Sorting by restricted-length-weighted reversals

Classical sorting by reversals uses the unit-cost model, that is, each reversal consumes an equal cost. This model limits the biological meaning of sorting by reversal. Bender and his colleagues extended it by assigning a cost function f（1） = l^a for all a≥ 0, where l is the length of the reversed subsequence. In this paper, we extend their results by considering a model in which long reversals are prohibited. Using the same cost function above for permitted reversals, we present tight or nearly tight bounds for the worst-case cost of sorting by reversals. Then we develop algorithms to approximate the optimal cost to sort a given 0/1 sequence as well as a given permutation. Our proposed problems are more biologically meaningful and more algorithmically general and challenging than the problem considered by Bender et al. Furthermore, our bounds are tight and nearly tight, whereas our algorithms provide good approximation ratios compared to the optimal cost to sort 0/1 sequences or permutations by reversals.     

Sorting by weighted reversals, transpositions, and inverted transpositions.

During evolution, genomes are subject to genome rearrangements that alter the ordering and orientation of genes on the chromosomes. If a genome consists of a single chromosome (like mitochondrial, chloroplast, or bacterial genomes), the biologically relevant genome rearrangements are (1) inversions--also called reversals--where a section of the genome is excised, reversed in orientation, and reinserted and (2) transpositions, where a section of the genome is excised and reinserted at a new position in the genome; if this also involves an inversion, one speaks of an inverted transposition. To reconstruct ancient events in the evolutionary history of organisms, one is interested in finding an optimal sequence of genome rearrangements that transforms a given genome into another genome. It is well known that this problem is equivalent to the problem of "sorting" a signed permutation into the identity permutation. In this paper, we provide a 1.5-approximation algorithm for sorting by weighted reversals, transpositions and inverted transpositions for biologically realistic weights.     

Sorting by reversals and block-interchanges with various weight assignments

Background  

A classical problem in studying genome rearrangements is understanding the series of rearrangement events involved in transforming one genome into another in accordance with the parsimonious principle when two genomes with the same set of genes differ in gene order. The most studied event is the reversal, but an increasing number of reports have considered reversals along with other genome rearrangement events. Some recent studies have investigated the use of reversals and block-interchanges simultaneously with a weight proportion of 1:2. However, there has been less progress towards exploring additional combinations of weights.     

Genetic counseling in reciprocal translocations

Segregation modes of human reciprocal translocations are briefly described. Risk figures and mode of imbalance at term differ greatly from one translocation to another. The causes of these variations are analysed and a genetic counselling proposed for each case.     

Structural differences in reciprocal translocations

Summary Interchange segment sizes and the sizes of chromosome imbalance arising from the different modes of meiotic segregation were measured in a selected sample of 20 reciprocal translocations (Rcp). The Rcp were selected by two modes of ascertainment: (I) neonates with an unbalanced form of the translocation, and (II) couples with recurrent spontaneous abortions without evidence of full-term translocation aneuploid offspring.The measurements (% of haploid autosomal length: %HAL) were plotted as the observed or potential chromosomal imbalance with monosomy (abscissa) and trisomy (ordinate). It was found that (a) the interchange segments were larger in the spontaneous abortion Rcp, (b) that all of the imbalances observed in full-term neonates plotted close to the origin and to the left of the line joining 4% trisomy to 2% monosomy, and (c) the imbalances observed in the neonates in each individual Rcp were of the smallest size possible arising by any segregation mode.It was concluded that a major factor in the survival to term of aneuploid conceptuses is the size (proportion of genome) of the chromosome abnormality, irrespective of the origin of the chromosome regions. These results are discussed in relation to their use as a model to evaluate the risk of abnormal offspring in the progeny of translocation heterozygotes (the Chromosome Imbalance Size-Viability Model).     

Topology of constitutional reciprocal translocations in metaphase

We studied in 39 carriers of 26 reciprocal translocations (including five de novo and seven of indeterminate occurrence) the metaphase localization of the derivative chromosomes, their normal non-homologous counterparts (here called A and B), and two control pairs (C and D). In eight familial translocations, we analysed two to five carriers. We digitally captured 10 G-banded lymphocyte metaphases per individual and measured in microns the largest diameter (d) of the metaphase and six intercentromeric distances: (1) der A<-->der B (problem distance 1, pd1), (2) der A<-->B (pd2), (3) der B<-->A (pd3), (4) A<-->B (control distance 1, cd1), (5) the smaller distance between C and D (cd2) and (6) the largest distance between C and D (cd3); in addition, the average between C and D (cd4) was calculated. We used the formula Delta = 100(cd - pd)/d 12 times per metaphase, compared each pd vs. each cd, and tested the differences by the Wilcoxon matched-pair test. Although, in the whole sample there were not significant differences respect to cd1, this distance emerged as the proper control. In the eight familial translocations, the three pd vs. cd1 comparisons revealed that in 19/24 times the pd was smaller but only once reached significance (cd1 vs. pd2 in t[3;4]). In the analysis per individual the pd was smaller than cd1 in 19 (pd1), 22 (pd2) and 22 (pd3) cases although only twice reached significance. We conclude that in some translocations, the derivative chromosomes actually lie close from each other or from a normal non-homologous counterpart.     

Analysis of reciprocal translocations by chromosome painting: applications and limitations of the technique.

Fluorescent in situ hybridization with chromosome-specific DNA libraries (chromosome painting) is an important new method for assessing chromosome rearrangements. In the research presented in this paper, two familial reciprocal translocations have been studied in the balanced and unbalanced forms, using both traditional G-banding techniques and chromosome painting. Although for each case two chromosomes were involved in the rearrangement, we found that only one chromosome library was suitable for detecting the translocation. These findings illustrate both the potential and the limitations of chromosome painting as a diagnostic tool in cytogenetics.     

Radiation-induced inversions and reciprocal translocations in Anopheles atroparvus

Inversions and reciprocal translocations were induced in Anopheles atroparvus by irradiation of males with X-rays. 22 aberrations were produced in stocks and were identified as follows: 6 paracentric, 6 pericentric inversions and 10 reciprocal translocations (9 autosomal and 1 sex-linked). Partial sterility in the offspring of this stock is demonstrated. The practical significance of constructing stocks with inversions and translocations for genetic control of pest insects is considered.     

Meiotic analysis of two human reciprocal X-autosome translocations

Two cases of human reciprocal X-autosome translocation, t(X;12) and t(X;2), are described in sterile males, along with meiotic findings. Each carrier had inherited the translocation from his mother. Both showed azoospermia and germ-cell maturation arrest at the primary spermatocyte level, with most cells being arrested at the pachytene stage. A few metaphase I (MI) divisions were found, with occasional metaphase II cells being seen in the t(X;2) carrier. MI air-dried preparations gave clear evidence of chain quadrivalent formation. In the t(X;2) heterozygote, the pairing characteristics of the quadrivalent at pachytene were also analyzed in electron microscopic spreads. Disturbance of pairing around the breakpoints characterized most quadrivalents, and there was evidence in about 20% of the cells that nonhomologous pairing had taken place between the translocated chromosomes and the normal chromosome 2. Comparisons are made with similar nonhomologous pairing configurations seen at pachytene in quadrivalents of male reciprocal X-autosome translocations of the mouse.     

Heterosynapsis in two fertile but hypoprolific boars carriers of reciprocal translocations.

The authors report here two new cases of reciprocal translocations in two fertile and hypoprolific boars. Silver stained synaptonemal complexes in surface-spread pachytene nuclei from a boar heterozygous for a reciprocal translocation, and from another one carrying two different reciprocal translocations, were analyzed by electron microscopy. In such heterozygotes, cross-shaped quadrivalent configurations are expected to form in order to allow homologous pairing. In the same boar, the lengths of the fully synapsed arms of the quadrivalent varies from one quadrivalent to the other and heterosynapsis was obvious. Heterosynapsis was also observed with asymmetrical pairing of the non-homologous partners of the quadrivalent. This heterosynapsis is assumed to be a mechanism preventing spermatocyte loss, but inducing a secondary segregational type of impairment of fertility due to foetal wastage leading to reduced prolificacy.     

Unbalanced reciprocal translocations in cases of Prader-Willi syndrome

Summary A case of Prader-Willi syndrome (PWS) associated with a de novo unbalanced 15q;17q reciprocal translocation presumptively resulting from the tertiary monosomic form of 3:1 meiotic disjunction is described. Twenty-three similar unbalanced translocations have been identified from the literature. The 24 karyotypes are characterised by having 45 chromosomes, monosomy for the pericentromeric region of chromosome 15 (range pter»q11 to q21), and little monosomy of the recipient (non-15) chromosome. Two-thirds of the cases with these karyotypes have phenotypic features of PWS. It seems probable that (i) where unbalanced reciprocal translocations are associated with PWS, they will almost invariably be presumptive segregants of the tertiary monosomic form of 3:1 disjunction and (ii) the majority of cases found with this type of karyotype, particularly it appears when de novo in origin, will be associated with phenotypic features of PWS.     

Meiotic products of two reciprocal translocations studied by multicolor fluorescence in situ hybridization

The sperm products of two male carriers of reciprocal translocations were studied by fluorescence in situ hybridization (FISH) using a combination of three probes for each translocation. One patient carried a t(2;18)(p21;q11.2), the other a t(8;9)(q24.2;q32). The probes selected included a centromeric marker for each chromosome involved in the translocation plus a third probe distal to the translocation breakpoint of one of the translocation chromosomes. This assay identifies alternate, adjacent 1, adjacent 2, and 3:1 types of meiotic products. It allows the identification of recombination events and also estimation of the frequency of diploidy. For the t(2;18), the frequency of normal and balanced sperm and of adjacent 1, adjacent 2, and 3:1 products was 43.6%, 29. 8%, 10.5%, and 12.8%, respectively. Similar segregation patterns had been reported for this donor by direct sperm karyotyping of sperm cells. For the t(8;9), the frequency of normal and balanced sperm and of adjacent 1, adjacent 2, and 3:1 products was 44.4%, 41%, 3.1%, and 9.4%, respectively. The frequency of complementary adjacent 1 products was statistically different in both the t(2;18) (P < 0. 0001) and the t(8;9) (P < 0.0001) carrier. When the number of adjacent 2 products with one translocation chromosome (regardless of normal or derivative) was compared to the number of adjacent 2 products with the second translocation chromosome (again, regardless of normal or derivative), no statistical difference was noted for either the t(2;18) (P = 0.32) or the t(8;9) (P = 0.69). Recombination events within the interstitial segment of chromosome 2 were statistically higher than those seen in chromosome 18 (P < 0. 0001), whereas in chromosomes 8 and 9, recombination in the interstitial segments was similar (P = 0.64). The rate of diploidy was similar in both the t(2;18) (0.5%) and the t(8;9) (0.6%). Thus, FISH provides chromosome information on the sperm products produced by translocation carriers, although it cannot provide an assessment of the full chromosome complement of the spermatozoon.     

Causes and consequences of reciprocal translocations on sex chromosomes

Under XY sex determination, the Y chromosome is only inherited via males, whereas the X chromosome is predominantly found in females. Thus, it is favourable when alleles with high male fitness become associated with the Y chromosome and when alleles with high female fitness become associated with the X chromosome. These favourable associations can be strengthened through linkage. Rearrangements, such as inversions and sex chromosome–autosome fusions, can increase linkage and thereby become favoured (Charlesworth, 2017). In a From the Cover article in this issue of Molecular Ecology, Toups, Rodrigues, Perrin, and Kirkpatrick (2019) present the first genomic analysis of a sex chromosome reciprocal translocation, a particularly dramatic chromosomal rearrangement that modifies linkage with the sex chromosome. As a result of reciprocal translocation, one studied population of the common frog (Rana temporaria, Figure 1) displays a remarkable sex‐determining system in which there are two physically unlinked sex chromosomes that are exclusively cotransmitted (Figure 2a).     

Possible excess of mental handicap and congenital malformations in autosomal reciprocal translocations.

The most pertinent data of the 32 cases of reciprocal autosomal translocations detected in the laboratory since the initiation of the banding-techniques are summarized. One third of these balanced translocations were detected in patients presenting with mental handicap and/or malformative symptoms.     

A woman carrier of two apparently unrelated reciprocal translocations

Summary Two reciprocal balanced translocations involving chromosomes 2, 9, 12, and 18 were found in the karyotype of a woman with a child showing several congenital malformations at birth.Prenatal cytogenetic diagnosis, performed when a second pregnancy occurred, showed a normal chromosome constitution in the foetus.     

Autosomal reciprocal translocations in newborn children and their relatives

Summary In an incidence study of chromosome abnormalities among 5049 consecutive liveborn children at a Danish maternity hospital we found 43 with major chromosome abnormalities; 7 had a balanced reciprocal autosomal translocation. Segregation rate in the families of the 6 cases, who were inherited, showed no significant deviation from unity between carriers of balanced translocations and individuals with normal chromosome constitution. The sex distribution of carriers and those with normal chromosome constitution was normal, and segregation and distribution between carriers and non-carriers was independent of whether the carrier was a male or a female.No relatives with the unbalanced form of the translocations were found in the 6 families studied, but there was a significant increase of stillborns and children, who died during the first week of life, among carriers of the translocation compared with near relatives with a normal chromosome constitution as well as with all children born in the hospital, where they were found, and in the Danish population. The frequency of abortions was higher among the carriers than among the individuals with normal karyotypes. When the figures of stillborn children, children who died during the first week of life as well as spontaneous abortions were added to those who lived, we found that the average number of pregnancies per family was the same in carriers as in their relatives with normal chromosomes. Children with presumptive unbalanced form of translocations thus seem to have ended up as abortions, stillborns or children, who died during their first week of life.None of the probands or their relatives with reciprocal translocations had physical or mental abnormalities which could be associated with the translocation. There was no obvious difference in the physical and mental appearance or health between carriers and relatives with normal chromosome constitutions. Reciprocal autosomal translocations do not seem to have any deleterious effect on mental or physical development.

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Zusammenfassung In einer Untersuchung über Chromosomenanomalien bei 5049 auslesefrei gewonnenen Neugeborenen einer dänischen Entbindungsklinik wurden 43 mit gröberen Anomalien der Chromosomen gefunden; 7 hatten eine balancierte autosomale reziproke Translokation. Eine Untersuchung der Aufspaltung zwischen Trägern balancierter Translokationen und Personen mit normalem Karyotyp zeigte in den 6 Familien, in denen die Translokation ererbt war, keine signifikante Abweichung von der Gleichverteilung. Auch das Geschlechtsverhältnis zwischen Überträgern und Normalen weicht nicht von der Erwartung ab. Die Überträgereigenschaft war vom Geschlecht unabhängig. In den 6 untersuchten Familien fanden sich keine Verwandten mit der unbalancierten Form der Translokation; jedoch war die Zahl der Totgeburten sowie der Kinder, die in der 1. Lebenswoche starben, signifikant erhöht bei den Trägern der Translokation im Vergleich mit nahen Verwandten mit normalem Karyotyp sowie mit den im gleichen Hospital geborenen Kindern und der dänischen Gesamtbevölkerung. Bei den Überträgern war auch die Abortrate höher als bei Personen mit normalem Karyotyp. Wenn man die Zahl der Totgeburten, der Todesfälle der 1. Lebenswoche sowie der spontanen Aborte zu den lebenden hinzuzählt, so erwies sich die durchschnittliche Zahl der Schwangerschaften/Familie in den Überträgerfamilien als gleich hoch wie bei den Verwandten mit normalem Karyotyp. Es sieht also so aus, als ob Früchte mit unbalancierten Translokationen als Aborte, Totgeburten und Todesfälle in der 1. Lebenswoche geendet wären.Keiner der Probanden oder ihrer Verwandten mit reziproker Translokation hatte physische oder geistige Anomalien, die auf die Translokation zurückgeführt werden konnten, und es fand sich kein physischer, geistiger oder geburtsbedingter Unterschied zwischen Überträgern und Verwandten mit normalen Chromosomen. Demnach haben autosomale reziproke Translokationen offenbar keinen schädlichen Effekt auf die geistige oder körperliche Entwicklung.

     

Origin of reciprocal translocations and their effect in Clarkia speciosa

Reciprocal translocations occur in high frequencies in Clarkia speciosa and closely related species. Observations from C. speciosa suggest this species is predisposed to translocations involving breaks in or adjacent to the centrochromatin (centromeric chromatin) due to the characteristic association of all nonhomologous centrochromatin in the genome during early meiotic prophase. Translocation heterozygote multiples involving six different breaks were examined for homologous pairing and in each case the euchromatic arms were completely paired, the change in homologous pairing occuring within the nonhomologous centrochromatic association. Such a proximal exchange point precludes the possibility of a structurally determined interstitial or differential region and, therefore, any genetically differential regions that might exist must be maintained solely by means of distal localization of crossing over. — The frequency of chromosomal nondisjunction (adjacent segregation) was found to be positively correlated with the number of chromosomes in the translocation multiple. Rings of four chromosomes had an average disjunction of over 99% and therefore had little affect on fertility whereas the largest multiples of 16 chromosomes had an average disjunction of about 10% and correspondingly low fertility.