the proposed genomic distance, it is possible to correct this problem adopting a surcharge on the number of non-common markers. We use our method to analyze six species of Rickettsia, a group of obligate intracellular parasites, and identify preliminary evidence of clusters of deletions.

Genomic distance with DCJ and indels

The double cut and join (DCJ) operation, introduced by Yancopoulos, Attie and Friedberg in 2005, allows one to represent most rearrangement events in genomes. However, a DCJ cannot perform an insertion or a deletion and most approaches under this model consider only genomes with the same content and without duplications, including the linear time algorithms to compute the DCJ distance and to find an optimal DCJ sorting sequence. In this work, we compare two genomes with unequal content, but still without duplications, and present a new linear time algorithm to compute the genomic distance, considering DCJ and indel operations. With this method we find preliminary evidence of the occurrence of clusters of deletions in the Rickettsia bacterium.

The Potential of Family-Free Genome Comparison

Many methods in computational comparative genomics require gene family assignments as a prerequisite. While the biological concept of gene families is well established, their computational prediction remains unreliable. This paper continues a new line of research in which family assignments are not presumed. We study the potential of several family-free approaches in detecting conserved structures, genome rearrangements and in reconstructing ancestral gene orders.

Restricted DCJ Model: Rearrangement Problems with Chromosome Reincorporation

We study three classical problems of genome rearrangement--sorting, halving, and the median problem--in a restricted double cut and join (DCJ) model. In the DCJ model, introduced by Yancopoulos et al., we can represent rearrangement events that happen in multichromosomal genomes, such as inversions, translocations, fusions, and fissions. Two DCJ operations can mimic transpositions or block interchanges by first extracting an appropriate segment of a chromosome, creating a temporary circular chromosome, and then reinserting it in its proper place. In the restricted model, we are concerned with multichromosomal linear genomes and we require that each circular excision is immediately followed by its reincorporation. Existing linear-time DCJ sorting and halving algorithms ignore this reincorporation constraint. In this article, we propose a new algorithm for the restricted sorting problem running in O(n log n) time, thus improving on the known quadratic time algorithm. We solve the restricted halving problem and give an algorithm that computes a multilinear halved genome in linear time. Finally, we show that the restricted median problem is NP-hard as conjectured.

On the weight of indels in genomic distances

BackgroundClassical approaches to compute the genomic distance are usually limited to genomes with the same content, without duplicated markers. However, differences in the gene content are frequently observed and can reflect important evolutionary aspects. A few polynomial time algorithms that include genome rearrangements, insertions and deletions (or substitutions) were already proposed. These methods often allow a block of contiguous markers to be inserted, deleted or substituted at once but result in distance functions that do not respect the triangular inequality and hence do not constitute metrics.ResultsIn the present study we discuss the disruption of the triangular inequality in some of the available methods and give a framework to establish an efficient correction for two models recently proposed, one that includes insertions, deletions and double cut and join (DCJ) operations, and one that includes substitutions and DCJ operations.ConclusionsWe show that the proposed framework establishes the triangular inequality in both distances, by summing a surcharge on indel operations and on substitutions that depends only on the number of markers affected by these operations. This correction can be applied a posteriori, without interfering with the already available formulas to compute these distances. We claim that this correction leads to distances that are biologically more plausible.

On the family-free DCJ distance and similarity

Structural variation in genomes can be revealed by many (dis)similarity measures. Rearrangement operations, such as the so called double-cut-and-join (DCJ), are large-scale mutations that can create complex changes and produce such variations in genomes. A basic task in comparative genomics is to find the rearrangement distance between two given genomes, i.e., the minimum number of rearragement operations that transform one given genome into another one. In a family-based setting, genes are grouped into gene families and efficient algorithms have already been presented to compute the DCJ distance between two given genomes. In this work we propose the problem of computing the DCJ distance of two given genomes without prior gene family assignment, directly using the pairwise similarities between genes. We prove that this new family-free DCJ distance problem is APX-hard and provide an integer linear program to its solution. We also study a family-free DCJ similarity and prove that its computation is NP-hard.

Genomic distance under gene substitutions

BackgroundThe distance between two genomes is often computed by comparing only the common markers between them. Some approaches are also able to deal with non-common 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.ResultsHere we propose a new model that sorts non-common 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.ConclusionsComparing genomes with unequal content, but without duplicated markers, we give a linear time algorithm to compute the genomic distance considering substitutions and double-cut-and-join (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.